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\date{ Dedicated to Professor Jacob Palis on his sixtieth's birthday  }
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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: The contracting case.
\footnote{Partially supported by Fondecyt grants \# 1970720 and  \#
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, the  meteorologist E.N. Lorenz
\cite{Lo} exhibited numerical evidence for  the existence of a
strange attractor in a quadratic system of ordinary
differential equations in three variables. Some time later
Afrajmovich, Bykov and Shilnikov( \cite{ABS1}, \cite{ABS2}) and
Guckenheimer, Williams( \cite{G}, \cite{W}, \cite{GW}) proposed
the so called geometrical models for the behavior observed by
Lorenz. An important feature of these models is the existence
of a(partial) cross-section to the flow, as well as a smooth
invariant foliation by curves. Using this, one can reduce the
dynamics of the flow to that of an interval transformation
with a discontinuity.These transformation, generically, divide
into two disjoint classes: the expanding ones(those whose derivative,
from both sides, at the discontinuity is infinity) and the contracting
ones(those whose derivative,
from both sides, at the discontinuity is zero). As observed in \cite{GW}
and \cite{ABS2},there exist uncountably many conjugacy classes of such
transformations.
In fact the moduli space is  essentially  2-dimensional and can be
parameterized by the admissible kneading sequences(forward
itineraries of the discontinuity).

\vspace{0.2cm}
In view of these results, it is natural to look for a bifurcation
theory of these transformations and flows using symbolic dynamics.
In this direction, de Melo and Martens showed(\cite{dMM})
the existence of parameterized families of contracting Lorenz flows
that are topologically universal in the sense that given any
geometric Lorenz flow, its dynamics is ``essentially'' the
same as the dynamics of some element of the family. In the
present work we describe the complete bifurcation diagram
for an explicit natural  choice of such an essentially universal family,
namely the quadratic family
$$ F_{\mu,\nu}(x) = \left\{\begin{array}{l} - \mu + x^2 \,\,, \
x
> 0\\ \, \nu - x^2 \,\,, \ x < 0
\end{array}
\right\}, $$

Our geometrical construction has several important consequences.One of
them is that axiom A maps are dense in this family
(this is  also proved in \cite{dMM}; see \cite{GS} for similar result for the
logistic family). It also follows that the set of parameters whose
corresponding Lorenz map has entropy zero is arc connected( see
section 4.6) . Also, the
combinatorial bifurcation diagramme allow us to reobtain, in
a constructive way, the result of the Melo and Martens on the
universality of the quadratic family. Moreover, we completely describe
the exceptional combinatorics not realized in the model(a countable set).
Also, as lemma 4 in section 3.1  shows,secondary bifurcations do not
appear for injective  Lorenz maps( as defined in section  2.1)

\vspace{0.2cm}

After this paper had been completed we became aware of the work
of St. Pierre( \cite{MSP})about the bifurcation diagram
asociated to periodic kneading sequences, which obtains, in an
independent way, some of the results presented here.  Also,
Hubbard and Sparrow defined, in \cite{HS}, a set of pair of
sequences which model all the topological expanding dynamics
exhibited by  Lorenz maps.

\vspace{0.2cm}



On the other hand, starting with  Arnold
(\cite{Ar}) there has been considerable interest in the
bifurcation theory of the canonical family( see for instance
\cite{Bo} and the references there in). The most common approach,
based on the notion of rotation number, is very useful when
applied to homeomorphisms of the circle, but it seems delicate to
apply it to non injective maps . In fact, a bifurcation diagram for the
canonical family beyond the injectivity domain is still unknown. The best
approximation seems to be the work by Boyland(\cite{Bo}).See also the work
of Epstein, Keen and Tresser(\cite{EKT}).

\vspace{0.2cm}

Recently, in \cite{LM}, we extended 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 $\,LW\,$).
This model includes the dynamics of all the interpolated maps used by
Boyland (\cite{Bo}, page 359 )
in his construction.

\vspace{0.2cm}

It is clear that the structure of the Lexicographical world has
to reflect in the bifurcation theory associated to any
parameterized family of allowed maps. This is an extremely interesting
problem focused for several authors in this and other contexts,
see for instance \cite{MacT1},\cite{CGTU},\cite{BMT},\cite{GMT},
\cite{Ar}, \cite{Bo} .


\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 about the quadratic family $ F_{\mu,\nu}$ ,
in section 5 we extend  our
results to more general contracting families and, finally, in section 6
we will relate them to three
dimensional geometric vector fields.




\vspace{0.2cm}



\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 $\Sigma_{a_f,b_f}$.}

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 $$I_f(x)(i)=
0 \quad \mbox{ if } f^{i}(x)<0 \,\,\mbox{and}\,\,I_f(x)(i)= 1
\quad \mbox{ if } f^{i}(x)>0. $$

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}(\cite{L2}) $I_f=\bigcap_{n=0}^\infty
\sigma^{-n}([a_f,b_f])=\Sigma_{a_f,b_f}$.
\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{LM})
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 contracting 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_1 \,\,;\,\, \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 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 $a$-surjective 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 $b$-surjective family}
if $\forall b \in Max_2 \,$ 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 $LW$-surjective 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:} Are there
$LW$-surjective 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 quadratic
contracting Lorenz maps is $a$-surjective, is $b$-surjective but
it is not $LW$-surjective. Moreover, the respective $\,
a\,,b\,,LW\,$-bifurcation diagrammes 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{LM}. In the sequel we will denote by
$\underline{a}= \underline{a_0,a_1,...a_n}$ the periodic
sequence $(a_0,a_1,...,a_n,a_0,a_1,...,a_n,....)\in \Sigma_2$.
Assume that we have a countable set of sequences, $A$. We will
say that two elements $\,a < b \in A$ are {\em consecutive in
$A$}( or consecutive if the context is clear) if there not exist
an element $c \in A$ such that $ a < c < b \,$

\subsection{Dynamical properties for sequences in LW}

\vspace{0.2cm}

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

\vspace{0.2cm}

\noindent \underline{Example}: For $\,\,\underline{a_1} =
\underline{01}\,,\underline{a_2} = \underline{011}\,\,$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 $\,\, A_{n+1} = A_n \cup
\{\,\, m(a_1\,,a_2);\,\underline{a_1}\,,\underline{a_2} \in A_n
\,\,\mbox{are consecutive sequences}\,\,\}$. Set $\,\,
A_{\infty} = \bigcup_{n=0}^{ \infty}A_n$. 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}

To prove this lemma, let us 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}

\noindent {\bf Proof:} It is easy to see that
\begin{itemize}
\item[1.-] $\, \alpha \le \beta \,$ imply $\, R_{0,01}(\alpha)
\, \le  \,R_{0,01}(\beta)\,$and
\item[2.-] $\sigma_{0,01} \circ R_{0,01}(\alpha) \,=\,
R_{0,01}(\sigma(\alpha)) \,,$ any $\, \alpha \in \Sigma_2.$
\end{itemize}

Now it is clear that if $\, \sigma^i(a) \ge a\,$ then $\,
R_{0,01} \circ \,\sigma^i(a) \ge \, R_{0,01}(a) \,$ and $\,
\sigma^i_{0,01}(R_{0,01}(a))\,\ge R_{0,01}(a)\,$ any $ i \in \Na
\,$.

\vspace{.2cm}

If $\, \sigma(a) \ge \sigma(b) \,$ then $\,
R_{0,01}(\sigma(a))\,\ge R_{0,01}(\sigma(b))\,$and $\,
\sigma_{0,01}(R_{0,01}(a))\, \ge \sigma_{0,01}(R_{0,01}(b))$.

Clearly;$\, R_{0,01}(b) \,= \sup\{\sigma^k_{0,01}(R_{0,01}(a));k
\in \Na \}$. So we have the result announced in the proposition.

\noindent {\bf \underline{Proof of the Lemma}} \quad Assume $a$
 satisfy (*) and $a= 00 \cdots $.

 \vspace{0.2cm}

 Let $b = \sup \{ \sigma^k (a); k \in \Na \}. $  In this situation
 we cannot have $b = (11 \cdots $).

 \vspace{0.2cm}

 In fact, otherwise $\sigma(b)= (1 \cdots ) > \sigma (a) = (0 , \cdots ) ,
 $ a contradiction.

 \vspace{0.2cm}

 Since $\sigma ^k (b)  \leq b , $  any $k \in \Na , $ we have  that
 $b$ cannot have two consecutive $1's $ between its symbols . Therefore in this situation
 we have:

 $$a= 0^{k_1} (01)^{k_2}   0^{k_3}(01)^{k_4}\cdots = R_{0,01}( 0^{k_1} 1^{k_2}
 0^{k_3} 1^{k_4} \cdots    )= R_{0,01} (\tilde{a})$$


 and  $\, \tilde{a}\,$, clearly, satisfy (*).

 \vspace{0.2cm}

 Similarly , if $b = (11 \cdots )$ and $a$ satisfy (*) then $a = 01 \cdots
 $ and $b = R _{10,1} (\tilde{b})$ some $\tilde{b}$  such that $\tilde{a} =
 \inf \, \{ \sigma^k (\tilde(b), k \in \Na  \}$ satisfy (*).

 \vspace{0.2cm}

 Since, the periodic sequences that satisfy (*) are obtained as

 \begin{eqnarray*}
a&=& \inf \left\{\sigma^k \left( R_{0,01}^{i_1}  \circ
 R_{10,1}^{i_2} \circ \cdots \circ R_{10,1}^{i_j}(\ul{0})\right) ;
 k \in \Na \right\} \quad \mbox{or}\\
 && \\
a&=& \inf \left\{\sigma^k \left( R_{0,01}^{i_1}  \circ
 R_{10,1}^{i_2} \circ \cdots \circ R_{0,01}^{i_j}(1)\right) ;
 k \in \Na \right\} \quad \mbox{or}\\
 && \\
 a&=& \inf \left\{\sigma^k \left( R_{10,1}^{i_1}  \circ
 R_{0,01}^{i_2} \circ \cdots \circ R_{10,1}^{i_j}(\ul{0})\right) ;
 k \in \Na \right\} \quad \mbox{or}\\
 && \\
 a&=& \inf \left\{\sigma^k \left( R_{10,1}^{i_1}  \circ \cdots \circ
 R_{0,01}^{i_j}(\ul{1})\right) ;
 k \in \Na \right\} \\
 && \\
  \end{eqnarray*}

we get the result $\Box$.


