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\title{ON THE TOPOLOGY OF ARITHMETIC SUMS OF REGULAR CANTOR SETS}


\begin{document}

\maketitle


\author{
{ Carlos G. Moreira}\footnote{Instituto de
Matem\'atica Pura e Aplicada. Estrada Dona Castorina 110, Rio
de Janeiro, Brasil. gugu@impa.br\,. Partially supported
by PRONEX-Dynamical Systems, IMPA CNPq, Brasil.} 
\and
{Eduardo  Mu\~{n}oz Morales}\footnote{Universidad Cat\'olica del Norte.
Avenida Angamos \ 0610, Antofagasta, Chile.  
\newline emunoz@socompa.ucn.cl\,. Partially supported by Fondecyt Grant
1981241, IMPA CNPq and UCN}
   \and 
{Juan Rivera-Letelier}\footnote{  Universidad Cat\'olica del
Norte.Avenida Angamos \ 0610, Antofagasta, Chile.  
\newline juanrive@socompa.ucn.cl  and  Universit\'e de Paris-Sud 
Laboratorie de Topologie et Dinamique
Batiment 425, 91405 Orsay Cedex, Orsay,France. 
rivera@topo.math.u-psud.fr\,. Partially
supported by  IMPA CNPq and UCN. } 
}

\vspace{0.5cm}


\begin{abstract} The study of geometrical and topological properties of arithmetic
sums and differences of regular Cantor sets appears naturally in
distinct fields as dynamical systems (particularly, in the study of homoclinic bifurcations related to non-trivial hyperbolic sets) and number theory (particularly, in the study of geometrical properties of the Markov and Lagrange spectra, related to diophantine approximations). We study the topological structure of the sum of two regular
Cantor sets and we obtain some local results related to this
problem; more precisely, we give persistent examples of several
different topological types of these sums or differences.
\end{abstract}

\section{ Introduction}

\vspace{0.5cm}
The study of geometrical and topological properties of arithmetic
sums and differences of regular Cantor sets appears naturally in
distinct fields as dynamical systems (e.g. in the study of
homoclinic bifurcations associated to non trivial hyperbolic sets)
and number theory (e.g. in the study of the Markov and Lagrange
spectra associated to Diophantine approximations).

Regular Cantor sets are, roughly speaking, defined by an iterate system of
expanding differentiable functions on intervals. It is not difficult to see
that if the sum of the Hausdorff dimensions of two regular Cantor sets \ $%
K_{1}$ \ and \ $K_{2}$ \ is smaller \ than \ one \ then \ the \
arithmetic \ sum \ $K_{1}+K_{2}=\left\{ x+y\;\left| x\in
K_{1},\;\;y\in K_{2}\right. \right\} $ \ and also the arithmetic
difference $K_{1}-K_{2}=\left\{ x-y\left| x\in K_{1},\;\;y\in
K_{2}\right. \right\} $ \ both have zero Lebesgue measure. \
Recently, Moreira and Yoccoz proved a conjecture of J. Palis,
according to which, generically, if the sum of the Hausdorff
dimensions of \
$K_{1}$ \ and\ $\ K_{2}$\ is bigger than one then \ $K_{1}+K_{2}$ \ (and \ $%
K_{1}-K_{2})$ \ contains intervals. \ More precisely, it was shown that
there is an open, dense and in some sense a total measure subset ${\cal U}
$ of \ $\medskip $\newline
$\Omega =\{\left( K_{1},K_{2}\right) \mid K_{1}\ $and$\ K_{2}$ regular
Cantor sets, $HD\left( K_{1}\right) +HD\left( K_{2}\right) >1\}$ \medskip
\newline
(here \ $HD$ \ denotes Hausdorff dimension) which is dense in the \ $%
C^{\infty }$ \ topology such that if \ $\left( K_{1},K_{2}\right) \in \Omega
$ \ then \ $K_{1}+K_{2}$ \ and \ $K_{1}-K_{2}$ \ have stably dense interior
(see $\left[ MY\right] $).

Mendes and Oliveira studied in $\left[ MO\right] $ the topological structure
of the sum of some class of affine Cantor sets in the interval $\left[ 0,1%
\right] $. In that paper they pose the question of what can be said about
the topological structure of the sum of two regular Cantor sets. \ We will
give here some local results about this problem. \ Since \ $%
K_{1}-K_{2}=K_{1}+\left( -K_{2}\right) $ \ we will restrict our attention to
sums instead of differences.

If \ $HD\left( K_{1}\right) +HD\left( K_{2}\right) <1$ \ then \
$K_{1}+K_{2}$ \ has zero Lebesgue measure, and so is a Cantor set
(recall that any two Cantor sets on the real line are
homeomorphic). \ So we will be concerned about sums of regular
Cantor sets such that the sum of their Hausdorff dimensions is
bigger than one. \ As we mentioned before, these sums have
generically dense interior. \ This fact is of course not enough to
characterize its topological structure. \ However, there are some
regions in \ $\Omega $ \ where we can characterize or give partial
descriptions of the topological structure of \ $K_{1}+K_{2}$ \ for
generic pairs \ $\left( K_{1},K_{2}\right) $ \ in the considered
region. \ The simplest example is\ the region of pairs $\;\left(
K_{1},K_{2}\right) $ $\;$such that $\ \tau \left( K_{1}\right)
\tau \left( K_{2}\right) >1$, where\ $\tau $\ denotes the
thickness defined by Newhouse, see for instance \ $\left[ N\right] $
\ or \ $\left[ PT\right] $. \ In
this open region\ $K_{1}+K_{2}$\ is stably a finite union of closed
intervals (see example 1 in section 4).\ \ The same happens in a larger open
set\ $\mathcal{U}^{0}$\ defined in\ $\left[ M\right] $.

\vspace{0.5cm}

\section{Background and Global Facts}

\vspace{0.5cm}
In this section we introduce some preliminary concepts and definitions. \
For \ $X\subseteq \mathbb{R}$, $\widehat{X}$ \ denotes the {\it convex
hull} of \ $X$ \ and \ $\left| X\right| =\left| \widehat{X}\right| $ \ the
{\it length} of the interval \ $\widehat{X}$, which can be eventually
infinite. \ When \ $X$ \ is compact, a connected component of \ $\mathbb{R}%
\backslash X$ \ is called a \textit{gap} of \ $X$, and a non trivial
connected component of \ $X$ \ is called an {\it interval} of \ $X$.
A \textit{Cantor set }\ $K\subseteq \mathbb{R}$ \ is a compact perfect set
with empty interior. \ A \textit{regular Cantor set }is defined whenever:
\begin{enumerate}
\item[(i)]  \textit{there exist pairwise disjoint compact intervals \ }$%
I_{1},I_{2},\ldots ,I_{r}\subseteq I_{0}=\widehat{K}$ {\it  \ with \ }$%
K\subseteq \bigcup\limits_{i=1}^{r}I_{i}$\textit{,}

\item[(ii)]  \textit{there exist expanding functions \ }$\psi
_{i}:I_{i}\longrightarrow I_{0}$\textit{\ \ of class \ }$C^{1+\varepsilon }$%
\textit{\ \ whose images are convex hull of disjoint unions of the \ }$I_{i}$%
\textit{\ \ such that \ if \ }$\psi
:\bigcup\limits_{i=1}^{r}I_{i}\longrightarrow I_{0}$\textit{\ \ is defined
by \ }$\psi \left| _{I_{i}}=\psi _{i}\right. $\textit{\ \ then for each \ }$%
i $\textit{\ \ there exists \ }$n_{i}$\textit{\ \ with \ }$\psi
^{n_{i}}\left( K\cap I_{i}\right) =K\cap I_{0}$\textit{\ \ and besides, \ }$%
K=\bigcap\limits_{n\geqslant 0}\psi ^{-n}\left(
\bigcup\limits_{i=1}^{r}I_{i}\right) $\textit{.}
\end{enumerate}

A map \ $\psi $ \ is \ $C^{1+\varepsilon }$ \ if \ $\psi $ \ is \
$C^{1}$ \ and there exists \ $k>0$ \ such that \ $\left| \psi
^{\prime }\left( x\right) -\psi ^{\prime }\left( y\right) \right|
\leqslant k\left| x-y\right| ^{\varepsilon }$ \ for all \ $x,y$. \
$C$ $^{1+}$ \ denotes the union of the spaces \ $C^{1+\varepsilon
}, \varepsilon >0$, with the induced topology. \ In particular,
when the \ $\psi _{i}$ \ are affine functions we say that the
Cantor set is affine. \ Note that for a regular Cantor set \ $K$, $%
f(K)$ \ is a regular Cantor set for every \ $C^{1+}$ \ diffeomorphism \ $f$ \
of \ $\mathbb{R}$.

