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\def\a{\alpha}
\def\la{\lambda}
\def\La{\Lambda}
\def\diam{\operatorname {diam}}
\def\ov{\overline}
\def\ve{\varepsilon}
\def\loc{\text{ loc}}

\leftheadtext{Ch. Bonatti, L. J. D\'\i az and M. Viana}
\rightheadtext{Discontinuity of Hausdorff Dimension}

\topmatter   
\title
DISCONTINUITY OF THE HAUSDORFF DIMENSION OF  HYPERBOLIC SETS
\endtitle
\author
Ch. Bonatti, L. J. D\'\i az and M. Viana
\footnote"*"{L.J.D. and M.V. are partially
             supported by CNPq, Brazil.\newline}
\endauthor
\abstract
We prove that  the Hausdorff dimension of a hyperbolic
basic set may  vary discontinuously with the dynamics
if the dimension of the ambient manifold is bigger than
two. This loss of continuity is associated to the
occurrence of intersections between the stable (resp. 
unstable) manifold and the strong unstable (resp. strong
stable) manifold of some periodic point.
\endabstract
\endtopmatter

\heading  1. Introduction \endheading
We begin by recalling a few basic facts concerning
hyperbolicity and Hausdorff dimension, see e.g. \cite{HP}
and \cite{F} for details.

Given a $\Cal C^r$-diffeomorphism $f\colon M\to M$,
$1\le r \le \infty$, we say that a compact $f$-invariant
set $\La_f\subset M$ is {\it hyperbolic\/} if there is a splitting 
$E^s_{\La_f} \oplus E^u_{\La_f}$ of the tangent bundle $T_{\La_f}M$
and there are constants $C >0$, $\la<1$, such that
$$
|| D_x f^{n} (v)|| \le C \la^n ||v|| \text{\ \ \ and\ \ \ }
|| D_x f^{-n} (w)|| \le C \la^n ||w||
$$
for all $v\in E^s_x$, $w\in E^u_x$, $x \in \La_f$
and $n\ge 1$.
We call  $\La_f$  a {\it basic set\/} if it is  
{\it transitive\/} (i.e. there is  a dense orbit of $f$ in $\La_f$)
and {\it isolated\/} (i.e. $\La_f =\cap_{i\in \Bbb Z} f^i (U)$
for some neighbourhood $U$ of $\La_f$) and contains a dense 
subset of periodic points. 
An important feature of basic sets  is their persistence
under perturbation of the dynamics: there is a neighbourhood
$\Cal V^r$ of $f$ in  Diff$^r(M)$ such that for all $g\in\Cal V^r$
$$
\La_g = \bigcap_{i \in \Bbb Z} g^i (U)
\text{\ \ \ (the\ {\it continuation}\ of\ }\La_f\text{)}
$$
is a basic set of $g$ and, moreover, $g|\La_g$ is {\it conjugate\/}
to $f|\La_f$: there is a homeomorphism $h\colon\La_f\to\La_g$ such
that $(g|\La_g)\circ h=h\circ(f|\La_f)$.

Given $\a>0$, the {\it Hausdorff $\a$-measure\/} of a compact
metric space $X$ is
$$ m_\a (X) = \lim_{\ve \to 0^+} \inf 
\sum_{ U \in \Cal  U} \diam (U)^\a,
$$
where the infimum is taken over all finite coverings 
$\Cal U$ of $X$ by sets with diameter less than $\ve$.
Then there is a unique $d\in[0,\infty]$ such that
$m_\a (X) = \infty$ if $\a < d$ and 
$m_\a (X) = 0$ if $\a > d$. 
One calls $d$ the {\it Hausdorff dimension\/} of $X$ and writes
$d =HD (X)$. Here we make use of the (easy) fact that the
Hausdorff dimension  is non-increasing under Lipschitz maps.

The Hausdorff dimension of basic sets of surface
diffeomorphisms depends in a quite regular way on the
diffeomorphism: the function
$$
\Cal V^r\ni g \mapsto HD(\La_g)
$$
is continuous, McCluskey-Manning \cite{MM} (see also \cite{PV1}),
and even of class $\Cal C^{r-1}$, Ma\~n\'e \cite{M}.