\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}(\cite{LM})  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 $b_f=\varphi(a)$.Moreover,
this map satisfy $ f(f(0^+))=f(f(1^-))$.
\end{lemma}

\noindent {\bf Proof:} Let $a=a_0<a_1<\dots<
(a_{\mbox{last}}=a_p)<0\underline{1}<1\underline{0}<
(b_{\mbox{first}}=b_{p+1})< b_{p+2}<\dots<b_k=b(a)$, 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 an injective 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
closed intervals $\,L_s\,, s= 0, \dots k \,,$ such that $$ L_0
\le L_1 \le \dots \le L_p \le L_{p+1} \le \dots \le L_k, $$
$\,\, L_i \cap L_{i+1} = \mbox{one point}\,$ or $\, L_i \cap
L_{i+1} = \emptyset \,$, 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 satisfies $\,f(L_i)\,=\, L_{\eta(i)} \,$ and is
increasing between $L_i\,$and$\, L_{i+1} \,\,$ when $L_i \cap
L_{i+1}\,=\, \emptyset \,\Box\,$ \vspace{0.2cm}


\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 quadratic family of 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{ Proof of the Results.}

\bigskip

\subsection{The Contracting Family.}

Let us consider the two parameter family
$F_{(\la,\mu)}:(\R\setminus\{0\})\to\R$, $\,\, \mu \ge
0\,\,,\,\, \nu \ge 0\,,$ of maps in $DM_0$

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

In this section we will provide the bifurcation theory associated
to this family.

\vglue .2in

\subsection{Primary Bifurcations.}

We call the bifurcations in this section \underline{\em primary
bifurcations} because they appears as bifurcation curves for
injective maps  in $\, DM_0 \,$ and can be studied by using
rotation numbers( see for instance \cite{Bo}, \cite{Ar},
\cite{GT} and \cite{He}).
\medskip

\noindent {\bf 4.2.1 Fixed points:}

\vspace{.2cm}

The fixed points of the map $F_{(\mu,\nu)}$ are given by

\medskip
\begin{itemize}
\item[(a)] $y_{\pm}(\nu)= \frac{-1\pm\sqrt{1+4\nu}}2$
defined for $\nu\ge-1/4$ and

\medskip

\item[(b)] $x_{\pm}(\mu)=\frac{1\pm\sqrt{1+4\mu}}3$ defined for $\mu\ge-1/4$.

\end{itemize}

We have $y_-(\nu)\le y_+(\nu)\le0\le x_-(\mu)\le x_+(\mu)$.
\vglue .2in

\noindent {\bf 4.2.2 Preimages of 0}

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

\item[] $\nu-x^2=-\sqrt\nu$ imply $x=-\sqrt{\nu+\sqrt\nu}=y_2(\nu)$

\item[] $\nu-x^2=y_n(\nu)$ imply $x=-\sqrt{\nu-y_n(\nu)}=y_{n+1}(\nu)$, $n\ge2$;

\bigskip

\item[(b)] $-\mu+x^2=0$ imply $x=\sqrt\mu=x_1(\mu)$, $\mu\ge0$

\item[] $-\mu+x^2=\sqrt\mu$ imply $x=\sqrt{\mu+\sqrt\mu}=x_2(\mu)$

\item[] $-\mu+x^2=y_n(\mu)$ imply $x=\sqrt{\mu + x_n(\mu)}\,=\, x_{n+1}\,, n \ge 2$.

\end{itemize}

\noindent{\bf 4.2.3 Curves of bifurcation (first generation)}

\begin{itemize}
\item[(a)] $F(0^-)=$ fixed point of the right hand side
$$ \nu=\frac{1\pm\sqrt{1+4\mu}}2 $$

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

\begin{figure}[h]
\begin{center}
\includegraphics[width=8cm]{16.eps}
\caption{}\label{f1}
\end{center}
\end{figure}

We note(see figure 2 and 3) that: $$ (\mu,\nu)\in\R^2\mbox{ such
that } \nu\ge\frac{1+\sqrt{1+4\mu}}{2},\quad \mu\ge0\mbox{ imply
} b(F_{(\mu,\nu)})=\underline{1} $$

\begin{figure}[h]
\begin{center}
\includegraphics[width=7cm]{figura7.eps}
\caption{}\label{f2}
\includegraphics[width=7cm]{figura8.eps}
\caption{}\label{f3}
\end{center}
\end{figure}


$$ (\mu,\nu)\in\R^2 \mbox{ such that }
\mu\ge\frac{1+\sqrt{1+4\nu}}{2},\quad \nu\ge0 \mbox{ imply }
a(F_{(\mu,\nu)})=\underline{0} $$


Given $a\in\ Min_2$ and $b\in\ Max_2$ let denote by $$
R_b=\{(\mu,\nu)\in\R_0^+\times\R_0^+;\quad b(F_{(\mu,\nu)})=b\}
$$ $$ L_a=\{(\mu,\nu)\in\R_0^+\times\R_0^+;\quad
a(F_{(\mu,\nu)})=a\} $$




\begin{itemize}
\item[(c)] $F(0^+)=n$-th preimage of zero $=y_n(\nu)$.
\end{itemize}

These equation define the curves $\mu_n(\nu)=-y_n(\nu)$ that
satisfies: $$ (\mu,\nu)\in\{(\mu_n(\nu);
\nu);\quad\nu\ge0\}\quad\mbox{ imply }\quad
a(F_{(\mu,\nu)})=\underline{0_n1}. $$



These curves converges to the curve $\, (\mu(\nu)\,,\,
\nu)\,$(see figure 4) where $\mu(\nu)\,$ is given by $$\,
\mu(\nu)\,=\, \displaystyle\frac{1+\sqrt{1+4\nu}}{2} \,, \nu \ge
0\,.$$

\begin{figure}[h]
\begin{center}
\includegraphics[width=9cm]{17.eps}
\caption{}\label{f4}
\end{center}
\end{figure}

In fact, we note that $\, \mu_1(\nu)\,=\, \sqrt{\nu} \,,\,
\mu_2(\nu)\,=\,-y_2(\nu)\, =\,\sqrt{\nu + \sqrt{\nu}}\,,\dots\,,
\mu_{n+1}(\nu)\,=\,-y_{n+1}(\nu) \, =\, \sqrt{\nu +
\mu_n(\nu)}\,$.

\vspace{0.2cm}

Hence, taking the limit when $\, n \to \infty \,$, we get $\,
\mu(\nu)\,=\, \sqrt{\nu+\mu(\nu)} \,$, so $\, \mu^2\,=\, \nu +
\mu \,\,$and, consequently,$\,
\mu(\nu)\,=\,\displaystyle\frac{1+\sqrt{1+4\nu}}{2} \,$ as
announced.

\vspace{0.2cm}

We also observe that the curves $\, (\mu_n(\nu),,\,\nu)\,$
transversally intersect the curve
$\,(\mu\,,\,\displaystyle\frac{1+\sqrt{1+4\mu}}{2}\,)\,$, any
$\, n \in \Na$.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=9cm]{figura10.eps}
\caption{}\label{f5}
\end{center}
\end{figure}

\begin{itemize}
\item[(d)] $F(0^-)=n$-th preimage of zero $=x_n(\mu)$.
\end{itemize}




These equation define the curves $\nu_n(\mu)=x_n(\mu)$ that
satisfies: $$
(\mu,\nu)\in\{(\mu,\nu_n(\mu));\quad\mu\ge0\}\quad\mbox{ imply
}\quad b(F_{(\mu,\nu)})=\underline{1_n0}. $$ These curves
converges to the curve $\,(\mu\,,\nu(\mu))\,$(see figure 5 )
where$\,\nu(\mu)\,$ given by $$\,
\nu(\mu)\,=\,\displaystyle\frac{1+\sqrt{1+4\mu}}{2}\,, \mu \ge
0\,.$$




Let us now prove the following

\vspace{0.2cm}
\begin{proposition}
The curve $\, C_{\underline{10}}\,=\,\{(\mu\,,\nu_1(\mu))\,,\,\,
\mu \in ]0\,,\infty[\,\}\,$transversally intersect the curve
$\,C_{\underline{0_n1}}\,=\,\{\mu_n(\nu)\,,\,\nu)\,,\,\, \nu \ge
0\,\} \,\,, \forall n \in \Na.$
\end{proposition}

\noindent{\bf Proof:} For $C_{\ul{10}}$ we have  $C_{\ul{10}} =
\{ (\mu,\sqrt{\mu)}; \mu > 0 \}.$ Hence $C_{\ul{10}}$ is
described by $\mu(\nu) = \nu^2 \,,\, \nu> 0.$

 Now $C_{\ul{01}}$ is described as $\mu_1 (\nu) = \sqrt{\nu}.$ We have
 $C_{\ul{01}} \cap C_{\ul{10}} = \{ (1,1)\} .$ and $\mu_1^{'} (\nu)=
  \displaystyle\frac{1}{2\sqrt{\nu}}; (\nu^2)^{'} = 2\nu \,$.  So $\, \mu_1(\nu) $ is
  transversal  to $\mu(\nu)= \nu^2 $  at $\nu=1.$
  Note $ \mu_1^{'} (\nu)\leq \displaystyle\frac{1}{2}$ and $\mu'(\nu)\geq 2,
  $ all  $ \nu \geq 1=\nu_1.$


\vspace{0.2cm}

Inductively, assume that $C_{\ul{10}}$ transversally intersect
$C_{\ul{0n1}}$ at a point $(\nu_n^2, \nu_n)$ with $\mu_n^{'}
(\nu) \leq 1/2 \,$ and $\, \mu^{'} (\nu) \geq 2, \,$  all $\,\nu
\geq \nu_n \geq \nu_{n-1} \,$.

