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\date{ Dedicated to the memory of Professor Camilo Bl\'azquez }
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\title{Bifurcations of the Essential Dynamics of Lorenz  maps on the
real line and the bifurcation scenario for the Linear Family
\footnote{Partially supported by Fondecyt grants \# 1970720, \#
1961212 DICYT-USACH- Chile and PRONEX on Dynamical Systems.
Brazil} }
\author{Rafael Labarca and Carlos Moreira}

\begin{document}

\maketitle
\section{Introduction}

In a remarkable contribution  a meteorologist, E.N. Lorenz
\cite{Lo}, showed numerical e\-vi\-den\-ce of the existence of a
strange attractor for a quadratic system of ordinary
di\-ffe\-ren\-tial equations in three variables. Some time later
J. Guckenheimer, \cite{G}, produced a work where he introduced
symbolic dynamics in order to understand the to\-po\-lo\-gi\-cal
equivalence classes for nearly similar attractors. At that time
R.F. Williams,\cite{W}, introduced a geometrical model in order
to understand the dynamics of these Lorenz attractors. Using
this geometrical model the dynamical behavior of the three
dimensional vector field can be reduce to the dynamical behavior
of a one-dimensional map with one discontinuity and Guckenheimer
and Williams,\cite{GW}, used this fact to shown uncountable many
classes of non-equivalent geometric Lorenz attractors. The
evidence of the non-equivalence where the kneading sequences
associated to these one-dimensional maps. The class of maps
introduced in this way is contained in the class of maps $\,DM_0
\,$ that we work with(see section 2.1).Associated to any $f \in
DM_0\,$ we have two kneading sequences $\, (a_f\,,b_f) = I(f)\,$
(see section 2.3 for the definition of these sequences) that
satisfies $ a_f = \inf\{\sigma^k(a_f)\,, k \in \Na \,\} \,\,,
b_f = \sup\{\sigma^k(b_f) \,, \, k \in \Na \, \} \,$ and $ \,
\{a_f\,,b_f \,\} \subset \Sigma_{a_f\,,b_f} \,$(here
$\Sigma_{a,b}\,$denote the set $ \,
\cap_{i=0}^{\infty}\sigma^{-i}([a,b])\,$, see section 2.3 for
details). These properties inspired ( \cite{RT})the following
definitions. A sequence of two symbols $a=(0\,,\dots) \in
\Sigma_2\,$ (resp. $b=(1\,,\dots) \in \Sigma_2)\,$ is called
{\em minimal} ( resp. {\em maximal}) if $\, a =
\inf\{\sigma^k(a)\,,\, k \in \Na \}\,$ (resp. $b=
\sup\{\sigma^k(b)\,,\, k \in \Na \,\}\,$. Here $\sigma: \Sigma_2
\to \Sigma_2 \,$ denotes the usual shift map and $\, \Sigma_2 \,
$ denote the set of sequences $\, \theta : \Na \rightarrow
\{0,1\} \,$ . We will denote by $Min_2\,$(resp. $Max_2\,$) the
set of minimal(resp. maximal) sequences in $\Sigma_2\,$.  The
properties that satisfies the kneading sequences allow us to
define the {\em Le\-xi\-co\-gra\-phi\-cal World} as $ LW=\,
\{(a\,,b) \in Min_2 \times Max_2 \,, \, \{a\,,b\} \subset
\Sigma_{a\,,b} \,\}$. The itinerary $\, I: DM_0 \to LW \,,\, f
\to (a_f\,,b_f)\,$ defines a continuous and surjective map(see
section 2.4). We will say that the map $f \in DM_0\,$ has {\em
essentially} the same dynamics as the map $\, g \in DM_0 \,$ if
$\,I(f)=I(g)\,$. It is clear that two topologically equivalent
maps are essentially equivalent. Therefore, the lexicographical
world provide a {\em universal model} for (essentially)
equivalent dynamics in this context. This mean the following:
given $\,(a\,,b) \in LW \,$ then there is $ f \in DM_0 \,$ such
that $\, \Sigma_{a\,,b} = \Sigma_{a_f,b_f} \,$and a surjective
map $ \, I_f : \Gamma_f \to \Sigma_{a,b} \,$ such that $\, I_f
\circ f = \sigma \circ I_f \,$ and reciprocally ( see section
2.3 for the definition of the set $ \Gamma_f \,$ and the map $\,
I_f\,$). So, all the possible essential dynamics produced by
elements in $\, DM_0\,$ are modeled by elements in $\, LW\,$ and
it is natural to ask if there is a parameterized family $\,
\alpha : U \subset {\Bbb R}^k \to DM_0 \,$ with the following
property : given any $\,(a,b) \in LW \,$ the set $\,
\alpha^{-1}(\{(a,b)\}) \,$ is non empty.  We believe that the
correct answer to this question is no.

\vspace{0.2cm}

In these directions are the main results of the present paper: we
describe, in a precise way, which part of the lexicographical
world is realized by the piecewise linear family


\vspace{.2cm}
 $ \, F :Q \to DM_0 \,,\,F(\la,\mu)=F_{(\la,\mu)}$ given by

 $$ F_{\la,\mu}(x)=\left \{
\begin{array}{lc}
(1+\la)x+1,\quad & x<0 \\ ( 1+\mu)x-1, &  x>0
\end{array} \right \} $$
answering a question posed by Hubbard and Sparrow in \cite{HS}.
Here $\, Q\,=\,\{(\lambda,\mu)\in\R^2;\quad -1\le\mu\le 1$ and $-1
\le\lambda\le1\,\}$. Our main results are that there is an
uncountable set of sequences $(a,b) \in LW \,$ such that $\, (I
\circ F)^{-1}(\{(a,b)\}) \,$ is empty( it is interesting to note
that, in doing this, we prove that period doubling solenoid do not
exists for this family of maps in an independent way from
\cite{JS1} and \cite{MT}) and the exact description of these sets
when they are non empty.

\vspace{0.2cm}

This paper is organized as follows: in section 2 we state our
results, in section 3 we introduce some symbolic dynamics and the
lexicographical world and in section 4 we prove the results.


Some comments are in order : de Melo and Martens announced, in an
unpublished paper \cite{dMM}, the existence of parametrized
families of elements in $\, DM_0 \,$ that are topologically
universal in the sense that given any geo\-me\-tric Lorenz flow
then its dynamics is " essentially " the same as the dynamics of
some element of the family. The exact way in which these families
realized the dynamics is given in \cite{LM2}.

\vspace{0.2cm}

Finally, we have to mention that there are many  authors which
have investigated these families from the same and from another
points of view (combinatorial, differentiable, geometrical or
topological)( see the list of references). In particular, compare
our results with those of Glendinning in \cite{Gl}.

\vspace{0.5cm}
\section{Statement of the results}

\subsection{ The Set $ DM_0$.}

In the sequel $DM_0$ will denote the set of maps $f:(\R
\setminus\{0\})\to \R$ such that:

\begin{itemize}
\item[(1)] The restriction maps $f|_{(-\infty,0)}:(-\infty,0)\mapsto \R$
and $f|_{(0,\infty)}:(0,\infty)\to \R$ are continuous and
non-decreasing maps.

\item[(2)]
$$ f(0^+)=\lim_{x\downarrow0}f(x)\in(-\infty,0] $$ and $$
f(0^-)=\lim_{x\uparrow0}f(x)\in[0,\infty[ $$

\end{itemize}

An element in $\,DM_0\,$ will be called \underline{\em injective}
if its restriction to the interval $\, ]f(0^+)\,,\,f(0^-)[\,$is an
injective map.

\vspace{0.2cm}

We will say that $\, f \in DM_0 \,$ is \underline{\em increasing}
if the restriction maps $f|_{(-\infty,0)}:(-\infty,0)\mapsto \R$
and $f|_{(0,\infty)}:(0,\infty)\to \R$ are increasing.

\vspace{0.2cm}

We call the elements in $\, DM_0 \,$ \underline{\em Lorenz maps}.

