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\controldates{10-OCT-2003,10-OCT-2003,10-OCT-2003,10-OCT-2003}
 
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\begin{document}


\title[Rigidity properties of $\zd$-actions on tori and solenoids]{Rigidity
properties of $\zd$-actions\linebreak[1] on tori and solenoids}

\author{Manfred Einsiedler}

\address{Department of Mathematics, Box 354350, University of Washington, 
Seattle, WA 98195}
\email{einsiedl@math.washington.edu}

\author{Elon Lindenstrauss}


\address{Department of Mathematics, Stanford University, Stanford, CA 94305 }
\curraddr{Courant Institute of Mathematical Sciences, 251 Mercer St., 
New York, NY 10012}
\email{elonbl@member.ams.org}


\thanks{E.L. is supported in part by NSF grant DMS-0140497. The two authors
gratefully acknowledge the hospitality of Stanford University and the
University of Washington, respectively}

\commby{Klaus Schmidt}

\date{July 12, 2003}

\keywords{Entropy, invariant measures, invariant
$\sigma$-algebras, measurable factors, joinings, toral
automorphisms, solenoid automorphism}

\subjclass[2000]{Primary 37A35; Secondary 37A45}




\begin{abstract}
We show that Haar measure is a unique measure on a torus or more generally a
solenoid $X$ invariant under a not virtually cyclic totally irreducible
$\mathbb Z^d$-action by automorphisms of $X$ such that at least one element of the action
acts with positive entropy. We also give a corresponding theorem in the
non-irreducible case. These results have applications regarding measurable
factors and joinings of these algebraic $\mathbb Z^d$-actions.
\end{abstract}

\maketitle

\section{Introduction and main results}\label{sec: Intro}

The map $T_p:x \mapsto px$ on $\TT=\RR/\ZZ$ has many closed
invariant sets and many invariant measures. Furstenberg
initiated
the study of jointly invariant sets in his seminal paper
\cite{Furstenberg-disjointness-1967}. A set $A \subset \TT$ is
called {\em jointly invariant} under $T_p$ and $T_q$ if $T_p
(A)\subset A$ and $T_q(A)\subset A$. Furstenberg proved that if
$p$ and $q$ are multiplicatively independent integers, then any
closed jointly invariant set is either finite or all of $\TT$.

Furstenberg also raised the question of what the jointly invariant
measures are, that is, which probability measures $\mu$ on $\TT$ satisfy
$(T_p)_*\mu=(T_q)_*\mu=\mu$. The obvious ones are Lebesgue
measure, atomic measures supported on finite invariant sets, and
(non-ergodic) convex combinations of these.

Here we give a partial answer to this question in the following
more general setting of $\zd$-actions on solenoids.

In the following a {\em solenoid} $X$ is a compact, connected,
abelian group whose Pontryagin dual $\widehat{X}$ can be embedded
into a finite-dimensional vector space over $\QQ$. The simplest
example is a finite-dimensional torus. A $\zd$-action $\alpha$ by
automorphisms of a solenoid $X$ is called {\em irreducible} if
there is no proper infinite closed subgroup which is invariant
under $\alpha$, and {\em totally irreducible} if there is no
finite index subgroup $\Lambda \subset\zd$ and no proper infinite
closed subgroup $Y \subset X$ which is invariant under the induced
action $\alpha_\Lambda$. A $\zd$-action is {\em virtually cyclic}
if there exists $\n \in\zd$ such that for every element $\bm \in
\Lambda$ of a finite index subgroup $\Lambda \subset\zd$ there
exists some $k \in \ZZ$ with $\alpha^\bm=\alpha^{k\n}$.

\begin{theorem}\label{thm: easy-case}
Let $\alpha$ be a totally irreducible, not virtually cyclic
$\zd$-action by automorphisms of a solenoid $X$.
Let $\mu$ be an $\alpha$-ergodic measure. Then either $\mu=\lambda$
is the Haar measure of $X$, or the entropy $\h_\mu(\alpha^\n)=0$
vanishes for all $\n \in\zd$.
\end{theorem}

We summarize the history of this problem. The topological
generalization of Furstenberg's result to higher dimensions was
given by Berend
\cite{Berend-invariant-tori,Berend-invariant-groups}: An action on
a torus or solenoid has no proper, infinite, closed, and invariant
subsets
if and only if it is totally irreducible, not
virtually cyclic, and contains a hyperbolic element. One direction
of this theorem is easy to see: if either of these properties
fails, one can construct a proper, infinite, closed, invariant
subset. For example, if a $\zd$-action on a torus does not contain
a hyperbolic element, then it can be shown that there exists a
common eigenspace $W \cong \CC$ of the matrices defining the action
so that the corresponding eigenvalues $\xi$ satisfy $|\xi|=1$.
Therefore, the unit ball $B$ in $W$ gives an infinite closed
invariant subset. Notice that we do not assume any hyperbolicity
in Theorem \ref{thm: easy-case}.