\vspace{0.2cm}

For the curve $C_{\underline{0_{n+1}1}}$ we have
$\mu_{n+1}(\nu)= \sqrt{\nu+\mu_n (\nu)}$ and $\,
\mu_{n+1}^{'}(\nu)=
\displaystyle\frac{1+\mu'_n(\nu)}{2\mu_{n+1}(\nu)} \,$

\vspace{0.2cm}

Let  $\,(\nu_{n+1}^2, \nu_{n+1}) \in   C_{\ul{0_{n+1}1}} \cap
C_{\ul{10}} \,$. We have $ \mu^{'} (\nu) = 2\nu \geq 2\nu_{n+1}
> 2\nu_n \geq  2\nu_{n-1}
\geq \cdots \geq 2 $ and $ \mu_{n+1}^{'}(\nu)=
\displaystyle\frac{1+\mu_n^{'}(\nu)}{2 \mu_{n+1}(\nu)} \leq
\displaystyle\frac{3/2}{4}< 1/2 $, any $ \nu \geq \nu_{n+1} >
\nu_n  $. So, we get the result. $\,\Box$

\vspace{0.2cm}

Let us now prove

\vspace{0.2cm}

\begin{proposition}
The curve $\,
C_{\underline{1_m0}}\,=\,\{(\mu\,,\,\nu_m(\mu))\,;\,\, \mu \in
]0\, ,\infty[\, \}\,$transversally intersect the curve
$\,\,C_{\underline{0_n1}}\,\,, \forall n \in \Na $.
\end{proposition}

\noindent{\bf Proof:} For $\,m\,=\,2\,$ we have $\,
\nu_2(\mu)\,=\, \sqrt{\mu+\sqrt{\mu}}\,$then $\,
\nu_2^{'}(\mu)\,=\,
\displaystyle\frac{1+\displaystyle\frac{1}{2\sqrt{\mu}}}{2\nu_2(\mu)}\,
=\,\displaystyle\frac{1+\nu_1^{'}(\mu)}{2\nu_2(\mu)}\,$.

\vspace{0.3cm} Since $\,\,\displaystyle\frac{1+
\displaystyle\frac{1}{2\sqrt{\mu}}}{2\nu_2(\mu)}\,< 1\,$ we get
$\, \nu_2'(\mu) < 1\,$. Let denote by $\, \tilde{\mu_2}(\nu)\,$
its inverse map. We have $\, \tilde{\mu_2}'(\nu)\,>1 \,$. Since
$\, \mu_n'(\nu) < 1 \,$ we obtain the result in this case.

\vspace{0.3cm}

Inductively, assume$\, \nu_m'(\mu) < 1\,$ and that $\,
\{(\mu\,,\nu_m(\mu))\,\}\,$ transversally intersect the curve
$\,\,C_{\underline{0_n1}}$.

\vspace{0.3cm}

Since
$\,\nu_{m+1}^{'}(\mu)\,=\,\displaystyle\frac{1+\nu_m^{'}(\mu)}{2\nu_{m+1}(\mu)}\,<\,
\displaystyle\frac{1+1}{2 \nu_{m+1}(\mu)}\,< 1$, we get the
result. $\,\, \Box$



\begin{itemize}
\item[(e)] Let us now assume that $F(0^+)<y_1(\nu)$.
\end{itemize}

 In this situation there is
$\overline{ x_1}(\mu,\nu)$ such that $-\mu+
\overline{x_1}^2=y_1(\nu)$; that is
$\overline{x_1}(\mu,\nu)=\sqrt{\mu-\sqrt\nu}$.


\vspace{0.3cm} If we ask for the condition
$F(0^-)=\overline{x_1}(\mu,\nu)$ we get the curve
$\overline{\mu_1}(\nu)=\nu^2+\sqrt\nu$(see figure 6). This curve
is tangent to $\mu_1(\nu)$ at $\nu=0$ and transversally
intersect $\mu_n(\nu)$, $n\ge2$ and $\mu(\nu)$.


\vspace{0.3cm}

In the same way: if we assume that $F(0^+)<y_n(\nu)$, $n\ge2$ we
will find $\overline{x_n}(\mu,\nu)$ such that $-\mu+\overline{
x_n}^2=y_n(\nu)$ that is
$\overline{x_n}(\mu,\nu)=\sqrt{\mu+y_n(\nu)}$.

If we look for the condition $F(0^-)=\overline{x_n}(\mu,\nu)$ we
get a curve $\overline{\mu_n}(\nu)=\nu^2-y_n(\nu)$. This curve
is tangent to $\mu_n(\nu)$ at $\nu=0$ and transversally
intersect $\mu_j(\nu)$, $j\ge n+1$.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=9cm]{figura11.eps}
\caption{} \label{f6}
\end{center}
\end{figure}

We note that
$(\mu,\nu)\in\{(\overline{\mu_n}(\nu),\nu);\quad\nu\ge0\}$ imply
$b(F_{(\mu,\nu)})=\underline{10_{n+1}}$.

\bigskip
\begin{itemize}
\item[(f)] In a similar way if we assume $F(0^-)>x_1(\mu)$ we will find
$\overline{y_1}(\mu,\nu)$ such that
$\nu-\overline{y_1}^2=x_1(\mu)$ that is:
$\overline{y_1}(\mu,\nu)=\,-\sqrt{\nu-x_1(\mu)}$.
\end{itemize}

\begin{figure}[ht]
\begin{center}
\includegraphics[width=9cm]{figura12.eps}
\caption{} \label{f7}
\end{center}
\end{figure}

 The condition $F(0^+)=\overline{y_1}(\mu,\nu)$ imply the existence
 of a curve, $\overline{\nu_1}(\mu)$, such that
  $\overline{\nu_1}(\mu)=\mu^2+x_1(\mu)=\mu^2+\sqrt\mu$,
$\mu\ge0$. This curve(see figure 7) is tangent to $\nu_1(\mu)$
at $\nu=0$ and transversally intersect $\nu_n(\mu)$; $n\ge2$ and
$\nu(\mu)$.



In a similar way, if we assume that $F(0^-)>x_n(\mu)$, $n\ge2$
we will find $\overline{ y_n}(\mu,\nu)$ such that
$\nu-\overline{ y_n}^2=x_n(\mu)$, that is $\overline{
y_n}(\mu,\nu)=\sqrt{\nu-x_n(\mu)}$.


If we ask for the condition $F(0^+)=\overline{ y_n}(\mu,\nu)$ we
get the curve $\overline{\nu_n}(\mu)=\mu^2+x_n(\mu)$, $\mu\ge0$.
This curve is tangent to $\nu_n(\mu)$ at $\nu=0$ and
transversally intersect $\nu_j(\mu)$; $j\ge n+1$.

\vspace{.2cm}

We note that
$(\mu,\nu)\in\{(\mu,\overline{\nu_n}(\mu));\quad\mu>0\}$ imply
$a(F_{(\mu,\nu)})=\underline{01_{n+1}}$.


\begin{note}  In this section we have proved :

\begin{itemize}
\item[1)] That associated to any
sequence $a\in A_0=\{\underline{0_n1};\quad \underline{01_n} /
n\in\Bbb N\}$, we have two curves, $\,C(a)\,$and $\,C(b(a))\,$,
such that $(\mu,\nu)\in C(a)$ imply $a(F_{(\mu,\nu)})=a\,$ and
$\,(\mu,\nu)\in C(b(a))\,$ imply $\,b(F_{(\mu,\nu)})=b(a)$.

Moreover, $\,C(a)\,$ and $\,C(b(a))\,$ transversally intersect
at the point $\{P(a,b)\}=C_a\cap C(b(a))$.

\item[2)] For $(\mu,\nu)\in C(a)$ we have that
$0$ is a fixed point of $F_{(\mu,\nu)}^{\#per(a)}$ that attracts
a small right neighborhood of it.

\item[3)] For $(\mu,\nu)\in C(b(a))$ we observe that $x=\nu$ is a fixed
point of $F_{(\mu,\nu)}^{\# per(b(a))}$ that attracts a small
left neighborhood of it.