\subsection{The Lexicographical Order}

Let $\, \Sigma_2 \, $ denote the set of sequences $\, \theta :
\Na \rightarrow \{0,1\} \,$ endowed with the topology given by
the metric $$ d(\alpha\,, \beta ) = \sum_{i=0}^{\infty}
\frac{\overline{d}(\alpha_i\,, \beta_i)}{2^i}, $$ where $$
\overline{d}(\alpha_i\,, \beta_i) = \left\{
\begin{array}{l}
0\,\,, \alpha_i = \beta_i\\ 1 \,\,, \alpha_i \ne \beta_i
\end{array}
\right. $$ Let $\, \sigma: \Sigma_2 \rightarrow \Sigma_2 \,$ be
the shift map $\, \sigma(\theta_0 \,,\theta_1 \,,\theta_2 \,,
\dots ) = (\theta_1\,,\theta_2\,, \dots)\,$. Let $\,
\Sigma_0\,\,$and$\,\Sigma_1\,\,$ denote the sets $\{ \theta \in
\Sigma_2 \,\,;\,\, \theta_0 = 0 \} \,$and$\, \{\theta \in \Sigma_2
\,\,;\,\, \theta_0 = 1 \}\,$respectively. It is clear that $ \,
\Sigma_2 = \Sigma_0 \cup \Sigma_1 \,$.

\vspace{0.5cm}

In $\, \Sigma_2 \,$ we consider the \underline{\em
lexicographical order}: $\,\, \theta < \alpha \,\,$ for any$\,
\theta \in \Sigma_0\,$and$\, \alpha \in \Sigma_1 \,\,$or $\,\,
\theta < \alpha \,\,$if there is $\, n \in \Na \,\,$such that
$\,\, \theta_i = \alpha_i \,\,$for$\, i = 0\,,1\,,2\,,\dots
\,,n-1 \,$and$\, \theta_n = 0 \,\,\,\mbox{and}\,\, \alpha_n =
1\,$. \vspace{0.2cm}

For $\, a \leq b \,$ in $\Sigma_2 \,$ let $\,[a\,,b] \,$ denote
the interval $\,\{ \theta \in \Sigma_2 |\, a \leq \theta \leq b
\,\}. \quad  \Sigma_{a\,,b}\,$will denote the set $\,
\bigcap_{n=0}^{ \infty} \sigma^{-n} ([a\,,b\,])\,$.

\subsection{The Set $\Sigma_{a_f,b_f}$ and the Essential Dynamics.}

For $ f \in DM_0 \,$ let $\Gamma_f=(\R \setminus
\bigcup_{j=0}^\infty f^{-j}(0))$ denote the set of ``continuity''
of the map $f$. \vspace{0.2cm}

For $x\in\Gamma_f$ we define $I_f(x)\in\Sigma_2$ by $$I_f(x)(i)=
0 \quad \mbox{ if } f^{i}(x)<0 \,\,\mbox{and}\,\,I_f(x)(i)= 1
\quad \mbox{ if } f^{i}(x)>1. $$

For $x=0$ we define: $$ I_f(0^+)=\lim_{ x\downarrow 0 \,,
x\in\Ga_f} I_f(x) $$ and $$ I_f(0^-)=\lim_{ x\uparrow 0 \,,
x\in\Ga_f} I_f(x). $$

In the same way to any $x\in\bigcup_{j=0}^\infty f^{-j}(0)$ such
that $f^i(x)\ne 0$, $0\le i<n$; $ f^n(x)=0$ we associate the
sequences :

$$ I_f(x^+)=(I_f(x)(0),\dots,I_f(x)(n-1),\, I_f(0^+)) $$ and $$
I_f(x^-)=(I_f(x)(0),\dots,I_f(x)(n-1),\, I_f(0^-)). $$

For $x\in\Ga_f$ we define $I_f(x^+)=I_f(x^-)=I_f(x)$.
\vspace{.2cm}

Let $I_f=\{I_f(x^+);\quad x\in[f(0^+), f(0^-)[ \} \cup
\{I_f(x^-);\quad x\in] f(0^+), f(0^-)]\}$.

\vspace{0.2cm}

Clearly $\sigma(I_f)\subset I_f$. Let us denote$
\,\,a_f=I_f((f(0^+))^+) $ and $ b_f=I_f((f(0^-))^-) $

\vspace{.2cm}

\begin{lemma} $I_f=\bigcap_{n=0}^\infty
\sigma^{-n}([a_f,b_f])=\Sigma_{a_f,b_f}$.
\end{lemma}

\vspace{0.2cm} We observe that associated to any  $f \in DM_0
\,$ we can define a continuous map $$ h : [f(0^+)\,,f(0^-)] \cap
\Gamma_f \to \Sigma_{a_f\,,b_f} \subset \Sigma_2,$$ such that $
h \circ f = \sigma \circ h $. The map $h$ is given by $ h(x) =
I_f(x) $and  collapses intervals into points. This map cannot be
extended, continuously, to the set $ \cup_{i=0}^{\infty}
f^{-i}(0) $. There are two kinds of intervals that the map $h$
can collapses: The wandering intervals and the intervals that
are contained in the stable manifold of periodic sinks. An
interval $I \subset  [f(0^+)\,,f(0^-)] \,$  is called a
\underline{\em wandering interval }, for the map $f$,  if for
any $ x \in I $ we have that $\,x\,$ is a wandering point. We
will call a point $x$ a \underline{\em nonwandering} point if
for any neighborhood $U_x$ of $x$ and any positive integer $N$
we can find $\, n \ge N $ such that $ f^n(U_x) \cap U_x \ne
\emptyset\,$. The set of nonwandering points of the map $f\,$ is
denoted by $ \Omega_f \,$. A point $x \notin \Omega_f\,$ is
called a \underline{\em wandering point}. Given any interval
,$I$, \underline{\em the orbit} of this interval is the sequence
of iterations $(f^n(I) \,, n \in \Na )$. Concerning the
existence of wandering intervals we have the following:

\begin{lemma}(\cite{LM1})
Let $\{\varphi_{\lambda}\,, \lambda \in \R \} \subset DM_0\,$be a
one parameter family of $C^2$ increasing maps such that for each
$\, \lambda \,$ there are  sequences $ \lambda_n \to \lambda \,$
and $ \mu_n \to \lambda \,$ with $ \varphi_{\lambda_n}(x) >
\varphi_{\lambda}(x) $ and $\, \varphi_{\mu_n}(x) <
\varphi_{\lambda}(x) \,, \forall x$ then there is a residual set
of parameters $\lambda \,$ for which $ \varphi_{\lambda} \,$ has
no wandering intervals.
\end{lemma}
\vspace{0.2cm} We observe that our two parameter family of linear
maps satisfy this property. \vspace{0.2cm}


\begin{definition} Given $f,g\in DM_0$. We will say that $f$ has
\underline{\em essentially} the same dynamics as $g$ if $I_f=I_g$.
\end{definition}

We note that in this situation, up to the existence of some
intervals where the itineraries of the points are the same, the
dynamics of the maps $f\,$and$\,g\,$ are topologically
equivalent(see \cite{dMVS}).

\subsection{The Lexicographical World}
Let $\,\, Min_2 \,=\, \{ a \in \Sigma_0 \,\,;\,\, \sigma^k(a)
\ge a  \,\,, k \in \Na \, \} \,$and $\,\, Max_2 \,=\, \{ b \in
\Sigma_b \,\,;\,\, \sigma^k(b)  \le b \,\,, k \in \Na \, \} $.

\vspace{0.2cm}

\begin{definition} The set $\,\, LW \,=\, \{ (a\,,b) \in Min_2
\times Max_2 \,\,; \{a\,,b \} \subset \Sigma_{a\,,b} \}\,\,$will
be called the \underline{\em lexicographical world}.
\end{definition}

\vspace{0.2cm}

For $\,\, a \in Min_2 \,\,$ its $\,LW$-fiber is the set $\,\,
LW_0(a)\,=\, \{ b \in Max_2 \,\,;\,\, (a\,,b) \in LW \} \,$. For
$\,\, b \in Max_2 \,\,$ its $\,LW$-fiber is the set $\,\, LW_1(b)
\,\,=\, \{ a \in Min_2 \,\,;\,\, (a\,,b) \in LW \, \} \,$.

\begin{remark} It is clear that given $(a\,,b) \in LW $ then
$ \Sigma_{a,b} \ne \emptyset .$\end{remark}


Let us now consider $(a,b)\in LW$.

\begin{lemma}  There is $f\in DM_0$ such that
$I_f=\Sigma_{a,b}$. \end{lemma}

\noindent {\bf Proof:} Let us consider the map
$g:(\R\setminus\{0\})\to\R$ given by $$ g(x)= \left \{
\begin{array}{lc}
2x-1, & x>0 \\ 2x+1, & x<0 \end{array} \right. $$ In this case
$I_g=\Sigma_2$. Let $x_a<0$ and $x_b>0$ be the points such that
$I_g(x_a^+)=a$, $I_g(x_b^-)=b$.

