text:
\section{Introduction}\label{sec:intro}

A celebrated  result proved
by Poincaré in the beginning  of the last century
put on evidence the relation intertwining  the {\em (linear and exponential) instability\/} of a
closed geodesic (as a critical point of the geodesics energy functional on the free loop space) of an oriented Riemannian two-dimensional manifold and its  Morse index. The literature on this criterion is quite broad.  We refer the interested reader to \cite{Poi99,  HS10, Bol88, BT10} and references therein.

Loosely speaking,  a closed geodesic $\gamma$
on $\MM$ is termed {\em linearly stable\/}  if the monodromy matrix  associated to
$\gamma$ splits into two-dimensional rotations. Accordingly, it is diagonalizable and  all Floquet multipliers
belong to  the unit circle $\U$  of the complex plane $\C$. Additionally,
if $1$ is not a Floquet multiplier, we term $\gamma$  {\em non-degenerate\/}. Thus, if $\gamma$ is a stable closed geodesic, then all orbits of the geodesic flow
near $\dot \gamma$ in $T_\gamma\MM$ stay near $\dot \gamma$ for all times. (Cf. Figure \ref{fig:geodetiche}).
\begin{figure}[ht]
 \centering
 \includegraphics[scale=0.15]{Closed_Geodesic.pdf}
 \caption{{\color{blue}{Stable}} closed geodesics:{\color{blue}{ nearby
geodesics}} stay in a neighborhood
 of the {\color{red}{closed geodesic $\gamma$}}. Then the
linearized  Poincaré map is diagonalizable and any eigenvalue $\lambda$ is on the unit circle.
 {\color{blue}{Unstable}} closed geodesics: {\color{blue}{nearby geodesics}} have the tendency to diverge. (Image credit to  Hans-Bert Rademacher for the picture available at pag.57 of  \url{http://www.math.uni-leipzig.de/~rademacher/tianjin.pdf}). }\label{fig:geodetiche}
\end{figure}

In  1988 the  Poincaré instability criterion for closed geodesics
was generalised  in several interesting directions by
D. V. Treschev in \cite{Tres88}.  In this paper the author describes  the connection intertwining
the {\em Morse index\/} of a closed non-degenerate Riemannian geodesic
$\gamma$ and the spectrum of the Poincaré map.  More precisely, denoting by $\iota_{\scriptscriptstyle{\mathrm{Mor}}}(\gamma)$
the Morse index of $\gamma$ as a critical
point of the geodesic energy functional on the free loop space of the
$(n+1)$-dimensional Riemannian manifold $\MM$,    the author proved that
if either  $\gamma$ is  a non-degenerate oriented closed geodesic such that
$n+\iota_{\scriptscriptstyle{\mathrm{Mor}}}(\gamma)$ is odd or  $\gamma$ is nonoriented non-degenerate closed
geodesic and $n+\iota_{\scriptscriptstyle{\mathrm{Mor}}}(\gamma)$ is even then $\gamma$ is linearly unstable. Several years later,  the first author and his collaborator in \cite{HS10} proved  a
generalization of the aforementioned  result, dropping the non-degeneracy assumption.  Of the same flavor is a
very recent instability
result proved by the first author and his collaborator in the case of
Hamiltonian systems. (We
refer the interested reader to \cite[Subsection 4.3, pag. 762]{HS09}).
Due to its interest in dynamical systems, a big effort has been given in the
investigation of
stability properties of closed geodesics on Riemannian manifolds as well as of
periodic solutions of more general Lagrangian  systems (cf. \cite{LL02,BJP14,
BJP16, HS09}
and references therein) under the standard Legendre convexity condition.

Dropping the positivity
assumption of the metric tensor is a quite challenging task. The first
problem  is that the critical points of the geodesic energy functional, have in general infinite Morse index and  co-index.  However, in this {\em strongly indefinite\/} situation a natural substitute of the Morse index is represented by
a topological invariant known in literature as the {\em spectral flow\/}.

The spectral flow is naturally associated to a path of  selfadjoint Fredholm operators  arising from the second variation along the geodesic $\gamma$. This invariant was introduced by Atiyah, Patodi and Singer in \cite{APS76} in their study of index theory on manifolds
with boundary and since then many interesting properties and applications have been established. (Cf., for instance, \cite{PPT04, MPP07, MPW17, HP17b, PW16} and references therein).  In general the
spectral flow depends on the homotopy class of the whole path and not only on its ends. However in the special case of geodesics on semi-Riemannian manifolds things are simpler since it depends only on the endpoints of the path and therefore it can be considered a relative form of Morse index known in literature as  {\em relative Morse index.\/}

It is well-known that this invariant is strictly related to a  symplectic
invariant known in literature as
{\em Maslov index\/}; by using these two integers, several  generalization of
the celebrated
Morse Index Theorem are available. (We refer the interested reader to
cf. \cite{APS08, MPP05, MPP07, HS09, GPP03, LZ00a, LZ00b,Por08, RS95} and references
therein). Very recently new spectral flow formulas has been established in the study of heteroclinic and
homoclinic orbits of Hamiltonian systems. (Cfr. \cite{BHPT17, HP17a, HPY17}).
{{\subsection*{Acknowledgements}

  The second named author warmly thanks  all faculties and staff at the Mathematics Department of the Shandong  University (Jinan)  for providing excellent working conditions during his stay.

  We are deeply grateful to the anonymous referees for their careful reading of our manuscript and their insightful comments and suggestions.
}}
\vskip1truecm

\section{Description of the problem and main
results}

The aim of this section is to describe the problem, the main results and to introduce some basic definitions and finally to fix our  notation.

