text:
\section{Introduction}
In multifractal analysis, the interest is in the behaviour of the local dimension map
\[x\mapsto\dimloc(\mu,x)=\lim_{r \downarrow 0} \log\mu(B(x,r))/\log r,\]
for some, often dynamically defined, fractal type measures $\mu$. From the mathematical point of view, the ultimate goal is to understand the size of the level sets
\[E_\alpha=\{x\,:\,\dimloc(\mu,x)=\alpha\}.\]
It is common to say that ``$\mu$ satisfies the
multifractal formalism'' if for all $\alpha\ge0$ the Hausdorff and packing dimensions of $E_\alpha$ are given by the Legendre transform of the $L^q$-spectrum $\tau_q$, that is,
\begin{equation}\label{eq:formalsim}
\dimh(E_\alpha)=\dimp(E_\alpha)=\inf_{q\in\mathbb{R}}\{q\alpha-\tau_q(\mu)\}.
\end{equation}
See Section \ref{sec:not} below for the precise definitions.

Since its origins in physics literature in the 80's (e.g.\ \cite{HentschelProcaccia1983, FrischParisi1985}), the multifractal analysis has gained a lot of interest. For many relevant works related to multifractal formalism, see e.g.\ references in \cite{FengLau2009}. However, it seems that most of the studies take place in Euclidean spaces, or in spaces having a Euclidean type manifold structure.

In this paper, our goal is to study the local dimensions of measures in doubling metric spaces. Perhaps the most classical situation in which the multifractal
formalism is known to hold is the case of self-similar measures
in Euclidean spaces under the strong separation condition; see
e.g.\ \cite{CawleyMauldin1992, Falconer1997}. Our main results can be viewed as a
generalisation of this result into metric spaces, but our method using local versions of the $L^q$-spectrum and dimensions is useful also in the Euclidean setting.

In a general doubling metric space, there are usually no nontrivial self-similar maps, but often there is still a large class of Moran constructions sharing many of the geometric properties of self-similar iterated function systems. We will consider measures on the limit sets of these Moran constructions and investigate the behaviour and multifractality of $\dimloc(\mu,x)$ for these measures.
To determine $\dimloc(\mu,x)$, we consider the local $L^q$-spectrum of $\mu$.
As for the classical (global) spectrum, the definition involves sums of the form $\sum_{B\in\mathcal{B}}\mu(B)^q$ over packings or partitions of the space $X$. However, in order to make the notion local, only those $B\in\mathcal{B}$ are taken into account which are ``sufficiently close to $x$''. It turns out that in many cases, the local spectrum gives more precise information on $\dimloc(\mu,x)$ than its global counterpart. The local spectrum was introduced in \cite{KaenmakiRajalaSuomala2012} as a tool to study local homogeneity properties of measures. Although the definition seems very natural, we were not able to track a definition of a local spectrum for measures in the existing literature. In \cite{Barraletal2010}, a local spectrum for functions is defined in order to study their H\"older regularity. After the completion of our work, the paper \cite{Barraletal2012} was made public. The paper deals with local multifractal analysis in Euclidean spaces for functions, measures, and distrib!
 utions.

The paper is organised as follows. In Section \ref{sec:not}, we set up some notation and define the necessary concepts. Further, in Section \ref{sec:par} we consider partitions of the space $X$, and show how the various dimensions and dimension spectra can be calculated using these partitions. We also relate the $L^q$-spectra and dimensions to the local entropy dimensions defined in \cite{KaenmakiRajalaSuomala2012}. Our main results are presented in Section \ref{sec:results}. We first give a series of conditions for Moran constructions in doubling metric spaces and measures defined on their limit sets. Then we show how the local $L^q$-spectrum can be used to calculate the local dimensions of these measures, and finally study their multifractal properties.

\section{Notation and preliminaries}\label{sec:not}

In this paper, we always assume the metric space $(X,d)$ to be \emph{doubling},
meaning that there is a constant $N=N(X)\in\mathbb{N}$, called the \emph{doubling constant} of $X$,
such that any closed ball $B(x,r) = \{ y\in X : d(x,y) \le r \}$ with centre $x \in X$ and radius $r>0$ can be
covered by $N$ balls of radius $r/2$.

For $M>0$ and a ball $B=B(x,r)$, we will use the abbreviation $MB=B(x,Mr)$. For this to make sense, we always assume that the radius and centre of the ball $B$ have been fixed, even if these are not explicitly mentioned.

We call any countable collection $\mathcal{B}$ of pairwise disjoint closed balls a \emph{packing}. It is called a \emph{packing of $A$} for a subset $A \subset X$ if the centers of the balls of $\mathcal{B}$ are in the set $A$, and it is a $\delta$-packing (for $\delta > 0$) if all the balls in $\mathcal{B}$ have radius $\delta$. A $\delta$-packing $\mathcal{B}$
of $A$ is termed \emph{maximal} if for every $x \in A$ there is $B \in \mathcal{B}$ so that $B(x,\delta) \cap B \ne \emptyset$. Note that if $\mathcal{B}$ is a maximal $\delta$-packing of $A$, then $2\mathcal{B} = \{ 2B : B \in \mathcal{B} \}$ covers $A$.

The following lemma will be frequently used in this paper.

\begin{lemma} \label{thm:covering_thm}
  For a metric space $X$, the following statements are equivalent:
\begin{enumerate}
  \item $X$ is doubling. \label{covering1}
  \item There are $s>0$ and $c>0$ such that for all $R>r>0$ any ball of radius $R$ can be covered by $c(r/R)^{-s}$ balls of radius $r$. \label{covering2}
  \item There are $s>0$ and $c>0$ such that if $R>r>0$ and $\mathcal{B}$ is an $r$-packing of a closed ball of radius $R$, then the cardinality of $\mathcal{B}$ is at most $c(r/R)^{-s}$. \label{covering3}
  \item For every $0<\lambda<1$ there is a constant $M=M(X,\lambda) \in
  \mathbb{N}$, satisfying the following: If $\mathcal{B}$ is a collection of closed
  balls of radius $\delta > 0$ so that $\lambda\mathcal{B}$ is pairwise
  disjoint,
  then there are $\delta$-packings $\{ \mathcal{B}_1,\ldots,\mathcal{B}_M \}$ so that
  $\mathcal{B}=\bigcup_{i=1}^M\mathcal{B}_i$. \label{covering4}
  \item There is $M=M(X) \in \mathbb{N}$ such that if $A\subset X$ and $\delta>0$, then there are $\delta$-packings of $A$, $\mathcal{B}_1,\ldots,\mathcal{B}_M$ whose union covers $A$. \label{covering5}
\end{enumerate}
\end{lemma}

The \emph{upper and lower local dimensions of a measure $\mu$ at $x$} are defined by
\begin{align*}
  \udimloc(\mu,x) &= \limsup_{r \downarrow 0} \log\mu(B(x,r))/\log r, \\
  \ldimloc(\mu,x) &= \liminf_{r \downarrow 0} \log\mu(B(x,r))/\log r,
\end{align*}
respectively. If the upper and lower dimensions agree, we call their
mutual value the \emph{local dimension of the measure $\mu$ at $x$}
and write $\dimloc(\mu,x)$ for this common value.
In this article, a \emph{measure} exclusively refers to a nontrivial Borel regular (outer) measure defined on all subsets of $X$ so that bounded sets have finite measure.

For estimating the local dimensions we will use the local $L^q$-dimensions defined in \cite{KaenmakiRajalaSuomala2012}.
Although the local definitions are obtained via their global counterparts in small balls, their behaviour can be quite different; see \cite[Examples 5.1--5.2]{KaenmakiRajalaSuomala2012}.

Let $\mu$ be a measure on $X$, $A\subset X$ a bounded set and $q\in\mathbb{R}$.
The \emph{(global) $L^q$-spectrum of $\mu$ on $A$} is defined by
\begin{equation*}
  \tau_q(\mu,A)=\liminf_{\delta \downarrow 0}
  \frac{\log S_{q}(\mu,A,\delta)}{\log\delta},
\end{equation*}
where
\begin{equation}\label{Sdef}
  S_{q}(\mu,A,\delta) = \sup\Bigl\{ \sum_{B \in \mathcal{B}} \mu(B)^q :
  \mathcal{B} \text{ is a $\delta$-packing of } A\cap\spt(\mu) \Bigr\}
\end{equation}
is the \emph{$L^q$-moment sum of $\mu$ on $A$ at the scale $\delta$}. Note that if $q\ge 0$, the definition of $\tau_q(\mu,A)$ does not change if $A\cap\spt(\mu)$ is replaced by $A$ in the right-hand side of \eqref{Sdef}.
If $q \ne 1$, then we define the \emph{(global) $L^q$-dimension of $\mu$ on $A$} by setting
\begin{equation*}
  \dim_q(\mu,A) = \tau_q(\mu,A)/(q-1).
\end{equation*}
We also denote $\tau_q(\mu) = \tau_q(\mu,X)$ and $\dim_q(\mu) = \dim_q(\mu,X)$ provided that $X$ is bounded.

In the case $q = 1$ the above definition makes no sense. Thus we define for every $A\subset X$ with $\mu(A)>0$ the \emph{(global) upper and lower entropy dimensions of $\mu$ on $A$} as
\begin{equation*}
\begin{split}
  \udim_1(\mu,A) &= \limsup_{\delta \downarrow 0} \fint_{A} \frac{\log\mu(B(y,\delta))}{\log\delta}\,d\mu(y), \\
  \ldim_1(\mu,A) &= \liminf_{\delta \downarrow 0} \fint_{A} \frac{\log\mu(B(y,\delta))}{\log\delta}\,d\mu(y),
\end{split}
\end{equation*}
respectively. If they agree, then their common value is denoted by $\dim_1(\mu,A)$.
Here and hereafter, for $A\subset X$ and a $\mu$-measurable $f\colon
X\rightarrow\overline{\mathbb{R}}$, we use the notation $\fint_A f(y)\,d\mu(y) =
\mu(A)^{-1} \int_A f(y)\,d\mu(y)$ whenever the integral is well defined.

From the above global definitions we then derive their local versions.
The \emph{local $L^q$-spectrum of $\mu$ at $x\in\spt(\mu)$} is defined as
\begin{equation*}
  \tau_q(\mu,x) = \lim_{r \downarrow 0} \tau_q(\mu, B(x,r))
\end{equation*}
and the \emph{local $L^q$-dimension of $\mu$ at $x$} as
\begin{equation*}
  \dim_q(\mu,x) = \lim_{r \downarrow 0} \dim_q(\mu,B(x,r)) = \tau_q(\mu,x)/(q-1).
\end{equation*}
Correspondingly, the \emph{local upper and lower entropy dimensions at $x\in\spt(\mu)$} are defined as
\begin{equation*}
\begin{split}
  \udim_1(\mu,x) &= \limsup_{r \downarrow 0} \udim_1(\mu,B(x,r)),\\
  \ldim_1(\mu,x) &= \liminf_{r \downarrow 0} \udim_1(\mu,B(x,r)).
\end{split}
\end{equation*}
For the basic properties of $\dim_q$, we refer to \cite{KaenmakiRajalaSuomala2012}.

The following theorem lists the main relationships between the different local dimensions. Recall that a measure $\mu$ has the density point property, if
\begin{equation*}
  \lim_{r \downarrow 0} \frac{\mu(A \cap B(x,r))}{\mu(B(x,r))} = 1
\end{equation*}
for $\mu$-almost all $x \in A$ whenever $A \subset X$ is $\mu$-measurable.
Note that although in Euclidean spaces the density point property is satisfied for all measures, this is not necessarily the case in doubling metric spaces; see \cite[Example 5.6]{KaenmakiRajalaSuomala2012}.

\begin{theorem} \label{thm:pointwise_dimq}
  If $\mu$ is a measure on a doubling metric space $X$, then
  \begin{equation} \label{eq:local_dimq}
    \lim_{q \downarrow 1} \dim_q(\mu,x) \le \ldimloc(\mu,x) \le \udimloc(\mu,x) \le \lim_{q \uparrow 1} \dim_q(\mu,x)
  \end{equation}
  for $\mu$-almost all $x \in X$ and
  \begin{equation} \label{eq:dim1_dimq}
    \lim_{q \downarrow 1} \dim_q(\mu,x) \le \ldim_1(\mu,x) \le \udim_1(\mu,x) \le \lim_{q \uparrow 1} \dim_q(\mu,x)
  \end{equation}
  for every $x \in \spt(\mu)$.

  Furthermore, if the measure $\mu$ has the density point property, then
  \begin{equation} \label{eq:local_dim1}
    \ldimloc(\mu,x) \le \ldim_1(\mu,x) \le \udim_1(\mu,x) \le \udimloc(\mu,x)
  \end{equation}
  for $\mu$-almost all $x \in X$.
\end{theorem}

The claims \eqref{eq:local_dimq} and \eqref{eq:local_dim1} are proved in \cite[Theorem 3.1 and Theorem 3.11]{KaenmakiRajalaSuomala2012} and \eqref{eq:dim1_dimq} follows immediately from Proposition \ref{prop:dim1} below. It is worthwhile to notice that the density point property is not needed in the global version of \eqref{eq:local_dim1} whereas in the local case, it is a necessary assumption; see \cite[Remark 3.12 and Examples 5.7--5.8]{KaenmakiRajalaSuomala2012}.

\section{Entropy and $L^q$-dimensions using partitions}\label{sec:par}

In this section, we reformulate the main definitions using partitions of the space $X$ and show that these definitions are consistent with the ordinary definitions presented above.
Concerning the global $L^q$-spectrum on $X$ and global entropy dimensions on $X$, this is of course a known result; see for instance \cite{Cutler1990, Olsen1995, Pesin1997}. We already saw in the inequality \eqref{eq:local_dim1} of Theorem \ref{thm:pointwise_dimq} that local and global definitions do not necessarily have the same basic properties. In Proposition \ref{1proposition} and Example \ref{1example}, we will see that global entropy dimensions on $A$ can be defined via partitions only when $A$ is compact.

The use of the partitions is motivated by the fact that the original definition of the $L^q$-dimension using packings often causes technical problems if $q<0$. Moreover, the measures that we are interested in usually have some additional a priori structure for which the partition definition suits well. For instance, see Lemma \ref{partitionlemma}. We also use the partition definitions to relate the $L^q$- and entropy dimensions in Proposition \ref{prop:dim1}.

