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\begin{document}

\centerline{\large\bf SUMS OF CANTOR SETS WHOSE SUM OF}
\centerline{\large\bf DIMENSIONS IS CLOSE TO ONE}

\thispagestyle{empty}

\vskip .5in

\centerline{\sc Carlos Gustavo Tamm de Araujo Moreira\footnote{Partially supported by CNPq and FAPERJ.}}
\centerline{\sl Instituto de Matem\'{a}tica Pura e Aplicada}
\centerline{\sl Estrada Dona Castorina 110, Jardim Bot\^{a}nico}
\centerline{\sl Rio de Janeiro-RJ, Brasil}
\vskip .1in

\centerline{\sc Eduardo M. Mu\~{n}oz Morales\footnote{Partially supported by IMPA-CNPq and U.C.N.}}
\centerline{\sl Departamento de Matem\'{a}ticas}
\centerline{\sl  Universidad Cat\'{o}lica del Norte}
\centerline{\sl Antofagasta, Chile}


\vskip .5in


\begin{abstract}
We prove that the arithmetic sum of two Cantor sets whose sum of Hausdorff
dimensions is slightly bigger than one is generically an M-Cantorval.\ We
conclude that, in most cases, homoclinic bifurcations in a surface
associated to non-trivial basic sets with Hausdorff dimension slightly
bigger than one present hyperbolicity and stable tangencies for positive
values \ of the parameter, both with positive density at the initial
bifurcation parameter.
\end{abstract}

\begin{center}
\bigskip\ \ \ 
\end{center}

\ \ \ \ \ \ \ \ \newpage\ \ 

\section{\textbf{Introduction}}

\bigskip

The study of arithmetic sums of regular Cantor sets plays a special role in
Dynamical Systems, mainly in the study of homoclinic bifurcations, as well
as in certain aspects of Number Theory related with the study of Markov and
Lagrange spectra (see \textbf{[Mo2]}). In this article, we study sums of Cantor sets whose sum of
Hausdorff dimensions is slightly bigger than one. More precisely, we will
study families at one parameter of pairs of regular Cantor sets in the real
line $(K_{\mu },K_{\mu }^{\prime })$ with $HD\left( K_{0}\right) +HD\left(
K_{0}^{\prime }\right) =1$\ (where $HD$ denotes the Hausdorff dimension),
and we shall characterize topologically sums of the type:

\begin{center}
$K_{\mu }+\lambda K_{\mu }^{\prime }=\{x+\lambda y\,:\,x\in K_{\mu },\,y\in
K_{\mu }^{\prime }\}$
\end{center}

with $\lambda \in \mathbb{R}^{\ast }$ at values of $\mu $ close to 0 for
whose $HD\left( K_{\mu }\right) +HD\left( K_{\mu }^{\prime }\right) >1$ (we
recall that, under these conditions, if $HD\left( K_{\mu }\right) +HD\left(
K_{\mu }^{\prime }\right) <1$ then for all $\lambda \in \mathbb{R}$, $K_{\mu
}\mathbb{+\lambda }K_{\mu }^{\prime }\mathbb{\,}$\ is a Cantor set with
Hausdorff dimension less than one, hence of zero Lebesgue measure - see \textbf{[PT]}, chapter IV).

Newhouse introduced the concept of thickness in order to give persistent examples of pairs of regular Cantor sets whose arithmetic difference (or sum) is a finite union of intervals ( see \textbf{[PT]}). It was proved later in \textbf{[MY]} that arithmetic sums of generic pairs of regular Cantor sets whose sum of Hausdorff dimensions is bigger than one have dense interior (see Section 2), solving a conjecture by Palis.
 
In \textbf{[MMR]} were exhibited distinct open sets of pairs of regular
Cantor sets, where the arithmetic sum presents different topological
properties. One of them, introduced by Mendes and Oliveira, see \textbf{%
[MO], }was of an M-Cantorval: we say that a compact set $C\subset \mathbb{R}$%
, is an \textit{M-Cantorval} if it has dense interior (i.e., $C=\overline{%
int(C)}$) and any \textit{gap} (connected component of the complement) of $C$
\ has its extremes accumulated by others gaps of $C.$ It is known that all
M-Cantorvals are homeomorphic (see \textbf{[MO]}).

We are going to prove that for families, as were described above, typically,
the sum of two regular Cantor sets whose sum of Hausdorff dimensions is
slightly greater than one is an M-Cantorval\ (see Theorem 1 below).

Furthermore, in section 3, we use this result to conclude that for
homoclinic bifurcations on surfaces associated to horseshoes with Hausdorff
dimension slightly bigger than one, there are open sets of positive values
of the parameter corresponding, respectively, to hyperbolicity and to stable
tangencies, both having positive density at the initial parameter of
bifurcation.

