# Up, down, two-sided Lorenz attractor, collisions, merging and switching

Diego Barros111partially supported by CAPES, Christian Bonatti and Maria José
Pacifico222partially supported by FAPERJ, CNPq, and Brazilian-French Network
in Mathematics

###### Abstract

We present a slightly modified version of the well known "geometric Lorenz
attractor". It consists in a $C^{1}$ open set ${\mathcal{O}}$ of vector fields
in ${\mathbb{R}}^{3}$ having an attracting region ${\mathcal{U}}$ containing:

  * •

a unique singular saddle point $\sigma$;

  * •

a unique attractor $\Lambda$ containing the singular point;

  * •

the maximal invariant in ${\mathcal{U}}$ contains at most $2$ chain recurrence
classes, which are $\Lambda$ and (at most) one hyperbolic horseshoe.

The horseshoe and the singular attractor have a collision along the union of
$2$ co-dimension $1$ submanifolds which divide ${\mathcal{O}}$ in $3$ regions.
By crossing this collision locus, the attractor and the horseshoe may merge in
a two-sided Lorenz attractor, or they may exchange their nature: the Lorenz
attractor expel the singular point $\sigma$ and becomes a horseshoe and the
horseshoe absorbs $\sigma$ becoming a Lorenz attractor.

##  1 Introduction

Lorenz presented in [13] an example of a parameterized 2-degree polynomial
system of differential equations as a very simplified model for the convection
of thermal fluid, motivated by an attempt to understand long-term weather
forecasting. Numerical simulations for an open neighborhood of the chosen
parameters suggested that almost all points in phase space tend to a strange
attractor, called the Lorenz attractor. However, Lorenz’s equations proved to
be very resistant to rigorous mathematical analysis, from both conceptual
(existence of an equilibrium point accumulated by regular orbits prevents the
attractor to be hyperbolic) as well numerical ( solutions slow down as they
pass near the equilibrium, which means unbounded return times and, thus,
unbounded integration errors) point of view. A very successful approach was
proposed by Guckenheimer and Williams [8]. They constructed a geometric Lorenz
attractor that reproduces the behavior observed by Lorenz.

As an abstract object, the geometric Lorenz attractor is the inverse limit of
a semiflow on a branched $2$-manifold (with boundary). The flow has a single
saddle-like rest point on the boundary of the surface, and orbits leaving a
neighborhood of this saddle follow either of two arms which return to (and are
glued together along) an interval of branch points transverse to the stable
manifold of the saddle. From the geometrical point of view, geometric Lorenz
attractors are flows on $3$ dimensional space that contain a rest point
accumulated by regular trajectories. Lorenz’s equations proved to be very
resistant to rigorous mathematical analysis, from both conceptual (existence
of a rest point accumulated by regular orbits prevents the attractor to be
hyperbolic) as well numerical (solutions slow down as they pass near the rest
point, which means unbounded return times and, thus, unbounded integration
errors) point of view. Moreover, for almost every pair of nearby initial, the
corresponding solutions move apart from each other exponentially fast as they
converge to the attractor, that is, the attractor is sensitive to initial
conditions. This unpredictability is a characteristic of chaos.

In the 90’s a breakthrough was obtained by Morales, Pacifico and Pujals, [18],
following the very original ideas developed by Mañé in the proof of the
$C^{1}$ stability conjecture: they provide a characterization of robustly
transitive attractors for 3-dimensional flows, of which the Lorenz attractor
is the more significant example: they are partially hyperbolic invariant sets
with volume expanding central direction. Moreover, robustly transitive
attractors without equilibria were proved to be hyperbolic. Thus these results
extend the classical hyperbolic theory for flows with isolated equilibria and
this characterization placed this class of attractors within the realm of a
weak form of hyperbolicity.

After this seminal work, significant advances in this theory for the
topological as well ergodic point of view where recently obtained through the
work of many authors, see [3] and references therein. Despite all this
progress, we still are far from a topological classification of singular
hyperbolic attracting sets in dimension $3$, and there is also a huge gap in
the understanding of unfolding parameterized families of singular hyperbolic
attractors, in any dimension.

The classical Lorenz attractor has a unique hyperbolic saddle singularity,
whose strong stable manifold $W^{ss}(\sigma)$ cuts the stable manifold
$W^{cs}(\sigma)$ in two _stable separatrices_ called the upper and lower
separatrices. One of these separatrices is _disjoint_ from the attractor.
These properties are shared by the geometrical Lorenz model.

This paper was motivated by a question of A. da Luz to the second author. She
wanted to know if a singular hyperbolic attractor, in dimension $3$,
containing a unique Lorenz-like singular saddle $\sigma$, could intersect
robustly both stable separatrices of $W^{cs}(\sigma)\setminus W^{ss}(\sigma)$.
There are many such examples which can be build easily, but one of the
simplest example display a very intriguing behavior under perturbation, and
that is what we want to present in this paper. This work aims to build an open
set of vector fields, for which the attractors have intersections with the
upper, lower and both stable separatrices.

We describe a $C^{1}$-open set ${\mathcal{O}}_{1}$ of vector fields on a
closed $3$-manifold having a common attracting region $U$ which contains a
unique saddle singularity $\sigma$ of Lorenz type, see Definition 2.3. As in
the classical geometric Lorenz attractor, each flow $X\in{\mathcal{O}}_{1}$
has a global cross section $\Sigma$, which is a topological annulus. The
intersection of the stable manifold of $\sigma$ with the cross section splits
$\Sigma$ into two connected components, $\Sigma^{1}$ and $\Sigma^{2},$ and the
intersection of the upper (lower) stable component of $W^{cs}(\sigma)$ contain
each a segment, $\gamma^{s}_{+}$ and $\gamma^{s}_{-}$, respectively,
transverse to the boundary $\partial\Sigma$ and connecting the two boundary
components of the annulus $\Sigma$. We denote the upper (lower) stable
component of $W^{cs}(\sigma)$ by $W^{s}_{+}(\sigma)$ ($W^{s}_{-}(\sigma)$ )
respectively. The first results give the topological nature of the class of
flows in $\mathcal{O}_{1}$, that intersect just one or both component of the
stable separatrices of $\sigma$, proving the existence of three disjointed and
non-empty sets in $\mathcal{O}_{1}:\,\,\mathcal{L}^{-}$, whose attractors
intersect just the lower component of $W^{cs}(\sigma)\setminus
W^{ss}(\sigma)$, $\mathcal{L}^{+}$, whose attractors intersect just the upper
component of $W^{cs}(\sigma)\setminus W^{ss}(\sigma)$ and $\mathcal{L}^{-,+}$,
whose attractors intersect both components of $W^{cs}(\sigma)\setminus
W^{ss}(\sigma)$. We call the attractors in $\mathcal{L}^{-}$ as down-Lorenz,
the attractors in $\mathcal{L}^{+}$ as up-Lorenz and the attractors in
$\mathcal{L}^{-,+}$ as two-sided Lorenz.

######  Theorem A.

Any vector field $X\in{\mathcal{L}}^{+}$ admits exactly $2$ chain recurrence
classes: one is a up-Lorenz attractor, and the other is a hyperbolic basic
set, topologically equivalent to the suspension of a fake horseshoe. The
symetric statement holds in ${\mathcal{L}}^{-}$, interchanging the up for the
down.

Here, recall that a fake horseshoe map consists of a sequence of operations on
the unit square, quite similar to the usual horseshoe, the only difference
being the way back to the square of the resulting folded rectangle: the bottom
of the folded rectangle fits back like the top of the starting square. See
Section 2.2 for a precise definition.

The topological nature of the attractors for flows in $\mathcal{L}^{+,-}$ is
described in the next result.

######  Theorem B.

For any $X\in{\mathcal{L}}^{+,-}$ the maximal invariant set in $U$ is a
transitive singular hyperbolic attractor meeting both stable separatrices of
$\sigma$, that is, it is a two-sided Lorenz attractor.

From our construction, we get that every
$X\in\mathcal{L}^{-,+}\cup\mathcal{L}^{+}\cup\mathcal{L^{-}}$ contains a
robustly attractor (that is, there exists an open neighborhood $\mathcal{U}$
of $X$ in the space of vector fields, such that every $Y\in\mathcal{U}$ has an
attractor). Besides that, although the complement of
$\mathcal{L}^{-,+}\cup\mathcal{L}^{+}\cup\mathcal{L^{-}}$ in $\mathcal{O}_{1}$
is nonempty it has empty interior. Thus, the next step is to understand the
reasons why an attractor stops intercepting just one component of
$W^{cs}(\sigma)\setminus W^{ss}(\sigma)$ and suddenly starts to intersect both
(and vice versa).

We will see that this phenomenon occurs due to the appearance of certain types
of homoclinic and heteroclinic connections caused by the displacement of the
intersection points $W^{u}(\sigma)\cap W^{cs}(\sigma)$ from one stable
separatrix of $W^{cs}(\sigma)\setminus W^{ss}(\sigma)$ to another. Recall that
the unstable manifold of $\sigma$ is one dimensional and
$W^{u}(\sigma)\setminus\\{\sigma\\}$ consists of two separatrices
$W^{u}_{i}(\sigma),\,1\leq i\leq 2.$ A homoclinic loop arises when the
intersection point $q_{i}$ of the unstable separatrix $W^{u}_{i}(\sigma)$ of
$\sigma$ belongs to $\gamma^{s}_{+}\cup\gamma^{s}_{-}$. Consider
${\mathcal{H}}^{i}\subset{\mathcal{O}}_{1}$, $i=1,2$, the hypersurface
corresponding to the vector fields $X$ for which the point $q_{i}$ (first
intersection with $\Sigma$ of the unstable separatrix $W^{u}_{i}(\sigma)$)
belongs to $\gamma^{s}_{+}\cup\gamma^{s}_{-}$. In other words, $X$ belongs to
${\mathcal{H}}^{i}$ if $\sigma$ admits a homoclinic loop for the unstable
separatrix $W^{u}_{i}(\sigma)$ so that the homoclinic loop cuts $\Sigma$ at a
unique point $q_{i}$. We split each ${\mathcal{H}}^{i}$ in two
${\mathcal{H}}^{i}={\mathcal{H}}^{i}_{+}\cup{\mathcal{H}}^{i}_{-}$ according
to $\\{q_{i}\in\gamma^{s}_{+}\\}$ and $\\{q_{i}\in\gamma^{s}_{-}\\}$
respectively. We then prove the following result, that characterizes the
topological nature of the attractors for flows in the hypersurfaces
${\mathcal{H}}^{i}_{+}$ and ${\mathcal{H}}^{i}_{-}$.

######  Theorem C.

For any $X\in{\mathcal{O}}_{1}$ in one of the hypersurfaces
${\mathcal{H}}^{1}$ or ${\mathcal{H}}^{2}$, the maximal invariant in $U$ is a
transitive singular attractor meeting both stable separatrices
$W^{s}_{\pm}(\sigma)$.

The statement of the theorem above does not announce two-sided Lorenz
attractor, because the hypersurface ${\mathcal{H}}^{i},i=1,2,$ is not an open
subset, and then the robustness of the transitivity in not ensured. However
Theorem D establishes that $X\in{\mathcal{H}}^{i}$ exhibits a two-sided Lorenz
attractor excepted for $X$ in a codimension $2$ submanifold.

There are still the case of a double homoclinic loop, that occurs when both
intersection points of the unstable manifold of $\sigma$ with the cross
section fall in the same segment of the intersection of the upper (lower)
stable component of $W^{cs}(\sigma)$ with the cross section. Let
${\mathcal{H}}^{1,2}_{+}$ and ${\mathcal{H}}^{1,2}_{-}$ the codimension $2$
submanifolds included in ${\mathcal{H}}^{1}\cup{\mathcal{H}}^{2}$ consisting
in vector fields $X$ so that $q_{1},q_{2}\in\gamma^{s}_{+}$ or
$q_{1},q_{2}\in\gamma^{s}_{-}$, respectively, see Figure 16. Thus both
unstable separatrices of $\sigma$ are homoclinic connections and are included
in the same stable separatrix of $\sigma$. Next result ensures that, for
$X\in{\mathcal{H}}^{i}$ out of ${\mathcal{H}}^{1,2}_{+}$ and
${\mathcal{H}}^{1,2}_{-}$ the transitivity of the attractor is robust so that
the attractor is a two-sided Lorenz attractor.

######  Theorem D.

If
$X\in{\mathcal{H}}^{i}\setminus({\mathcal{H}}^{1,2}_{+}\cup{\mathcal{H}}^{1,2}_{-})$,
there is a neighborhood ${\mathcal{U}}(X)$ of $X$ such that the maximal
invariant set for every $Y\in{\mathcal{U}}(X)$ is a two-sided Lorenz
attractor.

The appearance of a heteroclinic connection again is related to the position
of the intersections points $q_{1},\,q_{2}$ of $W^{u}_{i}(\sigma)$ with the
cross section, falling at a stable manifold of fixed points of the first
return map.

Recall that $\Sigma$ is splitted into two connected components $\Sigma^{1}$
and $\Sigma^{2}$, determined by the the intersections of $W^{cs}(\sigma)$ with
$\Sigma$. Assuming that the first return map has an expansion rate greater
then the golden number $\varphi$, when $q_{i}$ falls into $\Sigma^{j}$ with
$i\neq j$, the Poincaré map has a fixed point $p_{i}\in\Sigma^{j}$. Typically,
a heteroclinic cycle arises when $q_{i}$ belongs to the stable leaf
$W^{s}_{j}$ through $p_{j}$. We denote by ${\mathcal{O}}_{\varphi}$ the subset
of flows in ${\mathcal{O}}_{1}$ for which the first return map has an
expansion bigger that $\varphi$.

Let ${\mathcal{H}}{\mathcal{E}}_{i}\subset{\mathcal{O}}_{\varphi}$ be the
subset of flows corresponding to the vector fields $X$ for which $q_{i}\in
W^{s}_{j}$. In other words for $X\in{\mathcal{H}}{\mathcal{E}}_{i}$ the
unstable separatrix of $\sigma$ corresponding to $q_{i}$ is a heteroclinic
connection with $p_{j}$. The subsets ${\mathcal{H}}{\mathcal{E}}_{i}$ are
codimension $1$ submanifolds of ${\mathcal{O}}_{\varphi}$.

The case when both $q_{1}$ and $q_{2}$ belong to $W^{s}_{1}\cup W^{s}_{2}$
corresponds to
$X\in{\mathcal{H}}{\mathcal{E}}_{1}\cap{\mathcal{H}}{\mathcal{E}}_{2}$ which
is a codimension $2$ submanifold and we prove the following result:

######  Theorem E.

For any
$X\in{\mathcal{H}}{\mathcal{E}}_{1}\cap{\mathcal{H}}{\mathcal{E}}_{2}$, there
is a unique chain recurrence class, which is not transitive. Both $\Sigma_{-}$
and $\Sigma_{+}$ are invariant by $P$. The maximal invariant of the
restriction of $P$ to $\Sigma_{i}$ is transitive: every unstable segment in
$\Sigma_{i}$ has its iterates which cut every segment in $\Sigma_{i}$. The
open set $U$ splits in $2$ regions , each containing a _full Lorenz attractor_
(that is, a Lorenz crossing all stable foliations over each region splited on
$U$), and intersecting along $W^{s}(p_{1})\cup W^{s}(p_{2})\cup
W^{u}(\sigma)$. See Figure 17.

The goal now is to describe the drastic changes of the behavior appearing in
the topological dynamics when a family
$X_{\mu},\,\mu=(\mu_{1},\mu_{2})\in[-1,1]^{2},$ crosses the boundary of the
regions determined above. For flows in ${\mathcal{O}}_{\varphi}$ the first
return map $P$ has two fixed points $p_{1}$ and $p_{2}$, and the union of the
stable manifolds $W^{s}_{j}$ splits $\Sigma$ into two components $\Sigma^{+}$
and $\Sigma^{-}$, with $\gamma^{+}\subset\Sigma^{+}$ and
$\gamma^{-}\subset\Sigma^{-}$, and we can refine the analysis of the
topological behavior of these flows, based on the position of $q_{1}$ and
$q_{2}$ in $\Sigma^{\pm}$. This introduces a new region, denoted by
$\widetilde{{\mathcal{O}}_{\varphi}}^{+,+}$ and explicit defined at Section
4.5.

The behavior of the topological dynamics in all these regions is established
in the next theorem.

######  Theorem F.

With the previous notations, we obtain the following:

  * •

the vector field $X_{\mu}$ belongs to ${\mathcal{L}}^{+}$ iff $\mu$ belongs to
the quadrant $\mu_{1}>0,\mu_{2}>0$. In other words, the up full Lorenz
attractor of $X_{0,0}$ becomes a true (robust) Lorenz attractor, and the down
full Lorenz attractor of $X_{0,0}$ becomes a fake horseshoe.

  * •

the vector field $X_{\mu}$ belongs to ${\mathcal{L}}^{-}$ iff $\mu$ belongs to
the quadrant $\mu_{1}<0,\mu_{2}<0$. In other words, the up full Lorenz
attractor of $X_{0,0}$ becomes a true (robust) Lorenz attractor, and the down
full Lorenz attractor of $X_{0,0}$ becomes a fake horseshoe.

  * •

the vector field $X_{\mu}$ belongs to
$\widetilde{{\mathcal{O}}_{\varphi}}^{+,+}$ iff $\mu$ belongs to the quadrant
$\mu_{1}<0,\mu_{2}>0$ or to the quadrant $\mu_{1}>0,\mu_{2}<0$. In other
words, the two full Lorenz attractors of $X_{0,0}$ merge into a two-sided
Lorenz attractor.

Consider a $1$ parameter family $X_{\mu_{1},\alpha\mu_{1}}$ for some
$\alpha>0$. Then

  * •

for $\mu_{1}<0$, the vector fields admits a down Lorenz attractor and an up
fake horseshoe.

  * •

when $\mu_{1}=0$ the fake horseshoe become a full Lorenz attractor and,
simultanuously, enters in collision with the up Lorenz attractor, which
becomes a full Lorenz attractor

  * •

for $\mu_{1}>0$, the up full Lorenz attractor become a fake horseshoe when the
down full Lorenz attractor becomes a true robust Lorenz attractor.

Figure 1 below describes the main features stablished at Theorem F.

Figure 1: (a) Local bifurcation at
${\mathcal{H}}{\mathcal{E}}_{1}\cap{\mathcal{H}}{\mathcal{E}}_{2}$ and (b)
Global bifurcation

Next, we study deeper the topological nature of a flow in $\mathcal{O}_{1}$.
As for the classical geometric model of Lorenz attractor, we will see that to
any vector field in ${\mathcal{O}}_{1}$, we can associate a combinatorial
data, called the itinerary of the discontinuities. These itineraries induce,
in some sense, a semi-conjugacy of $(\Sigma,P)$ with
$({\mathbb{X}},\mathfrak{S})$, where $\mathfrak{S}$ is the shift map on
${\mathbb{X}}$, and ${\mathbb{X}}$ is an appropriated alphabet chosen to
represent the crossing points of the unstable manifold of the rest point with
the cross section. Being rigorous, $P$ is defined on
$\Sigma\setminus(\gamma^{s}_{+}\cup\gamma^{s}_{-})$ which is not invariant
under $P$. Thus we get conjugacies in the restriction of $\Sigma\setminus
W^{s}(\sigma)$. But, since the Poincaré map preserves a stable foliation and
the itineraries for the Poincaré map are functions of the stable leaf, they
passes to the quotient on the leaves space, inducing itineraries for the one
dimensional quotient map. We use this symbolic analysis to show that each
neighborhood of a flow in $\mathcal{O}_{1}$, in the $C^{1}$ topology, contains
flows whose attractor exhibits all of an uncountable family of topological
types. As in the case of the classical geometric Lorenz attractor, we prove
that this phenomenon is stable of co-dimension two, in the sense that the
topological types exhibited by various versions of this example can be
distinguished by two parameters, and in fact a two-parameter family of
examples can be constructed which is $C^{1}$ stable among such families.

######  Theorem G.

The restrictions to the maximal invariant sets of $X$ and
$Y\in\mathcal{O}_{\varphi}$ are topologically equivalent by a homeomorphism
close to identity if, and only if, $X$ and $Y$ have the same itineraries.