Let $(\MM, \mathfrak g)$ be a $(n+1)$-dimensional semi-Riemannian
manifold, where $\mathfrak g$ is a metric
tensor of  index,  $\mathrm{n_-}(\mathfrak g)=: \nu \in \{0, \dots, n+1\}$. If $\nu=0$ (resp. $\nu=1$) the
pair $(\MM, \mathfrak g)$  defines a Riemannian (resp. Lorentzian) manifold.
Let  $\nabla$ denote the Levi-Civita connection of $\mathfrak g$ and  let
$\mathcal R$ be the corresponding curvature tensor chosen with the following
sign convention
\begin{equation}
\mathcal R(X,Y)=[\nabla_X, \nabla_Y]- \nabla_{[X,Y]}
\end{equation}
where $[\nabla_X, \nabla_Y]\= \nabla_X \nabla_Y-\nabla_Y \nabla_X$. Denoting  by $\tfrac{\mathrm{D}}{\mathrm{d}t}$ the covariant derivative, we recall that
a {\em closed geodesic\/} on $M$ is a smooth solution $\gamma\colon
[0,T] \rightarrow M$ of  the following  problem
\begin{equation}\label{eq:geo}
\begin{cases}
 \tfrac{\mathrm{D}}{\mathrm{d}t} \dot \gamma(t)=0, \qquad t \in [0,T]\\
 \gamma(0)= \gamma(T)\\
 \dot\gamma(0)=\dot \gamma(T)
\end{cases}
 \end{equation}
 where $\cdot $ denotes the time derivative. It is well-known that, if $\gamma:[0,T]\to M$ is a geodesic, then there exists a
constant $e_\gamma$ such that
\begin{equation}\label{eq:energia-intro}
e_\gamma \= \mathfrak g\big(\dot  \gamma, \dot \gamma).
\end{equation}
The value of $e_\gamma$ given in Equation \eqref{eq:energia-intro} determines the causal
character of the geodesic; more precisely,
$\gamma$ is termed {\em timelike, lightlike\/} or {\em spacelike\/} if
$e_\gamma$ is negative, zero or
positive, respectively. Given a (closed) geodesic $\gamma$, a  {\em Jacobi field\/} is a
smooth  vector field $\xi$
along $\gamma$  that satisfies the second order linear differential equation
\begin{equation}\label{eq:jacobi-intro}
-\tfrac{\mathrm{D}^2}{\mathrm{d}t^2} \xi  (t) +\mathcal R(\dot \gamma (t) , \xi  (t)) \dot \gamma (t) =0, \qquad
t\in [0,T].
\end{equation}
We denote by
$\mathfrak P_\gamma: T_{\gamma(0)}M \oplus T_{\gamma(0)}M \to  T_{\gamma(0)}M
\oplus T_{\gamma(0)}M$ the map
 defined by
$
\mathfrak P_\gamma(v, v')= \left(\xi(T), \tfrac{\mathrm{D}}{\mathrm{d}t} \xi(T)\right)
$
where $\xi$ is the unique Jacobi field along $\gamma$ such that $\xi(0)=v $ and $\tfrac{\mathrm{D}}{\mathrm{d}t} \xi(0)=v'$.
We  assume that $\gamma$ is a closed spacelike (resp. timelike) geodesic, namely for
any $t\in[0,T]$, $\mathfrak g(\dot{\gamma}(t),\dot{\gamma}(t))=\abs{\dot \gamma(t)}^2$
(resp. $\mathfrak g(\dot{\gamma}(t),\dot{\gamma}(t))=-\abs{\dot \gamma(t)}^2$ ) . We let $e_{0}(0)\=\frac{\dot{\gamma}(0)}{|\dot{\gamma}(0)|}$ if $\gamma$ is spacelike
and $e_{n-\nu+1}(0)\=\frac{\dot{\gamma}(0)}{|\dot{\gamma}(0)|}$ if $\gamma$ is timelike. Then we can find $n$ linearly
independent  $\mathfrak g$-orthonormal vectors in $T_{\gamma(0)}M$ such that the following set
 \begin{multline}
 \big\{e_{1}(0),\ldots,e_{n}(0)\big\} \textrm{ in the spacelike case } \\ \big(\textrm{resp. } \big\{e_{0}(0),\ldots,e_{n-\nu}(0), e_{n-\nu+2}(0),\ldots,e_{n}(0)\big\}
\textrm{ in the timelike case}\big)
 \end{multline}
   is a basis of the subspace $\displaystyle
N_{\gamma(0)}M=\Set{v\in T_{\gamma(0)}M\mid \mathfrak g(v,\dot{\gamma}(0))=0}.$
Since parallel transport along $\gamma$ is
a $\mathfrak g$-isometry, then it follows that $\{e_{1}(t),\ldots,e_{n}(t)\}$ is a $\mathfrak g$-orthonormal basis of $N_{\gamma(t)}M$ in the spacelike case whilst  $\{e_{0}(t),\ldots,e_{n-\nu}(t), e_{n-\nu+2}(t), \ldots, e_n(t)\}$ is a $\mathfrak g$-orthonormal basis of $N_{\gamma(t)}M$ in the timelike case.
Let us denote by $\mathfrak P_\gamma^{\perp_\mathfrak g}:
N_{\gamma(0)} \oplus N_{\gamma(0)}\to N_{\gamma(0)} \oplus N_{\gamma(0)}$ the restriction to the
normal subspace at $\gamma(0)$ of  the
linearized {\em Poincaré map\/} defined by $
 \mathfrak P_\gamma^{\perp_\mathfrak g}(v, v')= \left(\xi(T), \tfrac{\mathrm{D}}{\mathrm{d}t} \xi(T)\right)$
where $\xi$ is the unique Jacobi field $\mathfrak g$-orthogonal to $\dot \gamma$
such that
$\xi(0)=v$ and $\tfrac{\mathrm{D}}{\mathrm{d}t} \xi(0)= v'$.
\begin{defn}\label{def:stability-intro}
A closed semi-Riemannian geodesic $\gamma$  is termed
{\em linearly stable\/} if the linearized Poincaré map $\mathfrak P_\gamma^{\perp_\mathfrak g}$
is semisimple and its spectrum $\sigma(\mathfrak P_\gamma^{\perp_\mathfrak g}) $ lies on the unit circle $\U$ of the
complex plane. Otherwise it is termed {\em linearly unstable\/}.
\end{defn}
 If
 $\gamma \in \Lambda\MM$  is a (non-constant) closed geodesic in $(\MM, \mathfrak g)$
we can introduce, by means of the parallel transport,  a  trivialization of the
pull-back bundle $\gamma^{*}(T\MM)$ of $T\MM$ along $\gamma$
by choosing a parallel $\mathfrak g$-orthonormal frame $\mathfrak E$
along $\gamma$ given by $(n+1)$-parallel linearly independent vector fields
$ e_0, \dots  e_n$ and by means of this parallel
trivialization,  the metric tensor $\mathfrak g$ reduces to the constant
indefinite scalar product $g$ in $\R^{n+1}$ of constant index $\nu$.
Writing the Jacobi  vector field along $\gamma$ in local coordinates as $\xi(t)=\sum_{i=0}^{n}u_{i}(t)e_{i}(t)$, inserting the above expression
into the Equation \eqref{eq:jacobi-intro} and by taking the $\mathfrak g$-scalar product with $e_j$, we reduce
it to the linear second order system of ordinary differential equations
\begin{equation}\label{eq:Space-Original-intro}
 -\bar{G}\ddot u (t) + \bar{R}(t)u(t)=0, \qquad t \in [0,T]
\end{equation}
where $\bar R_{ij}\= \mathfrak g\big(\mathcal R(\dot \gamma, e_i) \dot \gamma, e_j\big)$ and $\bar{G}= \begin{bmatrix}I_{n+1-\nu} & 0\\0 &-I_\nu \end{bmatrix}$. Being the map $\mathcal R(\dot \gamma, \cdot) \dot\gamma $
$\mathfrak g$-symmetric, it follows that the matrix $\bar R=[\bar R_{ij}]_{i,j=0}^n$ is symmetric; moreover in the spacelike case,
$ \bar{R}_{0i}(t)=\bar{R}_{i0}(t)=0$ for any $i=0,\dots,n$.
Since the Jacobi field $\xi$ is
$T$-periodic, inserting the local expression of $\xi$ into the equation $\xi(0)=\xi(T)$, we get the following boundary
condition for $u$:
\begin{equation}
u(0)=\bar A\,u(T)   \textrm{ where }  \bar{A}=[a_{ij}]_{i,j=0}^n.
\end{equation}
It is worth to observe that, since $\dot{\gamma}(0)=\dot{\gamma}(T)$, then we have $a_{00}=1$ and $a_{0j}=0$ for any $j=1,\dots,n$. Furthermore, by a direct
computation it follows also  that $\bar A$ is $\mathfrak g$-orthogonal.
By construction, the Morse-Sturm system  given in Equation \eqref{eq:Space-Original-intro} decouples into a scalar differential
equation (corresponding to the Jacobi field along $\dot \gamma$) and a differential system in $\R^n$
(corresponding to the restriction of the Jacobi deviation equation to vector fields $\mathfrak g$-orthogonal
to $\gamma$). Similarly in the timelike case, we have
\begin{multline}
\bar R_{(n-\nu+1)i}(t)=\bar R_{i(n-\nu+1)}(t)=0  \textrm{ and }\\
a_{(n-\nu+1)(n-\nu+1)}=1, \ a_{(n-\nu+1)j}=0 \textrm{ for } j \neq n-\nu+1.
\end{multline}
Depending on the causal character of $\gamma$, we can distinguish two cases:
\begin{itemize}
\item if $\gamma$ is {\em spacelike\/} then we have
\begin{equation}\label{eq:Space-Reduced-space-intro}
-G \ddot u(t)+\widehat{R}(t)u(t)=0, \qquad t \in [0,T]
\end{equation}
where $u(t)=\trasp{(u_{1}(t),\ldots,u_{n}(t))},  G\= \begin{bmatrix}I_{n-\nu} & 0\\0 &-I_\nu \end{bmatrix}$
and $\widehat{R}(t)=[ \bar{R}_{ij}(t)]_{i,j=1}^n$. Correspondingly the matrix $\bar A$ reduces to
\[
A=[a_{ij}]_{i,j=1}^n.
\]
\item if $\gamma$ is {\em timelike\/} then we have
\begin{equation}\label{eq:Space-Reduced-time-intro}
-G \ddot  u(t)+\widehat{R}(t)u(t)=0, \qquad t \in [0,T]
\end{equation}
where $u(t)=\trasp{(u_{0}(t),\dots, u_{n-\nu}(t), u_{n-\nu+2}, \dots, u_{n}(t))},$
$G\ =\begin{bmatrix}I_{n+1-\nu} & 0\\0 &-I_{\nu-1} \end{bmatrix}$ and $\widehat{R}(t)=[ \bar{R}_{ij}(t)]_{i,j=0}^n$ and
$i,j \neq n-\nu+1$. In this case the matrix $\bar A$ reduces to
\[
A=[a_{ij}]_{i,j}^n \textrm{ where } i,j =0, \dots, n \textrm{ and } i,j \neq n-\nu+1.
\]
\end{itemize}
Now, we are entitled to introduce the following definition.
\begin{defn}\label{def:oriented}
A closed geodesic $\gamma$ is termed oriented if $\det A= 1$; non-oriented if $\det A= -1$.
\end{defn}
\begin{rem}
We observe that the orientation of a geodesic $\gamma$ is independent either on the signature of $\mathfrak g$ or on the causal character of $\gamma$.
\end{rem}
\begin{note}
In short-hand notation, in the rest of the paper we will denote either a spacelike or a  timelike geodesic with
the same symbol, if not explicitly stated.
\end{note}
Given a spacelike (resp. timelike) closed geodesic $\gamma$, let
$t \mapsto \Psi(t)$ be the {\em flow\/} of the Morse-Sturm system
given in Equation \eqref{eq:Space-Reduced-space-intro} (resp. Equation \eqref{eq:Space-Reduced-time-intro});
thus  for every $t \in [0,T]$, $\Psi$ is the unique linear isomorphism of $\R^n \oplus \R^n$ such that $\displaystyle
 \Psi(t)\big(G\dot u(0), u(0)\big)= \big(G\dot u(t), u(t)\big),$
where $u$ is a solution of Equation \eqref{eq:Space-Reduced-space-intro}
(resp. Equation \eqref{eq:Space-Reduced-time-intro}).
We observe that $\Psi$ is a
smooth curve in the general linear
group of $\R^n \oplus \R^n$ satisfying the matrix differential equation $\dot
\Psi (t)= K(t)\Psi(t)$ with
initial condition $\Psi(0)=I$ where $K$ is given by $
K(t)\=\begin{bmatrix}
       0 & \hat{R}(t) \\
       G& 0
      \end{bmatrix}.$
The symmetry of $\hat{R}$ implies that $\Psi$ is actually a (smooth) curve in $\mathrm{Sp}(2n,\R)$.
We denote by $J$  the standard complex structure, given by $\displaystyle
J\= \begin{bmatrix}
       0 & -I\\
       I&0
      \end{bmatrix}$
where $I$ denotes the identity in the appropriate dimension.
Observing that $\trasp{A}G A= G$,  it follows  that the operator
$\displaystyle
A_d\=\begin{bmatrix}
\traspm{A} & 0 \\
0 & A
\end{bmatrix}$
lies in $\mathrm{Sp}(2n, \R)$ being, in fact, $\trasp{A_d}JA_d= J$.
By taking into account the $\mathfrak g$-orthonormal periodic trivialization of
$\gamma^*(TM)$, the induced linearized Poincaré map is given by
\begin{equation}\label{eq:poincare-1}
\mathcal P(T)\=   A_d\Psi(T) \in \mathrm{Sp}(2n, \R).
\end{equation}
\begin{rem}
We observe that, in terms of the the operator $A_d$, a spacelike (resp. timelike) geodesic $\gamma$ is {\em linearly stable\/} if the symplectic  matrix $\mathcal P(T)$ is linearly stable.
\end{rem}
In order to introduce both the geometric and analytic indices, we start to embed the second order self-adjoint differential operator coming out from the Jacobi deviation equation, into a one parameter family of operator. Now,   for any $s \in [0,+\infty)$, we   introduce  the closed (unbounded) operator
$
 \mathcal A_s: \mathscr D(\mathcal A_s) \subset L^2([0,T]; \R^n) \to L^2([0,T];
\R^n)
$
having  domain $\mathscr D(\mathcal A_s)\= \{u\in W^{2,2}([0,T],\R^{n}):
u(0)=Au(T),\dot{u}(0)=A\dot{u}(T)\}$
(independent on $s$) and  defined by
\begin{equation}\label{eq:gli-operatori}
 \mathcal A_s(u)(t)\= -G\dfrac{d^2}{dt^2} + \widehat R(t)+s\,G.
\end{equation}
It is well-known (cf., for instance, \cite{GGK90}) that,
for each $s \in [0,s_0]$ the operator $\mathcal A_s$ is
a closed Fredholm operator and selfadjoint in $L^2([0,T]; \R^n)$.
Since the domain $\mathscr D(\mathcal A_s)$ doesn't depend
on $s$, the path $s \mapsto \mathcal A_s$  can be seen as a path of bounded
Fredholm operators from
$\mathcal W\= W^{2,2}([0,T];\R^n) \cap W^{1,2}_A([0,T];\R^n)$ into $L^2([0,T],
\R^n)$ which  are selfadjoint when
regarded as operators on $L^2$, where
$
 W^{1,2}_A([0,T]; \R^n)\= \{u \in W^{1,2} ([0,T], \R^n): u(0)= A\,u(T)\}.
$
For any $c \in [0,1]$,  $s \in [0,+\infty)$ and for any $\omega \in \U$, we define the operator
\begin{equation}\label{eq:Acs}
\mathcal A^\omega_{c,s}=-G \dfrac{d^2}{dt^2} + c \widehat R(t)+ sG, \qquad t \in [0,T]
\end{equation}
on  the Hilbert  space
\begin{equation}\label{spazio-Acs}
E^2_\omega([0,T])\=\Set{u\in W^{2,2}([0,T], \C^n)| u(0)=\omega A u(T), \  \dot u(0)=\omega A \dot u(T)}.
\end{equation}
As we will prove in Corollary \ref{thm:lemma3}, there exists $s_0$ sufficiently large such that for $s \geq s_0$, the operator
$\mathcal A^\omega_{1,s}$ is non-degenerate.  Now we are entitled to define the spectral index of a
closed geodesic $\gamma$.
\begin{defn}\label{def:spectral-index}
Under the previous notation, we term {\em $\omega$-spectral index\/} of the closed non-lightlike geodesic $\gamma$, the integer
$\ispecomega(\gamma)$ defined by
\begin{equation}\label{eq:spectral-index}
 \ispecomega(\gamma)\=  \spfl(\mathcal A^\omega_{1,s}; s \in [0,s_0]).
\end{equation}
\end{defn}
\begin{rem}
As already observed, since  for $s \geq s_0$, the operator
$\mathcal A^\omega_{1,s}$ is non-degenerate, the spectral index given in Definition \ref{def:spectral-index} is well-defined, i.e. it is independent on $s_0$.
\end{rem}
We now introduce the Hamiltonian system
\begin{equation}\label{eq:Dcs}
\dot z(t)= J D_{c,s}(t) z(t), \qquad t \in [0,T]
\end{equation}
for $\displaystyle
D_{c,s}(t)\=
\begin{bmatrix}
G & 0 \\0 & - c\widehat R(t) -sG
\end{bmatrix}$, $s \in [0,+\infty)$
and we denote by $\Psi_{c,s}$ its fundamental solution.
\begin{defn}\label{def:Maslov-geodesic}
Let $\gamma$ be a closed non-lightlike geodesic and let $\mathcal P_\omega:[0,T] \to \mathrm{Sp}(2n)$ be the path pointwise given by $\mathcal P_\omega(t)\= \omega A_d\Psi_{1,0}(t)$.
We define the {\em $\omega$-geometric index of $\gamma$\/}  as follows
 \begin{equation}\label{eq:indice-geometrico}
  \igeo^\omega(\gamma)\=\iota_1(\mathcal P_\omega(t); t\in [0,T])
 \end{equation}
 where $\iota_1$ is the Maslov-type index (cf. Appendix \ref{sec:appendix}  and references therein for the
definition and the main properties of $\iota_1$).
 \end{defn}
 \begin{note}
 In shorthand notation, we will denote
 \begin{itemize}
\item  the path $\mathcal P_1$ (obtained by setting $\omega=1$) by $\mathcal P$;
\item $\igeo^1(\gamma) $ by $\igeo(\gamma)$;
\item $\ispec^1(\gamma)$ by $\ispec(\gamma)$.
\end{itemize}
 \end{note}
Our first main result reads as follows.
\begin{mainthm}\label{thm:Lemma6}({\bf An $\omega$-spectral flow formula\/})
Under the previous notation, the following spectral flow formula holds:
\[
 \ispecomega(\gamma) +\dim\ker ( A -\omega I) =  \igeo^\omega(\gamma).
 \]
\end{mainthm}
As a  direct consequence of the $\omega$-spectral flow formula, we immediately get  a new Morse-type Index Theorem.
\begin{maincor}\label{thm:Lemma7}({\bf A Morse Index Theorem \/})
Under the previous notation, we have
\[
\ispec(\gamma)+\dim\ker(A-I)= \igeo(\gamma).
\]
\end{maincor}
\begin{proof}
The proof of this result immediately follows by setting $\omega=1$. \end{proof}
We denote by $\gamma^{(m)}:[0,mT]\to M$ the $m$-th iteration of the  geodesic $\gamma$, defined by
\[
\gamma^{(m)}(t)\= \gamma(t-jT), \quad jT \leq t \leq (j+1)T, \ j=0, \dots, m-1.
\]
Another  contribution of the present paper is  represented by a sort of  semi-Riemannian version of the Bott-type iteration formula which plays a crucial role in the instability criteria that we shall prove. It is worth to
note that, with respect to the classical case,  in our framework the  Legendre convexity condition does not hold.
\begin{mainthm}({\bf Semi-Riemannian Bott-type formula\/}) \label{thm:Bott-SR}
Let $(M, \mathfrak g)$ be a semi-Riemannian manifold and $\gamma$ be a closed
non-lightlike closed geodesic. For any $m \in \N$, the
following iteration formula holds
\begin{equation}\label{eq:Bott-equation}
\ispec\big(\gamma^{(m)}\big)=\sum_{\omega^m=1} \ispecomega(\gamma)
\end{equation}
\end{mainthm}
By using the spectral flow formula stated in Theorem \ref{thm:Lemma6} and by taking into account the homotopy  properties of $\mathrm{Sp}_\omega^*(2n)$ (cf. Subsection \ref{subsec:Maslov}), we get our main linear instability result for spacelike and timelike  closed semi-Riemannian geodesics.
\begin{mainthm}\label{thm:main-deg}{\bf(Semi-Riemannian Instability criterion)}
Let $(M, \mathfrak g)$ be a $(n+1)$-dimensional semi-Riemannian manifold
of index $\nu$ and let $\gamma:[0,T] \to M$ be a  closed non-lightlike geodesic. If
\begin{itemize}
\item[]{\bf (OR)} $\gamma$   is oriented and $\ispec(\gamma) +n$   is odd
\item[]{\bf (NOR)} $\gamma$   is nonoriented and $\ispec (\gamma) +n$ is even
\end{itemize}
then $\gamma$ is linearly unstable.
\end{mainthm}
{\begin{rem}
It is worth to observe that the spectral flow techniques were successfully developed for proving  Morse type index theorem  (cfr. \cite[Theorem 5.6]{BP10} which holds even for  closed non-lightlike geodesic) and Bott's iteration formula (cf. \cite[Theorem 5.3]{JP08}) for closed semi-Riemannian geodesics. Please note that here we choose a different operator path to define the spectral flow. The choice made by authors in  the \cite{BP10} is in the direction of extending the classical Morse index theorem for closed Riemannian geodesics whilst our choice   comes from the original purpose of investigating the instability of closed semi-Riemannian geodesics.
\end{rem}

An immediate consequence of Theorem
\ref{thm:main-deg} is the following instability criterion for closed Riemannian (resp. Lorentzian)  geodesics (resp.timelike geodesics).
\begin{maincor}\label{thm:Riemann}
Let $(M, \mathfrak g)$ be a $(n+1)$-dimensional
Riemannian manifold (resp. Lorentzian) and let $\gamma:[0,T] \to M$ be a closed (resp. timelike closed)
geodesic.
If one of the following two alternatives hold
\begin{itemize}
\item[]{\bf (OR)} $\gamma$   is oriented (resp. oriented and timelike) and $
\iota_{\scriptscriptstyle{\mathrm{Mor}}}(\gamma) +n$   is odd
\item[]{\bf (NOR)} $\gamma$   is non-oriented (resp. non-oriented and
timelike) and $
\iota_{\scriptscriptstyle{\mathrm{Mor}}}(\gamma) +n$  is even
\end{itemize}
then the geodesic  is linearly unstable.
\end{maincor}
\begin{proof}
 The proof of this result readily follows by  Theorem
 \ref{thm:main-deg} once observed that for closed Riemannian  (resp. timelike closed Lorentzian) geodesics, $\ispec(\gamma)=\iota_{\scriptscriptstyle{\mathrm{Mor}}}(\gamma)$(cf. \cite[Section $3$]{HS09} or \cite[Section $3$]{HS10}).
\end{proof}
\begin{rem}
It is worth noting that the celebrated Poincare's instability criterion, can be recovered by  Corollary \ref{thm:Riemann}.  In fact,  if $\gamma$ is
a minimizing closed geodesic, then
$\iota_{\scriptscriptstyle{\mathrm{Mor}}}(\gamma)=0$ and on a Riemannian surface $n=1$; thus  $\iota_{\scriptscriptstyle{\mathrm{Mor}}}(\gamma)+n =1$.
\end{rem}
Before describing the last  result, we introduce the notion of strong stability for closed geodesic on Riemannian or Lorentzian
manifolds.
\begin{defn}\label{def:strong stable}
Let $\gamma$ be a  closed (resp. timelike closed) Riemannian (resp. Lorentzian)
geodesic and let $\mathcal P(T)$ be the corresponding monodromy matrix. We say that
$\gamma$ is termed {\em strongly stable\/} if,  there exists
$\varepsilon>0$ such that any symplectic matrix $M$ with
$\norm{ M-\mathcal P(T)}\leq\varepsilon$ is linearly stable.
\end{defn}

\begin{mainthm}\label{cor:non-strong stable}
Let $(M, \mathfrak g)$ be a  $(n+1)$-dimensional Riemannian (resp. Lorentzian) manifold and let $\gamma$ be a  closed (resp. timelike closed) Riemannian (resp. Lorentzian)
geodesic. We assume that
\[
\iota_{\scriptscriptstyle{\mathrm{Mor}}}\big(\gamma^{(m)}\big)=0, \qquad  \textrm{ for any } m\in \N.
\]
Then $\gamma$ is not strongly stable.
\end{mainthm}

\section{Variational preliminaries}\label{sec:geometric-variational-preliminaries}

The aim of this section is to introduce the basic  geometric
and variational setting  and to fix our notations.
Our basic references are \cite{Kli78}.