To define the \textit{thickness} of a Cantor set \ $K$, denoted by $\tau
\left( K\right) ,$\ we consider a gap \ $\mathcal{U}$ \ of \ $K$, and let \ $%
I_{\mathcal{U}}$ \ be the interval bounded by \ $\mathcal{U}$ \ and the
closest gap \ $\mathcal{V}$ \ that is larger than \ $\mathcal{U}$.
\ Then the thickness of \ $K$ is
\begin{equation*}
\tau \left( K \right) =\inf \left\{ \frac{\left| I_{\mathcal{U}%
}\right| }{\left| \mathcal{U}\right| } \ \text{\ ;
}\mathcal{U}\text{ \ is a bounded gap of\ }K \right\} .
\end{equation*}
Note that $\ \tau \left( K \right) =\tau \left( -K \right) $.\ The
importance of the thickness is the following lemma: \

\vspace{0.5cm}

\textbf{Gap-lemma:\ }\textit{For given}\textbf{\ }$K_{1}$\ \textit{and}\ $%
K_{2}$\ \textit{Cantor sets with}\ $\tau \left( K_{1}\right) \tau \left(
K_{2}\right) >1$ \ \textit{then exactly one of the following three
possibilities holds:} \
\begin{itemize}
\item  $\widehat{K}_{1}$ \ \textit{is contained in a gap of }\ $K_{2}.$
\item  $\widehat{K}_{2}$ \ \textit{is contained in a gap of} \ $K_{1}.$
\item  $K_{1}\cap K_{2}$ \ \textit{is non empty.}
\end{itemize}\

Thus, if $\tau \left( K_{1}\right) \tau \left( K_{2}\right) >1$, then $%
K_{1}+K_{2}$ is a finite union of intervals. Finally, we recall that the
thickness, as well as the Hausdorff dimension of a regular Cantor set \
$K$ \ \ depends\textit{\ }continuously on \ $K$ \ (see $\left[ PT\right] $ ).

Given a Cantor set \ $K\subseteq \mathbb{R}$ \ and a finite
covering of \ $K$ \ by open intervals \ $\mathcal{U}=\left\{
{U}_{i}\right\} _{i\in \mathcal{A}}$, put
$\,\,|\mathcal{U}|=max\left\{ \left| U_{i}\right| :i\in
\mathcal{A}\right\}. \,
$
Given \ $\alpha \in \mathbb{R}$, we set \ $H_{\alpha }\left( \mathcal{U}%
\right) =\sum_{i\in \mathcal{A}}\left| U_{i}\right| ^{\alpha }$, and we
define the Hausdorff $\alpha -$measure of \ $K$,
\begin{equation*}
m_{\alpha }\left( K\right) =\underset{\epsilon \longrightarrow 0}{\lim }%
(\inf \left\{ H_{\alpha }\left( \mathcal{U}\right) \mid \mathcal{U}\ \text{%
is a covering of\ K\ with\ }\left| \mathcal{U}\right| <\epsilon \right\} ).
\end{equation*}

The {\it \ Hausdorff dimension of }\ $K$ \ is defined as being
the unique real number \ $HD\left( K\right) $ \ such that

\begin{equation*}
m_{\alpha }\left( K\right) =\left\{
\begin{array}{ll}
+\infty \text{,} & \text{for \ \ }\alpha <HD\left( K\right). \\
0\text{,} & \text{for \ \ }\alpha >HD\left( K\right).
\end{array}
\right.
\end{equation*}




\section{ Local Results}

\vspace{0.5cm}
In this section some local results are presented.\ To start with,
it is considered a simple case where the arithmetic sum is stably
an interval.
\medskip

\noindent {\bf Example 0:} For \ $\alpha \in \left(
0,\dfrac{1}{2}\right) $ \ let \ $K_{\alpha }$ \ be the affine
Cantor set which is the maximal
invariant set contained in \ $\left[ 0,1\right] $ \ of \ $\psi _{1}$ and $%
\psi _{2}$ \ which are the orientation preserving affine functions such that
\ $\psi _{1}\left( \left[ 0,\alpha \right] \right) =\psi _{2}\left( \left[
1-\alpha ,1\right] \right) =\left[ 0,1\right] $. \ Then, for \ $\alpha >%
\frac{1}{3}\,$and $K_{1},K_{2}$ close to $K_{\alpha }$,\ $K_{1}+K_{2}$ \ is
an interval.

Note that \ $\tau \left( K_{\alpha }\right) =\dfrac{\alpha
}{1-2\alpha }$ \ so, if \ $\alpha >\dfrac{1}{3}$ , $\tau \left(
K_{\alpha }\right) >1$, and since \ $\tau $ \ is continuous, there
is a neighbourhood \ $\mathcal{U}$ \ of \ $K_{\alpha }$ \ such
that \ $\tau \left( K\right) >1$ \ for all \ $K\in \mathcal{U}$. \
Moreover, by reducing \ $\mathcal{U}$ if necessary, assume that
for all \ $K_{1}$, $K_{2}\in \mathcal{U}$, every gap of \ $K_{2}$
\ is shorter than \ $\left| K_{1}\right| $. Then, by the
Gap-lemma, $K_{1}+K_{2}$
\ is an interval, for all \ $K_{1}$, $K_{2}\in \mathcal{U}$.\vspace{0.5cm}

\noindent  \textbf{Remark: \ }This example can be generalized, so
that the arithmetic sum is stably the union of \ $n>1$ \ disjoint
intervals. In fact, let \ $n>1$, $\epsilon \in \left( 0,1\right) $
\ and \ $K_{\varepsilon }$ \
be the Cantor set generated by the orientation preserving affine maps \ $%
\psi _{0},\ldots ,\psi _{n-1}$ \ such that \ $\psi _{i}\left( \left[
4i,4i+2+\epsilon \right] \right) =\left[ 0,4n-2+\epsilon \right] $. \ Thus, $%
\tau \left( K_{\varepsilon }\right) =\dfrac{2+\epsilon
}{2-\epsilon }>1$ \ and there is a neighbourhood \ $\mathcal{V}$ \
of \ $K_{\varepsilon }$, such that for all \ $K_{2}\in
\mathcal{V}$, $\tau \left( K_{2}\right) >1$. \ Moreover, by
reducing \ $\mathcal{V}$ \ and \ $\mathcal{U}$ neighbourhoods as
in the previous example, if necessary, assume that for all \
$K_{1}\in \mathcal{U}$,$\;\;\emph{K}_{2}\in \mathcal{V} $, $\
\left| \emph{K}_{1}\right| <\left| \emph{K}_{2}\right| $, and \
$K_{2}$ has exactly \ $n-1$ \ gaps longer than \ $\left|
K_{1}\right| $. \ Then, by the Gap-lemma, $K_{1}+K_{2}$ \ is a
disjoint union of \ $n$ \ intervals, for all \ $K_{1}\in
\mathcal{U}$ \ and \ $K_{2}\in \mathcal{V}$.\

\vspace{0.5cm}

In general, the arithmetic sum of pairs of regular Cantor sets on
the line is more complex, and it can be classified according to
the nature of its different types of gaps. \ It can be
distinguished four kind of bounded gaps :
\begin{itemize}
\item  $LR-gap$ is a gap, between intervals,
\item  $R-gap$ is a gap with an interval only at its left extreme,
\item  $L-gap$ is a gap with an interval only at its right extreme,
\item  $C-gap$ is a Cantor-like gap, i.e. it does not contain
intervals at its extremes.
\end{itemize}


\noindent {\bf Remark: \ }In this description an interval means
a non-trivial interval contained in the sum of the Cantor sets.

\vspace{0.5cm}

$[ MO ]$ introduced the concept of an
$M$-Cantorval. \ It is a compact perfect set on the real line with
dense interior, whose all
bounded gaps are $C$-gaps. \ We will give here an example of an open set \ $%
\mathcal{U}_{1}\subset \Omega $ \ such that, for generic pairs \ $\left(
K_{1},K_{2}\right) \in \mathcal{U}_{1}$, $K_{1}+K_{2}$ \ is an \ $M$%
-Cantorval. \ It is proved in $\left[ MO\right] $ that all $M$-Cantorvals
are homeomorphic, so it characterizes the topology of \ $K_{1}+K_{2}$ \ for
generic \ $\left( K_{1},K_{2}\right) $ \ in \ $\mathcal{U}_{1}$. \ Suppose
that \ $K_{1}$ \ and \ $K_{2}$ \ are regular Cantor sets defined by increasing surjective
differentiable expanding functions \ $\psi _{j}^{\left( i\right)
}:I_{j}^{\left( i\right) }\longrightarrow I_{j}$ \ \ such that if
\newline  $\psi
_{j}:I_{j}^{\left( 1\right) }\cup I_{j}^{\left( 2\right) }\cup ...\cup
I_{j}^{\left( n_{j}\right) }\longrightarrow I_{j}$ \ \ \ is defined by \ $%
\;\;\psi _{j}\left| _{I_{j}^{\left( i\right) }}\right. =\psi _{j}^{\left(
i\right) }$ \ then
\begin{eqnarray*}
K_{j} &=&\bigcap\limits_{n\geqslant 0}{\psi_j}^{-n}\left(
I_{j}^{\left( 1\right) }\cup I_{j}^{\left( 2\right) }\cup ...\cup
I_{j}^{\left( n_{j}\right) }\right) \\ &=&\left\{ x\in
I_{j}^{\left( 1\right) }\cup ...\cup I_{j}^{\left( n_{j}\right)
}\left| \psi _{j}^{n}\left( x\right) \in I_{j}^{\left( 1\right)
}\cup ...\cup I_{j}^{\left( n_{j}\right) },\;\;\;\forall
\;n\geqslant 0\right. \right\}.
\end{eqnarray*}
We can order the intervals \ $I_{j}^{\left( i\right) }$ \ such that the left
extreme of \ $I_{j}$ \ belongs to \ $I_{j}^{\left( 1\right) }$ \ and the
right extreme of \ $I_{j}$ \ belongs to \ $I_{j}^{\left( n_{j}\right) }$.

