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\controldates{23-AUG-2004,23-AUG-2004,23-AUG-2004,23-AUG-2004}
 
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\dateposted{August 26, 2004}
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\PII{S 1079-6762(04)00133-7}
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\begin{document}
\title{Dimension product structure of hyperbolic sets}
\author{Boris Hasselblatt}
\address{Department of Mathematics,
Tufts University,
Medford, MA 02155}
\email{bhasselb@tufts.edu}

\author{J\"org Schmeling}
\address{Lund Institute of Technology, Lunds Universitet, 
Box 118,  SE-22100 Lund, Sweden}
\email{Jorg.Schmeling@math.lth.se}

\date{June 8, 2004}
\commby{Svetlana Katok}
\subjclass[2000]{Primary 37D10; Secondary 37C35}
\keywords{Hyperbolic set, fractal dimension, Hausdorff dimension,
Eckmann-Ruelle conjecture, holonomies, Lipschitz continuity, product structure} 

\begin{abstract}
We conjecture that the fractal dimension of hyperbolic sets can be computed
by adding those of their stable and unstable slices. This would facilitate
substantial progress in the calculation or estimation of these dimensions,
which are related in deep ways to dynamical properties. We prove the
conjecture in a model case of Smale solenoids.
\end{abstract}

\maketitle

\section{Introduction}
Since the 1980s pictures of various ``fractal'' sets have become part of
our popular culture, and their fractal dimension has been associated with
the smoothness or thickness of these sets. In recent years the rigorous and
systematic study of these dimensions (see Section \ref{Sdiment} and
\cite{Falconer}) and their relation to characteristic properties of the
dynamical systems that give rise to these sets has made substantial
progress. Much of this progress was initiated by physicists, but concerns
of this type have become an important part of mainstream mathematics
\cite{PesinBook}. While this has had an effect on fields such as number
theory, the present work is about invariant sets in hyperbolic dynamical
systems.
\subsection{Invariant sets in hyperbolic dynamical systems}
In this paper a dynamical system is a diffeomorphism or an embedding of a
manifold into itself, and the principal interest in the theory is to study
the long-term behavior of points under repeated iteration of the
map. Often, it is of interest to concentrate on particular invariant sets,
and here we concentrate on those that are hyperbolic (see 
Definition~\ref{D:hypset}),
i.e., at each point of the set the tangent space is a direct sum of two
subspaces that are contracted and expanded, respectively, by the
differential of the map. Simple examples are given by the \emph{horseshoe}
and the ``\emph{paper clip}'', where the map (of the plane or a disk)
compresses a rectangle vertically, expands it horizontally, folds it as in
Figure
\ref{Fhorseshoe} and places it over the original rectangle. It is not hard
to see that the invariant set is a planar Cantor set; if the map is linear
and respects the coordinate directions on the pieces that fall across the
original rectangle, then the invariant set is a Cartesian product of two
Cantor sets.
In this context it has become interesting
to obtain (or at least estimate) the fractal dimension (Section
\ref{Sdiment}) of a
hyperbolic invariant set from information about the dynamics on that
set. This is often difficult to do, and most progress appears to have been
made when the dynamics on the hyperbolic set is strictly expanding or
contracting. The general case combines both contraction and expansion, but
topologically, these sets have a (local) product structure, that is, they
can locally be represented as products of a representative contracting
slice and a representative expanding slice. The techniques that have helped
with contracting or expanding hyperbolic sets have been adapted to dealing
with these slices, and therefore great progress would be achieved by
establishing that the dimension of the hyperbolic set is the sum of those
of the contracting (``stable'') and expanding (``unstable'') slices, even
though the known product structure is only topological and certainly 
cannot be achieved with Lipschitz charts \cite{HasselblattWilkinson}. Charts
that are not Lipschitz-continuous can alter fractal dimensions (by a factor
given by the H\"older exponent of the chart).

\begin{figure}[h]
\includegraphics[scale=.97]{era133el-fig-1}
\caption{The horseshoe (with second iterate) and the paper clip.}
\flabel{Fhorseshoe}
\end{figure}

