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\title{Bifurcation of the Essential Dynamics of Lorenz maps on the
real line and the bifurcation scenario for Lorenz like flows :
Contributions to the study of the Expanding Case
\footnote{Partially supported by Fondecyt grants \# 1970720, \#
1990903,  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 evidence of the existence of a
strange attractor for a quadratic system of ordinary
differential equations in three variables. Some time later J.
Guckenheimer, \cite{G}, produced a work where he introduced
symbolic dynamics in order to understand the topologically
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 geo\-me\-tric Lorenz
attractors. The evidence of the non-equivalence where the
kneading sequences associated to these one-dimensional maps.
Later Afrajmovich, Bykov and Shilnikov,\cite{ABS},studied a two
parameter family of three dimensional vector fields that unfolds
a codimension two bifurcation which appears in a vector field
defined in a neighborhood of the origin $\, 0 \in \R^3 $. This
vector field has an hyperbolic singu\-la\-ri\-ty, at $\, 0 \in
\R^3 \,$,which has a two dimensional stable manifold and a one
dimensional unstable manifold. For the connected components $\,
\Lambda_1 \,\,$and$\, \Lambda_2 \,$of the set$\, W^u_0 - \{0 \}
$, the vector field $\,X\,$ satisfies $\, \Lambda_1 \subset
W^s_0 \,$ and $\, \Lambda_2 \subset W^s_0\,$. In that work they
described part of the bifurcation theory that appears in a
generic two parameter unfolding of the vector field $\,X \,$.

So, its seem to be natural to obtain a bifurcation theory for a
generic unfolding of $\, X \, $ using symbolic dynamics. In this
direction de Melo and Martens announced, in an unpublished paper
\cite{dMM}, the existence of parameterized families of
contacting Lorenz-like flows that are topologically universal in
the sense that given any geometric Lorenz flow then its dynamics
is " essentially " the same as the dynamics of some element of
the family. In \cite{LM3} we describe, exactly, the way in which
such a contracting family realize all the allowed dynamics for
Lorenz-like maps. The same universality is not true when we work
with families of expanding Lorenz-like flows as de Melo and
Martens knew and as we will show in the present paper.


Also we have to mention the work by  Hubbard and Sparrow
\cite{HS}, were they defined a set of pair of sequences which
model all the topological dynamics exhibed by expanding Lorenz
maps. Recently, in \cite{LM1}, we extend the Hubbard-Sparrow
model for expanding maps  to an Universal Model for the "
essential dynamics " of Lorenz maps and we called it  the {\em
Lexicographical World} (in the sequel denoted by $\,LW\,$). Now,
it is clear that any parameterized family of Lorenz maps has a
bifurcation theory induced by the lexicographical world ( an
extremely interesting problem focused for several authors in
this an another contexts, see for instance
\cite{CGTU},\cite{BMT},\cite{GMT}, \cite{Ar},
\cite{Bo}) . So, it seems to be natural to develop a programme
to obtain the bifurcation theories associated to some typical
parameterized families of Lorenz maps. We developed such a
programme. In \cite{LM2} and \cite{LM3} we describe the
bifurcation theory associated to a two parameter family of
linear maps and to a two parameter family of contracting maps,
respectively. In certain form, both bifurcation theories are
similar. In the present paper we describe the bifurcation theory
associated to a typical two parameter family of expanding Lorenz
maps and the associated bifurcation theory is dramatically
different from the previous one's.  \vspace{0.2cm}

Of course, there are other points of view which may be used to
described the bifurcation theory associated to parameterized
families of Lorenz maps. In fact,  starting with the work of
Arnold (\cite{Ar})a lot of work have been produced with res\-pect
to the bifurcation theory of the canonical family( see for
instance \cite{Bo} and the references there in). The
approaches,in this case, was done by using the rotation number.
This approach seems to be useful when applied to homeomorphisms
of the circle but looks very complicated when we deal with non
injective maps and, in fact, a bifurcation diagram for the
dynamics (of the canonical family) is unknown (at least for us).
The better approximation to the bifurcation theory of the
canonical map seems to be the work by Boyland ( \cite{Bo}). We
have to mention that the lexicographical world includes the
dynamics of all the interpolated maps used by Boyland (\cite{Bo}
page 359 )in its construction and, then, it is also a model for
these maps.

\vspace{0.2cm}

To be more specific,in the present paper we study the two
parameter family of cuadratic expanding Lorenz maps :

$$ G_{\mu,\nu}(x) = \left\{\begin{array}{l} - \mu + \sqrt{x}
\,\,, \ x > 0\\ \, \nu - \sqrt{-x} \,\,, \ x < 0
\end{array}
\right. $$ and we give the corresponding bifurcation theory in
the parameter space.

\vspace{0.2cm}


\vspace{0.2cm}

This paper is organized as follows : In section 2 we state our
results. In section 3 we describe the lexicographical world, in
Section 4 we prove our results, in section 5 we generalize our
results and, finally, in section 6 we will relate them to three
dimensional geometric vector fields.


Some other comments are in order: there are many other authors
that have investigated these families from the same and from
other points of view (combinatorial, differentiable, topological
or geometrical)( see the list of references).

\section{Statement of our 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 $\Sigma_{a_f,b_f}$.}

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 $f^i(x)\ne0$, $0\le i<n$; $ 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
expanding 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} (\cite{LM2} There is $f\in DM_0$ such that
$I_f=\Sigma_{a,b}$. \end{lemma}


\vspace{.2cm}

We will call this result the \underline{\em realization lemma}.
We observe that the maps, in this lemma, may be obtained as
increasing maps.

\vspace{0.2cm}

Therefore, we have a surjective map $I: DM_0\to LW$,
$I(f)=(a_f,b_f)$ and $DM_0=\bigcup_{(a,b)\in LW}
I^{-1}(\{(a,b)\})$.

\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 \alpha(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 \alpha(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 expanding
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 are proved in \cite{LM1} or in \cite{LM3}.

\vspace{0.2cm}

\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}

\noindent \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 the next section, primary
sequences are associated with primary bifurcations.
\vspace{0.2cm}
The elements in $ \overline{A_{\infty}}\,$are characterized by
the following property:

\begin{lemma}
$a \in \overline{A_{\infty}}\,$ if and only if

$$(*)\qquad a \in Min_2 \,\, \mbox{and}\,\, \sigma(a) \ge
\sigma(b),$$ for $\,b\,=\,\sup\{\sigma^k(a)\,; k \in \Na \}\,$.
\end{lemma}

In the proof of this lemma( see \cite{LM3})we define the renormalization map $
R_{a,b}: \Sigma_2 \to \Sigma(a,b)\,,$ by$ \, R_{a,b}(a_0,a_1,
\cdots)\,=\,(\tilde{a_0},\tilde{a_1},\cdots)\,$where
$\tilde{a_i}\,=\,a \,
\mbox{if}\,\,a_i=0\,\,$and$\,\,\tilde{a_i}\,=\,b \,\, \mbox{if}
\,\, a_i\,=\,1\,.$ Here $\Sigma(a,b)\,=\{\theta:\Na \to
\{a,b\}\}.$ We have

\begin{proposition}
Assume $ a \in \Sigma_0 $ satisfy $(*)$ then $R_{0,01}(a)$
satisfy $(*)$ in $\Sigma_{0,01}\,$.
\end{proposition}




\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 $$ A_2^1(a)=\,\{m(a_1\,,a_2)\,;\,\,a_1\,,a_2
\in A_1^1(a) \mbox{are consecutive sequences}\}\, \cup A_1^1(a)
$$ 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}


At this stage we have to point out the following:  Assume  $a
\in A_\infty^0 \,$.  Let  $\tilde{a} = \sup\{\sigma^k (a); k \in
\Na \quad \mbox{and}\quad  \sigma^k (a) \in \Sigma_0 \}$ and
  $\tilde{b} = \inf \,\{\sigma^k (a); k \in \Na
 \quad \mbox{and}\quad  \sigma^k (a) \in \Sigma_1 \}$.  Consider
 $\Sigma_{\tilde{a}, \tilde{b}}= \{  \theta  : \Na \to \{ \tilde{a} ,
 \tilde{b}\}\}$ be the set of sequences of the two symbols
 $\tilde{a}$ and $\tilde{b}$.  Replace $0=\tilde{a}$ and  $\tilde{b}= 1$
  and define  $A_0(a)= \{ \ul{0_{n}1}  , \ul{01_m}; n, m \in  \Na \setminus \{ 0 \} \}$
 and $A_{n+1} (a) = A_n(a) \cup \{ m(a_1, a_2); a_1, a_2 \in  A_n (a)
\quad \mbox{are consecutive sequences} \}$.  Set $A_\infty (a)
=\displaystyle \bigcup_{n=0}^\infty A_n(a)  $  and
$A_\infty^\star (a)  = \{\inf\{ \sigma^k (\alpha); k \in \Na \}
;  \alpha = R_{\tilde{a},\tilde{b}}(\eta) \,;\, \eta \in
A_\infty (a) \}$.

