Let's say you want to live a healthy life. You eat organic food to avoid pesticides, and you buy free-range chicken to steer clear of antibiotics. You stay away from swordfish because of the mercury warnings. You move out of the smoggy downtown.
But hard as you try, you will not be safe, says Carl F. Cranor, author of an unnerving new book, Legally Poisoned: How the Law Puts Us at Risk from Toxicants. Since 1999, the Centers for Disease Control and Prevention has measured 219 environmental chemicals in the bodies of Americans. Most of the population carries around measurable levels of lead from automobile exhaust, flame retardants from carpets and epoxy resins in water bottles, like those pictured on the cover of Cranor's book. Most Americans over 20 have been exposed to PCBs, dangerous coolants that were formerly used in electrical transformers.
No amount of "self-help" can save people from being chemically invaded, Cranor says. A lawyer and philosopher by training, he straddles science and law at the University of California, Riverside, studying the convoluted and contradictory patchwork of U.S. regulations on toxics and carcinogens.
"It is simply morally outrageous to treat citizens as experimental subjects by contaminating them with untested substances," Cranor writes. "This stands in stark contrast to the way people are treated under the ethical guidelines for medical experiments and the laws concerning pharmaceuticals and pesticides." In a recent interview, he said the U.S. needs to recognize "this deep inconsistency."
Drugs and pesticides undergo rigorous laboratory experiments before they can be sold, but there's almost no public data showing whether 82,000 industrial chemicals registered for sale in the United States were tested for potential health risks before entering the market, Cranor says. Because the Toxic Substances Control Act of 1976 doesn't require such testing, it effectively turns Americans into "guinea pigs for reckless free enterprise," he says: "I think it's a disastrous piece of legislation."
Instead of requiring manufacturers to conduct and pay for pre-market testing, Cranor says, the law puts the burden on the U.S. Environmental Protection Agency to police the industry after products are offered for sale, opening the gates for companies to bog down the system with court challenges.
To date, the EPA has required post-market testing for only 200 industrial chemicals. In 1998, the manufacturers of 2,800 chemicals of high production volume, defined as more than 1 million pounds a year, agreed voluntarily to test their products for potential risks to humans and the environment, but the program has fallen far behind schedule.
In this sort of enforcement system, few chemicals ever get the "death penalty." It was Congress, not the EPA, that banned PCBs. Today, studies of laboratory animals suggest that flame retardants known as polybrominated diphenyl ethers, or PBDEs, found in carpets and chair cushions, may cause damage to the liver, thyroid and nervous system, especially in children. Two types of these chemicals were phased out in 2004, and the EPA has announced a voluntary phase-out of another type by the end of 2013, but they have not been banned.
Earlier this year, citing an "uncertain business environment" stemming from a "fundamental lack of confidence in our nation's chemicals management system," the head of the American Chemistry Council told a congressional committee that the industry was in favor of modernizing the Toxic Substances Control Act. But the council's proposals do not include broad pre-market testing. "If we didn't believe our products were already safe for their intended uses, we wouldn't be making them," Cal Dooley, the council president, said.
Legally Poisoned is a frightening book. It reminds readers that lead poisons the nervous system and PCBs disrupt the reproductive system and that both have a long half-life, afflicting both humans and the environment decades after lead was banned from gasoline and PCB use was outlawed. The book cites a number of studies on laboratory animals that suggest a link between adult disease and early exposure to ubiquitous chemical substances, including perfluorinated compounds in Scotchgard and nonstick cookware, and bisphenol A, or BPA, a synthetic estrogen that hardens the plastic in water bottles, baby bottles and the liners of cans.
Taking the opposite tack, another new book, Scared to Death: How Chemophobia Threatens Public Health, seeks to debunk these studies and reassure the public. Published as a "position statement" by the American Council on Science and Health, an industry-friendly nonprofit group, it says the "regulation of chemicals is better and more effective than it's ever been," and chemicals used in consumer products "have undergone enough testing so that there is a reasonable likelihood that they will cause no harm when used properly." The book calls BPA "relatively innocuous" and questions whether studies on laboratory animals can be used to draw conclusions about risks to humans.
"The only significant science-based question is whether a particular substance is harmful at the trace levels to which humans are exposed," says author Jon Entine, a journalist and visiting fellow at the American Enterprise Institute, a conservative think tank in Washington, D.C.
Nonetheless, the troubling rise of autism, obesity, attention deficit hyperactivity disorder, type 2 diabetes and prostate and breast cancers in recent years has spread concern in the scientific community about the potential effects of low-dose exposure to hazardous chemicals such as BPA, especially early in life. The concept of the "developmental origins of disease," based on animal studies and human data, has gained broad acceptance, and it supplies the underpinnings for Cranor's arguments in Legally Poisoned — the idea that many illnesses may develop through a series of "hits" or exposures to hazardous chemicals, beginning in the womb. In this view, an exposure to toxics during a critical developmental window early in life might alter how a person's DNA functions, leading to cerebral palsy, leukemia, dementia, Parkinson's or other maladies later on.
Cranor believes a less-risky world is within reach. As an example, he points to the European Union, which in 2007 began requiring 30,000 existing and all new chemicals to be tested for potential health and environmental hazards. European manufacturers and importers must submit technical data on their products showing that they can be used safely, or they will be restricted or banned. The operative slogan is, "No data, no market."
In the U.S., two groundbreaking bills — the Safe Chemicals Act and Toxic Chemicals Safety Act — were introduced by Democrats in the House of Representatives last year. Both would have required pre-market testing; both died without a vote. In the vacuum, 18 states have taken matters into their own hands, passing 71 chemical safety laws during the past eight years. Some of these laws have banned BPA from baby and toddler products and phased out PBDEs in the home. In a recent example of "retail regulation," Wal-Mart Stores Inc. earlier this year banned PBDEs in hundreds of consumer goods, including couches, cameras and child car seats.
Among state laws, California's Proposition 65, a voter-approved initiative, is the strongest: It lists nearly 900 toxic substances (though not BPA or PBDEs) and requires companies to warn the public if their products contain them. Many companies have opted to take toxics off the market instead.
The American Chemistry Council says state and local governments lack the scientific expertise and resources to make such regulatory decisions. "In practice," Dooley, the council president, told members of Congress, "multiple state and local laws regarding chemicals create confusion among manufacturers, retailers and consumers, hamper the development of new products, close off markets, and ultimately prevent business growth and new hiring, all without significantly improving public safety."
Cranor disagrees: "As a tactic, I think we ought to encourage going jurisdiction by jurisdiction to deal with these problems. It's a headache for industry."
Legally Poisoned gets bogged down in the details of chemicals and their health effects, but its regulatory message comes through loud and clear. During a long career, Cranor has been an adviser to the Congressional Office of Technology Assessment and California's Proposition 65, and he has served on committees of the National Academy of Sciences and the Institute of Medicine. Along the way, he's been pulled into personal injury cases as an expert witness.
But tort law cannot prevent contamination; it can only compensate people after they get sick. In his book, Cranor reviews the cautionary tale of the "DES daughters," one of the most egregious regulatory failures of the last century. Beginning in the 1970s, thousands of women in the U.S. were diagnosed with rare vaginal cancers and other disorders decades after their mothers took diethylstilbestrol, a synthetic estrogen that was mistakenly believed to prevent miscarriages. The DES scandal prompted a huge wave of lawsuits against the manufacturers of the drug.
"The history of toxic contamination should worry us," Cranor says. "We have a world that's filled with industrial chemicals. We should recognize the seriousness of the problem and not wait for a catastrophe."


text:
\section{Introduction}

In general relativity, the nonexistence of an absolute time parameter introduces new aspects in the characterization
of chaos with respect to the well-known Newtonian dynamics \cite{book}.
Some relativistic systems in which the existence of chaos has been reported include charged particles in a magnetic
field interacting with gravitational waves, spinning particles orbiting rotating and nonrotating black holes,
gravitational waves from spinning compact binaries, as well as particles in Majumdar-Papapetrou geometries.

Most of the studies of chaos around black holes have focused on the analysis of the qualitative changes in the
dynamics of an isolated black-hole spacetime caused by a small perturbation due to external mass distributions
\cite{chandrasekhar1989,contopoulos1990a,dettmann1994,cornish1997,cornish2003a}, gravitational waves
\cite{bombelli1992}, spin-orbit and spin-spin coupling \cite{suzuki1997}, or magnetic fields \cite{karas1992}.
Recently, the projects to startup operative ground-base gravitational wave detectors (LIGO, VIRGO, GEO600, TAMA300, AIGO) and
a planned laser interferometer space antenna (LISA) \cite{observ}, which will be able to detect the signals from gravitational
wave sources, such as inspiralling compact binary systems of neutron stars or black holes, have increased the interest in
the presence of chaos in the dynamics of binary black holes and its effects on the outgoing gravitational radiation
\cite{cornish2003}.

In this work we use the Majumdar-Papapetrou metric \cite{MPmetric} to analyze the dynamics of photons in the gravitational
field of two extreme Reissner-Nordstr\"om black holes that are fixed in space due to the balance of their gravitational
attraction and electrostatic repulsion. Although it is unlikely that this metric describes any astrophysical system, since
in real universe black holes tend to rotate around their center of mass producing gravitational waves and do not possess
overall electric charge, the chaotic scattering of photons in the Majumdar-Papapetrou static spacetime of nonrotating
black holes with extreme electric charge still provides an interesting formal model that can be used to illustrate many
of the features expected in more realistic systems.

In \cite{chandrasekhar1989} Chandrasekhar studied the scattering of radiation by two extreme Reissner-Nordstr\"om
black-holes that are at finite distance apart. In contrast to the two center problem in Newtonian gravitation,
in general relativity this two center gravitational problem is generally not integrable. An appendix in
\cite{chandrasekhar1989} displays a set of null geodesics (photon's trajectories) in the meridian plane
of the system and concludes that the dynamics is probably not separable.

Shortly after, Contopoulos \cite{contopoulos1990a} systematically studied the set of photon trajectories with zero
angular momentum component along the axis that goes through the black-holes. In that case the motion is
confined to a plane (the meridian plane). The study concluded that there are three types of non
periodic motions. There are orbits that fall into one of the black-holes, with mass $M_1$ or mass $M_2$, these are
orbits of type $I$ and $II$, and there are orbits that escape to infinity, these are orbits of type $III$.
The orbits of different types are separated by orbits that tend asymptotically to three
main types of unstable periodic orbits. One kind of periodic orbits goes around one of the black-holes, either $M_1$ or
$M_2$. A third type of unstable periodic orbits goes around the two black-holes.
Between two non periodic orbits of two different types there are orbits of a third type.
In a related work Contopoulos {\sl et at.} \cite{contopoulos1993} analyze in detail the different types of periodic
orbits in the two fixed center system and compare families of these orbits in the Newtonian and the relativistic
problems. In this system chaos appears explicitly as the initial conditions of these types of orbits form a
Cantor set. Further numerical evidence indicated that all the photon periodic orbits are unstable.
In the same direction it was investigated the phase space trajectories in a multi-black-hole spacetime
\cite{dettmann1994}, it was found that the chaotic geodesics are well described by Lyapunov exponents.
All these works strongly indicate that the scattering of photons by two Reissner-Nordstr\"om black-holes held fixed
is chaotic and that the set of periodic orbits is unstable.
The geometric analysis of the flow also showed the chaotic behavior of the relativistic null-geodesic motion in the
two black hole spacetime \cite{yurtsever}.
More recently Contopoulos {\sl et al.} have studied in detail the asymptotic curves from the periodic orbits, their
homoclinic and heteroclinic intersections and the basins of attraction of two black holes \cite{contopoulos2004}.

Our aim is to give a full description of the chaotic escape dynamics of photons in the two black-hole system in terms
of the different physical indicators of chaos. We shall analyze the linear stability of the dynamics, in particular
the unstable periodic orbits, and from that information we evaluate the escape rate associated with the equations of
motion using the well known trace formula for hyperbolic flows of Cvitanovic and Eckhardt \cite{cvitanovic1991}.
It turns out that the escape rate is given by the leading eigenvalue of the spectrum of the evolution operator
\cite{gaspard1989}. In addition, we shall compare our results with numerical simulations that consider the time
evolution of statistical ensembles of photons.