Given two such Cantor sets \ $K_{1}$ \ and \ $K_{2}$ \ there are
diffeomorphisms \ $\varphi _{j}$ \ of \ $I_{j}$, $j=1,2$ \ such that \ $%
\varphi _{1}\circ \psi _{1}^{\left( n_{1}\right) }\circ \varphi _{1}^{-1}$ \
and \ $\varphi _{2}\circ \psi _{2}^{(1)}\circ \varphi _{2}^{-1}$ \ are
affine, with slope \ $\lambda _{1}$ \ and \ $\lambda _{2}$ \ respectively. \
We define the \textit{right linearized Cantor set} of \ $K_{1}$ \ as \ $\widetilde{K}%
_{1}^{R}:=\varphi _{1}\left( K_{1}\right) $, and the \textit{left linearized Cantor
set} of \ $K_{2}$ \ as \ $\widetilde{K}_{2}^{L}:=\varphi _{2}\left(
K_{2}\right) $, so that \ $\widetilde{K}_{1}^{R}$ \ and \ $\widetilde{K}%
_{2}^{L}$ \ are defined by the functions \ $\widetilde{\psi }%
_{1}^{R}:=\varphi _{1}\circ \psi _{1}\circ \varphi _{1}^{-1}\,$and $%
\widetilde{\psi }_{2}^{L}:=\varphi _{2}\circ \psi _{2}\circ \varphi
_{2}^{-1} $.

An interval \ $J_{1}=\left[ a,b\right] $ \ is \textit{left linked} with \ $J_{2}=%
\left[ c,d\right] $ \ if \ $a<c,\;b<d$ \ and \ $c<b$. \ We say
that a pair of Cantor sets \ $\left( K_{1},K_{2}\right) $ \ as
before satisfies the \ \textit{*$L$ \ property} if there are \ $\lambda
>0,\;t\in \mathbb{R}$ \ such that the support interval of \
$\widetilde{K}_{1}^{R}$ \ is left-linked with the
support interval of \ $\lambda \widetilde{K}_{2}^{L}+t$ \ and \ $\widetilde{K%
}_{1}^{R}\cap \left( \lambda \widetilde{K}_{2}^{L}+t\right)
=\varnothing $. The \textit{*$R$ \ property} is defined in the symmetric
way:\ $\left( K_{1},K_{2}\right) $ \ satisfies the \ *$R$ \
property if and only if \ $\left( K_{2},K_{1}\right) $ \ satisfies
the \ *$L$ \ property .\

It was proved in \ $\left[ M\right] $ \ (Theorem IV. 1) that if \ $\dfrac{%
\ln \lambda _{1}}{\ln \lambda _{2}}\notin Q$, then for \ $\lambda >0$, $%
K_{1}+\left( -\lambda K_{2}\right) $ \ does not contains any
interval containing the left extreme of \ $K_{1}+\left( -\lambda
K_{2}\right) $. \ As a consequence we can prove the proposition
below.

\vspace{0.5cm}
\noindent {\bf Remark:} The results of the all following propositions hold with
total measure in generic 1-parameter families.

\begin{proposition}
\textit{In the open set \ }$\mathcal{U}_{1}=\left\{ \left( K_{1},K_{2}\right) \right. $
\textit{\ pairs of regular Cantor sets defined by increasing
functions such that}\ \ $ \left( K_{1},-K_{2}\right) $ \textit{\ \
and } $ \left( K_{2},-K_{1}\right) $ \ \ \textit{ \ satisfy the } \ *$L$ \
 \textit{property and}\ \ $ \left. HD\left( K_{1}\right) +HD\left(
K_{2}\right) >1\right\} $.
\textit{\ Then
for pairs \ }$\left( K_{1},K_{2}\right) $\textit{\ \ in a residual subset \ }%
$\mathcal{U}_{1},\;K_{1}+K_{2}$\textit{\ \ is an }$M$\textit{%
-Cantorval.\bigskip }
\end{proposition}

\noindent {\bf Proof}.
Suppose that \ $\left( K_{1},\;-K_{2}\right) \in \mathcal{U}$, \ the open
and dense subset \ of \ $\Omega =\left\{ \left( K_{1},K_{2}\right) \left|
K_{1},K_{2}\,\text{regular Cantor sets, }HD\left( K_{1}\right) +HD\left(
K_{2}\right) >1\right. \right\} $ \ defined in the introduction. \ Then $%
K_{1}+K_{2}$ \ have dense interior, i.e. $K_{1}+K_{2}=\overline{%
int(K_{1}+K_{2}}$). \ Suppose moreover that for any \ $a,b$ \ extremes of
gaps of \ $K_{1}$ \ and \ $c,d$ \ extremes of gaps of \ $K_{2}$ \ we have \ $%
a+c\neq b+d$. \ This last property is generic, since for any \ $\left(
K_{1},K_{2}\right) $ \ there is only a countable subset of \ $\lambda \in
\mathbb{R}$ \ such that the property fails for \ $\left( K_{1},\lambda
K_{2}\right) $. \ Suppose moreover that the ratio of the logarithms of the
slopes of \ $\widetilde{\psi }_{1}^{L}\left| _{\widetilde{I}_{1}^{\left(
1\right) L}}\right. $ \ and \ $\widetilde{\psi }_{2}^{L}\left| _{\widetilde{I%
}_{2}^{\left( 1\right) L}}\right. $ \ is irrational and the same for \ $%
\widetilde{\psi }_{1}^{R}\left| _{\widetilde{I}_{1}^{\left( n_{1}\right)
L}}\right. $ \ and \ $\widetilde{\psi }_{2}^{R}\left| _{\widetilde{I}%
_{2}^{\left( n_{2}\right) L}}\right. $ \ \ $\left( \text{in our previous
notation}\ \widetilde{I}_{1}^{\left( n_{1}\right) R}:=\varphi _{1}\left(
I_{1}^{\left( n_{1}\right) }\right) \right. $ \ and \ $\left. \widetilde{I}%
_{2}^{\left( 1\right) L}:=\varphi _{2}\left( I_{2}^{\left( 1\right) }\right)
\right) $.

We claim that under these conditions \ $K_{1}+K_{2}$ \ is an $\
M$-Cantorval.

Indeed, an extreme of a gap of \ $K_{1}+K_{2}$ \ is always of the form \ $%
a+b $ \ where \ $a\in K_{1}$ \ and \ $b\in K_{2}$ \ are extremes of gaps. \
If we consider a left extreme, then \ $a$ \ and \ $b$ \ are left extremes of
gaps in \ $K_{1}\;\;$and \ $K_{2}$ \ respectively, so there are intervals of
the construction of \ $K_{1}$ \ and \ $K_{2}$, say \ $J_{1}$ \ and \ $J_{2}$
\ of which \ $a$ \ and \ $b$ \ are its left extremes, and if \ $\overline{K}%
_{1}=K_{1}\cap J_{1}$ \ and \ $\overline{K}_{2}=K_{2}\cap J_{2}$ \ then \ $%
\overline{K}_{1}$ \ and \ $\overline{K}_{2}$ \ are regular Cantor sets
conjugated to \ $K_{1}$ \ and \ $K_{2}$, and so their linearized Cantor sets
are similar to the linearized of \ $K_{1}$ \ and \ $K_{2}$.