The purpose of this article is to prove that such a regularity
of the Hausdorff dimension breaks down in higher dimensional
manifolds:

\proclaim{Theorem} Suppose $M$ is an $m$-manifold, $m\ge 3$.
Then, for any $1\le r\le \infty$, the function
$$
\Cal V^r\ni g \mapsto HD(\La_g)
$$
introduced above is, in general, not continuous.
\endproclaim

Let us give a brief sketch of the proof of this result,
details being provided
in the next section. Clearly, it is no restriction to consider
$M=\Bbb R^{n+1}$, $n\ge 2$, and we do so from now on.
We begin by taking a $\Cal C^r$-diffeomorphism $F$ of
$\Bbb R^n$ with a basic set $\Sigma$ (a {\it horseshoe\/})
such that $F|\Sigma$ is conjugate to the full shift on two
symbols. We assume that $HD (\Sigma)< 1$. Next, we let $1<\la <2$
and consider the diffeomorphism
$$
f\,\colon\, \Bbb R^n \times \Bbb R
\to \Bbb R^n \times \Bbb R, \quad (X,x)\mapsto (F(X), \la x).
$$
Note that $f$ also has a horseshoe $\La_f =\Sigma \times \{0\}$
and $HD (\La_f) = HD (\Sigma )<1.$
Let $P$ be some fixed point of $F$ in $\Sigma$. Then $(P,0)$ is
a hyperbolic fixed point of $f$ and
$$
W^{s}((P,0),f)= W^{s}(P,F)\times \{0\} \text{ and }
W^{u}((P,0),f)= W^{u}(P,F)\times \Bbb R.
$$ 
For simplicity, we assume that every expanding eigenvalue
of $DF(P)$ is larger than $2$ and then the strong unstable
manifold of $(P,0)$ is
$$
W^{uu}((P,0),f)= W^{u}(P,F)\times \{0\}.
$$
Since $W^{s}(P,F)$  and $W^{u}(P,F) $ meet transversely at
some $Q\in \Bbb R^n$, the manifolds
$W^{s} ((P,0),F)$ and $W^{uu} ((P,0),F)$ have
a quasi-transverse intersection at $(Q,0)$. 
Now we consider arcs of $\Cal C^r$-diffeomorphisms
$\{f_t\}_{t \in [-1,1]}$, with $f_0= f$,
unfolding generically this intersection
and we prove that the continuation $\La_t$ of
$\La_0=\La_f$ satisfies 
$$
HD (\La_t)\ge 1
\text{ for every small $t\ne 0$}
$$
(actually, the strict inequality holds).
Clearly this implies the Theorem.

It is interesting to contrast this construction with some of
the results in \cite{PV2},
where geometric invariants of hyperbolic basic sets in
any dimension were considered, in a context of bifurcations
of diffeomorphisms. Indeed, by Section 4 in that paper,
invariants such as the Hausdorff dimension, the
limit capacity or the thickness of basic sets, do vary
continuously with the dynamics, {\it if one avoids homoclinic
trajectories in strong stable or strong unstable
manifolds\/} (such as we are making use of here).
On the other hand, our present arguments, see also the
construction of {\it cs-blenders\/} in \cite{BD, Section 2},
suggest that explosion of the Hausdorff dimension is a 
fairly general phenomenon in situations involving
such {\it strong\/} homoclinic trajectories. In this
direction we state

\proclaim{Conjecture}
Given any $m$-dimensional manifold $M$, $m\ge 3$, and
$1\le r\le \infty$, there is a codimension $1$ submanifold
$\Cal W^r$ of {\rm Diff\,}$^r(M)$ such that every
$f\in\Cal W^r$ has a hyperbolic basic set $\La_f$
containing some strong homoclinic intersection
and $f$ is a point of upper semi-discontinuity of the
Hausdorff dimension of (the continuation of)
$\La_f$. 
\endproclaim

We close this section by posing the following natural question:
\proclaim{Question}
Is the Hausdorff dimension of basic sets always
a lower semi-continuous function of the dynamics?
\endproclaim

\heading 2. Proof of the Theorem\endheading

Now we fill-in the details of our argument. In order to
keep the exposition as transparent as possible we deal with
a fairly simple example even if, clearly, the present
construction has a rather more general scope.
As we already said, we start with a diffeomorphism $F$
of $\Bbb R^n$, $n\ge 2$, exhibiting a horseshoe $\Sigma$
$$
\Sigma =\bigcap_{i\in \Bbb Z} F^i(R),
\quad\text{ with } R=[-1,1]^n \text{ say, } 
$$
and $F^{-1}(R)\cap R$ consisting of two connected
components $\hat D_1$, $\hat D_2$.
We take $F$ to be affine on each of these components:
there are $s,\, u\ge 1$, with $s+u=n$, and linear maps
$S_i\colon\Bbb R^s\to\Bbb R^s$,
$U_i\colon\Bbb R^u\to\Bbb R^u$, such that
$$
DF|\hat D_i=\left(\matrix S_i & 0 \\ 0 & U_i\endmatrix\right)
\text{ and } ||S_i||, ||U_i^{-1}|| < 1
\text { for } i=1,\,2.
$$
In particular, $\hat D_i=[-1,1]^s\times D_i$, $D_i\subset[-1,1]^u$,
for $i=1,\,2$. In what
follows $(x^s, x^u)$ are the usual coordinates in
$R= [-1,1]^s\times [-1,1]^u$  and we suppose the fixed point $P$
of $F$ to be located at $(0^s, 0^u)\in\hat D_1$.