Let $f:(\R\setminus\{0\})\to\R$ be the map defined by: $$
f(x)=\left \{ \begin{array}{lc} g(x), \quad & x\le \overline{x_b}
\\ x_b,\quad & \overline{x_b}\le x<0\\ x_a, \quad & 0<x\le \overline{x_a} \\
g(x),\quad & \overline{x_a}\le x \end{array} \right. $$
here  $\overline{x_b}<0<\overline{x_a}$ are given by
$g(\overline{x_b})=x_b$ and $g(\overline{x_a})=x_a$.

\vspace{.2cm}

We will call this result the \underline{\em realization lemma}. It
is clear that the maps, in this construction, are non-decreasing
but we want a realization lemma  with increasing maps. In section
3.2 we prove a realization lemma with increasing maps.

\vspace{0.2cm}

Therefore, we have a surjective map $I: DM_0\to LW$,
$I(f)=(a_f,b_f)$ and $DM_0=\bigcup_{(a,b)\in LW}
I^{-1}(\{(a,b)\})$. Also, using $C^0\,$ proximity on compact sets,
this map is continuous.

\vspace{0.2cm}

In this context the next definition is natural. \vspace{0.2cm}


\begin{definition} Let $ \alpha : U\subset\R^k\to DM_0$ be a map.
\begin{itemize}
\item[1)] We will say that $\alpha$ is an
\underline{\em $a$-universal family}
if $\forall a \in Min_2 \,$ there is a nonempty set ,$\, A(a) \subset U \,$,
such that $ \, a_f = a \,, \forall f \in A(a) \,$.

\item[2)]  We will say that $\alpha$ is an \underline{\em $b$-universal family}
if $\forall a \in Max_2 \,$ there is a nonempty set ,$\, B(b) \subset U \,$,
such that $ \, b_f = b \,, \forall f \in B(b) \,$.

\item[3)] We will say that $\alpha$ is an \underline{\em $LW$-universal family} if
$\forall\,\, (a,b)\in LW$ there exists a nonempty set
$\,A(a,b)\subset U \,$ such that $I\circ \alpha(A(a,b))=(a,b)$.
\end{itemize}
\end{definition}
\bigskip

It is clear that associated to any map $\, \alpha \,$, as above,
we have an $a$($b\,,LW\,$)-decomposition of its domain. We will
call this $a$(resp.$\,b\,,LW\,$)-decomposition the
$a$(resp.$\,b\,,LW\,$) bifurcation theory defined by$\,\alpha\,$.

\vspace{0.2cm} \noindent {\bf Open Problem:} There are
$LW$-universal families?.

\vspace{0.2cm} Certainly, this is a very hard problem and we
believe that there is not a finite $\, k \in \Na \,$ with this
property. In the present paper we prove the following

\vspace{0.2cm}
\begin{theorem} The given two parameter family of linear Lorenz maps
is $a$-universal, is $b$-universal but it is not
$LW$-universal. Moreover, the respective $\,
a\,,b\,,LW\,$-bifurcation theories are given.
\end{theorem}



\section{Symbolic Dynamics and the Lexicographical World}

Here we introduce some results and notations that are necessary
for our results. The results which we do not prove in this section
are proved in \cite{LM1}.


\subsection{Dynamical properties for sequences in LW}

\vspace{0.2cm}

Let $\,\, a_1 \leq  a_2 \,$be two periodic sequences in $\,
\Sigma_0$. The sequence$\,\, m(a_1\,,a_2) =
\underline{a_1\,a_2}\,$will be called the \underline{\em average}
of the sequences $\, a_1 \,$and$\,\,a_2$.

\vspace{0.2cm}

\no \underline{Example}: For $\,\,a_1 = \underline{01}\,,a_2 =
\underline{011}\,\,$we have$\,\,m(a_1\,,a_2) =
\underline{01011}\,$.

\vspace{0.2cm}

Let $\,\, A_0 = \{ \underline{0_n1}\,\,,\underline{01_m} \,\,;
n,m \in \Na \setminus \{0 \}\,\,\}$ and $\,\, A_{n+1} = A_n \cup
\{\,\, m(a_1\,,a_2);\,a_1\,,a_2 \in A_n  \,\,\mbox{are
consecutive sequences}\,\,\}$. Set $\,\, A_{\infty} =
\bigcup_{n=0}^{n=\, \infty}A_n$. The elements in $\,\,
A_{\infty} \,\,$will be called \underline{\em primary
sequences}. As we will see in section 4.1, primary sequences are
associated with primary bifurcations.

\vspace{0.2cm}


Let us denote $\,\, A_{\infty}^0\,=\,A_{\infty}\,$and define,
for any $\, a \in A_{\infty}^0\,\,$the set: $$ \,\,A_1^1(a) =
\{c \in
\Sigma_0\,;\,c\,=\,\underline{a_-b_+a^n}\,\,\mbox{or}\,\,c\,=\,\underline{a_-b^mb_+}\,,
\mbox{for}\,\,n\,,m \in \Na \}. $$ Here for $\,\,
b\,=\,\underline{b_0b_1\dots b_k0}\,\,$we
have$\,\,b_+\,=\,b_0b_1\dots b_k1 \,$.

\vspace{0.2cm}

Observe that if $\, c_n\,=\,\underline{a_-b_+a^n}\,\,$then$\,\,c_n
\rightarrow a_-b_+\underline{a}\,\,$and
if$\,\,d_m\,=\,\underline{a_-b^mb_+}\,$then $\,d_m \rightarrow
a_-\underline{b}\,$.

\vspace{0.2cm} Set $$ A_2^1(a)=\,\{m(a_1\,,a_2)\,;\,\,a_1\,,a_2
\in A_1^1(a) \mbox{are consecutives sequences}\}\, \cup A_1^1(a)
$$ and in general $$
A_{n+1}^1(a)=\,\{m(a_1\,,a_2)\,;\,\,a_1\,,a_2 \in A_n^1(a)
\mbox{are consecutive sequences}\}\, \cup A_n^1(a)\,\,,n \geq 2.
$$ Let $\,\, A_{\infty}^1(a)\,=\,
\bigcup_{n=1}^{n=\infty}A_n^1(a) \,$and$\,\,
A_{\infty}^1\,=\,A_{\infty}^0 \cup \bigcup_{a \in
A_{\infty}^0}A_{\infty}^1(a)$ \vspace{0.5cm} Inductively, for
any $\,\, a \in A_{\infty}^n(a)\,\,$,let $$ A_1^{n+1}(a)\,=\,\{
c \in \Sigma_0\,;\,
c\,=\,\underline{a_-b_+a^j}\,\,\mbox{or}\,\,c=\underline{a_-b^kb_+}\,\mbox{for}\,\,j,k
\in \Na\,\} $$ Now , we define $$
A_2^{n+1}(a)\,=\,\{m(a_1\,,a_2)\,;\,\,a_1\,,a_2 \in
A_1^{n+1}(a)\,\,\mbox{are consecutive sequences} \} \bigcup
A_1^{n+1}(a) $$ and $$
A_{m+1}^{n+1}(a)\,=\,\{m(a_1\,,a_2)\,;\,\,a_1\,,a_2 \in
A_m^{n+1}(a)\,\,\mbox{are consecutive sequences} \} \bigcup
A_m^{n+1}(a)\,\,,m \geq 2. $$ \vspace{0.2cm}

As before, define $\,\, A_{\infty}^{n+1}(a) \,=\,
\bigcup_{m=1}^{\infty}A_m^{n+1}(a)\,\,;\,\,A_{\infty}^{n+1}\,=\,\bigcup_{a
\in A_{\infty}^n}A_{\infty}^{n+1}(a) \cup A_{\infty}^n \,\,$and
finally,$\,\,A_{\infty}^{\infty}\,\,=\,\,\bigcup_{j=0}^{\infty}A_{\infty}^j\,$.
\vspace{0.2cm}

The elements in $\,\, (A_{\infty}^{\infty}\,\setminus
A_{\infty}^0)\,\,$ will be called \underline{secondary sequences},
as we will see in section 4.2 secondary sequences are associated
with secondary bifurcations.