In Section II we describe the Chandrasekhas's model to analyze the photon dynamics under the gravitational field
of two extreme Reissner-Nordstr\"om black holes.  In Section III we discuss the linear stability of the dynamics and
the general methods that we will consider to obtain the escape dynamics of photons.
Section IV introduces the formalism to study the time evolution of statistical ensembles of photons and their escape
dynamics. In Section V we present the set of periodic orbits of the system in the meridian plane $(L_z=0)$ and give
their periods and stretching factors. In this section we also illustrate the fractal repeller associated with the
dynamics of photons in the two black-hole field. In Section VI we calculate the main quantities of chaos derived from
the analysis of the linear stability of the unstable periodic orbits. The escape rate obtained from the periodic orbit
theory is compared with the value obtained from numerical simulations that consider statistical ensembles of photons.
In Section VII we analyze the dynamics in the proximity of a perturbed black hole and give an analytic estimate of
the escape rate in this system. In Section VIII the main conclusions are put together.

\section{Dynamical equations}

The model studied by Chandrasekhar in \cite{chandrasekhar1989} derives from the solutions of the Einstein-Maxwell equations
to describe a problem analogous to the Newtonian arrangement of charged mass-points in which the mutual Coulomb repulsions
are balanced with the gravitational attraction. These solutions are known as the Majumdar-Papapetrou solutions \cite{MPmetric}.
They are obtained from an static solution of the Einstein-Maxwell equations \cite{chandrasekhar1983}. The metric of the
Majumdar-Papapetrou solution of the Einstein-Maxwell equations is given by
\begin{equation}
\label{metric}
ds^2= dt^2/U^2
- U^2 (dx^2+dy^2+dz^2),
\end{equation}
where $(x,y,z)$ are the spatial coordinates. The function $U$, which depends only on the spatial coordinates, is a solution
of the three-dimensional Laplace's equation
\begin{equation}
\nabla^2 U= \left(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2} \right)U=0.
\end{equation}
Hartle and Hawking \cite{hartle1972} showed that for a function $U$ of the form
\begin{equation}
U=1+\sum_{i=1}^N \frac{M_i}{r_i},
\end{equation}
being $r_i=\sqrt{(x-x_i)^2+(y-y_i)^2+(z-z_i)^2}$, the Majumdar-Papapetrou solution corresponds to a system of $N$ extreme
Reissner-Nordstr\"om black holes with horizons at $(x_i,y_i,z_i)$ and with masses equal to their charges, $M_i=Q_i>0$.
The metric is everywhere regular except at the black-hole locations where there are coordinate singularities,
as argued in \cite{chandrasekhar1989,chandrasekhar1983}.

We will consider a configuration of two black holes located at $(0, 0, \pm z_{bh})$. Using geometrized units
(the speed of light in vacuum $c=1$ and the gravitational constant $G=1$) the function $U$ takes the form
\begin{equation}
\label{funcU}
U=1+\frac{M_1}{(x^2+y^2+(z-z_{bh})^2)^{1/2}}+\frac{M_2}{(x^2+y^2+(z+z_{bh})^2)^{1/2}}.
\end{equation}

The Lagrangian ${\cal L}$ associated with the metric (\ref{metric}) is defined by
\begin{equation}\label{lagran}
{\cal L}= \frac{\dot t ^2}{2U^2}-\frac{U^2}{2}(\dot x^2+\dot y ^2+\dot z^2),
\end{equation}
where the dot denotes the derivative with respect to the affine parameter $\mu$. The Hamiltonian ${\cal H}$ can be expressed
as
\begin{equation}
{\cal H}=\frac{1}{2} U^2 p_t^2-\frac{1}{2} U^{-2}(p_x^2+p_y^2+p_z^2),
\end{equation}
being the Hamilton's equations that dictate the geodesic motion associated with the metric (\ref{metric}):
\begin{eqnarray}
\label{eqnmotion}
&&\dot \alpha=\frac{d\alpha}{d\mu}=-U^{-2}p_\alpha \\
\nonumber
&&\dot p_\alpha=\frac{dp_\alpha}{d\mu}=-\frac{1}{2}\partial_\alpha(U^2-U^{-2}P^2)
\hspace*{1.5cm}(\alpha=x,y,z)
\nonumber.
\end{eqnarray}

According to the Hamilton's equations $p_t={\cal E}=constant$ and angular momentum along the $z$-axis
$L_z=x p_y-y p_x=constant$. In the case of photons it follows also that
${\cal H}=U^2 {\cal E}^2/2-U^{-2}(p_x^2+p_y^2+p_z^2)/2=0$ \cite{chandrasekhar1989,contopoulos1990a}, and therefore
$p_x^2+p_y^2+p_z^2=P^2=U^4 {\cal E}^2$. Hence the problem scales with ${\cal E}$ and without loss of generality we can
consider ${\cal E}=1$.

The equations of motion with respect to the killing time $t$ can be written as:
\begin{eqnarray}
\label{eqnmotion2}
&&\alpha'=\frac{d\alpha}{dt}=\dot \alpha\frac{d\mu}{dt}=-U^{-4}p_\alpha \\
\nonumber
&&p_\alpha'=\frac{dp_\alpha}{dt}=\dot p_\alpha\frac{d\mu}{dt}=-2U^{-1}\partial_\alpha U
\hspace*{1.5cm}(\alpha=x,y,z)
\nonumber.
\end{eqnarray}
The sets of equations (\ref{eqnmotion}) and (\ref{eqnmotion2}) are related by the transformation
\begin{equation}
\frac{d\,}{d\mu}=U^2\frac{d\,}{dt}
\end{equation}
between the affine parameter $\mu$ and the killing time $t$.
Although the solutions of these two systems induce different flows, $\Phi^{\mu}$ and $\Phi^{t}$ respectively, and therefore
lead to different values in some properties that depend on the flow, the spatial distribution of their corresponding
set of orbits do not change, and neither are modified the critical elements of the flow, in particular the periodic orbits.

Due to the conservation of $L_z$, the motion for $L_z=0$ is restricted to a plane (the meridian plane). We shall take this
simplification and study the dynamics in the meridian plane.

\section{Linear stability}

We are particularly interested in the stability analysis of the solutions of the equations of motion, i.e. how
is they behave after being slightly perturbed. To fix the notation and make the paper self contained, in this section
we introduce the basic concepts of linear stability. First we shall discuss this problem in a general context. Then the
application to the two black hole system is straightforward.

Let us consider a Hamiltonian system with $f$ degrees of freedom. The $2f$-dimensional phase-space can be denoted by
$X=(X_1,X_2,\cdots,X_{2f})$, being $(X_1,\cdots,X_{f})$ the generalized coordinates and $(X_{f+1},\cdots,X_{2f})$
the conjugated momenta. The Hamilton's equations for a Hamiltonian ${\cal H}$ can be expressed as
\begin{equation}
\dot X= \Sigma \cdot\frac{\partial {\cal H}}{\partial X}\,\,,
\end{equation}
being $\dot X=dX/d\tau$, with $\tau$ the affine parameter $\mu$ or the killing time $t$, and $\Sigma$ a $2f \times 2f$
antisymmetric matrix of the symplectic form
\begin{equation}
\Sigma= \left( \begin{array}{cc} {\sl 0} & {\sl 1} \\ -{\sl 1} & {\sl 0}
\end{array} \right),
\end{equation}
with ${\sl 0}$ and ${\sl 1}$ the $f \times f$ null and unit matrices respectively.

A solution $X_\tau$ of these differential equations is a curve in phase-space corresponding to a given initial
condition $X_0$. The stability analysis of this solution involves the study of the time evolution of a small
perturbation $\delta X$ with respect to $X_\tau$. Considering $Y_\tau=X_\tau+\delta X$, it follows that
$\delta X$ satisfies the initial value problem
\begin{equation}\label{eqL}
\delta \dot X= {\sf L}(\tau) \cdot \delta X,
\end{equation}
with ${\sf L}(\tau)= \Sigma \cdot \frac{\partial^2 {\cal H}}{\partial X^2}|_{X_\tau}$. This is a linear initial
value problem with time-dependent coefficients. For a given initial perturbation $\delta X_0$ it has the solution
\begin{equation}
\label{pereq}
\delta X_\tau={\sf M}(X_0,\tau) \cdot\delta X_0.
\end{equation}
The $2f\times 2f$ matrix ${\sf M}(X_0,\tau)$ is usually referred as the {\it fundamental matrix} and gives the time evolution
of an initial displacement $\delta X_0$. The fundamental matrix satisfies the differential equation
\begin{equation}\label{eqM}
\dot {\sf M}(X_0,\tau)= {\sf L}(\tau) \cdot {\sf M}(X_0,\tau)\,, \,\,\hbox{with} \,\,{\sf M}(X_0,0)={\sf 1}.
\end{equation}

The behavior of the perturbation $\delta X$ can be analyzed in terms of the Lyapunov exponents of $X_\tau$, which
measure the rate of exponential separation or approach of initially infinitely close trajectories. The Lyapunov exponent
associated with the unit vector ${\bf e}_j=\delta X_j/|\delta X_j|$ along the direction of $\delta X_j$ is given by
\begin{equation}
\lambda(X_0,{\bf e}_j)=\lim_{\tau \to \infty} \frac{1}{\tau}\ln | {\sf M}(X_0,\tau)\cdot{\bf e}_j |.
\end{equation}
The number of Lyapunov exponents is equal to the phase-space dimension. Positive Lyapunov exponents indicate
dynamical instability of trajectories in phase space, and therefore an extreme sensitivity to the initial conditions.

For a periodic orbit $X_\tau$, the Lyapunov exponents are directly related to the stability of this solution.
The Lyapunov exponents can be degenerate with multiplicity $m_i$, in the sense that several of them have the same value.
They are ordered as $\lambda_1 \ge\lambda_2\ge \cdots \lambda_{\nu}$ with  $\sum\limits_{i=1}^{\nu} m_i=2f$. The fundamental
matrix has a symplectic structure and therefore satisfies the relation ${\sf M}^{T} \Sigma {\sf M}=\Sigma$ \cite{arnold1968}.
Due to this property all the Lyapunov exponents in Hamiltonian systems are grouped in pairs of equal absolute value and
opposite sign $\{\lambda_i,-\lambda_i\}_{i=1,\cdots,f}$. Furthermore,
perturbations along the direction of the flow give a zero Lyapunov exponent, and due to the pairing property another one is
also zero, which is a consequence of the conservation of energy \cite{gaspard1998,gaspard1995}. Thus at least two exponents
are zero.

The matrix ${\sf M}(X_0,\tau)$ corresponding to a periodic orbit is also periodic with the same period.
In general $X_\tau=X_{\tau+T}$, with $T=rT_p$ (r=1,2,$\cdots$) and $T_p$ the primitive period of the orbit.
The linear stability of a periodic orbit is determined by the eigenvalues of ${\sf M}(X_0,\tau)$ over one
primitive period, i.e. ${\sf M}(X_0,T_p)$. These eigenvalues, called {\it stretching factors} or {\it stability eigenvalues},
are obtained from the secular determinant
\begin{equation}
{\hbox {det}} \left[ {\sf M}(X_0,T_p)-\Lambda {\sf 1} \right]=0.
\end{equation}

Due to the symplectic property of the fundamental matrix the eigenvalues $\Lambda_j$ are also grouped in pairs.
From the previous discussion two of them are equal to one. One of
them corresponds to perturbations along the periodic orbit and the other to perturbations that are perpendicular to the energy
shell. The Lyapunov exponents associated with a periodic orbit are given in terms of the stretching factors as
$\lambda_i=\ln |\Lambda_i|/T_p$.

The stability of a particular motion is determined by the location of the stretching factor in the complex plane:
$|\Lambda_i| >1$ is a signature of instability and corresponds to the unstable manifold of the periodic orbit,
$|\Lambda _i|<1$ is related to the stable manifold and $|\Lambda _i|=1$ corresponds to the center manifold of the orbit.

A straightforward application, which is directly related to our problem, is the study of the dynamics of two-dimensional
systems. In these systems a periodic orbit is called {\it hyperbolic} when $\Lambda > 1 >\Lambda^{-1} > 0$,
{\it hyperbolic with reflection} when $\Lambda <-1 <\Lambda^{-1}<0$, and  {\it elliptic} when $|\Lambda|=1$ \cite{arnold1968}.
The orbit stability changes whenever $\Lambda$ crosses the unit circle, and generally a bifurcation may occur in which
some periodic orbits can disappear or emerge. In autonomous systems the energy is the main bifurcation parameter. However,
in the two black-hole system no bifurcation is expected as the photon dynamics scales with the energy.

