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\title{Homoclinic tangencies and fractal invariants
\\ in arbitrary dimension
\thanks{Work partially supported by Pronex-Dynamical Systems, CNPq 001/2000, and Faperj.} 
\\
Tangences homoclines et invariants fractaux
\\ en dimension arbitraire}
\author{Carlos Gustavo Moreira, Jacob Palis, Marcelo Viana}
\date{ }

\begin{document}
\maketitle

\section*{Version fran\c caise abr\'eg\'ee:}

Les r\'esultats sur les tangences homoclines des diff\'eomorphismes des
surfaces, obtenus dans le courant des trois derni\`eres d\'ecennies par
Newhouse, Palis, Takens, Yoccoz et Moreira d\'evoil\`erent une relation
profonde entre des invariants fractaux d'ensembles hyperboliques et les
comportements dynamiques typiques, dans une famille param\'etr\'ee,
apr\`es la bifurcation homocline. Voir \cite{NP76,PT87,PY94,MY2}
et \cite[chapitre~7]{PT93} pour un panorama de la th\'eorie.

Voici, en quelques mots, comment se manifeste cette relation~: si la
dimension de
Hausdorff de l'ensemble hyperbolique associ\'e \`a la tangence est plus
petite que 1, les dynamiques hyperboliques sont pr\'edominantes (densit\'e
de Lebesgue totale) dans l'espace des param\`etres pr\`es de la tangence;
si la dimension de Hausdorff est plus grande que 1, l'hyperbolicit\'e
n'est pas pr\'edominante dans l'espace des param\`etres, par contre
l'union de l'hyperbolicit\'e et des tangences persistantes l'est.

L'extension de ces r\'esultats aux dimensions sup\'erieures pr\'esente des 
difficult\'es importantes, li\'ees au manque de r\'egularit\'e des feuilletages
invariants des ensembles hyperboliques.
Par example, on ne sait m\^eme pas si la dimension de Hausdorff locale
est bien d\'efinie pour les ensembles hyperboliques dans les vari\'et\'es
de dimension plus grande que 2.
Une partie de ces difficult\'es fut r\'esolue par Palis et Viana \cite{PV94},
\`a l'aide d'une notion de diff\'erentiabilit\'e \`a la Whitney,
dans leur g\'en\'eralisation en dimension arbitraire du th\'eor\`eme de
Newhouse \cite{Ne79} sur la coexistence d'une infinit\'e de puits.

Le but de ce travail est de d\'emontrer que, pourtant, le paradigme
$$
\text{\em l'hyperbolicit\'e est pr\'edominante } \Leftrightarrow
\text{\em la dimension de Hausdorff est plus petite que $1$}
$$
peut \^etre \'etendu aux bifurcations homoclines en dimension arbitraire.

\medskip

Plus pr\'ecis\'ement, nous consid\'erons des familles \`a un param\`etre de
diff\'eomorphismes $\varphi_\mu : M^n \to M^n$, $\mu \in (-1, 1)$ de classe
$C^2$ qui d\'eveloppent une tangence homocline associ\'ee \`a un point
p\'eriodique $p_0$ inclus dans un fer-\`a-cheval $\Lambda_0$ de $\varphi_0$.
Nous supposons que $\varphi_\mu$ est hyperbolique (Axiome A) pour $\mu < 0$,
ce qui implique que les valeurs propres les plus faibles, contractante et 
dilatante, de $p_0$ sont r\'eelles.

Soient $d_s$ et $d_u$ les dimensions de Hausdorff de $W^s(p_0) \cap \Lambda_0$
et $W^u(p_0) \cap \Lambda_0$, respectivement.
Nous montrons que, pour un sous-ensemble r\'esiduel de diff\'eomorphismes
$\varphi_0$, les dimensions $d_s$ et $d_u$ d\'ependent contin\^ument de $\varphi_0$
et on a~:

\begin{itemize}
\item[i)] Si ${d_s} + {d_u} < 1$ alors
$H = \{\mu > 0 \mid \varphi_\mu$ est hyperbolique$\}$ a densit\'e de
Lebesgue totale en $\mu = 0$.