\section{Families of pairs of Cantor sets}

We will work with families $\{(K_{\mu },K_{\mu }^{\prime })\,:\,\mu \in
(-1,1)\}$ of pairs of regular Cantor sets of class $C^{\infty }$ that depend
(i.e., whose dynamics depend) \ $C^{\infty }$\ on the parameter $\mu .$ Let $%
d\left( \mu \right) =HD\left( K_{\mu }\right) +HD\left( K_{\mu }^{\prime
}\right) $ and let us suppose that $\ d\left( 0\right) =1$. Then we have
that $d$ \ is differentiable (see \textbf{[Ma]}) and, generically, $%
d^{\prime }\left( 0\right) \neq 0.$ Besides, we will suppose, without loss
of generality, that $d^{\prime }\left( 0\right) >0.$ Hence, if $\mu <0$ and $%
\left| \mu \right| $ is small then $d\left( \mu \right) <1,$ and therefore $%
K_{\mu }+\lambda K_{\mu }^{\prime }$ is a Cantor set of zero Lebesgue
measure (with Hausdorff dimension less than one), for all $\lambda \in 
\mathbb{R}$ in this case.

On the other hand, it was proved in \textbf{[MY]} that, if

\begin{center}
$\Omega =\{(K,K^{\prime })\,:\,K,K^{\prime }$ are $C^{\infty }$-Cantor sets, 
$HD\left( K\right) +HD\left( K^{\prime }\right) >1\},$
\end{center}
equiped with the $C^{\infty }$ topology on the expansive functions that define
$K$ and $K^{\prime }$,
then, for most pairs $(K,K^{\prime })\in \Omega ,\,K+\lambda K^{\prime }$
has dense interior for all $\lambda \in \mathbb{R}^{\ast }.$ More precisely,
for given $(K,K^{\prime })\in \Omega ,$ we say \ that $(K,K^{\prime })$ have%
\textit{\ stable intersection} if there exists a neighbourhood $U_{0}$ of $%
(K,K^{\prime })$ in $\Omega $ such that for $(\overset{\sim }{K},\overset{%
\sim }{K^{^{\prime }}})\in U_{0}\Longrightarrow \overset{\sim }{K}\cap 
\overset{\sim }{K^{^{\prime }}}\neq \emptyset .$ For given $(K,K^{\prime
})\in \Omega ,$ we define

\begin{center}
$I_{s}$ $(K,K^{\prime })=\{t\in \mathbb{R\,\,}:\mathbb{\ (}K,K^{\prime }+t)$
has stable intersection$\}.$
\end{center}

It is \ not difficult to see that $I_{s}$ $(K,K^{\prime })$\ is an open set
which is a subset of $K-K^{\prime }=\{t\in \mathbb{R\,\,:\ }K\cap (K^{\prime
}+t)\neq \emptyset \}=\{x-y\,:\ x\in K,\,y\in K^{\prime }\}.$

The main result of \textbf{[MY]}\ is that there exists $\mathcal{U} \subset
 \Omega $ open and dense (and in certain sense, of ''total measure'') such
that $(K,K^{\prime })\in \mathcal{U}$ implies that, for all $\lambda \in 
\mathbb{R}^{\ast },\,I_{s}$ $(K,\lambda K^{\prime })$ is dense in $K-\lambda
K^{\prime }$ and,\ furthermore, $HD\left( \left( K-\lambda K^{\prime
}\right) \backslash I_{s}(K,\lambda K^{\prime })\right) <1.$

This result implies that for a residual set of families $(K_{\mu },K_{\mu
}^{\prime })$ as above, for $\mu >0$ small, it is true that $K_{\mu
}+\lambda K_{\mu }^{\prime }$ has dense interior (i.e., $K_{\mu }+\lambda
K_{\mu }^{\prime }=\overline{int(K_{\mu }+\lambda K_{\mu }^{\prime })}$) for
all $\lambda \in \mathbb{R}^{\ast }$$.$

Let $\ \mathcal{H}=\{(K,K^{\prime })\,:\,K,K^{\prime }$ are $C^{\infty }$%
-Cantor sets, $HD\left( K\right) +HD\left( K^{\prime }\right) =1\}.$ Then,
our main result is the following one.

\begin{theorem}
Let $\mathcal{F}\,$be the set of smooth families $\{(K_{\mu },K_{\mu
}^{\prime })\,:\,\mu \in (-1,1)\}$ of pairs of regular Cantor sets such that 
$(K_{0},K_{0}^{\prime })\in \mathcal{H}$ and $d^{\prime }(0)>0.$\ Then there
is a residual set $\mathcal{R}\subset \mathcal{F}$ \ such that, for $%
(K_{0},K_{0}^{\prime })\in \mathcal{R},$ there exits $\delta >0$ such that $%
0<\mu <\delta $ implies that $K_{\mu }+K_{\mu }^{\prime }$ is an M-Cantorval.
\end{theorem}

The proof of this theorem is based on the following result due to
Besicovitch \ (see \textbf{[F]} ).

\begin{theorem}
Let $\Lambda \subset \mathbb{R}^{2}$ be an irregular set of finite Hausdorff measure of
dimension 1. Then $\pi _{\theta }\left( \Lambda \right) $ has zero Lebesgue
measure for almost all $\theta \in (0,\pi ),$ where $\pi _{\theta }:\mathbb{R%
}^{2}\rightarrow \mathbb{R}$ is the orthogonal projection on the line that
makes an angle $\theta $ with the $\overline{OX}$ axis.
\end{theorem}