\subsection{Dimension product structure}
Even though this obstacle is real, we conjecture that the desired
\emph{dimension product structure} still prevails, at least generically, or
under mild additional hypotheses. It is clear that establishing this will
be difficult, and there are examples that suggest the possible failure of
such a result in some situations. But we demonstrate in a model example how
the essential difficulties can be addressed (Theorem \ref{thmmain},
p.~\pageref{thmmain} and Theorem \ref{thmmain3}, p.~\pageref{thmmain3}). The
central insight is that while \cite{HasselblattWilkinson} exhibits
\emph{open} sets of symplectic Anosov diffeomorphisms for which the local
product charts are non-Lipschitz \emph{on a large set}, the problem is in
an essential way due to the assumption of the Anosov property (that the
entire manifold is a hyperbolic set). In this case, of course, little needs
to be done in order to determine the fractal dimension of the hyperbolic
set, since it is given at the outset. In our work we point to a subtle
aspect of the local geometry of these sets that is responsible for a
reversal of the situation in the case of fractal hyperbolic sets. For these
we show a way of establishing Lipschitz continuity of local product charts
off a set that is small enough to be discarded without affecting the
dimension of the hyperbolic set or of its stable or unstable slices. This
example is different from even nonlinear versions of the maps in Figure
\ref{Fhorseshoe} because in planar situations we always have a dimension
product structure as remarked in Subsection \ref{SSregularity}.

We should point out that a dimension product structure of invariant
measures was established some time ago \cite{BarreiraPesinSchmeling} in the
course of proving the Eckmann--Ruelle conjecture.
\begin{conjecture}
The fractal dimension of a hyperbolic set is (at least generically or under
mild hypotheses) the sum of those of its stable and unstable slices, where
``fractal'' can mean either Hausdorff or upper box dimension.
\end{conjecture}
Here we describe our proof of this conjecture for a class of solenoids \cite{HasselblattSchmeling}.
Even in this situation 
the dimension decomposition has been an open problem for at least two
decades. Bothe \cite{B} gave conditions of a transversality type under
which he could show (assuming estimates on the contraction and expansion
rates) that all stable slices have the same dimension, and mild conditions
on the contraction and expansion rates guarantee $C^1$-genericity of this
transversality condition (see Remark \ref{REMBothe}).
\subsection{Hyperbolic sets}
We now introduce hyperbolic sets, their stable and unstable slices as well
as the holonomies between them. We begin with the invariant laminations.
\begin{definition}[\cite{KatokHasselblatt}]\dlabel{D:hypset}
Let $M$ be a manifold, $U\subset M$ open, $f\colon U\to M$ a smooth
embedding, $\Lambda\subset M$ a compact $f$-invariant set. Then $\Lambda$ is
said to be a \emph{hyperbolic set} if the tangent bundle $T_\Lambda M$ over
$\Lambda$ splits as $T_\Lambda M=E^u\oplus E^s$ in such a way that there are
constants $C>0,\ 0<\lambda<1<\eta$ such that
$
\|Df^{-n}\rest{E^u}\|\le C\eta^{-n}$ and $\|Df^n\rest{E^s}\|\le C\lambda^n\text{ for }n\in\N$.
\end{definition}
The subbundles are then invariant and continuous and have smooth integral
manifolds $W^u$ and $W^s$ that are coherent in that $q\in W^u(p)\Rightarrow
W^u(q)=W^u(p)$. If $\Lambda=M$ then $W^u$ and $W^s$ define laminations
(continuous foliations with smooth leaves) and $f$ is said to be an Anosov
diffeomorphism. Even if $\Lambda\neq M$ (the case on which we concentrate)
we use the term laminations.