Thus, by the hypothesis, \ $\overline{K}_{1}+\overline{K}_{2}$ \
does not contain any interval adjacent to \ $a+b$, and so \ $a+b$
\ is accumulated by gaps and intervals. \ Analogously the same is
true for right extremes. \
Since \ $\overline{K}_{1}+\overline{K}_{2}$ \ contains a neighbourhood of \ $%
a+b$ \ \ in \ $\ K_{1}+K_{2}$ \ (by uniqueness of \ $a$ \ and \ \ $b$), \ $%
K_{1}+K_{2}$ \ \ \ is an \ $M$-Cantorval, which completes the proof.
\bigskip

\noindent {\bf Example:} Consider the Cantor set \ $K$ \
which is the maximal invariant set contained in $\ \left[
0,23\right] $ \ of the orientation preserving affine functions \
$\psi _{i}$, $1\leqslant i\leqslant 4$ ,\ such that $\psi
_{i}(I_{i})=\left[ 0,23\right] \,$where
\begin{equation*}
I_{1}=\left[ 0,1\right] \,,I_{2}=\left[ 3,11\right] ,\,I_{3}=\left[ 12,20%
\right] \,\text{and}\,I_{4}=\left[ 22,23\right] .
\end{equation*}
Then, there is a neighborhood \ $\mathcal{U}$ \ of \ $K$ \ in the
\ $C^{1+}$ \ topology and a residual set \ $\mathcal{R}\subseteq
\mathcal{U}\times \mathcal{U}$ \ such that for all \ $\left(
K_{1},K_{2}\right) \in \mathcal{R} $, $K_{1}+K_{2}$ \ is an
$M$-Cantorval.

In fact, the maximal invariant set \ $\widetilde{K}\subseteq K$ \
of \ $\psi _{2}$ \ and \ $\psi _{3}$ \ has thickness \
$\dfrac{8}{7}>1$, so there is a neighbourhood \ $\mathcal{U}$ \ of
\ $K$, such that for each \ $K_{1}\in \mathcal{U}$ \ there exists
\ $\widetilde{K}_{1}\subseteq K_{1}$ \ a Cantor set with \ $\tau
\left( \widetilde{K}_{1}\right) >1$. \ Therefore, by the Gap-
lemma, for all \ $K_{1},\;K_{2}\in \mathcal{U}$, $K_{1}+K_{2}$ \
has dense interior$.$ Moreover, since \ $\left( K,K\right) $ \
satisfies the \ $\ast L$ \ and the
\ $\ast R$ \ properties defined above, there is a residual \ $\mathcal{R}%
\subseteq \mathcal{U}\times \mathcal{U}$\ \ such that every bounded gap of \
$K_{1}+K_{2}$ \ is a $C-$gap. \ It follows that for such \ $\left(
K_{1},K_{2}\right) $, $K_{1}+K_{2}$ \ is an $M-$Cantorval. 
\medskip

Similarly, as in $\ \left[ MO\right] $, an $R$-(\ resp. $L$-)
Cantorval is a perfect compact subset of the real line, with dense
interior, whose all its bounded gaps are $R$-(\ resp.
$L$-)$\,$gaps.

As in $\left[ M\right] $, we define {\it left extremal stable
intersection} as follows: \ $\left( K_{1},K_{2}\right) $ \ has left
extremal stable
intersection if there are \ $\delta >0$ \ and \ a \ neighbourhood \ $%
\mathcal{V}$ \ of \ $\left( K_{1},K_{2}\right) $ \ such that \ $\left(
\widetilde{K}_{1},\;\widetilde{K}_{2}\right) \in \mathcal{V}\Longrightarrow
\widetilde{K}_{1}+\left( -\widetilde{K}_{2}\right) $ \ $\ $contains an
interval of lenght \ $\delta $ \ to which the left extreme of \ $\widetilde{K%
}_{1}+\left( -\widetilde{K}_{2}\right) $ \ belongs. \ Then we have the
following proposition, whose proof is analogous to the previous one. 

\begin{proposition}
\textit{In the open set \ }$\mathcal{U}_{2}^{R\left( resp.\;L\right) }=\left\{ \left(
K_{1},K_{2}\right) \right. $\textit{\ \ pairs of regular Cantor sets defined
by increasing functions such that \ }$\left( K_{1},\;-K_{2}\right) $\textit{%
\ \ satisfy the \ *}$L$\textit{\ \ property and \ }$\left(
K_{1},\;-K_{2}\right) $\textit{\ \ have extremal stable intersection (resp. }%
$\left( -K_{2},\;K_{1}\right) $\textit{\ \ satisfy the \ *}$L$\textit{\ \
property and \ }$\left( -K_{2},\;K_{1}\right) $\textit{\ \ have extremal
stable intersection) \ and \ }$\left. HD\left( K_{1}\right) +HD\left(
K_{2}\right) >1\right\} $\textit{. \ Then for generic pairs \ }$\left(
K_{1},\;K_{2}\right) \in \mathcal{U}_{2}^{R(resp.\;L)}$\textit{, \ }$K_{1}+K_{2}$%
\textit{\ \ is an \ }$R(resp.\;L)$\textit{-Cantorval .}
\end{proposition}

\noindent {\bf Example:} If \ $K$ \ is defined by the
increasing affine functions $\psi _{i}$, $i=1,2,3$ \ such that \
\begin{equation*}
\psi _{1}\left( I_{i}\right) =\left[ 0,14\right] ,\;\text{where}\;I_{1}=%
\left[ 0,5\right] ,\;I_{2}=\left[ 6,11\right] \;\text{and}\;I_{3}=\left[
13,14\right].
\end{equation*}
\noindent Then there is a neighbourhood \ $\mathcal{U}$ \ of \ $K$ \ in the \ $%
C^{1+}$ \ topology such that for generic \ $\left(
K_{1},K_{2}\right) \in \mathcal{U}\times \mathcal{U}$ \ then \
$K_{1}+K_{2}$ \ is an $R-$Cantorval and for generic \ $\left(
-K_{1},-K_{2}\right) \in \mathcal{U}\times
\mathcal{U}$ \ then \ $K_{1}+K_{2}$ \ is an $R-$Cantorval. 
\bigskip

\begin{proposition}
\textit{Let \ }$K$\textit{\ \ be the affine Cantor set whose
convex hull is \ }$\left[ 0,24\right] $\textit{\ \ defined by the
affine functions \ }$\psi _{i}$\textit{, \ }$i=1,2,3$\textit{,
such that }
\begin{equation*}
\psi_{i}\left( I_{i}\right) =\left[ 0,24\right], \mathit{ \ where
}\;
 I_{1}=\left[ 0,10\right], \; I_{2}=\left[ 11,21 \right]
\; \mathit{ \ and } \ I_{3}=\left[ 23,24\right],
\end{equation*}
\noindent and only $\psi_{2}\ $ is decreasing.
\textit{\ \ Then there is a neighbourhood \ }$\mathcal{U}_3$\textit{\ \ of \ }$%
K$\textit{\ \ in the \ }$C^{1+}$\textit{\ \ topology such that if \ }$%
K_{1},K_{2}\in \mathcal{U}_3$\textit{\ \ then \ }$ \, K_{1}\, + \,
K_{2}$ \textit{\ \ is an infinite disjoint union of closed
intervals converging to a point.}
\end{proposition}

\noindent {\bf Proof}.
The maximal invariant set of \ $\psi _{1}$ \ and \ $\psi _{2}$ \ in \ $\left[
0,21\right] $ \ is an affine Cantor set \ $\widetilde{K}$ \ whose convex
hull is \ $\left[ 0,21\right] $ \ with thickness \ $\tau \left( \widetilde{K}%
\right) =\dfrac{5}{2}$, so \ $\tau \left( K\right) \tau \left( \widetilde{K}%
\right) =\dfrac{5}{4}>1$. \ This implies that \ $K$ \ has stable
intersection with \ $\lambda \widetilde{K}+t$ \ if \ $K$ \ and \ $\lambda
\widetilde{K}+t$ \ are linked.

If \ $I=\left[ 0,21\right] $ \ then \ $K\subset \left\{ 24\right\}
\cup \bigcup\limits_{n=0}^{\infty }\psi _{3}^{-n}\left( I\right)
$, and \ $\psi
_{3}^{-n}\left( I\right) \supset \psi _{3}^{-n}\left( \widetilde{K}\right) $%
, a Cantor set whose thickness is \ $\dfrac{5}{2}$ \ and whose convex hull
is \ $\psi _{3}^{-n}\left( I\right) $.