In our context, if $K$ and $K^{\prime }$ are regular Cantor sets with $%
HD\left( K\right) +HD\left( K^{\prime }\right) =1,~$then $K\times K^{\prime
}\subset \mathbb{R}^{2}$ $\ $is a set of Hausdorff dimension 1 and finite
measure of dimension 1. Moreover, $K\times K^{\prime}$ is an irregular set because of the following criterion (Corollary 6.14 of \textbf{[F]}
- see also \textbf{[PSS]}): a 1-set $E\subset \mathbb{R}^{2}$ is irregular if and only if it has projections of Lebesgue measure 0 in two distinct directions (in this case $K\times K^{\prime}$ has the two zero-measure projections $K$ and $K^{\prime }$). Since $\pi _{\theta }\left( K\times K^{\prime
}\right) $ is equivalent to $K+\lambda K^{\prime }$ \ where $\lambda =\tan
\theta $, we obtain the following corollary.

\begin{corollary}
If $K$ and $K^{\prime }$ are regular Cantor sets, such that \bigskip $%
HD\left( K\right) +HD\left( K^{\prime }\right) =1,$ then for almost all $%
\lambda \in \mathbb{R},\,K+\lambda K^{\prime }$ is a Cantor set with zero
Lebesgue measure$.$
\end{corollary}

The idea of the proof for the Theorem 1 is to combine the Corollary above
with an extension of Theorem IV.1 of \textbf{[Mo1]}\ as we describe next.
First of all, we will make some considerations about the geometry of regular
Cantor sets near the extremes of the gaps. Let $\ K$ be a regular Cantor set
defined by an expanding differentiable function $\Psi :I_{1}\cup I_{2}\cup
...\cup I_{r}\overset{C^{1+\epsilon }}{\longrightarrow }\,I,\,$where $I$ is
the convex hull of the $I_{1}\cup I_{2}\cup ...\cup I_{r}$\ and,\ for all $%
j\leq r,\,\Psi (I_{j})$ is the convex hull of some union of the intervals $%
I_{s}.$For $I_{j}=[a_{j},b_{j}],$ $1\leq j\leq r,$ we define $~B\left(
K\right) =\cup _{j=1}^{r}\{a_{j},b_{j}\}$, the set of the extreme points of $%
K$. Since $\Psi \left( B\left( K\right) \right) \subset B\left( K\right) $
and $\ B\left( K\right) $ is a finite set, then for given $p\in B\left(
K\right) $ there exist integers $k_{p}$ and $l_{p},$ $k_{p}\geq 0,\,l_{p}>0$
such that $\Psi ^{l_{p}}(\Psi ^{k_{p}}\left( p\right) )=\Psi ^{k_{p}}\left(
p\right) ,$ (where it had been choosen the smallest $l_{p\text{ }}$ and $%
k_{p}$) \ Let $\ P\left( K\right) =\{\Psi ^{l}(\Psi ^{k_{p}}\left( p\right)
)\,:\,0\leq l<l_{p},$ $p\in B\left( K\right) \}$ be the set of periodic
points of $B\left( K\right) .$ Thus, for a given $\ p\in P\left( K\right) $
there exists a diffeomorphism $g_{p}:I\longrightarrow I$ such that, if $\Psi
_{p}=g_{p}\circ \Psi \circ g_{p}^{-1}$ then $\Psi _{p}^{l_{p}}=g_{p}\circ
\Psi ^{l_{p}}\circ g_{p}^{-1}$ is linear on $I_{j_{p}},$ where $j_{p}$ is
such that $p\in I_{j_{p}}.$ Let $\ K_{p}=g_{p}\left( K\cap I_{j_{p}}\right)
, $ then we say that $K_{p}$ is the \textit{linearization of K relative to p.}
Next, for given $\ q\in B\left( K\right) $ and $n\in \mathbb{N}$ we denote
by $\ I_{q,n}$ the connected component of $\Psi ^{-n}\left( I\right) \,$\
that contains $q$ (i.e., the interval in the n-step at the construction of $%
K $ that contains $q$). Let $\ \alpha _{n}:I_{q,n}\longrightarrow g_{%
\widetilde{q}}\left( I_{j_{\widetilde{q}}}\right) $ an affine surjective
homeomorphism, where $\widetilde{q}=\Psi ^{k_{q}}\left( q\right) $ \ is the
point of $P\left( K\right) $ corresponding to $q$. Therefore, we have that $%
\alpha _{n}\left( K\cap I_{q,n}\right) $ converges exponentially in the
Hausdorff's metric to $K_{\widetilde{q}}=g_{\widetilde{q}}\left( K\cap I_{j_{%
\widetilde{q}}}\right) .$ In fact, $K_{\widetilde{q}}$ \ is \ the limit
geometry of $K$ associated to the (periodic) orbit of $\ \widetilde{q}.$