\vspace{0.2cm}

 \noindent {\bf \underline{Lemma 2}} \,  $\bar{A_\infty^\star} (a) =
 \bar{A}_\infty^1 (a). $

\vspace{0.2cm}

 As before, let $\sigma_{\tilde{a},\tilde{b}} : \Sigma_{\tilde{a},\tilde{b}}  \to
 \Sigma_{\tilde{a},\tilde{b}}$ be the shift map. We have



 \vspace{0.2cm}
 \noindent {\bf \underline{Lemma 3}} \,  $  \alpha \in
 \bar{A_\infty^\star}
 (a)$ if and only if

 \begin{itemize}
\item[($\star$)] $\alpha \in  Min_2 (\tilde{a},\tilde{b})$ and
 $\sigma_{\tilde{a},\tilde{b}}(\alpha)\geq
 \sigma_{\tilde{a},\tilde{b}}(\beta) $ for $\beta = \sup
 \{ \sigma_{\tilde{a},\tilde{b}} ^k (\alpha); k \in \Na \}$
 \end{itemize}



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}
 \noindent {\bf \underline{Note}} \, A similar construction; as
 we did in Lemma 2 and Lemma 3, for $a \in A_\infty^\circ ,$  we
 can do for any $a \in A_\infty^\infty$.


 \vspace{0.2cm}

The elements in $\,\, (A_{\infty}^{\infty}\,\setminus
A_{\infty}^0)\,\,$ will be called \underline{secondary
sequences}. As we will see in the next section 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$\, a(b) \in Min_2 \,$.


\vspace{0.3cm}

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}


\noindent \underline{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 Morse-Smale and the Entropy Zero cases}


\begin{definition}

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

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

\end{definition}

\vspace{0.2cm}

We will denote by $\, MS_0 \subset DM_0 \,$ the set of
Morse-Smale and 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 $f\in DM_0$ be a ``Morse-Smale''
map we have that\\
$[f(0^+),f(0^-)]\,=\bigcup_{i=0}^{\mbox{per}(a_f)-1}
\overline{I_i}$, 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
$0<j \le \mbox{per}(a_f)-1$.
\end{lemma} \vspace{.2cm}

\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}



\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}



\vspace{0.2cm}
 In a similar way 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}

\vspace{0.5cm}

In general we get

\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}



\vspace{0.2cm}

As a consequence of these lemmas we have

\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 result obtained for expansive maps in \cite{HS}.

\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{BJ} and the references there in ).
For our  family of expanding Lorenz maps we will prove some
results, in this direction,  in section 4.6.

\vspace{0.2cm}


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





\section { Expanding Maps}


\subsection{ The map $G$.}
Let us define $G_{(\mu,\nu)}\colon(\R
\setminus\{0\})\to \R$, $(\mu,\nu)\in \R^2$, the two parameter
family of maps in $DM_0$ :

$$ G_{(\mu,\nu)}(x)=\left\{\begin{array}{l} -\mu+x^{1/2},\quad
x>0\\ \nu-(-x)^{1/2},\quad x<0. \end{array} \right. $$

In this section we will provide the bifurcation theory associated
to this family of elements in $DM_0$.

\vglue .3in
\subsection{The Injective Maps}

Let us define the set $IM\,=\,\{(\mu,\nu);G_{(\mu,\nu)}|_{[-\mu,\nu]}:[-\mu,\nu] \to [-\mu,\nu] \,\, \mbox{ is an injective map} \}\,$.It is nor hard to see that $\,IM\,=\,\{(\mu,\nu);\,\mu \ge 0\,\,,\nu \ge 0\,\mbox{and} (\sqrt{\nu}-1/2)^2 + (\sqrt{\mu} - 1/2)^2 \ge 1/2 \, \}$.


\subsection{Fixed points.} The fixed points of the map
$G_{(\mu,\nu)}$ are given by:

a) $x_\pm(\mu)=\left[ \frac{1\pm\sqrt{1-4\mu}}2 \right]^2$
defined for $\mu\le 1/4$. \vspace{.2cm}

b)$y_\pm(\nu)= - \left[ \frac{1\pm\sqrt{1-4\nu}}2 \right]^2$
defined for $\nu\le 1/4$.

\vspace{.2cm}

We have $y_+(\nu)\le y_-(\nu)\le 0\le x_-(\mu)\le x_+(\mu)$.


\noindent {\bf Note:} The maps $x_-(\mu)$ and $y_-(\nu)$ are
defined  for $0\le\mu\le \frac{1}{4}$; $0\le\nu\le
\frac{1}{4}$.

\vglue .2in

\subsection{Preimages of $0$}

\bigskip
\begin{itemize}
\item[(a)] $\nu-(-x)^{1/2}=0$ imply $x=-\nu^2=y_1(\nu)$, $\nu\ge0$;

$\nu-(-x)^{1/2}=y_1(\nu)$ imply $
x=-(\nu-y_1(\nu))^2=-[\nu+\nu^2]^2=y_2(\nu)$;

$\nu-(-x)^{1/2}=y_n(\nu)$ imply $ x=-[\nu-y_n(\nu)]^2=
y_{n+1}(\nu)$, $\nu\ge0$.

\medskip

\item[(b)] $-\mu+x^{1/2}=0$ imply $x=\mu^2=x_1(\mu)$, $\mu\ge0$;

$-\mu+x^{1/2}=x_1(\mu)$ imply
$x=(\mu+x_1(\mu))^2=(\mu+\mu^2)^2=x_2(\mu)$;

\end{itemize}
$-\mu+x^{1/2}=x_n(\mu)$ imply $x=(\mu+x_n(\mu))^2=x_{n+1}(\mu)$,
$\mu\ge0$.

\vglue .3in

\subsection{Primary Bifurcations}

\bigskip

a) $G(0^+)=$ fixed point of the left hand-side $$ -\mu=-\left[
\frac{1\pm\sqrt{1-4\nu}}2 \right]^2 $$ that is: $$ \mu=\left[
\frac{1\pm\sqrt{1-4\nu}}2 \right]^2. $$

b) $G(0^-)=$ fixed point of the right hand-side $$ \nu=\left[
\frac{1\pm\sqrt{1-4\mu}}2 \right]^2 $$

\begin{figure}[ht]
\begin{center}
\includegraphics[angle=0,width=11cm]{figure232.eps}
\caption{(b)}\label{f23}
\end{center}
\end{figure}\begin{figure}[ht]
\begin{center}
\includegraphics[angle=0,width=11cm]{figura24.eps}
\caption{(a)}\label{f24}
\end{center}
\end{figure}

Let $L_{\underline{0}}=\{(\mu,\nu)\in\R_0^+\times\R_0^+;\quad
a(G_{(\mu,\nu)})=\underline{0}\}$ and
$R_{\underline{1}}=\{(\mu,\nu)\in\R_0^+\times\R_0^+;\quad
b(G_{(\mu,\nu)})=\underline{1}\}$.

We observe that

$L_{\underline{0}}=\{(\mu,\nu);\quad 0\le\nu\le\nu_-(\mu),\quad
0\le\mu\le 1/4$ or $0\le\nu\le 1/4$ for $\mu\ge 1/4\}$

and

$R_{\underline{1}}=\{(\mu,\nu);\quad 0\le\mu\le\mu_-(\nu),\quad
0\le\nu\le 1/4$ or $0\le\mu\le 1/4$ for $\nu\ge 1/4\}$

Here $\nu_-(\mu)$ is given by $\mu= \left(
\frac{1-\sqrt{1-4\nu_-(\mu)}}2 \right)^2$ and $\mu_-(\nu)$ is
given by $\nu=\left(\frac{1-\sqrt{1-4\mu_-(\nu)}}2 \right)^2$.

\begin{figure}[ht]
\begin{center}
\includegraphics[angle=0,width=11cm]{figura25.eps}
\caption{}\label{f25}
\end{center}
\end{figure}
\begin{figure}[ht]
\begin{center}
\includegraphics[angle=0,width=11cm]{figura26.eps}
\caption{}\label{f26}
\end{center}
\end{figure}

c) $G(0^+)=n$th preimage of zero $=y_n(\nu)$.