The first partial result for the measure problem on $\TT$ was
given by Lyons \cite{Lyons-2-and-3} under a strong additional
assumption. Rudolph \cite{Rudolph-2-and-3} weakened this
assumption considerably, and proved the following theorem.

\begin{theorem}[{\cite[Thm.\ 4.9]{Rudolph-2-and-3}}] \label{thm: Rudolph}
Let $p, q \geq 2$ be relatively prime positive integers, and let
$\mu$ be a $T_p$, $T_q$-ergodic measure on $\TT$. Then either
$\mu=\lambda$ is the Lebesgue measure, or the entropy of $T_p$ and
$T_q$ is zero.
\end{theorem}

Johnson \cite{Johnson-invariant-measures} lifted the relative
primality assumption, showing it is enough to assume that $p$ and
$q$ are multiplicatively independent. By the ergodic decomposition
every invariant measure $\nu$ can be written as a convex
combination of a family of ergodic measures $\mu_\tau$,
$\tau \in \mathfrak T$. If $\nu$ has positive entropy, the same
must apply for some $\mu_\tau$. So Theorem \ref{thm: Rudolph} also
shows that every positive entropy measure is a convex combination
of the Lebesgue measure and a zero entropy measure. Thus the only
restricting assumption here is positive entropy. Feldman
\cite{Feldman-generalization}, Parry \cite{Parry-2-3}, and Host
\cite{Host-normal-numbers} have found different proofs of this
theorem, but positive entropy remains a crucial assumption.

Katok and Spatzier
\cite{Katok-Spatzier,Katok-Spatzier-corrections} obtained the
first analogous results for actions on higher-dimensional tori and
homogeneous spaces. However, their method required either an
additional ergodicity assumption on the measure (satisfied for
example if every one parameter subgroup of the suspension acts
ergodically), or that the action is totally non-symplectic (TNS).
A careful and
readable account of these results has been
written by Kalinin and Katok \cite{Kalinin-Katok-Seattle}, which
also fixed some minor inaccuracies. Theorem \ref{thm: easy-case}
gives a full generalization of the result of Rudolph and Johnson
to actions on higher-dimensional solenoids.

Without total
irreducibility the Haar measure of the group is no longer the only
measure with positive entropy. Thus the general theorem below is
(necessarily)
longer in its formulation than
Theorem \ref{thm: easy-case}. It strengthens e.g.\ \cite[Thm.\ 
3.1]{Kalinin-Katok-Seattle},
which has a similar conclusion but stronger assumptions.

\begin{theorem}\label{thm: main}
Let $\alpha$ be a $\Zd$-action ($d \geq 2$) by automorphisms of
a solenoid $X$.
Suppose $\alpha$ has no virtually cyclic factors, and let
$\mu$ be an $\alpha$-ergodic measure on $X$.
Then there exists a subgroup $\Lambda \subset\Zd$ of finite index and a decomposition
$\mu=\frac{1}{M}(\mu_1+\dots+\mu_M)$ of $\mu$ into mutually singular measures
with the following properties for every $i=1,\ldots,M$.
\begin{enumerate}
\item The measure $\mu_i$ is $\alpha_\Lambda$-ergodic, where
$\alpha_\Lambda$ is the restriction of $\alpha$ to $\Lambda$.
\item There exists an $\alpha_\Lambda$-invariant closed subgroup
$G_i$ such that $\mu_i$ is invariant under translation with
elements in $G_i$, i.e.\ $\mu_i(A)=\mu_i(A+g)$ for all $g \in G_i$
and every measurable set $A$. \item For $\n \in\zd$,
$(\alpha^\n)_*\mu_i=\mu_j$ for some $j$ and $\alpha^\n(G_i)=G_j$.
\item The measure $\mu_i$ induces a measure on the factor $X/G_i$, also denoted by $\mu _i$,
with $\h_{\mu_i}(\alpha^{\n}_{X/G_i})\linebreak[0]=0$ for any $\n \in \Lambda$.
\textup{(}Here $\alpha_{X/G_i}$ denotes the action induced on 
$X/G_i$.\textup{)}\end{enumerate}
\end{theorem}

We note that even in the topological category, where Berend gave definitive results regarding the totally irreducible case, the situation for the reducible case is far from understood.

The proofs of Theorems \ref{thm: easy-case} and \ref{thm:
main} follow the outline of Rud\-olph's proof of Theorem \ref{thm:
Rudolph}. One of the main ingredients there was the observation
that $\h_\mu(T_p)/ \log p = \h_\mu(T_q)/\log q$ (and a relativized version of this equality).
This follows from the particularly simple geometry of this system where both $T_p$ and $T_q$ expand the
one-dimensional space $\TT$ with fixed factors. There is no simple 
geometrical reason why such an equality should be true for more 
complicated $\ZZ ^ d$-actions on solenoids, and indeed is easily seen to 
fail in the reducible case.
However, somewhat surprisingly, such an equality {\em is} true for irreducible $\ZZ ^ d$ actions, even though this is true from subtle number theoretical reasons (see Theorem~\ref{thm: shape} below). It is interesting to note that along the way we get new and nontrivial information about measures invariant under a single, even hyperbolic, solenoidal automorphism.

We apply Theorem \ref{thm: main} to obtain new
information about the measurable structure, with respect to the
Haar measure, of irreducible algebraic $\zd$-actions on tori and
solenoids. Our first application characterizes the measurable
factors of $\alpha$, and generalizes the isomorphism rigidity
results by A.\ Katok, S.\ Katok, and Schmidt
\cite{Katok-Katok-Schmidt}.

\begin{theorem}\label{thm: factors}
Let $\alpha$ be an irreducible, not virtually cyclic $\zd$-action
on a solenoid $X$, and let $\cA$ be an $\alpha$-invariant
$\sigma$-algebra. Then either $\cA=\{\emptyset,X\}$ (modulo
$\lambda$), or there is a finite group $G$ which acts on $X$ by
affine transformations and
\[
\cA=\{ A \in\cB_X:gA=A \text{ for all } g \in G \}\mbox{ (modulo
}\lambda\mbox{)}.
\]
\end{theorem}

In other words, every infinite measurable factor of $\alpha$ is a quotient of $X$ by the action of a finite affine group.
The simplest examples of such groups are
finite translation groups. However, more complicated examples are also possible; for example, let $w \in X$ be any
$\alpha$-fixed point. Then the action of $G=\{ \Id,-\Id+w \}$ on $X$
commutes with $\alpha$.