Define the smooth arc $\{f_t\}_{t\in [-1,1]}$ of diffeomorphisms
of $\Bbb R^{n+1}$ by
$$
f_t(x^s,x^u,x) =  \left\{
\aligned 
&(F(x^s,x^u), \la x) \text{ if } x^u \in D_1,\\
&(F(x^s,x^u), \la x-t) \text{ if }x^u \in D_2.
\endaligned\right.
$$
We let $1<\la<2$ and $||U_1^{-1}||<1/2$ so that the fixed
point  $\Cal O = (0^s,0^u,0)$ of $f_t$ has
$$
W^{uu}_{\loc} ( \Cal O, f_t)=\{0^s\}\times [-1,1]^u \times \{0\}
\text{ and }
W^{s}_{\loc} (\Cal O, f_t)=[-1,1]^s \times \{0^u\}\times \{0\}.
$$
On the other hand,  given any $x^s\in[-1,1]^s$ there are
$x^s_1$,  $x^s_2\in[-1,1]^s$ such that
$$\aligned
f_t (\{x^s\} & \times [-1,1]^u \times \{x\})
\supset
\\&\supset
(\{x^s_1 \} \times [-1,1]^u \times \{\la x\})
\cup (\{x^s_2 \} \times [-1,1]^u \times \{\la x-t \})
\endaligned
\tag 1
$$
for every $x\in\Bbb R$. Hence,
$$
f_t (W^{uu}_{\loc}(\Cal O, f_t))
\supset
\{0^s_2\}\times[-1,1]^u\times\{-t\}
$$
(note that $0^s_1=0^s$) and so the arc $\{f_t\}_{t \in [-1,1]}$  unfolds generically the quasi-transverse  intersection  of  $W^s(\Cal O,f_0)$ and $W^{uu}(\Cal O,f_0)$ at $(0^s_2, 0^u, 0)$.

Denote by $\La_t$ the continuation for $f_t$, small $t$, of the basic set
$\La_0=\Sigma\times\{0\}$ of $f_0$. Observe that $\La_t$ coincides
with the closure of the set
$H(\Cal O ,f_t)= W^s (\Cal O, f_t) \cap W^u(\Cal O, f_t)$
of all (transverse) homoclinic points of  $\Cal O $.

\proclaim{Lemma}
Let $t>0$ (resp. $t<0$) be close to zero and $J\subset [0,t]$
(resp. $J\subset [t,0]$) be an open interval.
Then there are $x^s \in [-1,1]^s$ and $j\ge0$ such that
$$
f_t^j (\{0^s\} \times [-1,1]^u \times J)\supset
\{x^s\} \times [-1,1]^u \times \{\la^{-1}t\}.
$$
In particular,
$(\{0^s \} \times [-1,1]^u \times J) \cap H(\Cal O, f_t)\ne \emptyset$.
\endproclaim

As a consequence of this Lemma,
$$
 \pi\left(\ov{H(0,f_t)}\right) \supset [0,t],
\ \text{ where }
\pi \,\colon \, [-1,1]^s\times[-1,1]^u\times \Bbb R \to \Bbb R,
\ \pi(x^s,x^u,x)=x.
$$
Since $\pi$ is a Lipschitz map, it follows that
$$HD(\La_t)\ge HD( \pi (\La_t) ) \ge 1,
$$
which proves the Theorem. We are left to give the