\vspace{.2cm} These equations define the curves
$\mu_n(\nu)=-y_n(\nu)$ that satisfies:
$(\mu,\nu)\in\mbox{Graph}(\mu_n)$ imply
$a(G_{(\mu,\nu)})=\underline{0_n1}\,\,$. Moreover, the inverses
$\hat \nu_n(\mu)=\mu_n^{-1}(\nu)$ converges (uniformly in the
$C^1$ topology  on compact intervals) to the curve $s(\mu)$
given by: $$ s(\mu)= \left\{\begin{array}{l}
\frac{1-(2\mu^{1/2}-1)^2}4,\quad 0\le\mu\le 1/4
\\ \frac14, \quad \mu\ge 1/4 \end{array} \right. $$

\begin{figure}[htb]
\begin{center}
\includegraphics[angle=0,width=11cm]{figura27.eps}
\caption{}\label{f27}
\end{center}
\end{figure}

d) $G(0^-)=n$th preimage of zero $=x_n(\mu)$.

\vspace{.2cm}

These equations define the curves $\nu_n(\mu)=x_n(\mu)$ that
satisfies: $(\mu,\nu)\in\mbox{Graph}(\nu_n)$ imply
$b(G_{(\mu,\nu)})=\underline{1_n0}$. Moreover, the inverses
$\hat\mu_n(\nu)=\nu_n^{-1}(\mu)$ converges (uniformly in the
$C^1$ topology in compact intervals) to the curve $s(\nu)$ given
by $$ s(\nu)= \left\{
\begin{array}{l}
\frac{1-(2\nu^{1/2}-1)^2}{4},\quad  0\le\nu\le 1/4 \\ \frac{1}{4},
\quad \nu\ge 1/4 \end{array} \right.$$

\begin{figure}[ht]
\begin{center}
\includegraphics[angle=0,width=11cm]{figura28.eps}
\caption{}\label{f28}
\end{center}
\end{figure}

\begin{proposition}
The graph of the curve$\,\mu_n(\nu)\,$ is transversal to graph of the curve$\, \nu_m(\mu)\,$ all $ m,n \in \Na \,$.
\end{proposition}
\noindent{\bf Proof} We have $\, \mu_1(\nu)\,=\,\nu^2\,$and $\, \nu_1(\mu)\,=\,\mu^2\,$.So, the result is true for $\, n=1\,$and$\, m\,=1\,$.

\vspace{0.2cm}
For $\, \nu_2(\mu)\,$ we have $\, \nu_2(\mu)\,=\, (\mu+\nu_1(\mu))^2\,$, then $\, \nu_2'(\mu)\,=\,2(\mu+\nu_1(\mu))(1+\nu_1'(\mu)) \ge 2(1/4 +1/2)(1+\nu_1(\mu))\,=\,\frac\displaystyle{3}{2}(1+\nu_1(\mu))\,=\,\frac\displaystyle{3}{2}(1+2\mu)\,\ge\,\frac\displaystyle{3}{2}(1+1/2)\,\ge\,(3/2)^2 \ge 2\,$. Hence, we conclude the result for $\, n=1\,,m\,=\,2\,$.

\vspace{0.2cm}
Assume the result is true for $\, n=1\,$and$\,m=\,p\,$ with$\, \nu_p'(\mu) > 1\,$. For $\, \nu_{p+1}(\mu) \,$we have $\, \nu_{p+1}(\mu)\,=\,(\mu+\nu_p(\mu))^2\,$and$\, \nu_{p+1}'(\mu)\,=\,2(\mu+\nu_p(\mu))(1+\nu_p'(\mu))\,\ge\,2(1/4+1/2)(1+1) >\,4(3/4)\,=3\,$. Therefore, we get the result for $\, n\,=\,1 \,$and any $\, m \in \Na \,$.

\vspace{0.2cm}

Assume that the result is true for $\, n\,=\,k\,$and any$\, m \in \Na \,$ with$\, \mu_k'(\nu)\,>\,1\,$and $\, \nu_m'(\mu)\,>\,1\,$. For $\, \mu_{k+1}(\nu)\,=\,(\nu+\mu_k(\nu))^2\,$ we have $\, \mu_{k+1}'(\nu)\,=\,2(\nu+\mu_k(\nu))(1+\mu_k'(\nu))\,>\,2(1/2+1/4)(1+1)\,>\,3\,$.So, we conclude the result for $\, n\,=k+1\,$and any $\,m \in \Na\, \Box$. 

e) Let us now assume that $G(0^+)<y_1(\nu)$. In this situation
there is a point, $\overline{ x_1}(\mu,\nu)\ge0$, such that
$-\mu+\overline{x}_1^{1/2}=y_1(\nu)$, that is
$\overline{x_1}(\mu,\nu)=(\mu+y_1(\nu))^2$.

If we look for the condition $G(0^-)=\overline{x_1}(\mu,\nu)$ we
get the curve $\overline{\mu_1}(\nu)=\nu^{1/2}-y_1(\nu)$. This
curve is tangent to the curve $C_{\underline{10}}$ at $\nu=0$;
transversally intersects the curves $\mu_n(\nu)$, $n\ge2$ and the
curve $s(\mu)$. We note that
$(\mu,\nu)\in\mbox{Graph}(\overline{\mu_1})$ imply
$b(G_{(\mu,\nu)})=\underline{100}$

\begin{figure}[ht]
\begin{center}
\includegraphics[angle=0,width=11cm]{figura29.eps}
\caption{}\label{f29}
\end{center}
\end{figure}


In the same way, if we assume that $G(0^+)<y_n(\nu)$, $n\ge2$, we
will find a point $\overline{
x_n}(\mu,\nu)\le\overline{x_{n-1}}(\mu,\nu)$ such that
$-\mu+\overline{x_n}^{1/2}= y_n(\nu)$, that is $$ \overline{
x_n}(\mu,\nu)=(\mu+y_n(\nu))^2. $$

If we look for the condition $G(0^-)=\overline{ x_n}(\mu,\nu)$
we obtain a curve $\overline{\mu_n}(\nu)=\nu^{1/2}-y_n(\nu)$,
that is: $\overline{\mu_n}(\nu)=\nu^{1/2}+\mu_n(\nu)$. This
curve is tangent to $\overline{\mu_{n-1}}(\nu)$ at $\nu=0$;
transversally intersects the curves $\mu_m(\nu)$, $m\ge n+1$ and
the curve $s(\mu)$.

We observe that $(\mu,\nu)\in\mbox{Graph}(\overline{\mu_n})$ imply
$b(G_{(\mu,\nu)})=\underline{10_{n+1}}$

\begin{figure}[ht]
\begin{center}
\includegraphics[angle=0,width=11cm]{figura30.eps}
\caption{}\label{f30}
\end{center}
\end{figure}

f) Let us now assume that $G(0^-)> x_1(\mu)$. In this situation we
can find a point $\overline{y_1}(\mu,\nu)$ such that
$\nu-(-\overline {y_1})^{1/2}=x_1(\mu)$, that is $\overline{
y_1}(\mu,\nu)=-[\nu-x_1(\mu)]^2$.


If we look for the condition $G(0^+)=\overline{y_1}(\mu,\nu)$ we
obtain a curve $\overline{\nu_1}(\mu)=\mu^{1/2}+x_1(\mu)$. This
curve is tangent to the curve $C_{\underline{01}}$ at $\mu=0$;
transversally intersects the curves $\nu_n(\mu)$, $n\ge2$ and the
curve $s(\nu)$.

Also we have $(\mu,\nu)\in\mbox{Graph}(\overline{\nu_1)}$ imply
$a(G_{(\mu,\nu)})=\underline{011}$

\begin{figure}[ht]
\begin{center}
\includegraphics[angle=0,width=11cm]{figura31.eps}
\caption{}\label{f31}
\end{center}
\end{figure}

Similarly: if we assume that $G(0^-)> x_n(\mu)$ we can find a
point $\overline{y_n}(\mu,\nu)$ such that $\nu-(-\overline{
y_n})^{1/2}=x_n(\mu)$ that is
$\overline�{y_n}(\mu,\nu)=-[\nu-x_n(\mu)]^2$.