Let $1\le \Lambda<\infty$.
A countable partition $\mathcal{Q}$ of $X$ is called a \emph{$(\delta,\Lambda)$-partition} (for $\delta>0$) if all the sets of $\mathcal{Q}$ are Borel sets and for each $Q \in \mathcal{Q}$ there exists a ball $B_Q$ so that $Q \subset \Lambda B_Q$ and the collection $\{ B_Q : Q \in \mathcal{Q} \}$ is a $\delta$-packing. The choice of $\Lambda$ is usually not important, and thus we simply talk about $\delta$-partitions and assume that $\Lambda$ has been silently fixed. Usually we consider $\delta_n$-partitions for a sequence of $\delta_n$ and in this case we assume that $\Lambda$ is the same for all $\delta_n$.

Let $(\delta_n)_{n \in \mathbb{N}}$ be a decreasing sequence of positive real numbers so that there is $0<c<1$ for which
\begin{equation}\label{eq:c}
  \delta_n<c^n
\end{equation}
for all $n$ and
\begin{equation}\label{log1}
\log\delta_n/\log\delta_{n+1}\longrightarrow 1
\end{equation}
as $n\rightarrow\infty$. For each $n\in\mathbb{N}$
we fix a $\delta_n$-partition $\mathcal{Q}_n$.
If $x \in X$, then we
denote the unique element of $\mathcal{Q}_n$ containing $x$ by $Q_n(x)$.
Furthermore, if $A \subset X$, then we set $\mathcal{Q}_n(A) =\{ Q \in \mathcal{Q}_n : A \cap
Q \ne \emptyset \}$ for all $n \in \mathbb{N}$.

Perhaps the most classical example of a $\delta$-partition is the dyadic cubes of the Euclidean space. We remark that in doubling metric spaces, it is possible to define similar kind of nested partitions sharing most of the good properties of dyadic cubes; see \cite{KaenmakiRajalaSuomala2012a} and references therein. But often in applications, the nested structure is inconvenient to work with. Since the $\delta_n$-partitions do not have to be nested, they are slightly more flexible than such generalised nested cubes.

Throughout this section, we assume that for each $n \in \mathbb{N}$ we have a fixed $\delta_n$-partition $\mathcal{Q}_n$, where $(\delta_n)_{n \in \mathbb{N}}$ is a decreasing sequence satisfying \eqref{eq:c} and \eqref{log1}.

\subsection{Local dimensions via partitions}
We include a proof of the following folklore result, Proposition \ref{thm:localdim},
since we have not been able to track a complete proof in the literature
(see e.g.\ \cite[Lemma
2.3]{Cutler1990} and \cite[Theorem 15.3]{Pesin1997}).

To simplify the notation, we set
\begin{align*}
  \uld(\mu,x) &= \limsup_{n\to\infty} \log\mu(Q_n(x))/\log \delta_n, \\
  \lld(\mu,x) &= \liminf_{n\to\infty} \log\mu(Q_n(x))/\log \delta_n
\end{align*}
for all measures $\mu$ on $X$ and $x \in X$. A
priori, the definitions of $\uld(\mu,x)$ and $\lld(\mu,x)$ depend on
the choice of the partition, but Proposition \ref{thm:localdim} implies
that almost everywhere these quantities equal the local
dimensions and hence, the choice of the partition does not play any
role.

\begin{proposition}\label{thm:localdim}
If $\mu$ is a measure on a doubling metric space $X$, then
\begin{align*}
  \udimloc(\mu,x) &= \uld(\mu,x), \\
  \ldimloc(\mu,x) &= \lld(\mu,x)
\end{align*}
for $\mu$-almost all $x\in X$.
\end{proposition}

\begin{proof}
The inequalities
  $\udimloc(\mu,x) \le \uld(\mu,x)$,
  $\ldimloc(\mu,x) \le \lld(\mu,x)$
are seen to hold for all $x\in X$ by using \eqref{log1} and the fact $Q_n(x)\subset B(x,(\Lambda+1)\delta_n)$ for all $x\in X$ and $n\in\mathbb{N}$.

To prove the estimates in the other direction,
fix a bounded set $A \subset X$, $0<t<s<\infty$ and define
\[A_n(t,s)=\{x\in A : \mu(Q_n(x))<\delta_{n}^{s}\text{ and }\mu(B(x,\delta_n))>\delta_{n}^t\}.\]
Now the set
\[\{x\in A : \udimloc(\mu,x)<\uld(\mu,x)\text{ or }\ldimloc(\mu,x)<\lld(\mu,x)\}\]
is contained in
\[\bigcup_{0<t<s<\infty}\bigcap_{k\in\mathbb{N}}\bigcup_{n=k}^\infty A_n(t,s)\,\]
where the union is over countably many (e.g.\ rational) $t$ and $s$. Thus,
by the Borel-Cantelli Lemma, it suffices to show that $\sum_{n\in\mathbb{N}}\mu(A_n(t,s))<\infty$ for any choice of $t$ and $s$.
To verify this, let $n\in\mathbb{N}$ and consider $x\in A_n(t,s)$. Since only at most $C=C(N,\Lambda)$ of the sets $Q\in\mathcal{Q}_n$ meet $B(x,\delta_n)$ (cf.\ Lemma \ref{thm:covering_thm}\eqref{covering3}), we have the estimate
$\mu(A_n(t,s) \cap B(x,\delta_n))\le C \delta_n^{s}=C\delta_n^{s-t}\delta_{n}^t\le C\delta_n^{s-t}\mu(B(x,\delta_n))$. Using Lemma \ref{thm:covering_thm}\eqref{covering5}, we may cover $A_n(t,s)$ by a union of at most $M=M(N)$ $\delta_n$-packings of $A_n(t,s)$ and thus
\[\mu(A_n(t,s))\le CM \delta_n^{s-t}\mu(B),\]
where $B$ is a ball centered at $A$ with radius $\diam(A)+1$.
By \eqref{eq:c}, the sum $\sum_{n\in\mathbb{N}}\delta_{n}^{s-t}$ converges and the claim holds for $\mu$-almost all $x\in A$. As this is true for any bounded $A\subset X$, this finishes the proof.
\end{proof}

\subsection{$L^q$-spectrum and entropy dimension via partitions} \label{sec:partitions}
The equivalence of different definitions of $L^q$-spectum has already been considered in the literature,
in particular in \cite{Olsen1995}. Nevertheless, we present here a short proof of Proposition \ref{qproposition} for the convenience
of the reader. After that we study the more subtle case of entropy dimension, where the results, to our knowledge, are new.

The following proposition shows that both the local and global $L^q$-spectrum and $L^q$-dimension can equivalently be defined by using partitions.
Later we will show that this is also the case for the local entropy dimension, see Proposition \ref{1proposition}. For the global entropy dimension the situation is slightly more complicated.

\begin{proposition}\label{qproposition}
  If $\mu$ is a measure on a doubling metric space $X$, $A \subset X$ is bounded with $\mu(A)>0$ and $q\ge0$, then
    \begin{equation*}
      \tau_q(\mu,A) = \liminf_{n\rightarrow\infty}
      \frac{\log\sum_{Q \in \mathcal{Q}_n(A)} \mu(Q)^q}{\log \delta_n}.
    \end{equation*}
\end{proposition}

\begin{proof}
Let $0<\delta<\delta_1$, and $n\in\mathbb{N}$ so that
$\delta_{n+1}\le\delta<\delta_n$.
Our first goal is to show that for a constant
$c_1=c_1(N,\Lambda,q)>0$, we have
  \begin{equation} \label{eq:S_first_goal}
    S_{q}(\mu,A,\delta) \le c_1\Bigl(\frac{\delta_n}{\delta}\Bigr)^s\sum_{Q \in \mathcal{Q}_n(A)} \mu(Q)^q,
  \end{equation}
  where $s = s(N) >0$ is the constant given by Lemma \ref{thm:covering_thm}\eqref{covering3}.
  Recall that $N$ is the doubling constant of
  $X$ and $\Lambda$ is the fixed constant used in defining the
  partitions $\mathcal{Q}_n$.
 To show \eqref{eq:S_first_goal}, we fix a $\delta$-packing $\mathcal{B}$ of
 $A$ and let
  \begin{equation*}
    \mathcal{C}_B = \{ Q \in \mathcal{Q}_n(A) : Q \cap B \ne \emptyset \}
  \end{equation*}
  for all $B \in \mathcal{B}$. Since $\mathcal{C}_B$ is a cover for $B$, we have
  \begin{equation*}
    \mu(B)^q \le \biggl( \sum_{Q \in \mathcal{C}_B} \mu(Q) \biggr)^q \le (\#\mathcal{C}_B)^q \sum_{Q \in \mathcal{C}_B} \mu(Q)^q,
  \end{equation*}
  where $\#\mathcal{C}_B$ is the cardinality of $\mathcal{C}_B$. Notice that all
  the sets of $\mathcal{C}_B$ are contained in a ball of radius
  $(1+2\Lambda)\delta_n$ which, on the other hand, has a $\delta_n$-packing of cardinality $\#\mathcal{C}_B$. Hence, Lemma \ref{thm:covering_thm}\eqref{covering3} implies that $\#\mathcal{C}_B \le c_2=c_2(N,\Lambda)$ for all $B\in\mathcal{B}$ and therefore
  \begin{equation*}
    \sum_{B \in \mathcal{B}} \mu(B)^q \le c_2^q \sum_{B \in \mathcal{B}} \sum_{Q \in \mathcal{C}_B} \mu(Q)^q.
  \end{equation*}
  Furthermore, by Lemma \ref{thm:covering_thm}\eqref{covering3} there exists a constant $c_3 = c_3(N,\Lambda)>0$
  so that the cardinality of the set $\{ B \in \mathcal{B} : Q \cap B \ne \emptyset \}$ is at most $c_3\left(\delta_n/\delta\right)^s$
  for all $Q \in \mathcal{Q}_n$. Thus, \eqref{eq:S_first_goal} follows with $c_1=c_2^q c_3$.

To find an estimate in the other direction, choose for each $Q \in \mathcal{Q}_n(A)$ a point $x_Q \in A \cap Q$ and a ball $B_Q$ so that $Q \subset \Lambda B_Q$ and the collection $\{ B_Q : Q \in \mathcal{Q}_n(A) \}$ is a $\delta_n$-packing. Notice that $Q \subset B(x_Q,2\Lambda\delta_n) \subset 3\Lambda B_Q$ for all $Q \in \mathcal{Q}_n(A)$. According to Lemma \ref{thm:covering_thm}\eqref{covering4} there exists $M = M(N,\Lambda) \in \mathbb{N}$ and $\mathcal{Q}_1,\ldots,\mathcal{Q}_M$ so that $\mathcal{Q}_n(A) = \bigcup_{i=1}^M \mathcal{Q}_i$ and $\{ 3\Lambda B_Q : Q \in \mathcal{Q}_i \}$ is a $3\Lambda\delta_n$-packing for all $i \in \{ 1,\ldots,M \}$. Thus $\{ B(x_Q,2\Lambda\delta_n) : Q \in \mathcal{Q}_i \}$ is a $2\Lambda\delta_n$-packing of $A$ for all $i \in \{ 1,\ldots,M \}$. Since $\bigcup_{Q \in \mathcal{Q}_n(A)} Q \subset \bigcup_{i=1}^M \bigcup_{Q \in \mathcal{Q}_i} B(x_Q,2\Lambda\delta_n)$, we may choose $i \in \{ 1,\ldots,M \}$ so that
\begin{equation} \label{eq:S_second_goal}
  \sum_{Q \in \mathcal{Q}_n(A)} \mu(Q)^q \le M^{-1}\sum_{Q \in \mathcal{Q}_i} \mu(B(x_Q,2\Lambda\delta_n))^q \le M^{-1}S_q(\mu,A,2\Lambda\delta_n).
\end{equation}
The proof now follows by combining \eqref{eq:S_first_goal} and \eqref{eq:S_second_goal} and taking logarithms and limits.
\end{proof}

Global entropy dimensions can be defined via partitions if $A$ is compact. Before showing this, we exhibit a small technical lemma.

\begin{lemma}\label{lma:smallentropy}
  Suppose $\mu$ is a measure on a doubling metric space $X$ and $A \subset X$ is bounded. Let $s>0$ and $c>0$ be as in Lemma \ref{thm:covering_thm}\eqref{covering3}. Then
  \[
    \int_A \log\mu(B(y,\delta))\,d\mu(y) \ge - \tfrac1e -\mu(A)\biggl( \log c + s\log\frac{4\diam(A)}{\delta} \biggr)
  \]
  for all $\delta > 0$.
\end{lemma}

\begin{proof}
  Let $\mathcal{B}'$ be a maximal $\delta/4$-packing of $A$ and let $\mathcal{B} = \tfrac14 \mathcal{B}' = \{ B_1,B_2,\ldots, B_k\}$. Define $Q_1 = 8B_1 \setminus \bigcup_{B \in \mathcal{B} \setminus \{ B_1 \}} B$ and
  \begin{equation*}
    Q_{n+1} = \biggl( 8B_{n+1} \setminus \bigcup_{B \in \mathcal{B} \setminus \{ B_{n+1} \}} B \biggr) \setminus \bigcup_{i=1}^n Q_i
  \end{equation*}
  for all $1\le n\le k-1$. Then $\mathcal{Q} = \{ Q_1 \cap A, Q_2  \cap A, \ldots, Q_k\cap A \}$ is a $(\delta/16,8)$-partition of $A$ in the relative metric.