The proof of Theorem \ref{thm: factors} uses the relatively
independent joining of the Haar measure with itself over the
factor $\cA$, which gives an invariant measure on $X \times X$
analyzable by Theorem \ref{thm: main}. This is similar to the
proof of isomorphism rigidity in \cite{Katok-Katok-Schmidt}, which
followed a suggestion by Thouvenot.

Finally, we characterize disjointness in the case of irreducible
actions, which generalizes the corresponding results for TNS
actions by Kalinin and Katok \cite{Kalinin-Katok}, and by Kalinin
and Spatzier \cite{Kalinin-Spatzier}.

\begin{theorem}\label{thm: joining}
Suppose $\alpha_1$ and $\alpha_2$ are irreducible,
not virtually cyclic
$\Zd$-actions on solenoids $X_1$ and $X_2$.
Then either they are disjoint, or there exists a finite
index subgroup $\Lambda \subset\Zd$ such that the
subactions $\alpha_{1,\Lambda}$ and $\alpha_{2,\Lambda}$
have a common algebraic factor.
\end{theorem}

In this announcement we give an essentially complete proof of Theorem~\ref{thm: shape} regarding the relationship between entropies of individual elements of an irreducible action. First we explain in \S\ref{sec: entropy-conditional} how the various Lyapunov exponents contribute to the entropy. A bound on each of these contributions is given in Theorem \ref{thm:
bound-contribution}, which is a theorem about measures invariant under 
a {\em single} automorphism. Then in \S\ref{sec: entropy-function} we conclude the proof of the entropy identity using a key lemma from \cite{Einsiedler-Katok} regarding the product structure of certain conditional measures. A sketch of how Theorem~\ref{thm: easy-case} is proved using Theorem~\ref{thm: shape} is given in \S\ref{sec: outline-easy}. Full details of the proofs of all
theorems announced in this note will be given in
\cite{Einsiedler-Lindenstrauss}.


\section{Arithmetic automorphisms and irreducible actions}\label{sec: solenoids}


Throughout this note, the term {\em local field} will denote a
locally compact field of characteristic zero; these include $\RR$
and $\CC$ as well as finite extensions $\KK$ of the field of
$p$-adic numbers $\QQ_p$. Let $\KK$ be any local field, and let
$\lambda_\KK$ be the Haar measure on $\KK$. Let $\delta(\KK)=1$
for $\KK \neq \CC$ and $\delta(\CC)=2$. For $a \in\KK$ the norm
$|a|_\KK$ is defined as the real number satisfying
\begin{equation}\label{eq: norm_norm}
\lambda_\KK(aC)=|a|_\KK^{\delta(\KK)}\lambda_\KK(C)
\end{equation}
for any measurable set $C \subset\KK$. Then $|\cdot|_\KK$ satisfies
the triangle inequality for all $\KK$.

The following follows easily from \cite[Thm.~29.2 and
Sect.~7]{Schmidt-book} (see also \cite{Einsiedler-Lind},
\cite{Einsiedler-Schmidt}).

\begin{proposition}\label{prop: localprod}
Let $\alpha$ be an irreducible algebraic $\zd$-action on a
connected group. Then there exists a finite product $\bAA=\KK_1
\times \dots \times \KK_m$ of local fields $\KK_j$, a $\zd$-action
$\alpha _ {\bAA}$ by automorphisms of~$\bAA$ whose restrictions to
$\KK_j$ are linear, $(\alpha _ {\bAA}^\n(x))_j=\zeta_{j,\n} x_j$,
and a $\alpha_{\bAA}$-invariant cocompact discrete subgroup
$\Gamma$ of~$\bAA$, such that $\alpha$ is conjugate to the induced
action of~$\alpha _ {\bAA}$ on~$\bAA/\Gamma$.

Furthermore, we have
\begin{equation}\label{eq: product-formula-zeta}
\prod_{j=1}^m|\zeta_{j,\n}|_j^{\delta(\KK_j)}=1\text{ for all }\n \in\zd,
\end{equation}
and one can choose $\Gamma$ such that
\begin{equation}\label{eq: prodform}
\prod_{j=1}^m|a_j|_{\KK_j}^{\delta(\KK_j)}\geq 1\mbox{ for
every }\ba \in \Gamma.
\end{equation}
\end{proposition}

We note that the local fields $\KK_j$ above are all the Archimedean and some non-Archimedean
completions $\KK_j$ of a number field $\kk$ which depends on the action; \eqref{eq: product-formula-zeta} and \eqref{eq:
prodform} follow from the elementary properties of number fields and their completions.

A $\zd$-action $\alpha$ by automorphisms of a solenoid $X$ is {\em
arithmetic} if the conclusions of Proposition \ref{prop:
localprod} hold. An automorphism of $X$ is arithmetic if it is
part of an arithmetic $\zd$-action for $d \geq 1$.

