2.1 Basic topological notions

Let $\mathfrak {X}^r(\mathbb {R}^3)$ , $\,r\geq 1,$ be the space of vector fields defined on $\mathbb {R}^3$ , with the $C^r$ topology. For $X \in \mathfrak {X}^r(\mathbb {R}^3) $ , 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 an $\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 $p=p_0,p_1,\ldots ,p_n=p$ . The recurrent chain 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 an $\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’s theory [Reference Conley, Cesari, Hale and Lasalle14] proves that chain-equivalence is an 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. A compact region U is attracting for a vector field X of $\mathbb {R}^3$ if its boundary is a codimension $1$ submanifold transverse to X and X is entering in U.

Notice that the existence of an attractor is not ensured a priori. However, Conley [Reference Conley, Cesari, Hale and Lasalle14] proves the existence of quasi-attractors in any attracting region (for a vector field on a compact manifold).

2.2 Singular points

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

Hartman Grobman’s theorem asserts that a vector field is locally topologically equivalent to its linear part in a small neighborhood of a hyperbolic singular point [Reference Hartman24]. Then Sternberg provides conditions to guarantee that this local topological equivalence is true 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 $N = N(n) \ge 2$ such that if A is a real non-singular $d \times d$ matrix with eigenvalues $\unicode{x3bb} _1, \ldots ,\unicode{x3bb} _d$ satisfying

for any choice of non-negative 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 $\psi (0)=0$ and $\partial _v\psi (0)=0$ , 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 depend continuously on the $C^2$ topology from the vector field X, see for instance [Reference Pacifico42, Corollary in the Appendix] which provides a stronger statement.

2.3 The shift map $\mathfrak {S}$

We consider ${\mathbb X}=\{A_0,A_1,B_0,B_1\}^{\mathbb N}$ the space of infinite positive 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

$$ \begin{align*} (w_j)_{j\geq 0} \in {\mathbb X} \mapsto \mathfrak{S} (w_j)_{j\geq 0}= (w_{j+1})_{j\geq 0}.\end{align*} $$

We also define the star map (denoted by $\star $ ) on ${\mathbb X}$ as follows: given a sequence ${w \kern1.3pt{=}\kern1.3pt(w_0, w_1,\ldots )\in {\mathbb X}}$ and a letter $L \kern1.3pt{\in}\kern1.3pt \{A_0,A_1,B_0,B_1\}$ , we set $ L \star w \stackrel {\scriptscriptstyle \mathrm {def}}{=} (L, w_0, w_1, \ldots ). $

2.4 Hyperbolic notions

A hyperbolic invariant compact set K is called a basic hyperbolic set if it 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 $E^{cu}= E^u \oplus {\mathbb R} X$ , and these bundles are called respectively the weak stable and weakly unstable bundles.

Note that any hyperbolic set is also partially hyperbolic. Notice that $\Lambda _X$ an invariant compact set disjoint from $\mathrm {Zero}(X)$ 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 [Reference Morales, Pacifico and Pujals37].

2.5 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

$$ \begin{align*} W^{s}_X(p)&=\big\{q \in M\,: \mbox{dist}(X^t(q),X^t(p)) \underset{t \to +\infty}{\longrightarrow} 0\big\},\\ W^{u}_X(p)&=\big\{q \in M\,: \mbox{dist}(X^t(q),X^t(p)) \underset{t \to -\infty}{\longrightarrow} 0\big\}. \end{align*} $$

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 $W^{s}_X(\mathcal {O})= {\bigcup _{t \in \mathbb {R}}}W^{s}(X^t(p))$ . Analogously, the unstable set of the orbit of p is $W^{u}_X(\mathcal {O})= {\bigcup _{t \in \mathbb {R}}}W^{u}(X^t(p))$ .

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 partially hyperbolic, 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 uniquely to U as an invariant bundle (still denoted by $E^s$ ) tangent to a topological foliation. Recently, Araùjo and Melbourne, [Reference Araújo and Melbourne7, Theorem 4.12 and Remark 4.13(b)] provided bunching conditions ensuring the smoothness of the stable foliation, stated as follows.

2.6 Cone fields

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

$$ \begin{align*} C^s_{\gamma}(x)&= \{(v_1,v_2) \in F^u(x) \oplus F^s(x)\,: \|v_1\|\le \gamma \|v_2\|\},\\ C^s_{\gamma}(x)&= \{(v_1,v_2) \in F^u(x) \oplus F^s(x)\,: \|v_2\| \le \gamma \|v_1\|\}. \end{align*} $$

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

Here, $\unicode{x3bb} _i \in \mathbb {R},\, i \in \{ss, s, u\},$ of $DX(\sigma )$ satisfy $0<-\unicode{x3bb} _s<\unicode{x3bb} _{u}<-\unicode{x3bb} _{ss}$ .