  If the unique $Q \in \mathcal{Q}$ containing $y \in A$ is denoted by $Q(y)$, then we have $Q(y) \subset B(y, \delta)$ for all $y \in A$. By Theorem \ref{thm:covering_thm}\eqref{covering3} there exist constants $s>0$ and $c>0$ depending only on the doubling constant of $A$ so that $\#\mathcal{Q} \le c(4\diam(A)/\delta)^{s}$. Now Jensen's inequality gives
 \begin{align*}
  \int_A \log\mu(B(y,\delta))\,d\mu(y) & \ge \int_A \log\mu(Q(y))\,d\mu(y) = \sum_{Q \in \mathcal{Q}}\mu(Q)\log\mu(Q) \\
  &\ge \mu(A)\log\frac{\mu(A)}{\#\mathcal{Q}} \ge \mu(A)\log \mu(A) - \mu(A)\log \frac{c(4\diam(A))^s}{\delta^s}
 \end{align*}
  and the claim follows.
\end{proof}

\begin{proposition}\label{1proposition}
  If $\mu$ is a measure on a doubling metric space $X$ and $A \subset X$ is compact with $\mu(A)>0$, then
    \begin{align*}
      \udim_1(\mu,A) &=
      \limsup_{n\rightarrow\infty} \frac{\sum_{Q \in \mathcal{Q}_n(A)}
      \mu(Q)\log\mu(Q)}{\sum_{Q \in \mathcal{Q}_n(A)}
      \mu(Q) \log \delta_n}, \\
      \ldim_1(\mu,A) &=
      \liminf_{n\rightarrow\infty} \frac{\sum_{Q \in \mathcal{Q}_n(A)}
      \mu(Q)\log\mu(Q)}{\sum_{Q \in \mathcal{Q}_n(A)}
      \mu(Q)\log\delta_n}.
    \end{align*}
\end{proposition}

\begin{proof}
Choose for each $Q \in \mathcal{Q}_n(A)$ a ball $B_Q$ such that $Q \subset \Lambda B_Q$ and $\{ B_Q : Q \in \mathcal{Q}_n(A) \}$ is a $\delta_n$-packing. If $Q \in \mathcal{Q}_n(A)$, then for every $y \in Q$ we have
\begin{equation*}
  Q \subset B(y,2\Lambda\delta_n) \subset 3\Lambda B_Q \subset
  \bigcup_{Q' \in \mathcal{C}_Q} Q',
\end{equation*}
where $\mathcal{C}_Q = \{ Q' \in \mathcal{Q}_n(A) : Q' \cap 3\Lambda B_Q \ne \emptyset \}$. Thus, letting $A_n = \bigcup_{Q \in \mathcal{Q}_n(A)} Q$, we get
\begin{equation*}
\begin{split}
  \sum_{Q \in \mathcal{Q}_n(A)} \mu(Q)\log\mu(Q) &\le \sum_{Q \in \mathcal{Q}_n(A)} \int_{Q} \log\mu(B(y,2\Lambda\delta_n))\,d\mu(y) \\
  &= \int_{A_n} \log\mu(B(y,2\Lambda\delta_n))\,d\mu(y) \le \sum_{Q \in \mathcal{Q}_n(A)} \mu(Q)\log\sum_{Q' \in \mathcal{C}_Q} \mu(Q').
\end{split}
\end{equation*}
Moreover, since each $Q' \in \mathcal{Q}_n(A)$ is contained in at most $c_4(3\Lambda)^{s}$ collections $\mathcal{C}_Q$ by Lemma \ref{thm:covering_thm}(3), where $c_4=c_4(N)<\infty$, we have
 \begin{align*}
  \sum_{Q \in \mathcal{Q}_n(A)} \mu(Q) \log&\sum_{Q' \in \mathcal{C}_Q} \mu(Q')
   -\sum_{Q \in \mathcal{Q}_n(A)} \mu(Q)\log\mu(Q) \\
  &= \sum_{Q \in \mathcal{Q}_n(A)} \mu(Q)\log\biggl( 1 + \frac{\sum_{Q' \in \mathcal{C}_Q \setminus \{ Q \}} \mu(Q')}{\mu(Q)} \biggr) \\
  &\le \sum_{Q \in \mathcal{Q}_n(A)} \sum_{Q' \in \mathcal{C}_Q \setminus \{ Q \}} \mu(Q') \le c_4(3\Lambda)^s\mu(B_0),
 \end{align*}
where $B_0$ is a ball centered at $A$ with radius $\diam(A) + 2\Lambda\delta_n$. Putting these estimates together, we get
 \begin{equation}\label{eq:comb}
 \begin{split}
   \sum_{Q \in \mathcal{Q}_{n}(A)} \mu(Q)\log\mu(Q) &\le \int_{A_n} \log\mu(B(y,\delta))\,d\mu(y) \\ &\le \sum_{Q \in \mathcal{Q}_{n-1}(A)} \mu(Q)\log\mu(Q) + c_4(3\Lambda)^s\mu(B_0)
 \end{split}
 \end{equation}
for all $2\Lambda\delta_n \le \delta \le 2\Lambda\delta_{n-1}$.

Since $A$ is compact we have $\lim_{n \to \infty} \mu(A_n \setminus A) = 0$ and therefore, by Lemma \ref{lma:smallentropy},
\[
  \lim_{n \to \infty}\frac{1}{\log \delta_n}\int_{A_n \setminus A} \log\mu(B(y,2\Lambda\delta_n))\,d\mu(y) = 0.
\]
From this, \eqref{log1} and \eqref{eq:comb} the claim follows easily.
\end{proof}

\begin{example} \label{1example}
  In this example, we show that the claim in Proposition \ref{1proposition} does not hold for non-compact sets. Equip $X = [0,1]$ with the Euclidean metric and let $\mathcal{Q}_n$ be the partition of $X$ to the dyadic intervals of length $2^{-n}$. Let $A = \mathbb{Q} \cap [0,1] = \{ q_1,q_2,\ldots \}$ and $\nu = \sum_{i=1}^\infty 2^{-i} \delta_{q_i}$, where $\delta_x$ denotes the Dirac unit mass located at $x$. Finally, set $\mu = \mathcal{L}^1|_{[0,1]} + \nu$, where $\mathcal{L}^1$ denotes the Lebesgue measure.

  Since $\sum_{Q \in \mathcal{Q}_n} \mu(Q)\log\mu(Q) \le \log 2^{-n}$ for all $n$ large enough,
  we have
  \begin{equation*}
    \liminf_{n\rightarrow\infty} \frac{\sum_{Q \in \mathcal{Q}_n(A)}
      \mu(Q)\log\mu(Q)}{\sum_{Q \in \mathcal{Q}_n(A)}
      \mu(Q)\log 2^{-n}} = \liminf_{n \to \infty} \frac{\sum_{Q \in \mathcal{Q}_n} \mu(Q) \log\mu(Q)}{2\log 2^{-n}} \ge \tfrac12.
  \end{equation*}
  Let $\varepsilon>0$, choose $k \in \mathbb{N}$ so that $\sum_{i=k+1}^\infty 2^{-i} < \varepsilon$, and define $A' = \{ q_1,\ldots,q_k \}$. According to Lemma \ref{lma:smallentropy}, there exists $c > 0$ so that
  \begin{align*}
    \int_{A'} \log\mu(B(y,\delta))\,d\mu(y) &\ge -\tfrac{1}{e} - \nu(A')k \log 2, \\
    \int_{A \setminus A'} \log\mu(B(y,\delta))\,d\mu(y) &\ge -\tfrac{1}{e} - \nu(A \setminus A')\bigl( \log c + \log(4/\delta) \bigr)
  \end{align*}
  for all $\delta>0$ small enough.
  Since $\nu(A \setminus A')<\varepsilon$, we get
  \begin{equation*}
    \udim_1(\mu,A) = \limsup_{\delta \downarrow 0} \int_A \frac{\log\mu(B(y,\delta))}{\log\delta}  \le \limsup_{\delta \downarrow 0} \frac{-\tfrac{2}{e} - \nu(A')k\log 2 - \varepsilon\log(4c) + \varepsilon\log\delta}{\log\delta} = \varepsilon.
  \end{equation*}
  Thus $\udim_1(\mu,A) = 0$.
\end{example}

\begin{remark}
In view of the definitions of $\dim_q$, it is natural to ask if the entropy dimensions could also be defined in terms of maximal packings. However, simple examples such as
$\mu = \mathcal{L}^1|_{[0,1]} + \delta_1$ on $[0,1]$ show that this is usually not possible.
\end{remark}

To finish this section, we show that the definition of the entropy dimension as $\dim_1$ is consistent with the monotonicity of the $L^q$-dimensions.
The proof is standard and it is presented for the convenience of the reader.

\begin{proposition} \label{prop:dim1}
  If $\mu$ is a measure on a doubling metric space $X$ and $A\subset X$ compact with $\mu(A) > 0$, then
  \begin{equation*}
    \lim_{q \downarrow 1} \dim_q(\mu,A) \le \ldim_1(\mu,A) \le \udim_1(\mu,A) \le \lim_{q \uparrow 1} \dim_q(\mu,A).
  \end{equation*}
\end{proposition}

\begin{proof}
The existence of the limits follows from \cite[Proposition 2.7]{KaenmakiRajalaSuomala2012}. Thus, the claims follow if we can show that
\begin{equation}\label{goal}
\tau_q(\mu,A)/(q-1) \ge \udim_1(\mu,A) \ge \ldim_1(\mu,A) \ge \tau_p(\mu,A)/(p-1),
\end{equation}
where $0<q<1<p$.
Define $h_n(q) = \log \sum_{Q \in \mathcal{Q}_n(A)} \mu(Q)^q$ for all $q\geq0$.
A simple application of H\"older's inequality shows that $h_n$ is
convex. As $\mathcal{Q}_n(A)$ has only a finite number of elements, $h_n$
is differentiable with $h_n'(1) = \bigl( \sum_{Q \in \mathcal{Q}_n(A)} \mu(Q) \bigr)^{-1} \sum_{Q \in \mathcal{Q}_n(A)} \mu(Q)\log\mu(Q)$. Thus
  \begin{equation*}
    \frac{h_n(q)-h_n(1)}{q-1} \le h_n'(1) \le \frac{h_n(p)-h_n(1)}{p-1}.
  \end{equation*}
Using these estimates and the fact that $h_n(1)=\log\sum_{Q \in \mathcal{Q}_n(A)}\mu(Q)$ does not depend on $n$, we calculate
\begin{align*}
\frac{1}{q-1}&\liminf_{n\to\infty}
\frac{\log\sum_{Q\in\mathcal{Q}_n(A)}\mu(Q)^{q}}{\log \delta_n}=\limsup_{n\to\infty}
  \frac{h_n(q)-h_n(1)}{(q-1)\log \delta_n}\\
&\geq\limsup_{n\to\infty}\frac{\sum_{Q \in \mathcal{Q}_n(A)}
\mu(Q)\log\mu(Q)}{\sum_{Q \in \mathcal{Q}_n(A)}\mu(Q)\log \delta_n}
\geq\liminf_{n\to\infty}\frac{\sum_{Q \in \mathcal{Q}_n(A)}
\mu(Q)\log\mu(Q)}{\sum_{Q \in \mathcal{Q}_n(A)}\mu(Q)\log \delta_n}\\
&\geq\liminf_{n\to\infty}
  \frac{h_n(p)-h_n(1)}{(p-1)\log \delta_n}=\frac1{p-1}\liminf_{n\to\infty}
  \frac{\log\sum_{Q\in\mathcal{Q}_n(A)}\mu(Q)^{p}}{\log \delta_n}.
\end{align*}
The desired estimate \eqref{goal} now follows from Propositions \ref{qproposition} and \ref{1proposition}.
\end{proof}

\section{Local dimension and multifractal analysis for Moran measures}\label{sec:results}

We now turn towards our final goal to study multifractality of measures in metric spaces.
We first introduce a class of Moran constructions
in a complete doubling metric space $X$ and show how
Theorem \ref{thm:pointwise_dimq} can be applied to calculate the local
dimensions for a large class of measures defined on these Moran
fractals. Then, under certain additional assumptions, we turn to study the multifractal spectrum of these measures. Our main aim is to show that using the technique
introduced in \cite{KaenmakiRajalaSuomala2012}, we can push the standard methods
used to calculate the local dimensions for self-similar measures on
Euclidean spaces (see \cite{CawleyMauldin1992, Riedi1995, Falconer1997})
to obtain analogous results in doubling
metric spaces with very mild regularity assumptions, see Remark \ref{rem:moran}.

\subsection{Moran constructions and measures}

Let $m\in\mathbb{N}$,
$\Sigma = \{ 1,\ldots,m \}^\mathbb{N}$, $\Sigma_n = \{
1,\ldots,m \}^n$ for all $n \in \mathbb{N}$, and $\Sigma_* = \{\varnothing\}\cup\bigcup_{n \in
  \mathbb{N}} \Sigma_n$. If $n \in \mathbb{N}$ and $\mathtt{i} \in \Sigma \cup
\bigcup_{j=n}^\infty \Sigma_j$, then we let $\mathtt{i}|_n =
(i_1,\ldots,i_n)$ (and $\mathtt{i}|_0=\varnothing$). The concatenation of two words $\mathtt{i} \in \Sigma_*$
and $\mathtt{j} \in \Sigma \cup \Sigma_*$ is denoted by $\mathtt{i}\mathtt{j}$. We also
set $\mathtt{i}^- = \mathtt{i}|_{n-1}$ for $\mathtt{i} \in \Sigma_n$ and $n \in
\mathbb{N}$. By $|\mathtt{i}|$, we  denote the length of a word $\mathtt{i}\in\Sigma_*$.
We assume that
$\{ E_\mathtt{i} : \mathtt{i} \in \Sigma_* \}$ is a collection of compact subsets of
$X$ that satisfy the following conditions for some constants $0<C_0,C_1<\infty$:
\begin{enumerate}
  \item[(M1)] $E_\mathtt{i} \subset E_{\mathtt{i}^-}$ for all $\varnothing\neq\mathtt{i} \in \Sigma_*$.\label{M1}
  \item[(M2)] $E_{\mathtt{i} i}\cap E_{\mathtt{i} j}=\emptyset$ if\label{M2}
    $\mathtt{i}\in\Sigma_*$ and $i\neq j$.
  \item[(M3)] For each $\mathtt{i}\in\Sigma_*$, there is $x\in E_\mathtt{i}$ such
    that $B\left(x,C_0 \diam(E_\mathtt{i})\right)\subset E_\mathtt{i}$.\label{M3}
  \item[(M4)] $\diam(E_{\mathtt{i}|_n})\rightarrow 0$ as
    $n\rightarrow\infty$, for each $\mathtt{i}\in\Sigma$.
  \item[(M5)] $\diam(E_{\mathtt{i}^-})/\diam(E_{\mathtt{i}})\leq C_1<\infty$ for all $\varnothing\neq\mathtt{i}\in\Sigma_*$.\label{M5}
\end{enumerate}
We define the limit set of the construction as
$E=\bigcap_{n\in\mathbb{N}}\bigcup_{\mathtt{i}\in\Sigma_n}E_\mathtt{i}$ and given
$\mathtt{i}\in\Sigma$, denote by $x_\mathtt{i}$ the point obtained as
$\{x_\mathtt{i}\}=\bigcap_{n\in\mathbb{N}}E_{\mathtt{i}|_n}$. If $\mathtt{i}\in\Sigma_*$, we denote $x_\mathtt{i}:=x_{\iii000\cdots}$.
We assume that for each
  $i\in\{1,\ldots,m\}$ there is a continuous function $r_i\colon
  E\to(0,1)$. Given $x\in E$, and $\mathtt{i}\in\Sigma_n$, we let
  $r_\mathtt{i}(x)=\prod_{k=1}^{n}r_{i_k}(x)$. Moreover, we assume that
\begin{enumerate}
  \item[(M6)]
    $\lim_{n\rightarrow\infty}\log \diam(E_{\mathtt{i}|_n})/\log r_{\mathtt{i}|_n}(x_\mathtt{i})=1$
    uniformly for all $\mathtt{i}\in\Sigma$.
\end{enumerate}