We shall identify $\KK_j$ with the corresponding subspace in
$\bAA$, and refer to these as the eigenspaces. We use the norms
$|\cdot|_j=|\cdot|_{\KK_j}$ to induce a norm $\| x
\|=\max_i|x_i|_{i}$ on $\bAA$, and furthermore a metric
$d_X(\cdot,\cdot)$ on $X$. We also write $\delta_j=\delta(\KK_j)$.
A ball of radius $r$ around $x \in X$ ($a \in\bAA$) will be denoted
by $B_r(x)$ ($B_r(a)$), if we wish to emphasize the space
$B_r^X(x)$ ($B_r^\bAA(a)$), and by $B_r^X$
($B_r^\bAA$) if the center is zero.



\section{Entropy, invariant foliations, and conditional
measures}\label{sec: entropy-conditional}

In this and the following section we consider a single arithmetic
automorphism. In other words, let $T$ be an automorphism of
$X=\bAA/\Gamma$, where $\Gamma$ satisfies \eqref{eq: prodform}, and
$T$ is induced by a $T_\bAA$ with $(T_\bAA(x))_j=\zeta_jx_j$ for
$j=1,\ldots,m$.
It turns out to be useful to study the following more general situation: let $S$ be an arbitrary homeomorphism of a
compact space $Y$ with metric $d_Y(\cdot,\cdot)$ and let
$\widetilde{T}=S \times T$ be the product map on $\widetilde
X=Y \times X$. Define $d_{\widetilde
X}((y,x),(y',x'))=d_Y(y,y')+d_X(x,x')$.
We let $\cB_Y$ denote the Borel $ \sigma$-algebras of $Y$ identified with a 
sub-$\sigma$-algebra of $\cB_{\widetilde X}$ in the obvious way, and we wish to study
the
relative entropy $\h_{\widetilde \mu}(\widetilde T|\cB_Y)$.

The eigenspace $\KK_j$ is expanded by $T$ if $|\zeta_j|_j>1$. Let
$V \subset\bAA$ be a sum of expanded eigenspaces, and let
$W^+=W^+(T)$ be the sum of all expanded eigenspaces. Then $V$
induces a foliation of $Y \times X$ by letting the leaf through
$x$ be $F_V(\widetilde x)=\widetilde x+V$ (for $\widetilde
x=(y,a+\Gamma)$ we set $\widetilde x+v=(y,a+v+\Gamma)$).

In the following we need the connection between entropy and
conditional measures \cite{Ledrappier-Young-I},
\cite{Ledrappier-Young-II}, and conditional measures on foliations
\cite[Sect.~4]{Katok-Spatzier},
\cite[Sect.~3]{Lindenstrauss-Quantum}. The former we have to adapt
slightly to our problem, and for the latter we will use the
notation of \cite[Sect.\ 3]{Lindenstrauss-Quantum}.

\begin{definition}\label{def: subordinate}
Let $V$ be as above. A $\sigma$-algebra $\cA$ of Borel subsets of $\widetilde X$ is {\em subordinate to
$V$} if $\cA$ is countably generated, for every $\widetilde x \in \widetilde X$
the atom $[\widetilde x]_\cA$ of $\widetilde x$ with respect to $\cA$ is contained
in the leaf $\widetilde x+V$, and for a.e.\ $\widetilde x$
\[
\widetilde x+B_\epsilon^V\subseteq [\widetilde x]_\cA\subseteq \widetilde x+B_\rho^V\mbox{ for some }
\epsilon>0\mbox{ and }\rho>0.
\]
A $\sigma$-algebra $\cA$ is {\em increasing} (with respect to $\widetilde T$) if
$\widetilde T\cA \subset\cA$.
\end{definition}


The conditional measures for the foliation $F_V$ can be
characterized in terms of $\sigma$-algebras subordinate to $V$;
see \cite[Thm.\ 3.6]{Lindenstrauss-Quantum}. Let $\mathcal{M} _
\infty (V)$ denote the space of locally finite Borel measures on
$V$ equipped with the weakest topology for which $\mu \mapsto \int
f d \mu$ is continuous for every $f \in C_c(V)$. For any $v \in V$
let  $+_v$ denote the map $w \mapsto w+v$. For two measures
$\nu,\nu'$ we write $\nu\propto\nu'$ if there exists $c>0$ with
$\nu=c\nu'$.

\begin{proposition}\label{prop: cond-meas}
There exists a Borel measurable map $\widetilde x \mapsto \widetilde \mu _ {\widetilde x,V}$ from $\widetilde X \to \mathcal{M} _ \infty (V)$ with the following properties:
\begin{enumerate}
\item There is a set $N_0$ of zero measure so that for every
$\widetilde x \in \widetilde X$ and $v \in V$ for which
$\widetilde x, \widetilde x +v \not\in N_0$, \label{item:
compatibility} $\widetilde \mu _ {\widetilde x,V} \propto (+_v)_*
\widetilde \mu _ {\widetilde x + v,V}$. \item For a.e. $\widetilde
x$ and $r>0$, $\widetilde \mu _ {\widetilde x,V}(B_r^V)>0$. \item
If $\cC$ is a $\sigma$-algebra subordinate to $V$ with conditional
measures ${\widetilde \mu}_{\widetilde x}^\cC$, then there is a
 Borel measurable function $c_V(\widetilde x, \cC)>0$ so that for a.e.
$\widetilde x$,  for all Borel $B \subset V$ with $\widetilde x+B
\subset [\widetilde x]_\cC$, ${\widetilde \mu}_{\widetilde
x,V}(B)=c_V(\widetilde x, \cC){\widetilde \mu}_{\widetilde
x}^\cC(x+B)$.
\end{enumerate}
\end{proposition}

These properties characterize ${\widetilde \mu}_{\widetilde x,V}$
up to a multiplicative constant a.e. In order to get rid of the
the remaining ambiguity we require that $\widetilde \mu _
{\widetilde x,V}(B_1^V)=1$ for all $x$.