\item[ii)] Si ${d_s} + {d_u} > 1$ alors
${T_s} = \{\mu > 0 \mid \varphi_\mu$ pr\'esente des tangences homoclines
persistantes associ\'ees \`a la continuation hyperbolique $\Lambda_\mu$
de $\Lambda_0 \}$  est un ensemble ouvert avec densit\'e de Lebesgue
positive en $\mu = 0$.
\end{itemize}

La demonstration est bas\'ee sur une extension des techniques introduites
par Newhouse-Palis \cite{NP76}, Palis-Takens \cite{PT87} et
Moreira-Yoccoz \cite{MY1,MY2} pour montrer les r\'esultats correspondants
en dimension 2, et par Palis-Viana \cite{PV94} pour \'etudier les bifurcations
homoclines de codimension $1$ en dimension arbitraire.

Un des lemmes les plus importants dans ce travail est la construction 
(peut-\^etre apr\`es une perturbation de $\varphi_0$) des feuilletages
stable-fort et instable-fort de codimension 1 pour des sous-ensembles
hyperboliques de  $\Lambda_0$ avec presque les m\^emes dimensions $d_s$
et $d_u$.
Ces feuilletages sont utilis\'ees pour r\'eduire l'\'etude des
g\'eometries des feuilletages stable et instable pr\`es de la tangence 
homocline initiale a une situation proche du cas bidimensionel.

Il est int\'eressant de remarquer que la dimension critique $1$ est independante
de la dimension $n$ de l'espace ambiant.
Donc, dans un certain sens, le deuxi\`eme cas devient relativement plus fr\'equent
quand $n$ cro\^\i t.


\section{Introduction}

Over the last three decades, a number of results have been proved by Newhouse,
Palis, Takens, Yoccoz and Moreira, unveiling a deep connection between fractal
invariants of hyperbolic sets and the typical dynamics observed in the unfolding
of a homoclinic tangency by a parametrized family of surface diffeomorphisms.
See \cite{NP76,PT87,PY94,MY2} and \cite[chapter~7]{PT93} for an overview.

In brief terms, if the Hausdorff dimension of the hyperbolic set involved in
the tangency is less than 1, then hyperbolicity prevails in parameter space
(full Lebesgue density at the tangency parameter); if the Hausdorff
dimension is larger than 1 then hyperbolicity alone is not prevalent, but the
union of hyperbolicity and persistent tangencies is.

The extension of these results to higher dimension involves considerable
difficulties, related to the lack of smoothness of invariant foliations of
hyperbolic sets of codimension larger than $1$.
For instance, it is not even known whether the local Hausdorff dimension is
well-defined for hyperbolic sets in high dimension manifolds.
Part of these difficulties could be handled by Palis and Viana \cite{PV94},
by means of a notion of differentiability \`a la Whitney,
in their generalization to arbitrary dimension of Newhouse's theorem \cite{Ne79}
on coexistence of infinitely many sinks.

Our goal in this work is to prove that, nevertheless, the paradigm
$$
\text{\em hyperbolicity prevails }\Leftrightarrow
\text{\em the Hausdoff dimension is smaller than $1$}
$$ 
does extend to homoclinic bifurcations in any dimension.

\medskip

More precisely we consider 1-parameter families of diffeomorphisms 
$\varphi_\mu : M^n\to M^n$, $\mu \in (-1, 1)$ of class $C^2$, unfolding a generic
homoclinic tangency associated to a periodic point $p_0$ contained in a horseshoe
$\Lambda_0$ of $\varphi_0$.
We suppose that $\varphi_\mu$ is hyperbolic (Axiom A) for $\mu < 0$, which implies
that the weakest contracting and weakest expanding eigenvalues of $p_0$ are real
numbers.

Let $d_s$ and $d_u$ be the Hausdorff dimension of $W^s(p_0) \cap {\Lambda_0}$ and
$W^u(p_0) \cap {\Lambda_0}$, respectively.
We show that, for a residual subset of diffeomorphisms, $d_s$ and $d_u$ are continuous
with respect to $\varphi_0$, and we have:

\begin{itemize}
\item[i)] If ${d_s} + {d_u} < 1$ then
$H = \{\mu > 0 \mid \varphi_\mu$ is hyperbolic$\}$ has full Lebesgue density at $\mu = 0$.