If we look for the condition $G(0^-)=\overline{y_n}(\mu,\nu)$ we
obtain a curve
$\overline{\nu_n}(\mu)=\mu^{1/2}+x_n(\mu)=\mu^{1/2}+\nu_n(\mu)$.
This curve is tangent to $\overline{\nu_{n-1}}(\mu)$ at $\mu=0$;
transversally intersects the curves $\nu_n(\mu)$; $m\ge n+1$ and
the curve $s(\nu)$. We note that
$(\mu,\nu)\in\mbox{Graph}(\overline {\nu_n})$ imply
$a(G_{(\mu,\nu)})=\underline{01_{n+1}}$.

\begin{figure}[ht]
\begin{center}
\includegraphics[angle=0,width=11cm]{figura32.eps}
\caption{}\label{f32}
\end{center}
\end{figure}

\vspace{.2cm}

\noindent {\bf Remark:} a) Associated to any $a\in
A_0=\{\underline{0_n1};\quad \underline{01_n};\quad n\in\Bbb
N\}$ we have constructed two curves $C_a$, $C_b$, $b=b(a)$, such
that $(\mu,\nu)\in C_a$ imply $a(G_{(\mu,\la)})=a$ and
$(\mu,\nu)\in C_b$ imply $b(G_{(\mu,\la)})=b$. These two curves
transversally intersects at a point $\{P(a,b)\}=C_a\cap C_b$;

\medskip

b) For $a_1,a_2\in A_0$ such that $C_{a_1}\cap
C_{b_2}\ne\emptyset$, where $b_2 = b(a_2)\,$, it is not hard to
prove that the intersection is transversal and contains a unique
point.

\vglue .2in

g) Let us now  ask for the sets $L_{a_-\underline{b}}$ and
$R_{b_+\underline{a}}$ for $a\in A_0$, $b=b(a)$.

\medskip

At this point we establishes one of the main differences between
contracting and expanding families.

Let $a\in A_0$, $R_{b_+\underline{a}}=\{(\mu,\nu);\quad
b(G_{(\mu,\nu)})=b_+\underline{a}\}$ and
$L_{a_-\underline{b}}=\{(\mu,\nu);\quad
a(G_{(\mu,\la)})=a_-\underline{b}\}$

\bigskip

\begin{lemma} $R_{b_+\underline{a}}$ is formed by

\medskip
\begin{itemize}
\item[a)] a noncompact set, $R_{b_+\underline{a}}^1$, whose boundary
is formed by two curves, $\gamma_1(b_+\underline{a})$ and
$\gamma_2(b_+\underline{a})$ such that:

\medskip
\begin{itemize}
\item{(i)} $\gamma_2(b_+\underline{a})\subset C_b$; $P(a,b)\in\gamma_2(b_+\underline{a})$

\item{(ii)} $P(a,b)\in\overline{\gamma_1(b_+\underline{a})}$ and
$\gamma_1(b_+\underline{a})$ is tangent to $C_b$ at $P(a,b)$ and
\end{itemize}
\medskip

\item[b)] a curve, $C_{b_+\underline{a}}$, which is tangent to
$\gamma_1(b_+\underline{a})$ at a point $Q(b_+\underline{a})\ne
P(a,b)$ that satisfy $C_{b_+\underline{a}}$ is tangent to $C_b$ at
$(0,0)\in\overline{C_{b_+\underline{a}}}$.
\end{itemize}
\end{lemma}
The figure (\ref{f33}) displays the geometry of this Lemma.


\begin{figure}[ht]
\begin{center}
\includegraphics[angle=0,width=11cm]{figura33.eps}
\caption{}\label{f33}
\end{center}
\end{figure}

\begin{lemma}
$L_{a_-\underline{b}}$ is formed by

\medskip
\begin{itemize}
\item[a)] a noncompact set, $L_{a_-\underline{b}}^1$, whose boundary
is formed by two curves, $\gamma_1(a_-\underline{b)}$ and
$\gamma_2(a_-\underline{b})$ such that:

\medskip
\begin{itemize}
\item{(i)} $\gamma_2(a_-\underline{b})\subset C_a$; $P(a,b)\in\gamma_2(a_-\underline{b})$

\item{(ii)} $P(a,b)\in\overline{\gamma_1(a_-\underline{b})}$ and $\gamma_1(a_-\underline{b})$
is tangent to $C_a$ at $P(a,b)$ and
\end{itemize}
\medskip

\item[b)] a curve, $C_{a_-\underline{b}}$, which is tangent to
$\gamma_1(a_-\underline{b})$ at a point $Q(a_-\underline{b})=
Q(b_+\underline{a})$, that satisfy $C_{a_-\underline{b}}$ is
tangent to $C_a$ at $(0,0)\in\overline{C_{a_-\underline{b}}}$.
\end{itemize}
\end{lemma}
The following figure (\ref{f34}) displays the geometry of this Lemma.

\begin{figure}[ht]
\begin{center}
\includegraphics[angle=0,width=11cm]{figura341.eps}
\caption{}\label{f34}
\end{center}
\end{figure}

\noindent {\bf Proof:} The proof of these two lemmas can be
carried out in the following steps.

\medskip
\begin{itemize}
\item[(i)] assume $a=\underline{0_n1}$. In this case consider the point
$P=P(\underline{0_n1}, \underline{10_n})\in\R^2$.

\medskip

\item[(ii)] let $U(P)$ be a small neighborhood of the point $P$.

\medskip

\item[(iii)] the intersection $C_a\cap U(P)$ and $C_b\cap  U(P)$
divides $U(p)$ into two disjoint regions $ U_a^+, U_a^-$; $ U_b^+$
and $U_b^-$, respectively (see figure (\ref{f35})).

\begin{figure}[ht]
\begin{center}
\includegraphics[angle=0,width=11cm]{figura35.eps}
\caption{}\label{f35}
\end{center}
\end{figure}

\item[(iv)] verify that there is a point $SN(b)\in C_a\cap U_b^+$
where we can find a saddle node for $F^{\# \mbox{per}(b)}$, near
$x\,=\,\nu$. Moreover, for $(\mu,\nu)\in ]SN(b), P(a,b)[ \subset C_a\cap
\overline{U_b^+}$ we have $b(F_{(\mu,\nu)})=b_+\underline{a}$.

\medskip

\item[(v)] verify that there is a point $SN(a) \in C_b \cap  U_a^+$
where we can find a saddle-node for $F^{\# \mbox{per(a)}}$ near
$\,x\,=\,0^+$. Moreover, for $(\mu,\nu)\in] SN(a), P(a,b)[ \subset
C_b\cap U_a^+$ we have $a(F_{(\mu,\nu)})=a_{-}\underline{b}$.

\medskip

\item[(vi)] there is a point $Q(a,b)\in U_b^+\cap U_a^+$ such that

\begin{itemize}
\item{a)} $Q(a,b)\in\overline{SN(a)}\cap \overline{SN(b)}$;

\item{b)} the map $F_{Q(a,b)}^{\#\mbox{per(a)}}$ has a fixed point with derivative
$1$ which is ex\-pan\-ding.
\end{itemize}
\medskip

\item[(vii)] there is a curve, $C_{b_+\underline{a}}$, tangent to $C_b$ at
$P(a,b)$; $P(a,b)\in(\overline{C_{b_+\underline{a}}} \setminus
C_{b_+\underline{a}})$; such that $(\mu,\nu)\in
C_{b_+\underline{a}}$ imply $b(F_{(\mu,\nu)})=b_+\underline{a}$.
This curve is unbounded and coincides with the saddle-node curve,
$SN(b)$, up to $Q(a,b)$.

\medskip

\item[(viii)] there is a curve, $C_{a_-\underline{b}}$, tangent to
$C_a$ at $P(a,b)$;
$P(a,b)\in(\overline{C_{a_-\underline{b}}}\setminus
C_{a_-\underline{b}})$; such that $(\mu,\nu)\in
C_{a_-\underline{b}}$ imply $a(F_{(\mu,\nu)})=a_-\underline{b}$.
This curve is unbounded and coincides with the saddle-node curve,
$SN(a)$, up to $Q(a,b)$.