Let $\cP$ be a finite partition of $\widetilde X$, which we identify with the corresponding finite algebra of sets. For any $\epsilon > 0$ let $\partial ^ V _ \epsilon \cP = \left\{ \widetilde x \in \widetilde X: \widetilde x + B _ \epsilon ^ V \not \subset [\widetilde x] _\cP \right\}$.

\begin{lemma}
For any probability measure $\widetilde \mu$ on $\widetilde X$,
there exists a finite
partition $\cP$ of $ \widetilde X$ into arbitrarily small sets such that for some fixed $C$, for
every $\epsilon>0$
\begin{equation}\label{eq: bound-on-boundary}
\mu\bigl(\partial _ \epsilon ^ V\cP\bigr) 0$ sufficiently small. Let $\cP$ be a finite partition
satisfying \eqref{eq: bound-on-boundary} so that the diameter of
every atom of $\cP$ is at most $s$, and let $\cC_V$ be as in
\S\ref{sec: entropy-conditional}. For any $N \geq 1$, let $E_N$ be
the set of $\widetilde x \in \widetilde X$ for which the atom of $
\widetilde x$ with respect to $\cC_V = \cP^{[0,N)}\vee\cC_V^N$ is
{\bf not} equal to the atom of $ \widetilde x$ with respect to
$\cA=\cP^{[0,M)}\vee\cC_V^N$ for $M=\lceil \kappa N \rceil$. Then
$ \widetilde \mu (E _ N) < C \exp (-\rho N)$ for some $C, \rho>0$.
\end{lemma}

In the proof of Lemma \ref{lem: nice-atoms} we will study the
atoms $[\widetilde x]_\cA$ more closely. By definition $\cA$ is
subordinate to $V$ so $[\widetilde x]_\cA$ is a `bounded subset'
of $\widetilde x+V$, but a priori this bound is not known.

\begin{proof}
Let $r=\inf_{a \in \Gamma \setminus \{ 0 \}}\| a \|$, and take $s<1/4$ to be small enough so that
\begin{equation} \label{definition of s}
\| T_\bAA^{-1}v \|+s< r \qquad \text{for every $\|v \| \leq s$}.
\end{equation}
Suppose $\widetilde x,\widetilde x'\in \widetilde X$ are in the
same atom with respect
to $\cA$.
Then by definition of $\cA$,
$d_{\widetilde X}(\widetilde T^{-i}\widetilde x,\widetilde T^{-i}\widetilde x')\leq s$ for $i=0,\ldots,M-1$.
Since $\cB_Y \subset\cA$, we can take $w \in\bAA$ with $\widetilde x'=\widetilde x+w$.
Then $\| T_\bAA^{-i}w-a_i \| \leq s$ for some $a_i \in \Gamma$ and
$i=0,\ldots, M-1$. By an appropriate choice of $w$ we can assume that $a_0=0$.
Applying \eqref{definition of s} to $W$ we see that
$\| a_1 \| \leq \| T_\bAA^{-1}w \|+\| a_1-T_\bAA^{-1}w \|0$, as claimed.
\end{proof}


\begin{proof}[Proof of Theorem \ref{thm: bound-contribution}]
We have $\cC_V=\cP \vee \widetilde T\cC_V$, and so
\begin{multline}\label{eq: split}
\h_{\widetilde \mu}(\widetilde T,V)=\frac{1}{N}\Hh_{\widetilde \mu}\bigl(\cC_V|\widetilde T^N\cC_V\bigr)=
\frac{1}{N}\Hh_{\widetilde \mu}\bigl(\cP^{[0,N)}\big|\cC_V^N\bigr)\\=
\frac{1}{N}\Hh_{\widetilde \mu}\bigl(\cP^{[0,M)}\big|\cC_V^N\bigr)+
\frac{1}{N}\Hh_{\widetilde \mu}\bigl(\cP^{[M,N)}\big|\cP^{[0,M)}\vee \cC_V^N\bigr).
\end{multline}
Clearly
\begin{equation} \label{eq: first part}
\frac{1}{N}\Hh_{\widetilde \mu}\bigl(\cP^{[0,M)}\big|\cC_V^N\bigr)
\leq \frac{M}{N}\frac{1}{M}\Hh_{\widetilde \mu}\bigl(\cP^{[0,M)}\big|\cB_Y\bigr),
\end{equation}
and the last expression tends to $\kappa \h_{\widetilde \mu}(\widetilde T|\cB_Y)$ for $N \rightarrow \infty$.
For the proof of the theorem we need to show that the second expression
on the right hand side of \eqref{eq: split} tends to zero.