A periodic orbit is characterized by its primitive period and its stretching factors $\Lambda_i$. We shall see below how
this information can be used to analyze in detail the escape dynamics of photons scattered by two black-holes.
In general the Lyapunov exponents and the periods of the periodic orbits are not invariant under time-transformations,
but the stretching factor are invariant, as noted in \cite{dettmann1994,cornish2003a,cornish2001,cornishgibbons}.
In particular Motter \cite{motter2003a} showed that if we consider a transformation of the form $d\mu=f(X)dt$, the Lyapunov
exponents transform according to $\lambda_i^{\mu}=\lambda_i^t/<f>_t$, where
$<f>_t=lim_{t \rightarrow\infty}(1/t)\int_0^t f(X(t')) dt'$. For a given periodic orbit $p$ the Lyapunov exponents over
a period $T_{\mu}$ in the $\mu$-time and over a period $T_t$ in the $t$-time are related by
$\lambda_i^{T_{\mu}}=\lambda_i^{T_t}/<f>^*_{T_t}$ with $<f>^*_{T_t}=(1/T_t)\int_0^{T_t} f(X_p(t')) dt'=T_{\mu}/T_t$.
From these relations it can be concluded that $\lambda_i^{T_{\mu}}T_{\mu} =\lambda_i^{T_t} T_t$ which implies
the invariance of the stretching factors, $\Lambda_i^{T_{\mu}}=\Lambda_i^{T_t}$.

\section{Evolution of the density of photons and their decay dynamics}

The motion of a single photon in the meridian plane is chaotic and it is useful, in order to characterize its escape
dynamics from the black holes, to consider statistical ensembles that provide a probabilistic description of the process.
Therefore, instead of analyzing a single photon orbit, we consider an ensemble described by some density $\rho$.
For a given Hamiltonian ${\cal H}$ the time evolution of a density $\rho$ is dictated by the Liouville equation
\begin{equation}
\label{liouville}
\partial_\tau \rho= \hat L \rho,
\end{equation}
with the Liouville operator $\hat L$ given by the Poisson bracket as $\hat L=\{{\cal{H}},\rho\}$. Assuming a
time-independent Hamiltonian, the solution of this linear equation can be expressed as
\begin{equation}
\label{liouville1}
\rho_\tau=e^{\hat L \tau} \rho_0,
\end{equation}
being $\rho_0$ the initial density. From (\ref{liouville1}), the average over the ensemble of any observable $A$ at time
$\tau$ can be computed as;
\begin{equation}
\left< A \right>_\tau=\int dX \, A(\Phi^\tau X) \rho_0(X),
\end{equation}
being $\Phi^\tau$ the flow that maps an initial condition $X_0$ into $X_\tau$, according to
$X_\tau=\Phi^\tau X_0$. This last expression can also be written in the form
\begin{equation}\label{frobenius}
\left< A \right>_\tau=\int dX dY\, A(X)\delta (X-\Phi^\tau Y)\rho_0(Y),
\end{equation}
where the Dirac distribution $\delta (X-\Phi^\tau Y)$ represents the conditional probability density of a
trajectory being at position $X$ at time $\tau$ given the initial position $Y$. Indeed the Dirac distribution defines
the kernel of the evolution operator. The equation (\ref{frobenius}) defines two self adjoint operators as
\begin{equation}
\left< A \right>_\tau=
\left< \hat P^{\dagger \tau} A | \rho_0 \right>=
\left< A| \hat P^{\tau}\rho_0 \right>.
\end{equation}
The evolution of a probability density is ruled by the {\it Frobenius-Perron} equation \cite{gaspard1998,gaspard1995}
\begin{equation}
\rho_\tau(X)=\hat P^\tau \rho_0(X) \equiv \int dY \,\delta (X-\Phi^\tau Y) \rho_0(Y)
\end{equation}
and the time evolution of the observable is dictated by the Koopman operator \cite{koopman1931} defined by
\begin{equation}
A_\tau(Y) = \hat P^{\dagger \tau } A(Y) \equiv \int dX \, \delta (X-\Phi^\tau Y) A(X).
\end{equation}

In the case of an invertible and conservative flow $\Phi^\tau$ the Frobenius-Perron operator reduces to
$\rho_\tau(X)=\hat P ^\tau \rho_0(X) = \rho_0(\Phi^{-\tau} X)$. In open systems the trajectories that are initially
confined in a bounded region in phase space tend to escape towards infinity with an exponential escape rate. The number
of particles $N(\tau)$ that remain inside the initial domain at instant $\tau$ decays exponentially with time as
$N(\tau)\sim N(0) e^{-\gamma \tau}$, being $\gamma$ the so called {\sl escape rate}.

The leading eigenvalue of $\hat P^\tau$ dominates the decay and determines the escape rate. The rest of the spectrum
({\it resonances}) describe further details of the dynamics and give other important time scales besides the one
associated with the escape rate. Thus the spectrum of the Frobenius-Perron operator $\hat P^\tau$ provides a way to
describe the evolution of a set of trajectories characterized by a density $\rho_\tau(X)$.

The spectral theory of $\hat P^\tau$ developed from the works of Koopman \cite{koopman1931} and von Neumann \cite{neumann1932}.
It assumes that the evolution operator acts on a functional space of square integrable densities ${{\cal {L}}^2}$. In that
case the evolution is unitary and the eigenvalues of $\hat P^\tau$ are located on the unit circle. The spectrum of a chaotic
system presents continuous components on the unit circle which describe the correlation function decay for these systems. New
methods have been developed to obtain the eigenvalues of $\hat P^\tau$ outside the unit circle and to analyze the escape
process or relaxation dynamics \cite{pollicot1985,ruelle1986}. These methods, which are valid for systems where all
the periodic orbits are unstable, have been successfully used in chaotic systems and the study of transport
phenomena among other applications \cite{gaspard1992,gaspard1998,gaspard1995,alonso1996,alonso1993}. We shall apply
them to the study of the escape dynamics of photons from the two black holes.

One of the methods developed to obtain the spectrum computes the trace of the Frobenius-Perron operator. Here we shall
outline this procedure and refer the reader to the specialized literature on the subject
\cite{pollicot1985,ruelle1986,isola1988,gaspard1998,gaspard1995} for further details.

The trace of $\hat P^\tau$ can be formally written as
\begin{equation}
\label{trace}
Tr \hat P ^\tau = \int dX \, \delta (X-\Phi^\tau X).
\end{equation}
The contributions to the trace are due to the fixed points of the flow, which are given by the condition
\begin{equation}
\label{fixedpoint}
X=\Phi^\tau X,
\end{equation}
at some $\tau$. In general this equation presents two types of solutions: {\it stationary points} that satisfy
(\ref{fixedpoint}) for all $\tau$, and {\it periodic orbits} which satisfy the fix-point condition at a discrete
set of values, given by $\tau=rT_p$, with $r=1,2,\cdots$. We assume that the solutions of (\ref{fixedpoint}) are
isolated, and therefore each stationary point or periodic orbit is locally unique. A periodic orbit may belong to a
continuous family provide that there exists some continuous symmetry or some constant of motion. When
this occurs the integration domain in (\ref{trace}) must be reduced to a less dimensional space until the periodic
orbit is completely isolated. In time-independent Hamiltonian systems the periodic orbits are rarely isolated,
indeed they tend to form continuous families as the energy changes. Hence the phase-space must be reduced
considering the energy conservation ${\cal H}=E$. Then the trace (\ref{trace}) is formally given by
\begin{equation}
\label{trace2}
Tr_{E} \hat P^\tau = \int_{E} d^{2f-1}x \,\delta(X-\phi_E^\tau x),
\end{equation}
where $\phi_E^\tau$ denotes the flow on the energy shell \cite{cvitanovic1991,gaspard1998}. If any additional symmetry is
present in the system then further reductions are required. In the two black system we are considering the dynamics on
the energy shell and we have removed the axial symmetry by choosing a particular meridian plane. Therefore all the periodic
orbits of the flow can be treated as isolated.

The trace (\ref{trace2}) can be written in terms of the unstable periodic orbits of the flow and their repetitions
\cite{artuso1990,cvitanovic1991}. In order to do so, the integral in (\ref{trace2}) is resolved considering at each point
a coordinate system with one axis fixed along the periodic orbit. The integral over the coordinate along the orbit is
trivial, being related to the period of the orbit. The integration over the transverse coordinates takes into account
the stability of the periodic orbit. The result for the trace is \cite{cvitanovic1991}
\begin{equation}
\label{trace3}
Tr \hat P^\tau =\sum_{p=p.p.o.} \sum_{r=1}^{\infty} T_p \frac{\delta (\tau-rT_p)}{|{\hbox{det}}({\sf 1}-{\sf m}_p^r)|}
\hspace*{0.4cm}(\tau>0),
\end{equation}
where the sums are over the {\it primitive periodic orbits (p.p.o.)} and their $r$ repetitions.
The matrix ${\sf m}_p$ is derived from the fundamental matrix ${\sf M}(X_0,T_p)$ once the perturbations along the orbit
and the perturbations perpendicular to the energy shell have been removed. Hence in a two-dimensional flow the matrix
${\sf m}_p$ only contains the stretching factors whose modulus is different from one. These are precisely the
ones that are related to the stable and unstable manifolds of the isolated unstable periodic orbit.

Formally the Laplace transform of the Frobenius-Perron operator gives the resolvent of the Liouville operator defined
in (\ref{liouville}), which using (\ref{trace2}) can be written as
\begin{equation}
Tr \frac{1}{\sigma - \hat L}= \int_0^{\infty} d\tau \, e^{-\sigma \tau} Tr \hat P= \frac{\partial}{\partial \sigma}
\ln Z(\sigma),
\end{equation}
where $Z(\sigma)$ is the so called Selberg-Smale Zeta function \cite{smale1980}, given by
\begin{equation}
Z(\sigma)\equiv \exp \left( - \sum_{p=p.p.o.} \sum_{r=1}^{\infty} \frac{1}{r}
\frac{e^{-\sigma r T_p}}{|{\hbox{det}}({\sf 1}-{\sf m}_p^r)|} \right).
\end{equation}
Denoting by $\Lambda_p$ and $\Lambda_p^{-1}$ the eigenvalues of ${\sf m}_p$, the zeta function can be expressed as
\begin{equation}\label{zf}
Z(\sigma)=\prod_{p=p.p.o} \prod_{k=0}^{\infty} \left( 1- \frac{e^{-\sigma T_p}}
{ | \Lambda_p |\, \Lambda_p ^k}\right)^{k+1}
\end{equation}
where $ \Lambda_p > 1$ since all the periodic orbits are unstable in the two black-hole system. The spectrum of the system
can be determined from the zeroes of the function $Z(\sigma)$, which is usually expressed as a product of the inverse
{\it Ruelle} $\zeta$-functions,
\begin{equation}
Z(\sigma)= \zeta_0^{-1}(\sigma) \zeta_1^{-2}(\sigma)\zeta_2^{-3}(\sigma) \cdots
\end{equation}
with the definition
\begin{equation}
\zeta_k^{-1}(\sigma)= \prod_{p=p.p.o}
\left(
1-\frac{e^{-\sigma T_p}}{|\Lambda_p|\Lambda_p^k}
\right).
\end{equation}

The relaxation times $\tau_i$ associated with the dynamics are given by the poles of the resolvent of the Liouvillian, as
$Re \, \sigma_i=\tau_i^{-1}$, or equivalently, by the zeroes of the zeta function or the zeroes of the inverse of the
Ruelle $\zeta$-functions. The leading part of the spectrum is given by the poles of the first Ruelle
$\zeta$-function $\zeta_0(\sigma)$ (or zeroes of $\zeta_0^{-1}$). So we are mainly interested in the part
of the spectrum that controls the long-time behavior of the dynamics.

The set of unstable periodic orbits can be used to describe in fine detail the escape dynamics of photons from the black
holes. The number of periodic orbits in the fractal repeller grows exponentially with the period, with a rate of proliferation
that is given by the topological entropy $h_{top}$. The amount of chaos in the system is given by the Kolmogorov-Sinai
entropy per unit time $h_{KS}$, which measures the minimal data accumulation rate required to reconstruct a trajectory on
the repeller without ambiguity. The repeller has a mean Lyapunov exponent $\lambda$. In a Poincare section that is transverse
to the flow on the repeller, the fractal repeller defines another fractal which is characterized by the partial generalized
fractal dimensions. Two representative partial dimensions are the Hausdorff dimension $d_H$, which characterizes the bulkiness
of the repeller in phase space, and partial information dimension $d_I$ that can be obtained from the Kolmogorov-Sinai entropy
and the mean Lyapunov exponent as $d_I= h_{KS}/\lambda$ \cite{gaspard1989}.