Let \ $K_{1},\;K_{2}$ \ be Cantor sets close to \ $K$ \ defined by
\ $\left( \psi _{1}^{\left( 1\right) },\psi _{2}^{\left( 1\right)
},\psi _{3}^{\left( 1\right) }\right) $ \ and \ $\left( \psi
_{1}^{\left( 2\right) },\psi _{2}^{\left( 2\right) },\psi
_{3}^{\left( 2\right) }\right) $ respectively. \ Let \ $%
\widetilde{K}_{1}$  and  $\widetilde{K}_{2}$ be constructed as \ $%
\widetilde{K}$. \ We may suppose \ $\tau \left(
\widetilde{K}_{1}\right) \tau \left( K_{2}\right) >1$ \ and \
$\tau \left( \widetilde{K}_{2}\right) \tau \left( K_{1}\right)
>1$, since they are close to \ $\dfrac{5}{4}$. It is not difficult
to check ( by self similarity, as in the Theorem IV.1 of \ $\left[
M\right] $ \ ) that the rightmost point of $K_{1}+K_{2}$ is
accumulated by a sequence of gaps of $K_{1}+K_{2}$.
Let \ $b_{1}$ \ and \ $b_{2}$ \ be the rightmost points of \ $%
K_{1\text{ \ }}$ and \ $K_{2}$, respectively. The results follows
from
\begin{eqnarray*}
K_{1}+K_{2} &\subset & \left\{b_{1} \ + \ b_{2} \right\}  \cup
\bigcup\limits_{n=0}^{\infty
}\left(\left(\psi_{3}^{(1)}\right)^{-n}\left( I^{(1)}\right)+
\left(\psi_{3}^{(2)}\right)^{-[\frac{n}{2}]}
\left(K_{2}\right)\right) \\
 & &  \cup \bigcup\limits_{n=0}^{\infty
}\left(\left(\psi_{3}^{(1)}\right)^{-[\frac{n}{2}]}\left(K_{1} \right)\ +
\left(\psi _{3}^{(2)}\right)^{-n}\left( I^{(2)}\right) \right)\ \\
&  =  & \left\{b_1 \ + \ b_2 \right\}
\cup\bigcup\limits_{n=0}^{\infty }\left(\left(\psi
_{3}^{(1)}\right)^{-n}\left(\widetilde{K}_{1}\right)+ \left(\psi
_{3}^{(2)}\right)^{-[\frac{n}{2}\ ]} \left(K_{2}\right)\right) \\
 &&
\cup \bigcup\limits_{n=0}^{\infty
}\left(\left(\psi_3^{(1)}\right)^{-[\frac{n}{2}\ ]}\left(K_{1}
\right)\ + \left(\psi
_{3}^{(2)}\right)^{-n}\left(\widetilde{K}_{2}\right) \right)
\subset \ K_1+ K_2 ,
\end{eqnarray*}
\noindent  where \ $ I^{(1)} $\
    and \ $ I^{(2)}\ $ are the convex hull of
$\widetilde{K}_{1}\ $ and $ \widetilde{K}_{2}$. The equality
holds by the Gap-lemma since the sums in the right hand side are
sums of Cantor sets whose product of thicknesses is close to
$\frac{5}{4}\ $, and $\left(\psi
_{3}^{(i)}\right)^{-[\frac{n}{2}]}\left(K_{i}\right) \ $  is never
contained in a gap of $ \left(\psi
_{3}^{(j)}\right)^{-n}\left(\widetilde{K}_{j}\right), \ i,j = 0,1 \ $. Moreover,
these sums are finite unions of intervals that converge to
$\left\{b_{1} \ + \ b_{2} \right\} $, which shows that $ b_{1} \ +
\ b_{2}$ is the only accumulation point of the boundary of
$K_{1}+K_{2}$. Thus, the proof has been completed.

\vspace{0.5cm}
\noindent {\bf Remark:} If $K_{1}$ and $K_{2}$ are near K and $\lambda>0$ is small (for instance, if $\lambda<\dfrac{1}{30})$ then the topological type of $K_{1}+\lambda K_{2}$ is a finite union of
(2 or more) copies homeomorphic to $K+K$, and the number of copies increases as $\lambda$ tends to 0.

\bigskip

\begin{proposition}
\textit{Given \ }$\varepsilon >0$\textit{\ \ consider the affine Cantor set
\ }$K_{\varepsilon }$\textit{\ \ which is the maximal invariant set,
contained in \ }$\left[ 0,11\right] $\textit{\ \ of the affine functions \ }$%
\psi _{i}$\textit{, \ \ such that }$\ \psi _{i}\left( I_{i}\right) =\left[
0,11\right] $\textit{, }$1\leqslant i\leqslant 4,$\ \textit{\ where}$\;$%
\begin{equation*}
I_{1}=\left[ 0,4\right] \mathit{,\ }I_{2}=\left[ 6,10\right] \mathit{,\ }%
I_{3}=\left[ 11-2\varepsilon ,\;11-\varepsilon \right] \mathit{\ ,}\ I_{4}=%
\left[ 11-\varepsilon ^{2},\;11\right], \mathit{\ }
\end{equation*}
\textit{and only }$\ \psi _{1}$\textit{\ \ preserves orientation. \ If \ }$%
\varepsilon >0$\textit{\ \ is small enough (for example \ }$\varepsilon \in
\left( 0,\dfrac{1}{300}\right) $\textit{) \ there is a neighbourhood \ }$%
\mathcal{U}_4$\textit{\ \ of \ }$K_{\varepsilon }$\textit{\ \ in the }$%
\;C^{1+} $\textit{\ \ topology such that for all \ }$K_{1},K_{2}\in \mathcal{%
U}_4$\textit{, every gap of \ }$K_{1}+K_{2}$\textit{\ \ is an \ }$RL$\textit{%
-gap \ and the accumulation points of the boundary of \ }$K_{1}+K_{2}$%
\textit{\ \ are not countable. \ (In particular \ }$K_{1}+K_{2}$\textit{\ \
is not a finite union of intervals).}
\end{proposition}

\noindent {\bf Proof}.
1.- Note that the maximal invariant set \
$\widetilde{K}_{\varepsilon }$ \ of \ $\psi _{i}$, $i=1,2$ \ is
contained in \ $K_{\varepsilon }$ \ and has thickness \
$\dfrac{4}{3}$. \ Moreover, since \ $\psi _{4}$ \ reverses
orientation, $K_{\varepsilon }$ \ contains another subset at the
right extreme which has thickness \ $\dfrac{4}{3}$. \ Therefore,
by the Gap-lemma
all \ $K_{1}$ \ and \ $K_{2}$ \ close to \ $K_{\varepsilon }$ \ in the \ $%
C^{1+}$ \ topology are such that every gap of \ $K_{1}+K_{2}$ \ is an $RL$%
-gap for any $\lambda >0$.

2.- Given a neighbourhood $\mathcal{V}$ of $K_{\varepsilon }$
there is a neighbourhood $\mathcal{U}_4$ of $K_{\varepsilon }$
such that for $K\in \mathcal{U}_4$ and for each interval $I$ of
the construction of $K\cap I$ is the image by an affine map of an
element of $\mathcal{V}$.

We shall prove that if $\mathcal{V}$ is small enough and
$\mathcal{U}_4$ is as above, then, for any $\lambda >0$ and
$K_{1},K_{2}\in \mathcal{U}_4$, we can find two disjoint open
intervals $I^{\prime },I^{\prime \prime }$ such that $\left(
K_{1}+\lambda K_{2}\right) \cap I^{\prime }=\mu ^{\prime
}K_{1}^{\prime }+\lambda ^{\prime }K_{2}^{\prime }$ and $\left(
K_{1}+\lambda K_{2}\right) \cap I^{\prime \prime }=\mu ^{\prime
\prime }K_{1}^{\prime \prime }+\lambda ^{\prime \prime
}K_{2}^{\prime \prime }$
where $K_{1}^{\prime },K_{2}^{\prime },K_{1}^{\prime \prime }$ and $%
K_{2}^{\prime \prime }$ belong to $\mathcal{U}_4$, and both $\mu
^{\prime }\lambda ^{\prime }$ and $\mu ^{\prime \prime }\lambda
^{\prime \prime }$ are positive.

Repeating the process for the sums $\mu ^{\prime }K_{1}^{\prime
}+\lambda ^{\prime }K_{2}^{\prime }$ and $\mu ^{\prime \prime
}K_{1}^{\prime \prime }+\lambda ^{\prime \prime }K_{2}^{\prime
\prime }$ and successively, we get our result.