\medskip

\item[(ix)] the existence of the other component of the curves,
$C_{b_+\underline{a}}$ and $C_{a_-\underline{b}}$, can be now
easily computed.
\end{itemize}
\bigskip

\begin{corollary} (The geometry of sets $L_a$ and $R_b$, $a\in A_0$,
$b=b(a)$).

\medskip
\begin{itemize}
\item[a)] $L_a=\{(\mu,\nu); a(F_{(\mu,\nu)})=a\}$ is unbounded and
$(\overline{ L_a}\setminus L_a)\subset C_a\cup
\gamma_1(b_+\underline{a})$ (see figure(\ref{f36})).

\medskip

\item[(b)] $R_b=\{(\mu,\nu); b(F_{(\mu,\nu)})=b\}$ is unbounded and
$(\overline{ R_b}\setminus R_b)\subset C_b\cup
\gamma_1(a_-\underline{b})$ (see figure(\ref{f37})).
\end{itemize}
\end{corollary}

\begin{figure}[ht]
\begin{center}
\includegraphics[angle=0,width=11cm]{figura361.eps}
\caption{}\label{f36}
\end{center}
\end{figure}\begin{figure}[ht]
\begin{center}
\includegraphics[angle=0,width=11cm]{figura371.eps}
\caption{}\label{f37}
\end{center}
\end{figure}

\subsection{The average}
Let $a_1,a_2\in A_0$ be two consecutive sequences,
$a_1<a_2$. Let $b_1=b(a_1)<b_2=b(a_2)$. Denote $a=m(a_1,a_2)$
and $b=m(b_1,b_2)=b(a)$.

\medskip

\begin{lemma} There are curves $C_a$ and $C_b$ such that:

\medskip

\item{(i)} $(0,0)\in\overline{ C_a}$, $(0,0)\in \overline{ C_b}$.

\item{(ii)} $C_a$ is tangent to $C_{a_i}$ at $(0,0)$, $i=1,2$ and $C_b$
 is tangent to $C_{b_i}$ at $(0,0)\,\,, i=1,2$.

\item{(iii)} $C_a\cap C_b=\{P(a,b)\}$ and $C_a$ transversally intersects $C_b$ at $P(a,b)$.
\end{lemma}
The figure(\ref{f38}) displays the geometry of the curves $C_a$
and $C_b$.

\begin{figure}[ht]
\begin{center}
\includegraphics[angle=0,width=11cm]{figura38.eps}
\caption{}\label{f38}
\end{center}
\end{figure}


\noindent {\bf Note:} (i) As in the previous section we obtain a
similar result for $L_{a_-\underline{b}}$, $R_{b_+\underline{a}}$,
$L_a$ and $R_b$.

(ii) In this way we have obtained the geometry of the sets
$L_{a_-\underline{b}}$, $R_{b_+\underline{a}}$, $L_a$ and $R_b$
for $a\in A_1=A_0\cup\{m(a_1,a_2);\quad a_1,a_2\in A_0$ are
consecutive sequences\}.

Inductively, we can obtain the geometry of the sets
$L_{a_-\underline{b}}$, $R_{b_+\underline{a}}$, $L_a$ and $R_b$
for $a\in A_{n+1}=A_n \cup \{m(a_1,a_2)\,;\quad a_1,a_2\in A_n$
are consecutive sequences\}.

\vglue .3in

\subsection{The Closure of the Primary Bifurcations}
Let $A_\infty=\bigcup_{n=0}^\infty A_n$. Assume $a\in
\overline{ A_\infty}$ is a sequence which is not periodic nor
eventually periodic. Let $b=\sup\{\sigma^k(a);\quad k\in\Bbb
N\}$ and $ a_n\in A_\infty$, $a_n\to a$. Set $b_n = b(a_n)$, we
have $b_n\to b$.

In this situation we can find values of the parameter
$(\mu_a,\nu_a)$ and curves $C_a,C_b$, such that
$C_a=\{(\mu,\nu);\quad\nu=\varphi_a(\mu)\}$ ;
$C_b=\{(\mu,\nu);\quad\nu=\psi_b(\mu)\}$, where
$\varphi_a\colon(0,\infty)\to\R$ and $\psi_b\colon(0,\infty)\to\R$
satisfies $$ \varphi_a(\mu)=\psi_b(\mu)\quad\mbox{ for }\quad
\mu\ge\mu_a; $$

The maps $\varphi_a|_{(0,\mu_a)}$ and $\psi_b|_{(0,\mu_a)}$ are
tangent at $\mu_a$. The curves $C_{a_n}$ converges to $C_a$ and
the curves $C_{b_n}$ converges to $C_b$.

The intersections $\{P(a_n,b_n)\}\,= C_{a_n}\cap C_{b_n}$
converges to $\{P(a,b)\}=\{(\mu_a,\varphi_a(\mu_a))\}$.
Moreover, $(\mu,\nu)\in C_a$ imply $a(F_{(\mu,\nu)})=a$ and
$(\mu,\nu)\in C_b$ imply $b(F_{(\mu,\nu)})=b$.

\vglue .3in

In fact, the curve $C_a\,$ is the limit of the sequence of
curves$\,C_{a_n}\,$ and the curve $\,C_b\,$ is the limit of the
sequence of curves $\, C_{b_n}\,$. 

\subsection{Secondary Bifurcations}
Let $a\in A_\infty$. Associated with $a$ define
$A(a)=\{c\in\Sigma_0;\quad c=\underline{a_-b_+a^m}$ or
$c=\underline{a_-b^mb_+}$; $m\in\Bbb N\}$. Denote $b=b(a)$.

\bigskip

\begin{lemma} For any $c\in A(a)$ we can find a curve, $C_a$, such
that

\item{(i)} $\{(0,0), P(a,b)\}\subset(\overline{C_c}\setminus C_c)$;

\item{(ii)} $C_c$ is located in the bounded region limited by $C_a$ and
$C_{a_-\underline{b}}$ and

\item{(iii)} $(\mu,\nu)\in C_c$ imply $a(F_{(\mu,\la)})=c$.
\end{lemma}
The following figures describe these properties of the curves
$C_c$.
\begin{figure}[ht]
\begin{center}
\includegraphics[angle=0,width=11cm]{figura39.eps}
\caption{}\label{f39}
\end{center}
\end{figure}

\begin{figure}[ht]
\begin{center}
\includegraphics[angle=0,width=11cm]{figura40.eps}
\caption{}\label{f40}
\end{center}
\end{figure}

Now define $B(b)=\{d\in\Sigma_1;\quad d=\underline{b_+a^ma_-}$ or
$d=\underline{b_+a_-b^m}$, $m\in\Bbb N\}$.

\bigskip

\begin{lemma} For any $d\in B(b)$ we can find a curve, $C_d$,
such that
\begin{itemize}
\item[(i)] $\{(0,0), P(a,b)\}\subset(\overline{C_d} \setminus C_d)$;

\item[(ii)] $C_d$ is located in the bounded region limited by
$C_b$ and $C_{b_+\underline{a}}$ and

\item[(iii)] $(\mu,\nu)\in C_d$ imply $b(F_{(\mu,\la)})=d$.
\end{itemize}
\end{lemma}

The figures (\ref{f41}) and (\ref{f42}) describe these
properties of the curves $C_d$.
\begin{remark}
\begin{itemize}
\item[1)] We observe that: given $\, c \in A(a)\,$and $\, d\,=\,\sup{\sigma^k(c); k \in \Na }\,$ then $\, C_c \cap C_d\,=\, \emptyset $.
\item[2)] For any $\,c \in A(a)\,$ define $\, A(c)\,=\, \{ \gamma \in \Sigma_0; \gamma\,=\, \ul{c_-d_+c^m}\,\, \mbox{or}\,\,\gamma\,=\,\ul{c_-d^md_+}\,,m\in \Na \}\,$, where $d\,=\,a(c)\,$. The same result is true for any $\, \gamma \in A(c)\,$(that is, $\, C_{\gamma} \,$ is a curve located in a region bounded by $\, C_c \,$and$\, C_{c_-\ul{d}}\,\,;\{(0,0)\,,\,P(a,b)\,\} \subset (overline{C_\gamma} \setminus C_\gamma)$).
\item[3)] For elements $\, c \in (\overline{A(a)} \setminus A(a))\,$we obtain the corresponding curve $\,C_c\,$, as the limit of the curves $\, C_{c_n}\, \, c_n \in A(a)\,$and$\,c_n \to c\,$.
\item[4)] Given $\, a \in A_{\infty}\,, c \in A(a)\,$and$\, \gamma \in A(c)\,$ we can continue as in 2) above.
\end{itemize}
\end{remark}


\begin{figure}[ht]
\begin{center}
\includegraphics[angle=0,width=11cm]{figura41.eps}
\caption{}\label{f41}
\end{center}
\end{figure}
\begin{figure}[ht]
\begin{center}
\includegraphics[angle=0,width=11cm]{figura42.eps}
\caption{}\label{f42}
\end{center}
\end{figure}

\subsection{Tongues}

\bigskip

Consider $a\in A_\infty$ and $c\in A(a)$ or $c=a$.