\end{itemize}
\end{note}

\noindent{\bf 4.2.4  Transversality of the bifurcation curves.}

\vspace{0.3cm}

Let us denote $$ \varphi_1(\mu) = \sqrt{\mu}, \varphi_2 (\mu) =
\sqrt{\mu+\varphi_1(\mu)},\cdots , \varphi_{n+1}(\mu) =\sqrt{\mu
+ \varphi_n (\mu)} $$ and

$$  \psi_1 (\nu) = \sqrt{\nu} \psi_2(\nu) =
\sqrt{\nu+\psi_2(\nu)}; \cdots ; \psi_{n+1} , (\nu) =
\sqrt{\nu_n + \psi_n(\nu)}. $$


\vspace{0.2cm}

Define $G_n (\mu,\nu)=-\mu+\varphi_n (\nu)$  and
$H_n(\mu,\nu)=\nu- \varphi_n(\mu).  $


\vspace{0.3cm}

As we saw, in 4.2.3, we have that $G_n^{-1} (0) = C_{\ul{0_n
1}}$ and  $H_n^{-1} (0) = C_{\underline{1_n 0}}.$


\vspace{.2cm}

Let $$  L_n(\mu,\nu)= -\mu+\left[  G_n (\mu, \nu)\right]^2=$$

$$ = -\mu+(-\mu+ \psi_n (\nu))^2 = -\mu+\mu^2-2\mu\psi_n(\nu)+
\psi_n^2(\nu).$$

and $$   M_n(\mu, \nu)= \nu- \left[ H_n(\mu,\nu)\right]^2 = \nu-
\left[ \nu-\varphi_n(\mu) \right]^2=$$

$$= \nu - \nu^2 + 2\nu \varphi_n(\mu) - \varphi_n^2 (\mu). $$


We have that $L_n^{-1}(0)$  is transversal  to $H_m^{-1}(0), $
any $n, m \in \Na.$

\vspace{0.2cm}


In fact, we have

\begin{eqnarray*}
\displaystyle\frac{\partial L_n}{\partial\mu}& =&
-1+2\mu-2\psi_n(\nu).\\ &&\\ \displaystyle\frac{\partial
L_n}{\partial\nu}& =& 2\psi_n(\nu) \cdot \psi'_n (\nu)\,-\, 2\mu
\psi'_n(v).\\ &&\\
 \displaystyle\frac{\partial H_n}{\partial\mu}&
=& -\varphi'_n(\mu).\\ &&\\ \displaystyle\frac{\partial
H_n}{\partial\nu}& =& 1
\end{eqnarray*}


Then

$$  \displaystyle\frac{-\displaystyle\frac{\partial
L_n}{\partial \mu }}{\displaystyle\frac{\partial L_n}{\partial
\nu}} = \displaystyle\frac{1+2( \psi_n(\nu) - \mu)}{2\psi'_n
(\nu) [\psi_n(\nu)-\mu]} $$ and

$$
 \displaystyle\frac{-\displaystyle\frac{\partial H_n}{\partial
\mu }}{\displaystyle\frac{\partial H_n}{\partial \nu}} =
\varphi'_n (\mu) < 1.$$


\vspace{0.2cm}

Since $ L_n (\mu,\nu) = 0 \,$implies $\,  \quad \sqrt{\mu} =
\psi_n(\nu) - \mu ,$ \quad  then

\vspace{0.2cm}

$$   \displaystyle\frac{\displaystyle\frac{\partial
L_n}{\partial \mu }}{\displaystyle\frac{\partial L_n}{\partial
\nu}} = \displaystyle\frac{1}{\psi'_n (\nu)} \cdot \left[  1+
\displaystyle\frac{1}{2\sqrt{\mu}}\right] >1.$$

\vspace{0.2cm}

Hence $L_n^{-1}(0) $  is transversal to $H_n^{-1}(0).$

\vspace{0.2cm}

Let us now see that $M_m^{-1} (0) $  is transversal to

$L_n^{-1}(0).$  In fact

\vspace{0.2cm}

$$   \displaystyle\frac{-\displaystyle\frac{\partial
M_m}{\partial \mu }}{\displaystyle\frac{\partial M_m}{\partial
\nu}} = \displaystyle\frac{2\varphi_m^{'} (\mu)
[\varphi_m(\mu)-\nu]}{1+2[\varphi_m(\mu)-\nu
]}=\displaystyle\frac{2\varphi_m^{'}(\mu) \cdot
\sqrt{\nu}}{1+2\sqrt{\nu}}.$$

\vspace{0.2cm}

And, consequently $M_m^{-1} (0)$  is transversal to
$L_n^{-1}(0)$ for any $n,m \in \Na $(see figure 8)

\begin{figure}[ht]
\begin{center}
\includegraphics[width=9cm]{figura55.eps}
\caption{}
\end{center}
\end{figure}



\vspace{0.2cm}

 \noindent {\bf \underline{Remark}} \,  We note that $L_n^{-1}(0)
 $  "begin" at  the intersection   $G_{n-1}^{-1} (0) \cap H_1^{-1}
 (0)$and it is tangent to $\, H_1=0\,$. In the same way, if  $N_n (\mu, \nu) = F
 (L_n)$, then $N_n  $ is transversal to $H_n^{-1} (0) $and$\, M_n^{-1}
 (0)$ any $ m\geq 0, $ and $N_n^{-1}(0) $  "begin"  in the intersection $G_{n-1}^{-1}(0)
  \cap H_2 (0) $, and so on.


\vspace{0.3cm}


\noindent {\bf 4.2.5 Primary saddle-nodes.}

In this section we will prove

\medskip

\begin {proposition}

Associated with any $a\in A_0$ and
$b=b(a)$ there is a curve $\, SN(a,b)\subset\R^2$ such that:
\begin{itemize}

\item[(i)] there are values $\nu_{a,b}$ and $\mu_{a,b}$ such that
$F_{(\mu_{a,b},\nu_{a,b})}^{\# per(a)}$ has a fixed point with
derivative equal to 1 and $(\mu_{a,b},\nu_{a,b})\in SN(a,b)$;

\medskip

\item[(ii)] $SN(a,b)\setminus\{(\mu_{a,b},\nu_{a,b})\}=SN(a)\cup SN(b)$
are two curves which satisfies:

\begin{itemize}
\item[(a)] $SN(a)$ transversally intersects $C(b)$ at a point
 $\{(\overline{\mu}_{a,b},\overline{\nu}_{a,b})\}=SN(a)\cap C(b)$.
 $\,SN(b)$ transversally intersects $C(a)$ at a point
 $\{(\tilde\mu_{a,b},\tilde\nu_{a,b})\}=SN(b)\cap C(a)$;

\item[(b)] For any $(\mu,\nu)\in SN(a)$ there is a
saddle-node periodic point which correspond to an unfolding of
the fixed point of $F_{(\mu,\la)}^{\# per(a)}$,
  $(\mu,\nu)\in C(a)$.

\item[(c)] For any $(\mu,\nu)\in SN(b)$ there
is a saddle-node periodic point which correspond to an unfolding
of the fixed point of $F_{(\mu,\nu)}^{\# per(b)}$,
 $(\mu,\nu)\in C(b)$.
\end{itemize}
\end{itemize}


\end{proposition}

\noindent{\bf Proof:} Initially, let us consider $\,
a\,=\,\underline{01}.$ In this situation the curve
$\,C_a\,=\,C_{\underline{01}}\,=\,\{(\mu\,,\,\mu^2)\,,\, \mu \ge
0 \}\,$ represent the fact that $\, F(0^+)\,=\,y_1(\nu)
\,$,hence $\,F(F(0^+))\,=\,F(y_1(\nu))\,=\,0\,$ that is, "$0^+$"
is a periodic point whose period is $\,\#(a)\,$. Also, for $\,
b\,=\,\underline{10}\,$ the curve
$\,C_b\,=\,C_{\underline{01}}\,=\,\{(\mu\,,\sqrt{\mu})\,,\, \mu
\ge 0\}\,$ represent the fact that $\, F(0^-)\,=\, x_1(\mu)\,$
and, then, $\,F(F(0^-))\,=\,F(x_1(\mu))\,=0\,$, that is "$0^-$"
is a periodic point (of the map $\,F\,$) whose period is
$\,\#(b)\,$.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=7cm]{1.eps}
\includegraphics[width=7cm]{2.eps}
\caption{}\label{f8}
\end{center}
\end{figure}

\vspace{0.2cm}

Let us consider the point $\,(1\,,\,1)\in C_{\underline{01}}
\cap C_{\underline{10}} \,$. The graph of the map $\,
F_{(1,1)}\,$is as in left hand side of figure 9 and the graph of
the restriction of the map $\,F^2_{(1,1)}\,$is as in the right
hand side of figure 9.


\vspace{0.3cm}
Now, let $\, H\,=\, \{(\mu\,,\,\nu)\,; \,\mu \ge
0\,,\, \nu \ge 0 \}\,$. We have that $\, H \setminus
C_{\underline{01}} \,=\, \{(\mu\,,\,\nu)\,,\,, \mu^2 < \nu \}
\cup \{ (\mu\,,\,\nu)\,,\, \mu^2 > \nu \}\,=\, A_1 \cup A_2
\,$and $\, H \setminus C_{\underline{10}}\,=\,
\{(\mu\,,\,\nu)\,,\, \sqrt{\mu} < \nu \} \cup \{(\mu\,,\nu)\,,\,
\sqrt{\mu} > \nu \} \,=\, B_1 \cup B_2 \,$. So, we have the
following components of the set $\, H \setminus(C_a \cup C_b)
\,=\, A_1 \cap B_1 \cup A_1 \cap B_2 \cup A_2 \cap B_1 \cup A_2
\cap B_2 \,$ where $\, A_1 \cap B_1\,=\, \{(\mu\,,\nu)\,,\, \nu
> \mu^2 \,\mbox{and} \,\, \nu
> \sqrt{\mu} \} \,,\, A_1 \cap B_2\,=\, \{ (\mu\,,\,\nu)\,,\,
\sqrt{\mu} > \nu
>\mu^2 \}\,\,,\,\, A_2 \cap B_1 \,=\,
\{(\mu\,,\,\nu)\,,\,\, \mu^2 >\nu > \sqrt{\mu} \}\,\,$ and$\,\,
A_2 \cap B_2 \,=\,\{(\mu\,,\,\nu)\,\,,\,\, \mu^2 > \nu \,\,
\mbox{and} \,\, \sqrt{\mu} > \nu \} \,$. The next figures
displays the first return map, to the interval $\,[0\,,\nu] \,$
for $\, (\mu\,,\,\nu) \in A_i \cap B_j \,, i=j=1,2 \,$ and
located near the point $\, (1,1)\,$.


\begin{figure}[ht]
\begin{center}
\includegraphics[width=6cm]{3.eps}\hspace{1cm}
\includegraphics[width=6cm]{4.eps}
\caption{$A_1 \cap B_1 $ \hspace{4.8cm}$A_1 \cap B_2.$}
\includegraphics[width=6cm]{5.eps}\hspace{1cm}
\includegraphics[width=6cm]{6.eps}
\caption{ $A_2 \cap B_1 $ \hspace{4.8cm}$A_2 \cap B_2$.}
\label{f9}
\end{center}
\end{figure}