It is well-known that,  closed semi-Riemannian  geodesics are critical
points of the {\em geodesic energy functional\/} defined
on  the {\em free loop space\/}
$\Lambda \MM$ of $\MM$ where $\Lambda \MM$ denotes the Hilbert
manifold of
all closed curves in $\MM$ of Sobolev class $W^{1,2}$. Let $\circo$ be the circle, viewed as the quotient  $[0,T]/\{0,T\}$,
$\Lambda \MM\= W^{1,2}(\circo,\MM)$ be the infinite dimensional
Hilbert manifold
(cf. \cite[Theorem 1.2.9, pag.13]{Kli78}) of all loops $\gamma: [0,T] \to
\MM$ (namely
$\gamma(0)=\gamma(T)$)
of Sobolev class $W^{1,2}$. For any
$\gamma \in \Lambda \MM$,
the tangent space $T_\gamma \Lambda \MM$ can be identified with the Hilbert
space of all sections of the
pull-back bundle $\gamma^*(T\MM)$ (namely the bundle of all periodic
vector fields along
$\gamma$) of
Sobolev class $W^{1,2}$ (cf. \cite[Theorem 1.3.6, pag.19]{Kli78}) that we'll
denote by $\mathcal H_\gamma$; thus
\[
\mathcal H_\gamma=\{\xi \in W^{1,2}(\circo, T\MM): \tau \circ \xi=
\gamma\},
\]
where $\tau:TM \to M$ denotes the canonical tangent bundle projection.
We consider the {\em geodesics energy functional\/}
$E: \Lambda \MM \to \R$ given by
\begin{equation}\label{eq:funzionale-energia}
E(\gamma)= \dfrac12 \int_0^T\mathfrak g\big(\dot \gamma(t), \dot \gamma(t) \big)\,
dt
\end{equation}
and we observe that $E$ is
differentiable and for any $\xi \in T_\gamma \Lambda \MM$
\begin{equation}
dE (\gamma ) [\xi ] =  \int_0^T \mathfrak g\left( \tfrac{\mathrm{D}}{\mathrm{d}t} \xi(t),
\dot \gamma(t)\right)dt.
\end{equation}
(Cf.  \cite[Lemma 1.3.9, pag.21]{Kli78}, for further details).
By standard regularity arguments and by performing integration by parts, it follows that critical points corresponds to closed geodesics.
\begin{lem}
$\gamma \in \Lambda  \MM$ is a closed geodesic (or a constant map) if
and only if  it
is a critical point of $E$, namely $\displaystyle
dE(\gamma)[\xi]=0  \textrm{ for all  } \xi \in T_\gamma \Lambda  M.$
\end{lem}
\begin{proof}
 For the proof we refer the interested reader to \cite[Theorem 1.3.11]{Kli78}.
\end{proof}
If $\gamma \in \Lambda    M$ is a
non-constant closed geodesic in $( \MM,\mathfrak g)$ then  the
second variation of the geodesic energy functional $E$  at $\gamma$
is the {\em index form\/} $\mathfrak I_\gamma$ given by
\begin{equation}\label{eq:index_form}
\mathfrak I_\gamma[\xi, \eta]= \int_0^T \Big[\mathfrak g\left(\tfrac{\mathrm{D}}{\mathrm{d}t} \xi (t) ,
  \tfrac{\mathrm{D}}{\mathrm{d}t} \eta(t)\right )+  \mathfrak g\big (\mathcal R(\dot \gamma(t) , \xi(t)) \dot
\gamma(t), \eta(t)\big)
\Big]dt.
\end{equation}
It is readily seen that
$\mathfrak I_\gamma$ is a bounded symmetric bilinear form on $T_{\gamma}\Lambda M$
whose associated quadratic form will be denoted by $\mathfrak  Q_\gamma$.
Let $ \mathcal H^\perp_{\gamma}\subset \mathcal H_{\gamma}$ be the
closed (real) codimensional one subspace of all $T$-periodic $W^{1,2}$-vector fields
along $\gamma$  that are everywhere orthogonal to
$\dot
\gamma$. We observe that  $\displaystyle
\dim \ker \mathfrak Q_\gamma = \dim\ker \left(\mathfrak Q_\gamma|_{\mathcal
H^\perp_{\gamma}}\right)+1 $.  We term  {\em nullity of the
geodesic $\gamma$\/} the (non-negative) integer defined by
$\displaystyle n_0(\mathfrak Q_\gamma)\= \dim \ker \left(\mathfrak Q_\gamma|_{\mathcal H^\perp_{\gamma}}\right).$ Thus the nullity of $\gamma$ is defined as the dimension of the space of
periodic Jacobi fields along $\gamma$ that are pointwise $\mathfrak
g$-orthogonal to $\dot \gamma$.

Since $e_i$ is a parallel vector field  along $\gamma$ and if $\xi$ is a Jacobi field,
then, by direct computation we have
\begin{equation}
\frac{d^{2}}{d^{2}t}\mathfrak{g}(\xi(t),\dot{\gamma}(t))
=\mathfrak{g}(\tfrac{\mathrm{D}^2}{\mathrm{d}t^2} \xi  (t),\dot{\gamma}(t))
=\mathfrak{g}(\mathcal R(\dot \gamma (t) , \xi  (t)) \dot \gamma (t),\dot{\gamma}(t))=0,
\end{equation}
so $\mathfrak{g}(\xi(t),\dot{\gamma}(t))=\alpha t+\beta$ for some  $\alpha,\beta \in \R$.
In particular, if $\xi(0),\xi'(0)\in N_{\gamma(0)}M$, it follows that $\alpha=\beta=0$.  which is equivalent to
$\mathfrak{g}(\xi(t),\dot{\gamma}(t))\equiv0$ for any $t\in[0,T]$. By this fact, we infer that
 $N_{\gamma(0)}M\oplus N_{\gamma(0)}M$ is invariant
under the linearized geodesics flow. (As we already observed, being the parallel transport a $\mathfrak g $-isometry).
In particular the Morse-Sturm system  given in Equation \eqref{eq:Space-Original-intro} decouples into a scalar differential
equation (corresponding to the Jacobi field)
 and a differential system in $\R^n$
(corresponding to the restriction of the Jacobi deviation equation to vector fields $\mathfrak g$-orthogonal to $\dot \gamma$).

\section{Geometric and spectral index  of a closed
geodesic}\label{sec:geometric-variational-index}

This section is devoted to show that the geometrical and the spectral index previously introduced are actually well-defined.

 \begin{lem}
 The geometric index of a closed non-lightlike semi-Riemannian geodesic $\gamma$ is well-defined, i.e.
 it is independent on the trivializing parallel frame along $\gamma$.
 \end{lem}
 \begin{proof}
 We prove the result only in the case of closed spacelike geodesics being the timelike case completely analogous.
 Let  $\{f_1(0),\ldots,f_n(0)\}$ be another
 $\mathfrak g$-orthonormal basis of $N_{\gamma(0)}M$
 such that $f_i(0)=\sum_{j=1}^n c_{ij}e_j(0)$ and we let $C\=[c_{ij}]_{i,j=1}^n$.
 Then, by direct computation,
 the Morse-Sturm system given in Equation \eqref{eq:Space-Reduced-space-intro} fits into the following
  \begin{equation}\label{eq:transformed M-system}
-G\ddot u(t)+\widetilde{R}(t) u(t)=0,\qquad t \in [0,T]
  \end{equation}
where $\widetilde{R}(t)=C\widehat{R}(t)\trasp{C}$ and the boundary condition of $u$ is given
by $u(0)=A\trasp{C}u(T)$. We let
 $C_d\=\begin{bmatrix}\traspm{C}&0\\0&C\end{bmatrix}$ and let $\widetilde \Psi$ be the fundamental solution
 of the corresponding Hamiltonian system given by
 $
\dot z(t)= \widetilde{K}(t)z(t),
$
with $\widetilde{K}(t)=C_dK(t)\trasp{C}_d$. It is easy to check that $\widetilde{\Psi}(t)=\traspm{C}_d\Psi(t)$ and that
\[
\iota_1\big(A_d\trasp{C}_d\widetilde{\Psi}(t); t \in [0,T]\big)=\iota_1(A_d\Psi(t); t \in [0,T]\big).
\]
 This concludes the proof.
 \end{proof}
\begin{lem}
For every $c \in [0,1]$ and $s \in [0,+\infty)$, the operator $\mathcal A^\omega_{c,s}$ is
formally selfadjoint.
\end{lem}
\begin{proof}
The proof of this result readily follows by integrating by parts.
\end{proof}
\begin{lem}\label{thm:lemma1}
For any $c_0>0$, there exists a sufficiently large $s_0$ such that  for any  $|c| \leq c_0$ and $s \geq s_0$,
the operator defined by
\[
\mathcal B_{c,s}\= -G \dfrac{d^2}{dt^2} + c I + s G, \qquad s \in [0,+\infty)
\]
is non-degenerate (meaning that has a trivial kernel) on
\[
E^2_\omega([0,T])=\Set{u\in W^{2,2}([0,T], \C^n)| u(0)=\omega A u(T), \  \dot u(0)=\omega A \dot u(T)} \textrm{ for every } \omega\in\, \U.
\]
\end{lem}
\begin{proof}
We consider the Morse-Sturm system
\[
\begin{cases}
-G \ddot u(t) + \big(c I + s G\big) u(t)=0, \quad t \in [0,T]\\
u(0)= \omega A u(T) , \quad  \dot u(0)= \omega A \dot u(T).
\end{cases}
\]
Then the corresponding Hamiltonian system is given by
\begin{equation}\label{eq:new-ham}
\dot z(t) = J B_{c,s} z(t), \qquad t \in [0,T]
\end{equation}
where $B_{c,s}\= \begin{bmatrix} G & 0 \\ 0 & -cI -sG \end{bmatrix}$
and the boundary condition is given  by $z(0)= A_d z(T)$. Denoting by $\Phi_{c,s}$ the fundamental
solution of the Hamiltonian system, it follows that the non-degeneracy of $\mathcal B_{c,s}$ is equivalent
to the following condition
\[
\det(\omega A_d \Phi_{c,s}(T)-I) \neq 0.
\]
If $s$ is sufficiently large, then $s \pm c$ is positive. We set
\[
\lambda \= \sqrt{s+c} \quad \textrm{ and } \quad \mu= \sqrt{s-c}.
\]
By a direct computation, we get
\[
\Phi_{c,s}(t)=
\begin{bmatrix}
\cosh(\lambda\, t)I & 0 & \lambda \sinh (\lambda\, t) I& 0\\
0 & \cosh( \mu \, t) I& 0 & -\mu \sinh(\mu\, t)I\\
\dfrac{\sinh(\lambda \, t)}{\lambda}I & 0 & \cosh(\lambda \, t)I & 0 \\
0 & -\dfrac{\sinh(\mu t)}{\mu}I & 0 & \cosh(\mu \, t)I
\end{bmatrix}, \qquad t \in [0,T].
\]
We observe that
\begin{multline}
(\omega A_d)^{-1}= \bar \omega A_d^{-1}= \bar \omega
\begin{bmatrix} \trasp{A} & 0 \\0 & A^{-1}\end{bmatrix}=
\begin{bmatrix} \bar \omega \trasp{A} & 0 \\0 & \bar \omega A^{-1}\end{bmatrix} \textrm{ and } \\
(\bar \omega \trasp{A}) G (\omega A) = G \Rightarrow \bar \omega \trasp{A} = G (\omega A)^{-1} G= G(\bar \omega A^{-1}) G.
\end{multline}
We let
\[
\bar \omega A^{-1}\= \begin{bmatrix} P & Q \\ R & S \end{bmatrix}, \quad \textrm{ we immediately get that }  \bar \omega \trasp{A}= \begin{bmatrix} P &- Q \\ -R & S \end{bmatrix}.
 \]
In shorthand notation, we let

\begin{multline}
U\= \begin{bmatrix}
\cosh(\lambda T)I - P & Q \\
R &\cosh(\mu T)I - S
\end{bmatrix}, \quad V\= \begin{bmatrix}
\lambda \sinh(\lambda T)I  & 0 \\
0 &  -\mu\sinh(\mu T)I
\end{bmatrix}\\
X\= \begin{bmatrix}
\dfrac{\sinh(\lambda T)}{\lambda}I & 0 \\
0 & -\dfrac{\sinh(\mu T)}{\mu}I
\end{bmatrix},\quad Y\= \begin{bmatrix}
\cosh(\lambda T)I -P  & -Q \\
-R & \cosh(\mu T)I - S
\end{bmatrix}
 \end{multline}
and we observe that $U,V,X,Y$ are $n \times  n $ matrices.
 Then $\displaystyle
 \Phi_{c,s}(T) - (\omega A_d)^{-1}= \begin{bmatrix}
 U & V\\ X & Y
 \end{bmatrix}$
  and hence
 \begin{align}
  \det \big(\omega A_d\Phi_{c,s}(T) - I \big)&= \det(\omega A_d) \cdot \det\big(\Phi_{c,s}(T)-(\omega A_d)^{-1}\big)\\
 &= \omega^{2n} \cdot \det A \cdot \det \traspm{A} \det\begin{bmatrix}U&V\\X&Y\end{bmatrix}\\
 &=(-1)^{n^2} \omega^{2n}\cdot \det \begin{bmatrix}U&V\\X&Y\end{bmatrix}=