This paper is organized as follows. Section 2 gives the concepts, definition
and results proved elsewhere that will be needed in the text. In Section 3 we
construct the open set of flows ${\mathcal{O}}_{1}$ that will be analyzed in
the remained of the text. Section 4 gives the background and auxiliary results
to the proof of the main results. In Section 5 we prove theorems A, B, C, D
and E. In Section 6 we prove Theorems F and G.

##  2 Preliminaries

In this section we recall some definitions and results proved elsewhere that
will be used in this text.

###  2.1 Basic topological notions

Let $\mathfrak{X}^{r}(M)$, $\,r\geq 1,$ be the space of vector fields defined
on a compact boundary less manifold $M$, with the $C^{r}$ topology. For
$X\in\mathfrak{X}^{r}(M)$ , we denote by $X^{t}$ the flow of $X$. If $U\subset
M$, $\overline{U}$ denotes the closure of $U$, $\operatorname{int}U$ the
interior of $U$ and $\partial U$ the border of $U$.

An invariant compact set $\Lambda$ of $X$ is _transitive_ if there is
$p\in\Lambda$ so that its positive orbit is dense in $\Lambda$ (this notion is
called _topological ergodicity_ by some authors).

Recall that a sequence $\\{p_{i}\\}$ is a $\varepsilon$-pseudo-orbit for $X$
if there is $t_{i}>1$ such that $d(p_{i+1},X^{t_{i}}(p_{i}))<\varepsilon$, for
all $i$. A point $p$ is _chain recurrent_ if for each $\varepsilon>0$ there is
a $\varepsilon$-pseudo orbit $p=p_{0},p_{1},\dots,p_{n}=p$.

The chain recurrent set ${\mathcal{R}}(X)$ is the set of all chain recurrent
points. It is a compact invariant set.

An invariant compact set $\Lambda$ is _chain recurrent_ or _chain transitive_
if for every $\varepsilon>0$ it admits a $\varepsilon$-pseudo orbit
$\\{x_{i}\\}$ , $x_{i}\in\Lambda$, dense in $\Lambda$.

Two points $p,q$ are chain equivalent if for any $\varepsilon>0$ there are
$\varepsilon$-pseudo orbits from $p$ to $q$ and from $q$ to $p$. Conley theory
[7] proves that the chain-equivalence is a equivalence relation whose
equivalence classes are called the _chain recurrence classes_. They define a
partition of ${\mathcal{R}}(X)$ by invariant pairwise disjoint compact sets.

Given a region $U$ the maximal ivariant set in $U$ is the set $\Lambda_{U}$ of
points $p$ whose whole orbit is contained in $U$, that is:

$\Lambda_{U}=\bigcap_{t\in{\mathbb{R}}}X^{t}(U).$

A region $U$ is _attracting_ for a vector field $X$ of $M$ if its boundary is
a codimension $1$ submanifold transverse to $X$ and $X$ is entering in $U$.

######  Definition 2.1.

  * •

A compact invariant set $\Lambda_{X}$ of $X^{t}$ is _attracting_ if there
exists an open neighborhood $U\supset\Lambda_{X}$ which is an attracting
region and such that $\Lambda_{X}$ is the maximal invariant set in $U$, that
is:

$\Lambda_{X}=\underset{t>0}{\bigcap}X^{t}(U).$

  * •

We say that $\Lambda_{X}$ is _an attractor_ if it is attracting and chain
transitive (in particular, $\Lambda_{X}$ is a chain recurrence class).

  * •

An invariant compact set $K$ is a _quasi-attractor_ if it is a chain
recurrence class and admits a basis of neighborhoods which are attracting
regions: there are a decreasing sequence $U_{i}\subset U_{i-1}$ of attracting
regions so that $K=\bigcap_{i}U_{i}$.

Notice that the existence of an attractor is not ensured a priori, but Conley
proves the existence of quasi-attractors in any attracting region (for a
vector field on a compact manifold).

######  Definition 2.2.

Two vector fields $X$ and $Y$ defined on $M$ are topologically equivalent, if
there exists a homeomorphism $h:M\to M$ satisfying:

  1. 1.

$h(\mathcal{O}_{X}(p))=\mathcal{O}_{Y}(h(p))$.

  2. 2.

for all $p\in M$ and $\varepsilon>0$, there exists $\delta>0$ such that for
$t\in(0,\delta)$ there is $s\in(0,\varepsilon)$ satisfying
$h(X^{t}(p))=Y^{s}(h(p))$.

###  2.2 Fake horseshoe

The fake horseshoe map consists of a sequence of operations on the unit
square. First, stretch in the $y$ direction by more than a factor of two, then
compress in the $x$ direction by more than a factor of two. Finally, fold the
resulting rectangle and fit it back onto the square, overlapping at the top
and bottom, and not quite reaching the ends to the left and right (and with a
gap in the middle), as illustrated in the Figure 2. The only difference with
the usual horseshoe is the way back to the square of the resulting folded
rectangle: the bottom of the folded rectangle fits back like the top of the
starting square.

Figure 2:  The usual horseshoe and a fake horseshoe

###  2.3 Singular points

A point $\sigma$ is singular if $X$ vanishes at $\sigma$. The set of singular
points of $X$ is denoted by $Zero(X)$. A point $\sigma\in Zero(X)$ is
hyperbolic if the real part of the eigenvalues of $DX(\sigma)$ does not
vanish.

######  Definition 2.3.

A singularity $\sigma$ of $X$ is Lorenz-like if the eigenvalues
$\lambda_{i}\in\mathbb{R},\,i\in\\{ss,s,u\\},$ of $DX(\sigma)$ satisfy
$\lambda_{ss}<\lambda_{s}<0<-\lambda_{s}<\lambda_{u}<-\lambda_{s}$.

######  Definition 2.4.

In addition, $\sigma$ is non resonant, if for all $N>2$ and any choice of
nonnegative integer numbers $m_{1},m_{2},m_{3}$ with
$2\leq\overset{3}{\underset{j=1}{\sum}}m_{j}<N$ we have
$\overset{3}{\underset{j=1}{\sum}}m_{j}\lambda_{j}\neq\lambda_{i}$.

Hartman Grobman theorem asserts that, in a small neighborhood of a hyperbolic
singular point, a vector field is locally topologically equivalent to its
linear part. Then Sternberg provides conditions to guarantee that this local
topological equivalence is indeed of class $C^{2}$:

######  Theorem 2.5.

Let $X$ a vector field and let $n\in\mathbb{Z}^{+}$ be given. Then there
exists an integer $N=N(n)\geq 2$ such that: if $A$ is a real non-singular
$d\times d$ matrix with eigenvalues $\gamma_{1},\ldots,\gamma_{d}$ satisfying

$2\leq\overset{d}{\underset{j=1}{\sum}}m_{j}\lambda_{j}<N\,\,\,\mbox{e}\,\,\,\overset{d}{\underset{j=1}{\sum}}m_{j}\lambda_{j}\neq\lambda_{i}$

for any choice of nonnegative integers $m_{1},m_{2}\ldots m_{d}$ and if the
application $X(v)=A(v)+\psi(v)$ and $\psi$ is of class $C^{N}$ for small
$||v||$ with $\psi(0)=0$ e $\partial_{v}\psi(0)=0$; then there exists a
$C^{N}$ diffeomorphism from a neighborhood of $v=0$ to a neighborhood of
$\xi=0$ that define a topologic conjugation between $X$ and $A$.

Furthermore the linearizing diffeomorphisms depends continuously in the
$C^{2}$ topology from the vector field $X$, see for instance [20, Corollary in
the Appendix] which provides a stronger statement.

###  2.4 The shift map $\mathfrak{S}$

We consider ${\mathbb{X}}=\\{A_{0},A_{1},B_{0},B_{1}\\}^{\mathbb{N}}$ the
space of positive infinite words in the alphabet
$\\{A_{0},A_{1},B_{0},B_{1}\\}$. We endow the alphabet
$\\{A_{0},A_{1},B_{0},B_{1}\\}$ with the total order

$A_{0}<A_{1}<B_{0}<B_{1}<1.$

We endow ${\mathbb{X}}$ with the corresponding lexicographic order, that we
denote by $\prec$ (and $\preccurlyeq$ for the non-strict order).

The shift map $\mathfrak{S}:{\mathbb{X}}\to{\mathbb{X}}$ is defined as

$(T_{j})_{j\geq 0}\in{\mathbb{X}}\mapsto\mathfrak{S}(T_{j})_{j\geq
0}=(T_{j+1})_{j\geq 0}.$

We also define the  star map (denoted by $\star$) on ${\mathbb{X}}$ as
follows: given a sequence $w=(w_{0},w_{1},\cdots)\in{\mathbb{X}}$ and a letter
$L\in\\{A_{0},A_{1},B_{0},B_{1}\\}$

$L\star w\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}(L,w_{0},w_{1},\cdots).$

###  2.5 Hyperbolic notions

######  Definition 2.6.

Let $X$ be a vector field of a manifold $M$. A compact invariant set
$\Lambda\subset M\setminus Zero(X)$ is hyperbolic if the tangent bundle
$TM|_{\Lambda}$ over $\Lambda$ admits a splitting

$TM|\Lambda=E^{s}\oplus{\mathbb{R}}\cdot X\oplus E^{u}$

so that

  * •

the bundles $E^{s}$ and $E^{u}$ are continuous and invariant under the natural
action of the derivative of the flow.

  * •

there is $T>0$ so that for any point $p\in\Lambda$ and for any unit vectors
$u\in E^{s}(p)$ and $v\in E^{u}(p)$ one has

$\|DX^{T}(u)\|<\frac{1}{2}\,\,\mbox{ and }\,\,\|DX^{T}(v)\|>2$

(one says that $E^{s}$ is _uniformly contracted_ and $E^{u}$ is _uniformly
expanded_).

An hyperbolic invariant compact set $K$ is called an _hyperbolic basic set_ if
is transitive and admits a neighborhood on which it is the maximal invariant
set.

If $K$ is a hyperbolic set we denote $E^{cs}=E^{s}\oplus{\mathbb{R}}X$ and
$E^{cu}\oplus{\mathbb{R}}X$, and these bundles are called, respectively, the
weak stable and weak unstable bundles.

######  Definition 2.7.

A compact invariant set $\Lambda\subset M$ for $X^{t}$ is _partially
hyperbolic_ if there exist a continuous and $DX^{t}$-invariant splitting
$T_{\Lambda}M=E^{s}\oplus E^{cu}$ and constants $\lambda\in]0,1[,K>0$ such
that for all $x\in\Lambda$ and $t\geq 0$, the following inequalities hold:

  1. (a)

$||DX^{t}|_{E^{s}_{x}}||\cdot\|DX^{-t}|_{E^{cu}_{X_{t}(x)}}\|\leq
K\lambda^{t}\quad\mbox{(domination)};$

  2. (b)

$\|DX^{t}|_{E^{s}_{x}}\|\leq K\lambda^{t}\quad\mbox{(uniform contraction). }$

In addition, if $E^{cu}_{\Lambda}$ is volume expanding, that is,
$|\det(DX^{t}|_{E^{cu}_{x}})|>Ke^{\lambda t}$ for all $x\in\Lambda$ and $t\geq
0$, $\Lambda$ is, by definition, a singular hyperbolic set.

Note that any hyperbolic set is also partially hyperbolic.

Notice that if $\Lambda_{X}$ an invariant compact set disjoint from $Zero(X)$
is partially hyperbolic if and only if it is hyperbolic. In dimension $3$, if
$\Lambda_{X}$ is partially hyperbolic, then every singular point in
$\Lambda_{X}$ is a Lorenz-like singularity, see [18].

###  2.6 Invariant manifold and foliations

Let $\Lambda$ be a compact invariant set for the flow $X^{t}$ and
$p\in\Lambda$. The stable $W^{s}_{X}(p$ and unstable $W^{u}_{X}(p)$ sets at
$p$ are defined by

$\displaystyle W^{s}_{X}(p)$ $\displaystyle=$ $\displaystyle\\{q\in
M\,:\mbox{dist}(X^{t}(q),X^{t}(p))\underset{t\to+\infty}{\longrightarrow}0\\}$
$\displaystyle W^{u}_{X}(p)$ $\displaystyle=$ $\displaystyle\\{q\in
M\,:\mbox{dist}(X^{t}(q),X^{t}(p))\underset{t\to-\infty}{\longrightarrow}0\\}.$

If $\mathcal{O}=\mathcal{O}_{X}(p)\subset\Lambda$ denotes the orbit of $p\in
M$, the stable set of the orbit of $p$ is
$W^{s}_{X}(\mathcal{O})=\underset{t\in\mathbb{R}}{\bigcup}W^{s}(X^{t}(p))$.
Analogously, the unstable set of the orbit of $p$ is
$W^{u}_{X}(\mathcal{O})=\underset{t\in\mathbb{R}}{\bigcup}W^{u}(X^{t}(p))$.

If $\Lambda$ is a hyperbolic set and $p$ is a point in an orbit
${\mathcal{O}}$ in $\Lambda$, then $W^{s}(p)$ and $W^{s}({\mathcal{O}})$ are
manifolds of the same regularity as $X$, and are tangent (at $p$) to the
stable bundle $E^{s}(p)$ and $E^{cs}(p)$, respectively.

If $\Lambda$ is a partially hyperbolic set the stable sets of the points in
$\Lambda$ are submanifolds of the same regularity as $X$, tangent to $E^{s}$
and varying continuously with the point.

Assume now $U$ is an attracting region and that $\Lambda=\Lambda_{U}$, the
maximal invariant in $U$ is a partially hyperbolic attracting invariant
compact set. Let $E^{s}$ denote the stable bundle defined over $\Lambda$. Then
the bundle $E^{s}$ always extends in a unique way to $U$ as an invariant
bundle (still denoted by $E^{s}$) tangent to a topological foliation. Recently
Araùjo and Melbourne, [5], provided bunching conditions ensuring the
smoothness of the stable foliation, as stated below:

######  Theorem 2.8.

Let be $M$ differentiable riemannian manifold of dimension 3, $X$ a $C^{r}$
vector field, $U$ an attracting region and $\Lambda\subset M$ , maximal
invariant set in $U$, a partial hyperbolic attracting set. Let
$\\{W^{s}_{x}\\}$ denote the stable foliation in $U$.

Let $q\in[0,r]$ and suppose that there exists $t>0$ such that

$\|DX^{t}|_{E^{s}(x)}\|\cdot\|DX^{-t}|_{E^{cu}(X^{t}(x))}\|\cdot\|DX^{t}|_{E_{x}^{cu}}\|^{q}<1\quad\mbox{
for all $x\in\Lambda$}.$ (1)

Then the foliation $\\{W^{s}_{x}\\}$ is of class $C^{q}$.

###  2.7 Cone fields

Let $f:M\to M$ be a $C^{1}$ diffeomorphism and $T_{x}M=F^{u}(x)\oplus
F^{s}(x)$ a continuous splitting. We define the stable and unstable cone
fields of size $\gamma<1$ as

$\displaystyle C^{s}_{\gamma}(x)$ $\displaystyle=$
$\displaystyle\\{(v_{1},v_{2})\in F^{u}(x)\oplus
F^{s}(x)\,:\|v_{1}\|\leq\gamma\|v_{2}\|\\},$ $\displaystyle C^{s}_{\gamma}(x)$
$\displaystyle=$ $\displaystyle\\{(v_{1},v_{2})\in F^{u}(x)\oplus
F^{s}(x)\,:\|v_{2}\|\leq\gamma\|v_{1}\|\\}.$

We say that $C^{s}_{\gamma}$ (reciprocally $C^{u}_{\gamma}(x)$) is strictly
invariant by $Df^{-1}$ (reciprocally by $Df$) if there is $\alpha<1$ such that
$Df^{-1}(C^{s}_{\gamma}(f(x)))\subset C^{s}_{\alpha\gamma}(x)$ (reciprocally
$Df(C^{u}_{\gamma}(x))\subset C^{u}_{\alpha\gamma}(f(x))$).

##  3 An open set of singular hyperbolic flows

Le $M$ be a compact $3$-manifold.

We consider an open set ${\mathcal{O}}_{1}$ of vector fields $X$ on $M$ with
the following properties described in the two next sections.

###  3.1 Topological conditions

  1. 1.

$X$ has a Lorenz-like singularity $\sigma=\sigma(X)$ varying continuously with
$X$, and satisfying the Sternberg of non-resonance conditions. We denote by
$W^{s}_{+}(\sigma)$ and $W^{s}_{-}(\sigma)$ the connected components of
$W^{s}(\sigma)\setminus W^{ss}(\sigma)$.

  2. 2.

the vector field $X$ is transverse to $3$ embedded, disjoint annuli denoted by
$\Sigma$ , $D^{2}$ and $D^{1}$.

  3. 3.

for any point $p\in D^{i}$ there is $t(p)$ depending smoothly on $p$ so that
$X^{t(p)}(p)\in\Sigma$ and $X^{s}(p)\notin\Sigma\cup D^{1}\cup D^{2}$ for
$s\in]0,t(p)[$. In other words, $X^{t(p)}(p)$ is the first return of the orbit
of $p$ on $\Sigma\cup D^{1}\cup D^{2}$, and we denote it $R(p)=X^{t(p)}(p)$.

Note $R$ is a diffeomorphism from $D^{1}\cup D^{2}$ to its image in $\Sigma$.
In particular $R(D^{1})$ and $R(D^{2})$ are annuli. We require that each of
them is _an essential annulus_ in $\Sigma$ (that is, is not homotopic to a
point).

Figure 3: The starting flow

  4. 4.

$\Sigma\cap W^{s}_{+}(p)$ and $\Sigma\cap W^{s}_{-}(\sigma)$ contain each a
segment, $\gamma^{s}_{+}$ and $\gamma^{s}_{-}$, respectively, tranverse to the
boundary $\partial\Sigma$ and connecting the two boundary components of the
annulus $\Sigma$.

The positive orbits of points in $\gamma^{s}_{+}$ and $\gamma^{s}_{-}$ go
directly to the singular point and is disjoint from $\Sigma\cup D^{1}\cup
D^{2}$.

Then $\Sigma\setminus(\gamma^{s}_{+}\cup\gamma^{s}_{-})$ consists in two
components $\Sigma^{1}$ and $\Sigma^{2}.$ See Figure 3.

  1. 5.

For any $p\in\Sigma^{1}\cup\Sigma^{2}$ there is $t(p)$ depending smoothly on
$p$ so that $X^{t(p)}(p)\in D^{1}\cup D^{2}$ and $X^{s}(p)\notin\Sigma\cup
D^{1}\cup D^{2}$ for $s\in]0,t(p)[$. Thus, $X^{t(p)}(p)$ is the first return
of the orbit of $p$ on $\Sigma\cup D^{1}\cup D^{2}$, and we denote it
$S(p)=X^{t(p)}(p)$.

Using item 3) one gets that $P=R\circ
S\colon\Sigma\setminus(\gamma^{s}_{+}\cup\gamma^{s}_{-})\to\Sigma$ is the
first return map of $X$ on $\Sigma$.

Figure 4: (a) The cross-sections $\Sigma\cup D^{1}\cup D^{2}$, (b) The image
$P(\Sigma^{1}\cup\Sigma^{2})$.

  2. 6.

Let $W^{u}_{1}(\sigma)$ and $W^{u}_{2}(\sigma)$ denote the unstable
separatrices of $\sigma$, that is, the connected components of
$W^{u}(\sigma)\setminus\sigma$.

Then the interior of $D^{i}$, $i=1;2$ contains the first intersection point
$\widetilde{q}_{i}$ of $W^{u}_{i}(\sigma)$ with $\Sigma\cup D^{1}\cup D^{2}$.
Furthermore, $S(\Sigma^{i})\cup\\{\widetilde{q}_{i}\\}$ is a annulus which is
pinched at $\widetilde{q}_{i}$, and this pinched annulus is essential in
$D^{i}$ (see Figure 3(a)) : by pinched we means that, in a neighborhood of
$\widetilde{q}_{i}$ the set $S(\Sigma^{i})\cup\\{\widetilde{q}_{i}\\}$
consists in two cuspidal triangles, with cusp at $\widetilde{q}_{i}$ and
tangent at this point to the same line but oriented in opposite direction.