\item[ii)] If ${d_s} + {d_u} > 1$ then
${T_s} = \{\mu > 0 \mid \varphi_\mu$ presents persistent homoclinic tangencies
associated to the hyperbolic continuation $\Lambda_\mu$ of $\Lambda_0 \}$ is an
open set with positive lower density at $\mu = 0$.
\end{itemize}

The proof is based on an extension of the techniques introduced by Newhouse-Palis 
\cite{NP76}, Palis-Takens \cite{PT87}, and Moreira-Yoccoz \cite{MY1,MY2} to prove 
the corresponding results in dimension 2, and by Palis-Viana \cite{PV94} to 
investigate homoclinic bifurcations of codimension one in arbitrary dimension.

One of our main lemmas is the construction (perhaps after perturbation of $\varphi_0$)
of strong-stable and strong-unstable foliations of 
codimension 1 for hyperbolic subsets of $\Lambda_0$ with almost the same 
dimensions $d_s$ and $d_u$. These foliations are used to (essentially) 
reduce the study of the geometries of the stable and unstable 
foliations near the initial homoclinic tangency to the bidimensional case.

It is interesting to note that the separating dimension $1$ is independent of the
dimension $n$ of the ambient space. Thus, in some sense, the second case above becomes
more frequent when $n$ increases.

\section{Geometry of horseshoes : the upper dimension}

Let $\Lambda$ be a horseshoe for a diffeomorphism $\varphi:M\to M$, and $P$
be some Markov partition of $\Lambda$.
We call {\em vertical $n$-cylinder\/} any subset of $\Lambda$ defined by 
prescribing the first $n$ symbols in the backward itinerary with respect to $P$.
Let $\V_n$ be the set of vertical $n$-cylinders.
There is a dual notion of {\em horizontal $n$-cylinder\/}, where one considers
forward itinerary.

Fix $\ve>0$ small.
For each $V \in \V_n$ let
$D_s(V) = \sup\{\diam (W^s_\ve (x) \cap V) \mid \ x \in \Lambda \cap V\}$.
We define $\widetilde\lambda_n$ by the relation
$$
\sum_{V\in\V_n} D_s(V)^{\widetilde\lambda_n} = 1.
$$
and we let the {\em upper stable dimension\/} be given by
$\ovd_s (\Lambda) = \lim_{n\to\infty} \widetilde\lambda_n$.
There is a dual notion of {\em upper unstable dimension\/}
$\ovd_u(\Lambda)$, dealing with $W^u_\ve$ instead of $W^s_\ve$.

It is not difficult to show that the limit always exists, and $\ovd_s$ is an
upper-semicontinuous function of $\varphi$. Moreover,
$$
\HD(W_\ve^s (x) \cap \Lambda) \leq \ovd_s (\Lambda)
\quad\text{for each} \ x \in \Lambda.
$$
The following result shows that the equality holds in most cases where $\ovd_s < 1$:

\begin{theorem}
\label{t.1}
Suppose $\ovd_s(\Lambda) < 1$.
Then there is a residual set of diffeomorphisms $\R$ in a neghbourhood of $\varphi$
such that, for all $\psi \in \R$ and all $x\in \Lambda_\psi$,
$$
\HD(\Lambda_\psi \cap W_\ve^s(x)) = \ovd_s(\Lambda_\psi).
$$
In addition, $\R$ may be taken so that its elements are points of continuity of $\ovd_s$.
\end{theorem}

The idea of the proof is to consider $k$-parameter families of diffeomorphisms $\psi$,
for some large $k$, and to show that for each $V \in \V_n$ we have
$\diam(W^s_\ve(x) \cap V) \geq D_s(V)^{1-\delta}$ for every $x \in \Lambda_\psi$,
with large probability in parameter space.
This gives that, for most parameters, most of the vertical cylinders have subexponential
variation of their horizontal diameters.
Using this fact, we obtain lower estimates for $\HD(W^s_\ve(x)\cap\Lambda_\psi)$
that imply the conclusion of the theorem.