The different quantities that characterize the repeller can be obtained from the so called topological pressure $P(\beta)$
\cite{ruelle1978}, which can be determined from the periodic orbits that form part of the repeller as the leading zero of a
Ruelle $\zeta$-function
\begin{equation}
\zeta_\beta^{-1}(\sigma) \equiv \prod_{p=p.p.o}
\left(
1-\frac{e^{-\sigma T_p}}{|\Lambda_p|^{\beta}}
\right).
\end{equation}

The mean Lyapunov exponent for the trajectories on the fractal repeller is $\lambda=-dP(1)/d\beta$, the escape rate is
$\gamma=-P(1)$ and the topological entropy is given by $h_{top}=P(0)$. In two-dimensional Hamiltonian systems the
Kolmogorov-Sinai entropy follows from $h_{KS}=\lambda-\gamma$ \cite{pesin}. The partial Hausdorff dimension is obtained
from $P(d_H)=0$ and the information dimension is given by  $d_I=1-\gamma / \lambda$.

\section{Periodic orbits and the repeller in the meridian plane $(L_z=0)$}

From now on we shall focus on the analysis of the escape dynamics of photons from the black holes in the meridian plane.
As we have just mentioned, there are some important objects that can be used to describe this dynamics.
First we need to identify the set of periodic orbits  and their corresponding primitive periods. To analyze the linear
stability of each unstable periodic orbit we must integrate its fundamental matrix in time up to one primitive period.
The eigenvalues of the resulting matrix give the stretching factors associated with the orbit.
Then, once we know the periods and the stretching factors of the unstable periodic orbits, the main quantities of
chaos that characterize the dynamics can be obtained  following the methods described in the previous section.

The restriction on the motion to the meridian plane $L_z=0$ simplifies the set of equations to be integrated. Without
loss of generality we can consider that the photons move in the $(x,z)-$plane. Then the equations of motion (\ref{eqnmotion})
with respect to the affine parameter $\mu$ reduce to
\begin{equation}
\dot \alpha=-U^{-2}p_\alpha
\hspace*{0.5cm}\hspace*{0.5cm}
\dot p_\alpha=-2U\partial_\alpha U
\hspace*{0.5cm}(\alpha=x,z),
\end{equation}
being the equations (\ref{eqnmotion2}) with respect to the killing time $t$,
\begin{equation}
\alpha^\prime=-U^{-4}p_\alpha
\hspace*{0.5cm}\hspace*{0.5cm}
p_\alpha^\prime=-2U^{-1}\partial_\alpha U
\hspace*{0.5cm}(\alpha=x,z).
\end{equation}
These equations of motion and the equations associated with the stability analysis were simultaneously integrated using a
Runge-Kutta-Fehlberg method. The search for periodic orbits was performed using the Newton-Raphson method.
As we just mentioned, to compute the stretching factors of the set of unstable periodic orbits, each orbit was numerically
integrated together with its corresponding fundamental matrix up to one primitive period of the orbit.
Considering that the dynamics is highly unstable, it is important to compute the unstable periodic orbit with high
accuracy in order to obtain the stability factors with good precision.
In our numerical results the required high accuracy in the period of the orbits and the stretching factors was achieved
by means of a Mathematica code in which a very high precision was demanded.

The periodic orbits in the system can be classified according to a symbolic coding. In this work,
taking as reference the symmetry axis that contains the two black holes, in our coordinate system the $z$-axis, we
introduce a series of symbols to label these orbits. An orbit can cross the $z$-axis either above the black-hole located
at $z=z_{bh}$, below the black-hole located at $z=-z_{bh}$ or in between the two black-holes.
We label each crossing that occurs at $z>z_{bh}$ with the  symbol ``$+$'', to indicate a crossing at $z<-z_{bh}$ we use the
symbol ``$-$''. When an orbit crosses the $z$-axis twice at the same point in between the two black holes, drawing an
``$\times$'' on the axis, we use the symbol ``$\circ$''. Any other crossing in between the two black holes is indicated with the
symbol ``$+$'' or the symbol ``$-$'' depending on the sign of the $z$ coordinate at the crossing point, $z>0\,(+)$ or $z<0\,(-)$.
Figure (\ref{po1}) displays some of the unstable periodic orbits numerically found and their corresponding symbolic
coding. The primitive period of each orbit is given in Table (\ref{table1}).

In \cite{cornishgibbons} a different coding to label the periodic orbits was introduced. This coding was used to compute the
topological entropy per unit symbol. We remark that one of our goals in this work is to compute the topological entropy per
unit time.

\begin{figure}[h]
\centerline{\includegraphics[width=0.93\textwidth]{all_periodic_orbits.eps}}
\vspace*{0.8cm}
\caption{Some of the unstable periodic orbits in a system with two black holes located at $z_{bh}=\pm 1$ and masses $M_1=M_2=1$.
The symbolic coding that labels each orbit is indicated. Due to the symmetry of the system with respect to equatorial line
(the $x$-axis), the change of sign of the coordinate $z$ in each orbit on the last two columns gives an additional periodic
orbit. The symbolic coding for these {\sl inverse} orbits is obtained exchanging the symbols "+" and "-" in the orbits depicted
in these columns. For instance, the inverse orbit of $\{++\}$ has symbolic coding $\{--\}$, the inverse of $\{+-++\}$ is
$\{-+--\}$, the inverse of $\{+-+\circ-\circ\}$ is $\{-+-\circ+\circ\}$ and so on.}
\label{po1}
\end{figure}

\begin{table}
\begin{tabular}{lrrr}\hline
Symbolic coding & \hspace*{0.5cm}Period $T_p\,(\mu)$ &  \hspace*{0.5cm}Period $T_p\,(t)$ &
\hspace*{0.5cm}Stretching factor $\Lambda_p$\hspace*{0.3cm}\\ \hline
$\{++\}$, $\{--\}$ & \hspace*{0.5cm}4.1180374 & \hspace*{0.5cm}37.9255026 & \hspace*{0.5cm}99.1244899\hspace*{0.3cm}\\
$\{+-\}$ & \hspace*{0.5cm}15.292055 & \hspace*{0.5cm}52.7275448 & \hspace*{0.5cm}215.721472\hspace*{0.3cm} \\
$\{+\circ-\circ\}$ & \hspace*{0.5cm}10.608412 & \hspace*{0.5cm}79.8770864 & \hspace*{0.5cm}16996.3144\hspace*{0.3cm} \\
$\{+-++\}$, $\{-+--\}$ & \hspace*{0.5cm}18.900782 & \hspace*{0.5cm}91.7239060 & \hspace*{0.5cm}30044.8517\hspace*{0.3cm}\\
$\{+++\circ-\circ\}$,  $\{---\circ+\circ\}$ & \hspace*{0.5cm}14.739760 &\hspace*{0.5cm}117.837331 &
\hspace*{0.5cm}$1.70990814 \times 10^6$\hspace*{0.3cm} \\
$\{+++---\}$& \hspace*{0.5cm}22.635603& \hspace*{0.5cm}130.594585 &  \hspace*{0.5cm}$3.97082132 \times 10^6 $\hspace*{0.3cm}\\
$\{+-+\circ-\circ\}$,  $\{-+-\circ+\circ\}$ & \hspace*{0.5cm}25.410381 & \hspace*{0.5cm}133.342257 &
\hspace*{0.5cm}$4.57097810 \times 10^6$\hspace*{0.3cm} \\
$\{+-+++-\}$,  $\{-+---+\}$ & \hspace*{0.5cm}34.185532 & \hspace*{0.5cm}144.455986 &  \hspace*{0.5cm}$6.49738275 \times 10^6$
\hspace*{0.3cm} \\
$\{+++\circ---\circ\}$ & \hspace*{0.5cm}18.871204 & \hspace*{0.5cm}155.798106 &  \hspace*{0.5cm}$1.72085241 \times 10^8$
\hspace*{0.3cm} \\
$\{+-----++\}$,  $\{-+++++--\}$ & \hspace*{0.5cm}26.748625 & \hspace*{0.5cm}168.528893 &  \hspace*{0.5cm}$ 3.95770189
\times 10^8$\hspace*{0.5cm} \\
$\{+-+\circ---\circ\}$, $\{-+-\circ+++\circ\}$ &
\hspace*{0.5cm}29.542461 & \hspace*{0.5cm}171.300173 &  \hspace*{0.5cm}$4.59225358 \times 10^8$\hspace*{0.5cm} \\
$\{+-+\circ---\circ+-\}$,  $\{-+-\circ+++\circ-+\}$ &
\hspace*{0.5cm}44.828729 & \hspace*{0.5cm}224.030938 &  \hspace*{0.5cm}$ 9.92347816 \times 10^{10}$\hspace*{0.5cm}\\ \hline
\end{tabular}
\caption{The periods and the stretching factors of the unstable periodic orbits depicted in figure (\ref{po1}). The periods
corresponding to both the affine parameter $\mu$ and the killing time $t$ are indicated.}
\label{table1}
\end{table}

In Table (\ref{table1}) are listed the non-trivial stretching factors of the periodic orbits depicted in figure (\ref{po1}).
We recall that these are the stretching factors that are related to the directions transverse to the periodic orbit.
The trajectories $\{++\} (\{--\})$, $\{+-\}$ and $\{+\circ-+\}$ have the lowest stability eigenvalues. The rest of the orbits
are much more unstable as they have larger stretching factors. In general the more stable periodic orbits should dominate
the asymptotic escape of particles from the two black-hole configuration.

In the two black-hole system the set of unstable periodic orbits are the only critical elements of the flow in the fractal
repeller that determines the escape dynamics of photons. This distinguished subset of solutions plays a crucial role as the
averages of the dynamical quantities can be obtained in terms of periodic orbits, provide that it is dense on the repeller
\cite{gaspard1998}.

The dynamics in the proximity of the invariant set that defines the repeller can be analyzed considering the time interval
that it takes to an orbit to escape from a finite spatial domain ${\cal U}$, when evolves both forwards and backwards in time.
To numerically study the escape dynamics we introduce the forward and backward escape time functions \cite{gaspard1998},
which are defined by:
\begin{eqnarray}
T_{\cal U}^+(x)&=&Max \lbrace T>0\,;\,\, \Phi^{\mu} x \in {\cal U}\,, \,\, \forall \mu \in [0, T[\,, \,\, \forall x
\in {\cal U} \rbrace \\
T_{\cal U}^-(x)&=&Min \lbrace T<0\,;\,\, \Phi^{\mu} x \in {\cal U}\,, \,\, \forall \mu \in \,]T, 0]\,, \,\, \forall x
\in {\cal U} \rbrace
\end{eqnarray}

The escape time functions provide an useful method to construct the repeller \cite{burghardt1994}. The ensemble of initial
conditions whose escape time coordinate is larger than a given value $T$ contains all the trajectories that still remain inside
the domain $\cal U$ at time $T$. In the long time limit this set contains the trapped trajectories of the repeller and its
stable manifolds in $\cal U$. As time evolves this set splits into smaller pieces and eventually becomes a fractal set.
Similarly it is possible to find the set of initial conditions that have already arrived in the domain $\cal U$ before
some time $-T$ in the past. The repeller is given by the intersection of these two sets.