\vspace{0.2cm}

 Let denote by $B_{\infty}^{\infty}\,$the set $\{
\sup\{\sigma^k(a)\,, k \in \Na\}\,, a \in A_{\infty}^{\infty} \,
\}$. We will denote by $b(a)\,$ the sequence $\,
\sup\{\sigma^k(a)\,,k \in \Na\,\} \,$ for $a \in \Sigma_0\,$ and
by $a(b)\, ,$ the sequence $\inf\{\sigma^k(b) \,,\,k\in \Na \,\}
\,$ for $ b \in \Sigma_1 \,$. Clearly, $b(a) \in Max_2
\,\,$and$\, a(b) \in Min_2 \,$.

\begin{lemma}
For any $\,\, a \in A_{\infty}^{\infty} \,\,$and~$\,\, d \in
B_{\infty}^{\infty} \cap LW_0(a) \,\,$we have that the map $\,\,
\sigma : \Sigma_{a \,,d} \rightarrow \Sigma_{a\,,d} \,\,$is
topologically transitive.
\end{lemma}

\vspace{0.2cm}

As before let $\, a \in A_{\infty}^{\infty} \,$ and $\, d \in
LW_0(a) \,\,$be a sequence which is not periodic nor
preperiodic. Set $\,\, a_{last} \,=\, \sup \{ \sigma^k(a)
\,\,;\,\, \sigma^k(a) \in [a \,, 0\underline{1}] \,, k \in \Na
\} \,$and$\,\, d_{first} \,=\, \inf \{\sigma^k(d) \,;\,
\sigma^k(d) \in [1\underline{0} \,, d] \,, k \in \Na \} \,$.

\vspace{0.2cm}

In the sequel $\, int(F) \,$ will denote the interior of the set
$\,F \,$.
\begin{lemma}
\begin{itemize}
\item[(1)] If there is $\, p \in \Na \,$ such that $\, a_{last} \in
int( \sigma^p([a_{last}\,,0d])) \,\,$then $\, \, \sigma_{a \,,d} :
\Sigma_{a\,,d} \rightarrow \Sigma_{a \,,d} \,$is topologically
transitive.
\item[(2)] If there is $\,\, p \in \Na \,$such that $\,\,\,
d_{first} \in int(\sigma^p([a_{last} \,0d])) \,\,$ then $\, \,
\sigma_{a \,,d} : \Sigma_{a\,,d} \rightarrow \Sigma_{a \,,d} \,$is
topologically transitive.

\item[(3)] If there is $\,\, p \in \Na \,\,$ such that $\,\,
a_{last} \in int(\sigma^p([1a\,,d_{first}])) \,\,$then
 $\, \, \sigma_{a \,,d} :
\Sigma_{a\,,d} \rightarrow \Sigma_{a \,,d} \,$is topologically
transitive.
\item[(4)] If there is $\,\, p \in \Na \,$ such that $\,\, d_{first} \in
 int(\sigma^p([1a\,,d_{first}])) \,\,$then
 $\, \, \sigma_{a \,,d} :
\Sigma_{a\,,d} \rightarrow \Sigma_{a \,,d} \,$is topologically
transitive.
\end{itemize}
\end{lemma}

\vspace{0.2cm}

\begin{lemma} Assume that $\,\, a \in Min_2 \,,\,\, d \in Max_2 $
satisfies
\begin{itemize}
\item[(1)] $\,\, a < a(d) \le a_{last} \,\, \mbox{or} \,\,
d_{first}  \le b(a) < d \,\,$,
\item[(2)] There are $\, p \,>\,1  \,\,\mbox{and} \,\, q \,>\,1
\,$such that $\,\, \sigma^p([a_{last} \,, 0d]) \subset [a_{last}
\,, d_{first}] \,$and \\$\,\, \sigma^q([1a \,, d_{last}) \subset
[a_{last} \,, d_{first}] \,\,$. Then there are :
\end{itemize}

\begin{itemize}
\item[-] an invariant set $\,\, \Lambda \subset \Sigma_{a\,,d}
\,$,
\item[-] an open set $\,\, U \subset \Sigma_2 \,$ and an integer
$\,\, n_1 \in \Na \,$such that :
\end{itemize}
$$ \sigma^{n_1}(U \cap \Sigma_{a\,,d}) \subset \Lambda
\,\,\mbox{and}\,\, U \cap \Lambda \,=\, \Phi \, $$

In particular, $\,\, \sigma_{a\,,d} : \Sigma_{a\,,d} \rightarrow
\Sigma_{a \,,d} \,$ is not a topological transitive map.

\end{lemma}


\begin{remark}
Given $ \,\, a \in A_{\infty}^{\infty} \,\,$ we have :

\begin{itemize}
\item[(i)] $\,\, \sigma_{a \,,b_+\underline{a}} : \Sigma_{a \,,b_+\underline{a}}
 \rightarrow \Sigma_{a \,, b_+\underline{a}} \,\,$is not
 topologically transitive,

 \item[(ii)] $\,\,\sigma_{a_-b_+\underline{a} \,,b_+\underline{a}} :
 \Sigma_{a_-b_+\underline{a} \,,b_+\underline{a}}
 \rightarrow \Sigma_{a_-b_+\underline{a} \,, b_+\underline{a}}
 \,\,$is not topologically transitive and
 \item[(iii)] $\,\, \sigma_{a_-\underline{b} \,,b_+a_-\underline{b}} :
 \Sigma_{a_-\underline{b} \,,b_+\underline{a}}
\rightarrow \Sigma_{a_-\underline{b} \,, b_+a_-\underline{b}}
\,\,$is not topologically transitive.

\end{itemize}

\end{remark}

It is clear that  $\, \Sigma_{a\,,\underline{1}} \ne \Phi
\,\,\,$for any
 $\, a \in \Sigma_0\,\,$.
 Hence we can define maps $\, \varphi\,,\psi \,, \chi  : \Sigma_0 \rightarrow \Sigma_1 \,$ by :
$$ \varphi(a)\,=\,\inf \{b \in \Sigma_1 |\, \Sigma_{a\,,b\,} \ne
\Phi \, \}, $$
 $$ \psi(a)\,=\, \inf\{ b \in
\Sigma_1 \,;\, \Sigma_{a\,,b}\,\,
\mbox{contains$\,\infty\,$-elements} \} $$ and $$ \chi(a)\,=\,
\inf\{b \in \Sigma_1 \,;\, \Sigma_{a\,,b}\,\, \mbox{is
uncountable}\}. $$

Clearly, $\,\, a_1 \leq a_2 \,\,$imply$\,\,\varphi(a_1) \leq
\varphi(a_2)\,\,,\psi(a_1) \leq \psi(a_2)\,\,$and$\,\, \chi(a_1)
\leq \chi(a_2) \,$ and for all$\,\, c \in \Sigma_1 \,$ such that
$\, c < \varphi(a) \,$ we have $\,\, \Sigma_{a\,,c} = \Phi $.

\vspace{0.2cm}


\no \underline{\bf Examples} : For any $\,0\underline{01} \leq a
\leq \underline{01} \,$ we have $\, \varphi(a) = \underline{10}
\,\,, \psi(\underline{01}) = \chi(\underline{01})=
1\underline{10}$. Also $\, \varphi(\underline{0}) =
\psi(\underline{0}) = \chi(\underline{0}) = 1\underline{0}
\,$;$\, \varphi(0\underline{1}) = \psi(0\underline{1}) =
\underline{1} \,; \varphi(\underline{0_n1}) =
\underline{10_n}\,\,, \psi(\underline{0_n1})=
\chi(\underline{0_n1}) = 1\underline{10_n}\,\, $and$\,
\,\varphi(\underline{01_m}) = \,\underline{1_m0} \,\,,
\psi(\underline{01_m}) = \chi(\underline{01_m}) =
\underline{11_m0}$.