Let $\cA$ and $E_N$ be as in Lemma \ref{lem: nice-atoms}. We wish
to estimate
\[
\Hh_{\widetilde \mu}\bigl(\cP^{[M,N)}|\cA\bigr)=\int\Ih_{\widetilde \mu}\bigr(\cP^{[M,N)}|\cA\bigr)\operatorname{d}\!{\widetilde \mu}
=-\int \log{\widetilde \mu}_{\widetilde x}^{\cA}\bigl([\widetilde x]_{\cP^{[M,N)}}\bigr)\operatorname{d}\!{\widetilde \mu}(\widetilde x).
\]
The entropy of a partition is less than the logarithm of the
cardinality of the partition. Applying this for ${\widetilde
\mu}_{\widetilde x}^\cA$ gives the estimate
\begin{equation}
\label{eq: cases}
h(\widetilde x)=-\int_{[\widetilde x]_\cA}\log{\widetilde \mu}_{\widetilde x}^{\cA}\bigl([\widetilde x]_{\cP^{[M,N)}}\bigr)
\operatorname{d}\!{\widetilde \mu}_{\widetilde x}^\cA \leq
\begin{cases}
(N-M)\log|\cP| & \text{if $\widetilde x \in E _ N$}, \\
0& \text{otherwise}.
\end{cases}
\end{equation}
Integrating \eqref{eq: cases} and applying Lemma~\ref{lem: nice-atoms}, we get
\begin{equation} \label{eq: second part}
\tfrac {1 }{ N}  \Hh_{\widetilde \mu}\bigl(\cP^{[M,N)}|\cA\bigr)=
\tfrac {1 }{ N}  \int h(\widetilde x)\operatorname{d}\!{\widetilde \mu}  \leq \tfrac {N - M }{ N} \log|\cP| {\widetilde \mu}(E_N) \to 0 \text{ as $N \to \infty$}.
\end{equation}
Combining \eqref{eq: first part} with \eqref{eq: second part} we
get \eqref{eq: entropy inequality}.
\end{proof}





\section{The entropy function, and coarse Lyapunov foliations}
\label{sec: entropy-function}

We now return to $\zd$-actions, and establish the following identity regarding the relation between the entropies of individual elements of the action. This identity is central to our approach.

\begin{theorem}\label{thm: shape}
Let $\alpha$ be an irreducible $\zd$-action on a solenoid $X$. Let
$\mu$ be an $\alpha$-invariant measure and let $\cA \subset\cB_X$
be an $\alpha$-invariant $\sigma$-algebra. Then there exists a
constant $s_{\mu,\cA}$ with
$\h_\mu(\alpha^\n|\cA)=s_{\mu,\cA}\h_\lambda(\alpha^\n)$ for every
$\n \in\zd$.
\end{theorem}

To see how this relates to the last sections, let $\beta$ be a
continuous $\zd$-action on a compact space $Y$ which is measurably
isomorphic via $\phi$ to the factor of $\alpha$ induced by $\cA$
(so that $\phi:X \rightarrow Y$ is $\cA$-measurable and
$\phi \circ \alpha^\n=\beta^\n \circ \phi$ a.e.\ for every
$\n \in\zd$). Let $\widetilde \alpha$ be the product action on
$\widetilde X=Y \times X$, equipped with the measure $\widetilde \mu=(\phi \times \Id)_*\mu$, so that $\h_\mu(\alpha^\n|\cA)=\h_{\widetilde \mu}(\widetilde \alpha^\n|\cB_Y)$.

For every eigenspace $\KK_j$ the corresponding {\em Lyapunov
vector} is the linear functional defined by
$\bv_j(\n)=\log|\zeta_j(\n)|_j$. For any non-zero linear function
$\bw$ the subspace $V_\bw=\sum_{\bv_j \in \RR^+\bw}\KK_j$ is a {\em
coarse Lyapunov subspace}. The coarse Lyapunov subspaces are the
biggest sums of eigenspaces which are as a whole contracted,
expanded or isometric for every element $\alpha^\n$ of the action.
This is reflected by the entropy contribution of a coarse Lyapunov
subspace.


\begin{lemma}\label{lem: linear-contribution}
Let $V=V _ {\mathbf w}$ be a nontrivial coarse Lyapunov subspace.
Then there exists some $s_{\widetilde \mu }(V ) \geq 0$ with
$\h_{\widetilde \mu}(\widetilde \alpha^\n,V)=
s_{\widetilde \mu}(V)\h_\lambda(\alpha^\n,V)$ for all $\n \in\zd$.
\end{lemma}

Indeed, if $\mathbf w (\mathbf n) \leq 0$, i.e. $\alpha ^ \mathbf n$ does not expand $V$, Lemma~\ref{lem: linear-contribution} is satisfied trivially. It is also clear that for every $\mathbf n$ and $N$,
\begin{equation} \label{eq: N-times}
\h_{\widetilde \mu}(\widetilde \alpha^{N\n},V)= N \h_{\widetilde
\mu}(\widetilde \alpha^{\n},V) .\end{equation} The interpretation
of $\h_{\widetilde \mu}(\widetilde \alpha^\n,V) $ in terms of the
volume growth of suitably scaled boxes given by
Proposition~\ref{prop: contribution} implies that if
$\bw(\n)\geq\bw(\bm)>0$, then $\h_{\widetilde \mu}(\widetilde
\alpha^{\n},V) \geq \h_{\widetilde \mu}(\widetilde
\alpha^{\bm},V)$; in conjunction with \eqref{eq: N-times} we get
that $\h_{\widetilde \mu}(\widetilde \alpha^\n, V)=c
\max(0,\bw(\n))$ for some constant $c$ depending on $\tilde \mu$,
$V$ and $\alpha$; since $\h_{\lambda}(\widetilde \alpha^\n, V)$
is given by a similar formula, we get the desired identity.