The compound escape function $T_{\cal U}^+(x)+|T_{\cal U}^-(x)|$ is very useful to illustrate the structure of the repeller.
To numerically compute this function we analyze the evolution, both backwards and forwards in time, according to the
equations of motion (\ref{eqnmotion}), of a large ensemble of photons initially distributed at random in a finite region of
space and certain fixed initial directions.
In order to distinguish the possible outcomes of the photons in the two black-hole field, we color code the initial position
black if the corresponding photon falls into any of the two black holes and gray if the photon escapes towards infinity.
Figure (\ref{repel1}) displays a portrait of the basins of attraction associated with these two possible outcomes in
the plane $(x,z)$ of initial positions, for an ensemble of photons with initial directions randomly fixed between
five selected values for the angle with respect to the $x$-axis. It is well known that for chaotic systems the interweaving
of outcomes leads to fractal basin boundaries; and in the two black-hole system, as expected, we can see them explicitly.
The blown up regions depicted in the different panels of figure (\ref{repel2}) exemplify the repeated fractal structure in
the basin boundaries that characterize the hyperbolic repeller.
In the following sections we compute different characteristic quantities of the dynamics related to this fractal repeller.

\begin{figure}[h]
\vspace*{1cm}
\centerline{\includegraphics[width=1.0\textwidth]{repeller_0b.eps}}
\caption{Outcomes for photons in the two-black-hole field. The black dots are the initial positions of the trajectories which
fall into any of the two black holes. The gray dots localize the initial positions of the photons that escape towards infinity.
The evolution according to the equations of motion (\ref{eqnmotion}) was considered, both backwards and forwards in time.
The initial positions $(x,z)$ were chosen at random in the finite region depicted, and the initial directions were randomly
selected with an angle with respect to the $x$-axis equals to $0$, $\pi/5$, $2\pi/5$, $3\pi/5$ or $4\pi/5$.}
\label{repel1}
\end{figure}

\begin{figure}[h]
\centerline{\includegraphics[width=1.0\textwidth]{repeller_1b.eps}}
\caption{Several blown up regions of figure (\ref{repel1}) showing the fractal structure of the repeller.}
\label{repel2}
\end{figure}

\section{Escape rate in the meridian plane and other characteristic quantities of Chaos}

Using the primitive periods $T_p$ and stretching factors $\Lambda_p$ of the set of unstable periodic orbits we can compute
the escape rate of the system as well as other characteristic quantities of the dynamics related to the fractal repeller,
following the methods described in Section IV. To this aim we consider the cycle expansion method
\cite{artuso1990,cvitanovic1991} which has been successfully applied to other hyperbolic systems \cite{gaspard1992}.
Due to the high instability of higher order periodic orbits (longer symbolic coding) only three main periodic orbits,
$\{++\} (\{--\})$, $\{+-\}$ and $\{+\circ-+\}$, contribute significantly to the escape rate.
Indeed they give an acceptable escape rate and the different quantities of chaos.

Figure (\ref{pressure}) shows the pressure function obtained from these three periodic orbits. The main quantities
of chaos obtained from this pressure function and its first derivative are listed in Table (\ref{table2}).
\begin{figure}[h]
\centerline{\includegraphics[width=0.7\textwidth]{pressure_2.eps}}
\caption{The pressure function (solid line) obtained considering the periodic orbits that include up to six elements
in the symbolic code, see figure (\ref{po1}). The dashed line is the first derivative of this pressure function. The
panel $(a)$ is the pressure associated with the affine parameter, and $(b)$ the pressure corresponding to the killing
time
\vspace*{0.5cm}
.}
\label{pressure}
\end{figure}
\begin{table}
\begin{tabular}{|c||c|c|c|c|c|c|} \hline
    & $\lambda$ &  $\gamma$ &  $h_{KS}$ &  $h_{top}$ &  $d_{H}$ &  $d_{I}$ \\ \hline\hline
\hspace*{0.3cm} Affine parameter $\mu$ \hspace*{0.3cm} &
\hspace*{0.3cm}  0.364  \hspace*{0.3cm} &\hspace*{0.3cm}  0.348  \hspace*{0.3cm}&
\hspace*{0.3cm} 0.016  \hspace*{0.3cm}&\hspace*{0.3cm}  0.162  \hspace*{0.3cm}&
\hspace*{0.3cm} 0.221  \hspace*{0.3cm}&\hspace*{0.3cm}  0.045 \hspace*{0.3cm} \\ \hline
\hspace*{0.3cm} Killing time $t$ \hspace*{0.3cm} &
\hspace*{0.3cm}  0.111  \hspace*{0.3cm} &\hspace*{0.3cm}  0.089  \hspace*{0.3cm}&
\hspace*{0.3cm} 0.022  \hspace*{0.3cm}&\hspace*{0.3cm}  0.024  \hspace*{0.3cm}&
\hspace*{0.3cm} 0.221  \hspace*{0.3cm}&\hspace*{0.3cm}  0.196 \hspace*{0.3cm} \\ \hline
\end{tabular}
\caption{Characteristic quantities of chaos related to the fractal repeller associated with the unstable periodic orbits
given in figure (\ref{po1}). $\lambda$ is the mean Lyapunov exponent of the repeller, $\gamma$ the escape rate, $h_{KS}$
the Kolmogorov-Sinai entropy, $h_{top}$ the topological entropy, $d_H$ the Hausdorff dimension and $d_I$ the partial
information dimension.}
\label{table2}
\end{table}
In particular, the escape rate which is equal to $\gamma_\mu=0.348$ for the affine parameter and $\gamma_t=0.089$ for the
killing time. From the analysis of the linear stability of the set of unstable periodic orbits, it could be assumed that
the escape dynamics of photons from the two black holes is mainly controlled by the outermost periodic orbit $\{+-\}$.
If only this orbit is considered it follows an escape rate
\begin{equation}
\gamma_\mu^{\{+-\}}=\frac{\ln |\Lambda_{+-}|}{T_{+-}(\mu)}=0.351
\hspace*{2cm}
\gamma_t^{\{+-\}}=\frac{\ln |\Lambda_{+-}|}{T_{+-}(t)}=0.102
\end{equation}
which is close to the escape rate obtained from the pressure function. Thus, we can conclude that the orbit $\{+-\}$
dominates the effective escape of photons from the two black-hole configuration, and even provides a fair estimate of
the escape rate.

In the following section we contrast the escape rate value derived from the analysis of the linear stability of the
periodic orbits with the escape rate obtained from numerical simulations that consider the time evolution of an
statistical ensemble of photons.

\subsection{Numerical simulations with statistical ensembles}

We consider an ensemble of photons with initial positions uniformly distributed along the $z$-axis and initial velocity
parallel to the x-axis. This together with the energy conservation completely defines the ensemble. Once all the initial
conditions have been fixed the time evolution of each photon of the ensemble is numerically integrated. Some of the
photons fall into one of the two black holes, while others escape to infinity. We are precisely interested in the second
subset of trajectories.
To compute the escape rate we consider a circle of control that covers the two black holes and  which is larger than the
outermost periodic orbit $\{+-\}$. Hence all the photons that cross out this circle escape towards infinity and do not come
back to the proximity of the black holes.

Let us denote by $N(\tau)$, with $\tau$ the affine parameter $\mu$ or the killing time $t$, the number of photons that
remain inside the circle of control at time $\tau$ (this includes the photons that fall into the black-holes). This
function decreases with $\tau$ until it reaches an asymptotic value $N(\infty)$, which gives the number of photons
that fall into the black holes. The escape rate $\gamma$ is defined by the exponential decrease to zero of the function
$[N(\tau)-N(\infty)] \approx [N(0)-N(\infty)]e^{-\gamma\tau}$ as $\tau\to\infty$.

\begin{figure}[h]
\centerline{\includegraphics[width=0.8\textwidth]{escape_mu_t.eps}}
\caption{The evolution of $[N(\mu)-N(\infty)]\,\,(\,[N(t)-N(\infty)]\,)$ with the affine parameter (killing time) for
three ensembles with an increasing number of photons, $5\times10^5$, $5\times10^6$ and $5\times10^7$. The dashed lines give
the best linear fitting for the ensemble with the highest number of photons.}
\label{nt}
\end{figure}

Figure (\ref{nt}) shows the results obtained from different simulations that include an increasing number of particles
in the statistical ensemble. The larger is the number of particles in the ensemble the longer persists the exponential
decrease with $\tau$.
From the slope of these curves we extract a numerical escape rate $\gamma_{\mu}^{(num)}=0.344$ for the affine parameter
and $\gamma_{t}^{(num)}=0.083$ for the killing time. The agreement with the escape rate obtained from the analysis of the
linear stability of the unstable periodic orbits is excellent, see Table (\ref{table2}).
This result confirms our expectation with respect to the subset of periodic orbits that plays the main role in the escape
dynamics. The small oscillatory component in the evolution of the function $[N(\tau)-N(\infty)]$ can be explained by
the lower resonances of the system. This point has been emphasized in \cite{gaspard1992}. To further illustrate this
aspect we have computed the power spectra of the numerical data $[N(\mu)-N(\infty)]e^{\gamma_{\mu}^{(num)}\mu}$ for the affine
parameter, and $[N(t)-N(\infty)]e^{\gamma_{t}^{(num)}t}$ for the killing time. The figures (\ref{nt2a}) and (\ref{nt2b}) show
a peak structure associated with the oscillations of both functions once the main exponential decay has been subtracted. The
positions of the peaks is compared with the locations of the imaginary part of the lower resonances computed from the zeta
function taking orbits that include up to six elements in the symbolic code. The resonance distribution explains the peak
structure observed in the power spectrum.

\begin{figure}[h]
\centerline{\includegraphics[width=0.8\textwidth]{spec_affine.eps}}
\caption{In the upper panel the oscillatory component of the function $[N(\mu)-N(\infty)]$ depicted in the left panel of
figure (\ref{nt}), once the main exponential decay has been subtracted. In the middle panel its power spectrum, being the
vertical dotted lines the imaginary part of the resonances given in the lower panel.}
\label{nt2a}
\end{figure}

\begin{figure}[h]
\centerline{\includegraphics[width=0.8\textwidth]{spec_killing.eps}}
\caption{The same as figure (\ref{nt2a}), but for the function $[N(t)-N(\infty)]\,)$ depicted in the right panel of
figure (\ref{nt}).}
\label{nt2b}
\end{figure}

The escape rate of photons depends on the masses of the black holes and the separation distance between them.
The larger are the masses the more extensive in space are the unstable periodic orbits that form part of the fractal repeller
that marks the boundary between dynamical stability and instability, and therefore the regions where the light rays fall into
the black holes. It could be expected then that the escape rate of photons decreases as the masses of the black holes become
larger. Figure (\ref{nt2}) illustrates this behavior and shows the decrease of the escape rate as the mass of one of the
black holes is increased. Figure (\ref{nt3}) shows the dependence of the escape rate on the separation distance between
the two black holes. In both cases, after an initial transient, the escape rate decays asymptotically with the distance $d$.
A similar behavior is observed in other scattering systems where the escape rate decays as $f(d)/d$, being $f(d)$ a function
determined by the dispersion of the scattering interaction potential and $1/d$ is the contribution of the time of flight
\cite{gaspard1992,gaspard1998}.

\begin{figure}[h]
\centerline{\includegraphics[width=0.73\textwidth]{different_mass.eps}}
\caption{The escape rate $\gamma_\mu\,\,(\,\gamma_t\,)$ from the two black holes located at $z_{bh}=\pm 1$ for different
values of the mass $M_1$ ($M_2=1$). The data were obtained considering ensembles of photons with initial velocity parallel
to the x-axis and initial positions uniformly distributed along an interval on the $z$-axis. The circles correspond to an
ensemble initially distributed in the interval $[1.1,4]$ and the crosses to an ensemble initially distributed in the interval
$[-4,-1.1]$. The dashed lines are the analytic estimate of the escape rate from a perturbed black hole,
$\gamma_\mu^{(0)}=1/\sqrt{2}(M_1+M_2)$ and $\gamma_t^{(0)}=1/4\sqrt{2}(M_1+M_2)$, see Section VII.}
\label{nt2}
\end{figure}
\begin{figure}[h]
\centerline{\includegraphics[width=0.73\textwidth]{different_d2.eps}}
\caption{Escape rate $\gamma_\mu\,\,(\,\gamma_t\,)$ from two identical black holes ($M_1=M_2=1$) for different separation
distance between them. The black holes are located at $z_{bh}=\pm d$. The data were obtained using ensembles of photons with
initial velocity parallel to the x-axis and initial positions uniformly distributed along the $z$-axis.}
\label{nt3}
\end{figure}

In figure (\ref{nt3}) we consider the scattering of photons from two black holes with masses  $M_1=M_2=M$, located at
different positions ($0,\pm d$). From (\ref{funcU}) and (\ref{eqnmotion},\ref{eqnmotion2}) it can be easily shown that
a system of two black holes with scaled masses ${\tilde m}_1={\tilde m}_2=M/d$ located at $(0,\pm 1)$ satisfies identical
dynamics with respect to an scaled time $\tilde\tau$ defined by $d\alpha/d\tau=d{\tilde\alpha}/d\tilde\tau\,
(d\tilde\tau=d\tau/d)$, being ${\tilde\alpha}=\alpha/d\,(\alpha=x,z)$ scaled coordinates. Hence if we denote $\gamma_d$ the
escape rate from two identical black holes of mass $M$ located at ($0,\pm d$), and ${\tilde \gamma}$ the escape rate from
two black holes of the same mass  ${\tilde m}=M/d$ located at $(0,\pm 1)$, it follows that $\gamma_d={\tilde \gamma}/d$.