The following further conditions on the collection $\{E_\mathtt{i} : \mathtt{i}\in\Sigma_*\}$ are needed only in subsection \ref{lmf}. For $n\in\mathbb{N}$, denote
$\mathcal{E}_n=\{E_\mathtt{i}\,:\,\diam(E_\mathtt{i})\le C_1/(C_0 2^{n})<\diam(E_{\mathtt{i}^-})\}$.
\begin{enumerate}
\item[(M7)] There is $c>0$ so that for each $Q\in\mathcal{E}_n$, there
  is $x\in E$ with $B(x,c2^{-n})\subset Q$.
\item[(M8)] $\lim_{r\downarrow 0}\log r/\log\bigl(\diam(E_{\mathtt{i}|_{n(\mathtt{i},r)}})\bigr)=1$ for all $\mathtt{i}\in\Sigma$, where $n(\mathtt{i},r)=\max\{n\in\mathbb{N} : B(x_\mathtt{i},r)\cap E \subset E_{\mathtt{i}|_n}\}$.
\end{enumerate}

Our next lemma shows how we can obtain a
$\delta$-partition of $X$ from the elements of $\{E_\mathtt{i}\}$ which are
roughly of size $\delta$.

\begin{lemma}\label{partitionlemma}
If $\{ E_\mathtt{i} : \mathtt{i} \in \Sigma_* \}$ is a collection of compact sets satisfying the conditions (M1)--(M5), then for every $n \in \mathbb{N}$ there is a $(2^{-n})$-partition $\mathcal{Q}_n$ of $X$ such that each $E_\mathtt{i}\in\mathcal{E}_n$ is a subset of some $Q\in\mathcal{Q}_n$ and all elements of $\mathcal{Q}_n$ contain at most one element of $\mathcal{E}_n$.
\end{lemma}

\begin{proof}
Consider a maximal collection $\mathcal{B}_n$ of disjoint balls of radius
$2^{-n}$ contained in $X\setminus\bigcup\mathcal{E}_n$. Define $\mathcal{A}_n=\mathcal{E}_n\cup\mathcal{B}_n=\{A_1,A_2,\ldots\}$. For each
$x\in X$,
we let
$i_x=\min\{j\in\mathbb{N}\,:\,\dist(x,A_j)=\min_{A\in\mathcal{A}_n}\dist(x,A)\}$
and
set $Q_{A_i}=\{x\in X\,:\, i_x=i\}$ for all $i\in\mathbb{N}$.
It is
then easy to see that $\mathcal{Q}_n=\{Q_A\,:A\in\mathcal{A}_n\}$ is the desired
$(2^{-n})$-packing. Observe that each $Q\in\mathcal{Q}_n$ is a Borel set
since $\bigcup_{i=1}^k Q_{A_i}$ is closed for all $k$. Moreover, the constant $\Lambda$ of this
partition depends only on the constants $C_0$ and $C_1$ as one may choose $\Lambda=C_0C_1+1$.
\end{proof}

Let $\mu$ be a probability measure on $X$ with $\spt(\mu)=E$. Then for each
$\mathtt{i}\in\Sigma_*$, $\mu$ induces a probability
vector $p_\mathtt{i}=(p_\mathtt{i}^{1},\ldots,p_\mathtt{i}^{m})$ with $p_\mathtt{i}^{i}>0$ for $i\in\{1,\ldots,m\}$ such that $\mu(E_{\mathtt{i}
i})=p_\mathtt{i}^{i}\mu(E_\mathtt{i})$ for $i\in\{1,\ldots,m\}$. Given $\mathtt{i}\in\Sigma_n$, we denote
$\mu_\mathtt{i}:=\mu(E_\mathtt{i})=\prod_{j=1}^{n}p_{\mathtt{i}|_{j-1}}^{i_j}$. In the next theorem, we assume that the weights $p_\mathtt{i}$ are controlled in terms of continuous probability functions $p(x)=\bigl( p_1(x),\ldots,p_m(x) \bigr)$. More precisely, we assume that
 for each $i\in\{1,\ldots,m\}$, the function $p_i\colon
E\rightarrow(0,1)$ is continuous with $\sum_{i=1}^m p_i(x)=1$ for
all $x\in E$.
Similarly to $r_i$, we define $P_\mathtt{i}(x)=\prod_{k=1}^{n}p_{i_k}(x)$ when $\mathtt{i}\in\Sigma_n$.

\subsection{Local $L^q$-spectrum for Moran measures}

\begin{theorem}\label{thm:localdimappl}
Let $\{E_\mathtt{i} : \mathtt{i}\in\Sigma_*\}$ be a collection of compact sets that satisfy the conditions (M1)--(M6). Suppose that $\mu$ is a probability measure on $E$ and let $p_\mathtt{i}$ and $p$ be as above. If
$p_{\mathtt{i}|_n}\rightarrow p(x_\mathtt{i})$ as $n\rightarrow\infty$ uniformly
for all $\mathtt{i}\in\Sigma$, then, for all $x\in E$ and all $q\ge0$,
$\tau_q(\mu,x)$ is the unique $\tau\in\mathbb{R}$ that satisfies
\begin{equation}\label{tauformula}
\sum_{i=1}^m p_i(x)^qr_i(x)^{-\tau}=1.
\end{equation}
Moreover,
\begin{equation}\label{dimformula}
  \dim_1(\mu,x) = \dimloc(\mu,x) = \frac{\sum_{i=1}^m p_i(x)\log
    p_i(x)}{\sum_{i=1}^m p_i(x)\log r_i(x)}
\end{equation}
for $\mu$-almost all $x \in E$.
\end{theorem}

\begin{proof}
We prove the claim \eqref{tauformula}. The identities \eqref{dimformula}
then follow from \eqref{tauformula} by implicit differentiation together with
Theorem \ref{thm:pointwise_dimq}.

For each $n\in\mathbb{N}$, let $\mathcal{Q}_n$ be as in
Lemma \ref{partitionlemma}. Given $\varnothing\neq\mathtt{i}\in\Sigma_*$, we denote by
$Q_\mathtt{i}$ the unique element of $\bigcup_{n\in\mathbb{N}}\mathcal{Q}_n$ that
contains $E_\mathtt{i}$ and does not contain $E_{\mathtt{i}^-}$ (we assume without
loss of generality that $\mathcal{E}_1 = \{ E_\varnothing \}$ so that this makes
sense for all $n$).
Let us fix $q\ge 0$, $x\in E$ and let $\mathtt{i}\in\Sigma$ so that $x=x_\mathtt{i}$. Let
$\tau$ be as in \eqref{tauformula}. We first prove that
$\tau_q(\mu,x)\ge\tau$. Let $0<c<1$.
Since
$p_{\mathtt{i}|_n}\rightarrow p(x_\mathtt{i})$ uniformly and $y\mapsto p(y)$ is
continuous, we may choose $n_0$ so large that
$p_{\mathtt{j}}^i>cp_i(x)$ whenever $i\in\{1,\ldots,m\}$, $\mathtt{j}\in\Sigma_*$,
 and $E_\mathtt{j}\subset E_{\mathtt{i}|_{n_0}}$. Making $n_0$ even larger if
necessary, we may also assume that
\begin{equation}\label{tahhti}
c r_i(y)\le r_i(x)\le
r_i(y)/c
\end{equation}
for all $y\in E_{\mathtt{i}|_{n_0}}$ and all $i \in \{1,\ldots,m\}$.

Now, for all $r>0$, we
choose $N_0\ge n_0$ so that $Q_{\mathtt{j}}\subset B(x,r)$ whenever
$\mathtt{j}\in\Sigma_*$,
and $E_\mathtt{j}\subset E_{\mathtt{i}|_{N_0}}$. Given
$n\ge N_0$, let
$Z_{n}=\{\mathtt{j}\in\Sigma_* : Q_\mathtt{j}\in\mathcal{Q}_n\text{ and
}E_\mathtt{j}\subset E_{\mathtt{i}|_{N_0}}\}$.
Let $\varepsilon_n=\min_{\mathtt{j}\in
  Z_{n}}\bigl(\diam(E_\mathtt{j})/r_\mathtt{j}(x)\bigr)^{-\tau}$. Now, denoting
$c_0=C_{1}^{-|\tau|}\mu_{\mathtt{i}|_{N_0}}^q$,
we get an estimate
\begin{equation}\label{alku}
\begin{split}
2^{n\tau}\sum_{\mathtt{j}\in Z_{n}}\mu(Q_\mathtt{j})^q&\ge C_{1}^{-|\tau|}\sum_{\mathtt{j}\in
  Z_{n}}\mu_{\mathtt{j}}^q \diam(E_{\mathtt{j}})^{-\tau}\ge c_0\varepsilon_n
\sum_{\mathtt{j}\in Z_{n}}c^{q|\mathtt{j}|}P_{\mathtt{j}}(x)^q r_{\mathtt{j}}(x)^{-\tau}.
\end{split}
\end{equation}
For each $\mathtt{j}\in Z_{n}$, pick $y\in E_\mathtt{j}$.
Using (M6),
we may assume that $\log r_\mathtt{j}(y)\ge
2 \log \diam(E_\mathtt{j}) $ by making $N_0$ larger if necessary. Letting
$r_{\max}=\max\bigl\{r_i(y) : y\in E \text{ and } i \in \{ 1,\ldots,m\} \bigr\}$, we have
\begin{equation*}
\log r_{\max}^{|\mathtt{j}|}\ge \log r_\mathtt{j}(y)\ge 2\log
\diam(E_\mathtt{j})\ge -2n\log 2,
\end{equation*}
and consequently,
\begin{equation}\label{kaks}
|\mathtt{j}|\le \frac{2\log \diam(E_{\mathtt{j}})}{\log r_{\max}} \le\frac{-2n \log 2}{\log r_{\max}}\le n C_2,
\end{equation}
for a constant $C_2<\infty$ independent of $n$.
On the other hand,
\begin{equation}\label{kol}
\sum_{\mathtt{j}\in
  Z_{n}}P_\mathtt{j}(x)^q r_{\mathtt{j}}(x)^{-\tau}=P_{\mathtt{i}|_{n_0}}(x)^q
r_{\mathtt{i}|_{n_0}}(x)^{-\tau}=: C_3
\end{equation}
by iterative use of \eqref{tauformula}.
Putting \eqref{alku}--\eqref{kol} together, we get
\begin{equation}\label{nel}
\log\sum_{\mathtt{j}\in Z_{n}}\mu(Q_\mathtt{j})^q\ge\log(2^{-n\tau} c_0 \varepsilon_n c^{qnC_2} C_3)
=\log 2^{-n\tau}+\log\varepsilon_n +q n C_2\log c+\log (c_0C_3).
\end{equation}
To estimate $\log \varepsilon_n$, we choose $\mathtt{j}\in Z_n$ such
that $\varepsilon_n=\bigl(\diam(E_\mathtt{j})/r_\mathtt{j}(x)\bigr)^{-\tau}$. Then
\begin{equation}\label{kuus}
\log\varepsilon_n=-\tau\log \diam(E_\mathtt{j})\bigl(1-\log r_\mathtt{j}(x)/\log \diam(E_\mathtt{j})\bigr).
\end{equation}
Moreover,
$\log r_{\mathtt{j}}(y)+|\mathtt{j}|\log c\le\log r_{\mathtt{j}}(x)\le\log r_{\mathtt{j}}(y)-|\mathtt{j}|\log c$
for all $y\in E_\mathtt{j}$ by \eqref{tahhti}.
Using \eqref{kaks}, this gives
\begin{equation}\label{viis}
\frac{\log r_\mathtt{j}(y)}{\log \diam(E_\mathtt{j})}+C_4\log c\le\frac{\log
  r_\mathtt{j}(x)}{\log \diam(E_\mathtt{j})}\le\frac{\log r_\mathtt{j}(y)}{\log \diam(E_\mathtt{j})}-C_4\log c
\end{equation}
for some constant $0<C_4<\infty$.

Using \eqref{nel}, \eqref{kuus}, \eqref{viis}, and (M6), we finally get
\begin{align*}
\liminf_{n\rightarrow\infty}\frac{\log \sum_{Q \in \mathcal{Q}_n(B(x,r))} \mu(Q)^q}{\log 2^{-n}}&\le\liminf_{n\rightarrow\infty}\frac{\log\sum_{\mathtt{j}\in
    Z_{n}}\mu(Q_\mathtt{j})^q}{\log 2^{-n}} \le\tau-(qC_2+|\tau|C_4)\log c/\log 2.
\end{align*}
As $c<1$ and $r>0$ can be chosen arbitrarily, we get, by recalling Proposition \ref{qproposition}, that $\tau_q(\mu,x)\le\tau$.

To prove that $\tau_q(\mu,x)\ge\tau$, we first fix $0<c<1$ and $r_0>0$ so
that $cp_\mathtt{j}^i<p_i(x)$ and $cr_i(x)<r_i(y)<\tfrac1c r_i(x)$ whenever $i\in\{1,\ldots,m\}$ and
$y\in E_\mathtt{j}\subset B(x,r_0)$. Then, if $0<r<r_0$, we may
find $n_0\in\mathbb{N}$ and finitely many elements
$E_\mathtt{k}\in\mathcal{Q}_{n_0}$, $E_\mathtt{k}\subset B(x,r_0)$
whose union covers $B(x,r)$. For each such
$E_\mathtt{k}$, and $n\ge n_0$, we put
$Z_{n,\mathtt{k}}=\{\mathtt{j}\in\Sigma_*\,:\,Q_\mathtt{j}\in\mathcal{Q}_n\text{ and
}E_\mathtt{j}\subset E_\mathtt{k} \}$. Putting $M_n=\max_{\mathtt{j}\in
  Z_{n,\mathtt{k}}}\bigl(\diam(E_\mathtt{j})/r_\mathtt{j}(x)\bigr)^{-\tau}$, we may estimate as in
\eqref{alku} to obtain
\begin{equation*}
2^{n\tau}\sum_{\mathtt{j}\in Z_{n,\mathtt{k}}}\mu(Q_\mathtt{j})^q\le C_5M_{n}
\sum_{\mathtt{j}\in Z_{n,\mathtt{k}}}c^{-q|\mathtt{j}|}P_\mathtt{j}(x)^qr_{\mathtt{j}}(x)^{-\tau}.
\end{equation*}
Calculating as above, this implies
\begin{equation*}
\liminf_{n\rightarrow\infty}\frac{\log \sum_{Q \in \mathcal{Q}_n(B(x,r))}
  \mu(Q)^q}{\log 2^{-n}}\ge\tau+(qC_2+|\tau|C_4)\log c/\log 2.
\end{equation*}
Letting $r\downarrow 0$ and then $c\uparrow 1$, and using Proposition \ref{qproposition} gives $\tau_q(\mu,x)\ge\tau$.
\end{proof}

\begin{remark}\label{rem:moran}
(1) One can find Moran constructions that satisfy (M1)--(M6) on doubling metric spaces satisfying only
mild regularity assumptions on the space $X$. For instance, it suffices to assume that the space is uniformly perfect.
Different types of Moran constructions in metric spaces have been recently
studied in \cite{RajalaVilppolainen2009}.