\medskip

Now we can state our

\begin{main}
There are open sets $\U$ and $\V$ of families of diffeomorphisms unfolding a homoclinic
tangency as in the Introduction, such that $\U \cup \V$ is dense and 
\begin{itemize}
\item[i)] For families in $\U$ we have
$\ovd_s(\Lambda) + \ovd_u(\Lambda) < 1$  and 
$\lim_{\delta \to 0} m(H\cap [0,\delta])/\delta = 1$,
where $H=\{\mu \mid \varphi_\mu$ is hyperbolic $\}$.

\item[ii)] For families in $\V$ we have $\ovd_s(\Lambda) + \ovd_v(\Lambda) > 1$ and
$\liminf_{\delta\to 0} m \ (T_s \cap \ [0,\delta])/\delta > 0$, where 
$T_s = \{\mu \mid \varphi_\mu$ presents a persistent homoclinic tangency associated
to $\Lambda_\mu\}$.
\end{itemize}
\end{main}

In the next sections we indicate main ideas of the proof of this theorem.

\section{Prevalence of hyperbolicity}

In this section we outline the proof of the first part of the main theorem:
hyperbolicity prevails in parameter space when the sum of the stable and
unstable upper dimensions is less than 1.
The first main step is a reduction to the case when the horseshoe ${\Lambda}$
does not intersect the strong-stable manifold, nor the strong-unstable manifold
of the point $p$.

\begin{theorem}
\label{t.2}
Assume $\ovd_s(\Lambda) < 1$ (respectively, $\ovd_u(\Lambda) < 1$).
For an open and dense subset of maps in some neighborhood of $\varphi$
we have
$$
\Lambda\cap W^{ss} (p) = \{p\}
\quad(\text{respectively,\ }\Lambda \cap W^{uu}(p) = \{p\}).
$$
\end{theorem}

By continuity, $\ovd_s(\Lambda_\psi) < 1$ for any $\psi$ in a whole 
neigborhood of $\varphi$.
The density statement in the theorem is proved by deformation.
We embed $\psi$ in a one-parameter family of diffeomorphisms $\psi_\nu$ such that
$\Lambda_\nu \cap W^s(p_{\psi_{\nu}})$ moves with respect to ${W^{ss}} (p_{\psi_{\nu}})$
when $\nu$ varies.
The fact that the upper Hausdorff dimension is less than 1 ensures that
$\Lambda_{\psi_{\nu}} \cap W^s(p_{\psi_{\nu}})$ intersects $W^{ss}(p_{\psi_{\nu}})$
only for a zero Lebesgue measure set of parameters $\nu$.
This implies density, and openness is clear from compactness of $\Lambda_{\psi_{\nu}}$
and of a fundamental domain of $W^{ss}(p_{\psi_{\nu}})$.



Theorem~\ref{t.2} has the important geometric consequence that
$\Lambda\cap W^s(p)$ is contained in a cuspidal region around the weak-stable 
direction of $p$.
Let $K^s$ be the projection of $\Lambda \cap W^s(p)$ to this weak-stable
direction along the strong-stable foliation of $p$.
Then, for any $\delta > 0$ there exists $\ve > 0$ such that each $x \in\Lambda\cap W^s(p)$
in the $\ve$-neighborhood of $p$ is within distance $\ve\delta$ from some $y \in K_s$.
We say that $\Lambda\cap W^s(p)$ is {\it well-represented} by $K_s$ near $p$.

\begin{figure}[htb]
\begin{center}
\includegraphics[height=1.2in]{fig.eps}
\end{center}
\end{figure}

Then a similar fact is true replacing $W^s(p)$ by any other cross-section $\Sigma$
to the unstable direction at any $q \in W^u(p)$:
the intersection $W^u(p) \cap \Sigma$ is well-represented by $K_s$ near $q$.
Using also smoothness of the unstable leaves, together with H\"older continuity of
their tangent bundle, we conclude that the intersection of $W^u(\Lambda)$ with a
neighborhood of $q$ is well-represented by the product of $K_s$ by $T_q W^u(p)$
near $q$.