If $A,B$ are two nearby embedded disks transverse to $W^u$ then there is a
\emph{holonomy map} defined on a (possibly empty) subset of $A\cap\Lambda$
by $p\mapsto W^u_{\text{loc}}(p)\cap B$ whenever this makes sense. Here
$W^u_{\text{loc}}(p)$ is a small disk in $W^u(p)$ that contains $p$. In
other words, we move from $A$ to $B$ along unstable leaves. If
$W^u_{\text{loc}}(p)\cap B\neq\varnothing$ then the holonomy is defined on
the intersection of a neighborhood of $p$ with $A\cap\Lambda$. These
holonomies are always continuous (this is easy to see), and, in fact,
H\"older continuous (\ie $d(f(x),f(y))\le Cd(x,y)^\alpha$ for some
$\alpha>0$). There would be no need for the present paper if it were known
that these holonomies are always Lipschitz continuous (that is, H\"older
continuous with exponent 1). Indeed, that this Lipschitz property is not
obviously true might be said to have held back the study of the smooth
ergodic theory of geodesic flows and Anosov systems by a quarter century
(between the work of Hopf \cite{Hopf} and that of Anosov--Sinai
\cite{AnosovSinai}). This is but one motivation for studying the
regularity of holonomies and local product charts.
\subsection{Regularity}\sbslabel{SSregularity}
The search for ergodic mechanical systems led to the study of geodesic
flows of negatively curved manifolds in the 1920s and '30s. Eberhard Hopf
\cite{Hopf} established ergodicity of the geodesic flow of surfaces of
negative (and not necessarily constant) curvature and in doing so provided
the first and still only argument to establish ergodicity in wide classes
of hyperbolic dynamical systems. He used in a critical way that geodesic
flows on surfaces have $C^1$ holonomies. There was no progress in higher
dimension until the 1960s, and Anosov brought out the technical issue at
hand: The holonomies need not be $C^1$ in that case, and they may indeed
fail almost everywhere to be Lipschitz continuous \cite{Anosov}. Anosov and
Sinai overcame this difficulty by establishing that the holonomies are
nevertheless absolutely continuous (with respect to the Riemannian measure
on leaves they send null sets to null sets) \cite{AnosovSinai,BrinStuck}, which
suffices for the Hopf argument and has become important also in situations
where volume is not preserved \cite{AnosovSinai,YoungSRB}.

Subtle issues of regularity of the invariant foliations have been studied
since, including in the study of smooth and geometric rigidity (see, \eg
\cite{Hasselblattsurvey}), where rather high degrees of smoothness play a
role. We now outline some of the results of interest that are pertinent to
our desire for a Lipschitz property of the holonomies. A careful
discussion of the relations between  regularity of holonomies and
foliation charts is given in \cite{PughShubWilkinson}. 

If one makes assumptions about particular relations between the various
expansion and contraction rates of the diffeomorphism then one can
guarantee particular H\"older conditions. Sufficiently stringent such
conditions even give $C^1$ holonomies with H\"older derivative, where the
H\"older exponent of the derivative can be arbitrarily close to 1. For the
holonomies between stable or unstable slices in hyperbolic systems
Schmeling and Siegmund-Schultze \cite{SchmSieg} found optimal estimates of
the H\"older exponent in terms of Lyapunov exponents, while Hasselblatt
\cite{thesis} produced analogous estimates for the H\"older exponent of the
subbundles in terms of more uniform rates. It is not necessary for us to
describe the needed conditions, but here are some examples where they
follow from assumptions that are simpler to state. If one of the stable and
unstable laminations is 1-dimensional then the other one is $C^1$; in
particular, in 2-dimensional systems both laminations are $C^1$, which is
the context within which Hopf worked (and the reason planar horseshoes are
not a useful example for our agenda here). Also, if one foliation is
1-dimensional and the dynamical system preserves volume, then both
laminations are $C^1$ (of course, in this case there are no
attractors). Thus, there are open sets of hyperbolic dynamical systems
where the dimension product structure is known, but one would like to know
it in complete generality. We now note that there are significant obstacles
to achieving this.\looseness-1
\subsection{Failure of Lipschitz continuity} 
One of our difficulties is that the regularity results mentioned above have
been shown to be sharp in a strong sense, beginning with the aforementioned
work of Anosov. In at least 3 dimensions the set of maps where the H\"older
exponent obtained by these results is less than 1 is open, and we
established that within this open set those maps that do not have holonomy
maps with a larger H\"older exponent form a residual set. Indeed,
Hasselblatt and Wilkinson found \emph{open} sets of symplectic Anosov maps
with the property that on a \emph{residual set of full measure} (with
respect to any invariant measure) the subbundles are \emph{not} Lipschitz,
and the holonomies are non-Lipschitz a.e.\ with respect to Lebesgue measure
\cite{HasselblattWilkinson}. Earlier, Schmeling found that solenoids often
lack regular holonomies but the set of non-Lipschitz points seemed to be
rather small in the measure sense \cite{Schmeling}.