\bigskip

\begin{lemma} Associated with any of these sequences there is a
tongue $L_{c_-b_+\underline{a}}=\{(\mu,\nu);\quad
a(F_{(\mu,\nu)})=c_-b_+\underline{a}\}$ contained in
$R_{b_+\underline{a}}$. Also, there are : a point $Q(c)\in C_c$,
a curve $C_{c_-b_+\underline{a}}$ and a point $\tilde
Q(c_-b_+\underline{a})\in C_{c_-b_+\underline{a}}$ such that
\begin{itemize}
\item[(i)] $\{P(a,b), (0,0)\}\subset \overline{C_{c_-b_+\underline{a}}}$;

\item[(ii)] $[Q(c),P(a,b)]\subset C_c$,
$[Q(c),\tilde Q(c_-b_+\underline{a})]\subset
R_{b_+\underline{a}}$, $[\tilde Q(c_-b_+\underline{a}),
P(a,b)]\subset C_{c_-b_+\underline{a}}$ and

$\partial L_{c_-b_+\underline{a}}=[Q(c),P(a,b)]\cup [Q(c),\tilde
Q(c_-b_+\underline{a})]\cup [\tilde Q(c_-b_+\underline{a}),
P(a,b)]$.
\end{itemize}
\end{lemma}

The figure (\ref{f43}) displays these facts

\begin{figure}[ht]
\begin{center}
\includegraphics[angle=0,width=11cm]{figura43.eps}
\caption{}\label{f43}
\end{center}
\end{figure}

Now, consider $d\in B(b)$ or $d=b$, $b=b(a)$.

\bigskip

\begin{lemma} Associated with any of these sequences there is a
tongue $R_{d_+a_-\underline{b}}=\{(\mu,\nu);\quad
b(F_{(\mu,\nu)})=d_+a_-\underline{b}\}$ contained in
$L_{a_-\underline{b}}$. Also, there are: a point $Q(d)\in C_d$,
a curve $C_{d_+a_-\underline{b}}$ and a point $\tilde
Q(d_+a_-\underline{b})\in C_{d_+a_-\underline{b}}$ such that
\begin{itemize}

\item[(i)] $\{P(a,b), (0,0)\}\subset C_{d_+a_-\underline{b}}$

\item[(ii)] $[Q(d),P(a,b)\subset C_d$, $[Q(d),\tilde
Q(d_+a_-\underline{b})]\subset C_{a_-\underline{b}}$,

$[\tilde Q(c_-b_+\underline{a}), P(a,b)]\subset
C_{c_-b_+\underline{a}}$ and

$\partial R_{d_+a_-\underline{b}}=[Q(d),P(a,b)]\cup [Q(d),\tilde
Q(d_+a_-\underline{b})]\cup [\tilde Q(d_+a_-\underline{b}),
P(a,b)]$.
\end{itemize}
\end{lemma}

\begin{figure}[ht]
\begin{center}
\includegraphics[angle=0,width=11cm]{figura44.eps}
\caption{}\label{f44}
\end{center}
\end{figure}


\subsection{Completing the secondary bifurcations}
Let $a\in A_\infty$, $A_1(a)=A(a)$ and define
$A_2(a)=A_1(a)\cup\{m(a_1,a_2);\quad a_1,a_2\in A_1(a)$ are
consecutive sequences\}.

Inductively, $A_{n+1}(a)=A_n(a)\cup\{m(a_1,a_2);a_1,a_2\in A_1(a)$
are consecutive sequences\} and $A_\infty(a)\,=\,\bigcup_{n=1}^\infty
A_n(a)$.

For any $c\in A_\infty(a)$ we have a curve, $C_c$, as in 4.8,
also a tongue, $L_{c_-b_+\underline{a}}$, as in 4.9.

Take $a\in A_\infty$, $b=b(a)$, $B_1(b)=B(b)$ and
$B_2(b)=B_1(b)\cup\{m(d_1,d_2);\quad d_1,d_2\in B_1(b)$ are
consecutive sequences\}. Inductively, $B_{n+1}(b)=
B_n(b)\cup\{m(d_1,d_2);\quad d_1,d_2\in B_n(b)$ are consecutive
sequences\} and $B_\infty(b)=\bigcup_{n=1}^\infty B_n(b)$.

For any $d\in B_\infty(b)$ we have a curve, $C_d$, as in 4.8 and
a tongue, $L_{d_+a_-\underline{b}}$, as in 4.9.

\vglue .3in

\subsection{ The Doubling Period Sequences}
 Let $a\in A_\infty$; $c\in A_\infty(a)$. Define $$
a_0=c,\quad b_0=b(c);\quad a_1=\underline{(a_0)_-(b_0)_+}, $$ $$
b_1=\underline{(b_0)_+(a_0)_-}\quad\mbox{ and }\quad
a_{n+1}=\underline{(a_n)_-(b_n)_+}, $$ $$
b_{n+1}=\underline{(b_n)_+(a_n)_-},\quad n\ge1. $$

For any $a_n$, we have a curve, $C_{a_n}$, as in 4.8 and a
tongue, $L_{(a_n)_-(a(a_n))_+} \underline{b_n}$, as in 4.9.

For any $b_n$, we have a curve, $C_{b_n}$, as in 4.8 and a
tongue, $R_{(b_n)_+(a(b_n))_-}\underline{b_n}$, as in 4.9.

For $a=\lim_{n\to\infty}a_n$ and $b=\lim_{n\to\infty}b_n$ we
have a curve as in 4.8.

\vglue .3in

\subsection{ \underline{$\underline{0}$ and $\underline{1}$
tongues. }}

\bigskip
Let $\, a \in Min_2 \,$be any periodic sequence. Associated with the sequence $\, \tilde{a}\,=\,a_-\ul{1}\,$there is a tongue $\, L_{\tilde{a}} \subset R_{\ul{1}}\,$a curve$\, C_{\tilde{a}} \subset R_{\ul{1}}\,$and an interval $\,I_{\tilde{a}} \subset \{(1/4\,, \nu); \nu \in [1/4\,,\,3/4]\, \}\,$such that:
\begin{itemize}
\item{} $\, \partial\overline{L_{\tilde{a}}}\,=\,C_a \cap L_{\tilde{a}} \cup C_{\tilde{a}} \cup I_{\tilde{a}}$.
\item{} $\, (\mu\,,\, \nu) \in L_{\tilde{a}}\,$imply$\,a(\mu,\nu)\,=\,\tilde{a}\,$.
\item{}  $\, (\mu\,,\, \nu) \in I_{\tilde{a}} \cup C_{\tilde{a}}\,$imply$\,a(\mu,\nu)\,=\,\tilde{a}\,$.
\item{} $\, (0,0) \in \overline{C_{\tilde{a}}}\,$and$\, \overline{C_a}\,$is tangent to $\, \overline{C_{\tilde{a}}}\,$at $\,(0,0)\,$.
\end{itemize}
The next figure(\ref{f45}) represent these facts.
\bigskip
Let $\, b \in Max_2 \,$be any periodic sequence. Associated with the sequence $\, \tilde{b}\,=\,b_+\ul{0}\,$there is a tongue $\, R_{\tilde{b}} \subset L_{\ul{0}}\,$a curve$\, C_{\tilde{b}} \subset L_{\ul{0}}\,$and an interval $\,I_{\tilde{b}} \subset \{(\mu\,, 1/4); \mu \in [1/4\,,\,3/4]\, \}\,$such that:
\begin{itemize}
\item{} $\, \partial\overline{R_{\tilde{b}}}\,=\,C_b \cap R_{\tilde{b}} \cup C_{\tilde{b}} \cup I_{\tilde{b}}$.
\item{} $\, (\mu\,,\, \nu) \in R_{\tilde{b}}\,$imply$\,b(\mu,\nu)\,=\,\tilde{b}\,$.
\item{}  $\, (\mu\,,\, \nu) \in I_{\tilde{b}} \cup C_{\tilde{b}}\,$imply$\,b(\mu,\nu)\,=\,\tilde{b}\,$.
\item{} $\, (0,0) \in \overline{C_{\tilde{b}}}\,$and$\, \overline{C_b}\,$is tangent to $\, \overline{C_{\tilde{b}}}\,$at $\,(0,0)\,$.
\end{itemize}