As a consequence of item 5), the closure of $P(\Sigma^{1}\cup\Sigma^{2})$
consists in $2$ parallel essential annuli in $\Sigma$, pinched at
$q_{i}=R(\widetilde{q}_{i})$. See Figure 3(b).

  1. 7.

As a consequence of the above items, there is an attracting region $U$
containing $\Sigma$, $D^{1}$, $D^{2}$, $\sigma$ so that every regular orbit in
$U$ crosses $\Sigma$. The maximal invariant set $\Lambda$ in $U$ is an
attracting invariant compact set (see Figure 4).

Figure 5: Attraction region for the flow.

###  3.2 Singular hyperbolic conditions

  1. 8.

The maximal invariant set $\Lambda$ in $U$ is singular hyperbolic with bundles
$E^{s}$ and $E^{cu}$. This is equivalent to request that the first return map
$P$ is hyperbolic. We will request more.

  2. 9.

There is a conefield ${\mathcal{C}}^{u}$ on the annulus
$\Sigma\simeq\SS^{1}\times[-1,1]$, transverse to the fibers
$\\{\theta\\}\times[-1,1]$, and strictly invariant by the derivative $DP$ of
$P$, and so that the vectors in ${\mathcal{C}}^{u}$ are uniformly expanded by
a factor $\lambda>1$.

The cone ${\mathcal{C}}^{u}(p)$ has two components: vectors cutting the fiber
in the positive, or negative orientation. We require that $DP$ preserves this
transverse orientation.

  3. 10.

there exists $t>0$ such that

$\|DX^{t}|_{E^{s}(x)}\|\cdot\|DX^{-t}|_{E^{cu}(X^{t}(x))}\|\cdot\|DX^{t}|_{E_{x}^{cu}}\|<1\quad\mbox{
for all $x\in\Lambda$}.$ (2)

As a consequence of item • ‣ 8.2) and Theorem 2.8 the stable foliation, well
defined in $U$ is of class $C^{1}$.

This foliation is not tangent to $\Sigma$. However, there is a $2$ dimensional
_center-stable foliation_ well defined on $U\setminus\sigma$, obtained by
considering the $X^{t}$ orbits of the stable foliation: this foliation is
$C^{1}$ too. This center stable foliation cuts the annulus $\Sigma$
transversely, along a $C^{1}$, one-dimensional foliation ${\mathcal{F}}^{s}$,
which is the stable foliation of the return map $P$. The segment
$\gamma^{s}_{+}$ and $\gamma^{s}_{-}$ are leaves of ${\mathcal{F}}^{s}$.

The foliation ${\mathcal{F}}^{s}$ is transverse to the unstable cone field
${\mathcal{C}}^{u}$. The leaves of the foliation ${\mathcal{F}}^{s}$ are
segments crossing $\Sigma$ (connecting the two boundary components of
$\Sigma$).

##  4 Topological dynamics: the attractor and the chain recurrence classes

###  4.1 Quasi-attractor, two-sided, up and down Lorenz attractors

######  Definition 4.1.

  * •

We say that $X\in{\mathcal{O}}_{1}$ exhibits a _two-sided(geometric model)of
Lorenz attractor_ if, for any $Y$ in a neighborhood of $X$ maximal invariant
set $\Lambda_{Y}$ in the attracting region $U$ is transitive and have a non-
trivial intersection with both stable separatrices $W^{s}_{+}(\sigma,Y)$ and
$W^{s}_{-}(\sigma,Y)$.

  * •

We say that $X$ exhibit a _up-Lorenz attractor_ if it admits a geometric model
of Lorenz attractor (in the usual meaning) disjoint from the component
$W^{s}_{-}$.

  * •

We say that $X$ exhibit a _down-Lorenz attractor_ if it admits a geometric
model of Lorenz attractor disjoint from the component $W^{s}_{+}$.

The aim of this section is to show that, under a certain condition on the
expansion in the unstable direction, the vector fields $X$ in an open and
dense subset exhibit either a two-sided or a up- or a down- Lorenz attractor,
and the $3$ cases appear.

###  4.2 Quasi-attractor

We started by noticing that there is non ambiguity on what can be the
attractor:

######  Proposition 4.2.

For any $X\in{\mathcal{O}}_{1}$ the chain recurrence class of the sigularity
$\sigma$ is the unique quasi attractor in the attracting region $U$

As $U$ is an attracting region, Conley theory implies that $U$ contains at
least one quase attractor. A quasi attractor admits by definition, arbitrarily
small invairant neighborhoods. Now Proposition 4.2 is a direct consequence of
nex lemma:

######  Lemma 4.3.

The stable manifold of $\sigma$ is dense in $U$.

This proof is identical to the proof of the similar statement for the
classical geometric model of Lorenz attractor, and is very simple. We include
it here for completeness.

###### Proof.

As any orbit cuts the cross-section $\Sigma$, we just need to prove that, for
any open set $O\subset\Sigma$ there is $n>0$ so that
$P^{n}(O)\cap(\gamma^{s}_{+}\cup\gamma^{s}_{+})\neq\emptyset$.

We consider a segment $S\subset O$ tangent to the unstable cone
${\mathcal{C}}^{u}$. By item 9) there is $\lambda>1$ so that the vector in
${\mathcal{C}}^{u}$ are expanded by a factor larger than $\lambda$. Thus:

  * •

either $S$ cuts $(\gamma^{s}_{+}\cup\gamma^{s}_{+})$ and we are done,

  * •

or $P(S)$ is a segment tangent to ${\mathcal{C}}^{u}$ whose length is at least
$\lambda$ times the length of $S$.

Iterating the process one gets that either there is $n$ so that $P^{n}(S)$
cuts $\gamma^{s}_{+}\cup\gamma^{s}_{+}$ or $P^{n}(S)$ is a segment tangent to
${\mathcal{C}}^{u})$ whose length thend to infinity. As $\gamma^{s}_{+}$ and
$\gamma^{s}_{+}$ are leaves of the foliation ${\mathcal{F}}^{s}$ which is a
fibration by segments on the annulus $\Sigma$, there is a bound for the length
of any segment in the unstable cone which do not cut every leaf, ending the
proof.

∎

###  4.3 Iterating unstable segments and transitive properties

We fix a vector field $X\in{\mathcal{O}}_{1}$, $U$ is the attracting region in
the definition, $\Sigma$ is the global cross section, and $P$ the first return
map.

First notice that the closures $\bar{P}^{n}(\Sigma)$, $n\geq 0$, is a nested
family of compact subsets of $\Sigma$. Let $\Lambda_{P}$ denote the
intersection of the $\bar{P}^{n}(\Sigma)$

$\Lambda_{P}=\bigcap_{n\geq 0}\bar{P}^{n}(\Sigma).$

######  Lemma 4.4.

The compact set $\Lambda_{P}$ is the intersection of the maximal invariant set
$\Lambda_{X}$ with $\Sigma$. Furthermore, $\Lambda_{X}$ is the union of
$\sigma$ with the saturated by the flow of $\Lambda_{P}$.

###### Proof.

Any orbit of $\Lambda_{X}$, but $\sigma$, cuts $\Sigma$. Any orbit but
$W^{u}(\sigma)$ cut $\Sigma$ for an infinite sequence of negative times
(tending to $-\infty$): indeed the $\alpha$-limit set of such an orbit is not
reduced to $\sigma$, and therefore cuts $\Sigma$.

Thus every orbit of $\Lambda_{X}$ not in $W^{u}(\sigma)$ cuts $\Sigma$ in
$\bigcap_{n\geq 0}P^{n}(\Sigma)$.

In $\Lambda_{P}$ we consider the closures $\bar{P}^{n}(\Sigma)$. This consists
in adding $\Sigma\cap W^{u}(\sigma)$ to $\bigcap_{n\geq 0}P^{n}(\Sigma)$. The
proof follows from these observations.

∎

Note that for every $n>0$, the closure $\bar{P}^{n}(\Sigma)$ consists in the
union of $2^{n}$ pinched essential annuli, each annulus admits a finite set of
pinched (cuspidal) points, the boundary is tangent to the unstable cone, and
the intersection of two annuli is contained in this finite set of pinched
points. Finally the thickness of this annuli is exponentially decreasing with
$n$.

For any nested sequence of such pinched annuli in $\bar{P}^{n}(\Sigma)$,
$n>0$, the intersection is an essential circle. In particular the topological
dimension of $\Lambda_{P}$ is $1$. Moreover the boundary of any annulus in
tangent to the unstable cone, and to the image by $DP^{n}$ of this unstable
cone. The intersection of the iterates of the unstable cones converges to a
well defined continuous, invariant line field $E^{u}$ on $\Lambda_{P}$. Each
of the circles is tangent to $E^{u}$.

######  Lemma 4.5.

Let $X$ be a vector field in ${\mathcal{O}}_{1}$ and $P$ the first return map
on the cross section $\Sigma$. If, for every segment $S\subset\Sigma$
transverse to the stable foliation ${\mathcal{F}}^{s}$, the union of stable
leaves through the iterates $P^{n}(S)$, $n\geq 0$, covers an open and dense
subset of $\Sigma$, then the maximal invariant set $\Lambda_{X}$ is
transitive.

One easily check that is enough to prove the following

######  Lemma 4.6.

Under the same hypotesis, given any non-empty open subsets $U\cap\Lambda_{P}$,
$V\cap\Lambda_{P}$ ($U$ and $V$ open sets of $\Sigma$) there is $n>0$ so that

$P^{n}(V\cap\Lambda_{P})\cap(U\cap\Lambda_{P})\neq\emptyset.$

###### Proof.

The open subset $U\cap\Lambda_{P}$ contains a segment $S_{U}$ tangent to the
unstable direction. There is $\varepsilon>0$ so that the segments of stable
leaves through $S_{U}$ (shrinking $S_{U}$ if necessary) are contained in $U$.
Let $W^{s}_{\varepsilon}(S_{U})$ be the union of these segments.

Note that for any $n>0$, $P^{-n}$ is defined on $S_{U}$ up to a finite set
(the first $n$ iterates of the unstable manifold of $\sigma$). Now
$P^{-n}(W^{s}_{\varepsilon}(S_{U}))$ contains the saturated by
${\mathcal{F}}^{s}$ of $P^{-n}(S_{U})$ which contains open segments in
$\Lambda_{P}$.

Consider now a segment $S_{V}$ contained in $V\cap\Lambda_{P}$. By assumtion
there is $m>0$ so that $P^{m}(S_{V})$ intersect stable leaves in
$P^{-n}(W^{s}_{\varepsilon}(S_{U}))$. In other words $P^{m+n}(S_{V})$ cuts
$W^{s}_{\varepsilon}(S_{U})$ at points which belong to $\Lambda_{\Sigma}$
(because $S_{V}$ is contained in $\Lambda_{\Sigma}$ which is positively
invariant by $P$). Thus these intersection points belong to
$U\cap\Lambda_{\Sigma}$ concluding the proof. ∎

We now present some tools ensuring that the orbit of an _unstable segment_
(i.e. a segment in $\Sigma$ tangent to the unstable cone ) cuts almost every
stable leaves.

######  Lemma 4.7.

Let $S$ be an unstable segment joining a cuspidal point $q_{i}$ in
$\bar{P}(\Sigma)$ to a point $q_{\pm}$ in $\gamma^{s}_{\pm}$ . Assume that
$q_{i}$ belongs to $\Sigma_{i}$.

Then there is $n>0$ so that the union of the iterates $P^{i}(S)$,
$i\in\\{0,\dots,n\\}$ cut every stable leaves but a finite number.

###### Proof.

The unstable cone is oriented, inducing an orientation over every unstable
segment. Assume for instance that $S$ is starting (for this orientation) at
$q_{i}\subset\Sigma_{1}$ and ends on $q_{-}\in\gamma^{s}_{-}$: in particular,
$S\subset\Sigma_{1}$. Now, $P(S)$ is an unstable segment ending at $q_{1}$ and
of length $\ell(P(S)\geq\lambda\ell(S)$.

So $S_{1}=P(S)\cup S$ is an unstable segment of length at least twice
$\ell(S)$ and ending at $q_{-}$.

We define by induction a finite sequence $S_{n}$ as follows: if $S_{n-1}$ is
contained in $\Sigma_{1}$ then $S_{n}=P(S_{n-1})\cup S$. Otherwize, the
sequence ends and $S_{n}$ is not defined. Thus for every $n$, $S_{n}$ is a
unstable segment ending at $q_{-}$ and of size at least $n$ times the size of
$S$. In particular, this sequence ends at some $n_{0}$, and $S_{n_{0}}$ is not
contained in $\Sigma_{1}$: it cuts $\gamma^{s}_{+}$.

Now $P(S_{n_{0}})$ cuts every stable leaf up to the one of $q_{1}$. ∎

######  Remark 4.8.

In Lemma 4.7 the unique stable leaf which may not intersects the iterates
$P^{n}(S)$, $n\geq 0$, is the leaf through $q_{1}$. Furthermore, if $q_{1}$ is
distinct from $q_{2}$, then further iterates of $S$ cut $q_{1}$, so that every
leaf intersect some iterate of $S$.

######  Lemma 4.9.

Assume now that $\Sigma_{i}$ contains a fixed point $p_{i}$ of $P$. Let $S$ be
an unstable segment whose interior cuts the stable leaf through $p_{i}$. Then
there is $n_{0}$ so that for any $n\geq n_{0}$, the iterate $P^{n}(S)$ cuts
every stable leaf but the one through $q_{i}$.

###### Proof.

By the inclination lemma (also known as $\lambda$-lemma), the positive
iterates $P^{n}(S)$ accumulate the unstable manifold $W^{u}(p_{i})$. Some of
these iterates will cross completely $\Sigma_{i}$ (crossing both
$\gamma^{s}_{+}$ and $\gamma^{s}_{-}$). Then the next iterate will cross the
whole essential pinched annulus $P(\Sigma_{i})$, ending the proof. ∎

###  4.4 Fixing the expansion rate larger than the golden number

For any flow $X\in{\mathcal{O}}_{1}$ the return map $P$ is defined on the
cross section $\Sigma$ but the two stable leaves $\gamma^{s}_{+}$ and
$\gamma^{s}_{-}$. So we get two rectangles and both of them are sent into
$\Sigma$ as a pinched essential annulus. Therefore the expansion rate
$\lambda$ by $P$ in the unstable cone cannot be required uniformly larger than
$2$. However we will see that we can require a uniform rate expansion
arbitrarily close to $2$. Figure 6 displays the main features of the return
map $P$.

Figure 6:

The proof of the main topological properties consists in iterating unstable
segments (tangent to the unstable cone) and to estimate the length of the
components of these iterates. In this section we will choose a rate $\lambda$
which will allow us to estimate these length.

Let us start with a very elementary observation: Consider a segment
$[a,c]\subset{\mathbb{R}}$ and pick $b\in]a,c[$. Then one of the length
$\lambda^{2}\ell([a,b])$ or $\lambda^{2}\ell([b,c])$ is strictly larger than
$\ell([a,c]$ if $\lambda>\sqrt{2}$. That is, the choice of $\lambda>\sqrt{2}$
ensures the increase of a segment splitted into two components after two
iterations, if those components are not split again. This is the traditional
rate of expansion to guarantee the transitivity of a Lorenz attractor.We note
that there are examples of one-dimensional Lorenz maps with rate of expansion
smaller than $\sqrt{2}$ that are not transitive. Note that this map has an
isolated periodic orbit. See Figure 7.

Figure 7: The map at left is transitive but not robustly, the map at right is
not transitive.

Consider now the maximum of the lengths $\lambda\ell([a,b])$ and
$\lambda^{2}\ell([b,c])$. Notice that the golden ration
$\varphi=\frac{1+\sqrt{5}}{2}\in]1,2[$.

######  Lemma 4.10.

For any $a<b<c$, for any $\lambda\geq\varphi$ on gets

$\max\\{\lambda\ell([a,b]),\lambda^{2}\ell([b,c])\\}\geq\frac{\lambda}{\varphi}\ell([a,c]).$

###### Proof.

Taking $\alpha=\frac{\ell([b,c])}{\ell([a,c])}$, we only have to show

$\max\\{(1-\alpha),\lambda\alpha\\}\geq\frac{1}{\varphi}.$

Case $\alpha\geq\frac{\varphi-1}{\varphi}$, we have
$\lambda\alpha\geq\lambda\left(\frac{\varphi-1}{\varphi}\right)\geq\varphi\left(\frac{\varphi-1}{\varphi}\right)\geq\frac{\varphi^{2}-\varphi}{\varphi}\geq\frac{1}{\varphi}$.
On the other hand, if $\alpha<\frac{\varphi-1}{\varphi}$, we get
$1-\varphi>1-\left(\frac{\varphi-1}{\varphi}\right)=\frac{1}{\varphi}$. ∎

We consider now the open subset
${\mathcal{O}}_{\varphi}\subset{\mathcal{O}}_{1}$ consisting of flows $X$ so
that the expansion rate $\lambda$ of the return map in the unstable cone is
larger than $\varphi$. See Section which explicits vector fields in
${\mathcal{O}}_{\varphi}$, showing that $O_{\varphi}$ is not empty.

###  4.5 Cutting ${\mathcal{O}}_{\varphi}$ in regions

Consider ${\mathcal{H}}^{i}\subset{\mathcal{O}}_{1}$, $i=1,2$, the subset
corresponding to the vector fields $X$ for which the point $q_{i}$ (first
intersection with $\Sigma$ of the unstable separatrix $W^{u}_{i}(\sigma)$)
belongs to $\gamma^{s}_{+}\cup\gamma^{s}_{-}$. In other words, $X$ belongs to
${\mathcal{H}}^{i}$ if $\sigma$ admits a homoclinic loop for the unstable
separatrix $W^{u}_{i}(\sigma)$ so that the homoclinic loop cuts $\Sigma$ at a
unique point $q_{i}$. We split each ${\mathcal{H}}^{i}$ is two
${\mathcal{H}}^{i}={\mathcal{H}}^{i}_{+}\cup{\mathcal{H}}^{i}_{-}$ according
to $\\{q_{i}\in\gamma^{s}_{+}\\}$ and $\\{q_{i}\in\gamma^{s}_{-}\\}$
respectively. Figure 8 displays the feature of a vector field
$X\in{\mathcal{H}}^{1}$.

It is well known that a homoclinic connection corresponds to a codimension $1$
phenomenon, as expressed below:

######  Lemma 4.11.

The subsets ${\mathcal{H}}^{i}$ are codimension $1$ submanifolds of
${\mathcal{O}}_{1}$.

Figure 8: (a) $X\in{\mathcal{H}}^{1}_{+}$ and (b) $X\in{\mathcal{H}}^{1}_{-}$.

The submanifolds ${\mathcal{H}}^{1}$ and ${\mathcal{H}}^{2}$ cut
${\mathcal{O}}_{1}$ in $4$ regions
${\mathcal{O}}_{\varphi}^{\omega_{1},\omega_{2}}$, $\omega_{i}\in\\{+,-\\}$,
so that $\omega_{i}=-$ if and only if $q_{i}\in\Sigma_{i}$. Then:

######  Lemma 4.12.

Let $X$ be a vector field in ${\mathcal{O}}_{1}$ out of ${\mathcal{H}}^{1}$
and ${\mathcal{H}}^{2}$. Then the first return map $P$ admits a fixed point in
$\Sigma_{1}$ (resp. $\Sigma_{2}$) if and only if $X$ belongs to
${\mathcal{O}}_{\varphi}^{+,-}\cup{\mathcal{O}}_{\varphi}^{+,+}$ (resp.
${\mathcal{O}}_{\varphi}^{-,+}\cup{\mathcal{O}}_{\varphi}^{+,+}$).