\begin{proposition}\label{prop: product-means-sum}
Let $W^+$ be the sum of all expanding eigenspaces for $\alpha^\n$
and let $W^+=V_1+\dots+V_e$ be its decomposition into coarse Lyapunov
subspaces. Then
$\h_{\widetilde \mu}(\widetilde \alpha^\n|\cB_Y)=\h_{\widetilde \mu}(\widetilde \alpha^\n,W^+)=
\h_{\widetilde \mu}(\widetilde \alpha^\n,V_1)+\cdots+\h_{\widetilde \mu}(\widetilde \alpha^\n,V_e)$.
\end{proposition}

We give no details here, but mention the main reason: ${\widetilde
\mu}_{\widetilde x,W^+}$ is the product measure of the
conditionals ${\widetilde \mu}_{\widetilde x,V_j}$ for
$j=1,\ldots,e$, which  together with Proposition \ref{prop:
contribution} implies the above proposition. The product structure
of the conditionals appeared first in a different context in
\cite{Einsiedler-Katok}; see also \cite[Sect.\
6]{Lindenstrauss-Quantum}. Proposition~\ref{prop:
product-means-sum} is related to a more general result regarding
commuting diffeomorphisms by Hu
\cite[Thm.~B]{Hu-commuting-diffeomorphisms}.

Lemma \ref{lem: linear-contribution} and Proposition \ref{prop:
product-means-sum} show that Theorem \ref{thm: shape} is
equivalent to the next lemma.

\begin{lemma}\label{lem: mult-factor-const}
If $\alpha$ is irreducible, there exists some $s_{\mu,\cA}$
such that $s_{\widetilde \mu}(V)=s_{\mu,\cA}$
for all coarse Lyapunov subspaces $V$.
\end{lemma}

\begin{proof} Since by definition every coarse Lyapunov subspace
is expanded by some $\alpha^\n$, there exists $T=\alpha^\n$ such
that no coarse Lyapunov subspace is isometric for $T$. Suppose
$V_1,\ldots,V_e$ are the expanded Lyapunov subspaces and
$V_{e+1},\ldots,V_f$ the contracted ones. We apply Theorem
\ref{thm: bound-contribution} for $T$ and some $V_j$ with $j \leq
e$. Note that the denominator and the numerator of the fraction in
this theorem are exactly $\h_\lambda(T)$ and $\h_\lambda(T,V_j)$.
Therefore $\h_{\widetilde \mu}(\widetilde T,V_j)\leq s
\h_\lambda(T,V_j)$ with $s=\h_\mu(T|\cA)/\h_\lambda(T)$. However,
by Proposition \ref{prop: product-means-sum} the sum over these
inequalities for $j \leq e$ gives the inequality
$\h_\mu(T|\cA)\leq s \h_\lambda(T)=\h_\mu(T|\cA)$. This shows,
that all the inequalities have to be equalities, i.e.
$\h_{\widetilde \mu}(\widetilde T,V_j)= s \h_\lambda(T,V_j)$.
Since $\h_\lambda(T,V_j)>0$, we conclude that $s_{\widetilde
\mu}(V_j)=s$ for $j=1,\ldots,e$.

Using $T^{-1}$ in the above argument does not change $s$,
and proves $s_{\widetilde \mu}(V_j)=s$ for $j=e+1,\ldots,f$, thus concluding the proof
of Theorem \ref{thm: shape}.
\end{proof}











\section{Outline of the proof of Theorem
\ref{thm: easy-case}}\label{sec: outline-easy}

Once Theorem~\ref{thm: shape} is proved, Theorem~\ref{thm:
easy-case} can be proved in a way similar to Rudolph's proof of
Theorem~\ref{thm: Rudolph}. We give a variant of this method,
which avoids the need to explicitly employ the suspension
construction of Katok and Spatzier.

Let $\alpha$ be a totally irreducible (hence arithmetic) $\ZZ ^
d$-action, and let  $\alpha_\bAA$ be the corresponding $\zd$-action
on the covering space $\bAA$.

\begin{lemma}
$\alpha$  is {\bf not} virtually cyclic if and only if there exist
at least two linear independent Lyapunov vectors.
\end{lemma}

\begin{proof}[Sketch of the proof] If $\alpha$ has no two linearly
independent Lyapunov vectors, then $H=\bigcap_j\ker\bv_j$ is a
hyperplane. Suppose $\n\in\zd$ is close to $H$, i.e.\ satisfies
 $\bv_j(\n)\in (-\epsilon,\epsilon)$ for $j=1,\ldots,m$. Recall that the
 numbers $\zeta_{j,\n}\in\KK_j$ in Theorem \ref{prop: localprod}
 are the images of a single algebraic number $\zeta$.
 For small enough $\epsilon$ it follows that $\zeta$ must be an
 algebraic unit whose real and complex embeddings have absolute
 value close to one. It follows from Dirichlet's unit theorem that
 $\zeta$ must be a unit root and $\n\in H$. Thus $\alpha$ is
 virtually cyclic.
\end{proof}

Let $V$ and $W$ be the coarse Lyapunov subspaces corresponding to
two linearly independent Lyapunov vectors. Then $V+W$ is
contracted by some $\alpha^\n$, $\n \in\zd$. As in the discussion
following Proposition \ref{prop: product-means-sum}, the
conditional measure $\mu_{x,V+W}=\mu_{x,V}\times \mu_{x,W}$ is a
product measure a.s., which implies the following.

\begin{lemma}\label{lem: cond-constant}
There exists a null set $N \subset X$ such that
$\mu_{x,V}=\mu_{x',V}$ if $x,x'\notin N$ and $x'\in x+W$.
\end{lemma}