(-1)^n \omega^{2n} \cdot \det\big( V\cdot(X-Y V^{-1} U)\big).
 \end{align}
 By a straightforward calculation, we get
  \begin{multline}
 V\cdot(X-Y V^{-1} U)\\=
 \begin{bmatrix}
-\big(\cosh(\lambda T) -P\big)^2-C_{\lambda,\mu}QR+\sinh^{2}(\lambda T)&
-\big(\cosh(\lambda T)-P\big)Q-C_{\lambda,\mu} Q\big(\cosh(\mu T)-S\big)\\
-C_{\lambda,\mu}^{-1} R \big(\cosh(\lambda T)-P\big)-\big(\cosh(\mu T)-S\big)R&
-\big(\cosh(\mu T)-S)^2-C_{\lambda,\mu}^{-1}RQ+\sinh^2(\mu T)
 \end{bmatrix}
 \end{multline}
 where we set $C_{\lambda,\mu}\=\dfrac{\lambda \sinh(\lambda T)}{\mu\sinh(\mu T)}$. Thus asymptotically, we get
 following behavior
 \begin{multline}
 -\big(\cosh(\lambda T) -P\big)^2-C_{\lambda,\mu}QR+\sinh^{2}(\lambda T )= -1 + 2 \cosh(\lambda T) P-P^2
  - C_{\lambda,\mu} QR \sim_{+\infty} 2\cosh(\lambda T)P\\
  -\big(\cosh(\lambda T)-P\big)Q-C_{\lambda,\mu} Q\big(\cosh(\mu T)-S\big) \sim_{+\infty}  -2\cosh(\lambda T)Q\\
  -C_{\lambda,\mu}^{-1} R \big(\cosh(\lambda T)-P\big)-\big(\cosh(\mu T)-S\big)R \sim_{+\infty} -2\cosh(\lambda T)R\\
  -\big(\cosh(\mu T)-S)^2-C_{\lambda,\mu}^{-1}RQ+\sinh^2(\mu T)\sim_{+\infty} 2\cosh(\lambda T)S.
   \end{multline}
   Summing up, we have
   \begin{equation}\label{eq:stima}
    V\cdot(X-Y V^{-1} U)\sim_{+\infty} 2 \cosh(\lambda T) \begin{bmatrix} P & -Q\\ -R & S \end{bmatrix} = 2\cosh(\lambda T) \bar \omega \trasp{A} .
   \end{equation}
   By taking into account Equation \eqref{eq:stima},  it holds that
   \begin{align}\label{eq:stima2}
   \det \big(\omega A_d \Phi_{c,s}(T)-I\big)&=(-1)^n \omega^{2n} \det\big(V(X-YV^{-1}U)\big) &\sim_{+\infty} (-1)^n\omega ^{2n}\det\big(2 \cosh(\lambda T)\bar \omega \trasp{A}\big)\\
   &= (-1)^n \omega^{2n} \big(2\cosh(\lambda T)\big)^n\bar\omega^n \det A\\& \neq 0.
   \end{align}
 By Equation \eqref{eq:stima2} the thesis readily follows. This concludes the proof.
 \end{proof}
In order to prove the non-degeneracy of the operator $\mathcal A_{1,s}^\omega$  for $s$ sufficiently large, we need the following stability result proved by Kato.
\begin{lem}\label{thm:lemma22}
Let $T$ be a selfadjoint operator and $A$ be symmetric. Then the operator $S\=T+A$ is
selfadjoint and
\[
\mathrm{dist}\big(\sigma(S), \sigma(T)\big) \leq \norm{A},
\]
where $dist(\cdot,\cdot)$ is the Hausdorff distance.
\end{lem}
\begin{proof}
For the proof, we refer the interested reader, to \cite[pag. 291]{Kat80}.
\end{proof}
\begin{cor}\label{thm:lemma3}
Let $c_0>0$ be such that $\norm{\widehat R}_{\mathscr L(\R^n)} \leq c_0$  and $s_0$ be the number related to $c_0$ by Lemma \ref{thm:lemma1}. Then for any $s \geq s_0$ it holds
\[
\mathcal A_{1,s}^\omega\=-G \dfrac{d^2}{dt^2} +\widehat R + s G
\]
is non-degenerate.
\end{cor}
\begin{proof}
The proof of this result  follows by Lemma \ref{thm:lemma1} and Lemma \ref{thm:lemma22}, just by setting (with a slight abuse of notation)
\[
T\= -G \dfrac{d^2}{dt^2}  + s G \textrm{ and } A\=\widehat R.
\]
This concludes the proof.
\end{proof}
 \begin{rem}
We observe that in the Riemannian case, directly proof of this result can be easily conceived. However in the semi-Riemannian setting the abstract  way (which works in the Riemannian world) for simplifying this proof breaks-down. The obstruction to carry over that proof  is essentially based on the fact that the matrix $A$ coming from the trivialization is a $\mathfrak g$-orthogonal and not just an orthogonal matrix, like in the Riemannain case.
 \end{rem}

\section{A generalized spectral flow formula}\label{sec:index-theorem}

This section is devoted to the relation intertwining  the spectral index
and the geometric index. As a direct consequence we get a spectral flow
formula involving the $\omega$-spectral index and the geometric index.
The following spectral flow formula holds.
\begin{prop}\label{thm:lemma4}
Let $s_0$ be given in Lemma \ref{thm:lemma1}. Then, we have
\[
\spfl(\mathcal A^{\omega}_{1,s}; s \in [0,s_0]) =-\iota_1\big(\omega A_d\Psi_{1,s}(T); s \in [0,s_0]\big).
\]
\end{prop}
\begin{proof}
For a given matrix $M\in L(\C^n)$, we denote its graph by $\mathrm{Gr}(M)\=\{\trasp{(x,\trasp{(Mx)})}\ |\ x\in \C^n\}$.
By Definition \ref{def:Maslov-index} of  the $\mu^{\scriptscriptstyle{\mathrm{CLM}}}$-index, we have
\begin{align}
\iota_1\big(\omega A_d\Psi_{1,s}(T); s \in [0,s_0]\big)&= \iota_{\bar \omega}\big( A_d\Psi_{1,s}(T); s \in [0,s_0]\big)=
 \iota_{\omega}\big( A_d\Psi_{1,s}(T); s \in [0,s_0]\big)\\
 &= \mu^{\scriptscriptstyle{\mathrm{CLM}}}\big(\mathrm{Gr}(\omega I), \mathrm{Gr}\big(A_d\Psi_{1,s}(T) \big); s \in [0,s_0]\big),
\end{align}
where the last equality follows from Equation \eqref{eq:non-citata} and Proposition \ref{prop:L-C}. By invoking \cite[Lemma 2.3 and Lemma 2.4]{HS09}, we have
\begin{equation}\label{eq:caju}
\mu^{\scriptscriptstyle{\mathrm{CLM}}}\big(\mathrm{Gr}(\omega I), \mathrm{Gr}\big(A_d\Psi_{1,s}(T) \big); s \in [0,s_0]\big)= -\spfl(\mathcal D^\omega_{1,s}; s \in [0,s_0])
\end{equation}
where $\mathcal D_{c,s}\=-J \dfrac{d}{dt}- D_{c,s}(t); s \in [0,s_0]$ on the Sobolev space
\[
E^1_\omega([0,T])\=\Set{u\in W^{1,2}([0,T], \C^{2n})| u(0)=\omega A_d u(T)}.
\]
In order to concludes the proof, it is enough to prove that
\[
\spfl(\mathcal A^\omega_{1,s}; s \in [0,s_0])=
\spfl(\mathcal D^\omega_{1,s}; s \in [0,s_0]).
\]
For, we start to assume that both paths are regular in the sense specified in Appendix \ref{subsec:spectral-flow}.
Under this regularity assumption it is enough to show that the local contribution of the
spectral flow of both paths $\mathcal A^\omega_{1,s}$ and $\mathcal D_{1,s}^\omega$ coincide. Let $s_* \in [0,s_0]$  and
we start to observe that
\[
\dim \ker \mathcal A_{1,s_*} \neq \{0\} \iff \dim\ker \mathcal D_{1,s_*} \neq \{0\}.
\]
In particular $s_*$ is a crossing instant for $s\mapsto\mathcal A_{1,s}$ if and only if it is a crossing instant for $s\mapsto\mathcal D_{1,s}$.
Let $s_*$ be a crossing instant. By a direct computation, it follows that  the crossing form of $\mathcal A$ at $s_*$ is the quadratic form
\begin{equation}\label{eq:crossing1}
\Gamma(\mathcal A_{1,s}, s_*): \ker \mathcal A_{1,s} \to \R \textrm{ defined by }
\Gamma(\mathcal A_{1,s}, s_*)[u]= \int_0^T \langle Gu, u\rangle\, dt, \quad u \in \ker \mathcal A_{1,s_*}.
\end{equation}
The crossing form of $\mathcal D_{1,s}$ at $s_*$ is given by
\begin{multline}\label{eq:crossing2}
\Gamma(\mathcal D_{1,s}, s_*): \ker \mathcal D_{1,s} \to \R \textrm{ defined by }
\Gamma(\mathcal D_{1,s}, s_*)[w]= \int_0^T \left\langle \begin{bmatrix}0 & 0 \\ 0 & G\end{bmatrix}\begin{bmatrix} G \dot w\\ w\end{bmatrix},
\begin{bmatrix} G \dot w\\ w\end{bmatrix}\right \rangle\, dt \\
=\int_0^T \langle Gw, w \rangle\, dt \quad w \in \ker \mathcal D_{1,s_*}.
\end{multline}
By Equations \eqref{eq:crossing1}-\eqref{eq:crossing2}, we immediately conclude that the crossing forms
coincide and hence also their signatures. In particular the local contribution at $s_*$ of both paths to the spectral flow coincide and
this concludes the proof, under the assumption that $s\mapsto\mathcal A_{1,s}$ and $s\mapsto\mathcal D_{1,s}$ are regular.

In order to concludes the proof in the general case (i.e. for non-regular paths), we observe that by standard perturbation results there exists $\varepsilon >0$ sufficiently small
(cf. Appendix \ref{subsec:spectral-flow} and references therein) such that the perturbed path
$s \mapsto\mathcal A_{1,s}^\varepsilon\= \mathcal A_{1,s}+ \varepsilon I$ is regular and, by the homotopy invariance property with fixed ends,
it  has the same spectral flow of $s\mapsto \mathcal A_{1,s}$. Through the perturbed path $\mathcal A_{1,s}^\varepsilon$, we can define the perturbed
path of first order operators given by
\[
\mathcal D_{1,s}^\varepsilon \= -J \dfrac{d}{dt} - D_{1,s}^\varepsilon(t)
\]
where $D_{1,s}^\varepsilon(t)\= \begin{bmatrix}
G & 0 \\ 0 & -\widehat R(t)-sG +\varepsilon I
\end{bmatrix}$. Repeating ad verbatim the same arguments for  the perturbed paths, we get the thesis. This concludes the proof.
\end{proof}
Our next step is to compute the integer $\iota_1\big(\omega A_d \Psi_{0,s_0}(t); t \in [0,T]\big)$. Before introducing a technical
result needed for this computation, we observe that
\begin{align}\label{eq:suigrafici}
\iota_1\big(\omega A_d \Psi_{0,s_0}(t); t \in [0,T]\big)&= \iota_\omega\big( A_d \Psi_{0,s_0}(t); t \in [0,T]\big)\\
&=\mu^{\scriptscriptstyle{\mathrm{CLM}}}\Big(\mathrm{Gr}\big(\omega A_d^{-1}\big), \mathrm{Gr}\big(\Psi_{0,s_0}(t)\big); t \in [0,T]\Big).
\end{align}
The  idea for computing the (RHS) of Equation \eqref{eq:suigrafici} is to transform the path of graphs of a symplectic matrix into a path of graphs of
a symmetric matrices. In this  way the computation of the $\mu^{\scriptscriptstyle{\mathrm{CLM}}}$ index can be performed through the spectral flow.