We consider $q$ to be the point of homoclinic tangency.
The previous paragraph, together with a dual statement for $W^s(\Lambda)$, mean 
that the stable and unstable foliations of $\Lambda$ are well-represented by
affine models
$$
K_s \times T_q W^u (p) \quad\text{and} \quad K_u\times T_q W^s(p)
$$
near $q$.
By continuity of the invariant foliations this remains true when the tangency is
unfolded along a one-parameter family $\varphi_\mu$:
if $\mu$ is close to zero the $W^s(\Lambda_\mu)$ and $W^u(\Lambda_\mu)$ are
well-represented by translates
$$
[K_s \times T_q W^u(p)] + \mu v_1
\quad\text{and}\quad
[K_u \times T_q W^s(p)] + \mu v_2
$$
for some vectors $v_1$, $v_2$.
So, along the lines of \cite{PT87} and \cite{T86}, to ensure that the two foliations
are transverse and, in fact, the angles at their intersection are bounded away from
zero for most parameters, it suffices to prove a corresponding fact for these affine
models:

\begin{lemma}
Let $K_1$, $K_2$ be subsets of $\re^+$ with sum of limit capacities $c (K_1) + c (K_2) <1$.
Let $V_1$ and $V_2$ be subspaces of $\re^n$ with dim $V_1 = u - 1$, dim $V_2 = s - 1$,
and $u + s = n$, and $w_1 \not\in V_1$ and $w_2\not\in V_2$ be vectors such that the
closed half-subspaces $\re^+ w_1 \oplus V_1$ and $\re^+ w_2 \oplus V_2$ intersect at
the origin.
Then most translates of $K_1 w_1 \oplus V_1$ and $K_2 w_2 \oplus V_2$ do not intersect:
for any vectors $v_1$, and $v_2$ of $\re^n$ in general position, and for any $\delta >0$
$$
d(K_1 w_1 \oplus V_1 + \mu v_1, K_2 w_2 \oplus V_2 + \mu v_2) > 3 \ve\delta
$$
for a fraction of parameters in $[- \ve, \ve]$ that goes to 1 when $\ve \to 0$.
\end{lemma}

\section{Global strong-stable and strong-unstable foliations}

In order to prove the second part of our main result, we consider compact subsets
of our basic set that are invariant by some iterate of the dynamics at the first
bifurcation parameter and have good geometric properties (perhaps after a small
perturbation of the family).
More precisely, in case ii) of the theorem we prove that, generically, there are
disjoint subsets $\Lambda_1$ and $\Lambda_2$ of $\Lambda$, invariant by some
iterate $\varphi^{n_0}_0$, with the following properties:

\begin{itemize}
\item[a)] $\Lambda_1$ has a globally defined strong-stable foliation,
invariant under $\varphi^{n_0}_0$.
\item[b)] $\Lambda_2$ has a globally defined strong-unstable foliation, 
invariant under $\varphi^{n_0}_0$.
\item[c)] $\ovd_s(\Lambda_1) < 1$, $\ovd_u(\Lambda_2) < 1$, 
and \ $\ovd_s(\Lambda_1) + \ovd_u(\Lambda_2) > 1$
\item[d)] For each $p_1 \in \Lambda_1$ and $p_2 \in \Lambda_2$, the projections 
$\pi_{ss} \mid (\Lambda_1 \cap W^s(p_1))$ along strong-stable leaves and 
$\pi_{uu} \mid (\Lambda_2 \cap W^u(p_2))$ are injective and have 
{\it intrinsically $C^{1 + \ve}$\/} inverses, in the sense of \cite{PV94}.
\item[e)] Either $\Lambda_1$ contracts area in the weak (bidimensional) direction
or $\Lambda_2$ expands area in that weak direction.
\end{itemize}

The main and most delicate points of the above statement are items a) and b).
Let us give some ideas of their proof.

\smallskip

First, we show that, perhaps after a small perturbation, $\Lambda$ has a periodic
point whose weakest, contracting and expanding, eigenvalues are real.
Next, we use the existence of this periodic point to show that a sizable
proportion of the vertical (and of the horizontal) $n$-cylinders in an advanced
stage $n\gg 1$ of the construction of $\Lambda$ are exponentially squeezed along
the corresponding direction.

Finally, we consider families of perturbations with a large number of parameters,
whose effect is to rotate the rectangles of a Markov partition of $\Lambda$ with
enough independence, in order to show that, with large probability, there are
invariant subsets of $\Lambda$ with almost the same stable and unstable upper
dimensions as $\Lambda$, and admitting invariant strong-stable (respectively,
strong-unstable) cone fields.