\section{Escape dynamics of photons from a perturbed black hole}

In this section we study the escape rate from an extreme Reissner-Nordstr\"om black hole of mass $M_1$ that is
slightly perturbed by the interaction with an extreme Reissner-Nordstr\"om black hole of mass $M_2$ located at distance $d$.
In polar coordinates, the Hamiltonian can be written as
\begin{equation}\label{pert1}
{\cal H}=-\frac{h}{U^2} = -\frac{1}{U^2}\left[
\frac{1}{2}\left(p_r^2+\frac{p_\theta^2}{r^2}\right)-\frac{U^4}{2}\right]
\end{equation}
with the function $U$ given by:
\begin{equation}\label{pert2}
U=1+\frac{M_1}{r}+\frac{M_2}{\sqrt{r^2+d^2-2rd\sin\theta}}
\end{equation}

The analysis of the dynamics of this system can be greatly simplified if we consider the evolution with respect to a {\sl time}
$\nu$ defined by the transformation
\begin{equation}
\frac{d\,}{d\mu}=-\frac{1}{U^2}\frac{d\,}{d\nu}.
\end{equation}
Thus we focus on the equations of motion associated with the Hamiltonian $h$,
\begin{equation}\label{pert4}
\frac{dX}{d\nu}=\Sigma\cdot\frac{\partial h}{\partial X}
\hspace*{1.3cm}X=(r,\theta,p_r,p_\theta)
\end{equation}
In these equations, $h$ can be analyzed as a kinetic energy term plus an interaction term $-U^4/2$.
The function $U^4$ may be expressed as an expansion in powers of $(d/r)$, with the first two terms given by:
\begin{equation}\label{pert5}
U^4=\left(1+\frac{M_1+M_2}{r}\right)^4+\frac{4M_2d}{r^2}\left(1+\frac{M_1+M_2}{r}\right)^3\sin\theta + {\cal O}
\left[\left(\frac{d}{r}\right)^2\right]
\end{equation}

In the limit $(M_1+M_2)^2>>2M_2d$ the dynamics of the system is dominated by the radial term in the expansion
(\ref{pert5}), which is perturbed by the remaining angular terms. Thus to zero order approximation the space time associated
with the two extreme black holes becomes spherically symmetric. There exists a single unstable periodic orbit which is a
circle of radius $r=M\equiv(M_1+M_2)$ and period $T_\nu=M\pi/2$. The time evolution along this orbit is given by
\begin{eqnarray}\label{cir_orb1}
r_s(\nu)=M \hspace*{0.6cm}&&\hspace*{0.6cm} \theta_s (\nu)=\theta_0+\frac{4\nu}{M}   \\  \nonumber
p_{rs}(\nu)=0 \hspace*{0.6cm}&&\hspace*{0.6cm}  p_{\theta s}(\nu) = L
\end{eqnarray}
with angular momentum $L=4M$. The periodic orbits associated with the affine parameter and the killing time
are also a circle of radius $r=M$, but with periods $T_\mu=2\pi M$ and $T_t=8\pi M$ respectively.

To zero order approximation the escape rate of photons from an {\sl effective} black hole of mass $M$ can be determined
from the analysis of the linear stability of the circular orbit (\ref{cir_orb1}),
$X_s\equiv (r_s, \theta_s, p_{rs},  p_{\theta s})$. This implies the study of the time evolution of a small perturbation,
$\delta X\equiv(\delta r, \delta\theta,\delta p_r,\delta p_\theta)$, with respect $X_s$. In terms of the
Hamiltonian $h$, the linear differential equations that describe this evolution, $d\,(\delta X)/d\nu={\sf l}(\nu)\cdot\delta X$
with ${\sf l}(\nu)= \Sigma \cdot \frac{\partial^2 h}{\partial X^2}|_{X_s}$, are:
\begin{eqnarray}\label{cir_orb2}
\frac{d\,}{d\nu}
\left(\begin{array}{c}
\delta {r} \\ \delta {\theta} \\ \delta {p_r} \\ \delta {p_\theta}
\end{array}\right)=
\left(\begin{array}{cccc}
0 & 0 & 1 & 0 \\
-\frac{8}{M^2} & 0 & 0 & \frac{1}{M^2} \\
\frac{8}{M^2} & 0 & 0 & \frac{8}{M^2} \\
0 & 0 & 0 & 0 \end{array} \right) \cdot
\left(\begin{array}{c}
 \delta r \\ \delta \theta \\  \delta p_r \\ \delta p_\theta
\end{array}\right)
\end{eqnarray}
and the fundamental matrix ${\sf M}(X_s,\nu)$, solution of ${\sf M}'(X_s,\nu)={\sf l}(\nu)\cdot{\sf M}(X_s,\nu)$ with
${\sf M}(X_s,0)={\sf 1}$, is given by
\begin{eqnarray}\label{matrix_M}
{\sf M}(X_s,\nu)=\left(\begin{array}{cccc}
C(\nu) & 0 & \frac{M}{2\sqrt{2}}S(\nu) & \left[C(\nu)-1\right] \\
-\frac{2\sqrt{2}}{M}S(\nu) & 1 & \left[1-C(\nu)\right] &
\left[\frac{9}{M^2}\nu-\frac{2\sqrt{2}}{M}S(\nu)\right] \\
\frac{2\sqrt{2}}{M}S(\nu) & 0 & C(\nu) & \frac{2\sqrt{2}}{M}S(\nu) \\
0 & 0 & 0 & 1
\end{array}\right)
\end{eqnarray}
with
\begin{equation}
C(\nu)=\cosh\left(\frac{2\sqrt{2}}{M}\nu\right)
\hspace*{1cm}{\mbox {and}}\hspace*{1cm}
S(\nu)=\sinh\left(\frac{2\sqrt{2}}{M}\nu\right)
\end{equation}
The stretching factors associated with the circular orbit are obtained from the eigenvalues
of the matrix ${\sf M}(X_s,\nu)$,
\begin{equation}
\left\{ 1 , 1 , e^{-\frac{2\sqrt{2}}{M}\nu} , e^{\frac{2\sqrt{2}}{M}\nu} \right\},
\end{equation}
evaluated at one primitive period $T_\nu$. That is,
\begin{equation}
\Lambda= \left\{ 1 , 1 , e^{-\sqrt{2}\pi} , e^{\sqrt{2}\pi} \right\}.
\end{equation}
We recall that these stretching factors are invariant under time transformations, and therefore are
identical for the affine parameter and the killing time.
As expected, two of them are equal to one. The values $e^{-\sqrt{2}\pi}$ and $e^{\sqrt{2}\pi}$
are related to the stable and unstable manifolds of the unique unstable periodic orbit respectively.
The Lyapunov exponents associated with this orbit are given by
\begin{equation}
\lambda_\mu=\frac {\ln |\Lambda|}{T_\mu} =
\left\{ 0 , 0 , -\frac{1}{\sqrt{2}M}, \frac{1}{\sqrt{2}M} \right\}
\end{equation}
in the time evolution dictated by the affine parameter, and
\begin{equation}
\lambda_t=\frac {\ln |\Lambda|}{T_t} =
\left\{ 0 , 0 , -\frac{1}{4\sqrt{2}M}, \frac{1}{4\sqrt{2}M} \right\}
\end{equation}
in the evolution with respect to the killing time.
Thus to zero order the escape rate of photons from the unstable circular periodic orbit around the effective
extreme black hole of mass $M$  is determined by the leading exponent
$\gamma_\mu^{(0)}=1/\sqrt{2}M$ and $\gamma_t^{(0)}=1/4\sqrt{2}M=\gamma_\mu^{(0)}/4$.

In this system there is a unique unstable periodic. Since the dynamics on the
repeller is regular, the Kolmogorov-Sinai entropy is zero, $h_{KS}=\lambda-\gamma^{(0)}=0$. Thus the system
is hyperbolic but nonchaotic. The Pollicott-Ruelle resonances $\sigma_{pr}$ are given by the zeroes of the Zeta
function (\ref{zf}) associated with the single unstable periodic orbit,
\begin{equation}
Z(\sigma_{pr})=\prod_{k=0}^{\infty}\left(1-\frac{e^{-\sigma_{pr} T}}{|\Lambda|\Lambda^k}\right)^{k+1}=0,
\end{equation}
being $T$ the period $T_\mu$ or $T_t$. That is,
\begin{equation}\label{resonances_mu}
\sigma_{pr}^{(\mu)} (k,n)=-\frac{(k+1)}{\sqrt{2}M}+i\frac{n}{M}
\end{equation}
for the affine parameter and
\begin{equation}\label{resonances_t}
\sigma_{pr}^{(t)} (k,n)=-\frac{(k+1)}{4\sqrt{2}M}+i\frac{n}{4M}
\end{equation}
for the killing time, with $k=0,1,2,...$ and $n=0,\pm 1, \pm 2,...$
The resonances belong to the lower half plane of the complex plane $\sigma$ (Re $\sigma_{pr}<0$) and,
as occurs in the two-disk scatterer \cite{gaspard1998,gaspard2002}, their spectrum forms a semi-infinity periodic
array. Here the spacing along the (Re $\sigma$)-axis is given by the escape rate $\gamma_\mu^{(0)}=1/\sqrt{2}M$
$(\gamma_t^{(0)}=1/4\sqrt{2}M)$, and by the frequency $w_{\mu}=1/M$ $(w_{t}=1/4M)$ along the (Im $\sigma$)-axis.
These complex resonances play an important role in the time evolution of an statistical ensemble of photons; they
determine the different decay modes and their frequencies in a typical scattering process.
The ensemble dynamics is ruled by the resonances which are the closest to the imaginary axis $(k=0)$. The real
part of these leading resonances controls the exponential decay on the longest time scale, which defines the
escape rate of the system; and their imaginary parts give the frequencies of the oscillations that appear
superimposed on the gross exponential decay.

We now study the effect of the first angular term in the expansion (\ref{pert5}) on the zero-order circular orbit
(\ref{cir_orb1}) associated with the leading radial term. We still assume the limit $M^2=(M_1+M_2)^2>>2M_2d$,
in which this term can be treated as an perturbation to the spherically symmetric motion.
To make a first order perturbative analysis we write the $U^4$ function in the interaction term in the form
\begin{equation}\label{pert6}
U^4=\left(1+\frac{M}{r}\right)^4+\varepsilon\frac{4M_2d}{r^2}\left(1+\frac{M}{r}\right)^3\sin\theta
\end{equation}
where we have introduced a perturbative parameter $\varepsilon$, which is set equal to one at the end of the
analysis. The perturbed trajectory can be written as
\begin{eqnarray}\label{pert7}
& &r_1(\nu)=r_s(\nu)+\varepsilon r_c(\nu) \hspace*{0.6cm},\hspace*{0.6cm}
p_{r1}(\nu)=p_{rs}(\nu)+\varepsilon p_{rc}(\nu)  \nonumber \\
& &\theta_1(\nu)=\theta_s(\nu)+\varepsilon \theta_c(\nu) \hspace*{0.6cm},\hspace*{0.6cm}
p_{\theta 1}(\nu)=p_{\theta s}(\nu)+\varepsilon p_{\theta c}(\nu)
\end{eqnarray}
with first-order corrections to the circular orbit (\ref{cir_orb1}) that satisfy the equations
\begin{eqnarray}\label{pert8}
& & \frac{dr_c}{d\nu}=p_{rc}   \nonumber \\
& & \frac{d\theta_c}{d\nu}=\frac{1}{M^2}\left(p_{\theta c}-8r_c\right) \nonumber \\
& & \frac{dp_{rc}}{d\nu}=-\frac{56M_2d}{M^3}\cos\left(\frac{4\nu}{M}\right)+\frac{8}{M^2}\left(p_{\theta c}+r_c\right)
\nonumber \\
& & \frac{dp_{\theta c}}{d\nu}=-\frac{16M_2d}{M^2}\sin\left(\frac{4\nu}{M}\right).
\end{eqnarray}
The periodic solution of these equations gives the first-order perturbed orbit,
\begin{eqnarray}\label{pert9}
& & r_1(\nu)=M+\frac{dM_2}{M}\cos\left(\frac{4\nu}{M}\right) \nonumber \\
& & \theta_1(\nu)=\theta_0+\frac{4}{M}\nu-\frac{dM_2}{M^2}\sin\left(\frac{4\nu}{M}\right) \nonumber \\
& & p_{r1}(\nu)=-\frac{4dM_2}{M^2}\sin\left(\frac{4\nu}{M}\right) \nonumber \\
& & p_{\theta 1}(\nu)=4M+\frac{4dM_2}{M}\cos\left(\frac{4\nu}{M}\right).
\end{eqnarray}