\begin{lem}\label{thm:lemma0-2}
Under the previous notation, we have
\[
\mu^{\scriptscriptstyle{\mathrm{CLM}}}\Big(\mathrm{Gr}\big(\omega A_d^{-1}\big), \mathrm{Gr}\big(\Psi_{0,s_0}(t)\big); t \in [0,T]\Big)= \mu^{\scriptscriptstyle{\mathrm{CLM}}}\Big(L_D, \mathrm{Gr}\big(M_{s_0}(t)\big); t \in [0,T]\Big)
\]
where $L_D= \Set{\begin{bmatrix}x\\ 0\end{bmatrix}| x \in \C^{2n}}$ is the (horizontal) Dirichlet Lagrangian subspace
 and $t \mapsto M_{s_0}(t)$ is the path pointwise defined by
\[
M_{s_0}(t)\=\begin{bmatrix}
\dfrac{\sinh(\sqrt{s_0}t)}{\sqrt{s_0}\cosh(\sqrt{s_0}t)} G & \dfrac{I}{\cosh(\sqrt{s_0}t)}-\omega A^{-1}\\
\dfrac{I}{\cosh(\sqrt{s_0}t)}-\bar \omega \traspm{A} & -\sqrt{s_0}\dfrac{\sinh(\sqrt{s_0}t)}{\cosh(\sqrt{s_0}t)} G
\end{bmatrix}.
\]
\end{lem}
\begin{rem}\label{rem:idimu}
We observe that  if $P\in \mathrm{Sp}(2n)$, then
\[
\mathrm{Gr}(P)=\Set{\begin{bmatrix}x \\ Px\end{bmatrix}|x \in \C^{2n}}
\]
is a Lagrangian subspace of $\C^{2n} \oplus \C^{2n}$ under the symplectic form $-J \oplus J$, where $J$ is the standard symplectic matrix of $\C^{2n}$. We recall
that a {\em Lagrangian  frame}  for a Lagrangian subspace $L$ is an injective linear map $Z: \C^n \to \C^{2n}$  whose image is $L$. Such a frame has the form $\displaystyle Z= \trasp{(X,Y)}$
where $X,Y$ are $n \times n$-matrices and $Y^*X=X^*Y$,
where $*$ denotes the conjugate transpose.
In the special case in which $L$ is a graph of a symmetric matrix, then $X=I$ and the  relation $Y^*X=X^*Y$ trivially holds. We observe also that if $X$ is
invertible, then another Lagrangian frame for $L$ with respect to which it is a graph, is given by $\displaystyle
W= \trasp{ (I, YX^{-1})}$.  In fact, being $L = \mathrm{rge}\,(Z)=\Set{\begin{bmatrix}Xu\\ Yu\end{bmatrix}|u \in \C^n}$, it follows that by changing coordinates by
setting $u=X^{-1}w$, we get $L = \mathrm{rge}\,(Z)=\Set{\begin{bmatrix}w\\ YX^{-1}w\end{bmatrix}|u \in \C^n}$. We start to observe that
if $L$ is a Lagrangian subspace  with respect to a  symplectic form $ \widehat \omega (\cdot, \cdot)\=\langle \widehat J \cdot, \cdot \rangle$,
then  for any orthogonal matrix $S$, the subspace $S^{-1}L$ is a Lagrangian subspace with respect to the symplectic
form  $\widetilde \omega (\cdot, \cdot)\=\langle \widetilde J \cdot, \cdot \rangle$
represented  by $\widetilde J\= \trasp{S}\widehat J S$. Moreover, if  $\begin{bmatrix}Y \\X\end{bmatrix}$ is a Lagrangian frame for $L$, then
$S^{-1}\begin{bmatrix}Y \\X\end{bmatrix}$ is a Lagrangian frame for $S^{-1}L$.
\end{rem}
\begin{proof}
We start to define the orthogonal matrix $S$ given by
$\displaystyle
S\=
\begin{bmatrix}
0 & 0 & I & 0 \\
0 & I  & 0 & 0 \\
0 & 0 & 0 & I\\
I  & 0 & 0 & 0
\end{bmatrix}.
$
Identifying  $\mathrm{Gr}\big(\Psi_{0,s_0}(t)\big)$ with its Lagrangian frame $\begin{bmatrix}
I & 0 \\
0 & I\\
\cosh(\sqrt{s_0}t) I& \sqrt{s_0} \sinh(\sqrt{s_0} t) G\\
\dfrac{1}{\sqrt{s_0}} \sinh(\sqrt{s_0}t) G & \cosh(\sqrt{s_0} t)I
\end{bmatrix}$, we get
\begin{equation}
S\mathrm{Gr}\big(\Psi_{0,s_0}(t)\big)=
\begin{bmatrix}
\cosh(\sqrt{s_0}t) I& \sqrt{s_0} \sinh(\sqrt{s_0} t) G\\
0 & I\\
\dfrac{1}{\sqrt{s_0}} \sinh(\sqrt{s_0}t) G & \cosh(\sqrt{s_0} t)I\\
I & 0 \\
\end{bmatrix}
\end{equation}
is a Lagrangian subspace with respect to the standard symplectic form $\widetilde J$ of $\C^{2n}\oplus \C^{2n}$. To do so,
it is enough to observe that by a direct computation, we have
\begin{equation}
\widetilde J = S \big(-J \oplus J\big)\trasp{S}=
\begin{bmatrix}
0 & 0 &-I&0 \\
0 & 0& 0 & -I \\
I & 0 & 0 & 0 \\
0 & I & 0 & 0\\
\end{bmatrix}.
\end{equation}
By taking into account Remark \ref{rem:idimu} and by observing that  the matrix
\[
\begin{bmatrix}
\cosh(\sqrt{s_0}t) I& \sqrt{s_0} \sinh(\sqrt{s_0} t) G\\
0 & I
\end{bmatrix}
\]
is invertible, we can re-write the Lagrangian subspace $S\mathrm{Gr}\big(\Psi_{0,s_0}(t)\big)$ as the graph of a symmetric matrix, simply by
change the Lagrangian  frame. To do so, we observe that
\begin{multline}
\begin{bmatrix}
\dfrac{1}{\sqrt{s_0}} \sinh(\sqrt{s_0}t) G & \cosh(\sqrt{s_0} t)I\\
I & 0 \\
\end{bmatrix}\cdot
\begin{bmatrix}
\cosh(\sqrt{s_0}t) I& \sqrt{s_0} \sinh(\sqrt{s_0} t) G\\
0 & I\\
\end{bmatrix}^{-1}\\
=\begin{bmatrix}
\dfrac{\sinh(\sqrt{s_0}t)}{\sqrt{s_0}\cosh(\sqrt{s_0}t)} G & \dfrac{1}{\cosh(\sqrt{s_0}t)}I\\
\dfrac{1}{\cosh(\sqrt{s_0}t)} I& -\sqrt{s_0}\dfrac{\sinh(\sqrt{s_0}t)}{\cosh(\sqrt{s_0}t)} G
\end{bmatrix}
\end{multline}
Thus the Lagrangian frame for $S\mathrm{Gr}\big(\omega \Psi_{0,s_0}(t)\big)$ fits into the following
\[
\begin{bmatrix}
I&0\\0&I\\
\dfrac{\sinh(\sqrt{s_0}t)}{\sqrt{s_0}\cosh(\sqrt{s_0}t)} G & \dfrac{1}{\cosh(\sqrt{s_0}t)}I\\
\dfrac{1}{\cosh(\sqrt{s_0}t)} I& -\sqrt{s_0}\dfrac{\sinh(\sqrt{s_0}t)}{\cosh(\sqrt{s_0}t)} G
\end{bmatrix}
\]
By the very same computations for the Lagrangian subspace $\mathrm{Gr}\big(\omega A_d^{-1}\big)$, we get
\[
S\mathrm{Gr}\big(\omega A_d^{-1}\big)= \begin{bmatrix}
0 & 0 & I & 0 \\
0 & I& 0 & 0 \\
0 & 0 & 0 & I\\
I & 0 & 0 & 0
\end{bmatrix}\cdot
\begin{bmatrix}
I & 0 \\
0 & I\\
\omega \trasp{A}&0\\
0 &\omega A^{-1}
\end{bmatrix} =
\begin{bmatrix}
\omega \trasp{A} &0\\
0 & I\\
0 & \omega A^{-1}\\
I & 0 \\
\end{bmatrix}.
\]
In this way the Lagrangian subspace $\mathrm{Gr}\big(\omega A_d^{-1}\big)$ can be re-written as follows
\[
\begin{bmatrix}
I&0\\
0&I\\
0&\omega A^{-1}\\
\bar \omega \traspm{A} & 0
\end{bmatrix}
\]
Summing up, we proved that
\[
\mu^{\scriptscriptstyle{\mathrm{CLM}}}\Big(\mathrm{Gr}\big(\omega A_d^{-1}\big), \mathrm{Gr}\big(\Psi_{0,s_0}(t)\big); t \in [0,T]\Big)= \mu^{\scriptscriptstyle{\mathrm{CLM}}}\Big(\mathrm{Gr}\big(Z_\omega\big), \mathrm{Gr}\big(Z_{s_0}(t)\big); t \in [0,T]\Big)
\]
where
\[
Z_\omega \=\begin{bmatrix}
0&\omega A^{-1}\\
\bar \omega \traspm{A} & 0
\end{bmatrix}\quad \textrm{ and } \quad Z_{s_0}(t)\=
\begin{bmatrix}
\dfrac{\sinh(\sqrt{s_0}t)}{\sqrt{s_0}\cosh(\sqrt{s_0}t)} G & \dfrac{1}{\cosh(\sqrt{s_0}t)}I\\
\dfrac{1}{\cosh(\sqrt{s_0}t)} I& -\sqrt{s_0}\dfrac{\sinh(\sqrt{s_0}t)}{\cosh(\sqrt{s_0}t)} G
\end{bmatrix}.
\]
Now the conclusion follows by
\[
\mu^{\scriptscriptstyle{\mathrm{CLM}}}\Big(\mathrm{Gr}\big(Z_\omega\big), \mathrm{Gr}\big(Z_{s_0}(t)\big); t \in [0,T]\Big)= \mu^{\scriptscriptstyle{\mathrm{CLM}}}\Big(L_D, \mathrm{Gr}\big(Z_{s_0}(t)- Z_\omega\big); t \in [0,T]\Big)
\]
once observed that $Z_{s_0}(t)- Z_\omega=:M_{s_0}(t)$. This concludes the proof.
\end{proof}
\begin{lem}\label{thm:lemma2-2}
For every $t_0 \in (0,T]$, there exists  $s_0>0$ sufficiently large such that $M_{s_0}(t)$ is non-degenerate and
\[
 \ncind{M_{s_0}(t)}= \nind{M_{s_0}(t)} \textrm{ for any } t \in [t_0, T]
\]
 where $\ncind{*}$ and  $\nind{*}$ denotes respectively  the coindex and the index of $*$.
\end{lem}
\begin{proof}
For every $t \neq 0$ the matrix $\dfrac{\sinh(\sqrt{s_0}t)}{\sqrt{s_0}\cosh(\sqrt{s_0}t)} G$ is invertible. We let
\[
N(t)\=\begin{bmatrix}
I & -\dfrac{\sqrt{s_0}\cosh(\sqrt{s_0}t)}{\sinh(\sqrt{s_0}t)} G \left(\dfrac{I}{\cosh(\sqrt{s_0}t)}-\omega A^{-1}\right)\\
0 & I
\end{bmatrix}
\]
and we observe that
\begin{multline}
N^*(t)M_{s_0}(t)N(t)\\=
\begin{bmatrix}
\dfrac{1}{\sqrt{s_0}}\tanh(\sqrt{s_0t}) G & 0 \\
0 &- \sqrt{s_0}\dfrac{1}{\sinh(\sqrt{s_0}t) }\Big(2 \cosh(\sqrt{s_0} t)G-\bar \omega \traspm{A} G -\omega G A^{-1}\Big)
\end{bmatrix}.
\end{multline}
Thus
\begin{multline}
\det M_{s_0}(t)=\det\big(N^*(t)M_{s_0}(t)N(t)\big)=
\det\left(-2G+ \dfrac{1}{\cosh(\sqrt{s_0}t) }\big(\bar \omega \traspm{A} G + \omega G A^{-1}\big)\right).
\end{multline}
Since for $t \in [t_0, T]$ it holds that
\[
\det M_{s_0}(t) \sim_{+\infty} \det(-2 G )\neq 0,
\]
then $M_{s_0}(t)$ is non-degenerate for $t \in [t_0,T]$. We observe also that
\[
N^*(t)M_{s_0}(t)N(t)\sim_{+\infty}
\begin{bmatrix}
\dfrac{1}{\sqrt{s_0}}\tanh(\sqrt{s_0t}) G & 0 \\
0 & - 2\sqrt{s_0}\coth(\sqrt{s_0} t) G
\end{bmatrix}.
\]
Since the index and the coindex of
the two matrices
$M_{s_0}(t)$ and $N^*(t)M_{s_0}(t)N(t)$ agree. This concludes the proof.
\end{proof}

\begin{lem}\label{thm:lemma1-2}
For any complex matrix $B$, we let
\[
M\= \begin{bmatrix}
0 & B\\ B^* & 0
\end{bmatrix}
\]
where we denoted by $B^*$ the conjugate transpose of $B$. Then we have
\[
\nnull{M}= 2 \dim \ker B\quad \textrm{ and }\quad \ncind{M}= \nind{M}
\]
and where $\nnull{*}$ denotes the nullity of $*$.
\end{lem}
\begin{proof}
The proof of this result, readily follows by a direct computation.
\end{proof}

\begin{prop}\label{thm:lemma5}
Under the previous notation, for every $\omega \in \U$, we have
\[
\iota_1(\omega A_d \Psi_{0, s_0}(t); t \in [0,T])= \dim \ker (A-\omega I).
\]
\end{prop}
\begin{proof}
We start to observe that
\[\iota_1(\omega A_d \Psi_{0, s_0}(t); t \in [0,T])= \mu^{\scriptscriptstyle{\mathrm{CLM}}}\Big(\mathrm{Gr}\big(\omega A_d^{-1}\big), \mathrm{Gr}\big(\Psi_{0,s_0}(t)\big); t \in [0,T]\Big).
\]
Then in order to conclude, it
is enough to compute $ \mu^{\scriptscriptstyle{\mathrm{CLM}}}\Big(\mathrm{Gr}\big(\omega A_d^{-1}\big), \mathrm{Gr}\big(\Psi_{0,s_0}(t)\big); t \in [0,T]\Big)$. By Lemma \ref{thm:lemma0-2}, we have
\[
 \mu^{\scriptscriptstyle{\mathrm{CLM}}}\Big(\mathrm{Gr}\big(\omega A_d^{-1}\big), \mathrm{Gr}\big(\Psi_{0,s_0}(t)\big); t \in [0,T]\Big)= \mu^{\scriptscriptstyle{\mathrm{CLM}}}\Big(L_D, \mathrm{Gr}\big(M_{s_0}(t)\big); t \in [0,T]\Big)
\]
 $M_{s_0}(t)$ has been defined in Lemma \ref{thm:lemma1-2}. Moreover
for $t=0$, the matrix $M_{s_0}(t)$ reduces to
\[
M_{s_0}(0)= \begin{bmatrix}
0 & I -\omega A^{-1}\\
I-\bar\omega \traspm{A} & 0
\end{bmatrix}.
\]
By Lemma \ref{thm:lemma1-2}, we have
\[
\nnull{M_{s_0}(0)}= 2 \dim \ker(A-\omega I), \qquad \ncind{M_{s_0}(0)}= \nind{M_{s_0}(0)}.
\]
By Lemma \ref{thm:lemma2-2}, there exists $t_0$ and $s_0$ such that $M_{s_0}(t)$ is non-degenerate for $t \in [t_0,T]$ and
$ \ncind{M_{s_0}(t)}= \nind{M_{s_0}(t)}$ for every $t \in [t_0,T]$. By this, we infer that
\[
\spfl(M_{s_0}(t); t \in [0,T]) = \dim \ker (A-\omega I).
\]
In order to conclude, it is enough  to observe that
\[
\mu^{\scriptscriptstyle{\mathrm{CLM}}}\Big(L_D, \mathrm{Gr}\big(M_{s_0}(t)\big); t \in [0,T]\Big)= \spfl(M_{s_0}(t); t \in [0,T]).
\]
This concludes the proof.
\end{proof}
By using Proposition \ref{thm:lemma5} we are now in position to prove Theorem \ref{thm:Lemma6}.
\paragraph{ Proof of Theorem \ref{thm:Lemma6}.}For $c \in [0,1]$ and $s \in [0,s_0]$
we start to consider the path $s \mapsto  \mathcal A^\omega_{c,s}$ given in Equation \eqref{eq:Acs}, namely
\begin{equation}
\mathcal A^\omega_{c,s}=-G \dfrac{d^2}{dt^2} + c \widehat R(t)+ sG, \qquad t \in [0,T]
\end{equation}
on  the Hilbert  space $E^2_\omega([0,T])$ and defined in Equation \eqref{spazio-Acs}. The corresponding Morse-Sturm
system is given by
\begin{equation}
-G \ddot u(t) + \big(c \widehat R(t) + sG\big) u(t)=0, \qquad t \in [0,T]
\end{equation}
and the associated Hamiltonian system has been  given in Equation  \eqref{eq:Dcs}; i.e.
\begin{equation}
\dot z(t) = J D_{c,s}(t)z(t), \qquad t \in [0,T]
\end{equation}
whose fundamental solution has been  denoted by $s \mapsto \Psi_{c,s}(t)$.
\begin{figure}[ht]
 \centering
 \includegraphics[scale=0.2]{Figura.pdf}
 \caption{In the blue rectangle (above) is depicted the homotopy with respect to the parameter $c$ whilst in the pink rectangle
 (below) the homotopy with respect to the $s$ parameter.}\label{fig:omotopia}
\end{figure}
In order to prove the result we observe that
\begin{enumerate}
\item $\omega A_d \Psi_{c,s}(0)=\omega A_d$ being $\Psi_{c,s}(0)=I$.  This in particular implies that
\[
\iota_1\big(\omega A_d \Psi_{1,s}(0);s \in [0,s_0]\big)=0 \quad \textrm{ and } \quad \iota_1\big(\omega A_d \Psi_{c,s}(0);c \in [0,1]\big)=0.
\]
\item By Corollary \ref{thm:lemma3}, if $s_0$  is sufficiently large, then $\mathcal A_{c,s_0}$ is non-degenerate for every $c \in [0,1]$.
This in particular implies that
\[
\iota_1\big(\omega A_d \Psi_{c, s_0}(T);c \in [0,1]\big) =0.
\]
\item For $c=0$, by Proposition \ref{thm:lemma5}, we infer that
\[
\iota_1(\omega A_d \Psi_{0, s_0}(t); t \in [0,T])= \dim \ker (A-\omega I).
\]
\item By the homotopy invariance property of the Maslov-type index, we have
\begin{multline}\label{eq:dainvocare}
\iota_1\big(\omega A_d\Psi_{1,0}(t); t \in [0,T]\big) +\iota_1\big(\omega A_d\Psi_{1,s}(T); s \in [0,s_0]\big)=
\iota_1\big(\omega A_d\Psi_{1,s_0}(t); t \in [0,T]\big) \\
\iota_1\big(\omega A_d\Psi_{0,s_0}(t); t \in [0,T]\big) = \iota_1\big(\omega A_d\Psi_{1,s_0}(t); t \in [0,T]\big) .
\end{multline}
By Proposition \ref{thm:lemma4}, it holds that
\[
\spfl(\mathcal A^{\omega}_{1,s}; s \in [0,s_0]) =-\iota_1\big(\omega A_d\Psi_{1,s}(T); s \in [0,s_0]\big).
\]
\end{enumerate}
By taking into account the relations given in Equation \eqref{eq:dainvocare}, we  get that
\[
\iota_1\big(\omega A_d\Psi_{1,0}(t); t \in [0,T]\big) = \spfl(\mathcal A^{\omega}_{1,s}; s \in [0,s_0])
+ \dim \ker \big( A-\omega I\big).
\]
This concludes the proof. \qed

\section{Semi-Riemannian Bott-type iteration formula}\label{subsec:Bott-SR}

The goal of this section is to prove the Bott-type iteration formula for semi-Riemannian geodesics.