\demo{Proof of the Lemma}
We suppose $t>0$, the case $t<0$ being completely analogous.
Consider the affine functions 
$$
\align
&\pi_{1,t} \,\colon\, [0, \la^{-1} t]\to [0,t] ,\  \pi_{1,t} (y) = \la y,\\
&\pi_{2,t} \,\colon\, [\la^{-1} t, t]\to [0,t] ,\  \pi_{2,t} (y) = \la y-t.
\endalign
$$
Note that $\pi_{2,t}$ is well defined since $1< \la <2$. By (1)
$$
\aligned
& 
f_t (\{x^s\} \times [-1,1]^u \times \{x\})
\supset
\{ x^s_1\} \times [-1,1]^u \times \{ \pi_{1,t} (x)\}
\quad \text { if } x \in [0, \la^{-1} t],\\
& 
f_t (\{x^s\} \times [-1,1]^u \times \{x\})
\supset
\{ x^s_2\} \times [-1,1]^u \times \{ \pi_{2,t} (x)\}
\quad \text { if } x \in [\la^{-1} t,t].
\endaligned
\tag 2
$$
Write $I_0=J$ and $z_0=0^s$. If $I_0$ contains $\la^{-1}t$
then there is nothing to prove. Hence,  we may assume that either
$I_0 \subset (0, \la^{-1} t)$ or  $I_0 \subset (\la^{-1} t,t)$. We
let $i=1$ in the first case and $i=2$ in the second one and we
write $I_1 =\pi_{i,t} (I_0)$ and $z_1=0^s_i$. Then, by (2),
$$
f_t (\{0^s\} \times[-1,1]^u \times I_0)
\supset
\{z_1\} \times[-1,1]^u \times I_1 
$$ 
As above, if $I_1$ contains $\la^{-1}t$ then we are done.
Otherwise we apply the previous procedure inductively:
for each $j\ge 1$, if
$\la^{-1} t\not\in I_{j-1}$ then we construct an open interval
$I_j\subset[0,t]$ and a point $z_j\in[-1,1]^s$ so that
$$
\align
& I_j =\pi_{1,t} (I_{j-1}) \text{ if } I_{j-1}\subset [0,\la^{-1} t]
\quad\text{and}\quad
I_j =\pi_{2,t} (I_{j-1}) \text{ if  } I_{j-1}\subset [\la^{-1} t, t]\\
\intertext{and}
& f_t (\{z_{j-1}\}\times [-1,1]^u \times I_{j-1})
\supset(\{z_j\}\times [-1,1]^u \times I_j).
\endalign
$$
Since $\text{length\,}(I_j) = \la\cdot\text{length\,}(I_{j-1})$ and $\la>1$
there must be a first $j$ such that $\la^{-1} t \in I_j$.
This ends the proof of the first part of the Lemma.

As for the second one, it is now a direct consequence. Observe that
$$
f_t( \{x^s\} \times D_2\times \{\la^{-1}t\} )
\cap W^s_{\loc}(\Cal O ,f_t) \ne \emptyset
$$
and so $ \{x^s\} \times[-1,1]^u\times \{\la^{-1}t\}$ intersects
$W^s(\Cal O, f_t)$ at some point $Q_t$. Then, taking $j\ge 0$ as above,
$$
f_t^{-j}(Q_t)\in(\{0^s \} \times [-1,1]^u \times J)\subset W^u(\Cal O, f_t),
$$
which also means that $f_t^{-j}(Q_t)\in H(\Cal O, f_t)$. This completes our
argument.
\hfill $\square$\enddemo

\Refs
\nofrills
{REFERENCES}
\widestnumber\key{MM8}

\ref
\key BD
\by Ch. Bonatti, L. J. D\'\i az
\paper Persistence of transitive diffeomorphisms
\jour preprint
\yr 1994
\endref

\ref
\key F
\by K.J. Falconer
\book The geometry of fractal sets
\yr 1985
\publ Cambridge Univ. Press
\endref

\ref
\key HP
\by M. Hirsch, C. Pugh
\paper Stable manifolds and hyperbolic sets
\jour Global Analysis, Proc. Symp. in Pure Math.
\publ Amer. Math. Soc.
\vol 14
\yr 1970
\endref

\ref
\key MM
\by H. McCluskey, A. Manning
\paper Hausdorff dimension for horseshoes
\yr 1983
\jour  Ergod. Th. \& Dynam. Sys.
\vol 3
\pages  231-260
\endref 

\ref
\key  M
\by R. Ma\~n\'e
\paper The Hausdorff dimension of horseshoes of diffeomorphisms 
            of surfaces
\jour Bull. Braz. Math. Soc.
\yr 1990
\pages 1-24
\vol 20
\endref

\ref
\key PV1
\by  J. Palis, M. Viana
\paper On the continuity of Hausdorff dimension and limit capacity of horseshoes
\vol 1331
\jour Lect. Notes math
\yr 1988
\publ Springer Verlag
\pages 150-160
\endref

\ref
\key PV2
\by  J. Palis, M. Viana
\paper High dimension diffeomorphisms displaying
             infinitely many periodic attractors
\jour Annals Math.
\yr 1994
\endref
\endRefs 

\vskip .5cm

Christian Bonatti ( bonatti\@satie.u-bourgogne.fr )\newline 
Lab. Topologie, URA 755, Dep. Math\'ematiques, Univ. Bourgogne\newline
B.P. 138, 21004 Dijon Cedex, France.

\vskip .2cm

Lorenzo J. D\'\i az ( lodiaz\@mat.puc-rio.br )\newline
Dto. de Matem\'atica, PUC-RJ,
Rua Marqu\^es de S\~ao Vicente 225, G\'avea\newline
22453-900 Rio de Janeiro RJ, Brazil.

\vskip .2 cm

Marcelo Viana  ( viana\@impa.br )\newline
IMPA, Estrada D. Castorina 110, Jardim Bot\^anico\newline
22460-010 Rio de Janeiro RJ, Brazil.

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