Let $\cA'$ be the smallest $\sigma$-algebra with respect to which $x \mapsto
\mu_{x,V} $ is measurable. Then invariance of $\mu$ under $\alpha$ implies that
$\mu_{\alpha^\n x,V} \propto (\alpha_\bAA^\n)_*\mu_{x,V}$ a.s.; indeed,
$\mu_{\alpha^\n x,V} = \frac { (\alpha_\bAA^\n)_*\mu_{x,V} } { 
(\alpha_\bAA^\n)_*\mu_{x,V} (B^{V}_1)}$, and so $\cA'$ is $\alpha$-invariant.
Using Lemma~\ref{lem: cond-constant} one can find a countably generated
$\alpha$-invariant $\sigma$-algebra $\cA$ which is equal modulo $\mu$ to $\cA'$
so that every $A \in \mathcal{A}$ is a union of full $F_{W}$-leaves.

As in the preceding section, let $\beta$ be a continuous
realization of this factor, and recall that the action $\alpha$ on
$(X,\mu)$ is isomorphic to $\widetilde \alpha$ on $(\widetilde
X,\widetilde \mu)$ via the map $\Phi=\phi \times \Id$.


\begin{proposition}\label{prop: s-again-constant}
$\mu_{x,W}=\widetilde \mu_{\Phi(x),W}$ for a.e.\ $x \in X$.
\end{proposition}

\begin{proof}
Suppose $\cC \subset \cB_X$ is a $\sigma$-algebra subordinate to $W$. Since for every $ x$,
$[ x]_\cC \subset x+W$, $[x]_\cC \subset [x]_\cA$ and so $\cA \subset\cC$. Since $\phi(x)$ is constant on atoms of $\cC$, it follows that there is a $\sigma$-algebra $\widetilde\cC$ subordinate to $W$ in $\widetilde X$ such that $\cC=\Phi^{-1}\widetilde\cC$.
Since $\Phi$ is a measurable isomorphism,
the conditionals satisfy
$\Phi_*\mu_x^\cC=\mu_{\Phi(x)}^{\widetilde\cC}$. It now follows from
Proposition \ref{prop: cond-meas} that indeed
$\mu_{x,W}=\widetilde \mu_{\Phi(x),W}$ a.s.
\end{proof}


\begin{proposition}\label{prop: final-step}
Suppose the entropy $\h_\mu(\alpha^\n)$ is {\bf positive} for some $\n \in\zd$.
Then for $\mu$ a.e. $x$,  there is a nonzero $v \in V$ so that
$\mu_{x,V} \propto (+_v)_* \mu_{x,V}$.
\end{proposition}


\begin{proof}
Let $N_0$ be the set from Proposition~\ref{prop:
cond-meas}\eqref{item: compatibility} applied to $X$. By
definition of $\cA'$ and since $\cA = \cA'$  (mod $\mu$) there is
a set $N'$  of measure zero (which we may as well assume contains
$N_0$) such that if $x,x' \not \in N'$ and $[x]_\cA=[x']_\cA$
(equivalently, $\phi (x) = \phi (x ')$), then $\mu _ {x,V}= \mu _
{x',V}$. Let $\widetilde N = \Phi (N')$.

Let $s_\mu (V)$ and $s_\mu (W)$ be as in Lemma~\ref{lem: mult-factor-const}. Then
$s_\mu(V)=s_\mu(W)>0$ and similarly $s_{\widetilde \mu}(V)=s_{\widetilde 
\mu}(W)$. Since $s_\mu(W)$ and $s_{\widetilde \mu}(W)$ are determined by 
the conditional measures $\mu _ {x, W}$ and $\widetilde 
\mu_{\widetilde x,W}$, respectively,
by Proposition \ref{prop: s-again-constant}
$s_\mu(W)=s_{\widetilde \mu}(W)$. We conclude that
$s_\mu(V)=s_{\widetilde \mu}(V)>0$, and in particular $\widetilde \mu_{\widetilde x,V} (\left\{ 0 \right\})=0$ a.s.

Let $\widetilde\cC$ be any $V$-subordinate $\sigma$-algebra. For
a.e.\ $x$ (in particular for $x\notin N'$), $\widetilde \mu_{\Phi
(x)}^\cC (\widetilde N )=0$, so a.s. there is a nonzero $v \in V$
such that $x,x+v \not\in N'$ but $\Phi (x+v) \in [\Phi(x)]_\cC$. In
particular, $\phi (x) = \phi (x ')$, which implies that $\mu _
{x,V} = \mu _ {x+v,V}$. Since $x, x+v \not\in N_0$,
Proposition~\ref{prop: cond-meas} gives that $\mu _ {x,V} \propto
(+v)_* \mu _ {x+v,V}$ and so $ \mu _ {x,V} \propto (+v)_* \mu _
{x,V}$.
\end{proof}

From Proposition \ref{prop: final-step} one can conclude with
standard techniques (see for instance
\cite[Lemma~3.4]{Kalinin-Katok} or
\cite[Lemma~5.6]{Katok-Spatzier}) that $\mu_{x,V}$ is actually
translation invariant in the strict sense under some nonzero
element of $V$ a.s.  Note that so far we have only used that
$\alpha$ is irreducible. Using the total irreducibility of
$\alpha$ it is not hard to conclude at this stage that $\mu$ is
Haar measure on $X$.


\pagebreak


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\end{document}