We start to observe that, since  $\widehat R$ satisfies the condition $\widehat R(T)= \trasp{A} \widehat R(0) A$, we can  extend it on the
interval $[0,mT]$. More precisely, for every $k=1, \dots, m$, we  define  the  associated Morse-Sturm system  as
\begin{equation}\label{eq:m-iteration-MS-system}
\begin{cases}
-G \ddot u+ \widehat R(t) u=0, \qquad t \in [0,mT]\\
u(0)=A^k u(kT) \textrm{ and } \dot u(0)= A^k \dot u(kT).
\end{cases}
\end{equation}
We now consider the Hamiltonian system $
\dot z(t)= JD_{c,s}(t) z(t), \qquad t \in [0,mT]
$
and we define the operator
$
\mathcal A_{c,s}^{(m)}\= -G \dfrac{d^2}{dt^2} + c \widehat R(t) + s G, \qquad t \in [0, mT]
$
on the Sobolev space
\begin{equation}
E_m^k\= \Set{u \in W^{2,2}([0,mT], \R^n)| u(0)= A^ku(k\,T), \dot u(0)= A^k \dot u(k\,T)},\quad \textrm{ for }  k=1, \dots, m.
\end{equation}
Then, by Theorem \ref{thm:Lemma6}, we have
\begin{equation}
\spfl(\mathcal A_{1,s}^{(m)}; s \in[0,s_0]) +\dim \ker (A^m-I)= \iota_1(A_d^m \Psi_{1,0}(t); t \in[0,m\,T]).
\end{equation}
By \cite[Theorem 1.1]{LT15}, we infer
\begin{equation}\label{eq:facili}
\iota_1(A_d^m \Psi_{1,0}(t); t \in [0,m\,T]) =\sum_{\omega^m=1} \iota_\omega(A_d \Psi_{1,0}(t); t \in [0,T])
\end{equation}
and we observe that the following equalities hold
\begin{multline}\label{eq:nullity}
\dim\ker (A^m-I)=\sum_{\omega^m=1} \dim\ker (A-\omega I),\\
\iota_\omega(A_d \Psi_{1,0}(t); t \in [0,T]) = \iota_1(\omega A_d \Psi_{1,0}(t); t \in [0,T]).
\end{multline}
By Proposition \ref{thm:Lemma6},  Equation \eqref{eq:facili} and Equation \eqref{eq:nullity}, we have
\begin{multline}
\spfl(\mathcal A_{1,s}^{(m)}; s \in [0,s_0]) + \sum_{\omega^m=1}\dim \ker (A-\omega I)
= \sum _{\omega^m=1} \iota_1(\omega A_d\Psi_{1,0}(t); t \in [0,T])\quad \Rightarrow \\
\spfl(\mathcal A_{1,s}^{(m)}; s \in [0,s_0])= \sum _{\omega^m=1}\Big( \iota_1(\omega A_d\Psi_{1,0}(t); t \in [0,T])-
\dim \ker (A-\omega I)\Big)\\= \sum _{\omega^m=1}\spfl(\mathcal A_{1,s}^{(\omega)}; s \in [0,s_0]),
\end{multline}
where the last equality follows by invoking Theorem \ref{thm:Lemma6}.

\section{A linear  instability criterion}\label{thm:maintheorem}

This Section  is devoted to the proof of the
{\em instability criteria\/} for closed non-lightlike semi-Riemannian geodesics.
\begin{note} \label{not:sp-ours}
We set
\begin{multline}
\mathrm{Sp}(2n,\R)^{+}\=\{M\in\mathrm{Sp}(2n,\R)\mid \det(M-I_{2n})>0\} \textrm{  and } \\
\mathrm{Sp}(2n,\R)^{-}\=\{M\in\mathrm{Sp}(2n,\R)\mid \det(M-I_{2n})<0\}.
\end{multline}
\end{note}
	\begin{rem}
We observe that Notation \ref{not:sp-ours} are non consistent with the standard notation used in literature. In fact, as the reader can easily realize by looking at  Subsection \ref{subsec:Maslov}, there is the missing factor  $(-1)^{n-1}$ in the definition of $\mathrm{Sp}(2n, \R)^+$ and of $\mathrm{Sp}(2n, \R)^-$. However the advantage of defining $\mathrm{Sp}(2n, \R)^\pm$ likes in Notation \ref{not:sp-ours} is for simplifying the discussion about the  linear stability.
	\end{rem}
\begin{lem}\label{thm:lemma2}
 Let $T\in \mathrm{Sp}(2n, \R) $ be a linearly stable symplectic matrix. Then, there
exists $\delta >0$ sufficiently small such that $e^{\pm \delta J}T \in \mathrm{Sp}(2n,\R)^+$.
\end{lem}
\begin{proof}
Let  us consider the (smooth) symplectic path pointwise defined by
 $T(\theta)\= e^{-\theta J}T$. By a direct computation we get that
\[
 \trasp{T(\theta)} J \dfrac{d}{d\theta} T(\theta)\Big\vert_{\theta=0} =
\trasp{T}T.
\]
We observe that $\trasp{T}T$ is symmetric and positive semi-definite; moreover  being $T
$ invertible it follows that
$\trasp{T}T$  is actually positive definite. Thus, in particular,
$\mathrm{n_-}(\trasp{T}T)=0$. By invoking Proposition \ref{thm:lemma3.2} it follows
that there exists $\delta >0$ such that
$T(\pm \delta) \in \mathrm{Sp}(2n, \R)^+ $. This concludes the proof.
\end{proof}
\begin{rem}
We observe that dropping the linear stability assumption on $T$  in Lemma \ref{thm:lemma2},  the perturbed matrix $e^{\pm \delta J}T$ belongs to $\mathrm{Sp}(2n,\R)^*$; however  we  can't a control in which path-connected component it will lie. (We refer the interested reader to  \cite[Equation (6) and (7), pag. 124]{Lon02} for more details).
\end{rem}

\paragraph{ Proof of Theorem \ref{thm:main-deg}.}
Here it is enough to  prove the
contrapositive, namely
\begin{itemize}
 \item if $\gamma$ is linearly stable and oriented then   $\ispec(\gamma) +n$
  is even;
 \item if $\gamma$ is linearly stable and nonoriented then   $\ispec(\gamma)
+n$   is odd.
\end{itemize}
As direct consequence of Proposition \ref{thm:lemma4}, we know that
 $\ispec(\gamma)=-\igeo(A_d\Psi_{1,s}(T); s \in [0,s_0])$.
 In order to concludes the proof, it is enough to consider the parity of $\igeo(A_d\Psi_{1,s}(T); s \in [0,s_0])+n$.
{\bf Case (OR)} If $\gamma$ is oriented, then $\det(A)=1$. Thus, if
$A_d\Psi_{1,0}(T)$ is linear stable, then by Lemma \ref{thm:lemma2},  it follows that
$e^{-\theta J}A_d\Psi_{1,0}(T)\in \mathrm{Sp}(2n,\R)^{+}$. By a direct computation , we infer that  $\det(A_d\Psi_{0,s_0}-I_{2n})=(-1)^n\big(2\cosh(\sqrt{s_0}T)\big)^n\det A$; thus we get
$\det(A_d\Psi_{0,s_0}-I_{2n})>0$ (resp. $\det(A_d\Psi_{0,s_0}-I_{2n})<0$) iff $n$ is even (resp. odd). Hence, $A_d\Psi_{0,s_0}\in\mathrm{Sp}(2n,\R)^{+}$(resp. $A_d\Psi_{0,s_0}\in\mathrm{Sp}(2n,\R)^{-}$) iff $n$ is even (resp. odd).
By invoking  Corollary \ref{thm:lemma3}, if $s_0$  is sufficiently large, then $\mathcal A_{c,s_0}$ is non-degenerate for every $c \in [0,1]$. Consequently, $\det(A_d\Psi_{c,s_0}-I_{2n})\neq0$ for every $c\in[0,1]$ and in particular if $A_d\Psi_{0,s_0}\in\mathrm{Sp}(2n,\R)^{+}$ then $A_d\Psi_{1,s_0}\in\mathrm{Sp}(2n,\R)^{+}$ too.  By these arguments, we get that $A_d\Psi_{1,s_0}(T)\in
\mathrm{Sp}(2n,\R)^{+}$ (resp. $A_d\Psi_{1,s_0}(T)\in \mathrm{Sp}(2n,\R)^{-}$)  iff $n$ is even (resp. odd). Now, the conclusion follows by taking into account
Lemma \ref{thm:parity} since   $\igeo(A_d\Psi_{1,s}(T), s \in [0,s_0])$  is even
(resp. odd) or which is
equivalent that $\ispec(\gamma)$ is even (resp. odd) iff $n$ is even (resp. odd). Therefore, we can conclude that
 $n+\ispec(\gamma)$ is always even.

{\bf Case (NOR)} If $\gamma$ is nonoriented then $\det(A)=-1$. Arguing as above, we
get that
 $n+\ispec(\gamma)$ is  odd.\qed