(2) The result is interesting already in $\mathbb{R}^n$. We remark that
a self-similar measure on a self-similar set
satisfying the strong separation conditions is
a model case for Theorem \ref{thm:localdimappl} in the special case
when $p_i$ and $r_i$ are constant, see \cite{Falconer1997}.
However, as $p_i$ and $r_i$ are allowed to vary depending
on the point, Theorem \ref{thm:localdimappl} can be applied in
more general situations.

(3) One further difference to the self-similar situation
is that in Moran constructions the location of $E_\mathtt{i}$ inside $E_{\mathtt{i}^-}$ can be chosen quite freely,
whereas with similitude mappings the location of $E_\mathtt{i}$ is strictly dictated by the maps.
Consequently, even in the simplest case where we would force a Moran construction in $\mathbb{R}^n$ to obey $\diam(E_\mathtt{i}) = r^{|\mathtt{i}|}$ for some $0<r<1$ and all $\mathtt{i} \in \Sigma_*$, the limit set $E$ would not necessarily be bi-Lipschitz
 equivalent to a self-similar set.
\end{remark}

\subsection{Some multifractal analysis}\label{lmf}

To approach \eqref{eq:formalsim}, we have to deal with $\tau_q$ for negative values
of $q$ and for this we use the following lemma. Observe that we
cannot use Proposition \ref{qproposition} when $q<0$.

\begin{lemma}\label{taulemmaq<0}
Suppose that in the setting of Theorem \ref{thm:localdimappl} also
(M7) holds. Then, for all $x\in E$, $\tau_q(\mu,x)$ is determined by \eqref{tauformula} also when $q<0$.
\end{lemma}

\begin{proof}
Let $q<0$, $x\in E$ and let $\tau\in\mathbb{R}$ be the unique solution of \eqref{tauformula}.
With trivial modifications to the proof of Theorem \ref{thm:localdimappl}, we
see that
\begin{equation}\label{aa}
\tau=\lim_{t \downarrow 0} \liminf_{n\rightarrow\infty}
      \frac{\log\sum_{Q \in \mathcal{Q}_{n,t}} \mu(Q)^q}{\log 2^{-n}}
\end{equation}
where $\mathcal{Q}_{n,t}=\{Q\in\mathcal{Q}_n : Q\subset B(x,t)\text{ and
}Q\cap E\neq\emptyset\}$. (Observe that $\spt(\mu)=E$.)

In order to prove that $\tau=\tau_q(\mu,x)$, let $t>0$, $2^{-n}\le \delta<2^{-n+1}<t$ and $y\in E \cap B(x,t)$. Then there is $n_0\in\mathbb{N}$ depending only on the numbers $C_0$ and
$C_1$ so that $B(y,\delta)\supset Q_\mathtt{i}$ for some
$Q_\mathtt{i}\in\mathcal{Q}_{n+n_0,2t}$. Thus, for any $\delta$-packing
$\{B_i\}$ of $B(x,t)\cap\spt(\mu)$, we have
\begin{equation}\label{pee}
\sum_{i}\mu(B_i)^q\le\sum_{Q\in\mathcal{Q}_{n+n_0,2t}}\mu(Q)^q.
\end{equation}

To get an estimate in the other direction, we fix $n$ and $t$ and use
the assumption (M7) to find for each $Q\in\mathcal{Q}_{n,t}$ a point
$y\in \spt(\mu)\cap B(x,t)$ such that for $B_Q=B(y,c 2^{-n})$, we have $B_Q\subset Q$.
Thus, for the $(c 2^{-n})$-packing $\{B_Q : Q\in\mathcal{Q}_{n,t}\}$, we have
\begin{equation}\label{see}
\sum_{Q\in\mathcal{Q}_{n,t}}\mu(B_Q)^q\ge\sum_{Q\in\mathcal{Q}_{n,t}}\mu(Q)^q.
\end{equation}
Combining \eqref{aa}--\eqref{see}, and taking logarithms, it follows
that $\tau_q(\mu,x)=\tau$.
\end{proof}

\begin{remark}
Inspecting the proofs of Theorem \ref{thm:localdimappl} and Lemma
\ref{taulemmaq<0}, we observe that in the setting of these results,
$\liminf_{n \to \infty}$ in the definition of $\tau_q(\mu,x)$ can actually be
replaced by $\lim_{n \to \infty}$.
\end{remark}

To complete the paper, we show how the local $L^q$-spectrum can be
used in the setting of Theorem \ref{thm:localdimappl}. We derive a
local multifractal formalism for the spectrum
\[
f_{\textrm{H}}(\alpha,x)=\lim_{r\downarrow
  0}\dimh\bigl(\{y\in B(x,r)\,:\,\dimloc(\mu,y)=\alpha\}\bigr)\]
for $x\in X$ and $\alpha\ge0$. The corresponding packing spectrum,
$f_{\textrm{p}}(\alpha,x)$ is defined by replacing $\dimh$ by $\dimp$ above.

Let $\alpha_{\min}(x)\le\alpha_{\max}(x)$ be the asymptotic derivatives of $q\mapsto\tau_q(\mu,x)$. Thus, $\alpha_{\min}(x) = \min\bigl\{ \log p_i(x)/\log r_i(x) : i \in \{ 1,\ldots,m \} \bigr\}$ and $\alpha_{\max}(x) = \max\bigl\{ \log p_i(x)/\log r_i(x) : i \in \{ 1,\ldots,m \} \bigr\}$. Then it is easy to check that
\begin{equation}\label{eq:min-max}
\bigcap_{r>0}\bigcup_{y\in B(x,r)\cap E}\left[\ldimloc(\mu,y),\udimloc(\mu,y)\right]\subset[\alpha_{\min}(x),\alpha_{\max}(x)]
\end{equation}
for
any $x\in E$.

\begin{theorem}\label{thm:localmultifractal}
Let $\{E_\mathtt{i} : \mathtt{i}\in\Sigma_*\}$ be a collection of compact sets that satisfy the conditions (M1)--(M8). If $p_{\mathtt{i}|_n}\to p(x_\mathtt{i})$ uniformly for all $\mathtt{i}\in\Sigma$, then
\[
  f_{\emph{H}}(\alpha,x) = f_{\emph{p}}(\alpha,x) = \inf_{q\in\mathbb{R}}\{\alpha q-\tau_q(\mu,x)\}
\]
for all $x\in E$ and $\alpha_{\min}(x)\le\alpha\le\alpha_{\max}(x)$.
\end{theorem}

\begin{corollary}\label{cor:spectrum}
Let $\{E_\mathtt{i} : \mathtt{i}\in\Sigma_*\}$ be a collection of compact sets that satisfy the conditions (M1)--(M8). Suppose that $r=(r_1,\ldots, r_m)$
and $p=(p_1,\ldots,p_m)$ are constant functions and $\mu$ is
a measure with $\spt(\mu)=E$ such that $p_{\mathtt{i}|_n}\rightarrow p$
uniformly for all $\mathtt{i}\in\Sigma$. If
$0\leq \alpha_{\min}\le\alpha_{\max}$ are the asymptotic
derivatives of the concave function $q\mapsto\tau_q(\mu)$,
then \eqref{eq:formalsim} holds for all $\alpha_{\min}\le\alpha\le\alpha_{\max}$.
\end{corollary}

\begin{remark}
For the inhomogeneous Bernoulli products on $[0,1]$, the claims of Theorem \ref{thm:localdimappl} and Corollary \ref{cor:spectrum} have been obtained independently by Batakis and Testud \cite[Corollary 1.3]{BatakisTestud2011}. In particular, our result shows that the result is true not only in higher dimensional Euclidean spaces, but also for all Moran constructions satisfying (M1)--(M8) in all doubling metric spaces.
\end{remark}

Theorem \ref{thm:localmultifractal} is derived from Lemma \ref{lemma:spectrum}
below.
The main idea is similar to that of the proof of \cite[Proposition
11.4]{Falconer1997}, but the definition of the auxiliary measures is perhaps more delicate in our setting.

Let $\sup_{x\in E}\alpha_{\min}(x)<\alpha<\inf_{x\in E}\alpha_{\max}(x)$.
In what follows, we use the following notation: Denote $f(\alpha,x)=\min_{q\in\mathbb{R}}\{\alpha q-\tau_q(\mu,x)\}$ for all $x\in E$
and by $q(x)=q_\alpha(x)$ the value of $q$ for which the minimum is attained. For $\mathtt{i}\in\Sigma_*\cup\Sigma$ and $1\le i\le \mathtt{i}$, we also set $r_\mathtt{i}=r_\mathtt{i}(x_\mathtt{i})$, $r_{\mathtt{i}}^{i}=r_i(x_\mathtt{i})$, $q_\mathtt{i}=q(x_\mathtt{i})$, and $\tau_\mathtt{i}=\tau_{q_\mathtt{i}}(\mu,x_\mathtt{i})$.

We make use of the following technical lemma.

\begin{lemma}\label{prop:tekninen}
Under the assumptions of Theorem \ref{thm:localmultifractal}, if
$\sup_{x\in E}\alpha_{\min}(x)<\alpha<\inf_{x\in E}\alpha_{\max}(x)$,
then
$r_{\mathtt{i}|_n}\rightarrow r_\mathtt{i}$, $q_{\mathtt{i}|_n}\rightarrow q_\mathtt{i}$ and $\tau_{\mathtt{i}|_n}\rightarrow\tau_\mathtt{i}$
 for all $\mathtt{i}\in\Sigma$ unifromly as $n\rightarrow\infty$. Moreover, given $\eta>0$, there are $\delta>0$, $C<\infty$, and $\gamma<1$ such that
\begin{align}\label{eq:gammaexists}
&\sum_{\mathtt{i}\in\Sigma_n}\mu_{\mathtt{i}}^{q_\mathtt{i}-\delta} r_{\mathtt{i}}^{\delta(\alpha+\eta)-\tau_\mathtt{i}}\le C\gamma^n, \\
&\sum_{\mathtt{i}\in\Sigma_n}\mu_{\mathtt{i}}^{q_\mathtt{i}+\delta}
\label{eq:gammaexists2} r_{\mathtt{i}}^{\delta(\eta-\alpha)-\tau_\mathtt{i}}\le C\gamma^n,
\end{align}
for all $n\in\mathbb{N}$.
\end{lemma}

\begin{proof}
From our definition of the Moran construction, it follows that $r_{\mathtt{i}|_n}\rightarrow r_\mathtt{i}$ uniformly as $n\rightarrow\infty$.
We first recall that if $x=x_\mathtt{i}$, then $\tau(q)=\tau_q(\mu,x)$ is given by the formula \eqref{tauformula} for all $q\in\mathbb{R}$ by Theorem \ref{thm:localdimappl} and Lemma \ref{taulemmaq<0}.
From this, the assumption $\sup_{x\in E}\alpha_{\min}(x)<\alpha<\inf_{x\in E}\alpha_{\max}(x)$ and the fact that all $p_i(x)$, $r_i(x)$ are bounded away from $0$ and $1$ by the compactness of $E$, it follows that there is a compact interval $I\subset\mathbb{R}$ such that $q_\mathtt{i}\in I$ for all $\mathtt{i}\in\Sigma$ (and thus also for all $\mathtt{i}\in\Sigma_*$). Moreover,
using implicit differentation, it follows that there is $c>0$ such that for all $x\in E$, $q\in I$,
\begin{equation}\label{eq:d2}
\frac{d^2}{dq^2}\left(-\tau_q(\mu,x)\right)\ge c.
\end{equation}

Since $r_i(y)\rightarrow r_i(x)$ and $p_i(y)\rightarrow p_i(x)$ uniformly as $y\rightarrow x$, it follows that for all $q\in I$
\begin{equation}\label{eq:tau_unif}
\tau_q(\mu,x_{\mathtt{i}|_n})\rightarrow \tau_q(\mu,x_\mathtt{i})
\end{equation}
for all $\mathtt{i}\in\Sigma$, uniformly as $n\rightarrow\infty$. From this and the definition of $\tau_\mathtt{i}$, it follows immediately that
\begin{equation}\label{eq:difference_converges}
  (\alpha q_{\mathtt{i}|_n}-\tau_{\mathtt{i}|_n}) \to (\alpha q_{\mathtt{i}}-\tau_{\mathtt{i}}),
\end{equation}
uniformly as $n\rightarrow\infty$.
Given $\varepsilon>0$, \eqref{eq:tau_unif} implies that there is $n_0\in\mathbb{N}$ independent of $\mathtt{i}$, such that $|\tau_{\mathtt{i}|_n}-\tau_{q_{\mathtt{i}|_n}}(\mu,x_\mathtt{i})|<\varepsilon$ if $n\ge n_0$. Using \eqref{eq:d2}, we also have
\[\alpha q_{\mathtt{i}|_n}-\tau_{q_{\mathtt{i}|_n}}(\mu,x_{\mathtt{i}})\ge\alpha q_{\mathtt{i}}-\tau_{\mathtt{i}}+\tfrac{c}{4}|q_{\mathtt{i}|_n}-q_\mathtt{i}|^2.\]
From these two estimates, we infer
\begin{align*}
\alpha q_{\mathtt{i}|_n}-\tau_{\mathtt{i}|_n}\ge \alpha q_{\mathtt{i}}-\tau_{\mathtt{i}}+\tfrac{c}{4}|q_{\mathtt{i}|_n}-q_\mathtt{i}|^2-\varepsilon,
\end{align*}
and combining with \eqref{eq:difference_converges}, we see that $q_{\mathtt{i}|_n}\rightarrow q_\mathtt{i}$ and $\tau_{\mathtt{i}|_n}\rightarrow\tau_\mathtt{i}$ uniformly for all $\mathtt{i}\in\Sigma$.