This means that one should expect (or at least allow for the possibility)
that in a typical hyperbolic dynamical system the invariant laminations
fail to be Lipschitz continuous on a residual set that is conull with
respect to any (interesting) invariant measure.
\subsection{Controlling the set of non-Lipschitz points}
Despite these serious obstacles we endeavor to control the size of the set
of non-Lipschitz points in a manner that is sufficient for purposes of
dimension calculations. We consider solenoids that carry an invariant
smooth hyperbolic coordinate system on the basic set. This implies that
local fast stable manifolds depend smoothly on the points. We show for our
class of maps that deletion of the set NL of non-Lipschitz points does not
diminish the Hausdorff or (upper) box dimension of the hyperbolic set
$\Lambda$. More to the point, we show that \emph{all} stable slices have
the same fractal dimension, and the sum of this and the unstable dimension
(which is 1 here) gives $\dim\Lambda$.\looseness-1

\subsection{The solenoid}\sbslabel{SSsolenoid}
The situation we consider is the Smale solenoid or ``derived from
expanding'' (DE-) map, a smooth realization of the natural extension of the
map $x\mapsto\eta x$ on $\Sb^1$ for $\eta\in\N$ \cite{Smale}. 
\begin{figure}[ht]
\includegraphics[scale=.97]{era133el-fig-2}
\hfill
\caption{The solenoid and local stable cross-sections.}\flabel{fig24}
\end{figure} 

Specifically, let $M\dfn\Sb^1\times\Db$ be the solid torus, where
$\Db=\{v\in\R^2\st|v|<1\}$ carries the product distance $d=d_{\myone} \times
d_{\mytwo}$, and suppose $f\colon M\to M$, $(x,y,z)\mapsto(\eta x,\lambda
y+u(x),\mu z+v(x))$ is a $C^2$ map, where $\mu,\lambda,\eta\colon
M\to\R^+$, $\mu<\lambda<1<\eta$.

Let $\Lambda\dfn\bigcap_{n\in\N}f^n(M)$ be the attractor and denote by NL
the set of non-Lipschitz points of the unstable lamination, \ie the set of
points $p\in\Lambda$ for which there is a $q\in W^u_{\text{loc}}(p)$ such
that the unstable holonomy from $W^s(p)$ to $W^s(q)$ is not Lipschitz at
$p$.
\section{Results}
The natural projection $\pi_x\colon(x,y,z)\mapsto x$ allows us to define
stable slices: For any set $A\subset M$ we write $A_x\dfn(\pi_x\rest
A)^{-1}(\{x\})$. Thus, $W^s(p)=\Lambda_{\pi_x(p)}$. We also often project along
fast-stable leaves using $\pi_{(x,y)}\colon(x,y,z)\mapsto(x,y)$.
\begin{theorem}\tlabel{thmmain}
Consider a $C^2$ map $f\colon M\to M$ as above, and to fix ideas assume 
that $\lambda,\mu,\eta$ are constant and $\eta=2$ as well as
\begin{itemize}
\item[(i)] The slow contraction $\lambda$ is less than $1/2$,
\item[(ii)] The unstable foliation is transverse as illustrated in Figure
  \ref{figure1}: If $p\in\Lambda$, $p'\in W^s(p)$, and $n\in\N$ is such that
  $\pi_x(f^{-i}(p))=\pi_x(f^{-i}(p'))$ for $0\le i\le n$, but not for
  $i=n+1$, then  $\pi_{(x,y)}(W^u_{\mathrm{loc}}(f^{-n}(p)))$ and
  $\pi_{(x,y)}(W^u_{\mathrm{loc}}(f^{-n}(p')))$ are transverse (necessarily
  uniformly so). 
\end{itemize}
Then $\dim\Lambda_x=\dim\Lambda_{x'}$ for all $x,x'\in\Sb^1$, and
$\dim\Lambda=1+\dim\Lambda_x$, where $\dim$ is either Hausdorff or upper
box dimension.
\end{theorem}
\begin{figure}[ht]
\includegraphics[height=1.2in,width=.7\textwidth]{era133el-fig-3.eps}
\caption{Transverse crossings.}\flabel{figure1}
\end{figure}