Now, it is clear that we can find, in the interior of the set
$\, A_1 \cap B_2 \,$, a point $\,(\mu_{a,b}\,,\, \nu_{a,b}) \,$
such that the first return map to the interval $\,[0\,,\,\nu]\,$
looks like in figure 11.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=9cm]{7.eps}
\caption{} \label{f11}
\end{center}
\end{figure}

Let us now consider the family $\, F_{(\mu\,,\,\mu^2 + \rho)}
\,, \rho \in \R \,$. Clearly, there is a unique $\, \rho_{a,b}
\,$such that, for $\, \mu\,=\, \mu_{a,b}\,$we have
$\,\,\nu_{a,b}\,=\,\mu^2 + \rho \,$. Now it is not hard to
prove, for the one-parameter family $\,  F_{(\mu\,,\,\mu^2 +
\rho)}\,, \rho\,$fixed, the following

\begin{lemma}
\begin{itemize}
\item[(i)] Assume $\, \rho < \rho_{a,b} \,$and near
$\, \rho_{a,b} \,$. In this case the first return map has the
following generic unfolds :

\begin{figure}[h]
\begin{center}
\includegraphics[width=4.3cm]{8.eps}\hspace{.5cm}
\includegraphics[width=4.3cm]{9.eps}\hspace{.5cm}
\includegraphics[width=4.3cm]{10.eps}
\caption{} \label{f12}
\end{center}
\end{figure}

\item[(ii)] Assume $\,\rho > \rho_{a,b} \,$ and near
$\, \rho_{a,b} \,$. In this case the first return map has the
following generic unfold :


\begin{figure}[h]
\begin{center}
\includegraphics[width=4.3cm]{11.eps}\hspace{.5cm}
\includegraphics[width=4.3cm]{12.eps}\hspace{.5cm}
\includegraphics[width=4.3cm]{13.eps}
\caption{} \label{f13}
\end{center}
\end{figure}



\item[(iii)] The curves $\, SN(b)\,=\, \mu_{sn}^{\underline{10}}(\rho)\,\,$
and $\,\,SN(a)\,=\, \mu_{sn}^{\underline{01}}(\rho)\,\,$are
given in  figure 15,

\begin{figure}[ht]
\begin{center}
\includegraphics[width=10cm]{14.eps}
\caption{} \label{f14}
\end{center}
\end{figure}

\item[(iv)] The curve $\, SN(b) \,$ intersects the curve
$\, \{(\mu\,,\, \mu^2 + \rho )\,,\, \mu \ge 0\}\,, \rho < \rho_{a,b} \,$
fixed , in a unique point.
\item[(v)] The curve $\, SN(a) \,$ intersects the curve
$\, \{(\mu\,,\, \mu^2 + \rho )\,,\, \mu \ge 0\}\,, \rho >
\rho_{a,b} \,$ fixed , in a unique point(see figure 16). $\,
\Box$

\begin{figure}[h]
\begin{center}
\includegraphics[width=8cm]{15.eps}
\caption{} \label{f15}
\end{center}
\end{figure}

\end{itemize}
\end{lemma}

This complete the proof in the case $\, a\, =\, \underline{01}
\,$. The other cases follows in a similar way.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=9cm]{figura13.eps}
\caption{}\label{f16}
\end{center}
\end{figure}


Figure 17 displays these essential facts concerning the curve
$SN(a,b). \Box$


\begin{remark} Given $a\in A_0$, $b=b(a)$ we define
$L_a=\{(\mu,\nu); a(F_{(\mu,\nu)})=a\}$ and
 $R_b=\{(\mu,\nu);\quad b(F_{(\mu,\nu)})=b\}$.

It is not hard to prove that the sets $L_a$ and $R_b$ looks like
as in figure 18
\end{remark}

\begin{figure}[ht]
\begin{center}
\includegraphics[width=7cm]{figura14.eps}\hspace{.5cm}
\includegraphics[width=7cm]{figura15.eps}
\caption{} \label{f17}
\end{center}
\end{figure}


\noindent {\bf 4.2.6} Let us consider

$$ A_1=\{m(a_1,a_2);\quad a_1,a_2\in A_0 \,\mbox{are consecutive
sequences}\,\} \cup A_0. $$

It is not hard to prove(see 4.2.4) that associated with
$a=m(a_1,a_2)$, $b=b(a)$ we can find two curves $C(a), C(b)$,
tangent to $C(a_2)$ and $C(b(a_2))\,$ respectively, such that:
$$ (\mu,\nu)\in C(a)\,\, \mbox{ imply }\,\, a(F_{(\mu,\la)})=a
\,\mbox{and}\, (\mu,\nu)\in C(b)\,\,\mbox{ imply }\,\,
b(F_{(\mu,\la)})=b. $$

Also, there are saddle-nodes associated to $C(a)$ and $C(b)$ as
in 4.2.5. The geo\-me\-try of the sets $R_b$ and $L_a$ is
similar to the geometry of the sets  described in  the remark in
4.2.5.

Similarly, we consider the set

$$ A_{n+1}=A_n\cup\{m(a_1,a_2); a_1<a_2 \,\mbox{are consecutive
sequences in} A_n \}. $$

As before associated with $a\in A_{n+1}$, $a=m(a_1,a_2)$,we are
able to find the curves $C(a)$, $C(b(a))$ and $SN(a,b)$(see
figure 19).

\begin{figure}[h]
\begin{center}
\includegraphics[width=10cm]{figura16.eps}
\caption{}\label{f18}
\end{center}
\end{figure}



\noindent {\bf 4.2.6} 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 $a_n\in A_\infty$, $a_n\to a$. let
$b_n=b(a_n)$. It is clear that we  have $\,b_n\to b$.

\vspace{0.2cm}

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
$\varphi_a:(0,\infty)\to\R$, $\psi_b:(0,\infty)\to\R$ satisfies

\begin{itemize}

\item[-]$\varphi_a(\mu)=\psi_b(\mu)$, $0<\mu\le\mu_a$.

\item[-]The maps $\varphi_a|_{(\mu_a,\infty)}$ and
$\varphi_b|_{(\mu_a,\infty)}$ are tangent at $\mu_a$.
\end{itemize}

Moreover

\begin{itemize}
\item[(i)] the curves $C(a_n)$ converges to $C(a)$ and
the curves $\,C(b_n)$ converges to $C(b)$;


\item[(ii)] the intersections $C(a_n)\cap C(b_n)$ converges to
$\{(\mu_a,\nu_a)\}$;

\item[(iii)] $(\mu,\nu)\in C(a) \,\mbox{imply} \,a(F_{(\mu,\la)})=a\,
\mbox{and}(\mu,\nu)\in C(b)\, \mbox{imply} \,
b(F_{(\mu,\la)})=b$.
\end{itemize}


\begin{figure}[h]
\begin{center}
\includegraphics[width=11cm]{figura17.eps}
\caption{}\label{f19}
\end{center}
\end{figure}

Maybe, the unique point which is not easy to agree in this result
is related with the coincidence of the curves $C(a)$ and $C(b)$
along $\{(\mu,\varphi_a(\mu))$; $0<\mu\le\mu_a \}$, but this is an
easy consequence of the next result.

\vglue .2in

Let $r>0$ be a small number. Let $B_r^+=\{(\mu,\nu)\in\R^2$,
$\sqrt{\mu^2+\nu^2}\le r$, $\mu\ge0,\quad \nu\ge0\}$. Let $m(A)$
denotes the Lebesgue measure of the set $A\subset\R^2$.

\vspace{.2cm}

\begin{lemma} $m(\{(\mu,\nu)\in B_r^+;\quad F_{(\mu,\nu)}\notin
MS_0\})=0$.
\end{lemma}

\noindent {\bf Proof:} In fact, the proof can be done in a
similar way as was done in \cite{L} or in \cite{PR}.

\vspace{0.2cm}
\begin{remark}
\begin{itemize}

\item[1)] In fact, Arnold (\cite{Ar} and Boyland ( \cite{Bo}),
obtained a similar result for the structure of the irrational
Arnold's Tongues of the canonical family
$\, f_{\lambda\,,b}(x)\,=\, x + \lambda +
\displaystyle\frac{b}{2\pi}\sin(2\pi x)\,$.
\item[2)] A similar result(as in lemma 13) was proved by
Swiatek in \cite{Sw} with respect to parameterized families of
homeomorphisms of the circle.

\end{itemize}

\end{remark}


\subsection{Secondary Bifurcations}

We call the bifurcations in this section \underline{secondary}
because this kind of changes in the dynamic cannot appear for
injective maps in $\, DM_0 \,$. Also, it is not possible to
understand this scenario only by using rotation numbers as it is
possible to see by comparing our results with those of
\cite{Bo}.

\vspace{0.2cm}


\noindent {\bf 4.3.1} Let $B_\infty=\{b(a);\quad a\in
A_\infty\}$. Let us now consider $a\in A_\infty$, $b\in
B_\infty$ such that $C(a)\cap C(b)\ne \emptyset$.It is not hard
to prove that ( see section 4.2.3) that:

\medskip
\begin{itemize}
\item[a)]
$C_a\cap C_b=\{P(a,b)\}$ is a unique point.