To prove the estimate \eqref{eq:gammaexists}, fix $\eta>0$. We first observe by implicit differentation (see \cite[Lemma 11.3]{Falconer1997}), that there is $\delta>0$ and $\gamma<1$ such that for all $\mathtt{i}$, we have
\begin{equation}
\sum_{i=1}^m (p_{\mathtt{i}}^i)^{q_\mathtt{i}-2\delta} (r_{\mathtt{i}}^i)^{2\delta(\alpha+\eta)-\tau_\mathtt{i}}\le\gamma.
\end{equation}
By the first part of the lemma, we may choose $n_0\in\mathbb{N}$, such that if $\mathtt{i}=\mathtt{i}_0\mathtt{j}\in\Sigma_*$, where $\mathtt{i}_0\in\Sigma_{n_0}$, then $(p_{\mathtt{i}}^ {i})^{q_\mathtt{i}-\delta}(r_{\mathtt{i}}^ {i})^{\delta(\alpha+\eta)-\tau_\mathtt{i}}\le (p_{\mathtt{i}_0}^{i})^{q_{\mathtt{i}_0}-2\delta}(r_{\mathtt{i}_0}^ {i})^{2\delta(\alpha+\eta)-\tau_{\mathtt{i}_0}}$. This leads to
\begin{equation*}
\sum_{\mathtt{i}=\mathtt{i}_0 \mathtt{j}\in\Sigma_n}\mu_{\mathtt{i}}^{q_\mathtt{i}-\delta} r_{\mathtt{i}}^{\delta(\alpha+\eta)-\tau_\mathtt{i}}
\le C_{n_0}\left(\sum_{i=1}^m(p_{\mathtt{i}_0}^{i})^{q_{\mathtt{i}_0}-2\delta} (r_{\mathtt{i}_0}^{i})^{2\delta(\alpha+\eta)-\tau_{\mathtt{i}_0}}\right)^{n-n_0}.
\end{equation*}
Since there are only finitely many words $\mathtt{i}_0\in\Sigma_{n_0}$, this yields \eqref{eq:gammaexists}.
The estimate \eqref{eq:gammaexists2} is proved in a similar manner.
\end{proof}

\begin{lemma}\label{lemma:spectrum}
In the setting of Theorem \ref{thm:localmultifractal}, let
$\underline{f}(\alpha)=\inf_{x\in E} f(\alpha,x)$ and $\overline{f}(\alpha)=\sup_{x\in E} f(\alpha,x)$.
Then
\begin{align}
\underline{f}(\alpha)\le\dimh(E_{\alpha})\le\dimp(E_{\alpha})\le
\overline{f}(\alpha),
\end{align}
for all $\sup_{x\in E}\alpha_{\min}(x)<\alpha<\inf_{x\in E}\alpha_{\max}(x)$.
\end{lemma}

\begin{proof}
Given $\sup_{x\in E}\alpha_{\min}(x)<\alpha<\inf_{x\in E}\alpha_{\max}(x)$, we define $q_\mathtt{i}$ and $\tau_\mathtt{i}$ using this $\alpha$ and define a probability measure $\nu$
on $X$ with $\spt(\nu)=E$ by setting
\begin{equation*
\nu(E_{\mathtt{i} i})=(p_{\mathtt{i}}^i)^{q_\mathtt{i}} r_{\mathtt{i} i}^{-\tau_\mathtt{i}}\nu(E_\mathtt{i})
\end{equation*}
for $\mathtt{i}\in\Sigma_*$ and $i\in\{1,\ldots,m\}$.
Recall that $\sum_{i=1}^m (p_{\mathtt{i}}^i)^{q_\mathtt{i}} r_{\mathtt{i} i}^{-\tau_\mathtt{i}}=1$ by Theorem \ref{thm:localdimappl} and Lemma \ref{taulemmaq<0}.

The condition (M8) implies that
\begin{align}
\ldimloc(\nu,x_\mathtt{i})&=\liminf_{n\rightarrow\infty}\log\nu(E_{\mathtt{i}|_n})/\log\diam(E_{\mathtt{i}|_n}),\label{ojan2}\\
  \udimloc(\nu,x_\mathtt{i})&=\limsup_{n\rightarrow\infty}\log\nu(E_{\mathtt{i}|_n})/\log\diam(E_{\mathtt{i}|_n}) \label{ojan}
\end{align}
for all $\mathtt{i}\in \Sigma$, and similar formulas apply for $\mu$. Also, using (M6) and Lemma \ref{prop:tekninen}, we have
\begin{align}\label{eq:cornudeff}
\frac{\log\nu(E_\mathtt{i})}{\log\left(\mu(E_\mathtt{i})^{q_\mathtt{i}}\diam(E_\mathtt{i})^{-\tau_{\mathtt{i}}}\right)}\longrightarrow 1,
\end{align}
uniformly as $|\mathtt{i}|\rightarrow\infty$.
Let $\eta,\delta>0$.
Then there is $n_0\in\mathbb{N}$ so that if $n\ge n_0$, we have
\begin{align*}
\nu\bigl( \{ x_\mathtt{i} \in E :\; &\mu(E_{\mathtt{i}|_n}) < \diam(E_{\mathtt{i}|_n})^{\alpha + 3\eta} \} \bigr) \\
 & = \nu\bigl( \{ x_\mathtt{i} \in E : \mu(E_{\mathtt{i}|_n})^{-\delta} \diam(E_{\mathtt{i}|_n})^{\delta(\alpha + 3\eta)} \ge 1 \} \bigr) \\
&\le \sum_{\mathtt{i}\in\Sigma_n}\mu(E_\mathtt{i})^{-\delta}\diam(E_\mathtt{i})^{\delta(\alpha+3\eta)}\nu(E_\mathtt{i})
\le\sum_{\mathtt{i}\in\Sigma_n}\mu_{\mathtt{i}}^{q_\mathtt{i}-\delta} r_{\mathtt{i}}^{\delta(\alpha+\eta)-\tau_\mathtt{i}}.
\end{align*}
In the last estimate we
used (M6) to conclude that
$\diam(E_\mathtt{i})^{\delta(\alpha+3\eta)}<r_{\mathtt{i}}^{\delta(\alpha+2\eta)}$
for all $\mathtt{i}\in\Sigma_n$ and (M6) and \eqref{eq:cornudeff} to guarantee that $\nu(E_\mathtt{i})\le\mu_{\mathtt{i}}^{q_{\mathtt{i}}}r_{\mathtt{i}}^{-\tau_\mathtt{i}-\delta\eta}$. Combining this with \eqref{eq:gammaexists} and choosing $\delta>0$ small enough, we conclude that for $n\ge n_0$,
\[\nu\bigl( \{ x_\mathtt{i} \in E :\; \mu(E_{\mathtt{i}|_n}) < \diam(E_{\mathtt{i}|_n})^{\alpha + 3\eta} \} \bigr)\le C\gamma^n,\]
where $\gamma<1$ is independent of $n$.

Summing the above estimate over all $n\ge n_0$, using the Borel-Cantelli lemma, the analogs of \eqref{ojan2} and \eqref{ojan} for $\mu$ and
letting $\eta\downarrow0$, this
implies that
$\udimloc(\mu,x)\le\alpha$
for $\nu$-almost
all $x\in X$.
A similar calculation (in particular using \eqref{eq:gammaexists2} in place of \eqref{eq:gammaexists}) gives
$\ldimloc(\mu,x)\ge\alpha$
for $\nu$-almost all $x$. Thus, in particular, we have
\begin{equation}\label{tays}
\nu(X\setminus E_{\alpha})=0.
\end{equation}

From \eqref{eq:cornudeff}, Lemma \ref{prop:tekninen} and (M6), it follows that
\[\liminf_{n\rightarrow\infty}\frac{\log\nu(E_{\mathtt{i}|_n})}{\log\diam(E_{\mathtt{i}|_n})}=q(x_\mathtt{i})\liminf_{n\to\infty}\frac{\log\mu(E_{\mathtt{i}|_n})}{\log\diam(E_{\mathtt{i}|_n})}-\tau_{q(x_\mathtt{i})}(\mu,x_\mathtt{i})\]
for all $\mathtt{i}\in\Sigma$ and similarly for $\limsup$.
Taking \eqref{ojan2}--\eqref{ojan} into account yields
$\dimloc(\nu,x)=q(x)\alpha-\tau_{q(x)}(\mu,x)=f(\alpha,x)$
for all $x\in E_\alpha$.
Together with \eqref{tays},
these estimates readily imply that
$\underline{f}(\alpha)\le\dimh(E_{\alpha})\le\dimp(E_{\alpha})\le
\overline{f}(\alpha)$.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm:localmultifractal}]
Let $x=x_\mathtt{i}\in E$ and $\alpha_{\min}(x)\le\alpha\le\alpha_{\max}(x)$. We first consider the case $\alpha\neq\{\alpha_{\min}(x),\alpha_{\max}(x)\}$.

For each small $r>0$, let $n(r)$ and $N(r)$ be the largest and smallest natural numbers such that $B(x,r)\subset E_{\mathtt{i}|_{n(r)}}$ and $E_{\mathtt{i}|_{N(r)}}\subset B(x,r)$, respectively. By (M7) these are well defined for all small $r>0$ and moreover, $n(r),N(r)\longrightarrow\infty$
 as $r\downarrow 0$. Let $\alpha_0(r)=\inf\{f(\alpha,y)\,:\,y\in E_{\mathtt{i}|_{N(r)}}\}$, $\alpha_1(r)=\sup\{f(\alpha,y)\,:\,y\in E_{\mathtt{i}|_{n(r)}}\}$. Recall that by Lemma \ref{prop:tekninen}, if
$r>0$ is small enough then  $\sup\{\alpha_{\min}(y)\,:\,y\in E_{\mathtt{i}|_{n(r)}}\}<\alpha<\inf\{\alpha_{\max}(y)\,:\,y\in E_{\mathtt{i}|_{N(r)}}\}$ thanks to the assumption $\alpha_{\min}(x)<\alpha<\alpha_{\max}(x)$. In particular, $f(\alpha,y)$ is well defined for all $y\in E_{\mathtt{i}|_{n(r)}}$.

Lemma \ref{lemma:spectrum} applied to $E_{\mathtt{i}|_{n(r)}}$ and $E_{\mathtt{i}|_{N(r)}}$ yields the estimates
\begin{align*}
 \alpha_0(r)& \le\dimh(\{y\in B(x,r)\,:\,\dimloc(\mu,y)=\alpha\})\\
            & \le\dimp(\{y\in B(x,r)\,:\,\dimloc(\mu,y)=\alpha\})\le\alpha_1(r).
\end{align*}
From Lemma \ref{prop:tekninen} we infer that $\alpha_0(r), \alpha_1(r) \longrightarrow f(\alpha,x)$ as $r\downarrow0$ and this gives the claim.

Finally, let us assume that $\alpha=\alpha_{\min}(x)$ (the case $\alpha=\alpha_{\max}(x)$ is symmetric). In the degenerate case $\alpha_{\min}(x)=\alpha_{\max}(x)$, we have $f(\alpha,x)=\dimloc(\mu,x)=\alpha$ and the claim follows using \eqref{eq:min-max}.
If $\alpha_{\min}(x)<\alpha_{\max}(x)$, then $f(\alpha_{\min}(x),x)=0$. Given $\varepsilon>0$, we may consider the set
\[E_{\varepsilon,r}=\{y\in B(x,r)\,:\,\udimloc(\mu,x)\le\alpha_{\min}(x)+\varepsilon\}.\]
Now a minor variation of (the proof of) Lemma \ref{lemma:spectrum} implies that there is a constant $c<\infty$ (independent of $r$ and $\varepsilon$) such that for all small $r>0$, we have $\dimp(E_{\varepsilon,r})\le c\varepsilon$. Letting $\varepsilon\downarrow 0$ finishes the proof.
\end{proof}

\begin{remark}
  Our aim in this paper was to present a simple situation where local multifractal analysis could be carried out in general metric spaces. For this reason, we assumed that the measure $\mu$ locally resembles a Bernoulli measure. Concerning possible generalisations, it is natural to ask if Theorem \ref{thm:localdimappl} (or a version of it) remains true when the measure $\mu$ is required to locally resemble a quasi-Bernoulli measure. For example, see \cite{BrownMichonPeyriere1992, Feng2007} for such results in the global setting.
\end{remark}

\subsection*{Acknowledgments}
We are grateful to the referees for carefully reading the manuscript and for their comments that lead to several improvements.