2.7 Geometric Lorenz attractor

Here we recall the construction of the geometric Lorenz model, following [Reference Guckenheimer and Williams23]. For more on this construction, the interested reader can consult [Reference Galatolo and Pacifico18] and [Reference Araújo and Pacifico9, §3.3.2] for a detailed construction of this model. See Figure 5.

Figure 5 A geometric Lorenz flow, projection on I through the stable leaves; a sketch of the image of one leaf under the return map; and the one-dimensional map defined in the space of leaves of ${\mathcal F}^s$ .

Let $S\subset \mathbb {R}^3$ and a parameterization such that $S = \{(x, y, 1)|\,\, x,y \in \mathbb {R}\,\,\, \mbox {with}\,\, |x|, |y| \le 1/2\}$ . Let $\Gamma =\{(0, y,1) \in S, |y| \le 1/2\}$ and $\Sigma ^\pm = \{(x, y, z)\,\,x=\pm 1 \}.$

Consider a $C^1$ -vector field X on $\mathbb {R}^3$ satisfying the following conditions.

1. (1) For any point $(x, y,z)$ in a neighborhood of the origin $(0, 0, 0)$ , X is given by $(x',y',z') = (\unicode{x3bb} _1 x, \unicode{x3bb} _{2} y, \unicode{x3bb} _3 z)$ , where $0<-\unicode{x3bb} _3<\unicode{x3bb} _{1}<-\unicode{x3bb} _{2}$ .

2. (2) For all $p \in S \setminus \Gamma $ , there exist the smaller positive $t_p$ such that $X^{t_p}(p) \in S$ .

3. (3) S is a cross-section to X and the first return map $P:S \setminus \Gamma \to S$ is a piecewise $C^1$ -diffeomorphism given by $P(x,y) = (f(x), g(x,y))$ , satisfying: $f:[-1/2,1/2]\setminus \{0\} \to [-1/2,1/2]$ is piecewise $C^1$ map, $|f'|>\sqrt {2}$ where it is defined, $f(0^+)=-1/2$ and $f(0^-)=1/2$ , $|f'(0^{\pm })|=+\infty $ , and $\partial _y g(x,y) = C x^{\beta }$ for $C>0$ and $\beta>1$ .

Let $X^t$ be the flow of X. The maximal invariant set $\Lambda = \bigcap _{t\leq 0} X^t(S)$ is called a geometric Lorenz attractor. Figure 5 describes the main features of the construction of a geometric Lorenz attractor.

3.1 A geometric model for $X \in {\mathcal O}_1$

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

2. (2) The vector field X is transverse to three embedded, disjoint annuli denoted by $\Sigma $ , $D^1$ , and $D^2$ , see Figure 6.

   

   Figure 6 Trajectories of X crossing transversally $\Sigma $ , $D^1$ , and $D^2.$

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 by $R(p)=X^{t(p)}(p)$ .

4. (4) 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 they are disjoint and each of them is an essential annulus in $\Sigma $ (that is, it is not homotopic to a point).

   Note that the union of the orbit segments $\bigcup _{p\in D^1}X^{[0,t(p)]}(p)$ is the product of the annulus $D^1$ by a segment. The same holds for $D^2$ .

5. (5) $\Sigma \cap W^s_+(p)$ and $\Sigma \cap W^s_-(\sigma )$ 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 $ . The positive orbits of points in $\gamma ^s_+$ and $\gamma ^s_-$ go directly to the singular point and are disjoint from $\Sigma \cup D^1\cup D^2$ . Then $\Sigma \setminus (\gamma ^s_+ \cup \gamma ^s_-)$ consists of two components $\Sigma ^1$ and $\Sigma ^2.$ See Figure 6.

6. (6) 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 by $S(p)=X^{t(p)}(p)$ . 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 $ .

7. (7) 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 $D^i$ , $1\leq i \leq 2,$ 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 an annulus pinched at $\widetilde {q}_i$ . This pinched annulus is essential in $D^i$ (see Figure 7(a)): by pinched, we mean that in a neighborhood of $\widetilde {q}_i$ , the set $S(\Sigma ^i)\cup \{\widetilde {q}_i\}$ consists of two cuspidal triangles, with a cusp at $\widetilde {q}_i$ and tangent at this point to the same line but oriented in the opposite direction. By construction, the closure of $P(\Sigma ^1\cup \Sigma ^2)$ consists of two parallel essential annuli in $\Sigma $ , pinched at $q_i=R(\widetilde {q}_i)$ . See Figure 7(b).