###### Proof.

If $q_{1}\in\Sigma_{2}$ this means that the cuspidal point of the piched
annulus $P(\Sigma_{1})$ belongs to $\Sigma_{2}$. Thus $P(\Sigma_{1})$ crosses
$\Sigma_{2}$ in a (hyperbolic) Markov way leading to a unique fixed point in
$\Sigma_{1}$, see Figure 9(a). If $q_{1}\in\Sigma_{1}$ then $\Sigma_{1}\cap
P(\Sigma_{1})$ has $2$ connected components which are cuspidal triangles
$T_{1}^{+}$ (bounded by $\gamma^{s}_{+}$) , $T_{1}^{-}$ (bounded by
$\gamma^{s}_{-}$). Assume that there is a fixed point $p$ in $T_{1}^{-}$ (for
instance, the other case being equivalent). Then the region $R$ bounded by
$\gamma^{s}_{+}$ and $W^{s}(p)$ is invariant by $P$, see Figure 9. But
$W^{u}(p)$ contains a segment joining $p$ to $\gamma^{s}_{+}$. This segment in
expanded by $P$ contradicting the invariance of $R$. This ends the proof.

Figure 9:
$X\in{\mathcal{O}}_{1}\setminus{\mathcal{H}}^{1}\cup{\mathcal{H}}^{2}$

∎

Consider now the region ${\mathcal{O}}_{\varphi}^{+,+}$ : every vector field
$X\in{\mathcal{O}}_{\varphi}^{+,+}$ has exactly $1$ fixed point $p_{1}$ in
$\Sigma_{1}$ and $1$ fixed point $p_{2}$ in $\Sigma_{2}$. Let $W^{s}_{i}$ be
the stable leaf through $p_{i}$.

Note that $q_{1}$ and $p_{2}$ are both in $\Sigma_{2}$, and in the same way
$q_{2}$ and $p_{1}$ are both in $\Sigma_{1}$. This remark allows, _a priori_ ,
that $q_{1}$ and $p_{2}$ belong to the same stable leaf, or that $q_{2}$ and
$p_{1}$ belong to the same stable leaf. This corresponds to our next splitting
of the region ${\mathcal{O}}_{\varphi}^{+,+}$.

Let us denote ${\mathcal{H}}{\mathcal{E}}_{i}$ the subset of
${\mathcal{O}}_{\varphi}^{+,+}$ corresponding to the vector fields $X$ for
which $q_{i}\in W^{s}_{j}$. In other words for
$X\in{\mathcal{H}}{\mathcal{E}}_{i}$ the unstable separatrix of $\sigma$
corresponding to $q_{i}$ is a heteroclinic connection with $p_{j}$, see Figure
10. As in Lemma 4.11 one has:

Figure 10: (a) $X\in{\mathcal{H}}{\mathcal{E}}^{1}$ and (b)
$X\in{\mathcal{H}}{\mathcal{E}}^{2}$

######  Lemma 4.13.

The subsets ${\mathcal{H}}{\mathcal{E}}_{i}$ are codimension $1$ submanifold
of ${\mathcal{O}}_{1}$.

Consider
$X\in{\mathcal{O}}_{\varphi}^{+,+}\setminus({\mathcal{H}}{\mathcal{E}}_{1}\cup{\mathcal{H}}{\mathcal{E}}_{2})$.
Then $\Sigma\setminus(W^{s}_{1}\cup W^{s}_{2})$ has exactly $2$ connected
components: one, denoted by $\Sigma_{+}$ contains $\gamma^{s}_{+}$, and the
other, denoted by $\Sigma_{-}$ contains $\gamma^{s}_{-}$.

We denoted by ${\mathcal{L}}^{+}$ the open subset of
${\mathcal{O}}_{\varphi}^{+,+}$ where both points $q_{1},q_{2}$ belong to
$\Sigma_{+}$ and ${\mathcal{L}}^{-}$ the open set where both points
$q_{1},q_{2}$ belong to $\Sigma_{-}$. We denote by
$\widetilde{{\mathcal{O}}_{\varphi}}^{+,+}$ the open subset where $q_{1}$ and
$q_{2}$ belong to different component $\Sigma_{\pm}$ and $\Sigma_{\mp}$.

We will denote ${\mathcal{L}}^{+,-}$ the union

${\mathcal{L}}^{+,-}={\mathcal{O}}_{\varphi}^{-,-}\cup{\mathcal{O}}_{\varphi}^{+,-}\cup{\mathcal{O}}_{\varphi}^{-,+}\cup\widetilde{{\mathcal{O}}_{\varphi}}^{+,+}\cup\left(({\mathcal{H}}^{1}\cup{\mathcal{H}}^{2})\setminus(({\mathcal{H}}^{1}_{+}\cap{\mathcal{H}}^{2}_{+})\cup({\mathcal{H}}^{1}_{-}\cap{\mathcal{H}}^{2}_{-})\right).$

Recall that ${\mathcal{H}}^{i}$ corresponds to a homoclinic loop and
${\mathcal{H}}^{i}_{\pm}$ distinguish the up of down stable seapratrix of
$\sigma$ involved in that loop.

##  5 The topological dynamics in the different regions of
${\mathcal{O}}_{\varphi}$

###  5.1 Up and down Lorenz attractor: the regions ${\mathcal{L}}^{+}$ and
${\mathcal{L}}^{-}$

The aim of this section is to prove

######  Theorem 5.1 (A).

With the notation of Section 4.5, any vector field $X\in{\mathcal{L}}^{+}$
admits exactly $2$ chain recurrence classes: one is a up-Lorenz attractor, and
the other is a hyperbolic basic set, topologically equivalent to the
suspension of a fake horseshoe.

The symetric statement holds in ${\mathcal{L}}^{-}$, interchanging the up and
down.

###### Proof.

The component $\Sigma_{+}$ is a rectangle. The stable leaf $\gamma^{s}_{+}$
cuts $\Sigma_{+}$ in two component. One is $\Sigma_{1}\cap\Sigma_{+}$ and is
bounded by $W^{s}_{1}=W^{s}(p_{1})$ and the other is
$\Sigma_{2}\cap\Sigma_{+}$ and is bounded by $W^{s}_{2}$. Note that $q_{1}$
belongs to $\Sigma_{2}\cap\Sigma_{+}$ and $q_{2}$ belongs to
$\Sigma_{1}\cap\Sigma_{+}$.

Consider a stable leaf $L_{1}\in\Sigma_{1}\cap\Sigma_{+}$ separating
$W^{s}_{1}$ from $q_{2}$, and a stable leaf $L_{2}\in\Sigma_{2}\cap\Sigma_{+}$
separating $W^{s}_{2}$ from $q_{1}$, see Figure 11.

Then $L_{1}\cup L_{2}$ cut $\Sigma^{+}$ in $3$ components: one is bounded by
$W^{s}_{1}$ another by $W^{s}_{2}$ and the third, denoted by $R_{L}$ is a
rectangle bounded by both $L_{1}$ and $L_{2}$ and containing $q_{2}$,$q_{1}$
and $\gamma^{s}_{+}$. The leaf $\gamma^{s}_{+}$ cuts $R_{L}$ in two
components, $R_{L,1}\subset\Sigma_{1}$ and $R_{L_{2}}\subset\Sigma_{2}$.

Consider the restriction of $P$ to $R_{L}\setminus\gamma^{+}_{s}$. The images
of $R_{L_{1}}$ and $R_{L_{2}}$ are cuspidal triangle with cusps at $q_{1}$ and
$q_{2}$, and contained in $R_{L}$. Recall that $P$ is hyperbolic. Thus the
restriction or $P$ to $R_{L}$ satisfies all the properties of the return map
in the geometric model of Lorenz attractor with a rate expansion larger than
$\varphi>\sqrt{2}$. The rectangle $R_{L}$ is an attracting region for $P$. On
deduces that $U$ contains a sub-region which is attracting, and in which $X$
is a geometric model of Lorenz attractor.

Figure 11: The leaves $L_{1}$ and $L_{2}$, and the region $\Sigma_{+}$.

Consider now the component $R_{H}$ of $\Sigma\setminus(L_{1}\cup L_{2})$
disjoint from $R_{L}$, and containing $\gamma^{s}_{-}$. Consider the
restriction of $P$ to $R_{H}\setminus\gamma^{s}_{-}$. The image of each of
these components crosses $R_{H}$ in a Markovian way. Thus the maximal
invariant in $R_{H}$ is far from the discontinuity, and is conjugated to the
_fake horseshoe_ , as the map $P$ preserves the orientation of the unstable
cone field. This ends the proof. The down case is similar. ∎

###  5.2 Two-sided Lorenz attractor in ${\mathcal{O}}_{\varphi}^{-,-}$

######  Proposition 5.2.

For any $X\in{\mathcal{O}}_{\varphi}^{-,-}$ the maximal invariant set in $U$
is transitive and consists in a two-sided Lorenz attractor.

###### Proof.

According to Lemma 4.9, for proving the transitivity of the maximal invariant
it is enough to prove that the iterates of any unstable segment $S$ in
$\Sigma$ cut all the stable leaves with a possible exception of a set with
empty interior, see Figure 12.

Figure 12: $q_{1}\in\Sigma_{1}$ and $q_{2}\in\Sigma_{2}$

So consider a unstable segment $S\subset\Sigma$. Consider the set of lengths
of the connected components of all positive iterates $P^{n}(S)$. If this set
of lengths is not bounded, then some component cuts every stable leaf and we
are done.

Otherwise, given any $\delta>1$, up to replace $S$ by a segment in one of its
iterates, one may assume that any connected component $S^{\prime}$ in any
iterate $P^{n}(S)$ has a length bounded by $\delta\ell(S)$. We now fix
$\delta\in]1,\frac{\lambda^{2}}{2}[$.

If $S$ is disjoint from $\gamma^{s}_{+}\cup\gamma^{s}_{-}$ then
$\ell(P(S))>\lambda\ell(S)>\delta\ell(S)$ contradicting the choice of $S$. So
$S$ cuts $\gamma^{s}_{+}$ or $\gamma^{s}_{-}$. If it cuts both, then it
crosses completely $\Sigma_{1}$ or $\Sigma_{2}$ so that $P(S)$ cuts all the
stable leaves but $1$, and we are done.

So we may assume that $S$ cuts exactly one of $\gamma^{s}_{+}$ or
$\gamma^{s}_{-}$, say $\gamma^{s}_{+}$, for instance. Let $S_{1}$ be a
component of $S\setminus\gamma^{s}_{+}$ with length larger or equal to
$\frac{1}{2}\ell(S)$. The length of $P(S_{1})$ is at least
$\frac{\lambda}{2}\ell(S)$.

If $P(S_{1})$ is disjoint from $\gamma^{s}_{+}\cup\gamma^{s}_{-}$ then

$\ell(P(S_{1})>\lambda\ell(S_{1})>\frac{\lambda^{2}}{2}\ell(S)>\delta\ell(S).$

contradicting the choice of $S$. So $P(S_{1})$ cuts $\gamma^{s}_{+}$ or
$\gamma^{s}_{-}$. Once again if it cuts both of them we are done, so one may
assume that $P(S_{1})$ cuts exactly one of $\gamma^{s}_{+}$ or
$\gamma^{s}_{-}$. Note that one of the end points of $P(S_{1})$ is a cuspidal
point $q_{i}\in\Sigma_{i}$ as $X\in{\mathcal{O}}_{1}^{-,-}$. Thus Lemma 4.7
applies and concludes the proof of the transitivity of the maximal invariant
set $\Lambda_{X}$.

Note that the compact set $\Lambda_{P}$ consists in a union of essential
circles in $\Sigma$ and therefore allways cuts $\gamma^{s}_{+}$ and
$\gamma^{s}_{-}$, so that the maximal invariant set $\Lambda_{X}$ intersect
non-trivially both stable separatrices $W^{s}_{+}(\sigma)$ and
$W^{s}_{-}(\sigma)$. Thus $\Lambda_{X}$ is a two-sided Lorenz attractor,
ending the proof. See Figure 12.

∎

###  5.3 Two-sided Lorenz attractor in ${\mathcal{O}}^{+,-}_{\varphi}$ and
${\mathcal{O}}^{-,+}_{\varphi}$

######  Proposition 5.3.

For any $X\in{\mathcal{O}}_{\varphi}^{+,-}\cup{\mathcal{O}}_{\varphi}^{-,+}$
the maximal invariant set in $U$ is transitive and consist in a two-sided
Lorenz attractor.

The proof of the proposition for $X\in{\mathcal{O}}_{\varphi}^{-,+}$ is
identical to the proof for $X\in{\mathcal{O}}_{\varphi}^{+,-}$ interchanging
the componentns $\Sigma_{1}$ and $\Sigma_{2}$, so we will write the proof only
for $X\in{\mathcal{O}}_{\varphi}^{+,-}$. See Figure 13.

###### Proof.

Recall that the rate of exapansion of $X$ is $\lambda>\varphi$.

Figure 13: $q_{1}\in\Sigma_{+}$ for $X\in{\mathcal{O}}^{+,-}$

As for Proposition 5.2 it is enough to prove that the iterates of any unstable
segment $S$ in $\Sigma$ cut all the stable leaves with a possible exception of
a set with empty interior. See Figure 13.

So consider a unstable segment $S\subset\Sigma$. Consider the set of lengths
of the connected components of all positive iterates $P^{n}(S)$. If this set
of lengths is not bounded, then some component cuts every stable leaf and we
are done.

Otherwise, given any $\delta>1$, up to replace $S$ by a segment in one of its
iterates, one may assume that any connected component $S^{\prime}$ in any
iterate $P^{n}(S)$ has a length bounded by $\delta\ell(S)$. We now fix
$\delta\in]1,\frac{\lambda}{\varphi}[$.

If $S$ is disjoint from $\gamma^{s}_{+}\cup\gamma^{s}_{-}$ then
$\ell(P(S)>\lambda\ell(S)>\delta\ell(S)$ contradicting the choice of $S$. So
$S$ cuts $\gamma^{s}_{+}$ or $\gamma^{s}_{-}$. If it cuts both, then it
crosses completely $\Sigma_{1}$ or $\Sigma_{2}$ so that $P(S)$ cuts all the
stable leaves but $1$, and we are done.

So we may assume that $S$ cuts exactly one of $\gamma^{s}_{+}$ or
$\gamma^{s}_{-}$, say $\gamma^{s}_{-}$, for instance. Thus $S$ is cutted by
$\gamma^{s}_{-}$ in two components $S_{i}=S\cap\Sigma_{i}$.

Consider $P(S_{2})$. If
$P(S_{2})\cap(\gamma^{s}_{-}\cup\gamma^{s}_{+})\neq\emptyset$ then Lemma 4.7
applies (because $q_{2}\in\Sigma_{2}$ by our assumption
$X\in{\mathcal{O}}^{+,-}_{\varphi}$). Thus the iterate of $S_{2}$ cuts every
stable leaf but a finite number of them, so that we are done.

Thus we may assume that $P(S_{2})$ is disjoint from
$\gamma^{s}_{-}\cup\gamma^{s}_{+}$. Thus $P(S_{2})$ is an unstable segment of
length larger than $\lambda^{2}\ell(S_{2})$. On the other hand
$\ell(P(S_{1}))\geq\lambda\ell(S_{1})$. Now Lemma 4.10 implies that

$\max\\{\ell(P(S_{1})),\ell(P^{2}(S_{2}))\\}\geq\frac{\lambda}{\varphi}\ell(S)>\delta\ell(S),$

contradicting the choice of the segment $S$, finishing the proof.

∎

###  5.4 Two-sided Lorenz attractor in
$\widetilde{{\mathcal{O}}_{\varphi}}^{+,+}$

######  Proposition 5.4.

For any $X\in\widetilde{{\mathcal{O}}_{\varphi}}^{+,+}$ the maximal invariant
set in $U$ is transitive and consist in a two-sided Lorenz attractor.

Vector fields $X$ in $\widetilde{{\mathcal{O}}_{\varphi}}^{+,+}$ are
characterized by the fact that the points $q_{1}$ and $q_{2}$ belong to
different components $\Sigma_{+}$ (containing $\gamma^{s}_{+}$) and
$\Sigma_{-}$ (containing $\gamma^{s}_{-}$) of $\Sigma\setminus(W^{s}_{1}\cup
W^{s}_{2})$ where $W^{s}_{i}=W^{s}(p_{i})$ and $p_{i}$ is the fixed point of
$P$ in $\Sigma_{i}$.

Thus $\widetilde{{\mathcal{O}}_{\varphi}}^{+,+}$ is the union of two disjoint
open subsets, defined by $q_{1}\in\Sigma_{+}$ or $q_{1}\in\Sigma_{-}$.

The proof of the proposition is symetrical in these two open sets. We provide
here the proof where $q_{1}\in\Sigma_{+}$. See Figure 14.

Figure 14: $q_{1}\in\Sigma_{+}$ for
$X\in\widetilde{{\mathcal{O}}_{\varphi}}^{+,+}$

###### Proof.

Most of the proof is identical to the proofs of Propositions 5.2 and 5.3, and
allow us to consider a segment $S$ so that any component in any iterate
$P^{n}(S)$ has a length bounded by $\delta\ell(S)$ with
$1<\delta<\frac{\lambda}{\varphi}$ where $\lambda$ is the expansion rate of
$X$. Furthermore $S$ cuts exactly one of the stable leaves $\gamma^{s}_{+}$ or
$\gamma^{s}_{-}$.

Let us assume that $S$ cuts $\gamma^{s}_{-}$, and denote
$S_{i}=S\cap\Sigma_{i}$. If one of $P(S_{1})$ or $P(S_{2})$ is disjoint from
$\gamma^{s}_{+}\cup\gamma^{s}_{-}$ then one concludes in the same way as in
the proof of Proposition 5.3 that, using Lemma 4.10 one of the iterates
$P(S_{1})$, $P^{2}(S_{1})$, $P(S_{2})$ or $P^{2}(S_{2})$ contains a segment of
length larger than $\frac{\lambda}{\varphi}$ contradicting the choice of $S$.

Thus one may assume that both $P(S_{1})$ and $P(S_{2})$ cut
$\gamma^{s}_{+}\cup\gamma^{s}_{-}$.

For the orientation of ${\mathcal{C}}^{u}$, $S_{1}$ has its end point at
$\gamma^{s}_{-}$. Thus $P(S_{1})$ ends at $q_{1}\in\Sigma_{+}$. As seen above
$P(S_{1})$ cuts $\gamma^{s}_{+}\cup\gamma^{s}_{-}$ and its orientation implies
that indeed it cuts $\gamma^{s}_{-}$. In particular its goes out of
$\Sigma_{+}$. One deduces that $P(S_{1})$ cuts transversely
$W^{s}_{2}=W^{s}(p_{2})$, where $p_{2}$ is the fixed point in $\Sigma_{2}$
(recall that $X\in{\mathcal{O}}_{\varphi}^{+,+}$). Thus Lemma 4.9 ensures that
the iterates of $P(S_{2})$ cross all stable leaves but a finite number of
them, concluding that case.

The case where $S$ cuts $\gamma^{s}_{+}$ is similar, just replacing
$S_{1}=S\cap\Sigma_{1}$ and $W^{s}(p_{2})$ by $S_{2}=S\cap\Sigma_{2}$ and
$W^{s}(p_{1})$. This concludes the proof.

∎

###  5.5 Two-sided Lorenz attractor in hypersufaces ${\mathcal{H}}^{i}$ of
homoclinic loops

######  Proposition 5.5.

For any $X\in{\mathcal{O}}_{1}$ in one of the hypersurfaces
${\mathcal{H}}^{1}$ or ${\mathcal{H}}^{2}$, the maximal invariant in $U$ is a
transitive singular attractor meeting both stable separatrices
$W^{s}_{\pm}(\sigma)$.

###### Proof.

We assume that one of the points $q_{1}$,$q_{2}$, say $q_{1}$ belongs to one
of the stable leaves $\gamma^{s}_{+}$ or $\gamma^{s}_{-}$, say
$q_{1}\in\gamma^{s}_{+}$ (all the cases admits an identical proof, _mutatis
mutandi_). See Figure 15.

Figure 15: $X\in{\mathcal{H}}^{1}\cup{\mathcal{H}}^{2}$

In a similar way to the proof of the Propositions 5.2, 5.3 and 5.4, the proof
consist in considering an unstable segment $S$ which does not admit any
segment $\tilde{S}$ of length larger than $\delta\ell(S)$,
$1<\delta<\frac{\lambda}{\varphi}$, so that $\tilde{S}$, excepted a finite
subset, is contained in the union of the iterate $P^{n}(S)$, $n\geq 0$. One
needs to prove that the iterates of such a segment $S$ cut any stable leaves
but a finite number of them.