\begin{remark}\rlabel{REMBothe}
This situation was first considered by Bothe~\cite{B}, who obtained results
on the dimension of $\Lambda$ and $\Lambda_x$. In fact, the analysis in
\cite{B} implies Theorem \ref{thmmain} for the case $\lambda\le1/8$, and Bothe
proved that transversality is generic in $C^1$ when $\lambda<1/4$. Contrary
to Theorem \ref{thmmain}, Bothe does not take the contraction and expansion rates
to be constant. (We do so only in order to keep the notation light.)
\end{remark}
Our second (and more difficult) result does not require the transversality
assumption at all, although we assume analyticity in this case:
\begin{theorem}\tlabel{thmmain3}
Let $M\dfn\Sb^1\times\Db$, $\mu,\lambda\in\R$, $0<\mu<\lambda<1/2$, and let
$f\colon M\to M$, $(x,y,z)\mapsto(2x,\lambda y+u(x),\mu z+v(x))$ be
analytic. Then $\dim\Lambda_x=\dim\Lambda_y$ for all $x,y\in\Sb^1$, and
$\dim\Lambda=1+\dim\Lambda_x$, where $\dim$ is either Hausdorff or upper
box dimension and $\Lambda$ is the attractor.
\end{theorem}
While Theorem \ref{thmmain} has more robust hypotheses, which is interesting
with respect to  the prevalence of a dimension product structure, those of
Theorem \ref{thmmain3} are easily checked in concrete examples.

We expect that dropping the assumption of parallel (in particular, $C^1$)
fast directions immediately leads to technical problems, although these may
not be fundamental.
\section{Methods}\label{Sgeometry}
In this section we give a description of the phenomena that may give rise
to non-Lipschitz points. The discussion here requires only the assumptions
of Subsection \ref{SSsolenoid}, \emph{not} the full strength
of the hypotheses of Theorem \ref{thmmain} or Theorem \ref{thmmain3}.

Suppose $p\in\Lambda$, $q\in W^u_{\text{loc}}(p)$ are such that the holonomy
from $W^s(p)$ to $W^s(q)$ is not Lipschitz continuous at $p$, \ie for all
$C>2$ there is a $p'\in W^s(p)$ arbitrarily near $p$ such that $\Delta q\dfn
d(q,q')>Cd(p,p')=C\Delta p$, where $q'$ is the image of $p'$ under the
holonomy. Let $d_{pq}\dfn d(\pi_x(p),\pi_x(q))$.
\subsection{Pulling the failure of a Lipschitz condition back to unit scale}
We now show that when the ``bend'' in unstable manifolds that gives rise
to a failure of a Lipschitz property at a small scale is pulled back to
unit scale, one observes there a corresponding discrepancy in the
\emph{slow} stable direction.\looseness-1

Take $n\in\N$ such that $\pi_x(f^{-i}(p))=\pi_x(f^{-i}(p'))$ for $0\le i\le
n$, but not for $i=n+1$ (since $\Delta p$ may be small,
$n$ can be arbitrarily large). Then there exist $0C\Delta p$ then $\overbar{\Delta_{\myone} q}>C'\overbar{\Delta_{\myone} p}$, where
$C'\dfn\frac C{1+C_1d_{pq}^\alpha}$ for a moderate constant $C_1$.