In order to guarantee that certain sums of Cantor sets are M-Cantorvals, our
criterium will be the following one: we recall that an interval $\
J_{1}=[a,b]$\textit{\ is left-linked with }$\ J_{2}=[c,d]$ if $a<c,$ $b<d$
and $c<b.$ Let $\ K$ and $\ K^{\prime }$ be regular Cantor sets defined by $%
\Psi $ and $\overset{\sim }{\Psi },$ respectively, and suppose that for any $%
\ p\in P\left( K\right) $ and \ $p^{\prime }\in P\left( K^{\prime }\right) $
periodic extremes of \ $K$ and $K^{\prime },$ we have:

\begin{description}
\item[i)]  $\log \left| (\Psi ^{l_{p}})^{\prime }(p)\right| \left/ \log
\left| (\overset{\sim }{\Psi }^{l_{p^{\prime }}})^{\prime }(p^{\prime
})\right| \right. \notin \mathbb{Q.}$

\item[ii)]  There exist $\lambda _{1},\lambda _{2},\overset{\sim }{\lambda }%
_{1},\overset{\sim }{\lambda }_{2}\in \mathbb{R}^{\ast },$ $t_{1},t_{2}\in 
\mathbb{R}$ (with $\lambda _{1}\lambda _{2}<0,\overset{\sim }{\lambda }_{1}%
\overset{\sim }{\lambda }_{2}<0$) such that $\lambda _{1}\left( g_{p}\left(
I_{j_{p}}\right) \right) =[a_{1},\lambda _{1}g_{p}\left( p\right) ]$ is
left-linked with $\overset{\sim }{\lambda }_{1}g_{p^{\prime }}\left(
I_{j_{p^{\prime }}}^{^{\prime }}\right) +t_{1}=[\overset{\sim }{\lambda }%
_{1}(g_{p^{\prime }}(p^{\prime }))+t_{1},d_{1}],$ $\overset{\sim }{\lambda }%
_{2}g_{p^{\prime }}\left( I_{j_{p^{\prime }}}^{\prime }\right) +t_{2}=[a_{2},%
\overset{\sim }{\lambda }_{2}(g_{p^{\prime }}(p^{\prime }))+t_{2}]$ is
left-linked with $\lambda _{2}\left( g_{p}\left( I_{j_{p}}\right) \right)
=[\lambda _{2}(g_{p}\left( p\right) ),d_{2}]$ and $\lambda _{j}\cdot
K_{p}\cap \left( \overset{\sim }{\lambda }_{j}K_{p^{\prime }}^{\prime
}+t_{j}\right) =\emptyset ,$ $j=1,2.$
\end{description}

Then, for every $\lambda \in \mathbb{R}^{\ast },$ the infinite gaps of $%
K+\lambda K^{\prime }$ (i.e., the extreme points of the convex hull of $%
K+\lambda K^{\prime }$) are accumulated by gaps of $K+\lambda K^{\prime }$
(i.e., $K+\lambda K^{\prime }$ does not begin nor end by intervals). The
proof is analogue to that of Theorem IV.1 of \textbf{[Mo1]}.

We can use this fact (as in Proposition 1 of \textbf{[MMR]}) to show the
following: suppose that $\lambda \in \mathbb{R}^{\ast }$ is such that, if $%
a_{1}$ and $a_{2}$ are extremes of gaps of $K,$ $b_{1}$ and $b_{2}$ are
extremes of gaps of $K^{\prime },$ then $a_{1}+\lambda b_{1}\neq
a_{2}+\lambda b_{2}$. Then every gap of $K+\lambda K^{\prime }$ is
accumulated on both sides by gaps of $K+\lambda K^{\prime }$. In particular,
if $K+\lambda K^{\prime }$ has dense interior, then $K+\lambda K^{\prime }$
is an M-Cantorval.

Note that if $HD\left( K\right) +HD\left( K^{\prime }\right) =1$ then $%
HD\left( K_{p}\right) +HD\left( K_{p^{\prime }}^{\prime }\right) =1\,$for
all $p\in P(K),p^{\prime }\in P(K^{^{\prime }})$ (since $HD(K)=HD(K_{p})$
and $HD(K^{^{\prime }})=HD\left( K_{p^{\prime }}^{\prime }\right) $; in
fact, diffeomorphisms preserve the Hausdorff's dimensions). Therefore, in
this case by the Besicovitch Theorem, for given $p\in P(K),p^{\prime }\in
P(K^{^{\prime }})$ and $\mu \in \mathbb{R}^{\ast }$ we have that for almost
all $\lambda $ $\in \mathbb{R}$, $\mu K_{p}+\lambda K_{p^{\prime }}^{\prime
} $ has zero Lebesgue measure. Since $P(K)\times P(K^{^{\prime }})$ is a
finite set, then for almost all $(\mu ,\lambda )\in \mathbb{R}^{2}$, $\mu
K_{p}+\lambda K_{p^{\prime }}^{\prime }$ has zero Lebesgue measure for all $%
\ (p,q)\in P(K)\times P(K^{^{\prime }})$. This implies that the condition $%
ii)$ above is fullfilled inmediatly for the case $HD\left( K\right)
+HD\left( K^{\prime }\right) =1$ (and indeed in a neighbourhood, since the
condition $ii)$ is an open one). Because the condition $i)$ is clearly
generic, we conclude that for generic pairs of Cantor sets $(K,K^{^{\prime
}})$ with $HD\left( K\right) +HD\left( K^{\prime }\right) $ slightly bigger
than one, $\ K+K^{^{\prime }}$ is an M-Cantorval.