\subsection{ The Realization Lemma.}

In this section we will prove a realization lemma with increasing
maps.

\begin{lemma}  Let $a\in  A_{\infty}^0$. There is an injective
$f\in DM_0$ such that $a_f=a$ and $b_f=\varphi(a)$.
\end{lemma}

\noindent {\bf Proof:} Let $a=a_0<a_1<\dots<
(a_{\mbox{last}}=a_p)<0\underline{1}<1\underline{0}<
(b_{\mbox{last}}=b_{p+1})< b_{p+2}<\dots<b_k=b(a)$, denote the
consecutive elements in $\{\sigma^j(a);\quad j\in\Bbb N\}$. Let
$\ell<0<r$ and $I=[\ell,r]$. Consider closed intervals
$L_0,L_1,\dots,L_k$ such that $\bigcup_{i=0}^kL_i=[\ell,r]$;
$\mbox{left}(L_0)=\ell$,
$\mbox{right}(L_0)=\mbox{left}(L_1),\dots,\mbox{right}(L_p)=0=
\mbox{left}(L_{p+1}),\dots,\mbox{right}(L_{k-1})=\mbox{left}(L_k)$;
$\mbox{right}(L_k)=r$. Here $\mbox{left}[x,y]=x$ and
$\mbox{right}[x,y]=y$.

Now, define a map $f$ that sends $L_{p+1}\to L_0$, $L_{p+2}\to
L_1,\dots,L_k\to L_{k-p-1}$ and $L_0\to L_{k-p}$; $L_1\to
L_{k-p+1},\dots,L_p\to L_k$. This map satisfy the lemma.$\,\Box
\,$

\vspace{0.2cm}

The next result follows immediately from this lemma:

\vspace{0.2cm}
\begin{corollary}
Let $\, a \in \overline{A_{\infty}^0} \setminus A_{\infty}^0\,.$
There is an injective map $\, f \in DM_0 \,$ such that $\,
a_f\,=\,a \,$and$\, b_f\,=\,\varphi(a)\,\,\Box.$

\end{corollary}


We note that these maps can be considered as  bijective maps on
the circle.

\begin{lemma} Let $\, a \in A_{\infty}^{\infty} \,$. There is $\, f \in DM_0 \,$
an increasing map, such that $\, a_f = a\,$and$\, \varphi(a) = b_f\,$.
\end{lemma}

\noindent {\bf Proof:} Let $$ a= a_0 <a_1 < \dots <a_p
<0\underline{1} <1\underline{0} < a_{p+1} < \dots < a_k. $$
denote the consecutive elements in $\{ \sigma^j(a)\,,\, j \in
\Na \}\,$. Associated with $a_s \,, 0 \le s \le k\,\,,$consider
a closed interval $\,L_s\,$ such that $$ L_0 < L_1 < \dots <L_p
\le L_{p+1} < \dots < L_k, $$ $\,\, L_p \cap L_{p+1} = \{0\}\,$
and $\, L_i \cap L_{i+1} = \emptyset \,$, otherwise.
\vspace{0.2cm} Let $\, \eta : \{0\,,1\,, \dots\,,k\} \to
\{0\,,1\,, \dots\,,k\} \,$be the permutation defined by $\,
\sigma(a_j) =\, a_{\eta(j)}\,$. Clearly, $\, \eta(p) = k \,$and
$\, \eta(p+1) = 0 \,\,$. Now, define an increasing map $\, f \in
DM_0\,$ which satisfy $\,f(L_i)\,=\, L_{\eta(i)} \,$ and is
increasing between $L_i\,$and$\, L_{i+1} \,\,, i\ne p \,, 0 \le
i \le k-1\,\,\Box\,.$ \vspace{0.2cm}

Now we prove:

\begin{lemma}
For  $\, a \in  A_{\infty}^{\infty} \,$ and $\, d \in
B_{\infty}^{\infty} \cap LW_0(a) \,$. There is an increasing map
$\, f \in DM_0 \,$ such that $\, a_f \,=a \,,b_f = d \,$.
\end{lemma}
\noindent{\bf Proof:} Assume $\, \#(per(a)) \,=\, n \,$and$\,\,
\#(per(d)\,=\, m\,$. Let $ c_0=a<c_1 < \dots < c_p <
0\underline{1} < 1\underline{0} < c_{p+1} < \dots <
c_{m+n-1}\,$, denote the consecutive elements in $\,\{
\sigma^j(a)\,, \sigma^j(d)\,, \, j \in \Na \}\,$. \vspace{0.2cm}
Associated to $\, c_j\,$ consider closed intervals $\, L_s\,\,;
0 \le s \le m+n-1 \,\,,$ such that $$ L_0 < L_1 < \dots < L_p <
L_{p+1} < \dots < L_k, $$ $\, L_P \cap L_{p+1} = \{0\} \,$ and
$\, L_i \cap L_{i+1} = \emptyset \,$, otherwise.

\vspace{0.2cm} Let $\,\{ i_1\,,i_2 \,,\dots \,, i_n \} \,$ and $\,
\{j_1\,,j_2\,, \dots \,, j_m \}\,$, be the indices defined by $\,
c_{i_s} = \sigma^t(a) \,;$ for some $\,t\,$and $\, c_{j_r} =
\sigma^l(d)\,$for some $\,l\,$. Let $\, \eta: \{i_1\,, \dots \,,
i_n\} \to \{i_1\,, \dots\,,i_n\} \,$and $\, \rho: \{ j_1\,,
\dots\,, j_m\} \to \{ j_1\,, \dots \,, j_m\}\,$be the permutations
defined by $\, \sigma(a_{i_s})\,\,=\, a_{\eta(i_s)}\,$and $\,
\rho(d_{j_r})\,=\, d_{\rho(j_r)} \,$.

\vspace{0.2cm}

Now, define an increasing map $\, f \in DM_0 \,$ by $\,\,
f(L_{i_r}) \,=\,L_{\eta(i_r)} \,$and $\, f(L_{j_r})\,=\,
L_{\rho(j_r)}\, \Box .$


\vspace{0.2cm}
  By a limit process we obtain

\begin{lemma}
Associated to any $\, a \in Min_2 \,$ there is an increasing map
$\, f \in DM_0 \,$ such that $\, a_f = a \,$ and $\, b_f =
\varphi(a_f) \,$.
\end{lemma}

As a consequence of these lemmas we get

\vspace{0.2cm}

\begin{proposition}
Given $\, (a,b) \in LW \,$ there is an increasing map $\, f \in
DM_0 \,$ such that $\, I(f)\,=\,(a_f\,,b_f)\,=\,(a,b)\,$.
\end{proposition}



This result is a generalization of a similar one obtained for
expansive maps in \cite{HS}.
\subsection{The finite and the entropy zero cases.}

\begin{definition} We will call a map $f\in DM_0$
\underline{\em essentially Morse-Smale} if $a_f\in A^0_\infty$ and
$b_f= \varphi(a_f)$.
\end{definition}

\vspace{0.2cm}

We will denote by $\, MS_0 \subset DM_0 \,$ the set of essentially
Morse-Smale maps . We call these maps Morse-Smale because its
dynamics  essentially reduces to a periodic orbit.

\vspace{0.2cm}

\begin{lemma}  Given $f\in DM_0$ a ``essentially Morse-Smale''
map we have that\\
$[f(0^+),f(0^-)]\cap\Ga_f=\bigcup_{i=0}^{\mbox{per}(a_f)-1}
I_i$, where $x\in I_0$ implies $I_f(x)=a_f$ and $I_f|_{I_j}$ is
constant and equal to $a_j=\sigma^j(a_f)$ for
$0<j<\mbox{per}(a_f)-1$.
\end{lemma} \vspace{.2cm}

\begin{definition} We will call a map $\, f \in DM_0 \,$ an
\underline{\em entropy zero} map if $\, b_f \le \chi(a_f)\,$.
\end{definition}



One of the most interesting problems related with the bifurcation
theory associated to a parameterized family of dynamical systems
$\,\, \{ f_{\lambda}\,;\, \lambda \in U \subset \R^k\,\}\,$, is to
describe the set $\,\,\{\lambda \in U\,;\, f_{\lambda}
\,\,\mbox{is an entropy zero map} \,\}\,$(see for instance
\cite{M}, \cite{Tr} and the references there in ). Actually, see
\cite{LM1}, this problem reduces to find the region where the
previous definition applies. For our linear family of Lorenz maps
we will solve this problem in section 4.1.


\vspace{0.2cm}


We observe that every essentially Morse-Smale map is an entropy
zero map.