This is a quasi-circular orbit that nearly reproduces the periodic orbit $\{+-\}$ in two black-hole
systems in which $(M_1+M_2)>d$, and the periodic orbit $\{--\}\,(\{++\})$ in systems where $(M_1+M_2)<d$,
see figure (\ref{pert_orb}).
\begin{figure}[h]
\includegraphics[width=0.48\textwidth]{case_A.eps}
\hspace*{0.3cm}
\includegraphics[width=0.48\textwidth]{case_D.eps}
\caption{Some of the periodic orbits in two systems of a black hole of mass $M_1$ perturbed by the interaction with
a black hole of mass $M_2$ located at distance $d$. The solid lines give the periodic orbits $\{++\}$, $\{--\}$, $\{+-\}$,
$\{+-++\}$ and $\{+\circ -\circ\}$, and the dashed line is the first-order periodic orbit (\ref{pert9}). The system
$(a)$ corresponds to $(\,M_1=1,\,M_2=0.01,\,d=2\,)$ and $(b)$ to $(\,M_1=7,\,M_2=0.1,\,d=2)$. In both systems the
main black hole of mass $M_1$ is located at $(0,0)$ and the black hole of mass $M_2$ at $(0,d)$.}
\label{pert_orb}
\end{figure}
The more complex periodic orbits (longer symbolic coding) are out of scope of our perturbative analysis. However,
in the limit $(M_1+M_2)^2>>2M_2d$  these are highly unstable periodic orbits that play a secondary role in the escape
rate of photons from the black holes. Here the escape rate is controlled by the quasi-circular periodic orbit,
$\{+-\}$ or $\{--\}\,(\{++\})$, which results from the slight deformation of the circular orbit of radius $M$.
Hence the analysis of the linear stability of the zero- and first-order periodic orbits, (\ref{cir_orb1}) and
(\ref{pert9}), should provide a good estimate of the escape rate of photons from a perturbed black hole. Indeed, as the
results in Table (\ref{tab_pert}) indicate, both the analytical escape rate $\gamma_\mu^{(0)}\,\,(\,\gamma_t^{(0)}\,)$ derived
from the analysis of the linear stability of the zero-order circular orbit (\ref{cir_orb1}) and the escape rate
$\gamma_\mu^{(1)}\,\,(\,\gamma_t^{(1)}\,)$  obtained from the leading eigenvalue of the fundamental matrix associated with
the first-order quasi-circular orbit (\ref{pert9}) are very close to the escape rate $\gamma_\mu^{(num)}\,\,(\,\gamma_t^{(num)}\,)$
calculated from the numerical simulations that consider an statistical ensemble of photons, see figure (\ref{pert_osc}).
\begin{table}
\begin{tabular}{|c||c|c|c||c|c|c|} \hline
 System &  $\gamma_\mu^{(0)}$ &  $\gamma_\mu^{(1)}$ & $\gamma_\mu^{(num)}$ &
 $\gamma_t^{(0)}$ &  $\gamma_t^{(1)}$ & $\gamma_t^{(num)}$\\ \hline\hline
$\,\,(\,M_1=1,\,M_2=0.01,\,d=2\,)\,\,$ & $\,\,0.70011\,\,$ &  $\,\,0.69559\,\,$ & $\,\,0.69258\,\,$
& $\,\,0.17503\,\,$ &  $\,\,0.17390\,\,$ & $\,\,0.17624\,\,$ \\ \hline
$\,\,(\,M_1=7,\,M_2=0.1,\,d=2\,)\,\,$ & $\,\,0.09959\,\,$ &  $\,\,0.09957\,\,$ & $\,\,0.09901\,\,$
& $\,\,0.02490\,\,$ &  $\,\,0.02489\,\,$ & $\,\,0.02436\,\,$ \\ \hline
\end{tabular}
\caption{Escape rate values for two systems of perturbed black holes. $\gamma_\mu^{(0)}\,\,(\,\gamma_t^{(0)}\,)$ is the
escape rate derived from the zero-order circular orbit (\ref{cir_orb1}), $\gamma_\mu^{(1)}\,\,(\,\gamma_t^{(1)}\,)$ is the
escape rate obtained from the analysis of linear stability of the first-order orbit (\ref{pert9}) and
$\gamma_\mu^{(num)}\,\,(\,\gamma_t^{(num)}\,)$ is the escape rate calculated from the numerical simulations that
consider an statistical ensemble of photons.}
\label{tab_pert}
\end{table}
As figure (\ref{nt2}) shows, the zero-order analytic estimate $\gamma_\mu^{(0)}\,\,(\,\gamma_t^{(0)}\,)$ even provides an
approximated value for the escape rate from a system of two black holes with similar masses.

In the numerical simulations with an statistical ensemble of photons, see figure (\ref{pert_osc}), the escape from the perturbed
black hole presents a smooth oscillation superposed on the gross exponential decay determined by the escape rate
$\gamma_\mu^{(num)}\,\,(\,\gamma_t^{(num)}\,)$.
The zero-order Pollicott-Ruelle resonances (\ref{resonances_mu}) and (\ref{resonances_t}) account for this behavior of the decay
curves; the exponential decay determined by the escape rate $\gamma_{\mu}\simeq \gamma_\mu^{(0)}\,\,(\,\gamma_t\simeq \gamma_t^{(0)}\,)$
is given by the real part of the leading resonances $k=0$, and the frequency of the oscillation
$w_\mu\simeq 1/M\,\,(\,w_t\simeq 1/4M\,)$ is given by the imaginary part of the leading resonance given by $n=1\,(k=0)$.
This is illustrated in figure (\ref{fourier}) which displays the Fourier spectra of the signals in figure (\ref{pert_osc}). It
clearly shows that the most intense peaks in the spectra, associated with the frequency of the main oscillation in the escape rate,
are localized pretty close to these frequencies predicted by the leading zero-order Pollicott-Ruelle resonances. These peaks
correspond to the frequency of the leading orbits, the orbit $\{--\}(\{++\})$ in case A and the orbit $\{+-\}$ in case B, see figure
(\ref{pert_orb}).
This is another indication of the leading role of the quasi-circular orbit that results from the slight deformation of the
circular orbit of radius $r=M$ in the escape dynamics of photons.
\begin{figure}[h]
\includegraphics[width=0.75\textwidth]{escape_rate_A_D.eps}
\vspace*{0.1cm}
\caption{The evolution of $[N(\mu)-N(\infty)]\,\,([N(t)-N(\infty)])$ with the affine parameter (killing time) for two
statistical ensembles in the perturbed systems of figure (\ref{pert_orb}). Panels $(a)$ and $(c)$ correspond to the system
$(a)$ in figure (\ref{pert_orb}), and panels $(b)$ and $(d)$ to the system $(b)$ in figure (\ref{pert_orb}).
The solid line gives the escape from a circle of control centered in between the two black holes, at $(0,d/2)$, and the dashed
lines the escape from a circle of control centered on the main black hole, at $(0,0)$. See the Section VII for comments.}
\label{pert_osc}
\end{figure}

\begin{figure}[h]
\includegraphics[width=0.75\textwidth]{fourier_analysis2.eps}
\vspace*{0.1cm}
\caption{The Fourier transform of the four signals in figure (\ref{pert_osc}). The dashed lines localize the frequency of
the main oscillation in the escape rate predicted by the leading zero-order Pollicott-Ruelle resonances, $w_\mu\simeq 1/M\,\,
(\,w_t\simeq 1/4M\,)$.}
\label{fourier}
\end{figure}

\section{Conclusions}

We have studied the scattering of photons in the Majumdar-Papapetrou static spacetime of two extreme Reissner-Nordstr\"om black
holes held fixed in space due to the balance between the gravitational attraction of their masses and the electrostatic repulsion
of their charges. We have identified the set of unstable periodic orbits that form part of the fractal repeller which fully
characterizes the chaotic escape dynamics of the photons from the two black holes. These orbits were classified according to a
symbolic coding.

The linear stability of the dynamics, in particular the unstable periodic orbits, was analyzed using their stretching factors,
which are given by the eigenvalues of the fundamental matrix integrated up to one primitive period of the orbits. With the
primitive periods and stretching factors of the periodic orbits we determined the topological pressure, and from this function and
its first derivative the main quantities of chaos that characterize the fractal repeller. In particular the escape rate, which was
calculated using the trace formula for hyperbolic flows of Cvitanovic and Eckhardt. This escape rate derived from the periodic
orbit theory was in good agreement with the value obtained from numerical simulations that analyze the escape dynamics of
an statistical ensemble of photons.

From the study of the linear stability of the dynamics we also identified the periodic orbit that plays the leading role
in the escape of photons from the two black-hole configuration. In systems with two identical extreme Reissner-Nordstr\"om black
holes the escape is mainly controlled by the outermost periodic orbit $\{+-\}$ that encircles the two black holes.
In systems where a main black hole of mass $M_1$ is perturbed by the weak interaction with a black hole of mass $M_2$,
the escape of photons is dominated by the quasi-circular orbit that results from the slight deformation of the circular orbit
of radius $r=M=(M_1+M_2)$.
Depending on the separation distance between the two black holes this leading orbit can cover the whole system, and corresponds
to the periodic orbit $\{+-\}$, or only encircles the main black hole, and is given by the periodic orbit $\{++\}\,(\{--\})$.

In systems of a perturbed black hole, the analysis of the linear stability of the unique zero-order periodic orbit in a
perturbative treatment provides an analytic estimate of the escape rate,
$\gamma_\mu^{(0)}=1/\sqrt{2}M\,\,(\,\gamma_t^{(0)}=1/4\sqrt{2}M\,)$, which is in good agreement with numerical value derived
from the analysis of the linear stability of the first-order perturbative periodic orbit, and also with the escape rate obtained
from the time evolution of an statistical ensemble of photons.
This analytical value even provides an approximated value for the escape rate of photons from a system of two extreme
Reissner-Nordstr\"om black holes with similar masses. And therefore a way to estimate the mass of a black hole from the escape
rate of photons.

\begin{acknowledgments}

D. Alonso thanks C.P. Dettmann for fruitful discussions. We thank G. Contopoulos and N.J. Cornish for pointing out important
references. Financial support has been provided by Ministerio de Educaci\'on y Ciencia (FIS2004-05687, FIS2005-02886,
FIS2007-64018) and Gobierno de Canarias (PI2004/025).