Again, the choice of $S$ implies that $S$ cuts $\gamma^{s}_{+}$ or
$\gamma^{s}_{-}$ and if it cuts both then $P(S)$ already cuts all stable
leaves but one. So we assume that $S$ cuts only $1$ of these leaves.

Assume first that $S$ cuts $\gamma^{s}_{+}$. Consider $S_{1}=S\cap\Sigma_{1}$.
It is a segment starting at a point in $\gamma^{s}_{+}$ and contained in
$\Sigma_{1}$. Then $P(S_{1})$ is a segment a length larger than
$\lambda\ell(S)$ and starting at $q_{1}\in\gamma^{s}_{+}$. If $P(S_{1})$ is
not included in $\Sigma_{1}$ then it cuts both $\gamma^{s}_{+}$ and
$\gamma^{s}_{-}$ and $P^{2}(S_{1})$ cuts all leaves but one. If
$P(S^{1})\subset\Sigma_{1}$ then $P^{2}(S_{1})$ is a segment of length larger
than $\lambda^{2}\ell(S)$ and starting at $q_{1}$.

Iterating the process one gets that one of the iterates $P^{k}(S_{1})$ crosses
completely $\Sigma_{1}$, so that $P^{k+1}(S_{1})$ cuts all stable leaves but
one, and we are done.

Assume now that $S$ cuts $\gamma^{s}_{-}$. Consider $S_{i}=S\cap\Sigma_{i}$.
Then $P(S_{1})$ is a segment ending at $q_{1}\in\gamma^{s}_{+}$, and
$P(S_{1})$ either crosses completely $\Sigma_{2}$ (and we are done) or is
contained in $\Sigma_{2}$. Now $P^{2}(S_{1})$ is a segment ending at $q_{2}$.
On the other hand $P(S_{2})$ is a segment staring at $q_{2}$. Now
$P^{2}(S_{1})\cup\\{q_{2}\\}\cup P(S_{2})$ is a segment of length at least
$\lambda\ell(S)$. This contradicts the choice of $S$, ending the proof. ∎

The statement of Proposition 5.5 does not announce two-sided Lorenz attractor,
because the ${\mathcal{H}}^{i}$ are not open subsets, and then the robustness
of the transitivity in not ensured. However we will see below that
$X\in{\mathcal{H}}^{i}$ exhibits indeed a two-sided Lorenz attractor excepted
for $X$ in a codimension $2$ submanifold.

Let ${\mathcal{H}}^{1,2}_{+}$ and ${\mathcal{H}}^{1,2}_{-}$ the codimension
$2$ submanifolds included in ${\mathcal{H}}^{1}\cup{\mathcal{H}}^{2}$
consisting in vector fields $X$ so that $q_{1},q_{2}\in\gamma^{s}_{+}$ or
$q_{1},q_{2}\in\gamma^{s}_{-}$, repectively, see Figure 16.Thus both unstable
separatrices of $\sigma$ are homoclinic connections and are included in the
same stable separatrix of $\sigma$.

Figure 16: $X\in{\mathcal{H}}^{1,2}_{+}\cup{\mathcal{H}}^{1,2}_{-}$

Next lemma ensures that, for $X\in{\mathcal{H}}^{i}$ out of
${\mathcal{H}}^{1,2}_{+}$ and ${\mathcal{H}}^{1,2}_{-}$ the transitivity of
the attractor is robust so that the attractor is a two-sided Lorenz attractor.

######  Lemma 5.6.

If
$X\in{\mathcal{H}}^{i}\setminus({\mathcal{H}}^{1,2}_{+}\cup{\mathcal{H}}^{1,2}_{-})$,
a neighborhood of $X$ is contained in
${\mathcal{O}}^{-,-}_{\varphi}\cup{\mathcal{O}}^{+,-}_{\varphi}\cup{\mathcal{O}}^{-,+}_{\varphi}\cup\widetilde{{\mathcal{O}}^{+,+}_{\varphi}}\cup{\mathcal{H}}^{1}\cup{\mathcal{H}}^{2}$.

###### Proof.

The proof consists in unfolding the homoclinic loops and check that all the
possibilities lead to one of the sets listed. ∎

###  5.6 Two-sided Lorenz attractor in ${\mathcal{L}}^{+-}$

Note that Propositions 5.2, 5.3, 5.4 and 5.5 and Lemma 5.6 together prove

######  Proposition 5.7.

For any $X\in{\mathcal{L}}^{+,-}$ the maximal invariant set in $U$ is a
transitive singular hyperbolic attractor meeting both stable separatrices of
$\sigma$, that is, is a two-sided Lorenz attractor.

The region ${\mathcal{L}}^{+,-}$ has been defined as a union of many regions
and the propositions listed above prove the conclusion of Proposition 5.7 in
each of these regions. Finally, some of these regions are not open so Lemma
5.6 checks that the vector fields in these non-open regions admits
neighborhood contained it the union of the other regions.

###  5.7 The collisions of the Lorenz attractor and a fake horseshoe: vector
fields in ${\mathcal{H}}{\mathcal{E}}_{i}$

In this section we consider vector fields $X$ in
${\mathcal{O}}^{+,+}_{\varphi}$, that is, so that the return map $P$ have $2$
fixed point $p_{i}\in\Sigma_{i}$, $i=1,2$, and a heteroclinic connections
between $\sigma$ and one of the points $p_{i}$, more precisely $q_{i}$ belongs
to the stable leaf $W^{s}_{j}$ through $p_{j}$ ; note that $j\neq i$ because
$q_{i}$ is not in $\Sigma_{i}$ when $\Sigma_{i}$ contains a fixed point.

The case when both $q_{1}$ and $q_{2}$ belong to $W^{s}_{1}\cup W^{s}_{2}$
corresponds to
$X\in{\mathcal{H}}{\mathcal{E}}_{1}\cap{\mathcal{H}}{\mathcal{E}}_{2}$ which
is a codimension $2$ submanifold.

######  Proposition 5.8.

For any
$X\in{\mathcal{H}}{\mathcal{E}}_{1}\cap{\mathcal{H}}{\mathcal{E}}_{2}$, there
is a unique chain recurrence class, which is not transitive. Both $\Sigma_{-}$
and $\Sigma_{+}$ are invariant by $P$. The maximal invariant of the
restriction of $P$ to $\Sigma_{i}$ is transitive: every unstable segment in
$\Sigma_{i}$ has its iterates which cut every segment in $\Sigma_{i}$.

Figure 17:
$X\in{\mathcal{H}}{\mathcal{E}}_{1}\cap{\mathcal{H}}{\mathcal{E}}_{2}$

The open set $U$ splits in $2$ regions , each containing a _full Lorenz
attractor_ (that is, a Lorenz crossing all stable foliations over each region
splited on $U$), and intersecting along $W^{s}(p_{1})\cup W^{s}(p_{2})\cup
W^{u}(\sigma)$. See Figure 17.

###### Proof.

The first return map is illustrated in Figure… The study of the first return
map in each rectangle $\overline{\Sigma}_{i}$ is classical for the expansion
rate larger than $\sqrt{2}<\varphi$. ∎

The submanifold
${\mathcal{H}}{\mathcal{E}}_{1}\cap{\mathcal{H}}{\mathcal{E}}_{2}$ cuts
${\mathcal{H}}{\mathcal{E}}_{1}$ in two (relative) open subsets, as $q_{2}$
belongs either to $\Sigma_{+}$ or $\Sigma_{-}$ (connected components of
$\Sigma\setminus(W^{s}_{1}\cup W^{s}_{2})$.

Consider
$X\in{\mathcal{H}}{\mathcal{E}}_{1}\setminus({\mathcal{H}}{\mathcal{E}}_{1}\cap{\mathcal{H}}{\mathcal{E}}_{2})$,
and assume $q_{2}\in\Sigma_{+}$. Then the stable leaf through $q_{2}$ cuts
$\Sigma_{+}$ in two components, one, denoted as $\Sigma^{+}_{1}$ bounded by
the stable leaf $W^{s}_{1}$ trough $p_{1}$ and the other, denoted as
$\Sigma^{+}_{2}$ bounded by $W^{s}_{2}$ (which contains $q_{1}$ by definition
of ${\mathcal{H}}{\mathcal{E}}_{1}$); notice that $\Sigma^{+}_{2}$ contains
the stable leaf $\gamma^{s}_{+}$ , because $q_{2}\in\Sigma_{1}$.

######  Proposition 5.9.

Consider
$X\in{\mathcal{H}}{\mathcal{E}}_{1}\setminus({\mathcal{H}}{\mathcal{E}}_{1}\cap{\mathcal{H}}{\mathcal{E}}_{2})$,
and assume $q_{2}\in\Sigma_{+}$. Then :

  * •

$X$ has a unique chain recurrence class in $U$, but this class is not
transitive and consists in two (singular) homoclinic classes $K_{-},K_{+}$
containing $\sigma$.

  * •

The iterates by $P$ of any unstable segment in $\Sigma$ cut any stable leaf in
$\Sigma^{+}_{2}$ but finitely many of them. Further more the return map $P$
restricted to the rectangle $\overline{\Sigma^{+}_{2}}$ is a classical Lorenz
map (for the geometrical model of Lorenz attractor) for the parameter
corresponding to one homoclinic connection. See Figure 18.

Figure 18:

  * •

The homoclinic class $K_{-}$ is a _singular fake horseshoe_ which intersects
$\Sigma_{-}$ and is disjoint from $\Sigma_{+}$.

  * •

the intersection $K_{-}\cap K_{+}\cap\Sigma$ is contained in $W^{s}_{2}$ and
consists in $p_{2}$ and the orbit of $q_{1}$.

###### Proof.

We just refer the reader to the (very classical) pictures illustrating the
restriction of the return map $P$ to $\Sigma^{+}_{2}$ and to
$\Sigma\setminus\Sigma^{+}_{2}$. See Figure 18. ∎

A similar result holds for
$X\in\mathcal{HE}_{2}\setminus(\mathcal{HE}_{1}\cap\mathcal{HE}_{2})$ and
$q_{1}\in\Sigma_{-}$.

###  5.8 Fat Lorenz attractor: vector fields in ${\mathcal{H}}^{1,2}_{+}$ and
${\mathcal{H}}^{1,2}_{-}$

######  Proposition 5.10.

If $X\in{\mathcal{H}}^{1,2}_{+}$ then the iterates of every unstable segment
in $\Sigma\setminus\gamma^{s}_{+}$ cut every stable leaf excepted
$\gamma^{s}_{+}$. The maximal invariant set in $U$ is a homoclinic class which
is a singular attractor, that we call the fat Lorenz attractor.

A similar result holds for $X\in\mathcal{H}_{-}^{1,2}$, interchanging
$\gamma_{+}^{s}$ by $\gamma_{-}^{s}$.

###### Proof.

We just refer the reader to the pictures illustrating the return map $P$,
Figures 15 and 16. ∎

##  6 Collisions and collapses

The aim of this section is to describe the drastic changes of behaviour
appearing in the topological dynamics when a family $X_{\mu}$ crosses the
boundary of the regions we described in Section 5.

###  6.1 Collisions

Let us consider a $1$ parameter family
$X_{\mu}\in{\mathcal{O}}^{+,+}_{\varphi}$, $\mu\in[-1,1]$ with the following
properties:

  * •

The family is crossing transversely the hypersurface
${\mathcal{H}}{\mathcal{E}}^{1}_{+}$ at $\mu=0$: this means that $X_{0}$
exhibit a heteroclinic connection $q_{1}\in W^{s}_{2}$ and
$q_{2}\in\Sigma_{+}$.

  * •

for $\mu<0$ the vector field $X_{\mu}$ belongs to ${\mathcal{L}}^{+}$: it has
a up Lorenz attractor and $\Sigma_{-}$ contains a fake horseshoe for $P$.

  * •

for $\mu>0$ the vector field $X_{\mu}$ has a two-sided Lorenz attractor.

In other words, $X_{\mu}$ consists in a generic unfolding of the heteroclinic
connection $q_{1}\in W^{s}_{2}$. The cuspidal point $q_{1,\mu}$ is moving with
the parameter $\mu$ and is crossing transversely $W^{s}_{2,\mu}$ (stable leaf
through $p_{2,\mu}$) for $\mu=0$.

For $\mu<0$ , the up-Lorenz attractor is intersecting exactly every stable
leaf in $\Sigma^{+}$ between $q_{2,\mu}$ and $q_{1,\mu}$, and the horseshoe in
$\Sigma_{-}$ is bounded by the stable leaves $W^{s}_{1,\mu}$, $W^{s}_{2,\mu}$
and contains the periodic points $p_{1,\mu}$ $p_{2,\mu}$.

When $\mu$ tends to $0$ the point $q_{1,\mu}$ tends to a point $q_{1,0}$ in
$W^{s}_{2}$ and this point $q_{1,0}$ is also the limite of the intersection of
$W^{s}_{2}$ with one of the rectangle (the one containing $p_{1}$) of the
Markov partition of the fake horseshoe. This corresponds to a Cantor set in
$W^{s}_{2}$ of points of the fake horseshoe for $X_{\mu}$, $\mu<0$ whose
diameter tends to $0$ as $\mu\to 0$: all the Cantor set tends to $q_{1,0}$.

For the parameter $0$, the Lorenz attractor no more admits an attracting
neighborhood, and is no more robust. The fake horseshoe became singular, and
intersects the Lorenz attractor along $\sigma$ and the orbits of $p_{2,0}$ and
of $q_{1,0}$.

When $\mu>0$ the Lorenz attractor and the (singular) fake horseshoe merge into
a two-sided Lorenz attractor.

Notice that if all the vector fields $X_{\mu}$ are assumed to be of class
$C^{2}$ then all along the parameter, $X_{\mu}$ admits a unique SRB-measure
$\nu_{\mu}$ (see [2]). For $\mu\leq 0$ the support of $\nu_{\mu}$ is the
Lorenz attractor, and in particular does not intersect $\Sigma_{-}$. For
$\mu>0$ the support of $\nu_{\mu}$ intersects any stable leaf in $\Sigma$.

Thus the support of $\nu_{\mu}$ change drastically at the collision point.

######  Question 1.

Is the map $\mu\mapsto\nu_{\mu}$ continuous (for the weak topology) at
$\mu=0$?

###  6.2 Collapse of the horseshoe: parameter families crossing
${\mathcal{H}}^{1,2}_{+}$ or ${\mathcal{H}}^{1,2}_{-}$

Recall that ${\mathcal{H}}^{1,2}_{+}$ is the codimension $2$ submanifold
corresponding to the double homoclinic connection
$q_{1},q_{2}\in\gamma^{s}_{+}$. We consider here a $2$-parameter family
$X_{\mu}$, $\mu=(\mu_{1},\mu_{2})\in[-1,1]^{2}$ which is a generic unfolding
of this double homoclinic connection: in other words, the family cuts
transversely ${\mathcal{H}}^{1,2}_{+}$ at $\mu=(0,0)$. We assume furthermore
that, for any fixed $\mu_{2}$, the $1$ parameter family $X_{\mu_{1},\mu_{2}}$
is tranverse to ${\mathcal{H}}^{1}$ at $\mu_{1}=0$ and, reciprocally, for any
fixed for any fixed $\mu_{1}$, the $1$ parameter family $X_{\mu_{1},\mu_{2}}$
is tranverse to ${\mathcal{H}}^{1}$ at $\mu_{2}=0$. More precisely one may
assume that
$X_{\mu_{1},\mu_{2}}\in{\mathcal{O}}_{\varphi}^{\omega_{1},\omega_{2}}$ where
$\omega_{i}\in\\{-,+\\}$ is the sign of $\mu_{i}$.

When one unfolds this double homoclinic connection, if $q_{1,\mu}$ enters in
$\Sigma_{2}$ (that is, $\mu_{1}>0$), then a fixed point
$p_{1,\mu}\in\Sigma_{1}$ is created and tends to $q_{1,0}\in\gamma^{s}_{+}$ as
$\mu$ tends to $0$. In the same way, if $\mu_{2}>0$ then $q_{2}$ enters in
$\Sigma_{1}$ and the fixed point $p_{2,\mu}$ is created in $\Sigma_{2}$ and
tends to $q_{2,0}\in\gamma^{s}_{+}$. The stable leaves $W^{s}_{1,\mu}$ and
$W^{s}_{2,\mu}$ bound the small strip $\Sigma^{+}$ in $\Sigma$ containing
$\gamma^{s}_{+}$ (tending to the segment $\gamma^{s}_{+}$ as $\mu\to 0$), and
a large strip $\Sigma^{-}$.

The vector field $X_{\mu}$ in ${\mathcal{O}}^{+,+}_{\varphi}$ belongs to
${\mathcal{L}}^{+}\cup{\mathcal{L}}^{-}$ if and only if both $q_{1}$ and
$q_{2}$ belong to the same component $\Sigma^{+}$ or $\Sigma^{-}$.

######  Lemma 6.1.

For small $\mu$, $\\{q_{1},q_{2}\\}$ is not contained in $\Sigma^{+}$. In
other words the closure of ${\mathcal{L}}^{+}$ is disjoint from
${\mathcal{H}}^{1,2}_{+}$.

For the set $\\{\mu|\\{q_{1,\mu},q_{2,\mu}\\}\subset\Sigma_{-}\\}$ is an open
subset containing $(0,0)$ in its closure. More precisely, for any $\alpha>0$
there is $\mu_{0}$ so that for any $0<\mu<\mu_{0}$ the vector field
$X_{\mu,\alpha\mu}\in{\mathcal{L}}^{-}$.

###### Proof.

Assume (arguing by contradiction) that $\\{q_{1},q_{2}\\}\subset\Sigma^{+}$.
Thus $X\in{\mathcal{L}}^{+}$. Then $\Sigma^{+}$ is invariant under the action
of $P$. However, as $\Sigma^{+}$ is contained in an arbitrarily small
neighborhood of $\gamma^{s}_{+}$ the rate of expansion of $P$ in the unstable
cone is arbitrarily large, in particular $\gg 2$. So $\Sigma^{+}$ cannot be
invariant, ending the proof.

The proof of the second point is as follonws: consider one half line
$X_{\mu,\alpha\mu}$ and consider $\mu>0$ very small. Thus the point $q_{1}$
(resp. $q_{2}$) belongs to $\Sigma_{2}$ (resp. $\Sigma_{1}$) and its distance
to $\gamma^{s}_{+}$ is $\mu$ (resp. $\alpha\mu$).

Consider any point $p$ in $\Sigma_{1}$ at a small distance
$>\frac{1}{2}\alpha\mu$ de $\gamma^{s}_{+}$. Consider an unstable segment
$\gamma_{p}$ realizing this distance and so that $\gamma_{p}$ starts at
$\gamma^{s}_{+}$ and ends at $p$. Then $P(\gamma_{p})$ is an unstable segment
starting at $q_{1}$ and ending at $P(p)$. The length $\ell(P(\gamma_{p}))$ is
arbitrarily larger than $\ell(\gamma_{p})$ (say, larger than
$100(1+\alpha)\mu\ell(\gamma_{p})$ as $\mu$ tends to $0$ (as the expansion
rate close to $\gamma^{s}_{+}$ tends to infinity).

One deduce that the point $P(p)$ belongs to $\Sigma_{1}$ and is at a distance
larger than $99(1+\alpha)\mu$ of $\gamma^{s}_{+}$. One deduces that $P(p)$
(and thus $p$) does not belong to the stable leaf through the fixed point
$p_{1}\in\Sigma_{1}$. In particular $q_{2}$ belongs to $\Sigma_{-}$. One
proves in the same way than for $\mu>0$ small $q_{1}\in\Sigma_{-}$, and thus
the vector filed belongs to ${\mathcal{L}}^{-}$, ending the proof. Figure 1
displays a local bifurcation diagram in this case. ∎

The lemma implies that, when one consider a segment in the parameter plane
crossing $(0,0)$ and entering in ${\mathcal{O}}^{+,+}_{\varphi}$ transversely
to both ${\mathcal{H}}^{1}$ and ${\mathcal{H}}^{2}$, then one creates a down
Lorenz attractor, which cuts every stable leaves in $\Sigma^{-}$. When the
parameter tends to $(0,0)$ then $\Sigma^{-}$ tends to $\Sigma$ and the Lorenz
attractor tends to what we called the fat Lorenz attractor. The horseshoe,
corresponding to $\Sigma^{+}$ collapses into the double homoclinic connection.