Our previous statement follows from the following analysis:
\begin{enumerate}
\item[(i)]  If$\ \frac{1}{100}\leq \lambda \leq 100,$ $K_{1}+\lambda
K_{2}\supset \left( \widetilde{\psi }_{4}^{-1}\left( K_{1}\right) +\lambda
\overline{\psi }_{4}^{-1}\left( K_{2}\right) \right) \cup \left( \widetilde{
\psi }_{3}^{-1}\widetilde{\psi }_{4}^{-1}\left( K_{1}\right) + \right.
$\newline
$\left. \lambda \overline{\psi }_{3}^{-1}\overline{\psi }_{4}^{-1}\left( K_{2}\right) \right) $ we
denote the functions that define $K_{1}$ and $K_{2}$ by $\widetilde{\psi }$
and $\overline{\psi }$ respectively). Since support intervals of the two
previous sums are disjoint, we get the statement because $\widetilde{\psi }%
_{j},\overline{\psi }_{j},$ $j=3,4$ reverse orientation.
\item[(ii)]  If $\lambda \leq \frac{1}{100}$, $K_{1}+\lambda K_{2}\supset
\left( \widetilde{\psi }_{1}^{-1}\left( K_{1}\right) +\lambda K_{2}\right)
\cup \left( \widetilde{\psi }_{2}^{-2}\left( K_{1}\right) + \lambda
K_{2}\right) . $ As in the previous case, the support intervals of the sums
of the right side \ are disjoint, and both $\widetilde{\psi }_{1}$ and $%
\widetilde{\psi }_{2}^{2}$ preserve orientation, so again \ we get the
desired result. The case $\lambda \geq 100$ is proved in the same
way, changing the role of $K_{1}$ and $K_{2}.$  Therefore, the proof has been completed.
\end{enumerate}

\bigskip

\begin{proposition}
\textit{Given }$\varepsilon >0\,$ small, \textit{let \ }$K_{\varepsilon }$%
\textit{\ \ be the Cantor set defined by the affine increasing functions }$%
\psi _{i}$\textit{\ }$\;$\textit{such that }$\ \psi _{i}$\textit{, }$%
1\leqslant i\leqslant 6$\textit{,}$\ \psi _{i}\left( I_{i}\right)
=\left[ 0,25\right] $\textit{, where }
\begin{eqnarray*}
I_{1} &=&\left[ 0,10\right] \mathit{,\ \ }I_{2}=\left[ 11,11+\dfrac{%
\varepsilon ^{2}}{2}\right] \mathit{,\ \ }I_{3}=\left[ 12,12+2\varepsilon
^{2}\right] \mathit{,\ \ } \\
I_{4} &=&\left[ 13,13+\dfrac{\varepsilon ^{2}}{2}\right] \mathit{,\ }I_{5}=%
\left[ 14,14+2\varepsilon ^{2}\right] \mathit{\ \ and\ \ }I_{6}=\left[ 15,25%
\right] \mathit{.}
\end{eqnarray*}
\textit{If \ }$\varepsilon $\textit{\ \ small enough (for instance if \ }$%
0<\varepsilon <\dfrac{1}{100}$\textit{) \ then there is a neighbourhood \ }$%
\mathcal{U}_5$\textit{\ \ of \ }$K_{\varepsilon }$\textit{\ \ in the \ }$%
C^{1+} $\textit{\ \ topology such that if \ }$\left( K_{1},K_{2}\right) \in
\mathcal{U}_5\times \mathcal{U}_5$\textit{\ }$\ $\textit{and \ }$\lambda \in \left( \dfrac{%
\varepsilon }{3},3\varepsilon \right) $\textit{\ \ then \ }$K_{1}+\lambda
K_{2}$\textit{\ \ has non-countably many connected components and all of its
gaps are of the type \ }$RL$\textit{.}
\end{proposition}
\bigskip

\noindent {\bf Proof}.
Let $\mathcal{U}_5\ $ be a small neighbourhood of $K_{\varepsilon }\ $
and $\widetilde{K}$ \ the maximal invariant set of $\psi _{1}$ \ and \ $%
\psi _{6}$, then\ $\tau \left( \widetilde{K}\right) =2$ \ and the
convex hull of \ $\widetilde{K}$ \ is \ $[0,25]$. In a similar
way, given $K_{1}\in \mathcal{U}_5$ $\ $one defines\
$\widetilde{K}_{1}$, which has thickness close
to \ $2$, \ and whose convex hull is equal to that of $K_{1}$.Thus if $%
(K_{1},K_{2})\in \mathcal{U}_5\,\times \mathcal{U}_5\,\}$ then\ $\tau \left( \widetilde{K}%
_{1}\right) \tau \left( \widetilde{K}_{2}\right) >1$ since it is close to \ $%
4$, and therefore all gaps of $K_{1}+K_{2}\,$are of type $RL.$

Let\ $\widetilde{\mathcal{U}}$ \ be a small neighbourhood of \ $%
K_{\varepsilon },$ thus if  $\mathcal{U}_5$ \ is small enough,\
then every
sub-Cantor set of \ $K_{1}\in \mathcal{U}_5$ \ belongs to $\widetilde{\mathcal{%
U}}$\ \ after been properly reescaled.

Let \ $\left( K_{1},K_{2}\right) \in \mathcal{U}_5\times {U}_5$, \
and \ $\psi _{i}^{\left( j\right) }$, $1\leqslant i\leqslant 6$
,$1\leqslant j\leqslant
2 $\ be the functions that define \ $K_{1}$ \ and \ $K_{2}$. \ For \ $%
\lambda \in \left( \dfrac{\varepsilon }{3},3\varepsilon \right) $,
we have
\begin{equation*}
K_{1}+\lambda K_{2}=\bigcup\limits_{i=1}^{6}\left( \left( \psi
_{i}^{\left( 1\right) }\right) ^{-1}\left( K_{1}\right) +\lambda
K_{2}\right).
\end{equation*}

Since \ $\lambda \in \left( \dfrac{\varepsilon }{3},3\varepsilon
\right) $, the diameter of \ $\lambda K_{2}$ \ is smaller than the
lengh of the gaps of
\ $K_{1}$ in the first step of its construction, so the convex hulls of \ $%
\left( \psi _{i}^{\left( 1\right) }\right) ^{-1}\left( K_{1}\right) +\lambda
K_{2}$, $1\leqslant i\leqslant 6$ \ are disjoint.

Since \ $\lambda $ \ belongs to \ $\left( \dfrac{\varepsilon }{3}%
,3\varepsilon \right) $ \ and if $\mathcal{U}_5$ $\ $is small
enough, the ratio of the diameters of the Cantor sets \ $\left(
\psi _{i}^{\left( 1\right) }\right) ^{-1}\left( K_{1}\right) $ $\
$and $\ \lambda K_{2}$ \ belongs to \ $\left( \dfrac{2\varepsilon
}{5},\dfrac{5\varepsilon }{2}\right) \subset \left(
\dfrac{\varepsilon }{3},3\varepsilon \right) $ \ for at least two
values of \ $i$ \ between \ $2$ \ and \ $5$. \ For these values of
\ $j$, the sets \ $\left( \psi _{j}^{\left( 1\right) }\right)
^{-1}\left(
K_{1}\right) +\lambda K_{2}$ \ are open and closed in \ $K_{1}+\lambda K_{2}$%
, and $\left( \dfrac{25}{\left| K_{2}\right| }\cdot
K_{2},\dfrac{25}{\lambda \left| K_{2}\right| }\cdot \psi
_{j}^{\left( 1\right) }\left( K_{1}\right) \right) \in
\mathcal{U}_5\times \mathcal{U}_5$, so we can repeat the argument to these
pairs and conclude that \ $K_{1}+K_{2}$ \ has non-countably many
connected components. Therefore the proof has been completed.

\bigskip

\noindent {\bf Remark:}  If  $\mathcal{U}_5$  is small enough,  $%
\lambda $ \ is close to \ $1$ \ and \ $\left( K_{1},K_{2}\right)
\in \mathcal{U}_5\times \mathcal{U}_5$ \ then \ $\left( K_{1}\right)
+\lambda K_{2}$ \ is an
interval, so the topological structure of \ $K_{1}+\lambda K_{2}$ \ for \ $%
K_{1}$ \ and \ $K_{2}$ \ close to \ $K$ \ depends strongly on the size of \ $%
\lambda $, that is not the case of the Proposition 4.
\bigskip 

\begin{proposition}
Let \ $K$ \ be the Cantor set which is the maximal invariant set of
the maps \ $\psi _{i}$, $1\leqslant i\leqslant 4$, such that \ $\psi
_{i}\left( I_{i}\right) =\left[ 0,52\right] $, where
\begin{equation*}
\ I_{1}=\left[ 0,21\right] ,\;I_{2}=\left[ 22,43\right] ,\;I_{3}=\left[ 48,49%
\right] ,\;I_{4}=\left[ 51,52\right] ,
\end{equation*}
\noindent \ and only \ $\psi _{2}$ \ reverses orientation.Then,
there is a neighbourhood \ $\mathcal{U}_6$ \ of \ $K$ \ and \ a \
residual set \ $\mathcal{R}\subseteq \mathcal{U}_6\times \mathcal{U}_6$ \
such that for all $\left( K_{1},K_{2}\right)
\in
\mathcal{R}$ and  $%
\lambda >0, $
\begin{enumerate}
\item[(i)] the bounded gaps of \ $K_{1}+\lambda \ K_{2}$ \ are of type \ $RL$ \ or \ $L \ .$
\item[(ii)] $K_{1}+\lambda \ K_{2}$ \ has infinitely many gaps of type \ $RL$ \ and
of type \ $L$.
\end{enumerate}
\end{proposition}

\bigskip

\noindent {\bf Proof}.
1.-  Fix \ $\eta >1$ \ close to 1 and let \ $\mathcal{V}=\mathcal{V}\left(
\eta \right) $ \ be a neighbourhood of \ $K$ \ such that all the ratios in
the first step in the construction of a Cantor set in \ $\mathcal{V}$ \ are
like the ones in \ $K$ \ with a factor in \ $\left( \eta ^{-1},\eta \right) $%
.