\subsection{ The topologically expansive cases}

\bigskip



\medskip

\begin{definition} A map $f\in DM_0$ will be called \underline{
topologically expansive} if $\bigcup_{i=0}^\infty f^{-i}(0)$ is a
dense set in $[f(0^+), f(0^-)]$.
\end{definition}
\bigskip

Let denote by $TE_0=\{f\in DM_0$; $f$ is a topologically expansive
map\} and $TE=I(TE_0)\subset LW$.

\bigskip

\noindent {\bf Note:} It is clear that $EMS_0\cap
TE_0=\emptyset$ and, as a consequence, $\,(LW\setminus
TE)\ne\emptyset$.

\bigskip

In fact, we have the following result due to Hubbard and Sparrow
(\cite{HS}).


\bigskip

\begin{lemma} (\cite{HS}) $(a,b)\in TE$ if and only if $(a,b)\in LW$ and
there are not finite sequences $d=1d_1,\dots,d_k$; $c=0c_1\dots
c_p$ such that $a=cb$ or $b=da$.
\end{lemma}
\vglue .2in

\section{Proof of Our Results}

\medskip

Let $Q=\{(\lambda,\mu)\in\R^2;\quad -1\le\mu\le 1$ and $-1
\le\lambda\le1 \,\}$.

\vspace{.2cm}

Let $ \, F :Q \to DM_0$ be the two-parameter family,
$F_{\la,\mu}=F_{(\la,\mu)}$ given by $$ F_{\la,\mu}(x)=\left \{
\begin{array}{lc}
(1+\la)x+1,\quad & x<0 \\ ( 1+\mu)x-1, &  x>0
\end{array} \right. $$

\bigskip

\subsection{ Primary Bifurcations}

Assume $\, a \in A_{0}^{\infty}$.

\begin{lemma} There is a non empty region $\, L_a \subset Q \,$
such that $\, (\lambda, \mu ) \in Q \,$imply $\, a(\lambda,\mu)
\,=\,a(F_{\lambda,\mu})\,=\,a $. \end{lemma}

\noindent{\bf Proof:} It is not difficult to see that the
regions are given by the next figure.

\begin{figure}[hb]
\begin{center}
\includegraphics[width=12cm]{figura0.eps}
\caption{} \label{f1}
\end{center}
\end{figure}

\begin{corollary}
\begin{itemize}
\item[a)] For any $\, a \in (\overline{A_0^{\infty}} \setminus
A_0^{\infty}) \,$ the set $\, L_a \subset Q \,$ is a curve.

\item[b)] Let $\, G\,=\, \{(\lambda,\mu)\,;\, -1 \le \mu \le 1
\,,\, -\mu < \lambda \le 1 \,\}.$ Then $$ Q \setminus
G\,=\,\{(\lambda\,,\mu)\,;\,F_{(\lambda,\mu)} \mbox{is an entropy
zero map} \,\}.$$
\end{itemize}
\end{corollary}
\subsection{Secondary Bifurcations}

In the sequel we are only interesting in the bifurcation scenario
in $\, D\,=\, \overline{G}\,$.


\begin{lemma}  There is $(a,b)\in(TE\setminus I\circ F(D))$.
As a consequence: the linear family do not contains a universal
model for topologically expanding maps.
\end{lemma}


\noindent {\bf Proof:} Let us consider $a_0=\underline{01}$,
$b_0=\underline{10}$, $a_1=\underline{(a_0)_-(b_0)_+}=
\underline{0011}$,
$b_1=\underline{(b_0)_+(a_0)_-}=\underline{1100}$ and
$a_{n+1}=\underline{(a_n)_-(b_n)_+}$,
$b_{n+1}=\underline{(b_n)_+(a_n)_-}$; $n\ge1$. The sequences
$a=\lim_{n\to\infty}a_n$, $b=\lim_{n\to\infty}b_n$ satisfies the
conditions of the Lemma in III.5.1. It is easy to see that
associated to $a_n$ and $b_n$ there are curves $TE(a_n)$,
$TE(b_n)$ that satisfies

\begin{itemize}

\item[(1)] $(\la,\mu)\in TE(a_n)$ implies $a(F_{\la,\mu})=a_n$ and
 $(\la,\mu)\in TE(b_n)$ implies $b(F_{\la,\mu})=b_n$


In fact; for $a_0=\underline{01}$ we have
$TE(\underline{01})=\{(0,\mu);\quad 0\le\mu\le1\}$ and for
$b_0=\underline{10}$ we have $TE(\underline{10})=\{(\la,0);\quad
0\le\la\le1\}$.

Let $A_\mu^0=[0^+,\mu]$ and its second image
$F^2(A_\mu^0)=[-\la,(1+\la)(\mu(1+\mu)-1)+1]$. When
$(1+\la)(\mu(1+\mu)-1)+1=0$ we get the sequence
$b_1=\underline{1100}$. That is $$ TE(\underline{1100}) = \left\{
\left( \frac{\mu(\mu+1)}{1-\mu(1+\mu)};\quad \mu \right);\quad
0\le\mu\le - \frac{1+\sqrt3}2 \right\} $$

In a symmetric way we get the curve
$TE(\underline{0011})=\{(\la,\mu(\la)),\quad
0\le\la\le-\frac{1+\sqrt3}2\}$.

Clearly, $TE(\underline{1100})\subset\{(\la,\mu);\quad \mu<\la\}$
and $TE(\underline{0011})\subset\{(\la,\mu);\quad \mu>\la\}$.
Moreover $\la'(0)=\mu'(0)=1$.

Now, for $(1+\la)(\mu(1+\mu)-1)+1\le\mu$ we have
$F^2(A_\mu^0)\subset[-\la,\mu]$ and we can define
$F_1:[0^+,\mu]\to[-\la,\mu]$ by $$ F_1(x)=(1+\la)(1+\mu)x-\la. $$
Similarly, for $B_\la^0=[-\la,0^-]$ we have
$F^2(B_\la^0)=[(1+\mu)[1-\la(1+\la)]-1,\mu]$. For
$-\la\le(1+\mu)[1-\la(1+\la)]-1$ we have
$F^2(B_\la^0)\subset[-\la,\mu]$ and we can define
$F_1:[-\la,0^+]\to[-\la,\mu]$ by $$ F_1(x)=(1+\mu)(1+\la)x+\mu. $$

Using the map $F_1$ we can start again and we will obtain
$TE(\underline{11010010})$ which is a curve
$\{(\la(\mu);\mu);\quad 0\le\mu\le\mu_1\}$ where $\la(\mu)$
satisfy the equation $$
(1+\la)^3(1+\mu)^3\mu-(1+\la)^2(1+\mu)^2\la-(1+\la)(1+\mu)\la+\mu=0
$$ and $$ 8(1+\mu_1)^3\mu_1-4(1+\mu_1)^2-2(1+\mu_1)+\mu_1=0. $$

Inductively we get the result.


\medskip

\item[(2)] $(0,0) \in \overline{TE(a_n)}$ and $(0,0)\in\overline{TE(b_n)}$

\medskip

\item[(3)] $TE(a_n)\subset\{(\la,\mu)\in[0,1]\times[0,1];\quad\la<\mu\}$

and
\\$TE(b_n)\subset\{(\la,\mu)\in[0,1]\times[0,1];\quad\mu<\la\}$

\medskip

\item[(4)] $TE(a_n)\to TE(a)$ and $TE(b_n)\to TE(b)$ with
$$ TE(a)\subset\{(\la,\mu)\in[0,1]\times[0,1];\quad\la<\mu\} $$ $$
TE(b)\subset\{(\la,\mu)\in[0,1]\times[0,1];\quad\mu<\la\} $$

\medskip

\item[(5)] $(\la,\mu)\in TE(a)$ imply $a(F_{(\la,\mu)})=a \,\,
\mbox{and} (\la,\mu)\in TE(b)$ imply $b(F_{(\la,\mu)})=b$.

\medskip

So, we conclude $(a,b)\in[TE\setminus I\circ F(D)]$ as we
announced in the lemma.

\vglue .2in
\end{itemize}
\noindent {\bf 3.5.3 Some remarks concerning the family
$F_{\la,\mu}$.}

\begin{itemize}
\item[a)] Let $A_0, A_1,\dots,A_\infty$ be defined as in I.5.(c). Let $a\in
A_\infty$. There is a parameter value $(-\mu_a,\mu_a)\in D$ such
that $a(-\mu_a,\mu_a)=a$; $b(-\mu_a,\mu_a)=b(a)$.