Unfortunately, given a generic family $(K_{\mu },K_{\mu }^{\prime })$ of
pairs of regular Cantor sets, we have a non-empty (countable) set of
paramenters $\mu $ for which there are $a_{1}$,$a_{2}$ extremes of gaps of $%
K_{\mu }$ and $b_{1},b_{2}$ extremes of gaps of $K_{\mu }^{\prime }$ with $%
a_{1}+b_{1}=a_{2}+b_{2}.$ However, for generic one-parameter families $%
(K_{\mu },K_{\mu }^{\prime })$ of pairs of regular Cantor sets (contained in
the complement of a countable union of codimension 2 submanifolds), we never
have $a_{1}+b_{1}=a_{2}+b_{2}=a_{3}+b_{3}$ for $a_{1}$,$a_{2},a_{3}$
distinct extremes of gaps of $K_{\mu }$ and $b_{1},b_{2},b_{3}$ distinct
extremes of gaps of $K_{\mu }^{\prime },\,\forall \mu \in (-1,1).$

For these families, we may apply a variation of the above criterium, and in
order to do that, let us first make a definition.

Given a compact set $X$ and a point $a\in X,$ we define the \textit{%
logarithmic density} $\widehat{d}(X,a)$ of $X$ at $a$ as follows: $\widehat{d}(X,a)=%
\underset{M\rightarrow \infty }{\limsup }\frac{m(\{t\in \lbrack
0,M]\,;\,a+e^{-t}\in X\text{ or }a-e^{-t}\in X\})}{M},$

where $m$ denotes Lebesgue measure.

\begin{example}
The set $X = \bigcup_{n=1}^\infty [\frac{1}{n!} , \frac{2}{n!}]  $ does not have zero density at 0 but it has zero logarithmic density at 0.
\end{example}

Now, let $K,\overset{\sim }{K}$ be two regular Cantor sets with left extreme
points $a,\overset{\sim }{a},$ respectively, and invariant by contractions:
there are $\lambda _{1},\lambda _{2}>1$ such that, putting $g_{1}(x)=\lambda
_{1}^{-1}(x-a)+a$ and $g_{2}(x)=\lambda _{2}^{-1}(x-\overset{\sim }{a})+%
\overset{\sim }{a}$ then $g_{1}(K)$ and $g_{2}(\overset{\sim }{K})$ are
respectively open and closed neighbourhoods of $a$ and $\overset{\sim }{a}$
in $K$ and $\overset{\sim }{K}$ which are intersections of $K$ and $\overset{%
\sim }{K}$ with intervals. We claim that, given $\epsilon >0$, the set of
pairs of regular Cantor sets $K,\overset{\sim }{K}$ as before such that, for
every $\lambda >0$,$\,\widehat{d}(K+\lambda \overset{\sim }{K},a+\lambda 
\overset{\sim }{a})<\epsilon $ is open. Indeed, if $d$ and $\overset{\sim }{d}$ are
the diameters of $K$ and $\overset{\sim }{K}$ and $\delta >0 $ is such that $%
\delta <\lambda _{1}^{-m}d$ and $\delta <\lambda _{2}^{-n}\overset{\sim }{d}$ then $%
a+\lambda \overset{\sim }{a}+e^{-t}\in K+\lambda \overset{\sim }{K}$ if and
only if $a+\lambda \lambda _{1}^{m}\lambda _{2}^{-n}\overset{\sim }{a}%
+e^{m\log \lambda _{1}-t}\in K+\lambda \lambda _{1}^{m}\lambda _{2}^{-n}%
\overset{\sim }{K}.$

We may take $A=\max \{2\lambda _{1},2\lambda _{2}\}$ and $M\gg 1$ such that

\begin{center}
$\frac{m(\{t\in \lbrack 0,M]\,;\,a+\lambda \overset{\sim }{a}+e^{-t}\in
K+\lambda \overset{\sim }{K}\})}{M}<\epsilon ,$
\end{center}

$\forall \lambda \in \lbrack A^{-1},A].$

Since for every $\lambda >0,m\in \mathbb{N}$ it is possible to find $n\in 
\mathbb{N}$ such that $\lambda \lambda _{1}^{m}\lambda _{2}^{-n}\in
(A^{-1},A),$ it is not difficult to conclude our claim.

Notice that we can replace left extremes by right extremes in our previous
discussion.