redpajama_set_name: RedPajamaArXiv


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1200 FLIGHT International, 22 April 1978 /gMt*** Tarom—see Transporturile A e r i e n e Romane. TAT (France)—see Touraine Air Trans port. Texas International Airlines Inc, known until 1969 as Trans-Texas Airways, was formed in 1940 as Aviation Enterprises! and began scheduled local services in October 1947. Texas International now serves nine states along a 10,000-mile route system and offers international services to Monterey and Mexico City. Head Office: PO Box 12788, 8437 Lock heed, Houston, Texas 77017, USA. ^ 644-77. Executives: president, chief executive, Francisco A. Lorenzo; executive vice-presidents: Donald C. Burr; Robert J. Carney; vice-presidents: technical, James W. Arpey; flight operations, Donald J. Breeding; personnel relations, Edwin J. Cathell; planning, Gerald L. Gitner; sales/service, Frank H. Wheeler; general counsel, Sam Coats; controller, Robert J. McAdoo; treasurer, Robert D. Snedeker. Employees: 2,300. Fleet: seven DC-9-30, 16 DC-9-15, three DC-9-15F, five Convair CV-600. On order: five DC-9-30. Simulator: one DC-9. Thai Airways Ltd was formed on Novem ber 1, 1951, by the merger of Siamese Airways and Pacific Overseas Airlines (Siam). The Government holds 99-9% of the stock. Thai Airways operates domestic services and also flies to Vientiane and Penang. The airline is a majority shareholder in Thai Airways International, formed jointly with SAS. Head Office: 6 Lara Luang Road, Bangkok, Thailand. Executives: chairman, ACM Panieng Kantarat; managing director, AVM Thaveep Bunnag; managers: administra tion, Sirmsakdi Brohmsubha; opera tions, F/Lt Kiatnarong Sumongkol; traffic/sales, Prasad Sukhondabandhu; finance, Choob Itthikul; service, Som-pongse Noypongse; public relations, M. L. Ajcharaporn Suksawadi. Employees: 599. Fleet: one Boeing 737-200, nine HS.748, two DC-3. Thai Airways International Ltd (Thai International) was formed in August 1959 as a subsidiary of Thai Airways with technical, managerial and equip ment assistance from SAS. The original SAS shareholding was gradually re duced from 30% to nil by March 31, 1977. Operations began in May 1960 and greatly expanded the international routes of Thai Airways. An extensive network of scheduled passenger and cargo services now links Bangkok with points in South-east Asia, the Middle East, Australia and Europe. Head Office: 1043 Phaholyothin Road, Bangkok, Thailand. Executives: chairman, ACM Panieng Kantarat; president. Air Marshal Choo Suthichoti; executive vice-president, Capt Brom na Thalang; senior vice- president industry affairs, Niels Lum-holdt; vice-presidents: corporate plan ning/international relations, Nikorn Maneelert; finance / administration, Thamnoon Wanglee; marketing, Chat-rachai Bunya-Ananta; flight operations, Capt Jothin Pamon-Montri; technical, Air Marshal Soontorn Sundarakul. Employees: 5,096. Fleet: three DC-8-63, six DC-8-30, two DC-10-30, three Airbus A300B4. On order: three A300B4. Simulators: one Singer-Link DC-8. On order: one LMT A300. THY—see Turk Hava Yollari AO. TMA—see Trans Mediterranean Airways SAL. Toa Domestic Airlines Co Ltd (TDA) was formed on May 15, 1971, by a merger of Japan Domestic Airlines (formed 1960 by the amalgamation of North Japan, Nitto and Fuji Airlines) and Toa Airways (formed 1953). Stock holders include Toa, Japan Domestic, Japan Air Lines (9-1%), Tokyu Corp (26-2%), FujiSash Industries (10-5%) and Kinki Nippon Railway (8-7%). Operations began in May 1971 and now Cover an extensive network of scheduled domestic services to 35 cities through out Japan from Tokyo to> Osaka. Head Office: No 18 Mori Buildings, 20 Akefune-cho, Shiba, Minato-ku, Tokyo 105, Japan. ^222-5182. Executives: president, Isamu Tanaka; senior vice-presidents: Masao Hama, Yoshitaka Ohki, Toshihiko Kubota, Jinzo Azami, Michihisa Ihara; secretary, K. Tezuka. Employees: 2,601. Fleet: two DC-9-50, 18 DC-9-41. 38 YS- 11A, one Bell 214B, three Fuji-Bell 204B, five Hughes H-500, eight Kawasaki KH-4, six KH-47G. On order: two DC-9-41. Simulator: one DC-9-41. Touraine Air Transport (TAT) was formed by M. Marchais in 1968 as an air-taxi company. In 1973 the airline be came a subsidiary of Societe Auxiliare de Services et de Materiel Aero-nautiques (SASMAT), which also owns Rousseau Aviation. TAT subsidiaries in clude Air Paris, TAT Escale and TAT Express. Scheduled services began in 1968, and with the total integration of Rousseau Aviation, Taxi Avia France and Air Paris routes, operations now cover an extensive (55-point) network of domestic and international services from four main points: Paris (Orly), Lyons (Satolas), Lille and St Brieuc. A number of scheduled services are operated on behalf of Air France and Air Inter. Head Office: Aerodrome de Tours St Symphorien, 37100, Tours, France. ••'75876. Executives: president/director-general, Michel Marchais; secretary-general, Pierre Dugenet; directors: marketing, Roger Parodi; programmes, B. P. Bouquet; technical, Arsene Morel; development, J. M. Desindes; freight, Claude Jeantur; public relations, Antoine Houdiard. Employees: 446. Fleet: two Fokker F.28, two VFW 614, nine Fokker F.27, two Nord 262, three Corvette, five Beech 99, one Cessna 402. Trabajos Aereos y Enlaces (TAE) was formed in 1957 but did not begin opera tions until April 1967. Charter and inclusive-tour services were operated until 1970, when the airline ceased operations. Following lengthy refinanc ing negotiations by the owner (Spanish shipping company Cia Naveira Aznar), TAE was reorganised and resumed operations on April 1, 1973, through a technical-assistance agreement with the French airline UTA. Head Office: Aeropuerto de Palma-Majorca, Spain. >•-68669. Executives: founder/owner, Naveira Aznar; general manager, Santiago de Andres Solana; assistant general manager, Alfredo G. Espantaleon Navarro. Fleet: one DC-8-50, three DC-8-30, two Caravelle 10B. Tradewinds Airways Ltd was formed in November 1968 to operate cargo charter services. Major shareholders were Charles Hughesden (85%) and Stewart Wrightson & Co until January 1977, when the Lonrho conglomerate acquired 100% of the stockholding. Tradewinds Helicopters of Fairoaks is a subsidiary. Head Office: Timberham House, Gatwick Airport, Horley, Surrey, Great Britain. •»-" 87427. Executives: chairman. Charles F. Hughesden; managing director, N. F. Curtis; director: finance, P. G. Gibson; commercial. D. J. Harris; operations, Capt T. W. G. Harrison. Employees: 220. Fleet: three Boeing 707-320C. five Canadair CL-44. On order: one 707-320C. Trans European Airways (TEA) was formed in October 1970 to operate passenger and cargo charter and inclusive-tour services. Major share holders are the Belgian travel company TIFA, and G. P. Gutelman. Subsidiary companies are TEA-Holland (formed in August 1975) and TEA Duty-Free Shop SprL. Head Office: Building 117, Melsbroek Airport, Brussels, Belgium.


Dharmakshanti means something in Buddhism, Pali. If you want to know the exact meaning, history, etymology or English translation of this term then check out the descriptions on this page. Add your comment or reference to a book if you want to contribute to this summary article.
The Sanskrit term Dharmakṣānti can be transliterated into English as Dharmaksanti or Dharmakshanti, using the IAST transliteration scheme (?).
Dharmakṣānti (धर्मक्षान्ति) refers to "patience towards dharmas" according to the Mahāprajñāpāramitāśāstra chapter X.—Dharmakshanti represents a type of kṣānti (patience): a set of qualities acquired by the Bodhisattvas accompanying the Buddha at Rājagṛha on the Gṛdhrakūṭaparvata. The patience relating to dharmas (dharmakṣānti) is the unhindered mind (apratihata-citta) relating to profound dharmas (gambhīra-dharma).
According to the Mahāprajñāpāramitāśāstra (chapter XXV), "to endure sycophants and flatterers as well as violent and lustful people constitutes patience toward beings (satvakṣānti); to endure adulation (satkāra) and flattery (pūjā) as well as violence (viheṭhana) and lust (kāma-mithyācāra) constitutes patience in regard to the Dharma (dharmakṣānti)".


Another ridiculously busy month passes the number 1 Party Band Hampshire by - wedding season is well under way! We've had the pleasure of meeting some lovely couples and performing at some incredible venues.
Although our normal haunts are around Hampshire, Surrey and Sussex, we do also travel further afield and we've had a couple of band road-trips 'oooop Norf' recently. One for a wonderful Wirral wedding and the second near Yorkshire, for our first ever wedding in an Indian-style Teepee!
In between all that, we managed to find time to pop across to Wiltshire and the lovely village of Ramsbury, where the whole town seemed to turn up to party en-masse to celebrate Chloe's wedding in the family garden.


The 91 Butler Street runs daily.
Weekday trips most often run every 10 minutes. Saturday trips most often run every 25 minutes. Sunday trips most often run every 45 minutes.
Daily trips have at most 81 stops with a total travel time of about 39 minutes.
Weekdays travel includes, RIDC PARK TERMINAL, LAYOVER VA HOSPITAL TERMINAL (NO STOP), OLD FREEPORT RD AT FREEPORT RD (TERM), and PENN AVE AT GARRISON PL NS (#933).
Weekdays travel to RIDC PARK TERMINAL, LAYOVER VA HOSPITAL TERMINAL (NO STOP), OLD FREEPORT RD AT FREEPORT RD (TERM) starts at PENN AVE AT GARRISON PL NS (#933). Weekdays travel to PENN AVE AT GARRISON PL NS (#933) starts at Old Freeport Rd At Freeport Rd (term), Ridc Park Terminal, and Layover Va Hospital Terminal (no Stop).
Saturdays travel includes, OLD FREEPORT RD AT FREEPORT RD (TERM), LAYOVER VA HOSPITAL TERMINAL (NO STOP), and PENN AVE AT GARRISON PL NS (#933).
Saturdays travel to OLD FREEPORT RD AT FREEPORT RD (TERM), LAYOVER VA HOSPITAL TERMINAL (NO STOP) starts at PENN AVE AT GARRISON PL NS (#933). Saturdays travel to PENN AVE AT GARRISON PL NS (#933) starts at Layover Va Hospital Terminal (no Stop) and Old Freeport Rd At Freeport Rd (term).
Sundays travel includes, LAYOVER VA HOSPITAL TERMINAL (NO STOP), OLD FREEPORT RD AT FREEPORT RD (TERM), and PENN AVE AT GARRISON PL NS (#933).
Sundays travel to LAYOVER VA HOSPITAL TERMINAL (NO STOP), OLD FREEPORT RD AT FREEPORT RD (TERM) starts at PENN AVE AT GARRISON PL NS (#933). Sundays travel to PENN AVE AT GARRISON PL NS (#933) starts at Old Freeport Rd At Freeport Rd (term) and Layover Va Hospital Terminal (no Stop).
Good Friday 2019 and Memorial Day 2019 trips start at 4:32am with the last trip at 12:48am and most often run every 10 minutes. Easter Sunday 2019 trips start at 5:08am with the last trip at 12:38am and most often run every 45 minutes.


text:
Home The Nation Tanker drives into crowd of protesters in Minneapolis on I-35; driver injured...
Tanker drives into crowd of protesters in Minneapolis on I-35; driver injured and arrested
Wendy Miller
The Minnesota Highway Patrol said the driver was taken to a hospital with non-life-threatening injuries. TV footage from local news sources show protesters swarming the truck, and then law enforcement quickly moving in.
MINNEAPOLIS — A tanker truck has been reported to have driven into a group of thousands of protesters on the Interstate 35 west in Minneapolis today, Sunday, May 31.
The Minnesota Highway Patrol said the driver was taken to a hospital with non-life-threatening injuries. TV footage from local news sources show protesters swarming the truck, and then law enforcement quickly moving in. Other TV footage showed the tanker truck moving rapidly onto the bridge and protesters appearing to part ahead of it.
Video courtesy of FOX9 KMSP.
Officials in Minnesota say no protesters appear to have been hit. The Minnesota State Patrol said in a tweet that the action appeared deliberate.
The Minnesota Department of Public safety tweeted this message from DPS Commissioner John Harrington: "From the footage we are seeing on @MnDOT cameras, it appears the semi was on I-35W as authorities were closing the road. It didn't appear to drive through any barricades. Driver released from the hospital and under arrest."
The Minnesota Department of Public safety initially tweeted "very disturbing actions by a truck driver on I-35W, inciting a crowd of peaceful demonstrators. The truck driver was injured & taken to a hospital with non-life-threatening injuries. He is under arrest. It doesn't appear any protesters were hit by the truck."
The protesters were demonstrating against the death of George Floyd. The Minnesota Department of Public Safety has asked everyone to clear the area near the I-35W bridge on University and Washington avenues. People are being given the lawful order to clear the area. If that order is refused, police may use tear gas as a dispersal mechanism.
[The Associated Press contributed to this report.]
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Wendy Miller is the managing editor of The Trucker Newspaper and TheTrucker.com. She holds a bachelor's degree in journalism and a master's degree in strategic communications. Wendy has been a journalist and editor for 12 years and has specialized in niche publications for the past eight years. Wendy draws her love for the trucking industry from growing up as a trucker's daughter.
Tanker driver who drive into I-35 protest did not intentionally try to 'intersect the protest' | TheTrucker.com June 2, 2020 At 10:08 am
[…] The Trucker first reported on this incident on Sunday, May 31. To view the original story, which includes local news footage of the incident, click here. […]
https://www.thetrucker.com/trucking-news/the-nation/tanker-drives-into-crowd-of-protesters-in-minneapolis-on-i-35-driver-injured-and-arrested