\item[b)] Associated to any $a\in A_\infty$ there are two curves $TE(a)$, $TE(b(a))$
 such that
$$ (\la,\mu)\in TE(a)\quad\mbox{ imply }\quad a(F_{(\la,\mu)})=a
$$ and $$ (\la,\mu)\in TE(b(a))\quad\mbox{ imply }\quad
b(F_{(\la,\mu)})=b(a) $$

Moreover $$ TE(a)\cap TE(b(a))=\{(-\mu_a,\mu_a)\} $$
\end{itemize}

For example, take $a=\underline{01_n}$, $b=\underline{1_n0}$. In
this case we have $$ 1 \stackrel{f}\to \mu \stackrel{f}\to
(1+\mu)\mu-1 \stackrel{f}\to   (1+\mu)^2\mu-(1+\mu)-1
\stackrel{f}\to\qquad \dots $$
$$\stackrel{f}\to(1+\mu)^k\mu-(1+\mu)^{k-1}-\dots-(1+\mu)-1. $$
Therefore $$
f^{k+1}(1)=\xi_{k+1}(\mu)=(1+\mu)^k\mu-(1+\mu)^{k-1}-\dots-1 $$

\medskip

\noindent {\bf Observe:}

1) $\xi_{k+1}(0)=0 - \underbrace{1-1 \dots-1}_{k} = - k
$

\medskip

2)
$\xi_{k+1}(1)=2^k-(2^{k-1}+\dots+1)\,=\,2^k-\frac{2^k-1}{2-1}=1$.

\medskip

3)
$\xi_{k+1}'(\mu)=k(1+\mu)^{k-1}\mu+(1+\mu)^k-(k-1)(1+\mu)^{k-2}-1$
or, since \\

$\xi_{k+1}(\mu)=(1+\mu)^k\mu-\frac{(1+\mu)^k-1}{1+\mu-1}=(1+\mu)^k\mu-\frac{(1+\mu)^k-1}\mu$,
then $$
\xi_{k+1}'(\mu)=k(1+\mu)^{k-1}\mu+(1+\mu)^k-\frac{k(1+\mu)^{k-1}\mu-[(1+\mu)^k-1]}{\mu^2},
\quad\mu \ne0. $$

\medskip

4) Let $\mu_k \pitchfork \xi_k(\mu_k)=0$. Clearly$\,\, \mu_k\to1$.

\medskip

The equation $h_n \,=\, \mu_{n-1}$ correspond to
$b=\underline{1_no}$. These curve  satisfies $h_n \to-1$ with
$n\to\infty$ and so on.

\medskip
\begin{itemize}
\item[(c)] The curve $TE(a)$ transversally intersects the curves $\mu=\mu_0$
for $\mu_a\le\mu_0\le1$.

The curve $TE(b(a))$ transversally intersects the curves
$\la=\la_0$ for $\,\,\\ -\mu_a\le \la_0 \le1$.

Moreover,for $a_n=0_n1\dots$, we have that $TE(a_n)$ converges to
$\{(1,\mu);\quad -1\le\mu\le 1\}$ in the $C^1$-topology. For
$b_n=1_n0\dots$, we have that $TE(b_n)$ converges to
$\{(\la,1);\,\, -1\le\la\le1\}$ in the $C^1$-topology.

\medskip

\item[(d)] The curves $TE(10_n\dots)$ converges to $\{(1,-1)\}$ and the curves
$TE(01_n\dots)$ converges to $\{(-1,1)\}$ with $n\to\infty$.

\medskip

\item[(e)] Given $a\in A_\infty$, $b=b(a)$, we can find curves $TE(b_+\underline{a})$,
$TE(b_+ a_-\underline{b})$, $TE(a_-b_+\underline{a})$ and
$TE(a_-\underline{b})$ such that:
\end{itemize}
\begin{itemize}

\item[(1)] $(-\mu_a,\mu_a)\in\overline{TE(c)}$ for
$c=b_+\underline{a},\, b_+ a_-\underline{b},\,
a_-b_+\underline{a},\, a_-\underline{b}$;

\medskip

\item[(2)] $TE(a_-b_+\underline{a})$ and $TE(b_+\underline{a})$ are tangents to
$TE(a)$ at $(-\mu_a,\mu_a)$;

\medskip

\item[(3)] $TE(a-\underline{b})$ and
$TE(b_+a_-\underline{b})$ are tangents to $TE(b)$ at $(-\mu_a,\mu_a)$

\medskip

Figure (\ref{f1}) represents these essential facts concerning the
geo\-me\-try of these curves.
\begin{figure}[hb]
\begin{center}
\includegraphics[angle=0,width=11cm]{figura1.eps}
\caption{} \label{f1}
\end{center}
\end{figure}


\item[(4)] For $\, a \in  A_\infty$ let
$$\,\, A_1^1(a)\,=\, \{c \in \Sigma_0;\quad
c=\underline{a_-b_+a^n},\quad n \in \Bbb N \mbox{ or }
c=\underline{a_-b^nb_+},\quad n \in \Bbb N \}$$.

\begin{itemize}

\item[(i)] Associated with any $c=\underline{a_-b_+a^n}$ we can
find a direction $v_n\in\Bbb R^2$, $v_n\to v_a$, $v_a=$ the
direction of the curve $TE(a)$ at $(-\mu_a,\mu_a)$;

\medskip

\item{} Associated with any $c=\underline{a_-b^nb_+}$ we can find a direction
$v_n\in\R^2$, $v_n\to v_b$, $v_b=$ the direction of the curve $TE(b)$ at $(-\mu_a,\mu_a)$

\medskip

\item[(ii)] Associated with any $c=\underline{a_-b_+a^n}$ $(c=\underline{a_-b^mb_+})$
we can find two curves $TE(c)$ and $TE(\underline{b_+a^na})$
(resp. $TE(c)$, $TE(\underline{b_+a_-b^m}))$ tangents to the
direction $v_n$ (resp. $v_m$) and located is different sides of
the direction $v_n$ (resp. $v_m$) that satisfies $$ (\la,\mu)\in
TE(c) \quad\mbox{ imply }\quad a(F_{(\la,\mu)})=c $$ $$
(\la,\mu)\in TE(\underline{b_+a^na})\quad\mbox{ imply } \quad
b(F_{(\la,\,\mu)})=\underline{b_+a^na} $$ $$ \Bigl[\mbox{resp.
}\quad (\la,\mu)\in TE(c)\quad \mbox{ imply }\quad
a(F_{(\la,\mu)})=c\qquad \mbox{ and} $$ $$ (\la,\mu)\in
TE(\underline{b_+a_-b^m}) \quad\mbox{ imply }\quad
b(F_{(\la,\mu)})= \underline{b_+a_-b^m} \Bigl ] $$

Figure (\ref{f2}) represents these essential facts.

\begin{figure}[ht]
\begin{center}
\includegraphics[angle=0,width=11cm]{figura2.eps}
\caption{} \label{f2}
\end{center}
\end{figure}

\vglue .2in


\item[(iii)] Given $c \in A_1^1(a), \,, b=b(c)$, we can find curves
$TE(b_+\underline{c})$, $TE(b_+c_-\underline{b})$,
$TE(c_-b_+\underline{c})$ and $TE(c_-\underline{b})$ such that:

\begin{itemize}
\item[(a)] $(\la,\mu)\in TE(b_+\underline{c})$ imply $ b(F_{(\la,\mu)})=b_+\underline{c}$;

\item{} $(\la,\mu)\in TE(b_+c_- \underline{b})$ imply
$ b(F_{(\la,\mu)})=b_+c_-\underline{b}$;

\item{} $(\la,\mu)\in TE(c_-b_+\underline{c})$ imply $ a(F_{(\la,\mu)})=c_-b_+\underline{c}$;

\item{} $(\la,\mu)\in TE(c_-\underline{b})$ imply $ a(F_{(\la,\mu)})\,=\, c_-\underline{b}$;

\bigskip

\item[(b)] These four curves are tangents to the corresponding direction associated to $c$;

\bigskip

\item[(c)] $(-\mu_a,\mu_a)\in \overline{TE(d)}$ for $d=b_+\underline{c},
\,b_+c_-\underline{b},\, c_-b_+\underline{c}$ and
$c_-\underline{b}$.
\end{itemize}
\end{itemize}

 Figure (\ref{f3}) represents these essential facts.

\begin{figure}
\begin{center}
\includegraphics[angle=0,width=11cm]{figura3.eps}
\caption{} \label{f3}
\end{center}
\end{figure}

\vglue .2in


\noindent {\bf Remark:} $\forall\,\, a\in A_\infty$ we have that
$\, \inf A_1^1(a)=a_-b_+\underline{a}$ and $\sup
A_1^1(a)=a_-\underline{b}$.