Notice also that, if $K$ and $\overset{\sim }{K}$ are as before, $g,\overset{%
\sim }{g}$ are orientation-preserving diffeomorphisms, and $\widehat{d}%
(K+\lambda \overset{\sim }{K},a+\lambda \overset{\sim }{a})<\epsilon $ for
every $\lambda >0$ then \linebreak $\widehat{d}(g(K)+\overset{\sim }{g}(\overset{\sim }{%
K}),g(a)+\overset{\sim }{g}(\overset{\sim }{a}))<\epsilon ,$ by the
self-similarity of $K$ and $\overset{\sim }{K},$ and the fact that rescaling
of small neighbourhoods of $g(a)$ in $g(K)$ (resp. of $\overset{\sim }{g}(%
\overset{\sim }{a})$ in $\overset{\sim }{g}(\overset{\sim }{K})$) converge
in the Hausdorff metric to $K$ (resp. to $\overset{\sim }{K}$ $).$

Now we have the following

\begin{lemma}
: Suppose that $K$ and $\overset{\sim }{K}$ are as before, $HD(K)+HD(%
\overset{\sim }{K})=1$ and $\frac{\log \lambda _{1}}{\log \lambda _{2}}%
\notin \mathbb{Q}.$Then $\widehat{d}(K+\lambda \overset{\sim }{K},a+\lambda 
\overset{\sim }{a})=0$ for every $\lambda >0.$
\end{lemma}

\begin{proof}
: We know that $m(K+\lambda \overset{\sim }{K})=0$ for almost every $\lambda
.$ Given $m\in \mathbb{N}$ and $\lambda >0$ there is a unique $n(m)\in 
\mathbb{Z}$ such that $\lambda \lambda _{1}^{m}\lambda _{2}^{-n(m)}\in $ [1,$%
\lambda _{2})$ (indeed $n(m)=\left\lfloor \frac{\log \lambda +m\log \lambda
_{1}}{\log \lambda _{2}}\right\rfloor ),$ and $\frac{\log (\lambda \lambda
_{1}^{m}\lambda _{2}^{-n(m)})}{\log \lambda _{2}}=\left( \frac{\log \lambda 
}{\log \lambda _{2}}+m\frac{\log \lambda _{1}}{\log \lambda _{2}}\right)
-\left\lfloor \frac{\log \lambda }{\log \lambda _{2}}+m\frac{\log \lambda
_{1}}{\log \lambda _{2}}\right\rfloor $ is uniformly distributed in [0,1)
when $m$ varies.Given $\epsilon >0$ and $M_{0}\gg 1,$ there is an open set $%
L\subset \lbrack 1,\lambda _{2})$ with $m([1,\lambda _{2})\backslash
L)<\epsilon $ and $M>M_{0}$ such that $\frac{m(\{t\in \lbrack
0,M]\,;\,a+\lambda \overset{\sim }{a}+e^{-t}\in K+\lambda \overset{\sim }{K}%
\})}{M}<\epsilon ,$ for every $\lambda \in L.$ This implies that, for every $%
\lambda >0,\widehat{d}(K+\lambda \overset{\sim }{K},a+\lambda \overset{\sim 
}{a})<2\epsilon $ (since for most values of $m\in \mathbb{N}$, $\lambda
\lambda _{1}^{m}\lambda _{2}^{-n(m)}\in $ $L$), and this clearly implies the
Lemma.
\end{proof}

Now let $(K_{\mu },K_{\mu }^{\prime })\,$be a family of pairs of regular
Cantor sets as in the statement of Theorem 1. We can apply the above Lemma
to the pairs of Cantor sets $(K_{0,p}$,$K_{0,p^{\prime }}^{\prime }),$ where 
$(p,p^{\prime })\in P(K_{0})\times P^{\prime }(K_{0}^{\prime })$, and the
previous claim and remarks in order to show that, if $a+b,a\in K_{\mu },b\in
K_{\mu }^{\prime },$ then taking open and closed neighbourhoods $\overset{%
\sim }{K_{\mu }}$ and $\overset{\sim }{K_{\mu }^{\prime }}$ of $a$ and $b$
in $K_{\mu }$ and $K_{\mu }^{\prime }$ respectively, diffeomorphic to some
sets $K_{\mu ,p}$ and $K_{\mu ,p^{\prime }}^{\prime },$ where $(p,p^{\prime
})\in P(K_{\mu })\times P^{\prime }(K_{\mu }^{\prime })$, we have $\widehat{d%
}(\overset{\sim }{K_{\mu }}+\overset{\sim }{K_{\mu }^{\prime }},a+b)<\frac{1%
}{2}$ for $\mu $ small enough.