redpajama_set_name: RedPajamaCommonCrawl


text:
National Women's Soccer League faces challenges and opportunities as World Cup looms
Updated: Apr. 09, 2015, 10:05 a.m. |
Published: Apr. 09, 2015, 9:05 a.m.
Jamie Goldberg | The Oregonian/OregonLive
When the National Women's Soccer League (NWSL) commenced play in April of 2013, it became the third iteration of a top-flight women's pro soccer league in the United States in just 13 years.
The Women's United Soccer Association (WUSA) was formed in 2000 after the rousing success of the 1999 FIFA Women's World Cup, but lasted just three seasons before folding, citing losses of $100 million. The Women's Professional Soccer League (WPS) followed in 2009, but was gone by 2012 due to systemic organizational problems.
Due in large part to the failure of its predecessors, the NWSL has followed a different blueprint.
The league is bolstered by the support of the U.S. Women's National Team as well as support from the Canadian and Mexican Federations. This is deliberately designed to limit costs, a model that has put the NWSL on stable footing heading into its own critical third season.
"The structural differences have been key," NWSL commissioner Jeff Plush said. "Now, it's not about whether we will survive, because I have no doubt about that. It's about how are we going to grow as a league?"
Fan support will be critical to the NWSL as attempts to evolve into a sustainable and lasting operation. Both the WUSA and the WPS struggled to retain fans and saw precipitous drops in attendance before folding.
Over the NWSL's first two seasons, attendance numbers have remained fairly consistent, but uniformly low. The outlier franchise has been the Portland Thorns, who have averaged an incredible 13,341 fans per game over the last two years.
The overall league attendance dropped slightly from 2013 to 2014, going from an average of 4,271 to 4,139 fans per game. Not including the Thorns, the rest of the league drew a modest average of just 2,982 fans per game over the league's first two seasons.
2013 NWSL home attendances by team
Total Attendance
Average Attendance
Portland Thorns 11 146,521 13,320
FC Kansas City 11 50,884 4,626
Western New York Flash 11 49,334 4,485
Washington Spirit 11 39,877 3,625
Boston Breakers 11 26,697 2,427
Seattle Reign 11 25,365 2,306
Chicago Red Stars 11 18,817 1,711
Sky Blue FC 11 18,329 1,666
League Totals 88 375,824 4,271
Houston Dash 12 55,801 4,650
League Totals 108 447,013 4,139
"I think that the players need to continue to be in the community and continue to be role models for kids and keep trying to expand the league," said Thorns midfielder Allie Long, who also played in the WPS before it folded. "I also think after the World Cup this summer, the league will see a huge bump in attendance as well."
The NWSL could see increased attendance numbers immediately following this summer's 2015 FIFA Women's World Cup, but will be tested early in the season as it attempts to maintain support before and during the iconic international tournament.
Across the league, about 50 of the NWSL's roughly 180 rostered players will be competing in the World Cup this summer, leaving the league with the difficult task of going on without its most talented and recognizable stars. The league's allocated players from the U.S., Canada and Mexico are expected to miss 7-8 games of the short 20-game NWSL season, while national team players from other countries could miss even more. The league will face a similar test next season with the 2016 Olympics.
Adding to the difficulty, U.S. Women's National Team star Abby Wambach, who is arguably one of the most recognizable faces in women's soccer, announced recently that she would forgo the entire NWSL season to focus on her preparations for the World Cup.
"Clearly there are challenges whenever you have an event of the scale of the World Cup, but it's a great situation to have," Plush said. "The best players in the world are going to be playing in our time zone. Our teams are going to be doing things in their markets to build off the excitement of the World Cup."
Plush said that league and its clubs would be working in collaboration with the U.S. Soccer Federation, Nike and other sponsors to put on events this summer surrounding the World Cup in a concerted effort to draw more fans to the league.
The NWSL is also hoping to announce a national television sponsor in the near future, according to Plush, which should give the league some visibility during a summer where the World Cup will be creating a buzz around women's soccer.
"We are focused on the opportunity to connect with the fan that's paying attention to the national team more than the club teams and pull them into our league as well," Plush said. "It is our job to take advantage of the excitement that will build during the World Cup as we look to continue to grow fans for women's soccer long-term."
While its predecessors were already weathering changes in ownership groups, clubs folding and significant drops in attendance after just two seasons, the NWSL is seeing some encouraging signs. One of its teams - the Portland Thorns - is already turning a profit and the league added another MLS-affiliated expansion team in the Houston Dash in 2014.
The league is also generating interest from ownership groups in other new markets. There have been rumors that Real Salt Lake could be interested in forming an NWSL side in the future. Investors in Atlanta and Indianapolis have also expressed interest in forming NWSL teams.
"We are on a very positive path," Plush said. "You can see that by the fact that there is significant interest in many markets for expansion."
As the league turns its focus away from simply surviving to growing into a sustainable enterprise, it will inevitably face other questions about its current structure.
While the U.S., Canadian and Mexican federations pay the salaries of the players that they allocate to the NWSL, teams were allowed a salary cap of just $200,000 in 2014 to pay the rest of their players, meaning that the majority of players in the NWSL earned between just $6,000 and $30,000 a season.
The unsustainable salaries have forced many players in the NWSL to balance outside jobs with their playing careers and convinced a slew of young and talented players to retire early. Among those players is former Thorn Nikki Marshall, who made the decision to retire during the offseason at the age of 26, citing the low salaries in the NWSL as a contributing factor in her decision.
"We need to give these players something to play for," Thorns coach Paul Riley said. "There needs to be a minimum monthly [salary] that they can live on and that they can make a career off of for five, six, eight years because to lose this many young players doesn't make any sense."
Plush agreed that the league must look to address salaries in the future, though there are no immediate plans to make changes to the salary cap and roster rules.
"Certainly we need to grow the league and continue to grow the business of the league and that includes raising player salaries," Plush said. "That has to happen over time and it has to happen in a proven path that's sustainable."
The next two years will provide a stern test for the NWSL as it looks to grow, evolve and perhaps expand, even as its top players are forced to miss significant time for the World Cup and then the Olympics.
But it will also present an opportunity and challenge for the league to take the next steps towards sustainability as it attempts to parlay the nationwide enthusiasm around the sport's two most prestigious tournaments into a loyal and enduring fan base for years to come.
"Soccer has continued to evolve and grow in the U.S.," Plush said. "Just look at what Major League Soccer continues to do. Just look at the continued growth in the interest of the international game. Soccer continues to grow and have a place in our culture and we want our league to grow with that."
-- Jamie Goldberg | jgoldberg@oregonian.com
503-853-3761 | @jamiebgoldberg


Dr Richard Fortey and Professor Dame Janet Nelson were awarded honorary degrees by the University of St Andrews yesterday (21 June 2007).
Dr Fortey, Honorary Research Associate of the Natural History Museum London, was made an Honorary Doctor of Science (Hon DSc) and is pictured below with (left-right) Dr Charles Warren (School of Geography & Geosciences), Principal and Vice-Chancellor Dr Brian Lang, Chancellor Sir Menzies Campbell and Rector Simon Pepper.
Dame Janet Nelson, Professor of Mediaeval History at Kings College, London, was awarded an Honorary Doctor of Letters (Hon DLitt). She is pictured below with Dr Robert Bartlett (School of History).


Leave it to basketball legend Shaq to give us something none of us knew we needed: A "Fade" parody!
SEE ALSO: Misty from 'The Amanda Show's' 'Moody's Point' is super good looking now!
During the MTV Video Music Awards a few weeks ago, Kanye West premiered his music video for "Fade," featuring the now Queen of Body Goals, Teyana Taylor. You could check that out here, but remember NSFW.
Since the premiere, there have been countless parodies of the music video, most notably, James Corden's on "The Late Late Show," which is pretty great.
However, we most defintiely have a winner for "Fade" parodies now that Shaq has arrived! The former NBA player took to Instagram to debut his hilarious version of "Fade," complete with boxer briefs and all! Take a look below.
We think it's safe to say, Teyana Taylor has a run for her money now that Shaq has entered the picture!
Check out more from Shaq in the gallery below!


Remember those dark nights spent terrified of monsters under the bed? Well now it's time to strike back!
Stop the deceivingly cute Squabies before they reach your bed in this classic tower defense game. Place three different weapons strategically in the right spots along the monster's path to your bed. Earn gold for each stopped Squaby stepping out of your closet to build new towers and upgrade existing ones.
Squaby is an easy to play TD game and usable by everyone.
Super easy and fast way to create sprite sheets to increase performance and loading time of the game.
Definitely dithering. Gives you the ability to save so much memory while sprites still look awesome. And also it's cross platform, what more could you possibly want?!


text:
Top 10 Spectacular Birds of Antarctica
Sukhdev Shukla
Flickr - Photo Credit: Anne Froehlich
Colourful Spectacular Birds
While talking about Antarctica, which bird immediately comes to mind? Most people would say the penguin.
But there are about 45 other species of fantastic birds living and breeding in Antarctica. Against the backdrop of blue hues reflected by icebergs, numerous spectacular birds thrive there and some of them are really colourful birds.
Antarctica, containing about 70% of world's freshwater, offers fantastic opportunities to experience seabird life. This happens in spite of extremely cold conditions prevailing where these birds have to adapt to live and breed.
Let's discover the 10 most beautiful birds of Antarctica.
Map of Antarctica
Flickr - Photo credit: BlatantWorld.com (In Public Domain)
Emperor penguins - Aptenodytes forsteri
Flickr - Photo credit: lin padgham
1. Emperor Penguin
Penguins are flightless birds and the emperor penguin is the tallest and heaviest of all living species of penguins. They can grow up to 4 ft. and weigh between 22 to 45 kgs, males normally weigh more than females. They stay in very harsh environment with sub-zero temperature but are quite capable of adapting themselves to these conditions.
There are similarities in colours of the male and female penguins. Their dorsal feathers are deep black which cover their back, dorsal flippers and tail while also their head, throat and chin. Big portion of their bellies are white with pale yellow upper breast and bright yellow ear patches.
Female penguins lay one egg but the incubating job is with the males. In the mean time females move to the sea for about two months for hunting and return with big belly full of food for the chicks. Surprisingly, while keeping the eggs warm males do not eat anything but move to sea after the females return. Fantastic, how faithful they are to each other.
The emperor penguin is now a nearly threatened category of conservation status due to decline of food as a result of climate changes.
Wikimedia Commons: Photo credit: jammach_uk
Wandering albatross - Diomedea exulans
Flickr - Photo Credit: renke.luehken
2. Wandering Albatross
The wandering albatross is one of the largest birds in the world and known for spending most of their lifespan in flight. The only time they are seen on land is for mating, feeding and raising their young ones, large colonies are seen during this period.
Also, they have the largest wingspan, may cross even 11 feet and they are long-lived bird, going to even 50 years. The large wings help them to fly for miles over ocean without flapping their wings and even glide for hours together.
They have white bodies with white and black wings. Males have faint peach spot on the side of the head, pink large bill and pink feet. Their salt gland above the nasal passage helps to desalinate their bodies. They feed themselves on small fish and crustaceans. They are capable of following ship hoping in search of food in garbage.
They are hunted for their feathers and are vulnerable for conservation status.
Flickr - Photo Credit: brian.gratwicke
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Southern giant petrel - Macronectes giganteus
Flickr- Photo Credit: sussexbirder
3. Southern Giant Petrel
The southern giant petrel is a large seabird with a wingspan nearing 200 cms. Out of the two morphs, the darker one has a grey-brown body but the head, breast and neck are whitish. It has very large yellow bill with greenish tip. It gives a hunchbacked appearance while in flight.
It has a large range from Antarctica to Chile, Africa and Australia. Nesting is done in ice-free coastal areas and many of them are known for returning to the same nest every breeding season. They prefer krill, squid and fish for feeding themselves. The conservation status of the species is of the least concern.
Other Names: Antarctic giant petrel, giant fulmar, stinker, stinkpot
Flickr - Photo Credit: NOAA Photo Library
Imperial Shag - Phalacrocorax atriceps
Wikimedia Commons - Photo credit: Jerzy Strzelecki
4. Imperial Shag
With 40 species of cormorants and shags in the Phalacrocoracidae family, there is no consistency in their distinction. Imperial shag is covered with glossy black feathers and the belly and neck are white. Legs and feet are pinkish and the eyes have a ring of blue skin. With a large global population, they are native to Antarctic Peninsula, Sub-Antarctic Islands and southern parts of Southern America.
They breed in colonies which are relatively small and shared with others. Their nests are made of grass and seaweeds. Their 2 to 3 chicks are vulnerable to predators.
They feed on small fish, crustaceans, squid and they are capable of catching its food by pursuit-diving for which they may travel some distance even.
Other names: blue-eyed shag, blue-eyed cormorant, king comorant
In flight
Flickr - Photo credit: Liam Quinn
Flickr - Photo credit: chrispearson72
Snowy Sheathbill - Chionis albus
5. Snowy Sheathbill
Snowy sheathbill is the only land bird without webbed feet at Antarctica. It is called snowy because it is pure white in colour except its face, which is pink. It is found on Sub-Antarctic islands and Antarctic peninsula. Migration to South America and Falkland Islands happens in winter.
It regularly roosts on one leg and finds its food on land. It nests under a rock or in a crevice and fights during courtship periods. Normally 2 to 3 eggs are laid. It is omnivorous and believes in stealing food like fish, eggs, small chicks of breeding penguins.
It is not a threatened species globally.
Other names: greater sheathbill, pale-faced sheathbill, paddy
Standing on one leg
Wikimedia Commons - Photo credit: David M. Jensen
Wilson's storm petrel - Oceanites oceanicus
6. Wilson's Storm Petrel
Wilson's storm petrel is a small gregarious seabird. With dark brown plumage excepting the flanks and the rump which are white. It has long thin legs and feet which are black but the feet has yellow webbing. It is one of the most abundant species breeding in Antarctic coastlines and nearby islands. It migrates and spends autumn in Northern Hemisphere as a non-breeding season.
While hovering over water with quite low with its feet skimming, it picks up food from the water's surface. It feeds on crustaceans and fish.
It is species with least concern on conservation status.
Other names: Wilson's petrel, flat-clawed or yellow-webbed storm petrel
Wikimedia Commons - Photo credit: Patrick Coin
Brown skua - Stercorarius antarcticus
Wikimedia Commons - Winfried Bruenken
7. Brown Skua
Brown skua is a large sea bird which breeds in some areas of the Antarctica, South Georgia and New Zealand. They are very heavy, noisy and known for belligerent postures and attacking penguins.
They feed on fish, small mammals, eggs and other birds, and also serve as scavengers. They stay near sea for securing their food but move to dry lands for nesting and breeding. They are opportunistic birds and there is no consensus about their classification. They are not threatened birds.
Other names: Antarctic skua, subantarctic skua, southern skua, southern great skua
Kelp gull - Larus dominicanus
Flickr- Photo credit: Bring on the Photog
8. Kelp Gull
Kelp gull is distinguished from other gulls because of upper parts and wings which are very black and pale yellow-green legs. The head, tail and underparts are white but the bill is yellow with a red spot. The eye is yellow and surrounded by with orange-red ring.
Kelp gull is not only spread over Antarctica and many sub-antarctic islands but found on coasts and islands throughout the Southern Hemisphere. Breeding occurs between September and January. They are monogamous and breed in colonies. Their diet includes fish, worms, arthropods, reptiles, amphibians and they also believe in scavenging.
Other names: Dominican gull
Flickr - Photo credit: Ian N. White
Flickr - Photo credit: angrysunbird
Cape petrel - Daption capense
Wikimedia Commons - Photo credit: Henrike Mühlichen
9. Cape Petrel
Cape petrel has distinctive pattern of colours with black-and-white chequered plumage. It has black head and neck but the upper parts are black and white. The legs, feet and bill are black.
They are quite common seabirds in the Southern Sea and they breed in colonies of the Antarctic continent and near New Zealand sub-temperate islands. They are quite quarrelsome and noisy on food and nest issues. The species is not threatened. A single large egg is laid in November and incubation period is about one and a half month. They live on small fish, squid and amphipods.
Other names: Cape pigeon, Pintado petrel
Flickr - Photo credit: dfaulder
Macaroni penguin - Eudyptes chrysolophus
10. Macaroni Penguin
Macaroni penguin is markedly different because of the yellow crest arising out of the central part of the forehead. Its beak is also peculiar, bulbous with orange-brown colour. Its legs and feet are pink and eyes are red. In contrast with the white underparts it has black chin, throat and head along with some underparts differentiating from white.
It used to be the most numerous among penguin species, but it is reduced to threatened status of conservation. It ranges from from Sub-antarctic to Antarctic Peninsula and lives in large and dense colonies on rocky coasts and lower cliffs during breeding.
They lay two eggs, the first one is smaller than the second which comes 4-5 days later. This large and flightless species eats small fish and cephalopods.