This means that after pulling back to unit scale the failure of the
Lipschitz condition shows up precisely as a corresponding discrepancy in the
size of the \emph{slow}-stable components, which also means that these have to be
very small.
\subsection{Principal case distinction---crossings}\sbslabel{SBSeasycase}
The small size of $\overbar{\Delta_{\myone} p}$ combined with the substantial
angle between the projections of the two leaves that is implied by
$\overbar{\Delta_{\myone} q}>C'\overbar{\Delta_{\myone} p}$ suggests that the fast-stable
projections of the local unstable leaves in the scaled-up preimage should
cross rather near $f^{-n}(p)$. Indeed, the transversality assumption in
Theorem \ref{thmmain} implies this: For a point $p\in\text{NL}$, there are
infinitely many $n\in\N$ such that $f^{-n}(p)$ is within $d_{pq}/\eta_n(q)$
of a crossing of fast-stable projections of local unstable leaves of
$f^{-n}(p)$ and $f^{-n}(p')$ for some $p'$ such that
$\pi_x(f^{-i}(p))=\pi_x(f^{-i}(p'))$ for $i\le n$ but not for $i=n+1$.

We show that the set of such $p$ (and hence $\text{NL}$) is benign with
respect to Hausdorff dimension. (In higher dimension the sets of points
with crossings may not be negligible because of the Ledrappier--Young
entropy formula, but that very formula bounds the dimension of fast-stable
leaves.)
\subsection{Absence of crossings}\sbslabel{SBShardcase}
Points $p$ not of the above kind have the property that if $n$ is
sufficiently large then $f^{-n}(p)$ is \emph{never} within
$d_{pq}/\eta_n(q)$ of such a crossing. Absent the transversality condition,
this failure together with simultaneous failure of a Lipschitz condition
imposes additional constraints on the local geometry, which we can control
using analyticity.
\subsection{A sample of the dimension arguments}
As noted above, transversality implies that every $p\in\text{NL}$ has a
subsequence of preimages that accumulate exponentially fast to the set of
crossings. To suggest why one should expect the ``bad'' set of points to
have smaller Hausdorff dimension than $\Lambda$ we outline the proof of the
following result.
\begin{proposition}
Under the assumptions of Theorem \ref{thmmain}, NL carries less entropy than
$\Lambda$.
\end{proposition}
\begin{proof}
The dynamical system in Theorem \ref{thmmain} acts affinely on slices, so a
cross-section $\{x=\const\}$ of $f(M)$ consists of two convex sets in
$\Db$. For some $x$ these must have disjoint projections (this is the only
place where $\eta\equiv2$ is used, and we do not need $\lambda$ and $\mu$
to be constant). Therefore there is a Markov partition such that the
$(x,y)$-projections of some of its elements do not contain any
crossings. The subshift without any of the corresponding symbols has
strictly smaller entropy $h$ than the shift over $f$ obtained from the
Markov partition, whose entropy we denote by $H$ for the moment.\looseness-1

For a sequence that projects to a point $p$ in NL and an $N\in\N$ there is
an $n\ge N$ such that $f^{-n}(p)$ is within $\const\eta^{-n}$ of a crossing
point. This means that for any  $\beta\in(0,1)$ the
next $\beta n$ iterates $f^{-i}(p)$ lie in a Markov rectangle whose
projection contains crossings, \ie after some $e^{Hn}$ possible choices for
the first $n$ entries there are only some $e^{h\beta n}$ choices for the
next $\beta n$ entries. This allows for some $e^{Hn}\cdot e^{h\beta
n}$ initial strings of length $n+\beta n$. Since
$$
\frac{\log(e^{Hn}\cdot e^{h\beta n})}{(1+\beta)n}=\frac{H+\beta h}{1+\beta},
$$
we obtain entropy at most $(H+h)/2\dim_{\text{H}} Y\ .
\end{cases}
\end{gather*}
\end{definition}
Standard arguments give $\dim_{\text{H}} Y\leq\underline\dim_{\text{B}} Y\leq
\overline\dim_{\text{B}} Y$, and these inequalities may be strict.

The box dimension can also be defined in terms of covering sums, as is the
Hausdorff dimension, with the only change being that the covering balls all
have the same size. In order to estimate the box dimension, it suffices
that the sizes of the covering balls tend to $0$ along a geometric
sequence.


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\end{document}