Since each extreme $x\,$\ of a gap of $K_{\mu }+K_{\mu }^{\prime }$ can be
written in at most two ways as $a+b$, where $a\in K_{\mu },b\in K_{\mu
}^{\prime }$, say $x=a_{1}+b_{1}=a_{2}+b_{2},$ there is $\delta >0$ such
that $(K_{\mu }+K_{\mu }^{\prime })\cap (x-\delta ,x+\delta )=$

\begin{center}
$\left( (\overset{\sim }{K_{\mu }}^{(1)}+\overset{\sim }{K_{\mu }^{\prime }}%
^{(1)})\cap (a_{1}+b_{1}-\delta ,a_{1}+b_{1}+\delta )\right) \cup \left( (%
\overset{\sim }{K_{\mu }}^{(2)}+\overset{\sim }{K_{\mu }^{\prime }}%
^{(2)})\cap (a_{2}+b_{2}-\delta ,a_{2}+b_{2}+\delta )\right) ,$
\end{center}

where $\overset{\sim }{K_{\mu }}^{(1)},\overset{\sim }{K_{\mu }}^{(2)},%
\overset{\sim }{K_{\mu }^{\prime }}^{(1)},\overset{\sim }{K_{\mu }^{\prime }}%
^{(2)}$ are the above mentioned open and closed neighbourhoods of $%
a_{1},b_{1},a_{2}$ and $b_{2}$ in $K_{\mu }$ or $K_{\mu }^{\prime }.$ Now,
since for every sets $\ X,Y$ and $a\in \mathbb{R}$ we have $\widehat{d}%
(X\cup Y,a)\leq \widehat{d}(X,a)+\widehat{d}(Y,a),$ we have \linebreak $\widehat{d}%
(K_{\mu }+K_{\mu }^{\prime },x)<\frac{1}{2}+\frac{1}{2}=1,$ so $x$ is
accumulated by gaps of $K_{\mu }+K_{\mu }^{\prime }$, and this is enough to
conclude the proof of Theorem 1.$\square $

\begin{remark}
The above proof of Theorem 1 gives the existence of an open set $\overset{%
\sim }{\mathcal{U}}$ of pairs of regular Cantor sets such that $\overset{%
\sim }{\mathcal{U}}\cap \mathcal{H}$ is (open and) dense such that if $(K,%
\overset{\sim }{K})\in \overset{\sim }{\mathcal{U}}$ and $K,\overset{\sim }{K}$
are linearized, i.e., self-similar near their left extreme points $a,%
\overset{\sim }{a}$ and $g,\overset{\sim }{g}$ are orientation-preserving
diffeomorphisms then $\widehat{d}(g(K)+\overset{\sim }{g}(\overset{\sim }{K}%
),g(a)+\overset{\sim }{g}(\overset{\sim }{a}))<\frac{1}{2}.$ We may also
change $\frac{1}{2}$ by any fixed $\epsilon >0$ and get a similar result.
\end{remark}

\begin{remark}
We can adapt this proof in order to show an analogous version of Theorem 1
for families with $k\,$\ parameters for any $k\in \mathbb{N}$: for a
residual set of families of pairs of regular Cantor sets $(K_{\mu },K_{\mu
}^{\prime }),\mu \in (-1,1)^{k},$ if $d(\mu )=HD(K_{\mu })+HD(K_{\mu
}^{\prime }),$ there is a neighbourhood $V$ of $\{\mu \in
(-1,1)^{k}\,:\,d(\mu )=1\}$ such that $\mu \in V,\,d(\mu )>1$ implies
that $K_{\mu }+K_{\mu }^{\prime }$ is an M-Cantorval. Indeed, it is enough
to take $\epsilon =\frac{1}{k+1}$ in the previous remark and to use the fact
that for generic $k$-parameter families of regular Cantor sets (contained in
the complement of a countable union of codimension $k+1$ submanifolds) we
never\ have distinct extremes of gaps $a_{1},a_{2},..,a_{k+2}$ of $K_{\mu }$
and $b_{1},b_{2},..,b_{k+2}$ of $K_{\mu }^{\prime }$ with $%
a_{i}+b_{i}=a_{j}+b_{j},\,\forall \,i,j\leq k+2,$ for any $\mu \in
(-1,1)^{k}.$
\end{remark}

\section{Homoclinic Bifurcations}

We will prove that in homoclinic bifurcations in surfaces associated to horseshoes with
Hausdorff dimension close to 1, one has positive density of hyperbolicity at
the first parameter of bifurcation:

Consider a first homoclinic bifurcation associated to a hyperbolic set
$\La$, i.e.,  
a family of diffeomorphisms $\vr_\mu\colon M^2 \to M^2$, $\mu\in(-1,1)$
such that:

$\bullet$ $\vr_\mu$ is persistently hyperbolic for $\mu<0$;

$\bullet$ $\vr_0$ presents a homoclinic tangency associated to the periodic
saddle point $p_0$ belonging to the hyperbolic set $\La$, which unfolds into 
transversal homoclinic orbits for $\mu>0$.