From now on, if not differently stated, the pair $(M,\mathfrak g)$
will denote a $(n+1)$-dimensional Riemannian  (resp. Lorentzian) manifold and $\gamma:[0,T]\rightarrow M$ a closed (resp. timelike closed) geodesic.
Once trivialized the pull-back of the tangent bundle along the closed Riemannian. (resp. timelike Lorentzian) geodesic $\gamma$, the index form
reduces to the symmetric bilinear form
$I:E\times E\rightarrow \R$ given by
\begin{equation}
I(u,v)=\int^T_0\langle\dot{u}(t),\dot{v}(t)\rangle+\langle\widehat{R}(t)u(t),v(t)\rangle dt,
\end{equation}
where $E\=\{u\in W^{1,2}([0,T],\mathbf{R}^n)\mid u(0)=Au(T)\}$, $A\in \mathrm{O}(n)$ is an orthogonal matrix. We observe that, in this case, as observed in Section  \ref{subsec:Index-Theorem}, the Morse index $\iota_{\scriptscriptstyle{\mathrm{Mor}}}(\gamma)$ of $\gamma$ (meaning the
dimension of the maximal  subspace such that  $I$ is negative definite),  is well-defined.
Given  $\omega\in \U$, we let $E_{\omega}=\Set{u\in W^{1,2}([0,T],\mathbf{C}^n)|u(0)=\omega Au(T)}$ we define the {\em $\omega$-index form\/}  on $E_\omega$
as follows
\begin{equation}\label{eq:omegaindexform}
\iota_\omega(u,v)=\int^T_0\langle\dot{u}(t),\dot{v}(t)\rangle+\langle\widehat{R}(t)u(t),v(t)\rangle dt,
\end{equation}
where $\langle\cdot,\cdot\rangle$ in Equation \eqref{eq:omegaindexform} denotes the standard Hermitian product.
Following \cite{BTZ82} and denoting by
$\iota_{\scriptscriptstyle{\mathrm{Mor}}}(\omega,\gamma)$ the Morse index of $\iota_\omega$ on $E_{\omega}$, we are in position to give the following definition.
\begin{defn}\label{def:splitting-number-geo} ({\bf \cite[pag. 244]{Lon02}\/} )
The splitting numbers of the closed geodesic $\gamma$ at $\omega\in \U$ are defined by
\begin{equation}\label{eq:S-N}
\mathcal{S}^{\pm}(\omega,\gamma)=\lim_{\theta\to 0^\pm}\iota_{\scriptscriptstyle{\mathrm{Mor}}}(\omega e^{\sqrt{-1}\theta},\gamma)-\iota_{\scriptscriptstyle{\mathrm{Mor}}}(\omega,\gamma).
\end{equation}
\end{defn}
Following the discussion given above, to  the geodesic $\gamma$ we associate the
Morse-Sturm system given by
 \begin{equation}\label{eq:R-M-S}
\begin{cases}
-\ddot{u}(t)+\widehat{R}(t)u(t)=0, \qquad t \in [0,T]\\
u(0)=Au(T).
\end{cases}
\end{equation}
By using the Legendre transformation, the second order system given in Equation \eqref{eq:R-M-S} corresponds to the linear
Hamiltonian system
 \begin{equation}\label{eq:R-H-S}
\dot{z}(t)=JB(t)z(t),  \qquad t \in [0,T]\\
\end{equation}
where $B(t)\=\begin{bmatrix}I&0\\0&-\widehat{R}(t)\end{bmatrix}$.
The next result point out the relation intertwining the splitting numbers of a closed geodesic $\gamma$ and the
splitting numbers of the  symplectic matrix (linearized Poincaré map) given in Definition. \ref{defn:S-S-N}.
(We refer the interested reader to \cite[pag. 191]{Lon02}).
 \begin{lem}\label{thm:R-S-N}
 Under the above notations, we have
 \begin{equation}\label{eq:R-S-N}
 \mathcal{S}^{\pm}(\omega,\gamma)=S^\pm_{\mathcal P(T)}(\omega), \forall \ \omega\in\U.
 \end{equation}
 \end{lem}
\begin{rem}
In \cite[pag.226]{BTZ82},  authors proved that the splitting numbers $\mathcal{S}^{\pm}(\omega,\gamma)$ only depend on the conjugacy class of
$P$ and in \cite[pag. 247]{Lon02}, Long  generalized this result. Here we provide a different proof with respect to
the one given by authors in \cite[pag.252]{Lon02}.
\end{rem}
\begin{proof}
For any  $\bar{\omega}\in \U$, we let $S\=\bar{\omega} A$. By Proposition \ref{Thm:R-O-M-M} and Remark \ref{Rem:C-C}, we have
\begin{equation}\label{eq:R-M-M}
\iota_{\scriptscriptstyle{\mathrm{Mor}}}(\bar{\omega},\gamma)+\nu(\bar{\omega} A)=\mu^{\scriptscriptstyle{\mathrm{CLM}}}(\mathrm{Gr}(\bar \omega \trasp{A}_d),\mathrm{Gr}(\Phi(t)); t \in [0,T])
=\mu^{\scriptscriptstyle{\mathrm{CLM}}}(\mathrm{Gr}(\bar \omega),\mathrm{Gr}(A_d\Phi(t)); t \in [0,T]).
\end{equation}
Moreover, by taking into account  Proposition \ref{prop:L-C}, we have
\begin{equation}\label{eq:R-M-M-4}
\begin{aligned}
&\mu^{\scriptscriptstyle{\mathrm{CLM}}}(\mathrm{Gr}(\bar \omegaI),\mathrm{Gr}(A_d\Phi(t)); t \in [0,T])\\&=\mu^{\scriptscriptstyle{\mathrm{CLM}}}(\mathrm{Gr}(\bar \omegaI),\mathrm{Gr}(A_d\Phi(t))*\xi(t)); t \in [0,T])-\mu^{\scriptscriptstyle{\mathrm{CLM}}}(\mathrm{Gr}(\bar \omegaI),\mathrm{Gr}(\xi(t))); t \in [0,T])\\&=\begin{cases}
(\iota_{\bar \omega}(A_d\Phi(t)*\xi(t); t \in [0,T])+n)-(\iota_{\bar \omega}(\xi(t); t \in [0,T])+n) \quad \omega=1\\
\iota_{\bar \omega}(A_d\Phi(t)*\xi(t); t \in [0,T])-\iota_{\bar \omega}(\xi(t); t \in [0,T])\quad \omega\neq 1\\
\end{cases}
\\&=\iota_{\bar \omega}(A_d\Phi(t)*\xi(t); t \in [0,T])-\iota_{\bar \omega}(\xi(t); t \in [0,T])
\quad \forall \, \omega\in\U,
\end{aligned}
\end{equation}
where $\xi$ is any symplectic path joining  $I$ to $A_d$. Summing up Equations \eqref{eq:R-M-M}-\eqref{eq:R-M-M-4},
we get
 \begin{equation}\label{eq:R-M-M-2}
\iota_{\scriptscriptstyle{\mathrm{Mor}}}(\bar{\omega},\gamma)=\iota_{\bar \omega}(A_d\Phi(t)*\xi(t))-\iota_{\bar \omega}(\xi(t))-\nu(\bar{\omega} A).
\end{equation}
According to Definition \ref{defn:S-S-N}, we infer that
\begin{equation}\label{eq:R-M-M-3}
\mathcal{S}^{\pm}(\bar \omega,\gamma)=S^\pm_{\mathcal P}(\bar \omega)-S^\pm_{A_d}(\bar \omega)+\nu(\bar{\omega} A)
\end{equation}
and being $A$  an orthogonal matrix, then by direct computation, we get
$S^\pm_{A_d}(\bar \omega)=\nu(\bar{\omega} A)$.
 This complete the proof.
\end{proof}
We let $\widetilde J\=-\sqrt{-1}J$. For any symplectic matrix $M\in\mathrm{Sp}(2n,\R)$, we define the $\widetilde J$-invariant (generalized eigenspace)
subspace $E_\lambda$ as
\begin{equation}
E_\lambda\=\bigcup_{m\geq1}\ker(M-\lambda I)^m.
\end{equation}
and we observe that the following $\widetilde J$-orthogonal splitting holds
\begin{equation}
\C^{2n}=\bigoplus_{\lambda\in \sigma(M)}E_{\lambda}.
\end{equation}
\begin{defn}\label{def:krein-type}
For any $\lambda\in\sigma(M)\cap\U$, the restriction of  $\widetilde J$ to the subspace $E_{\lambda}$ is non-degenerate.
We define the {\em Krein-type\/} of $\lambda$ by $(p,q)$,
where $p$ (resp. $q$) denotes respectively the total multiplicity of the positive (resp. negative) negative eigenvalues of $\widetilde J\mid_{E_{\lambda}}$.
 If $p=0$ (resp. $q=0$) the eigenvalue  $\lambda$ is termed {\em Krein-negative\/} (resp. {\em Krein-positive\/});
 otherwise, $\lambda$ is called {\em Krein-indefinite\/} or of {\em mixed-type.\/}
\end{defn}
For short, we will  refer to   the case $p=0$ or $q=0$ simply as {\em Krein-definite.\/}
Before recalling  the definition of {\em strongly stability\/}, we  introduce the following new definition.
\begin{defn}\label{def:trival geodesic}
A closed geodesic $\gamma$ is called {\em index hyperbolic\/} if, for any $\lambda\=e^{\sqrt{-1}2\pi \theta}\in\sigma(\mathcal P(T))\cap \U$ (eigenvalue of
the  Poincaré map $\mathcal P(T)=A_d\Phi(T)$, here $\Phi$ denotes the fundamental solution) it holds
\begin{multline}
S^+_{\mathcal P}(e^{\sqrt{-1}2\pi\theta}) =S^-_{\mathcal P}(e^{\sqrt{-1}2\pi\theta}) =0 \quad \textrm{ if }   \theta\in \Q,  \\
S^+_{\mathcal P}(e^{\sqrt{-1}2\pi\theta}) =S^-_{\mathcal P}(e^{\sqrt{-1}2\pi\theta}) \quad \textrm{ if  }  \theta\notin \Q.
\end{multline}
\end{defn}

\begin{prop}\label{thm:trivial}
Let $(M,\mathfrak g)$ be a Riemannian (resp. Lorentzian) closed  (resp. timelike and  closed) geodesic. We assume that
for any $m\in\N$, $\iota_{\scriptscriptstyle{\mathrm{Mor}}}\big(\gamma^{(m)}\big)=0$. Then  $\gamma$ is index hyperbolic.
\end{prop}
\begin{proof}
We start to observe that  by Proposition \ref{eq:I-F-C} we have
\begin{equation}\label{eq:above}
\mu^{\scriptscriptstyle{\mathrm{CLM}}}\big(\mathrm{Gr}((\trasp{A}_d)^m),\mathrm{Gr}(\Phi(t));t\in[0,mT]\big)=\sum_{i=1}^{m}\mu^{\scriptscriptstyle{\mathrm{CLM}}}\big(\mathrm{Gr}(\exp(\dfrac{i}{m}2\pi\sqrt{-1})\trasp{A}_d),\mathrm{Gr}(\Phi(t));t\in[0,T]\big).
\end{equation}
By invoking Proposition \eqref{eq:M-M},  Equation \eqref{eq:above} fits into the following
 \begin{equation}\label{eq:mancante}
\iota_{\scriptscriptstyle{\mathrm{Mor}}}(\gamma^{(m)})+\nu(A^m)=\sum_{i=1}^{m}\iota_{\scriptscriptstyle{\mathrm{Mor}}}\left(\exp(\dfrac{i}{m}2\pi\sqrt{-1}),\gamma\right)+\sum_{i=1}^{m}\nu(\exp(\dfrac{i}{m}2\pi\sqrt{-1}) A).
\end{equation}
By Equation \eqref{eq:nullity} we infer that $\nu(A^m)=\displaystyle{\sum_{i=1}^{m}}\nu(\exp(\dfrac{i}{m}2\pi\sqrt{-1}) A)$ and by substituting into Equation \eqref{eq:mancante},
we immediately get that
\begin{equation}
\iota_{\scriptscriptstyle{\mathrm{Mor}}}(\gamma^{(m)})=\sum_{i=1}^{m}\iota_{\scriptscriptstyle{\mathrm{Mor}}}\left(\exp(\dfrac{i}{m}2\pi\sqrt{-1}),\gamma\right).
\end{equation}
By assumption, for every $m \in \N$,   $\iota_{\scriptscriptstyle{\mathrm{Mor}}}(\gamma^{(m)})=0$. By definition,  $\iota_{\scriptscriptstyle{\mathrm{Mor}}}(\exp(\frac{i}{m}2\pi\sqrt{-1},\gamma)\geq0$ this immediately  implies that
\[
\iota_{\scriptscriptstyle{\mathrm{Mor}}}\left(\exp(\dfrac{i}{m}2\pi\sqrt{-1}),\gamma\right)=0 \quad \textrm{ for any } i, m\in\N
\]
which is  equivalent to assert that $\iota_{\scriptscriptstyle{\mathrm{Mor}}}(e^{\sqrt{-1}2\pi\theta},\gamma)=0$ for
any $\theta\in\Q$. So, for any $\lambda=e^{\sqrt{-1}2\pi\theta}\in\sigma(\mathcal P(T))$ and $\theta\in\Q$, by invoking Lemma \ref{thm:R-S-N}, we get
that
\[
S^\pm_{\mathcal P(T)}(\lambda)= \mathcal{S}^{\pm}(\lambda,\gamma)=0.
\]
This conclude the first claim. In order to prove the second claim,
if $\theta\notin\Q$, let $\theta_1<\theta<\theta_2$ be such that  $\theta_1,\theta_2$ are in $\Q$, $|\theta_j-\theta|$ is small enough and
$e^{\sqrt{-1}2\pi\theta_j}\notin\sigma(\mathcal P(T))$ for $j=1,2$. Then $\iota_{\scriptscriptstyle{\mathrm{Mor}}}(e^{\sqrt{-1}2\pi\theta_j},\gamma)=0$ and $\mathcal{S}^{\pm}(e^{\sqrt{-1}2\pi\theta_j},\gamma)=0$.  By Definition \ref{def:splitting-number-geo},  we know that in fact the splitting numbers $\mathcal{S}^{\pm}(\omega,\gamma)$ measure the jumps between $n_-(\omega,\gamma)$ and $n_-(\lambda,\gamma)$
 for $\lambda\in \U$ in a neighborhood of  $\omega$. By invoking once again Lemma \ref{thm:R-S-N} as well as Proposition \ref{prop:splitting number and ultimate type}, we infer that $\mathcal{S}^{\pm}(\omega,\gamma)=0$ if $\omega \notin \sigma(\mathcal P(T))$. In conclusion, we have
\begin{equation}
 \iota_{\scriptscriptstyle{\mathrm{Mor}}}(e^{\sqrt{-1}2\pi\theta_2},\gamma)=\iota_{\scriptscriptstyle{\mathrm{Mor}}}(e^{\sqrt{-1}2\pi\theta_1},\gamma)+\mathcal{S}^{+}(e^{\sqrt{-1}2\pi\theta_1},\gamma)+
 \mathcal{S}^{+}(\lambda,\gamma)-\mathcal{S}^{-}(\lambda,\gamma)-\mathcal{S}^{-}(e^{\sqrt{-1}2\pi\theta_2},\gamma),
\end{equation}
 so $\mathcal{S}^{+}(\lambda,\gamma)=\mathcal{S}^{-}(\lambda,\gamma)$. The conclusion, readily follows, by invoking once again
 Lemma \ref{thm:R-S-N}. This concludes the proof.
\end{proof}
We recall that the strong stability of a symplectic matrix can be characterized through its eigenvalues. For the
sake of the reader, we recall the following result proved by author in \cite{Eke90}.
\begin{lem}\label{lem:strong stability}({\bf  \cite[Thm 10, pag.11]{Eke90}\/})
$M$ is strongly stable if and only if it is linearly stable and all of its eigenvalues are Krein-definite.
\end{lem}
\begin{rem}\label{rem:strong stable}
We observe that since the eigenvalues $\{-1,1\}$ have  Krein-type $(p,p)$, by this it readily follows that  $\pm1$ cannot be in the spectrum  of a
strongly stable symplectic matrix.
\end{rem}
As a direct consequence of Proposition \ref{thm:trivial} and of the basic normal forms of a symplectic matrix, we get the
following strongly instability result for closed geodesics.
\paragraph{Proof of Theorem \ref{cor:non-strong stable}.}
Under  the assumptions of Theorem \ref{cor:non-strong stable}, by invoking Proposition \ref{thm:trivial},
we can conclude that the geodesic is  index hyperbolic. By taking into account Definition \ref{def:trival geodesic} on
index hyperbolicity applied to  the monodromy $\mathcal P(T)=A_d\Phi(T)$, we get that,
for every eigenvalue $e^{\sqrt{-1}\theta}\in \sigma(\mathcal P(T))$, $S^+_{\mathcal P(T)}(e^{\sqrt{-1}\theta}) -S^-_{\mathcal P(T)}(e^{\sqrt{-1}\theta}) =0 $.
In order to concludes the proof, we argue by contradiction. For, we assume that $\mathcal P(T)$ is strongly stable.
Thus, by the characterization given in Lemma \ref{lem:strong stability}, $\mathcal P(T)$ is linearly stable and  all of its
 eigenvalues  are Krein-definite. By this fact, it immediately  follows that, for any
eigenvalue $e^{\sqrt{-1}\theta}\in \sigma(\mathcal P(T))$ having Krein type $(p,q)$, it holds that $p-q\neq0$. By this we get a contradiction once invoked
\cite[Corollary 8, pag.198]{Lon02}, being
\begin{multline}
S^+_{\mathcal P}(e^{\sqrt{-1}\theta}) -S^-_{\mathcal P}(e^{\sqrt{-1}\theta})=p-q\neq 0 \textrm{ and } \\
S^+_{\mathcal P}(e^{\sqrt{-1}\theta}) -S^-_{\mathcal P}(e^{\sqrt{-1}\theta})=p-q= 0
\end{multline}
at the same time.  This concludes the proof.
\qed


text:
Rick Herrington will replace brother Mike as football coach at Hart
Rick Herrington (right) has been named football coach at Hart, replacing brother Mike (middle). His other brother, Dean, remains coach at Paraclete.
(Eric Sondheimer / Los Angeles Times)
When Mike Herrington announced he was retiring after 31 years as football coach at Hart, one of his longtime assistants, brother Rick, indicated he was not interested in replacing him.
So how did it happen that Hart named Rick Herrington its new football coach on Thursday?
"Old people sometimes change their mind," Rick, 60, said. "I was done coaching. I was fine. I was sitting around for a week or two with people in my ear. I said, 'Shoot, I might as well try it.' Now I'm excited."
Rick has never been a varsity head coach. He was defensive coordinator at Hart and also helped coach with brother Dean at Alemany. His head coaching experience was as a freshman coach at Hart from 1978-79 and sophomore coach from 1980-87. He has been coaching for 42 years.
"All those interviews I went through in the past and never got one," he said. "Maybe this is something I can do. This is a good place to be a head coach. I'll have a pretty good mentor to ask questions."
Rick, who graduated from Hart just like his brothers, is a teacher at the Newhall school.
He said Mike will not be on the staff. "He'll probably be on the sideline second guessing," he said.

redpajama_set_name: RedPajamaCommonCrawl


text:
Côte-Vertu may refer to:

Côte-Vertu Boulevard, in Saint-Laurent and Dorval (Montreal)
Côte-Vertu station, a Montreal Metro station in the borough of Saint-Laurent
Terminus Côte-Vertu, an Exo bus terminus partly north and partly south of the Côte-Vertu Metro station in the borough of Saint-Laurent