The next figure(\ref{f46}) represent these facts.

\begin{figure}[ht]
\begin{center}
\includegraphics[angle=0,width=11cm]{figura451.eps}
\caption{}\label{f45}
\end{center}
\end{figure}



\begin{figure}[ht]
\begin{center}
\includegraphics[angle=0,width=11cm]{figura461.eps}
\caption{}\label{f46}
\end{center}
\end{figure}

%aquitoy

\subsection{Completing the bifurcation diagram}

Assume $a\in Min_2$, $b\in Max_2 \,,\,b\ne b(a)$ and $C_a\cap
C_b\ne\emptyset$.
\begin{itemize}
\item[(i)] Let us assume that $a$ and $b$ are periodic sequences.
In this situation we have:
\begin{itemize}
\item{a)} there are two curves, $C_{a_-\underline{b}}$ and
$C_{\underline{a_-b_+}}$, tangents to $C_a$ at $P(a,b) ((0,0))$
such that $P(a,b)\in \overline{C_{a_-\underline{b}}}$ (or
$C_{\underline{a_- b_+}})$ and
$(0,0)\in\overline{C_{a_-\underline{b}}}$ (or
$C_{\underline{a_-b_+}}$).

\item{b)} there are two curves, $C_{b_+\underline{a}}$ and $C_{\underline{b_+a_-}}$,
tangents to $C_b$ at $P(a,b)((0,0))$ such that $P(a,b)\in
\overline{ C_{b_+\underline{a}}}$ (or
$P(a,b)\in\overline{C_{\underline{b_+a_-}}}$) and
$(0,0)\in\overline{ C_{b_+\underline{a}}}$ (or
$\overline{C_{\underline{b_+a}}}$).

\item{c)} there is a curve, $C_{a_-b_+\underline{a}}$, tangent to $C_a$ at
$P(a,b)$ and $(0,0)$ such that $P(a,b)\in
\overline{C_{a_-b_+\underline{a}}}$ and
$(0,0)\in\overline{C_{a_-b_+\underline{a}}}$.

\item{d)} there is a curve, $C_{b_+a_-\underline{b}}$, tangent to $C_b$ at
$P(a,b)$ and $(0,0)$ such that
$P(a,b)\in\overline{C_{b_+a_-\underline{b}}}$ and $(0,0)\in
\overline{C_{b_+a_-\underline{b}}}$.
\begin{figure}[ht]
\begin{center}
\includegraphics[angle=0,width=11cm]{figura47.eps}
\caption{(a) and (c)}\label{f47}
\end{center}
\end{figure}\begin{figure}[ht]
\begin{center}
\includegraphics[angle=0,width=11cm]{figura48.eps}
\caption{(b) and (d)}\label{f48}
\end{center}
\end{figure}

\end{itemize}


\item[(ii)] Let us now assume that $b$ is a periodic sequence and $a\,$
it is not a periodic sequence. In this situation there is a
curve, $C_{b_{+}a}$, tangent to $C_b$ at $P(a,b)$ and $(0,0)$.
The figure (\ref{f49}) represent these facts.


\item[(iii)] Let us assume that $a$ is a periodic sequence and $b$
is not a periodic sequence. In this situation there is a curve,
$C_{a_-\underline{b}}$, tangent to $C_a$ at $P(a,b)$ and
$(0,0)$. The figure (\ref{f50}) represent these facts.

\begin{figure}[ht]
\begin{center}
\includegraphics[angle=0,width=9cm]{figura49.eps}
\caption{(ii)}\label{f49}
\end{center}
\end{figure}\begin{figure}[ht]
\begin{center}
\includegraphics[angle=0,width=9cm]{figura50.eps}
\caption{(iii)}\label{f50}
\end{center}
\end{figure}
\end{itemize}

\subsection{ Measure of the bifurcation set}

Set $\pi=\{(\mu,\nu);\quad \mu\ge0, \nu\ge 0\}:$
$AA=\{(\mu,\nu)\in\pi; G_{(\mu,\nu)}$ satisfy the Axiom $A$\} and
$B=\pi\setminus AA$.

For $a\in A_\infty$ let $\Delta_a=L_a\cup L_{a_-\underline{b}}\cup
R_{b_+\underline{a}}$, where $b=b(a)$ and
$\Delta_\infty=\bigcup_{a\in A_\infty} \Delta_a\cup
L_{\underline{0}}\cup R_{\underline{1}}$.

\begin{lemma} a) $m(\Delta_\infty\setminus AA)=0$

b) $B$ contains open sets in $\pi$.

\end{lemma}

This result follows as in lemma ...in \cite{L}.

\subsection{Entropy zero}


For $a\in A_\infty$ let $\Ga_a=L_a\cup R_{b}\cup
L_{a_-b_+\underline{a}}\cup R_{b_+a_-\underline{b}}$, where
$b=b(a)$ and $\Ga_\infty=\bigcup_{a\in A_\infty} \Ga_a\cup
L_{0\underline{1}}\cup R_{1\underline{0}}$.


Let $\pi^+=\{(\mu,\nu)\in\pi; \mu\ne0\quad \mbox{ and } \quad
\nu\ne 0\}$.


\begin{lemma} $EZ\cap \pi^+=\Ga_\infty$, here $EZ$ denotes the
set of parameter values $(\mu\,,\,\nu)$ where the respective map
$G_{(\mu,\nu)}\,$ has entropy zero.
\end{lemma}

This result follows from theorem 1 in \cite{LM1}

\subsection{The Global Picture}

Taking together all the results in this section the global
picture of the bifurcation diagramme is the following:


\section {The general case}

Let us consider the five parameter family of maps
$G_{(\alpha,c_1,c_2,\mu,\nu)}\colon (R\setminus\{0\})\to\R$,
$\mu\ge0 \,\,,\nu\ge0 \,,\,\alpha \,<1$, $c_1>0 \,\,, c_2>0$ in
$DM_0\,$ given by: $$ G_{((\alpha,c_1,c_2,\mu,\nu)}(x)=
\left\{\begin{array}{l} -\mu+c_1x^{\alpha},\quad  x>0
\\ \nu-c_2(-x)^\alpha,\quad  x<0 \end{array} \right. $$

We observe that all the  previous results, for the expanding
family, are (still) true when we fix the parameters $\, \alpha
\,, c_1 \,$ and $c_2\,$. That is, we can obtain the
$\,a\,\,$and$\,\,b\,\,$-decomposition,  associated to any two parameter family
$F_{(\mu,\nu)}= F_{(\alpha,c_1,c_2,\mu,\nu)}$ in  the same way
as for the previous family. Nevertheless, the situation for
topological equivalence is dramatically different as we will
show in the next subsection.


\subsection{Topologically equivalence for families}

Assume $F=\{F_{(\mu,\nu)}; \quad (\mu,\nu)\in\pi\}$ and
$G=\{G_{(\mu,\nu)};\quad (\mu,\nu)\in\pi\}$ are two-parameter
families of maps in $DM_0$.

\bigskip

\begin{definition} We will say that $F$ and $G$ are \underline{equivalent}
if there is a bijection $\phi\colon\pi\to\pi$ such that $\forall
(\mu,\nu)\in\pi$ the map $F_{(\mu,\nu)}$ is topologically
equivalent to $G_{\phi(\mu,\nu)}$.
\end{definition}

\vspace{0.2cm} 
For the five parameter family of
maps$\,G_{(\alpha,c_1,c_2,\mu,\nu)}\,$ we have

\begin{lemma}
There are parameter values
$\,(\alpha,c_1,c_2)\,(\tilde{\alpha},\tilde{c_1},\tilde{c_2})\,$such
that the two parameter families
$\,F_{(\mu,\nu)}\,=\,G_{(\alpha,c_1,c_2,\mu,\nu)}\,$and$\,H_{(\mu,\nu)}\,=\,
G_{(\tilde{\alpha},\tilde{c_1},\tilde{c_2},\mu,\nu)}\,$are not
equivalent.
\end{lemma}

\noindent{\bf Proof.}

\begin{itemize}
\item[(i)]  We have $G'(x) = 1 $  for $x > 0 $  if and only if
$x(\alpha, c_1)= (\alpha c_1
)^{\displaystyle\frac{1}{1-\alpha}}$.