\item[b)] define $\,A(a,b)=\{c\in\Sigma_0\,;
\quad c=\underline{a_-b_+a^n};\,\, n\in\Na \,\,\mbox{or} \,\,
c=\underline{a_-b^mb_+}; \,m\in \Na\}\,$ and\\
$B(a,b)=\{d\in\Sigma_1;\,\, d=\underline{b_+a^na_-},\quad
n\in\Bbb N$ or $d=\underline{b_+a_-b^m}; \,\, m\in\Na\}$.

\begin{itemize}

\item[((i)] Associated with $c=\underline{a_-b_+}$ there is a curve
$C_c$, tangent to $C_b$; $P(a,b)\in\overline{ C_c}$ and such
that $(\mu,\nu)\in C_c$ imply $a(F_{(\mu,\nu)})=c$;

\item[((ii)] Associated with $d=\underline{b_+a}$ there is a curve
 $C_d$, tangent to $C_a$; $P(a,b)\in\overline{C_d}$ and such that
 $(\mu,\nu)\in C_d$ imply $b(F_{(\mu,\nu)})=d$;
\begin{figure}[h]
\begin{center}
\includegraphics[width=10cm]{figura18.eps}
\caption{(b)(i) and (b)(ii)}\label{f20}
\end{center}
\end{figure}


\item[(iii)] Associated with $c=\underline{a_-b_+a^n}$, $n\ge1$,
there is a curve $C_c$, tangent to $C_a$; $P(a,b)\in
\overline{C_c}$ and such that $(\mu,\nu)\in C_c$ imply
$a(F_{(\mu,\nu)})=c$;

\item[(iv)] Associated with $d=\underline{b_+a^na_-}$, $n\ge1$,
there is a curve $C_d$, tangent to $C_a$; $P(a,b)\in
\overline{C_c}$ and such that $(\mu,\nu)\in C_d$ imply
$b(F_{(\mu,\la)})=d$;

\begin{figure}[ht]
\begin{center}
\includegraphics[width=10cm]{figura19.eps}
\caption{(b)(iii) and (b)(iv)}\label{f21}
\end{center}
\end{figure}

\item[(v)] Associated with $c=\underline{a_-b^nb_+}$, $n\ge1$,
there is a curve $C_c$, tangent to $C_b$; $P(a,b)\in
\overline{C_c}$ and such that $(\mu,\la)\in C_c$ imply
$a(F_{(\mu,\la)})=c$;

\item[(vi)] Associated with $d=\underline{b_+a_-b^n}$, $n\ge1$,
there is a curve,$ C_d$, tangent to $C_b$; $P(a,b)\in \overline{
C_d}$ and such that $(\mu,\la)\in C_d$ imply
$b(F_{(\mu,\la)})=d$;

\begin{figure}[ht]
\begin{center}
\includegraphics[width=9cm]{figura20.eps}
\caption{(b)(v) and (b)(vi)}\label{f22}
\end{center}
\end{figure}

\end{itemize}

In fact, let denote by $\xi_b:[\mu_b,\infty[\to\R \,$and
$\,\xi_a:[\mu_a,\infty[\to\R\,$ the maps that satisfies:

\medskip
\begin{itemize}
\item[(i)] $\{(\mu; \xi_b(\mu)\};\quad \mu\ge\mu_b\}$ is the
curve which represents the fact that $x=v$ is a periodic point
with itinerary $b$;

\item[(ii)] $\{(\mu; \xi_a(\mu)\};\quad\mu>\mu_a\}$ is the curve
which represents the fact that $x=-\mu$ is a periodic point with
itinerary $a$.
\end{itemize}
Let denote $C_a\cap
C_b=\{(\overline{\mu_b},\overline{\nu_b})\}$. \vspace{0.2cm}

Now,  associated with $c_n=\underline{b_+a^na_-}$, $n\ge0$ we
can obtain  maps $\xi_n:[\overline{\mu_b},\infty[\to\R$ such
that


\item[(i)] $\xi_n'(\overline{\mu_b})=\xi_a'(\overline{\mu_b})-\xi_b'(\mu_b)$;
$\xi_n(\overline{\mu_b})=0$

\item[(ii)] $\{(\mu; \xi_b(\mu)+\xi_n(\mu));\quad \mu\ge \overline{\mu_b}\}$
is the curve which represents the fact that $x=v$ is a periodic
point with itinerary $c_n$.

\vspace{0.2cm}

In a similar way, associated with $d_m=\underline{b_+a_-b^m}$, $m\ge1$,
there are maps $\phi_m:[\overline{\mu_b},\infty[\to\R$ such that

\begin{itemize}
\item[(i)] $\phi_m(\overline{\mu_b})=0$; $\phi_m(\overline{\mu_b})=0$.

\item[(ii)] $\{(\mu,\xi_b(\mu)+\phi_m(\mu));\quad \mu\ge \overline{\mu_b}\}$
is the curve which represents the fact that $x=v$ is a periodic point with itinerary $d_m$.
\end{itemize}

\item[(c)] Associated with $c=a_-\underline{b}$ there is a curve
$C_c$, tangent to $SN(b)$ at the point $P_c\in C_c$, transversal to
$C_b$ at the point $\,Q_c$, such that $(\mu,\la)\in C_c$ imply $a(F_{(\mu,\la)})=c$.

Moreover let $L_c$ be the open rectangle bounded by $C_a,SN(b)$,
$C_c$ and $C_b$. We have $L_c\subset R_b$ and $(\mu,\nu)\in L_c$
imply $a(F_{(\mu,\la)})=c$. Also, $(\mu,\la)\in \overline{
L_c}\cap[C_b\cup SN(b)\setminus C_a]$ imply
$a(F_{(\mu,\la)})=c$;

\item[(d)] Associated with $d=b_+a_-\underline{b}$ there is a curve
 $C_d$, tangent to $C_{a_-\underline{b}}$ at
the point  $\, Q_{a_-\underline{b}}\in \overline{C_d}\cap
C_{a_-\underline{b}}$ and such that $(\mu,\nu)\in C_d$ imply
$b(F_{(\mu,\la)})=d$;

\begin{figure}[h]
\begin{center}
\includegraphics[width=10cm]{figura21.eps}
\caption{(c) and (e)}\label{f23}
\includegraphics[width=10cm]{figura22.eps}
\caption{(d) and (f)}\label{f24}
\end{center}
\end{figure}


\item[(e)] Associated with $d=b_+\underline{a}$ there is a curve, $C_d$,
tangent to $SN(a)$ at the point $\,P_d\in C_d$, transversal to
$C_a$ at the point $\,Q_d$ such that $(\mu,\nu)\in C_d$ imply
$b(F_{(\mu,\la)})=d$. Moreover, let $R_d$ be the open rectangle
bounded by $C_a$, $SN(a)$, $C_b$ and $C_d$. We have $R_d\subset
L_a$ and $(\mu,\nu)\in R_d$ imply $b(F_{(\mu,\la)})=d$. Also, $$
(\mu,\la)\in \overline{R_d}\cap [C_a\cup SN(a)\setminus
C_b]\mbox{ imply } b(F_{(\mu,\la)})=d; $$


\item[(f)]Associated with $c=a_-b_+\underline{a}$ there is a curve
 $C_c$, tangent to $C_{b_+\underline{a}}$at the point  $\,Q_{b_+\underline{a}}\in \overline{C_c}\cap
 C_{b_+\underline{a}}$, and such that $(\mu,\nu)\in C_c$ imply $ a(F_{(\mu,\la)})=c$.

\end{itemize}

Figures (\ref{f20}),(\ref{f21}),(\ref{f22}),(\ref{f23}) and (25)
displays the properties in 4.3.1.


\begin{remark} (i) Let $\gamma(a,a_-b_+\underline{a})$
be the curve which start in $P(a,b)$, follows $C_a$ up to
$Q_{b_+\underline{a}}$ and then follows
$C_{a_-b_+\underline{a}}$. We have that the sequence of curves
$$ \{C_{\underline{a_-b_+a^m}};\,\, m\in\Na\}\quad \mbox{
converges to }\quad \gamma(a,a_-b_+\underline{a}); $$

(ii) Let $\gamma(a,b_+\underline{a})$ be the curve which start
in $\,P(a,b)$, follows $C_a$ up to $Q_{b_+\underline{a}}$ and
then follows $C_{b_+\underline{a}}$. We have that the sequence
of curves $\{C_{\underline{b_+a^ma_-}},\,\, m\in\Na\}$ converges
to $\gamma(a,b_+\underline{a})$; \vspace{0.2cm}


(iii) Let $\gamma(b,a_-\underline{b})$ be the curve which start
in $P(a,b)$, follows $C_b$ up to $Q_{a_-\underline{b}}$ and then
follows $C_{a_-b}$. We have that the sequence of curves
$\{C_{\underline{a_-b^mb_+}}; \,\, m\in \Na\}$ converges to
$\gamma(b,a_-\underline{b})$; \vspace{0.2cm}

(iv) Let $\gamma(b,b_+a_-\underline{b})$ be the curve which
start in $P(a,b)$, follows $C_b$ up to $Q_{a_+\underline{b}}$
and then follows $C_{b_+ a_-\underline{b}}$. We have that the
sequence of curves $\{C_{\underline{b_+a_-b^m}},\,\, m\in\Na\}$
converges to $\gamma(b,b_+a_-\underline{b})$; \vspace{0.2cm}

(v) The same construction that we did here can be done for any
$a\in A_\infty^\infty$, $b\in B_\infty^\infty$ such that
$C_a\cap C_b\ne \emptyset$.
\end{remark}
\vglue .2in

\noindent {\bf 4.3.2} We will say that $f\in DM_0$ satisfy the
\underline{axiom $\,A$} if:

(i) All the attracting periodic point of $f$ are hyperbolic and

(ii) The complement of the stable manifolds of the attracting
periodic orbits, in $[f(0^+), f(0^-)]$, is an hyperbolic set.

\bigskip

\begin{lemma}  Let $a\in\cal A_\infty^\infty$ and
$b\in\be_\infty^\infty$ such that $C_a\cap C_b \ne \emptyset$.
There is a neighborhood $U(a,b)\subset\R^2$ of the point
$P(a,b)$ such that $$ m(\{(\mu,\la)\in U(a,b);\,\, F_{(\mu,\nu)}
\,\,\mbox{ do not satisfy axiom} A\})=0. $$ That is; the measure
of the bifurcation in $U(a,b)$ in zero.
\end{lemma}

\vspace{0.2cm}

\noindent{\bf Proof} This result follows as similar results in
\cite{L} or \cite{PR}.