See also
Place Vertu, a shopping centre in the borough of Saint-Laurent (Montreal)

redpajama_set_name: RedPajamaWikipedia


text:
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Home » How COVID-19 changed Miami athletics this fall
Athletics, Sports
How COVID-19 changed Miami athletics this fall
In the classroom, in training and in games, the fall semester was unlike any other for student athletes at the University of Miami and throughout the NCAA.
Back in June, long before classes had begun or the majority of students were welcomed back to campus, Miami's football team resumed training, starting with 65 athletes, as others returned in phases throughout the summer. At that time, Florida had begun experiencing its first major spike of COVID-19 cases, and it was still unknown whether the ACC would have a fall football season.
The athletic department instituted various guidelines to ensure a safe return and prevent the spread of COVID-19 among athletes, coaches and athletic staff. These protocols included regular COVID-19 testing, contact-tracing, quarantine and isolation when necessary, daily temperature checks and screening before practices.
Manny Diaz celebrates a touchdown against Florida State during the first half of Miami's game versus Florida State on Sept. 26, 2020. This season, all sideline personnel wear face coverings to prevent the spread of Covid-19. Photo credit: Michael Laughlin/ACC
Miami's football players were the first to put these protocols to the test.
Jaelan Phillips, a redshirt junior defensive lineman, returned to campus in July to begin training for the 2020 football season.
"We weren't able to operate like we are used to," Phillips said. "We have had limited access to the locker room, training room, and practice facilities. So, it has been challenging, to say the least."
Phillips said he did not worry so much about contracting the virus.
"I am confident in the ACC protocols, and I believe that we are doing everything we can to be safe."
UM's athletic department did not make its COVID numbers public.
Once it was confirmed that the season would take place, Phillips said many doubted the team's and the season's success.
Miami had just come off a disappointing 2019 season where the Hurricanes went 6-7 and ended on a three-game losing streak. This included a gut-wrenching loss to Miami-Dade County rival Florida International University on the resting grounds of the Orange Bowl, the former stadium where the Miami football program saw some of its best years. The Canes topped off their season with a 14-0 loss to Louisiana Tech at the Independence Bowl.
Fortified by a young, new offensive coordinator, Brett Lashlee, and a new transfer starting quarterback, D'Eriq King, Miami faced a highly anticipated season, further complicated by the pandemic.
The team and Phillips proved many of the predictions wrong, said Phillips, who earned his first ACC Defensive Lineman of the Week honor this season.
"Many people thought that we wouldn't get past four games," he said. Miami, now 8-1 and ranked No. 10 in the country by the AP Top 25 Poll, had three College GameDays back to back this season and is a current contender for a shot at the ACC championship and a top bowl game.
More than halfway through the season, Miami faced its first major setback due to COVID-19.
Even though the team reportedly followed strict protocols, three games were postponed after an internal coronavirus outbreak, during which both head coach Manny Diaz and head trainer Vinny Scavo tested positive.
However, Miami came back from the three-week break with a resounding 48-0 victory over the Duke Blue Devils on Dec. 5.
Diaz said he is encouraged by how his team and the entire university handled every challenge that came their way this semester.
"We worked together tirelessly to ensure that we could operate safely during a pandemic," Diaz said. "Many challenges still lie ahead, but I am confident we will continue to rise to meet them."
Miami will end its season at home, facing University of North Carolina, ranked No. 17, on Saturday.
Miami's volleyball team finishes a successful shortened season
In what may be the program's best performance yet, the women's volleyball team ended 5-3 and ranked No. 12 by the American Volleyball Coaches Association after an eight-game season shortened due to the pandemic.
"This semester was something completely different and new as an athlete," said Emily Damon, a defensive specialist on the team. "The athletic department would be very cautious and test us three times a week during season."
Members of the Canes volleyball team huddle during their match against NC State on Nov. 15, 2019 in the Knight Sports Complex. Photo credit: Jared Lennon
The volleyball team was still able to travel and competed in regional pods and two-game weekend series.
Toward the end of the season, a volleyball student tested positive for COVID-19, putting the whole team in quarantine for two weeks, said Damon, a junior public health major.
"At first I was terrified because I had just seen my parents who came in town to see our match," Damon said. "But I ended up testing negative, and continued to quarantine for 14 days."
Damon said the university did a great job of taking care of the volleyball athletes while they were in quarantine.
"I was getting three meals a day, sometimes it was even too much food," Damon said. "I think I gained 5 pounds because I wasn't working out either. But I am very grateful the university took good care of us; I felt safe at all times."
Student athletes on UM's COVID-19 protocol
Sydnee Roby, a center on the women's basketball team, arrived on campus in July to begin preseason workouts with the women's basketball team.
"There have been numerous changes since I got back to campus, and they've kept the number of cases lower than I expected them to be," said Roby, a sophomore sports administration major. "But I'm still not comfortable with the numbers that have been posted."
UM reports more than 1,500 confirmed cases of COVID-19 since Aug. 16, which includes positive tests for students, faculty and staff.
Roby, complimented the university's and the athletics department's efforts to keep students safe.
"The school has done a fairly good job with regulations and protocols," said Roby. "And the athletic department has really been keeping an eye on us as athletes to make sure we do our part in keeping numbers down."
Selena Fortich, a defender on UM's women's soccer team, said she was glad UM gave students the option to complete the semester remotely.
"I think administrators did well in giving students the option to stay at home and do classes on Zoom if they felt uncomfortable, but also provided an option for those who did want in-person classes," said Fortich, a senior political science and criminology double major. However, with the hybrid format in which some students could attend class remotely, attendance became more flexible for in-person classes, she said.
"Not every class was held to the same standard," Fortich said. "I felt like some classes were more lenient in terms of showing up on the day you were assigned to be on campus compared to others that were stricter about it."
Andrew Barnes, a redshirt freshman safety on the football team, said despite the number of COVID-19 cases, the university was able to successfully remain open and keep outbreaks at a minimum.
"I feel more safe on campus than being at home in my community," Barnes said.
Barnes said he thinks fewer students will opt to be on campus in the spring because of the impact of COVID-19.
"The whole point of college is to get the college experience, but because of COVID-19, students aren't getting that true experience, so it's no real reason to be here. You can earn your grades from home," Barnes said.
Althea Castro, Jarius Howard, Brianna Jackson, Dante Johnson and Morgan Ledenko contributed to the reporting of this story.
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Artist's life, legacy honored on new USPS postage
The United States Postal Service features Emilio Sanchez, the first Cuban American visual artist to be recognized in a new series of Forever stamps. ...
Inaugural Frost Institute takes shape
With the first elevated concrete floor deck poured last week, the Frost Institute of Chemistry and Molecular Science is on pace to open in the summer of 2022. ...
Scholars link diet, dentition, and linguistics
In a new study that scrutinized the speech patterns of Freddie Mercury, Michael Phelps, and other celebrities, University of Miami researchers offer insights on how diet ultimately reshapes language. ...
Symposium to explore climate change in the Caribbean, South Florida
The multidisciplinary forum in April will explore lessons learned from the regions. There is a call for papers to be presented, but abstracts must be submitted by Jan. 17. ...
Remembering the man who 'saved Miami football'
Former executive vice president John L. Green Jr., who helped launch the University's football dynasty with two key coaching hires, passed away this month. ...
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text:
Jan. 9, 2014 / 12:01 AM
One in five workers has plans to vamoose, survey finds
CHICAGO, Jan. 9 (UPI) -- One-in-five U.S. workers expects to change jobs in 2014, employment firm CareerBuilder said Thursday.
In an online survey conducted mostly in November, 21 percent of respondents indicated they had plans to change jobs in 2014, up from 17 percent who indicated they would make a job switch in 2013.
It is the highest percentage of workers in the post-recession era indicating they have plans to change jobs.
CareerBuilder said the percentage of workers satisfied with their jobs had dropped from 66 percent in the previous survey to 59 percent in the survey conducted Nov. 6-Dec. 2.
RELATED Hiring intentions slightly stronger on global scale
Eighteen percent expressed dissatisfaction with their jobs, up from 15 percent in the poll taken in late 2012.
Among those who indicated they were dissatisfied, 66 percent cited salary and 65 percent cited feeling they were not valued by their employer.
Among those who are unhappy with their jobs, 58 percent indicate they would make a switch in 2014.
RELATED Poll: Huge gap between personal, national satisfaction
Thirty-nine percent of those planning on making a change indicated they were not happy with their work and home life balance, underemployed or highly stressed (39 percent). Thirty-seven percent indicated they had a poor opinion of their boss' performance and 36 percent indicated they were overlooked for a promotion.
But 79 percent of workers indicated they had no plans to leave their jobs.
In that group, 54 percent indicated they liked their co-workers, 50 percent indicated they had a good balance between life and work, and 49 percent indicated they had good benefits.
RELATED Poll: Americans' satisfaction with life high despite slow recovery
Slightly fewer -- 43 percent -- indicated they made a good salary and some -- 35 percent -- indicated an uncertain job market was keeping them at their current job.
The survey, conducted by Harris Interactive, involved 3,008 responses. The results carries a margin of error of 1.79 percentage points and 95 percent certainty.
Defense News // 1 year ago
Austin visits Ukraine, voices U.S. support against Russian aggression
Oct. 19 (UPI) -- U.S. Defense Secretary Lloyd Austin said Ukraine continues to have U.S. support against Russian aggression and is free to decide its own foreign policy during a visit to Kyiv Tuesday.
Citing Russian occupation, U.S. signs new defense pact with Georgia
Oct. 18 (UPI) -- U.S. Defense Secretary Lloyd Austin signed a new security agreement with Georgia Monday, seeking to buttress relations with the Black Sea nation as it continues to face Russian troops in two of its regions.
Pentagon orders all civilian employees to be fully vaccinated by Nov. 22
Oct. 4 (UPI) -- The Pentagon on Monday ordered all Department of Defense civilian employees to be fully vaccinated against COVID-19 by no later than Nov. 22.
Army investigating suspected suicides of three Fort Drum soldiers
Sept. 29 (UPI) -- The U.S. Army is investigating the suspected suicides of three soldiers based at New York's Fort Drum.
Four Marines remain hospitalized month after suicide bombing at Afghan airport
Sept. 24 (UPI) -- Four U.S. Marines remain hospitalized after an attack near Hamid Karzai International Airport in Kabul, Afghanistan, last month as service members were evacuating foreign nationals and refugees.
South Korean firm to supply rapid COVID-19 test kits to U.S. military
SEOUL, Sept. 24 (UPI) -- South Korean biopharmaceutical firm Celltrion has agreed to supply its DiaTrust rapid COVID-19 test kit to the U.S. military.
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text:
David Letterman Reveals Bar Refaeli Is New Sports Illustrated Swimsuit Cover Girl
David Letterman unveiled a new billboard in Times Square in New York City last night.
But it wasn't for his show. It was for Sports Ilustrated.
The late night talk show host pulled the curtain on Sports Illustrated's 2009 Swimsuit Issue cover, revealing Leonardo DiCaprio's main squeeze, Bar Refaeli, as this year's lucky supermodel.
PHOTO EXCLUSIVE: Click for 3 exclusive photos of Bar's SI spread!
(For those keeping score, this is DiCaprio's second Sports Illustrated sueprmodel girlfriend, the first being Gisele Bundchen. Grrr.)
So why all the hoopla?
MORE PHOTOS: 10 years of Swimsuit Issue cover girls.
Over 10 percent of Sports Illustrated's annual income comes from its swimsuit issue. And this year, with magazines closing left and right due to contracting advertising dollars, the magazine is trying to wring its venerable swimsuit issue for everything its worth.
In addition to the annual issue itself, Sports Illustrated is launching a "24/7 Sports Illustrated Swimsuit Experience" at the Luxor casino in Las Vegas, a Southwest Airlines flight from New York to Vegas emblazoned with Refaeli's likeness, and an online component complete with additional photos and behind the scenes videos of each of the swimsuit models.
RELATED: Check out SportsIllustrated.com for the all of the 2009 Swimsuit Issue's online bells and whistles.
What's next, Swimsuit Issue credit cards? Swimsuit Issue cereal? Swimsuit Issue on Broadway?
Oh yeah, they did that last night.


text:
Guillermo Gorostiza Paredes (* 15. Februar 1909; † 23. August 1966) war ein spanischer Fußballspieler.

Karriere

Verein
Gorostiza begann seine Karriere 1927 bei Arenas Club de Getxo. Nach einem kurzen Zwischenspiel in Galicien bei Racing de Ferrol unterschrieb der Baske zur Saison 1929/30 einen Vertrag bei Athletic Bilbao und prägte dort das erfolgreichste Jahrzehnt der Vereinshistorie. Gemeinsam mit seinem kongenialen Sturmpartner Bata lief Gorostiza schnell zur Höchstform auf und sicherte sich mit insgesamt 19 Toren umgehend den Titel des Torschützenkönigs. Ferner gewann er mit seiner Mannschaft die Copa del Rey sowie die erste spanische Meisterschaft der Vereinsgeschichte. Dieses sogenannte Double konnte in der Saison 1930/31 erfolgreich verteidigt werden. Während in der Liga anschließend zwei Vize-Meisterschaften folgten, feierte man 1932 und 1933 zwei weitere Pokalsiege. Weitere Meistertitel folgten 1934 und 1936.

Da der Spielbetrieb während des Spanischen Bürgerkrieges eingestellt wurde, schloss sich Gorostiza 1937 der baskischen Fußballauswahl an und bestritt daraufhin mit dieser Spiele in Europa und Mexiko.

Nach Beendigung des Bürgerkrieges kehrte er nach Spanien zurück und unterschrieb einen Vertrag beim FC Valencia. Auch hier ließen die Erfolge nicht lange auf sich warten. 1941 gewann der Club erstmals in der Vereinsgeschichte den spanischen Pokal, 1942 und 1944 die ersten spanischen Meistertitel.

1946 wechselte Gorostiza schließlich für eine Saison zum FC Barakaldo, ehe er seine Karriere als Spielertrainer bei CD Logroñés ausklingen ließ.

Nationalmannschaft
Gorostiza feierte am 14. Juni 1930, im Rahmen einer 0:2-Niederlage gegen die Tschechoslowakei, sein Debüt in der spanischen Nationalmannschaft. Bis zu seinem letzten Auftritt am 28. Dezember 1941 gegen die Schweiz bestritt er insgesamt 19 Länderspiele und nahm an der Weltmeisterschaft 1934 teil.

Erfolge
Mit dem Verein
 Spanischer Meister (6): 1930, 1931, 1934, 1936, 1942, 1944
 Spanischer Pokalsieger (5): 1930, 1931, 1932, 1933, 1941

Persönliche Auszeichnungen
 Pichichi-Trophäe: 1930, 1932

Weblinks

 Einsatzdaten auf Athletic-Club.net
 Einsatzdaten (Nationalmannschaft)

Fußballnationalspieler (Spanien)
Fußballspieler (Arenas Club)
Fußballspieler (Athletic Bilbao)
Fußballspieler (CD Logroñés)
Fußballspieler (FC Barakaldo)
Fußballspieler (FC Valencia)
Fußballspieler (Racing de Ferrol)
Teilnehmer an einer Fußball-Weltmeisterschaft (Spanien)
Spanischer Meister (Fußball)
Fußballtrainer (CD Logroñés)
Geboren 1909
Gestorben 1966
Mann
Person (Baskenland)
Spanier


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