###  6.3 Switching from up to down: family crossing
${\mathcal{H}}{\mathcal{E}}_{1}\cap{\mathcal{H}}{\mathcal{E}}_{2}$

In this section we consider a two parameter family
$X_{\mu}\in{\mathcal{O}}_{\varphi}^{+,+}$, $\mu=(\mu_{1},\mu_{2})$ crossing
transversely
${\mathcal{H}}{\mathcal{E}}_{1}\cap{\mathcal{H}}{\mathcal{E}}_{2}$ at $\mu=0$.

In other words $X_{(0,0)}$ exibits two heteroclinic connections $q_{1}\in
W^{s}_{2}$ and $q_{2}\in W^{s}_{1}$ (recall that $W^{s}_{i}$ is the stable
leaf through the fixed point $p_{i}$). We have seen that this behavior implies
that $X_{(0,0)}$ has two _full Lorenz attractors_ which intersect along
$W^{s}(p_{1})\cup W^{s}(p_{2})\cup W^{u}(\sigma)$. See Figure 17.

Up to reparametrize the family one may assume that

$\begin{array}[]{c}\left(\mu_{1}=0\Leftrightarrow q_{1}\in
W^{s}_{2}\right)\mbox{ and }\left(\mu_{2}=0\Leftrightarrow q_{2}\in
W^{s}_{1}\right)\\\ \left(\mu_{1}>0\Leftrightarrow
q_{1}\in\Sigma^{+}\right)\mbox{ and }\left(\mu_{2}>0\Leftrightarrow
q_{2}\in\Sigma^{+}\right)\end{array}$

Figure 19: Local bifurcation: (a) $(\mu_{1},\mu_{2}),\mu_{i}>0$,
$(\mu_{1},\mu_{2}),\mu_{i}=0$, (c) $(\mu_{1},\mu_{2}),\mu_{i}<0$.

With these notations, note that Theorem F is just a reformulation of the
results in the previous sections.

##  7 Parameter families $X_{\mu}\in{\mathcal{O}}_{\varphi}$, with parameters
in the torus

###  7.1 ${\mathbb{T}}^{2}$-parameter families

######  Definition 7.1.

Let
$\pi\colon{\mathbb{R}}^{2}\to{\mathbb{T}}^{2}={\mathbb{R}}^{2}/{\mathbb{Z}}^{2}$
be the canonical projection. Let $V\subset{\mathbb{R}}^{2}$ be an open subset
so that the projection $\pi(V)$ is the whole torus ${\mathbb{T}}^{2}$.

We say that a family $\\{X_{\mu}\in{\mathcal{O}}_{1}\\}_{\mu\in V}$ is a
$C^{r}$ family, $r\geq 0$ of vector fields in ${\mathcal{O}}_{1}$ parametrized
by ${\mathbb{T}}^{2}$ if

  * •

the map $\mu\to X_{\mu}$ is continuous for the $C^{1}$-topology.

  * •

the map $(p,\mu)\mapsto X_{\mu}(p)$ is of class $C^{r}$.

  * •

for any $\mu,\mu^{\prime}$ so that $\mu^{\prime}-\mu\in{\mathbb{Z}}^{2}$ one
has: the return maps $P,P^{\prime}$ of $X,X^{\prime}$ on the transverse cross
section $\Sigma$ coincide: $P=P^{\prime}$. In particular, $X$ and $X^{\prime}$
are smoothly topologically equivalent in restriction to the attracting region
$U$, by an equivalence whose restriction to the cross section $\Sigma$ is the
identity map.

Shortly, we say that $X_{\mu}$ is a ${\mathbb{T}}^{2}$-parameter family.

###  7.2 Essential families

The aim of this section is to define the notion of essential
${\mathbb{T}}^{2}$ families. In a rough way of speaking, we don’t want that
the family $X_{\mu}$ is contained in a small neighborhood of a given vector
field $X_{0}$. We want that, when $\mu$ follows a closed simple path in
${\mathbb{T}}^{2}$, non homotopic to a point, then the images $P(\Sigma_{1})$
or $P(\Sigma_{2})$ give a turn in $\Sigma$ in an essential way.

Let us firt present a non-intrinsecal definition of these phenomenon, and we
will see then an intrinsecal definition, showing that the definition does not
depend of the choices.

Consider a ${\mathbb{T}}^{2}$-parameter family $\\{X_{\mu}\\}$ and let
$\gamma^{s}_{+,\mu}$ and $\gamma^{s}_{-,\mu}$ the associated stable leaves
corresponding to the discontinuities of the first return map. The leaves
$\gamma^{s}_{\mu}$ vary continuously with $\mu$. This allow us to choose a
parametrization of $\Sigma$, depending on ${}_{m}u$ and so that
$\gamma^{s}_{+,\mu}$ is the segment $\\{0\\}\times[-1,1]$ in the annulus
$\Sigma={\mathbb{R}}/{\mathbb{Z}}\times[-1,1]$.

Consider now the points $q_{1,\mu}$ and $q_{2,\mu}$ (first intersection points
of the unstable separatrices of $\sigma_{\mu}$ with $\Sigma$). This defines
two continuous maps
$q_{1}\colon{\mathbb{T}}^{2}\to{\mathbb{R}}/{\mathbb{Z}}\times[-1,1]\mbox{ and
}q_{2}\colon{\mathbb{T}}^{2}\to{\mathbb{R}}/{\mathbb{Z}}\times[-1,1]$.

Then, composing $q_{1}$ and $q_{2}$ by the projection
$\psi\colon\SS^{1}\times[-1,1]\to\SS^{1}$ one gets two continuous maps
$\psi\circ q_{i}\colon{\mathbb{T}}^{2}\to\SS^{1}$, $i=1,2$.

Let us denote

$Q=\left(\psi\circ q_{1},\psi\circ
q_{2}\right)\colon{\mathbb{T}}^{2}\to{\mathbb{T}}^{2}.$

######  Definition 7.2.

With the notation above, we say that the family $X_{\mu}$ is essential if the
topological degree of $\mu$ is $1$ (or else, if $Q$ is homotopic to an
orientation preserving homeomorphism).

Let us convince you now that this notion does not depend on the choice of the
parametrization of $\Sigma$.

Consider the hypersuperface
$\Gamma^{s}_{+}\subset\Sigma\times{\mathbb{T}}^{2}$ defined by
$\Gamma^{s}_{+}=\bigcup_{\mu}\gamma^{s}_{m}u\times\\{\mu\\}$. Is it
homeomorphic to a $3$-torus in $\Sigma\times{\mathbb{T}}^{2}$.

To every closed path $c\colon\SS^{1}\to{\mathbb{T}}^{2}$ let us consider the
closed pathes $q_{i,c}\colon\SS^{1}\to\Sigma\times{\mathbb{T}}^{2}$,
$i\in\\{1,2\\}$ so that
$q_{i,c}(\theta)=\left(q_{i,c(\theta)},c(\theta)\right)$ for
$\theta\in\SS^{1}$.

Notice that $q_{i,c}$ depends continuouly on the closed path $c$. In
particular, its algebraic intersection number with the hypersurface
$\Sigma^{s}_{+}$ only depends on the homology class $[c]$. We denote it
$[q_{i,c}]\cdot\Gamma^{s}_{+}$.

One gets a maps

$[c]\mapsto\left([q_{1,c}]\cdot\Gamma^{s}_{+}],[q_{2,c}]\cdot\Gamma^{s}_{+}\right)\in{\mathbb{Z}}^{2}$

defined on $H_{1}({\mathbb{T}}^{2},{\mathbb{Z}})\simeq{\mathbb{Z}}^{2}$ with
values in ${\mathbb{Z}}^{2}$, and this map is a linear map given by a $2$ by
$2$ matrix with ${\mathbb{Z}}$-entries. The topological degree in the first
presentation of the notion is just the determinant of this linear map: the
family is essential if and only iff the determinant is $1$.

###  7.3 Building ${\mathbb{T}}^{2}$-families

Consider a vector field $X\in{\mathcal{O}}_{1}$ on a $3$-manifold $M$. Thus,
by definition, it is transversal to the annuli $\Sigma$ $D_{1}$ and $D^{2}$.
Furthermore, the first return maps of $D_{i}$ to $\Sigma\cup D_{1}\cup D_{2}$
is a smooth map $R\colon D_{1}\cup D_{2}\to\Sigma$ , mapping $D_{1}$ and
$D_{2}$ on two disjoint essential annuli in the interior of $\Sigma$.

Consider a annuli $\tilde{D}_{i}$, $i=1,2$, containing $D_{i}$ in its
interior, and so that $R$ extends in a diffeomorphism
$\tilde{R}\colon\tilde{D}_{1}\cup\tilde{D}_{2}\to\Sigma$ which is the first
return maps from $\tilde{D}_{i}$ to
$\Sigma\cup\tilde{D}_{1}\cup\tilde{D}_{2}$.

We denote by $\Delta_{i}$ the union of the $X$-orbit segment joining points
$p\in\tilde{D}_{i}$ to $\tilde{R}(p)$. Thus $(\Delta_{i},X)$ is smoothly
orbitally equivalent to $(\tilde{D}_{i}\times[0,1],\frac{\partial}{\partial
t})$

Next lemma expresses that one can realize by a continuous family of vector
fileds in $M$ any continuous deformation of the return map $R$.

######  Lemma 7.3.

Consider a continuous family $R_{\mu}\colon D_{1}\cup
D_{2}\to\tilde{R}(\tilde{D}_{1}\cup\tilde{D}_{2})$, $\mu\in V$, where
$V\subset{\mathbb{R}}^{2}$ is an open disk containing $0$. One assume
$R_{0}=R$.

Then there is a family of vector fields $X_{\mu}$ with the following
properties:

  * •

$X_{0}=X$

  * •

for any $\mu$, $X_{\mu}$ satifies all the topological properties (items 1 to
7)) of the definition of the set ${\mathcal{O}}_{1}$.

  * •

for any $\mu$ , $X_{\mu}$ coincides with $X$ out of
$\Delta_{1}\cup\Delta_{2}$.

  * •

for any $\mu$, the restriction of $X_{\mu}$ to $\Delta_{i}$ is transverse to
the fibers $\tilde{D}_{i}\times\\{t\\}$

  * •

the return map of $X$ from $D_{1}\cup D_{2}$ to $\Sigma$ is $R_{\mu}$.

###### Proof.

One extends $R_{\mu}$ to $\tilde{D}_{i}$ in a diffeomorphism $\tilde{R}_{\mu}$
so that all $\tilde{R}_{\mu}$ coincide with $\tilde{R}$ in a neighborhood of
the boundary $\partial\tilde{D}_{i}$.

Then we replace the retriction of $X$ in $\Delta_{i}$ by a vectorfield which
coincides with $X$ in a neighborhood of $\partial\Delta_{i}$ and which
entrance exit map is $\tilde{R}_{\mu}$. ∎

Assume now that the projection of $V$ on ${\mathbb{R}}^{2}/{\mathbb{T}}^{2}$
covers the whole torus ${\mathbb{T}}^{2}$, and assume that the family
$R_{\mu}$ is ${\mathbb{Z}}^{2}$-periodic in the following sense:

  * •

for any $\mu_{1},\mu_{2}\in V$ so that $\mu_{2}-\mu_{1}\in{\mathbb{Z}}^{2}$
then $R_{\mu_{1}}=R_{\mu_{2}}$.

Then the family $X_{\mu}$ defined in Lemma 7.3 is a ${\mathbb{T}}^{2}$
parameter family of vector fields having $U$ has an attracting region.

###  7.4 An essential ${\mathbb{T}}^{2}$-family of vector fields in
${\mathcal{O}}_{\varphi}$ obtained by rotating the return maps $R|_{D_{i}}$

Consider $\Sigma\simeq\SS^{1}\times[-1,1]$ with the coordinates $\theta,t$ and
consider the constant cone field ${\mathcal{C}}$ defined by

${\mathcal{C}}(p)=\\{u=\alpha\frac{\partial}{\partial\theta}+\beta\frac{\partial}{\partial\theta}\in
T_{p}\Sigma\mbox{ so that }|\alpha|\geq|\beta|\\}$

We denote by ${\mathcal{R}}_{\alpha}\colon\Sigma\to\Sigma$ the rotation of
angle $\alpha\in\SS^{1}$, that is $(theta,t)\mapsto(\theta+\alpha,t)$.
Consider a vector field $X\in{\mathcal{O}}_{\varphi}$ with the extra following
properties

  * •

The constant cone field ${\mathcal{C}}$ is the unstable cone field
${\mathcal{C}}^{u}$

  * •

The images $R(D_{i})$ are product annulus $\SS^{1}\times
I_{i}\subset\Sigma=\SS^{1}\times[-1,1]$

Now consider the ${\mathbb{Z}}^{2}$-family of maps $R_{\alpha,\beta}\colon
D_{1}\cup D_{2}\to\Sigma$,
$\alpha,\beta\in{\mathbb{T}}^{2}={\mathbb{R}}^{2}/\SS^{2}$ defined by

$R={\mathcal{R}}_{\alpha}\circ R\mbox{ on }D_{1},\mbox{ and
}R={\mathcal{R}}_{\beta}\circ R\mbox{ on }D_{2}$

According to Lemma 7.3 one can realize the periodic family $R_{\mu}$ by a
${\mathbb{T}}^{2}$-parameter family of vector fields $X_{\mu}$ and one easily
check

######  Lemma 7.4.

The family $X_{\mu}$ is a ${\mathbb{T}}^{2}$-parameter essential family in
${\mathcal{O}}_{\varphi}$.

##  8 Reduction to a $1$-dimensional dynamics

###  8.1 Action of the return map on the space of stable leaves

Any $C^{1}$ vector field $X\in{\mathcal{O}}_{1}$ is singular hyperbolic in the
attracting region $U$, with a continuous strong stable direction. There is a
well defined _stable foliation_ (also called _strong stable foliation_ tangent
to the stable distribution, and with leaves having the same regularity as $X$.

It admits therefore a well defined $2$-dimensional _(weak)stable foliation_
(also called _center-stable folaition_) out of the strong stable manifold of
the singularity. The leaves of the weak stable foliation are the orbits for
the flow of the leaves of the strong stable: out of the strong stable manifold
of $\sigma$, the vector is not tangent to the stable direction, so that these
orbits are $2$-dimensional. Along the strong stable manifold of $\sigma$ the
vector field is tangent to the $1$ dimensional strong stable leaf, so the
foliation is singular.

The strong stable foliation is not tangent to $\Sigma$. However, the $2$
dimensional center-stable foliation cuts the annulus $\Sigma$ transversely,
along a one-dimensional foliation ${\mathcal{F}}^{s}$, which is the stable
foliation of the return map $P$. The segment $\gamma^{s}_{+}$ and
$\gamma^{s}_{-}$ are leaves of ${\mathcal{F}}^{s}$.

The foliation ${\mathcal{F}}^{s}$ is transverse to the unstable cone field
${\mathcal{C}}^{u}$. The leave of the foliation ${\mathcal{F}}^{s}$ are
segments crossing $\Sigma$ (connecting the two boundary components of
$\Sigma$). The leaves space $\Sigma/{\mathcal{F}}^{s}$ is a (topological)
circle $\SS^{1}_{X}$. The leaves $\gamma^{s}_{+}$ and $\gamma^{s}_{-}$ induce
each a point $c_{+}$ and $c_{-}$ respectively, on $\SS^{1}_{X}$.

Note that the flow $X^{t}$ preserves the center-stable foliation, and thus the
first return map $P$ preserves the foliation ${\mathcal{F}}^{s}$.

As a consequence $P$ passes to the quotient as a map $f=f_{X}$ defined from
$\SS^{1}_{X}\setminus\\{c_{+},c_{-}\\}$ to $\SS^{1}_{X}$.

As $P(\Sigma_{i})$ is an essential pinched annulus in $\Sigma$ we deduce that
$f_{X}$ restricted to each interval of
$\SS^{1}_{X}\setminus\\{c_{+},c_{-}\\}$, is a diffeomorphism on a punctured
circle.

###  8.2 Increasing the regularity of the foliation

In a recent work [5] Araújo and Melbourne adapt to our setting a condition
from [11] ensuring the smouthness of the strong stable foliation of $X$.

  * •

there exists $t>0$ such that

$\|DX^{t}|_{E^{s}(x)}\|\cdot\|DX^{-t}|_{E^{cu}(X^{t}(x))}\|\cdot\|DX^{t}|_{E_{x}^{cu}}\|<1\quad\mbox{
for all $x\in\Lambda$}.$ (3)

As a consequence of item • ‣ 8.2), the weak stable foliation of $X$, and ,
also the stable foliation $F^{s}$ of the first return map $P$ are of class
$C^{1}$. Therefore the circle $\SS^{1}_{X}$ is endowed with a natural
$C^{1}$-structure, and $f_{X}$ is of class $C^{1}$.

##  9 Symbolic dynamics and topological classification

As for the classical geometric model of Lorenz attractor, we will see in that
section that, to any vector field in ${\mathcal{O}}_{1}$ one, can associate a
combinatorial data, called the itinerary of the discontinuities. Furthermore,
these itineraries provide a topological classification of the vector fields in
the attracting region $U$.

In this section we fix a vector field $X\in{\mathcal{O}}_{1}$, and its return
map $P\colon\Sigma\to\Sigma$.

###  9.1 Itineraries for the return map $P$ on $\Sigma$

Recall that $\Sigma$ is endowed with two specific stable leaves
$\gamma^{s}_{+}$ and $\gamma^{s}_{-}$ which split $\Sigma$ in $\Sigma_{1}$ and
$\Sigma_{2}$.

Note that $\gamma^{s}_{+}$ cuts both pinched annuli $P(\Sigma_{i})$ along one
stable leaf, excepted the case where $q_{i}\in\gamma^{s}_{+}$.

Thus $P^{-1}(\gamma^{s}_{+})$ cut $\Sigma_{i}$ in two components (one of them
being empty if $q_{i}\in\gamma^{s}_{+}$):

  * •

we denote by $A_{0}$ and $A_{1}$ the two components of $\Sigma_{1}\setminus
P^{-1}(\gamma^{s}_{+})$ where $A_{0}$ is starting at $\gamma^{s}_{+}$ (for the
positive orientation of the circle $\SS^{1}$ of $\Sigma=\SS^{1}\times[-1,1]$.
If $q_{1}\in\gamma^{s}_{+}$ then $A_{1}=\emptyset$.

  * •

we denote by $B_{0}$ and $B_{1}$ the two components of $\Sigma_{2}\setminus
P^{-1}(\gamma^{s}_{+})$ where $B_{0}$ is starting at $\gamma^{s}_{-}$. If
$q_{1}\in\gamma^{s}_{+}$ then $B_{1}=\emptyset$.

We consider ${\mathbb{X}}=\\{A_{0},A_{1},B_{0},B_{1}\\}^{\mathbb{N}}$ the
space of positive infinite words in the alphabet
$\\{A_{0},A_{1},B_{0},B_{1}\\}$.

A finite word of length $k$ is an element
$\\{\omega_{0},\dots,\omega_{k-1}\\}$ of $\\{A_{0},A_{1},B_{0},B_{1}\\}^{k}$.

Given $\omega=\\{\omega_{i}\\}_{i\in{\mathbb{N}}}\in{\mathbb{X}}$ we denote by
$[\omega]_{k}$ the initial word $\\{\omega_{0},\dots,\omega_{k-1}\\}$ of
length $k$ of $\omega$.

######  Remark 9.1.

For any $k>0$ and any $\varepsilon\in+,-$, $P^{-k}(\gamma^{s}_{\varepsilon})$
consists in at most $2^{k}$ stable leaves.