Then, by bounded distortion, there is a neighbourhood \ $\mathcal{U}_6=%
\mathcal{U}_6\left( \eta \right) $ \ of \ $K$ \ such that every
sub-Cantor set of \ $K\in \mathcal{U}_6$ \ belongs to \
$\mathcal{V}$ \ after been properly
reescaled. \ Note that \ $\tau \left( K\right) =\dfrac{1}{2}$ \ and, if \ $%
\widetilde{K}$ \ denotes the maximal invariant set contained \ $\left[ 0,52%
\right] $ \ of \ $\psi _{1}$ \ and \ $\psi _{2}$, then \ $\tau \left(
\widetilde{K}\right) =\dfrac{21}{10}$, so \ $\tau \left( \widetilde{K}%
\right) \cdot \tau \left( K\right) >1$.

Therefore, if $\eta >1$ is close to 1 for all $K_{1},K_{2}\in \mathcal{U}_6$, $%
\tau \left( \widetilde{K}_{1}\right) \cdot \tau \left(
K_{2}\right) >1$, where \ $\widetilde{K}_{1}$ \ is the analogous \
set of \ $\widetilde{K}$ \ for \ $K_{1}$. \ Thus, by the
Gap-lemma, \ $\widetilde{K}_{1}+K_{2}$ \ is a
finite union of intervals. \ In particular, the leftmost point of \ $%
K_{1}+K_{2}$ \ belongs to an interval of \ $K_{1}+K_{2}$, since it belongs
to \ $\widetilde{K}_{1}+K_{2}$.

2.-  Let \ $t\in K_{1}+K_{2}$ \ be the right boundary point of a gap of \ $%
K_{1}+K_{2}$, so \ $t=t_{1}+t_{2}$ \ where \ $t_{i}$ \ is the left extreme
of a sub-Cantor set \ $K_{i}^{\prime }$ $\ $of $\ K_{i}$, $i=1,2$. \ By the
observation in \ $1$ \ and considering that \ $\tau \left( K\right) =\tau
\left( -K\right) $, it follows that if \ $K_{1}^{\prime }$ \ or \ $%
K_{2}^{\prime }$ \ is an oriented copy of \ $K_{1}$ \ or \ $K_{2}$ ,
 respectively, then \ $t$ \ belongs to an interval of \ $K_{1}+K_{2}$.

On the other hand, if \ $K_{1}^{\prime }$ \ and \ $K_{2}^{\prime
}$ \ are reversed copies of \ $K_{1}$ \ and \ $K_{2}$ \ let \
$n_{i}$ \ be the least
integer such that \ $t_{i}$ \ is an extreme of an interval in the \ $%
n_{i}^{th}$ \ step of the construction of \ $K_{i}$, $i=1,2$. \ Note that \ $%
n_{i}>1$ \ and \ let \ $I_{i}$ \ be the interval in the \ $\left(
n_{i}-1\right) ^{th}$ \ step in the construction of \ $K$ \ that contains \ $%
t_{i},\;i=1,2$.

Thus \ $I_{1}\cap K_{1}$ \ is an oriented copy of \ $K_{1}$ \ and \ $%
t_{1}\in intJ\subseteq I_{1}$, where \ $J$ \ is the support of two thick
Cantor set, i.e. $J\cap K_{1}$ \ is a Cantor set such that \ $\widehat{%
J\cap K_{1}}=J$ \ and \ $\tau \left( J\cap K_{1}\right) \tau \left(
K_{2}\right) >1$.

Reducing \ $\mathcal{U}_6$,\ if necessary, one may suppose that for \ $%
t^{\prime }$ \ close to \ $t$, $-K_{2}+t^{^{\prime }}$ \ is not
contained in a gap of \ $J\cap K_{1}$. \ Since \ $t_{1}=t-t_{2}\in
int\left( J\right) \cap \left( -K_{2}+t\right) $, for \ $t^{\prime
}$ \ close to \ $t$, $J\cap K_{1}$ \ is not contained in a gap of
\ $-K_{2}+t^{\prime }$. \ Then, by the Gap-lemma \ $\left( J\cap
K_{1}\right) \cap \left( -K_{2}+t^{\prime }\right) \neq \phi $ \
for all \ $t^{\prime }$ \ close to \ $t$. \ Thus, \ $t$ \ belongs
to the interior of an interval of \ $K_{1}+K_{2}$, contradicting
its
definition. \ This proves that all gaps of \ $K_{1}+K_{2}$ \ are of type \ $%
RL$ \ or \ $L$, since they all have an interval to their right, so part $%
\left( \text{i}\right) $ has been proved.

3.-  If $K_{1},K_{2}\in \mathcal{V}$ and $\lambda >0$, there are
$m,n\in
\mathbb{N}$ (that can be choose $n$ arbitrarily large) such that $\frac{%
\left| \widetilde{\psi }_{4}^{-m}\left( K_{1}\right) \right|
}{\lambda \left| \overline{\psi }_{4}^{-n}\left( K_{2}\right)
\right| }$ is close to 1 (where $\widetilde{\psi }$,
$\overline{\psi }$ denote the functions that define $K_{1}$ and
$K_{2}$), provided that the ratio of the logarithms of the
eigenvalues of $\widetilde{\psi }_{4}$ and $\overline{\psi }_{4}$
at their fixed points is irrational (which is a generic condition,
called irrational condition).

Now, assuming the generic condition that no point can be written
in two or more different ways as the sum of extremes\ points \ of
\ gaps \ of \ $\ K_{1}$ \ and $\ K_{2}$ \ ( this \ is \ called \
the \ separation \ condition
), \ there \ is \ a \ gap \ between \ the \ convex hulls of \ $\widetilde{%
\psi }_{4}^{-m-1}\left( K_{1}\right) +\lambda \overline{\psi }%
_{4}^{-n-1}\left( K_{2}\right) $ \ and \ $\left( \widetilde{\psi }_{4}^{-m}%
\widetilde{\psi }_{3}^{-1}\left( K_{1}\right) +\lambda \overline{\psi }%
_{4}^{-n-1}\left( K_{2}\right) \right) \cup \left( \widetilde{\psi }%
_{4}^{-m-1}\left( K_{1}\right) +\lambda \overline{\psi }_{4}^{-n}\overline{%
\psi }_{3}^{-1}\left( K_{2}\right) \right) $ \ of type $L$ and there is a
gap between the convex hulls of \ $\widetilde{\psi }_{4}^{-m}\widetilde{\psi
}_{3}^{-1}\left( K_{1}\right) +\lambda \overline{\psi }_{4}^{-n}\overline{%
\psi }_{3}^{-1}\left( K_{2}\right) $ \ and \ $\left( \widetilde{\psi }%
_{4}^{-m}\widetilde{\psi }_{3}^{-1}\left( K_{1}\right) +\lambda \overline{%
\psi }_{4}^{-n}\overline{\psi }_{2}^{-1}\left( K_{2}\right)
\right) +\left( \widetilde{\psi }_{4}^{-m}\widetilde{\psi
}_{2}^{-1}\left( K_{1}\right) +\lambda \overline{\psi
}_{4}^{-n}\overline{\psi }_{3}^{-1}\left( K_{2}\right) \right) $ \
of type $RL$. \ Since $m,\;n$ \ can \ be \ taken \ arbitrarily
large, this gives our result.

\bigskip

\noindent {\bf Remark:} In fact is possible to adapt this
argument to show that there is an uncountable set $ {\cal C} \subset
K_{1}+K_{2}$ such that every neighbourhood of any point of {\cal C}
contains infinitely many gaps of both types $R$ and $RL$.
\bigskip

\begin{proposition}
\textbf{:} \ Consider the Cantor set \ $K$ \ defined by the affine functions
\ $\psi _{i}$, $1\leqslant i\leqslant 4$, such that \ $\psi _{i}\left(
I_{i}\right) =\left[ 0,52\right] $, where \
\begin{equation*}
I_{1}=\left[ 0,21\right] ,I_{2}=\left[ 22,43\right] ,I_{3}=\left[ 48,49%
\right] ,I_{4}=\left[ 51,52\right],
\end{equation*}
and \ only \ $\psi _{3}$ \ reverses orientation. \ Then, there is a
neighbourhood \ $\mathcal{U}$ \ of \ $K$ \ and a residual subset \ $\mathcal{%
R}$ \ of \ $\mathcal{U\times U}\ $\ such that for all \ $\left(
K_{1},K_{2}\right) \in \mathcal{R}$, $K_{1}+K_{2}$ \ has
infinitely many gaps of each four types as described at the
beginning of this section.
\end{proposition}

\noindent {\bf Proof}.
Let we first remark that if we denote by \ $\widetilde{K}$ \ the maximal
invariant set of \ $\psi _{1}$ $\ $and $\ \psi _{2}$ \ contained in \ $\left[
0,52\right] $. \ Note that \ $\widetilde{K}\subseteq K$ \ and \ $\tau \left(
\widetilde{K}\right) =\dfrac{21}{10}$ and so $\tau \left( K\right) \tau
\left( \widetilde{K}\right) =\frac{21}{20}>1.$

Let $K_{1},K_{2}$ be regular Cantor sets close to $K$ that satisfy
the irrational condition.