Let $(\overset{\sim }{K}_{1})_{0}$ and $(\overset{\sim }{K}_{2})_{0}$ be the linearizations of 
the stable and the unstable Cantor sets of $\La$. Suppose that $((\overset{\sim }{K}_{1})_{0}$, 
$(\overset{\sim }{K}_{2})_{0})$ belongs to the open set $\overset{\sim }{\mathcal{U}}$ described in Remark
6 above. 
Recall that $\overset{\sim }{\mathcal{U}} \cap \mathcal{H}$ is dense in $\mathcal{H}$. Then we have
the following

 \begin{theorem}
   Under the above conditions, $(\vr_\mu)$ presents positive density of hyperbolicity at $\mu=0$. 
More precisely, he have 

\begin{center}
$(\ast )$ \ \ \ \ $\underset{\delta \rightarrow 0}{\liminf }\frac{m(\{\mu
\in \lbrack 0,\delta ]\,:\,\varphi _{\mu }\text{ is persistently hyperbolic}%
\})}{\delta }>0.$
\end{center}

   If $((\overset{\sim }{K}_{1})_{0}$, $(\overset{\sim }{K}_{2})_{0})$ belongs to 
$\overset{\sim }{\mathcal{U}} \cap \mathcal{U}$, where 
$\mathcal{U} \subset \Omega$ was defined in Section 2 (so $\overset{\sim }{\mathcal{U}} \cap \mathcal{U}$
 is a generic set in the intersection of $\Omega$ with a neighbourhood of $\mathcal{H}$) then both $(\ast )$
 and 

\begin{center}
 \ \ \ \ $\underset{\delta \rightarrow 0}{\liminf }\frac{m(\{\mu
\in \lbrack 0,\delta ]\,:\,\varphi _{\mu }\text{ presents persistent homoclinic tangencies}%
\})}{\delta }>0.$
\end{center}


hold. 
 \end{theorem}

 \begin{proof}  

 We will use the Theorem V.1 of \textbf{[Mo1]}, which we describe next.

At the first moment of bifurcation (at the parameter $\mu =0$) there is a
line of tangencies which intersects the stable and unstable foliations of $\La$ at
 the Cantor sets $(K_{1})_{\mu }$ and $(K_{2})_{\mu }$,
whose extreme points ($p$ and $p^{\prime }$, respectively) coincide (in the
sense that their support intervals intersect at one point). The sets $(\overset{%
\sim }{K}_{1})_{0}$ and $(\overset{\sim }{K}_{2})_{0}$ are the linearizizations of 
$(K_{1})_{0}$ and $(K_{2})_{0}$ relative to $p$ and $p^{^{\prime }}$,
respectively. The mentioned Theorem says that, if there exists $t>0$ for
which $(\overset{\sim }{K}_{1})_{0}+t$ is left-linked with $(\overset{\sim }{%
K}_{2})_{0}$ and $\left( (\overset{\sim }{K}_{1})_{0}+t\right) \cap (%
\overset{\sim }{K}_{2})_{0}=\emptyset $, then

\begin{center}
$(\ast )$ \ \ \ \ $\underset{\delta \rightarrow 0}{\liminf }\frac{m(\{\mu
\in \lbrack 0,\delta ]\,:\,\varphi _{\mu }\text{ is persistently hyperbolic}%
\})}{\delta }>0.$
\end{center}

In particular, if $((\overset{\sim }{K}_{1})_{0}, (\overset{\sim }{K}_{2})_{0})$ 
(or, equivalently, $((K_{1})_{0}$,$(K_{2})_{0})$) belongs to the open set $%
\overset{\sim }{\mathcal{U}}$ described at Remark 6 of the last section, the condition
above about $(\overset{\sim }{K}_{1})_{0}$ and $(\overset{\sim }{K}_{2})_{0}$, and therefore, the $(\ast )$ condition are both satisfied, persistently.

If $((\overset{\sim }{K}_{1})_{0}$,$(\overset{\sim }{K}_{2})_{0})\in 
\mathcal{U\cap }\overset{\sim }{\mathcal{U}}$, then the $(\ast )$\ condition
and

\begin{center}
$(\ast \ast )$ \ \ \ \ $\underset{\delta \rightarrow 0}{\liminf }\frac{%
m(\{\mu \in \lbrack 0,\delta ]\,:\,(K_{1})_{\mu }\cap (K_{2})_{\mu }\neq
\emptyset ,\text{ persistently}\})}{\delta }>0,$
\end{center}

both hold by Theorem I.2 of \textbf{[Mo1]}. This concludes the proof.
\end{proof}

\begin{remark}
The same techniques can be used to show that for generic families $\varphi
_{\mu ,s},$ $(\mu ,s)\in (-1,1)^{2}$ of diffeomorphisms such that, for each $%
s,\,\varphi _{0,s}$ presents a first homoclinic tangency and $%
d(s)=HD((K_{1})_{0,s})+HD((K_{2})_{0,s})$ satisfying $d(0)=1,d^{\prime
}(0)>0,$ we have that $(\ast )$ and $(\ast \ast )$ hold for fixed small
positive values of $s$ (where $\varphi _{\mu }$ is replaced by $\varphi
_{\mu ,s}),$ and moreover
\end{remark}

\begin{center}
$~\underset{s\rightarrow 0}{\lim }\underset{\delta \rightarrow 0}{\liminf }$%
\bigskip\ $\frac{m(\{\mu \in \lbrack 0,\delta ]\,:\,\varphi _{\mu ,s}\text{
is persistently hyperbolic}\})}{\delta }=1.\ $
\end{center}

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\end{document}