\subsection{ Doubling Periods}

Let us consider $a\in  A_\infty^\infty$ and
$b\in\be_\infty^\infty$ such that $C_a\cap C_b \ne \emptyset$.
\vspace{0.2cm}

We will use $a_0=a$, $b_0=b$. Let us construct the following
sequences: $$ a_{n+1}=\underline{(a_n)_-(b_n)_+}\,;\, \mbox{and} \,\,
b_{n+1}=\underline{(b_n)_+(a_n)_-}. $$

We have $$ a_0>a_1>a_2>\dots> \lim_{n\to\infty}a_n=c $$ $$
b_0<b_1<b_2<\dots< \lim_{n\to\infty} b_n=d $$
We will call the sequence $(a_n,b_n)$ the doubling period sequence
with starting point $(a_0,b_0)$. The pair $(c,d)$ will be called
the \underline{ target} of the doubling period with starting point
$(a_0,b_0)$.

\medskip
\begin{lemma}
Associated with $\,c\,$(resp. $\,d\,$) there is a curve $\,C_c\,$( resp. $\,C_d\,$)
which is the limit of the curves $C_{a_n}$ (resp. $C_{b_n}$).
Moreover $\,C_{c} \cap C_d\,$ contains a unique point $P(c,d)$.


For $(\mu,\la)\in C_c$ we have $a(F_{(\mu,\la)})=c$ and for
$(\mu,\la)\in C_d$ we have $b(F_{(\mu,\la)})=d$.
\end{lemma}


\subsection{ Tails}

\begin{lemma}
\begin{itemize}
\item[a)] Assume $a\in A_\infty^\infty$ and $d\in Max_2$ is a
non-periodic sequence such that $C_a\cap C_d \ne \emptyset$.

There is a curve $C_{a_- d}$, which ``born'' in
$\{P(a,d)\}=C_a\cap C_d$, tangent to $C_d$ such that
$P(a,d)\in\overline{C_{a_-d}}$ and $(\mu,\la)\in C_{a_-d}$ imply
$a(F_{(\mu,\la)})=a_-d$.


\item[b)] Assume $b\in\be_\infty^\infty$ and $c\in Max_2$ is a
non-periodic sequence such that $C_c\cap C_b \ne \emptyset$.

There is a curve $C_{b_+c}$, which ``born'' in $\{P(c,d)\}=C_c\cap
C_b$, tangent to $C_c$ at this point such that $P(c,d)\in
\overline{C_{b_+c}}$ and $(\mu,\la)\in C_{b_+c}$ imply
$b(F_{(\mu,d)})=b_+c$.
\end{itemize}

\end{lemma}

We let the proof of this lemma to the reader.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=11cm]{figuraf1.eps}
\caption{}\label{f27}
\includegraphics[width=11cm]{figuraf2.eps}
\caption{}\label{f28}
\end{center}
\end{figure}


The figures (26)and (27) displays these facts.


\subsection{ Entropy zero}


In this section we will make (following \cite{LM}) some remarks
concerning the set
$$EZ=\{(\mu,\la);\,\, h(F_{(\mu,\la)})=0\}.$$

Here, $h(f)\,$ means the topological entropy of the map $\,f \in DM_0 \,$.


a) Assume $a\in A_\infty^\infty$, $b=b(a)$. We have $$
\mbox{int}(L_a\cap R_b)\subset\mbox{int}(EZ) $$

\vspace{0.2cm}

In fact, in this situation the set $ \Sigma_{a,b}\,$ is formed
by the iterations $ \sigma^i(a)\,\,, i= 1\,,
\cdots\,,\#per(a)-1\,$ and their preimages under the map $
\sigma$.

\vspace{0.2cm}

b) Assume $a\in A_\infty^\infty$, $b=b(a)$ and
$\{P(a,b)\}=C_a\cap C_b$ then $P(a,b)\in\mbox{int}(EZ)$.

\vspace{0.2cm}

In fact, there is a neighborhood of the point $P(a,b)$, which is
covered by the union $ L_a \cup L_{a_-\ul{b}}\cup \{ L_c; c \in
A(a,b) \cup B(a,b) \} \cup \{C_c; c \in \overline{A(a,b) \cup
B(a,b)} \}\,$. ( here $A(a,b)\,$ and $B(a,b)\,$ where defined in
4.3.1. It is clear that for any $ c \in \overline{A(a,b) \cup
B(a,b)} \}\,$ we have that the set $ \Sigma_{c,d}\,$ is formed
by the iterations $ \sigma^i(c)\, i= 1\,, \cdots\,,\#per(c)-1\,$
and their preimages under the map $ \sigma$, here $ d=
\sup\{\sigma^i(c); i \in \Na \} $.

\vspace{.2cm}


c) Assume $a \in A_\infty^\infty$, $b=b(a)$ then $$
int(L_{a_-\underline{b}})\cup R_{b_+\underline{a}})\subset
\mbox{int}(EZ). $$

\vspace{0.5cm}

In fact, if $ (\mu,\nu) \in \mbox{int}(L_{a_-\underline{b}})\,$
we have that $a(\mu,\nu)\,=\,a \,\,$and$\, \,
b(\mu,\nu)\,=\,\sup\{\sigma^i(a)\, i \in \Na \} $.

d) Assume $a\in A_\infty^\infty$, $b=b(a)$ then $$
\overline{L_a}\cap C_{b_+\underline{a}} \subset
EZ\setminus\mbox{int}(EZ) $$ and $$ \overline{R_b}\cap
C_{a_-\underline{b}} \subset EZ\setminus \mbox{int}(EZ) $$

\vspace{0.5cm}

In fact, given any $\, (\mu,\nu) \in \overline{L_a}\cap
C_{b_+\underline{a}} \,$ we have that there is a small
neighborhood $V(\mu,\nu)\,$ of the point $(\mu,\nu)\,$ which is
divided by the curve $\,C_{b_+\underline{a}}\,$ in two disjoint
pieces, one of them, included in $ \mbox{int}(EZ)\,$ and the
other included in the interior of the positive entropy set.

\vspace{0.2cm}


All the next results, in this section, follows in a similar way.

\vspace{0.2cm}


e) Let $a\in A_\infty^\infty$, $b=b(a)$ and $(P(a,b),
Q_{a_-\underline{b}})\subset C_b$ be the open interval whose
boundary points are $P(a,b)$ and $Q_{a_-\underline{b}}$ then

$$
(P(a,b), Q_{a_-b})\subset\mbox{int}(EZ).
$$

f) Let $a\in A_\infty^\infty$, $b=b(a)$ and $(P(a,b),
Q_{b_+\underline{a}})\subset C_a$ be the open interval whose
boundary points are $P(a,b)$ and $Q_{b_+\underline{a}}$ then
$(P(a,b), Q_{b_+\underline{a}})\subset\mbox{int}(EZ)$.
\vspace{0.2cm}

g) Also, we have $$ \{(\mu,0);\,\,
0<\mu<1\}\subset\mbox{int}(EZ) $$ and $$ \{(0,\nu);\,\,
0<\nu<1\}\subset\mbox{int}(EZ). $$ Moreover: $$
(1,0)\in[EZ\setminus\mbox{int}(EZ)] $$ and $$
(0,1)\in[EZ\setminus\mbox{int}(EZ)]. $$

\vspace{0.2cm} h) Let $a\in A_\infty^\infty$, $b=b(a)$. For any
$(\mu,\nu)$ such that $$ a(F_{(\mu,\la)})\le
a_-\underline{b}\quad \mbox{ and } \quad b(F_{(\mu,\la)})>b $$
or $$ a(F_{(\mu,\la)})< a_-\underline{b}\quad \mbox{ and } \quad
b(F_{(\mu,\la)})\ge b $$ we have that
$(\mu,\la)\in\mbox{int}\{(\mu,\la);\,\, h(F_{(\mu,\la)})>0\}$

\vspace{.2cm}

i) Let $a\in A_\infty^\infty$, $b=b(a)$. For any $(\mu,\nu)$
such that $$ a(F_{(\mu,\la)}) \le a\quad \mbox{ and } \quad
b(F_{(\mu,\la)}) > b_+ \underline{a} $$ or $$ a(F_{(\mu,\la)})<a
\quad\mbox{ and } \quad b(F_{(\mu,\la)}) \ge b_+ \underline{a}
$$

We have that $(\mu,\la)\in\mbox{int} \{(\mu,\la);\,\,
h(F_{(\mu,\la)})\ne 0\}$.

\vspace{0.2cm}

j) Let $a\in A_\infty^\infty$, $b=b(a)$. Denote by $B(a,b)\in\R^2$
the point where the curves $C_a$ and $C_b$ born. Let
$SN(a,b)=C_a\cap SN(b)$; $\overline{SN}(a,b)= C_b\cap SN(a)$. We
have that the intervals $[B(a,b), SN(a,b)[ \subset C_a$ and
$[B(a,b), \overline{SN}(a,b)[$ are contained in
$\mbox{int}(\{(\mu,\la);\,\, h(F_{(\mu,\la)})=0\})$.