\section{Positive density of persistent tangencies}

Conditions a) to d) of the previous section imply that the projection of
$\Lambda_1 \cap W^s (p_1)$ onto a weak-stable (1-dimensional) manifold,
along the strong-stable foliation, is a regular $C^{1 + \ve}$ Cantor set $K^s$,
whose conjugation class does not depend on $p_1 \in\Lambda_1$ nor on the
choice of the weak-stable leaf.
The same holds for $\pi_{uu}(\Lambda_2 \cap W^u (p_2)) = K^u$.

We adapt arguments of \cite{M} to show the next lemma. It is assumed that the
eigenvalues of $\varphi$ at the point $p$ satisfy a generic non-resonance
condition.

\begin{lemma}
If there are $\lambda > 0$ and $t \in \re$ such that $K^s$ intersects $\lambda K^u + t$
stably, then $(\varphi_\mu)_\mu$ persistently exhibits positive density of stable tangencies at
$\mu = 0$.
\end{lemma}

In order to get the stable intersection between $K^s$ and $\lambda 
{K^u} + t$, we first prove that condition e) implies that either $K^s$ or 
$K^u$ is a $C^2$-regular Cantor set on the central leaf.
This is important to show that, generically, $K^s$ and $K^u$ satisfy the Scale 
Recurrence Lemma, the main technical tool of \cite{MY1,MY2}.
Afterwards, we consider again a family with a large number of parameters of
local perturbations of the diffeomorphism $\varphi_0$ and show that for most 
parameters there are $\lambda$ and $t$ such that $K^s$ stably intersects 
$\lambda {K^u} + t$.
This final step is adapted from the perturbation arguments of \cite{MY1,MY2}.
This allows us to to use the previous lemma and conclude the proof of part ii)
of the main theorem.

\begin{thebibliography}{31}

\bibitem[NP]{NP76} S. Newhouse and J. Palis,
{\it Cycles and bifurcation theory},
Ast\'erisque 31 (1976), 44-140. 

\bibitem[M]{M} C. G. Moreira,
{\it Stable intersections of Cantor sets and homoclinic bifurcations},
Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 13 (1996), no. 6, 741-781.

\bibitem[MY1]{MY1} C. G. Moreira and J.-C. Yoccoz,
{\it Stable intersections of regular Cantor sets with large Hausdorff dimensions},
accepted for publication in Annals of Math.

\bibitem[MY2]{MY2} C. G. Moreira and J.-C. Yoccoz,
{\it Tangences  homoclines stables pour des ensembles hyperboliques de grande dimension fractale},
to appear.

\bibitem[Ne]{Ne79} S. Newhouse,
{\it The abundance of wild hyperbolic sets and nonsmooth stable
            sets for diffeomorphisms},
Publ. Math. I.H.E.S. 50 (1979), 101--151.

\bibitem[PT1]{PT87} J. Palis and F. Takens,
{\it Hyperbolicity and the creation of homoclinic orbits},
Annals of Math., vol. 125 (1987), 337-374. 

\bibitem[PT2]{PT93} J. Palis and F. Takens,
{\it Hyperbolicity and sensitive chaotic dynamics at homolinic bifurcations :
fractal dimensions  and infinitely many attractors},
Cambridge Univ. Press, (1992). 

\bibitem[PV]{PV94} J. Palis and M. Viana,
{\it High dimension diffeomorphisms displaying infinitely many periodic attractors},
Annals of Math., vol 140 (1994), 207-250.

\bibitem[PY]{PY94} J. Palis and J.C. Yoccoz,
{\it Homoclinic tangencies for hyperbolic sets
of large Hausdorff dimension,\/}
Acta Math., 172 (1994), 91-136.

\bibitem[T]{T86} F. Takens,
{\it Homoclinic bifurcations},
Proc. Ins. Congress of Math., Berkeley (1986), 1229-1236

\end{thebibliography}

\vspace{1cm}

\noindent
Carlos Gustavo Moreira (gugu\@@impa.br)

\noindent
Jacob Palis (jpalis\@@impa.br)

\noindent
Marcelo Viana (viana\@@impa.br)

\medskip

\noindent
IMPA,
Estrada D. Castorina 110,
22460-320 Rio de Janeiro, Brasil
\end{document}