\end{acknowledgments}

\section*{References}

redpajama_set_name: RedPajamaArXiv


text:
Edinburgh 9°c
Our Instagram feedsInstagram
Media / BBC
Scottish Greens
The Jouker
FMQs
Yes Groups
Yes DIY
Scotland Means Business
Big Enough, Rich Enough, Smart Enough
Power to the people: The significance of the Declaration of Arbroath explained
The significance and power of the Declaration of Arbroath
By Alan Taylor
One Celtic Connections performance will pay tribute to the Declaration
THE past, as we are regularly reminded, is a foreign country where things are done differently. Meanwhile, the distant past, the past far beyond living memory, is not so much foreign as ungraspable. This is how Scotland of 700 ago seems to me.
Historians tell us that it was largely tree-covered and uncultivated and that people were relatively few. Where they existed roads were rough, towns were more like hamlets, and when darkness descended only the intrepid and foolhardy ventured forth.
Glasgow, for example, in the 14th century, was a huddle of rudimentary buildings situated on the banks of the Clyde. Edinburgh, albeit already accepted as the country's capital, was centred on the Castle Rock around which the population huddled for protection. In contrast, say, to Florence and Tuscany, where the Renaissance was already in full swing, Scotland was a primitive backwater.
The reasons for this are many and complex, involving not only history but geography. For many foreigners, and indeed for many Scots, much of the place that came later to be known as "North Briton" was terra incognita. In my mind's eye, I imagine it as the Wild West was to trailblazers. Who knew what lay beyond the pockets of what passed for civilisation? Shakespeare, writing in the 16th century, may never have ventured across the Border but Macbeth gives a good impression of what others believed Scotland to be like: "Bleed, bleed, poor country!" howls the unfortunate Macduff. "Great tyranny, lay thou thy basis sure, For goodness dare not check thee."
READ MORE: Declaration of Arbroath to go on public display in Edinburgh
BUT whatever Scotland's failings, it did, even by the 13th century, have a clear sense of itself as a nation. William Wallace, for example, was described in 1297, as "a true champion of the kingdom for the independence of its people". The country was coalescing, becoming less factional and fractious and, consequently, more governable.
It was a kingdom, albeit one tightly controlled by a small group of people. As the historian Michael Lynch has remarked, Scotland had a "a distinct sense of its own identity", which was carefully scrutinised and examined. "Kings," added Lynch, "were tested as never before, in their dual roles as warlord and good lord of their subjects. The peoples of the kingdom – from magnates to freeholders, husbandmen or burgesses – were for the first time dragged on to the national stage for a long period."
Of course war with England was a constant threat, and a debilitating and distracting factor. For decades Scotland lived with the neurosis of attack from its nearest neighbour – as South Korea does today from North Korea.
This is the context in which the Declaration of Arbroath was composed. What are now known as the Wars of Independence raged for years. The initiator was Edward I, who, in Flower of Scotland, must be sent homeward to think again. Seizing the opportunity of a power vacuum, he laid claim to Scotland, wreaking havoc and killing wantonly.
The Stone of Destiny was seized and taken to Westminster Abbey. Scotland, as well as Wales and Ireland, was to be subjected to English rule. Subsequently, in 1305, Pope Clement V recognised Edward's claim of overlordship of Scotland and excommunicated Robert the Bruce for the murder of his rival John Comyn in Dumfries in 1306.
It was in a bid to have Bruce's excommunication lifted that the 39 signatories of the Declaration of Arbroath – eight earls and 31 barons – petitioned the new pope, John XXII, who was then based in Avignon in France. Written originally in Latin just six years after the Battle of Bannockburn, it eloquently expressed the heartfelt anxieties and wishes of the country's leaders. Who its principle author was we cannot be sure, though it has been popularly assumed to have been one Bernard de Linton, Abbot of Arbroath. Its style is ornate, as papal protocol demanded, but its sentiments are clear, as articulated in its most famous and sonorous passage: "Yet if he [Robert the Bruce] should give up what he has begun, seeking to make us or our kingdom subject the King of England or the English, we should exert ourselves at once to drive him out as our enemy and a subverter of his own right and ours, and make some other man who was well able to defend us our King; for, as long as a hundred of us remain alive, never will we on any conditions be subjected to the lordship of the English. It is in truth not for glory, nor riches, nor honours that we are fighting, but for freedom alone, which no honest man gives up but with life itself."
READ MORE: Declaration of Arbroath story 'comes to life' on National Trust site
In all, the Declaration runs to a little more than 1200 words. Like a newspaper editorial, it took a point of view and stuck to it.
The origins of the Scots, the signatories informed Pope John, could be traced "from Greater Scythia by way of the Tyrrhenian Sea and the Pillars of Hercules". For a while, they continued, our ancestors dwelt in Spain "among the most savage of peoples". Never, however, were they subdued.
Thereafter, 1200 years after the people of Israel crossed the Red Sea, the ancient Scots made their way north to what would become their homeland. Often "assailed" by would-be usurpers, including the Norwegians, the Danes and the English, they fought them all off, holding it "free of all servitude ever since", "through the reigns of 113 kings of their own royal stock, the line unbroken by a single foreigner".
What the pope made of all of this is anyone's guess. What we do know is that there was no rush to respond. Eight years passed before a reply, sanctioned by John XXII, lifting Bruce's excommunication, was received. The Declaration – today accepted as "the most celebrated document in Scottish history" and "a surrogate Scottish constitution" – had served its purpose but soon sank into obscurity. Centuries passed before it was disinterred and its near-mythic status was achieved.
Much of its enduring appeal, argues Lynch, is due to the simplicity of its language, condensing the mythology of the nation's past with "a compelling vision of the relationship of kings of Scots and the Scottish people" and summarising "the history of the present struggle" as one in which the key issue was liberty. It is an idea to which everyone in the country, irrespective of their position, can relate.
SIMILARLY, the tone of letter (it was not designated a Declaration until the 20th century), while respectful, demonstrates that no person is above reproach, not even a king, who owes his position to the will of the people.
It is they who will decide whether he is worthy of their support. They will not tolerate any treachery or decline in standards. There must be no wheeling and dealing for personal favour. It is a precursor of the age-old, homespun attitude to those who believe themselves superior, summed up in the phrase "ah kent your faither". Even kings, the Declaration implies, are subjects and must be judged as such.
READ MORE: Never forget the Scottish people are sovereign
Its sentiments run through Scottish life. At its core is the word freedom – libertas in Latin. Defining it is not easy. Freedom from what? Tyranny? Slavery? An unwelcome union? Societal constraints? Conventions? Expectations?
"Freedom's just another word for nothin' left to lose," sings Kris Kristofferson in Me and Bobby McGee. "A life without freedom to choose is not worth having," wrote the recently deceased Alasdair Gray. In Hamish Henderson's poem, The Freedom Come-All-Ye, it is seen as a universal right, in which the peoples of the world are united, irrespective of creed or colour or class: "So come all yi' at hame wi' freedom, Never heed whit the hoodies croak for doom.
In your hoose a' the bairns o' Adam Can find breid, barley-bree and painted room."
Bemused visitors to Sandy Bell's bar in Edinburgh would often be treated to its composer singing it a cappella and lustily like an anthem that belonged to all countries. In that regard Henderson was reimagining the Declaration for a century in which there had been two calamitous wars. It was a call not to arms but to lay them down and work in the common good. It also contained more than an echo of Robert Burns's A Man's A Man for A' That, which Sheena Wellington sang at the reconvening of the Scottish Parliament in 1999: A prince can mak a belted knight, A marquis duke, an' a' that; But an honest man's abon his might, Gude faith, he maunna fa' that!
For a' that, an' a' that, Their dignities an' a' that; The pith o' sense, an' pride o' worth, Are higher rank than a' that.
Burns may not have read the Declaration of Arbroath but Hamish Henderson most certainly did. The rise of nationalism in the twentieth century led to a revival of interest in such documents. Paul Henderson Scott, the nationalist commentator, described it in 1997 as "the foundation of Scottish political thought" and traced its influence on countless other pronouncements, including several Claims of Rights.
It was also believed to have helped shaped the American Declaration of Independence, given the contribution of Scots Americans to the drafting of that document. But while some historians believe this credible others dismiss it. Suffice it to say it divides opinion.
As, indeed, does the interpretation of the document. As time passes, the meaning of words change. What people thought and believed in the 14th century is very different from our perspective. To that extent, the past defies comprehension and resists imagination. It is the backdrop of historical novelists and professional historians. In his book The Scottish Enlightenment: The Scots' Invention of the Modern World, the American historian Arthur Herman notes that William Robertson makes no mention of the Declaration of Arbroath in his History of Scotland, published in 1759.
This may well be, of course, because the period covered by Robertson was not the 14th century but the second half of the 16th. Herman argues that Robertson ignored the Declaration "not because he was a brainwashed Anglophile, but because he saw it in a historical context, as a well-worded defence of the old Scottish feudal regime by its oligarchic regime." But if that was the case, then why didn't Robertson say so?
It all depends to some extent on which side of the independence argument you sit. History is as open to distortion and revision and manipulation as other human endeavours.
The latest event to take the Declaration as its inspiration is musical. Celebrating the 700th anniversary of its writing, the 80-strong GRIT orchestra will perform The Declaration on the opening night of the forthcoming Celtic Connections. GRIT's starting point is the music of the late Martyn Bennett, says the orchestra's conductor, Greg Lawson. The musicians he has gathered are drawn from all corners of the musical spectrum, encompassing jazz, folk, classical and others. To give it structure, Lawson has created a "loose narrative" through which he has threaded the Declaration's words. "I picked people I'd worked with in different genres," he says. "Who's into the idea and who has the curiosity." The result, he adds, will be both a celebration of difference and a recognition of commonality with "freedom" the pivotal concept. It will be a declaration for a new decade, free from constraints, fear and received ideas. It will also be one helluva noise.
The Declaration by The GRIT Orchestra is at Glasgow Royal Concert Hall, 7.30pm, January 16
Lesley Riddoch: Irish will carry on flourishing after we're dragged from EU
Andrew Tickell: Guantanamo Bay, where the ends justify the torturous means
Boris Johnson's Brexit deal reaches last hurdle but SNP plan protest
Tories scrap bill changes that gave legal protection to lone child refugees
Most Scots' disability payment claims are refused by Westminster
Top analyst calls for Scotland to 'fully embrace' the EU and euro
Scotland will have a vital part to play in the EU so must embrace the euro
ScotRail under fire for 'disappointing' increase in stop skipping
Spain criticised by Russia and other UN member states over civil rights abuses

redpajama_set_name: RedPajamaCommonCrawl


text:
Joseph Murray –conocido como Joe Murray– (Mánchester, 3 de enero de 1987) es un deportista británico que compitió por Inglaterra en boxeo. Ganó una medalla de bronce en el Campeonato Mundial de Boxeo Aficionado de 2007, en el peso gallo.

En marzo de 2009 disputó su primera pelea como profesional. En su carrera profesional tuvo en total 26 combates, con un registro de 23 victorias y 3 derrotas.

Palmarés internacional

Referencias

Boxeadores del Reino Unido
Boxeadores en los Juegos Olímpicos de Pekín 2008
Deportistas del Reino Unido en los Juegos Olímpicos de Pekín 2008
Nacidos en Mánchester
Británicos del siglo XXI

redpajama_set_name: RedPajamaWikipedia


JDL Strategies' 'Financial Wealth Check' combines decades of finance industry experience and employs proprietary software to analyse and present individuals' results in a uniquely powerful manner.
It only takes 5 minutes and the results will show you if you are on track to financial freedom.
This week, Julio de Laffitte, CEO of JDL Strategies which specialise in wealth management and property investing, has offered to waive the $220 fee to visitors from ShareTradingEducation.com.


text:
Korea to reduce securities transaction taxes in June
PUBLISHED :April 15, 2019 - 18:38
UPDATED :April 17, 2019 - 11:17
South Korea's Deputy Prime Minister and Finance Minister Hong Nam-ki has unveiled plans to cut the securities transaction taxes by up to 0.2 percentage point starting June to boost the sluggish stock market.
All eyes are now on how the administration plans to offset the projected 1.4 trillion won ($1.24 billion) tax losses following the first major stock transaction tax cut in over two decades.
From June 3, trading securities on Korea's main Kospi index and second-tier bourse Kosdaq will be lowered by 0.05 to 0.25 percent, Hong said on April 12 during his visit to Washington DC for meetings of G-20 finance ministers and central bank governors last week.
The rate for trading on the third-tier bourse Konex will be cut by 0.2 percentage point to 0.1 percent. Both the Kosdaq and Konex are dedicated to supporting fundraising for small and medium-sized companies, but Konex-listed firms have less rigorous listing requirements.
As for over-the-counter stock transactions, the proposed rate cut to 0.45 percent per transaction from the current 0.5 percent will take effect following a law revision next year.
The government is facing with growing calls for such tax reliefs amid dampened investor sentiment.
After the ruling Democratic Party proposed scrapping the securities transaction tax by 2024 to stimulate the market through a series of rate cuts, Financial Services Commission Chairman Choi Jong-ku promptly unveiled a relief package including stock transaction tax rate cuts.
In March, the Finance Ministry hinted it will come up with ways to impose capital gains tax on a wider range of investors by the first half of next year to help offset tax revenue losses from the securities transaction tax cut. Currently, capital gains tax from returns on stock investment equals the lower of the two -- 22 percent of net capital gains or 11 percent of gross sales proceeds.
However, the capital gains tax regime in Korea will be difficult to comply with the potential revision to increase tax revenue through tougher capital gains tax rules, without the centralized infrastructure system to track and calculate gains of foreign investors, said Patrick Pang, managing director and head of fixed income and compliance at Asia Securities Industry & Financial Markets Association.
"We do want to emphasize the importance of not letting a securities transaction tax reduction impact the treatment of capital gains tax as its levy is operationally very challenging," he said.
By Son Ji-hyoung (consnow@heraldcorp.com)