\vspace{.2cm}

We have $G' (x) = 1 $ for $x<0$ if and only if $y(\alpha, c_2)=
 -(\alpha c_2)^{ \displaystyle\frac{1}{1-\alpha}}$.

\item[(ii)] The equation for a fixed point of the saddle node
type, for $x<0$, are

$\nu - c_2 (-x)^\alpha = x $  and  $x = - (\alpha c_2)^{
\displaystyle\frac{1}{1-\alpha}}\,$, that is; $  \nu = \nu_{sn}=
c_2 (\alpha c_2)^{ \displaystyle\frac{\alpha}{1-\alpha}}
(1-\alpha),  $ is the parameter value where the map $G_{(\mu,
\nu)}$ presents a saddle-node fixed point at $x(\alpha, c_2)=
-(\alpha c_2)^{ \displaystyle\frac{1}{1-\alpha}}$.

\item[(iii)] Now we ask for the curve $G(0^+) = -\mu =\, \mbox{fixed
point in} \,x<0$.

We have the equations $-\mu = y_2 (v) $ and

$\nu  - c_2 (-y_2 (v))^\alpha = y_2 (v)$,

\vspace{.2cm} that is, $\nu - c_2 \mu^\alpha = -\mu $,

\vspace{0,2cm}

therefore; $\nu(\mu)  = c_2 \mu^\alpha -\mu$.

\vspace{.2cm}

This curve  is tangent to the curve $\nu = \nu_{sn}$ at the value
$\mu = \mu_{max} = (\alpha c_2
)^{\displaystyle\frac{1}{1-\alpha}}.$

\item[(iv)] Figure  \ref{f51} displays the region of the parameters
$ (\mu, \nu) $ such that $a(\mu, \nu) = \underline{0}$. This
region is given by $\{ (\mu, \nu);0 \leq \nu \leq \nu (\mu); 0
\leq \mu \leq \mu_{max}\}$ or $0 \leq \nu \leq \nu_{sn}$ for
$\mu \geq \mu_{max}$.


\begin{figure}[ht]
\begin{center}
\includegraphics[angle=0,width=11cm]{figura51.eps}
\caption{}\label{f51}
\end{center}
\end{figure}



\item[(v)]  The set $\{(\mu, \nu), b(\mu, \nu)= \underline{10}\}$
contains the curve $\nu_{\underline{10}}=
\left(\displaystyle\frac{\mu}{c_1} \right)^{1/\alpha}$. See
Figure  \ref{f52} for this picture.

\begin{figure}[ht]
\begin{center}
\includegraphics[angle=0,width=11cm]{figura52.eps}
\caption{}\label{f52}
\end{center}
\end{figure}




\item[(vi)]  We have $\nu_{\underline{10}} (\mu) = \nu_{sn}$
if and only of

$$ \mu = \bar{\mu}\,=\, c_1 c_2^\alpha (1 - \alpha)^\alpha \cdot
(\alpha c_2)^{\displaystyle\frac{\alpha^2}{1-\alpha}}. $$


\item[(vii)] In this situation


$$  \displaystyle\frac{\bar{\mu}}{\mu_{max}}=
\displaystyle\frac{c_1 c_2^\alpha (1-\alpha)^\alpha (\alpha
c_2)^{\displaystyle\frac{\alpha^2}{1-\alpha}}}{(\alpha c_2)^{
\displaystyle\frac{1}{1-\alpha}}} = \displaystyle\frac{c_1}{c_2}
 \displaystyle\frac{1-\alpha}{\alpha^{1+ \alpha}}= S (c_1, c_2, \alpha).$$


It is clear that there  are values $(c_1, c_2, \alpha)$ and $
(\tilde{C}_1 , \tilde{C}_2, \tilde{\alpha})$ such that

$$ S(c_1, c_2, \alpha) < 1 < S  (\tilde{c}_1 , \tilde{c}_2,
\tilde{\alpha}).$$

In the first case the curve $\nu _{10} (\mu) $ is like in figure
\ref{f53}


\begin{figure}[ht]
\begin{center}
\includegraphics[angle=0,width=11cm]{figura53.eps}
\caption{}\label{f53}
\end{center}
\end{figure}


 In the second  case the curve $\nu_{10}(\mu)$ is like in
figure \ref{f54}


\begin{figure}[ht]
\begin{center}
\includegraphics[angle=0,width=11cm]{figura54.eps}
\caption{}\label{f54}
\end{center}
\end{figure}



Clearly, each case define a two parameter families as announced
in the lemma .

\end{itemize}
\section{ Application to Geometric Vector fields}

\medskip

Assume $X_0\colon U\subset\R^3\to\R^3$ is a three dimensional
vector field that satisfies:
\begin{itemize}
\item[a)] $0\in U$ is an hyperbolic singularity of the vector field $X$,
 whose eigenvalues satisfies $-\la_1<-\la_2<0<\la_3$;

\item[b)] The components, $\gamma_1$ and $\gamma_2$, of the set $(W_0^u\setminus\{0\})$
satisfies $\gamma_1\subset W_0^s$, $\gamma_2\subset W_0^s$;

\item[c)] There is a transversal section, $\Sigma\subset U$, such that:
$\Sigma$ is transversal to $W_0^s$ and $\Sigma\setminus
W_0^s=\Sigma_1\cup\Sigma_2$ are two disjoint sets in $\Sigma$
that satisfies: $$ \pi_1(\Sigma_1)\subset\Sigma_1\quad\mbox{ and
}\quad \pi_2(\Sigma_2)\subset \Sigma_2 $$ where
$\pi_i\colon\Sigma_i\to\Sigma \,\,, i\,=\,1,2 \,$ is the first
return map associated to the respective cross section.
\end{itemize}
Let $\, {\cal U}_0  \subset\frak X^r(U,\R^3)$ be a neighborhood
of the vector field $X_0$ in $\frak X^r(U,\R^3)=\{X\colon
U\subset \R^3\to\R^3; \quad X$ is $C^r\}$ with the usual
$C^r$-topology.

In the above conditions there are codimension one submanifold,
${\cal N}_1,{\cal N}_2\subset {\cal U}_0$; such that:

\medskip
\begin{itemize}
\item[a)] $Y\in{\cal N}_i$ if and only if $\gamma_i(Y) \subset W_{\sigma(Y)}^s$ here
$\sigma(Y)$ is the hyperbolic singularity, near $0$, that
correspond to the vector field $Y$. Also,
$(W_{\sigma(Y)}�\setminus\{\sigma(Y)\})=\gamma_1(Y)\cup\gamma_2(Y)$,
where $\gamma_i(Y)$ is the natural extension of the component
$\gamma_i$;

\medskip

\item[b)] $({\cal U}_0\setminus {\cal N}_i)={\cal U}_1^i \cup {\cal U}_2^i$ where
$Y\in {\cal U}_1^1$ if and only if
$\pi_1(\Sigma_1)\subset\Sigma_1$ and $Y\in{\cal U}_2^2$ if and
only if $\pi_2(\Sigma_2)\subset\Sigma_2$ and
\medskip

\item[c)] ${\cal N}_1$ is transversal to ${\cal N}_2\,$.
\end{itemize}
\medskip

Finally we will assume that $\Sigma$ can be foliated by
one-dimensional submanifolds $\{{\cal F}_x;\quad x\in\Sigma\}$
such that

\medskip
\begin{itemize}
\item[(i)] $x\in W_Y^s\cap\Sigma $ imply ${\cal F}_x=W_Y^s\cap\Sigma$ and
\medskip
\item[(ii)] $\pi_i({\cal F}_x)\subset {\cal F}_{\pi_i(x)}$, $x\in\Sigma_i$.
\end{itemize}

\bigskip
In this situation the interesting part of the bifurcation
theory, for elements in ${\cal U}_0$, is located in ${\cal
U}_1^2\cap{\cal U}_2^1$. In this set we can apply the results in
Sections 3,4 and 5 to obtain, bis a bis, similar results for
generic two parameter families of vector fields in ${\cal U}_0$.

\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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\end{thebibliography}
\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}



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