######  Definition 9.2.

For every
$p\in\Sigma\setminus\bigcup_{j=0}^{k}P^{-j}(\gamma^{s}_{+}\cup\gamma^{s}_{-})$
we denote by $[\omega(p)]_{k}=\\{\omega_{0}(p),\dots,\omega_{k-1(p)}\\}$ the
word defined by $f^{i}(p)\in\omega_{i}(p)$ (recall that $\omega_{i}(p)$ is one
of the four regions $A_{0},A_{1},B_{0},B_{1}$).

$[\omega(p)]_{k}$ is called the _$k$ -itinerary_ of $p$.

Figure 20 displays the choice of the alphabet above for the $1$-dimensional
dynamics as in the picture.

Figure 20: The alphabet chosen.

######  Lemma 9.3.

Consider any point $p\in\Sigma$, and $S\colon[-1,1]\to\Sigma$ a positively
oriented unstable segment centered at $p$ (i.e., $S(0)=p$). Then for any
$k\geq 0$ the itinerary $[\omega(S(t))]_{k}$ is well defined and constant for
$t>0$ (resp. $t<0$) small enough. This itinerary is independent of the choice
of $S$. We denote them

$[\omega_{-}(p)]_{k}\quad\mbox{and}\quad[\omega_{+}(p)]_{k}.$

If $p_{1}$ and $p_{2}$ belong to the same stable leaf, then

$[\omega_{\pm}(p_{1})]_{k}=[\omega_{\pm}(p_{2})]_{k}.$

In other words, the itinerary depends only on the stable leaf, and not of the
point in the leaf.

For any $p\in\Sigma$ one denotes by $\omega_{-}(p)$ and $\omega_{+}(p)$ the
infinite words whose first segments of length $k$ are, respectively,
$[\omega_{-}(p)]_{k}$ and $[\omega_{+}(p)]_{k}$. They are called the _down-_
and _up -itinerary_ of $p$. The itinerary of $p$ is the pair of sequences
$\omega(p)=(\omega_{-}(p),\omega_{+}(p)).$

######  Remark 9.4.

If $p\notin W^{s}(\sigma)$ ( that is, $P^{k}(p)\notin\gamma^{s}_{\pm}$ for all
$k>0$) $k$ is then $\omega_{-}(p)=\omega_{+}(p)$ and the segment of length $k$
is _$k$ -itinerary_ $[\omega(p)]_{k}$ of $p$. This shows that our terminology
and notations are consistent.

Next remark says that the itineraries $\omega_{+}$ and $\omega_{-}$ induce, in
some sense, a semi-conjugacy of $(\Sigma,P)$ with
$({\mathbb{X}},\mathfrak{S})$, where $\mathfrak{S}$ is the shift on
${\mathbb{X}}$. Being rigorous, $P$ is defined on
$\Sigma\setminus(\gamma^{s}_{+}\cup\gamma^{s}_{-})$ which is not invariant
under $P$. Thus $\omega_{\pm}$ are conjugacies in restriction of
$\Sigma\setminus W^{s}(\sigma)$.

As $P$ is discontinuous along $\gamma^{s}+$ and $\gamma^{s}_{-}$ we also
explain the itinerary of these points.

######  Remark 9.5.

  * •

For any $x\in\Sigma\setminus(\gamma^{s}_{+}\cup\gamma^{s}_{-})$ (that is,
$P(x)$ is defined), then

$\omega_{-}(P(x))=\mathfrak{S}(\omega_{-}(x))\quad\mbox{and}\quad\omega_{+}(P(x))=\mathfrak{S}(\omega_{+}(x))$

  * •

For $x\in\gamma^{s}_{+}$ one has:

$\omega_{+}(x)=A_{0}\star\omega_{+}(q_{1})$

and

$\begin{array}[]{l}\omega_{-}(x)=B_{1}\star\omega_{-}(q_{2}),\mbox{ if
}q_{2}\notin\gamma^{s}_{+}\\\ \omega_{-}(x)=B_{0}\dots B_{0}\dots,\mbox{ if
}q_{2}\in\gamma^{s}_{+}\quad(\mbox{and then }B_{1}=\emptyset)\end{array}$

  * •

For $x\in\gamma^{s}_{-}$ one has:

$\omega_{+}(x)=B_{0}\star\omega_{+}(q_{2})$

and

$\begin{array}[]{l}\omega_{-}(x)=A_{1}\star\omega_{-}(q_{1}),\mbox{ if
}q_{1}\notin\gamma^{s}_{+}\\\ \omega_{-}(x)=A_{0}\star\omega_{-}(q_{1})\mbox{
if }q_{1}\in\gamma^{s}_{+}\quad(\mbox{and then }A_{1}=\emptyset)\end{array}$

###  9.2 Itineraries for the $1$-dimensional dynamics

According to Lemma 9.3, the itineraries $\omega_{-}$ and $\omega_{+}$ are
functions of the stable leaf. So they passes to the quotient on the leaves
space $\SS^{1}_{X}$. We still denote by $\omega_{-}$ and $\omega_{+}$ the
quotient maps

$\omega_{\pm}\colon\SS^{1}_{X}\to{\mathbb{X}}.$

###  9.3 Order and topology

We endow the alphabet $\\{A_{0},A_{1},B_{0},B_{1}\\}$ with the total order

$A_{0}<A_{1}<B_{0}<B_{1}<1$

which corresponds to the order that an unstable segment starting at
$\gamma^{s}_{+}$ crosses the corresponding regions in $\Sigma$.

We endow ${\mathbb{X}}$ with the corresponding lexicographic order, that we
denote by $\prec$ (and $\preccurlyeq$ for the non-strict order).

######  Proposition 9.6.

Let $S\colon[0,1]\to\Sigma$ be a unstable segment, positively oriented, whose
interior is contained in $\Sigma\setminus\gamma^{s}_{+}$. Then:

  * •

for any $t\in(0,1)$ one has $\omega_{-}(S(t))\preccurlyeq\omega_{+}(S(t)),$

  * •

for any $t_{1},t_{2}\in[0,1]$ so that $t_{1}<t_{2}$ one has
$\omega_{+}(S(t_{1}))\prec\omega_{-}(S(t_{2})).$

This proposition has a straightforward translation for the itineraries
associated to the $1$-dimensional dynamic $f$.

###### Proof.

For the first item, we already have seen that if $S(t)$ does not belong to
$W^{s}(\sigma)$ then $\omega_{-}(S(t))=\omega_{+}(S(t)$, and there is nothing
to prove.

Consider $t_{1}<t_{2}$ so that $S(t_{i})\notin W^{s}(\sigma)$. As $\omega_{+}$
and $\omega_{-}$ coincide on $S(t_{i})$ we just note
$\omega^{i}=\omega_{+}(S(t_{i}))=\omega_{-}(S(t_{i}))$. If the first letter is
not the same, that is $S(t_{1})$ and $S(t_{2})$ are not in the same region
$A_{0},A_{1},B_{0},B_{1}$, then $(\omega^{1})_{0}<(\omega^{2})_{0}$, by the
choice of the order on our alphabet, and so $\omega^{1}\prec\omega^{2}$.

Assume now that $(\omega^{1})_{j}=(\omega^{2}_{j})$ for $j=0,\dots,k-1$ but
$(\omega^{1})_{k}\neq(\omega^{2})_{k}$.

######  Claim 1.

With the hypotheses above

  * •

$P^{j}$, $0\leq j\leq k$ is defined on the unstable segment $S([t_{1},t_{2}])$

  * •

$P^{j}(S([t_{1},t_{2}])$ is contained in one of the regions
$A_{0},A_{1},B_{0},B_{1}$ for $0\leq j<k$,

  * •

$P^{k}(S(t_{1}))$ and $P^{k}(S(t_{2}))$ are not in same region

###### Proof.

The third item just says $(\omega^{1})_{k}\neq(\omega^{2})_{k}$ which is our
hypothesis.

The proof of the two first items goes togetehr and by induction. As $S(t_{1})$
and $S(t_{2})$ belongs to the same region, and as the interior of $S$ is
disjoint from $\gamma^{s}_{+}$ and $S$ is an unstable segment (transverse to
the fibration by stable leaves) this implies that $S([t_{1},t_{2}])$ is
contained in one of the region of the alphabet. As $P^{0}=id$ is clearly
defined on $S([t_{1},t_{2}])$, we proved both items for $j=0$.

We assume now that both items have been proved for $0,\dots,j-1$ and let us
prove them for $j$. Thus $P^{j-1}(S([t_{1},t_{1}]))$ is contained in one of
the regions. Thus $P$ is well defined on this interval meaning that $P^{j}$ is
well defined on $S([t_{1},t_{2}])$, proving the the first item.

If $j\neq k$, then $(\omega^{1})_{j}=(\omega^{2})_{j}$: in other words the end
points of the unstable segment $P^{j}(S([t_{1},t_{1}]))$ belong to the same
region. If the whole segment is contained in that region, we are done.
Otherwize, $P^{j}(S([t_{1},t_{1}]))$ crosses $\gamma^{s}_{+}$. This means that
$P^{j-1}(S([t_{1},t_{1}]))$ is crossing $P^{-1}(\gamma^{s}_{+}$, and this (by
definition of the regions $A_{0},A_{1},B_{0},B_{1}$) is contradicting the fact
that $P^{j-1}(S([t_{1},t_{1}]))$ is contained in one of these regions. This
ends the proof of the claim. ∎

######  Claim 2.

With the hypotheses above, $(\omega^{1})_{k}<(\omega^{2})_{k}$.

###### Proof.

According to Claim 1, the $P^{k-1}(S([t_{1},t_{1}]))$ is an unstable segment
contained in one of the regions, and we have seen in the proof that this
implies that $P^{k}(S([t_{1},t_{1}]))$ is an unstable segment which does not
cross $\gamma^{s}_{+}$. As already seen, the choice of the order on
$A_{0},A_{1},B_{0},B_{1}$ implies that, either $P^{k}(S([t_{1},t_{1}]))$ is
contained in one region (which contradicts
$(\omega^{1})_{k}\neq(\omega^{2})_{k}$) or
$(\omega^{1})_{k}<(\omega^{2})_{k}$, ending the proof of the claim. ∎

The claims above show that, for any $t_{1}<t_{2}$ so that $S(t_{i})\notin
W^{s}(\sigma)$ one has

$\omega^{1}\preccurlyeq\omega^{2}.$

######  Claim 3.

Given $t_{1}<t_{2}$ so that $S(t_{i})\notin W^{s}(\sigma)$, then
$\omega^{1}\neq\omega^{2}$.

###### Proof.

As in Claim 1, if $\omega^{1}=\omega^{2}$ then $P^{j}$ is well defined on
$S([t_{1},t_{2}])$ for any $j\geq 0$ and $P^{j}(S([t_{1},t_{2}])$ is contained
in $1$ of the regions $A_{0},A_{1},B_{0},B_{1}$. The length of these iterates
is increasing exponentially, and these forbids (for large iterates) these
segments to be contained in one region, ending the proof of the claim. ∎

Consider now any $t_{1}<t_{2}$. We need to prove

$\omega^{1}_{+}=\omega_{+}(S(t_{1}))\prec\omega_{-}(S(t_{2}))=\omega^{2}_{-}.$

By definition of $\omega_{-}$, there is a decreasing sequence $t_{1,n}<t_{2}$
tending to $t_{1}$ and so that:

  * •

$S(t_{1,n})\notin W^{s}(\sigma)$

  * •

$[\omega^{1}_{+}]_{n}=[\omega^{1,n}]_{n}$ (where $\omega^{1,n}$ is the
itinerary of $S(t_{1,n})$.

Note that the sequence $\omega^{1,n}$ is strictly descreasing for $\prec$ and
tends to $\omega^{1}_{-}$. In other words,

$\omega^{1}_{-}=\inf_{n\to+\infty}\omega^{1,n}.$

In the same way we fix an incresing sequence $t_{2,n}>t_{1,0}$ tending to
$t_{2}$ and so that

  * •

$S(t_{2,n})\notin W^{s}(\sigma)$

  * •

$[\omega^{2}_{-}]_{n}=[\omega^{2,n}]_{n}$ (where $\omega^{2,n}$ is the
itinerary of $S(t_{2,n})$.

Then

$\omega^{2}_{+}=\sup_{n\to+\infty}\omega^{2,n}.$

As $\omega^{1,n}\prec\omega^{2,n}$, we conclude

$\omega^{1}_{+}\prec\omega^{2}_{-}$

proving the second item of the proposition.

To end the proof of the proposition it remains to show that
$\omega_{-}(S(t))\preccurlyeq\omega_{+}(S(t))$ for any $t\in(0,1)$.

For that we consider sequences
$t_{-,n}<t_{-,n+1}<\dots<t<\dots<t_{+,n+1}<t_{+,n}$ tending to $t$ as
$n\to+\infty$ and so that

  * •

$S(t_{\pm,n})\notin W^{s}(\sigma)$

  * •

$[\omega_{-}(S(t))]_{n}=[\omega^{-,n}]_{n}$ (where $\omega^{-,n}$ is the
itinerary of $S(t_{-,n})$)

  * •

$[\omega_{+}(S(t))]_{n}=[\omega^{+,n}]_{n}$ (where $\omega^{+,n}$ is the
itinerary of $S(t_{-,n})$ )

We know that $\omega^{-,n}\prec\omega^{+,n}$ (the itinerary is strictly
increasing on point out of $W^{s}(\sigma)$). So for every $n$ one has
$[\omega_{-}(S(t))]_{n}\preccurlyeq[\omega_{+}(S(t))]_{n}$, ending the proof.
∎

###  9.4 Admissible itineraries

Given a vector field $X\in{\mathcal{O}}_{1}$, we associate four itineraries:
$\omega^{+}_{+}=\omega_{+}(\gamma^{s}_{+})$,
$\omega^{+}_{-}=\omega_{-}(\gamma^{s}_{+})$,
$\omega^{-}_{+}=\omega_{+}(\gamma^{s}_{-})$ and
$\omega^{-}_{-}=\omega_{-}(\gamma^{s}_{-})$.

######  Definition 9.7.

We say that a $\omega\in{\mathbb{X}}$ is admissible for the vector field $X$
(or, shortly, $X$-admissible), if it satisfies the following inequalities

  * •

$\omega^{+}_{+}\preccurlyeq\mathfrak{S}^{n}(\omega)\preccurlyeq\omega^{+}_{-}$.

  * •

If $(\omega)_{i}\in\\{A_{0},A_{1}\\}$ then
$\mathfrak{S}^{i}(\omega)\preccurlyeq\omega^{-}_{-}$.

  * •

If $(\omega)_{i}\in\\{B_{0},B_{1}\\}$ then
$\omega^{-}_{+}\preccurlyeq\mathfrak{S}^{i}\omega$.

We denote by ${\mathcal{A}}_{X}\subset{\mathbb{X}}$ the set of $X$-admissibles
itinerires.

######  Remark 9.8.

The subset ${\mathcal{A}}_{X}$ is a $\mathfrak{S}$-invariant compact set.

We note that if $X$ does not exhibit homoclinic loops then $\underset{x\to
c^{-}_{-}}{\lim}f^{i}_{X}(x)=\underset{x\to c^{+}_{+}}{\lim}f^{i}_{X}(x)$ and
$\underset{x\to c^{-}_{+}}{\lim}f^{i}_{X}(x)=\underset{x\to
c^{+}_{-}}{\lim}f^{i}_{X}(x)$, for all $i\in\mathbb{N}$, and therefore

$\begin{array}[]{c}\mathfrak{S}(\omega^{+}_{+})=\mathfrak{S}(\omega^{-}_{-})\\\
\mathfrak{S}(\omega^{+}_{-})=\mathfrak{S}(\omega^{-}_{+})\end{array}$ (4)

Remembering that $(\omega^{+}_{+})_{0}=A_{0}$ and $(\omega^{-}_{+})_{0}=B_{0}$
one gets that these $4$ itineraries are determined by $\omega^{+}_{-}$ and
$\omega^{-}_{-}$.

If $X$ exhibits a homoclinic loop, the equalities (4) are no longer true. For
instance, this is the case when $W^{u}_{+}(\sigma)\cap
W^{s}_{+}(\sigma)\neq\emptyset$ and $W^{u}_{-}(\sigma)\cap
W^{s}(\sigma)=\emptyset$, $\omega_{+}^{+}$ is periodic and $\omega_{-}^{-}$ is
not. See Figure 21. However, since $\underset{x\to
c^{-}_{-}}{\lim}f_{X}(x)=\underset{x\to c^{+}_{+}}{\lim}f_{X}(x)$ and
$\underset{x\to c^{-}_{+}}{\lim}f_{X}(x)=\underset{x\to
c^{+}_{-}}{\lim}f_{X}(x)$, the set ${\mathcal{A}}_{X}$ is still determined by
$\\{\omega_{+}^{+},\omega_{-}^{+}\\}$ or
$\\{\omega_{-}^{-},\omega_{+}^{-}\\}$.

Figure 21: (a) $\mathcal{H}_{-}^{1}\cap\mathcal{H}^{2}_{+}$, (b)
$\mathcal{H}_{-}^{1}\cap\mathcal{H}_{-}^{2}$.

$\displaystyle\mbox{Itineraries
of}\quad\mathcal{H}_{-}^{1}\cap\mathcal{H}_{-}^{2}$
$\displaystyle\mbox{Itineraries
of}\quad\mathcal{H}_{-}^{1}\cap\mathcal{H}_{+}^{2}$
$\displaystyle\omega_{+}^{+}=A_{0}B_{0}A_{0}B_{0}\ldots\,\,\,\,\,$
$\displaystyle\,\,\,\,\,\omega_{+}^{+}=A_{0}B_{0}B_{0}B_{0}\ldots$
$\displaystyle\omega_{-}^{-}=A_{1}A_{1}A_{1}A_{1}\ldots\,\,\,\,\,$
$\displaystyle\,\,\,\,\,\omega_{-}^{-}=A_{1}A_{1}A_{1}A_{1}\ldots$
$\displaystyle\omega_{-}^{+}=B_{0}A_{0}B_{0}A_{0}\ldots\,\,\,\,\,$
$\displaystyle\,\,\,\,\,\omega_{-}^{+}=B_{1}A_{1}A_{1}A_{1}\ldots$
$\displaystyle\omega_{+}^{-}=B_{0}B_{0}B_{0}B_{0}\ldots\,\,\,\,\,$
$\displaystyle\,\,\,\,\,\omega_{+}^{-}=B_{0}B_{0}B_{0}B_{0}\ldots$

######  Lemma 9.9.

If $p\in\Sigma$ then $\omega_{-}(p)$ and $\omega_{+}(p)$ are $X$-admissible.

###### Proof.

Consider an unstable segment $S\colon[0,1]\to\Sigma$ whose interior is
disjoint from $\gamma^{s}_{+}$ and so that $S(0),S(1)\in\gamma^{s}_{+}$ and
$p\in S([0,1])$. Then Proposition 9.6 applies and implies that, if
$p\notin\gamma^{s}_{+}$ then
$\omega^{+}_{+}\preccurlyeq\omega_{-}(p)\preccurlyeq\omega_{+}(p)\preccurlyeq\omega^{+}_{-}$.

In particular,

$\begin{array}[]{c}\omega^{+}_{-}=\max\\{\omega_{-}(p),\omega_{+}(p),p\in\Sigma\\}\\\
\omega^{+}_{+}=\min\\{\omega_{-}(p),\omega_{+}(p),p\in\Sigma\\}\end{array}$

In particular, this shows that $\omega_{-}(p)$ and $\omega_{+}(p)$ satisfy the
first item of the definition of $X$-admissibility.

The other two items correspond to several cases whose proof are very similiar,
let us just present one of these cases:

Let $p=S(t)$ so that $(\omega_{+}(p))_{0}\in\\{A_{0},A_{1}\\}$. Then there is
a decreasing sequence $t_{n}\to t$ so that $S(t_{i})\notin W^{s}(\sigma)$ and

$[\omega_{+}(p)]_{n}=[\omega^{n}]_{n}$

where $\omega^{n}$ denotes $\omega_{-}(S(t_{n}))=\omega_{+}(S(t_{n}))$.