\bigskip

\item[(5)] Given $\, a \in A_{\infty}$ let
$\, A_2^1(a)\,=\, A_1^1(a) \cup \{ m(a_1,a_2) \,; \, a_1,a_2 \in
 A_1^1(a)\\
 \mbox{are consecutive sequences} \} $.

\begin{itemize}

\item[(i)] Associated with $c=m(a_1,a_2)$ we can find a
direction, $v_c \in \R^2$, located between $v_{a_1}$ and
$v_{a_2}$;

\medskip

\item[(ii)] Associated with  $c=m(a_1,a_2)$ we can find two
curves, $TE(c)$ and $TE(b(c))$, tangents to the direction $v_c$
 and located in different sides of the direction $v_c$ that satisfies:
$$ (\la,\mu)\in TE(c) \mbox{ imply } a(F_{(\la,\mu)})=c; $$ $$
(\la,\mu)\in TE(b(c)) \mbox{ imply } b(F_{(\la,\mu)})=b(c); $$

\medskip

\item[(iii)] Given $c \in A_2^1(a)$ we can continuous as in (4.iii).

\bigskip

In the same way, as we did with $A_1^1$and$A_2^1$ we can
continuous with the set \\ $\, A_{n+1}^1(a)= \, A_n^1(a) \cup
\{m(a_1,a_2)\,; a_1,a_2 \in A_n^1 \,\, \mbox{ are consecutive
sequences} \} \,,n \ge 2$.

Let $\, A_{\infty}^1(a)=\bigcup_{n=1}^{\infty}  A_n^1(a) \,$ and $
\, A_{\infty}^1=\bigcup_{a \in A_{\infty}^0} A_{\infty}^1(a) \cup
A_{\infty}^0. $

Inductively, we make the same construction for elements in $\,
A_{\infty}^{n+1}$, where: $\, A_{\infty}^{n+1}=\bigcup_{a \in
A_{\infty}^n}  A_{\infty}^n(a) \cup  A_{\infty}^n. $

Let $\, A_{\infty}^{\infty} \,=\, \bigcup_{n=0}^{\infty}
A_{\infty}^n$. The same construction, as we did in (4.iii), holds
for any $c \in A_{\infty}^{\infty}$.

\end{itemize}

\item[(6)] Let us now consider $a\in  A_{\infty}^{\infty}$ and
$b\in B_{\infty}^{\infty}$ such that $TE(a)\cap
TE(b)\ne\emptyset$ we have:

\begin{itemize}
\item[(i)] The intersection between these curves is transversal
and contains a unique point $P(a,b)$;

\medskip

\item[(ii)] Define $\, A(a,b)=\{c \in \Sigma_0;\quad c=\underline{a_-b_+a^n};
\quad n\in\Bbb N  \mbox{or} \,c=\underline{a_-b^mb_+};\quad
m\in\Bbb N\}$. Associated with any $c \in A(a,b)$ we can make
the same construction  as we did in (4) and (5) before.

\end{itemize}
\item[(7)] Let $a\in A_{\infty}^{\infty}$ and
$d \in Max_2$ be a sequence which is eventually periodic such
that $TE(d)\cap TE(a)\ne\emptyset$. We have:

\begin{itemize}
\item[(i)] The intersection between these two curves is transversal
and contains a unique point $P(a,d)$;

\medskip

\item[(ii)] We can find a curve $TE(a_-d)$, tangent to $TE(d)$ and such that
 $P(a,d)\in\overline{TE(a_-d)}$ which satisfy:
$$ (\la,\mu)\in TE(a_-d)\quad\mbox{ imply }\quad
a(F_{(\la,\mu)})=a_-d. $$
\end{itemize}
Figure (\ref{f4}) represents these facts.

\begin{figure}[h]
\begin{center}
\includegraphics[angle=0,width=11cm]{figura4.eps}
\caption{} \label{f4}
\end{center}
\end{figure}
\vglue .2in

\item[(8)] Let $b\in B_\infty^\infty$ and $c\in Min_2$
be an eventually periodic sequence such that $TE(c)\cap
TE(b)\ne\emptyset$. We have that:

\begin{itemize}

\item[(i)] The intersection between these two curves is transversal and contains
a unique point $P(c,b)$;

\medskip

\item[(ii)] We can find a curve, $TE(b_+c)$, tangent to $TE(c)$  such that
$P(c,b)\in \overline{TE(b_+c)}$ which satisfy $$ (\la,\mu)\in
TE(b_+c)\quad\mbox{ imply }\quad b(F_{(\la,\mu)})=b_+c. $$
\end{itemize}
Figure (\ref{f5}) represents these facts

\begin{figure}[h]
\begin{center}
\includegraphics[angle=0,width=11cm]{figura5.eps}
\caption{} \label{f5}
\end{center}
\end{figure}

\vglue .2in

\item[(9)] Let $a \in A_{\infty}^{\infty}$ and $ d \in Max_2$ be a sequence which is
not periodic nor eventually periodic such that $TE(a)\cap
TE(d)\ne\emptyset$. We have that:

\begin{itemize}
\item[(i)] The intersection between these two curves is transversal and contains
 a unique point $P(a,d)$;

\medskip

\item[(ii)] We can find a curve, $TE(a_-d)$, tangent to $TE(d)$ and such that
 $P(a,d)\in \overline{TE(a_-d)}$ which satisfy:
$$ (\la,\mu)\in TE(a_-d)\quad \mbox{imply}\quad a(F_{(\la,\mu)
})=a_-d. $$

\end{itemize}

\item[(10)] Let  $\, b \in B_{\infty}^{\infty}$ and $c\in Min_2$
 be a sequence which is not periodic nor eventually periodic such that
  $TE(b)\cap TE(c)\ne\emptyset$. We have that:

\begin{itemize}
\item[(i)] The intersection between these two curves is transversal and
contains a unique point $P(c,b)$;

\medskip

\item[(ii)] We can find a curve, $TE(b_+c)$, tangent to $TE(c)$ and such that
$P(c,b)\in\overline{TE(b_+c)}$ which satisfy: $$ (\la,\mu)\in
TE(b_+c)\quad\mbox{ imply }\quad b(F_{(\la,\mu)})=b_+ c. $$

\end{itemize}
\item[(11)] $F_{(\la,\mu)}$ is conjugated to $F_{(\tilde\la,\tilde\mu)}$ if and only if
$$ a(F_{(\la,\mu)})=a(F_{(\tilde\la,\tilde\mu)}) $$ and $$
b(F_{(\la,\mu)})=b(F_{(\tilde\la,\tilde\mu)}). $$

In fact, any $(\la,\mu)\in D$ represents a unique equivalence
class of elements in $F(D)$.
\end{itemize}
\vglue .3in

\begin{remark} Theorem 1 is a consequence of the results in
section 4.
\end{remark}

\vspace{0.2cm}

\begin{center}
 {\bf ACKNOWLEDGMENTS}
\end{center}


Part of this paper is an outgrowth of research  during a visit of
the authors to ICTP (Trieste - Italy), a visit of RL to
IMPA-Brazil and a visit of CM to the Universidad de Santiago de
Chile. We thanks to these institutions for their support and
hospitality while preparing this paper. \vspace{0.2cm}



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\vspace{0.2cm}
\begin{flushleft}{\bf R. Labarca \\
Departamento de Matematica y CC\\ Universidad de Santiago de Chile
\\ Casilla 307 Correo 2\\ Santiago - Chile\\ and\\ C. Moreira\\
IMPA\\ Estrada Dona Castorina 110\\ CEP 22460-320\\Jardim
Botanico\\ Rio de Janeiro\\ Brazil}
\end{flushleft}








\end{document}