\vspace{0.2cm}

k) Let $$ C_{\underline{0}}= \left\{ \left(
\frac{1+\sqrt{1+4\nu}}2, \nu \right);\quad \nu \ge 0 \right\} $$
and $$ C_{\underline 1} = \left\{ \left( \mu,
\frac{1+\sqrt{1+4\mu}}2 \right); \quad \mu\ge0 \right\}. $$

We have, $$ (C_{\underline 0} \setminus \{(1,0)\}) \cup
(C_{\underline 1} \setminus
\{(0,1)\})\subset\mbox{int}(\{(\mu,z);\,\, h(F_{(\mu,\la)})\ne
0\}). $$

l) Let $$ L_{\underline 0}= \{(\mu,\nu);\,\, \mu\ge
\frac{1+\sqrt{1+4\nu}}2, \,\, \nu\ge0\}, $$ $$ R_{\underline
1}=\{(\mu,\nu);\,\, \nu\ge \frac{1+\sqrt{1+4\nu}}2, \,\,
\mu\ge0\} ,$$

$$ L_{0\underline{1}}\,=\, \{(\mu\,,\nu)\,,\, a(\mu\,,\nu)\,=\,
0\underline{1}\,\}, $$ and $$ R_{1\underline{0}} \,=\, \{
(\mu\,,\nu)\,,\, b(\mu\,,\nu)\,=\,1\underline{0}\,\}. $$ We have
$$ \mbox{int}(L_{\underline 0} \setminus
R_{1\underline{0}})\cup\mbox{int}(R_{\underline 1} \setminus
L_{0\underline{1}})\subset \mbox{int}(\{(\mu,\nu); \,\,
h(F_{(\mu,\la)})\ne 0 \}). $$

\vglue .3in


>From all of the above results it is clear that $EZ$ is an
arc-connected set. We do not know if it is locally connected.

\vspace{0.2cm}
\subsection{ Non-zero measure for the Bifurcation}


The  result in this subsection is a consequence of a result proved
by Rovella in \cite{Ro}. We established it here only for the
sake of completness.

\vspace{0.2cm}

 Let $a\in Min_2$ and $b\in Max_2$ be eventually periodic
sequences such that $C_a \cap C_b \ne \emptyset$. Let $C_a\cap
C_b=\{P(a,b)\}$ and $U(a,b)$ be any neighborhood of $P(a,b)$. We
have

\bigskip
\begin{lemma} $m(\{(\mu,\nu)\in U(a,b);\,\, F_{(\mu,\nu)}$ is not
$Ax A\})>0$.
\end{lemma}


As a consequence of this result we get

\medskip

\begin{corollary}  Let $(\mu,\nu)\in (EZ  \setminus \mbox{int}(EZ))$
and $U(\mu,\la)$ be any neighborhood of $(\mu,\la)$. We have $$
m(\{(\al,\beta)\in U(\mu,\la);\,\, F_{(\al,\beta)} \mbox{ is not }
Ax. A\})>0. $$
\end{corollary}

\noindent {\bf Proof:} $(\mu,\la)$ can be approximated by a
sequences $(\mu_n,\nu_n)$ as in the Lemma 17.

\vglue .3in

\subsection{ Tales in $L_{\underline 0}$ and $R_{\underline 1}$}

\medskip
a) Associated to $b=1\underline{0}$ there is a curve
$\,C_{1\underline{0}}\subset L_{\underline{0}} \,$ such that
$\,(1,0)\in \overline{C_{1\underline{0}}}\,$;this curve is
tangent to $\,C_{\underline{0}}\,$ at $\,(1,0)\,$ and
$\,(\mu,\nu)\in C_{1\underline{0}}\,$ imply
$\,b(F_{(\mu,\la)})=1\underline{0}$.
\medskip

b) Associated to $a=0\underline{1}$ there is a curve
$\,C_{0\underline{1}}\subset R_{\underline{1}}\,$ that satisfy
$\,(0,1)\in \overline {C_{0\underline{1}}}\,$; it is tangent to
$C_{\underline{1}}$ and $(0,1)$; $(\mu,\nu)\in
C_{0\underline{1}}$ imply $a(F_{(\mu,\la)})=0\underline{1}$;

\medskip

c) Let $\varphi_{1\underline{0}}:]1,\infty[\to\R$ denote the map
that satisfy
$\mbox{Graph}(\varphi_{1\underline{0}})=C_{1\underline{0}}$.

We have $$ \{(\mu,\nu);\,\,
b(F_{(\mu,\la)})\}=1\underline{0}\}=\{(\mu,\nu);\,\, 0 < \nu \le
\varphi_{1\underline{0}}(\mu);\,\, \mu >1\}. $$

\medskip

d) Let $\varphi_{0\underline{1}}:]1,\infty[\to\R$ denote the map
that satisfy
$\mbox{Graph}(\varphi_{0\underline{1}})=C_{0\underline{1}}$.

We have $$ \{(\mu,\nu);\,\,
a(F_{(\mu,\la)})=0\underline{1}\}=\{(\mu,\nu);
\,\,0<\mu\le\varphi_{0\underline{1}}(\nu),\quad \nu>1\}. $$

e) Let $b\in B_\infty^\infty$ and
$\{P(\underline{0},b)\}=C_{\underline{0}}\cap C_b$.

\vspace{.2cm}

There is a curve $C_{b_+\underline{0}}\subset L_{\underline{0}}$
such that $P(\underline{0},b)\in\overline{C_{b_+\underline{0}}}$;
$C_{b_+\underline{0}}$ is tangent to $C_{\underline{0}}$ at
$P(\underline{0},b)$ and $(\mu,\la)\in C_{b_+\underline{0}}$ imply
$b(F_{(\mu,\la)})=b_+\underline{0}$.

\medskip

f) Let $P(\underline{0},b)=(\mu_{\underline{0},b}, \nu_{u0,b})$
and $\varphi_{b_+\underline{0}}:
]\mu_{\underline{0},b},\infty[\to\R$ be the map that satisfy
$\mbox{Graph}(\varphi_{b_+\underline{0}})=C_{b_+\underline{0}}$.

\vspace{.2cm}


Let $\varphi_b:[\mu_{\underline{0}},\infty[\to\R$ be the map that
satisfy $\mbox{Graph}(\varphi_b)=L_{\underline{0}}\cap C_b$.
\vspace{.2cm}

 Clearly $\varphi_b(\mu) <
\varphi_{b_+\underline{0}}(\mu)$, $\mu> \mu_{\underline{0},b}$.

\vspace{.2cm} We have $\{(\mu,\nu);
b(F_{(\mu,\la)})=b_+\underline{0}\}=\{(\mu,\nu);\,\,
\varphi_b(\mu)<\nu \le \varphi_{b_+\underline{0}}(\mu)\}$.

\medskip

g) Let $a\in A_\infty^\infty$ and $\{P(a,\underline1)\}=C_a\cap
C_{\underline{1}}$. There is a curve $C_{a_- \underline{1}}\subset
R_{\underline{1}}$ such that
$P(a,\underline{1})\in\overline{C_{a_-\underline{1}}}$;
$C_{a_-\underline{1}}$ is tangent to $C_{\underline{1}}$ at
$\,P(a,\underline1)\,$
and $\,(\mu,\la)\in C_{a_-\underline{1}}\,$ imply
$\,a(F_{(\mu,\la)})=a_-\underline{1}$.

\medskip

h) Let
$P(a,\underline{1})=(\mu_{a,\underline{1}},\nu_{a,\underline{1}})$
and $\varphi_{a_-\underline{1}}:
]\nu_{a,\underline{1}},\infty[\to\R$, $\varphi_a:
[\nu_{a,\underline{1}},\infty[\to\R$ be the maps that satisfies:
$$
\mbox{Graph}(\varphi_{a_-\underline{1}})=C_{a_-\underline{1}}\quad
\mbox{ and } \quad \mbox{Graph}(\varphi_a)=R_{\underline{1}}\cap
C_a. $$

Clearly $\varphi_a(\nu)<\varphi_{a_-\underline{1}}(\nu)$,for any
$\,\nu> \nu_{a_-,\underline{1}}$.

\vspace{.2cm} We have $\{(\mu,\nu)$;
$a(F_{(\mu,\nu)})=a_-\underline{1}\}=\{(\mu,\nu);\,\,
\varphi_a(\nu)<\mu\le\varphi_{a_-\underline1}(\nu),\quad
\nu>\nu_{a,\underline1}\}$.

\vglue .3in

\subsection{ Bifurcations in $R_{\underline1}\cup L_{\underline0}$}

\medskip

In the same way as was proved in \cite{L} or as in \cite{PR} we
can prove.
\medskip
\begin{lemma} $m(\{(\mu,\nu)\in R_{\underline1}\cup
L_{\underline{0}}$; $F_{(\mu,\nu)}$ do not satisfy $Ax. A\})=0$.
\end{lemma}
\vglue .3in

\subsection{Density of Axiom A Systems.} A consequence of our's
and Rovella's result is:

\medskip

\begin{lemma} $\{(\mu,\nu);\,\, F_{(\mu,\nu)}$ satisfy $ Ax. A\}$
is a dense set in $\R_0^+\times \R_0^+$.

\end{lemma}

\subsection{The Global Picture}

Taking together all the results in this section the global
picture of the bifurcation diagramme is as in figure 28.

\begin{figure}[h]
\begin{center}
\includegraphics[width=13cm]{globalpicture.eps}
\caption{}\label{f27}
\end{center}
\end{figure}


\section {The general case}

Let us consider the five parameter family of maps
$F_{(\alpha,c_1,c_2,\mu,\nu)}\colon (R\setminus\{0\})\to\R$,
$\mu\ge0 \,\,,\nu\ge0 \,\,\alpha \,>1$, $c_1>0 \,\,, c_2>0$ in
$DM_0\,$ given by: $$ F_{((\alpha,c_1,c_2,\mu,\nu)}(x)=
\left\{\begin{array}{l} -\mu+c_1x^{\alpha},\quad  x>0
\\ \nu-c_2(-x)^\alpha,\quad  x<0 \end{array} \right. $$
We have to observe that all the  previous results, for the
quadratic family, are (still) true when we fix generic values of
the  parameter $\, \alpha \,, c_1 \,$ and $c_2\,$. That is, the
bifurcation diagramme associated  to any generic two parameter
family $F_{(\mu,\nu)}= F_{(\alpha,c_1,c_2,\mu,\nu)}$is,up to an
homeomorphism, the same as the given for the quadratic family.



\medskip


\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,
$(W_{\sigma(Y)}Û\setminus\{\sigma(Y)\})=\gamma_1(Y)\cup\gamma_2(Y)$,
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 Rafael
Labarca to IMPA-Brazil and a visit of Carlos Moreira 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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