As in proposition 6, we have, arbitrarily close to the right extreme of $%
K_{1}+K_{2}$, sums of regular Cantor sets which are close to $K$
after reescaling, that are open and closed in $K_{1}+K_{2}$ (and
indeed are intersections of $K_{1}+K_{2}$ with intervals). The
result follows from noting that if $K_{1}$ and $K_{2}$ are close
to $K$ and satisfy the
separation condition, then denoting by $\widetilde{\psi }$ and $\overline{%
\psi }$ the functions that define $K_{1}$ and $K_{2}$, we have:
\begin{enumerate}
\item[(i)]  The gap between the convex hulls of $\widetilde{%
\psi }_{4}^{-1}( K_{1}) +\overline{\psi }_{4}^{-1}(
K_{2}) $  and $( \widetilde{\psi }_{3}^{-1}( K_{1})\ +
\overline{\psi }_{4}^{-1}( K_{2}))  \cup ( \widetilde{%
\psi }_{4}^{-1}( K_{1}) +\overline{\psi }_{3}^{-1}(
K_{2}))$ is of type $RL$.
\item[(ii)]  The gap between the convex hulls of $
(\widetilde{\psi}_{3}^{-1}(K_{1}) +\overline{\psi }
_{4}^{-1}( K_{2}) ) \cup ( \widetilde{\psi }
_{4}^{-1}( K_{1}) +\overline{\psi }_{3}^{-1}( K_{2})$ and $\widetilde{\psi }_{3}^{-1}( K_{1}) +\overline{\psi
}_{3}^{-1}( K_{2})$ is of type $R$.
\item[(iii)]  The gap between the convex hulls of $\widetilde{
\psi }_{3}^{-1}( K_{1}) +\overline{\psi }_{3}^{-1}(
K_{2})$ and $(\widetilde{\psi }_{2}^{-1}(K_{1}) +
\overline{\psi }_{4}^{-1}(K_{2})) \cup (\widetilde{
\psi }_{4}^{-1}(K_{1}) +\overline{\psi }_{2}^{-1}(
K_{2}))$ is of type $C$.
\item[(iv)]  We have, arbitrarily close to the right end point of the gap
in ${\cal C}$ mentioned in last remark, sums of small Cantor sets which are close to
$-K$ after rescaling, that are open and closed in $K_{1}+K_{2}$
(this gives, by item (ii) gaps of type $L$).
\end{enumerate}
Thus, the proof has been completed.

\bigskip

\noindent {\bf Remark:} In fact one can prove, by slightly reducing $%
\mathcal{R}$, that every end point of a gap is the boundary of an
interval, or it is accumulated by infinitely many gaps of each one
of the four types. 

We can use this fact to show that there is an uncountable set $ {\cal C}
\subset K_{1}+K_{2}$ such that every neighbourhood of
any point of $ {\cal C}$ contains infinitely many gaps of each of the four types ($%
C,L,R$ and $RL$).


\vspace{0.5cm}

\section {Open Questions}

\vspace{0.5cm}

The following is a natural question.

{\bf Problem 1:} Does there exists  a generic set  $%
\mathcal{R}\subset \{\left( K_{1},K_{2}\right) \mid K_{1},K_{2}$ \ are $C^{1+}$ \ regular Cantor sets\} \ such that each  $( K_{1},K_{2}) \in\mathcal{R}$  has a neighbourhood  $\mathcal{U}$  such that  
$( \widetilde{K_{1}},\widetilde{K_{2}}
) \in \mathcal{R} \cap \mathcal{U} \implies  \widetilde{K_{1}}+\widetilde{K_{2}}$  is homeomorphic to $K_{1}+K_{2}$ ?

In particular, if generic is taken in the Baire sense, it would imply that there are countably many robust topologies of a sum of Cantor sets.
The remarks after Example 0 and Proposition 3 show that there are infinitely many robust topologies of sums of Cantor sets.

An interesting particular instance of Problem 1 is the problem of determining the typical topology (or topologies) for sums of pairs of Cantor sets in the open sets   $\mathcal{U}_{2}^{R\left( resp.\;L\right) }$ of Proposition 2.
For generic pairs  ${\left( K_{1},K_{2}\right)}\in \mathcal{U}^{L},
K_{1}+K_{2}$ is an L-Cantorval, which that not determine its topology. There are two natural possibilities for the topology of \ $K_{1}+K_{2}$ \ in this case: the  ${L}_{0}$ \ topology, obtained from a Cantor set by blowing up each right extreme of a gap creating intervals; and the  ${L}_{\infty}$ \ topology, obtained by blowing up each right extreme of a gap of an M-Cantorval. We do not know robust examples of sums of Cantor sets with the ${L}_{0}$ topology, (resp. with the ${L}_{\infty}$ \ topology), but it is reasonable to think that generically in $\mathcal{U}^{L}, K_{1}+K_{2}$ is either ${L}_{0}$ \ or ${L}_{\infty}.$


\bigskip





\begin{thebibliography}{MY1}
\bibitem[B]{B}  R. Bam\'{o}n, S. Plaza, C. G. Moreira and J. Vera,
Differentiable Structures of Central Cantor Sets, Ergod. Th. and Dynam.
Syst. 17, pp. 1027-1042, $\left( 1997\right) $.
\bibitem[H]{}  M. Hall, On the Sum and Product of Continued Fractions,
Annals of Math., Vol. 48, pp. 966-993, $\left( 1947\right) $.
\bibitem[M]{}  C. G. Moreira, Stable Intersections of Cantor Sets and
Homoclinic Bifurcations, Ann. Inst. H. Poincar\'{e} Anal. Non Lin\'{e}aire
13, pp. 741-781, $\left( 1996\right) $.
\bibitem[M1]{}  C. G. Moreira, Sums of Regular Cantor Sets, Periodica
Mathematica. Hungarica, Vol. 37, pp. 55-63 $\left( 1998\right) $ .
\bibitem[M2]{}  C. G. Moreira, Geometric Properties of the Markov and
Lagrange Spectra. \ To appear.
\bibitem[MO]{}  P. Mendes and F. Oliveira, On the Topological Structure of
the Arithmetic Sum of Two Cantor Sets, Nonlinearity, Vol. 7, pp. 329-343, $%
\left( 1994\right) $.
\bibitem[MY1]{}  C. G. Moreira and J. C. Yoccoz, Stable Intersections of
Regular Cantor Sets with Large Hausdorff Dimension. \ Submitted to Annals of
mathematics. $\left( 1998\right) $.
\bibitem[MY2]{}  C. G. Moreira and J. C. Yoccoz, Tangencies Homoclines
Stable Pour les Ensembles Hyperboliques de Grande Dimension Fractale. \ To
appear.
\bibitem[N1]{}  S. Newhouse, Non Density of Axiom \ $A\left( a\right) $ \ on
\ $S^{2}$, Proc. A. M. S. Symp. Pure Math., Vol. 14, pp. 191-202, $\left(
1970\right) $.
\bibitem[N2]{}  S. Newhouse, Diffeomorphisms with Infinitely Many Sinks,
Topology, Vol. 13, pp. 9-18, $\left( 1974\right) $.
\bibitem[P]{}  J. Palis, Homoclinic Orbits, Hyperbolic Dynamics and
Fractional Dimension of Cantor Sets, Contemporary Mathematics 58, pp.
203-216, $\left( 1987\right) $.
\bibitem[P1]{}  J. Palis, Homoclinic Bifurcations, Sensitive Chaotic
Dynamics and Strange Attractors, Dynamical Syst. and Related Topics. \ World
Scientific, pp. 466-473, $\left( 1991\right) $.
\bibitem[PT]{}  J. Palis and F.Takens, Hyperbolicity \& Sensitive Chaotic Dynamics at Homoclinic Bifurcations: Fractal Dimensions and Infinitely many Attractors. Cambridge University Press, $\left( 1993\right) $.
\end{thebibliography}
\end{document}