Then $\omega_{+}(p)=\inf\omega^{n}$. On hte other hand $S(t_{n})$ is a point
out of $W^{s}(\sigma)$ and contained in $A_{0}\cup A_{1}$, and thus in
$\Sigma_{1}$. Proposition 9.6 implies that $\omega^{n}\prec\omega^{-}_{-}$,
finishing this case, and the proof. ∎

###  9.5 Realizing $X$-admissible itineraries

The aim of this section is to prove

######  Proposition 9.10.

Given any $\omega\in{\mathcal{A}}_{X}$ there is $p\in\Sigma$ so that

$\omega\in\\{\omega_{+}(p),\omega_{-}(p)\\}.$

######  Remark 9.11.

Proposition 9.6 implies that any two points satisfying the conclusion of
Proposition 9.10 belong to the same stable leaf.

Proposition 9.10 is a direct consequence of Lemma 9.12 below:

######  Lemma 9.12.

Given any $\omega\in{\mathcal{A}}_{X}$, the set $\Omega_{n}$ of points $p$ in
$\Sigma$ so that

$[\omega]_{n}\in\\{[\omega_{+}(p)]_{n},[\omega_{-}(p)]_{n}\\}$

is a non empty compact subset of $\Sigma$.

Assuming that Lemma 9.12 is true, then the sequence $\Omega_{n}$ is a nested
sequence of non-empty compact sets and any $p\in\bigcap\Omega_{n}$ satifies
that $\omega\in\\{\omega_{+}(p),\omega_{-}(p)\\}.$ Note that, indeed, this
intersection is exactly the stable leaf through $p$, according to Proposition
9.6.

It remains to prove Lemma 9.12.

###  9.6 Proof of Lemma 9.12

For any itinerary $\omega\in{\mathcal{A}}_{X}$ we denote by
$\Omega_{n}(\omega)$ the set of points $q\in\Sigma$ so that
$[\omega]_{n}\in\\{[\omega_{-}(q)]_{n},[\omega_{+}(q)]_{n}\\}$.

######  Lemma 9.13.

Let $\omega\in{\mathcal{A}}_{X}$ so that there is some $p\in\Sigma$ for which
$\omega\in\\{\omega_{-}(p),\omega_{+}(p)\\}$.

Fix $n\in{\mathbb{N}}$ and denote $\Omega_{n}(\omega)$ the set of points
$q\in\Sigma$ so that $[\omega]_{n}\in[\omega_{-}(q)]_{n},[\omega_{+}(q)]_{n}$.

Then $\Omega_{n}(\omega)$ is the closure of a connected component of

$\Gamma_{n}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\Sigma\setminus\bigcup_{i=0}^{n-1}P^{-i}(\gamma^{s}_{+}\cup
P^{-1}(\gamma^{s}_{+})\cup\gamma^{s}_{-}).$

###### Proof.

First notice that $\Gamma_{n}$ consist in the union of finitely many stable
leaves.

Consider an unstable segment $S\colon[0,1]\to\Sigma$ whose interior is
disjoint from $\Gamma^{n}$, and having its end points on $\Gamma_{n}$. Let
$\Omega_{n}$ be the closure of the connected component of
$\Sigma\setminus\Gamma_{n}$ containing $S((0,1))$. Then for any $0\leq i<n$,
$P^{i}(S((0,1)))$ is well defined and disjoint from $\gamma^{s}_{\pm}$ and of
$P^{-1}(\gamma^{s}_{+})$. In other word, $P^{i}(S((0,1)))$ is contained in one
of the region defining the alphabet. Thus $[\omega_{-}(S(t))]_{n}$ does not
depend on $t\in(0,1)$ and is equal to $[\omega_{+}(S(0))]_{n}$ and
$[\omega_{-}(S(1))]_{n}$.

One deduces that $[\omega_{-}]_{n}$ and $[\omega_{+}]_{n}$ are equal and
constant on the interior of $\Omega_{n}$, and $[\omega_{-}]_{n}$ takes the
same value on one of the boundary stable leaf, and $[\omega_{+}]_{n}$ on the
other boudary stable leaf.

This proves that $\Omega_{n}(\omega)$ is a union of such closures of connected
components $\Omega_{n}$.

Fix $\Omega_{n}\subset\Omega_{n}(\omega)$ and let $q\notin\Omega_{n}$. If $q$
is not is the same region $\\{A_{0},A_{1},B_{0},B_{1}\\}$ as $\Omega_{n}$ then
$[\omega_{\pm}(q)]_{n}$ is not $[\omega]_{n}$. Otherwise, there is an unstable
segment ( still denoted by $S$) in this region (hence disjoint from
$\gamma^{s}_{+}\cup\gamma^{s}_{-}\cup P^{-1}(\gamma^{s}_{+})$, joining $q$ to
a point $p$ in the interior of $\Omega_{n}$.

The interior $\operatorname{int}(S)$ of segment $S$ crosses the boundary of
$\Omega_{n}$ that is crosses $\Gamma_{n}$. Let $i$ the smallest integer so
that $\operatorname{int}(S)\cap P^{-i}(\gamma^{s}_{+}\cup\gamma^{s}_{-}\cup
P^{-1}(\gamma^{s}_{+}))$.

Then $P^{i-1}(S)$ is a contained in the closure of one of the regions
$\\{A_{0},A_{1},B_{0},B_{1}\\}$ but not $P^{i}(S)$. This implies that the two
end points of $P_{i}(S)$ are not in the same region
$\\{A_{0},A_{1},B_{0},B_{1}\\}$. This implies that $[\omega_{+}(q)]_{i}$ and
$[\omega_{-}(q)]_{i}$ are different from $[\omega]_{i}$, proving that
$q\notin\Omega_{n}(\omega)$, ending the proof. ∎

###### Prof of Lemma 9.12.

. The proof goes by induction.

We want to prove that, for every $n\geq 0$ and every
$\omega\in{\mathcal{A}}_{X}$ , $\Omega_{n}(\omega)$ is the closure of one
connected component of $\Sigma\setminus\Gamma_{n}$.

Let us check this is true for $n=0$. Each itineraries of length $0$ is one
letter of our alphabet, which corresponds to a connected component of
$\Sigma\setminus\Gamma_{0}$, and its closure is the one we announced.

We now prove it also holds for $n=1$. Assume for instance that
$(\omega)_{0}=A_{0}$. Thus $\Omega_{0}(\omega)=\bar{A}_{0}$, and
$P(\Omega_{0}(\omega))$ is a cuspidal triangle starting at $q_{1}$ and ending
at $\gamma^{s}_{+}$.

Now, by definition of ${\mathcal{A}}_{X}$ one has

$\omega_{+}(\gamma_{+}^{s})\preccurlyeq\omega\preccurlyeq\omega_{-}(\gamma^{s}_{-}).$

As the first letter of $\omega$ is the same as the first letter of
$\omega_{+}(\gamma^{s}_{+})$ one gets

$\mathfrak{\omega_{+}}(\gamma^{s}_{+})=\omega_{+}(q_{1})\preccurlyeq\mathfrak{S}(\omega).$

In particular $(\omega)_{1}=(\mathfrak{S}(\omega))_{0}$ either is strictly
bigger than or is equal to $(\omega_{+}(q_{1}))_{0}$. In both cases
$P(\Omega_{0}(\omega))$ intersects $\Omega_{0}(\mathfrak{S}(\omega))$ proving
that $\Omega_{2}(\omega)$ is not empty. Now Lemma 9.13 asserts that it is the
closure of a connecting component of $\Sigma\setminus\Gamma_{2}$ proving the
induction hypothesis in that case. The case $(\omega)_{0}=A_{1},B_{0},B_{1}$
are very similar.

We assume now that the induction hypothese have been proved for $i=0\dots n$.
Consider $\omega\in{\mathcal{A}}_{X}$. By the induction hypothesis,
$\Omega_{n}(\omega)$ is the closure of one connected component of
$\Sigma\setminus\Gamma_{n}$.

We split the proof in cases.

Case 1: $\gamma^{s}_{+}$ and $\gamma^{s}_{-}$ are not contained in the compact
set $\Omega_{n}(\omega)$.

Then $P$ is defined on $\Omega_{n}(\omega)$ and the boundary $\partial
P(\Omega_{n}(\omega))$ is contained in $\Gamma_{n-1}$.

This implies that $P(\Omega_{n}(\omega))$ crosses every stable leaf in the
closure of a connected component of $\Sigma\setminus\Gamma_{n-1}$ , which has
to be $\Omega_{n-1}(\mathfrak{S}(\omega))$ (by Lemma 9.13).

By the induction hypothesis $\Omega_{n}(\mathfrak{S}(\omega))$ is a connected
component of $\Sigma\setminus\Gamma_{n}$, and is contained in
$\Omega_{n-1}(\mathfrak{S}(\omega))$.

This implies that $P(\Omega_{n}(\omega))$ intersects
$\Omega_{n-1}(\mathfrak{S}(\omega))$. Thus $\Omega_{n+1}(\omega)$ is not
empty, and therefore is a connected component of $\Sigma\setminus\Gamma_{n+1}$
by Lemma 9.13.

Case 2: We now assume that $1$ of the boundary components of
$\Omega_{n}(\omega)$ is $\gamma^{s}_{+}$ and the other is not
$\gamma^{s}_{-}$. Up to reverse the orientation we assume that the positively
oriented unstable segments starting at $\gamma^{s}_{+}$ enter in
$\Omega_{n}(\omega)$.

Then $P(\Omega_{n}(\omega))$ is a cuspidal triangle starting at $q_{1}$ and
ending on a stable leaf in $\Gamma_{n-1}$.

Now $\Omega_{n-1}(\mathfrak{S}(\omega))$ is (induction hypothesis) the closure
of a connected component in $\Sigma\setminus\Gamma_{n-1}$, which contains
$P(\Omega_{n}(\omega))$ and thus contains $q_{1}$.

Now $\Omega_{n}(\mathfrak{S}(\omega))$ is (induction hypothesis) the closure
of a connected component in $\Sigma\setminus\Gamma_{n}$.

######  Claim 4.

$P(\Omega_{n}(\omega))\cap\Omega_{n}(\mathfrak{S}(\omega)\neq\emptyset$

###### Proof.

Note that the first letter of $\omega$ is $A_{0}$. As
$\omega\in{\mathcal{A}}_{X}$ on has
$\omega_{+}(\gamma^{s}_{+})\preccurlyeq\omega$. As their first letters are
equal, this implies

$\mathfrak{\omega}_{+}(\gamma^{s}_{+})=\omega_{+}(q_{1})\preccurlyeq\mathfrak{S}(\omega).$

Consider a positively oriented unstable segment $S\colon[0,1]\to\Sigma$
crossing every stable leaf in $\Omega_{n-1}(\mathfrak{S}(\omega))$ and
containing $q_{1}=S(t_{0})$. Then $S$ is crossing
$\Omega_{n}(\mathfrak{S}(\omega))$ at point $S(t)$.

Recall that $\omega_{+}(q_{1})\preccurlyeq\mathfrak{S}(\omega)$.

Recall that the fonction $[\omega_{+}(S(t))]n$ is non decreasing with $t$, so
that $S^{-1}(\Omega_{n}(\mathfrak{S}(\omega)))\subset[0,1]$ is not inferior to
$S^{-1}(\Omega_{n}(\omega_{+}(q_{1}))$.

As a consequence $S^{-1}(\Omega_{n}(\mathfrak{S}(\omega)))\subset[0,1]$ either
coincide with $S^{-1}(\Omega_{n}(\omega_{+}(q_{1}))$ or is strictly larger
than $t_{0}$.

In both cases $P(\Omega_{n}(\omega))$ intersects
$\Omega_{n}(\mathfrak{S}(\omega))$, concluding.

∎

This implies that $\Omega_{n+1}(\omega)$ is not empty, and Lemma 9.13
concludes that case.

Case 3: We now assume that $1$ of the boundary components of
$\Omega_{n}(\omega)$ is $\gamma^{s}_{-}$ and the other is not
$\gamma^{s}_{+}$.

Up to reverse the orientation we assume that the positively oriented unstable
segments starting at $\gamma^{s}_{+}$ enter in $\Omega_{n}(\omega)$.

Then $P(\Omega_{n}(\omega))$ is a cuspidal triangle starting at $q_{2}$ and
ending on a stable leaf in $\Gamma_{n-1}$.

Note that the first letter of $\omega$ is $B_{0}$. As
$\omega\in{\mathcal{A}}_{X}$ on has
$\omega_{+}(\gamma^{s}_{-})\preccurlyeq\omega$. As their first letters are
equal, this implies

$\mathfrak{S}(\omega)\preccurlyeq\mathfrak{S}\omega_{+}(\gamma^{s}_{+})=\omega_{+}(q_{2}).$

The proof follows now in a similar way to Case 2.

Case 4: Finally, we assume that both boundary components of
$\Omega_{n}(\omega)$ are $\gamma^{s}_{-}$ and $\gamma^{s}_{+}$. This implies
that $\Omega_{n}(\omega)$ is the closure of $\Sigma_{1}$ or of $\Sigma_{2}$,
and thus $P(\Sigma)$ crosses $\Omega_{n}(\mathfrak{S}(\omega))$ concluding.

Now the proof of Lemma 9.12 (and thus of Proposition 9.10) is complete. ∎

###  9.7 Itineraries, conjugacy, and topological equivalence

######  Theorem 9.14.

Consider $X,Y\in{\mathcal{O}}_{1}$ ad let $f_{X},f_{Y}$ be the corresponding
$1$-dimensional dynamics. Assume that the itineraries
$\omega_{-}(\gamma^{s}_{-},X)=\omega_{-}(\gamma^{s}_{-},Y)$ and
$\omega_{-}(\gamma^{s}_{+},X)=\omega_{-}(\gamma^{s}_{+},Y)$. Then there is a
orientation preserving map of $\SS^{1}$ which is a conjugation between $f_{X}$
and $f_{Y}$.

###### Proof.

We have seen that the sets ${\mathcal{A}}_{X}$ and ${\mathcal{A}}_{Y}$ of
admissible itineraries for $X$ and $Y$ are completely determined by
$\omega_{-}(\gamma^{s}_{\pm},X)$ and $\omega_{-}(\gamma^{s}_{\pm},Y)$,
respectively. Thus

${\mathcal{A}}_{X}={\mathcal{A}}_{Y}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}{\mathcal{A}}.$

Now for any $\omega\in{\mathcal{A}}$, Proposition 9.12 and Remark 9.11 imply
that there are a unique point $x_{\omega}\in\SS^{1}_{X}$ and
$y_{\omega}\in\SS^{1}_{Y}$ so that
$\omega\in\\{\omega_{-}(x_{\omega},f_{X}),\omega_{+}(x_{\omega},f_{X})\\}$ and
$\omega\in\\{\omega_{-}(y_{\omega},f_{Y}),\omega_{+}(y_{\omega},f_{Y})\\}$.

We define $h(x_{\omega})=y_{\omega}$. This defines a bijection from
$\SS^{1}_{X}$ to $\SS^{1}_{Y}$, which sends $\gamma^{s}_{i,X}$ on
$\gamma^{s}_{i,Y}$.

The punctured circle is an interval endowed with an order (frome the positive
orientation of the unstable segments), and Proposition 9.6 implies that
$\omega\mapsto x_{\omega}$ and $\omega\mapsto y_{\omega}$ are increasing. This
implies that $h\colon x_{\omega}\mapsto y_{\omega}$ is an increasing bijection
from $\SS^{1}_{X}\setminus\\{\gamma^{s}_{+,X}\\}$ onto
$\SS^{1}_{Y}\setminus\\{\gamma^{s}_{+,Y}\\}$. An increasing bijection between
to intervals is aa homeomorphism, so that $h$ is a homeomorphism.

The fact that $h$ is a conjugacy now comes from the fact that
$x_{\mathfrak{S}(\omega)}=f_{X}(x)$ for $x\notin\\{\gamma^{s}_{\pm,X}\\}$. ∎

Recall that the discontinuities are fixed points for the conjugacy $h$
contructed above.

We finish proving Theorem G, which establishes that the restriction to the
maximal invariant set of $X,\,Y\in\mathcal{O}_{1}$ are topologically
equivalent by a conjugacy close to identity if, and only if, $X$ and $Y$ have
the same itineraries.

###### Proof.

(of Theorem G) ($\Rightarrow$) A conjugation $\mathbb{H}$ between
$X|_{\Lambda_{X}}$ and $Y|_{\Lambda_{Y}}$ induces a topological conjugation
$h:\SS^{1}\to\SS^{1}$ between $f_{X}:\SS^{1}\to\SS^{1}$ and
$f_{Y}:\SS^{1}\to\SS^{1}$. Let
$\varepsilon=\frac{1}{2}\min\\{d(D_{1},\Sigma;d(D_{2},\Sigma))\\}$ (see
Section 3.1 for the definitions of $D_{1}$ and $D_{2}$). If
$d(\mathbb{H},id)<\varepsilon$, $h$ is orientation preserving and
$h(c_{i,X})=c_{i,Y}$ for $i\in\\{-,+\\}$, and therefore $X$ and $Y$ have the
same itineraries.
($\Leftarrow$ ) Fix $p\in\Sigma\cap\Lambda_{X}$. Because $X$ and $Y$ have de
the same itineraries, Lemma 9.12 together with Theorem 9.14 ensure the
existence of $q\in\Sigma\cap\Lambda_{Y}$ such that for all $n\in\mathbb{N}$,
points in $\gamma^{s}_{X}(P_{X}^{-n}(p))$ and $\gamma^{s}_{Y}(P_{Y}^{-n}(q)$
have the same itineraries. Thus
$C_{n}=\\{P_{Y}^{n}(\gamma^{s}_{Y}(P_{Y}^{-n}(q))\\}_{n\in\mathbb{N}}$ is a
sequence of compact sets in $\gamma^{s}_{Y}(q)$ converging to a single point
and we define $H(p)=\underset{n\in\mathbb{N}}{\cap}C_{n}$. Lema 9.9 implies
that $H$ is onto. For $p_{1},p_{2}\in\Sigma$ such that
$\gamma^{s}_{X}(p_{1})\neq\gamma^{s}_{X}(p_{2})$, it is easy to see that
$H(p_{1})\neq H(p_{2})$. For $p_{1}$ and $p_{2}$ in the same leave, there
exists $n_{1}\in\mathbb{N}$ for which $P^{-n_{1}}(p_{1})$ and
$P^{-n_{1}}(p_{2})$ belongs to different connected component of
$\Sigma\setminus\\{\gamma^{s}_{-,X},\gamma^{s}_{+,X}\\}$, which implies that
$\gamma^{s}_{X}(P_{X}^{-n_{1}}(p_{1}))\cap\gamma^{s}_{X}(P_{X}^{-n_{1}})(p_{2}))=\emptyset$
and hence $H_{n}(p_{1})\cap H_{n}(p_{2})\neq\emptyset$ for all $n>n_{1}$,
providing $H$ injectivity. The existence of unstable cone fields around
$\Sigma$ and the continuity of $h$ and $h^{-1}$ give the continuity of $H$ and
$H^{-1}$. To finish, for $p\in\Sigma\cap\Lambda_{X}$, consider
$\alpha\subset\mathcal{O}(p)$ and $\beta\subset\mathcal{O}(H(p))$, curves
parametrized by the arc length, joining $p$ to $D^{1}\cup D^{2}$ and $H(p)$ to
$D^{1}\cup D^{2}$ respectively, in away that $\alpha(t_{1}),\beta(t_{2})\notin
D^{1}\cup D^{2}$ for all $0<t_{1}<\ell(\alpha)$ and $0<t_{2}<\ell(\beta)$).
For $\rho$ being the ratio of length of $\alpha$ to the length of $\beta$, we
define $\mathbb{H}(t)=\beta(\rho t)$. Extend this map to segments of
trajectories leaving $D^{1}\cup D^{2}$ and returning to $\Sigma$ in the same
way as before. $\mathbb{H}$ defines a topological equivalence. Note that
$\mathbb{H}(D_{i})=D_{i}$ and $\mathbb{H}(\Sigma)=\Sigma$, which implies that
$d(\mathbb{H},id)<\varepsilon$ and we have the result.

∎

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