   

   Figure 7 (a) The cross-section $\Sigma =\Sigma ^1 \cup \Sigma ^2,$ and the images $S(\Sigma ^1)\subset D^1$ and $S(\Sigma ^2)\subset D^2$ . (b) The image $P(\Sigma ^1\cup \Sigma ^2)\subset \Sigma $ .

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 the 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 $\Sigma \simeq {\mathbb S}^1\times [-1,1]$ , 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 $\unicode{x3bb}>1$ . The cone ${\mathcal C}^u(p)$ has two components: vectors cutting the fiber positively or negatively.

3. (10) There exists $t>0$ such that $\|DX^t|_{E^s_x}\|\cdot \|DX^{-t}|_{E^{cu}_{X^t(x)}}\|\cdot \|DX^t|_{E_x^{cu}}\|<1 \mbox { for all} x \in \Lambda .$

As a consequence of item (10) and [Reference Araújo and Melbourne7, Theorem 4.12], the stable foliation is well defined in U and it is of class $C^1$ . This foliation is not tangent to $\Sigma $ . However, there is a two-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 segments $\gamma ^s_+$ 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 $ ).

3.3 $X\in {\mathcal O}_1$ can be constructed in any three-dimensional manifold

To see that X satisfying items (1) to (10) can be realized in any $3$ -manifold, we first construct an attracting region ${\mathbb U}\subset {\mathbb R}^3$ containing $\Sigma $ , $D^1$ , $D^2$ , and the singularity $\sigma $ so that every regular orbit of X in ${\mathbb U}$ crosses $\Sigma $ transversally. The maximal invariant set $\Lambda $ in ${\mathbb U}$ is an attracting invariant compact set (see Figure 8).

Figure 8 Determining the attracting region.

To construct ${\mathbb U}$ , we proceed as follows. Let ${\mathbb D}\subset {\mathbb C}$ be the unit disk in the complex plane and consider the solid cylinder ${\mathbb A}_0={\mathbb D} \times [-1,1]$ with coordinates $z\in {\mathbb C}$ and $ t \in [-1,1]$ . Define the solid cylinders

$$ \begin{align*}&{\mathbb C}_1= \{ (z, t); \|z\|< 1-\varepsilon, t \in (-1+\varepsilon, -\varepsilon) \} \quad\mbox{and}\\ &\quad{\mathbb C}_2= \{ (z, t); \|z\|< 1-\varepsilon, t\in (\varepsilon, 1-\varepsilon) \}. \end{align*} $$

Let ${\mathbb A}_1={\mathbb A}_0\setminus ({\mathbb C}_1 \cup {\mathbb C}_2)$ and pick the points $x=1/2, y=-1/2 \in {\mathbb C}$ . Define

$$ \begin{align*}&\widetilde{{\mathbb D}_1}=\{(z,t); \|z-x\|< 1/4,\, t \in (-1 + \varepsilon,0)\} \quad\mbox{and}\\ &\quad \widetilde{{\mathbb D}_2}=\{(z,t); \|z-y\|< 1/4,\, t \in (0,1-\varepsilon)\}. \end{align*} $$

Consider ${\mathbb A}_2= {\mathbb A}_1 \cup (\widetilde {{\mathbb D}_1} \cup \widetilde {{\mathbb D}_2}),$ and define

$$ \begin{align*} {\mathbb C}_3=\{(z,t); \|z-x\|< 1/8,\, t \in (-\varepsilon,1)\}, \,\, {\mathbb C}_4=\{(z,t); \|z-y\|< 1/4,\, t \in (-1,\varepsilon)\}. \end{align*} $$

Consider

(2) $$ \begin{align} {\mathbb U}={\mathbb A}_2 \setminus ({\mathbb C}_3 \cup{\mathbb C}_4). \end{align} $$

We define X in the interior of the region ${\mathbb U}$ as follows. We can assume that the singularity $\sigma $ is in the interior of ${\mathbb U}$ and that its stable set $W^s(\sigma )$ intersects $\Sigma = \{(z,t) \in {\mathbb U} \times [-\varepsilon ,\varepsilon ]\,\,;\,\, \|z\|=1-\varepsilon \}$ into the two disjoint curves $\gamma ^s_-$ and $\gamma ^s_+$ that split $\Sigma $ into two regions $\Sigma ^1$ and $\Sigma ^2$ . Let $D^1=\{(z,t); \|z\|= 1/4, \,\,t \in (-\varepsilon /2,\varepsilon /2)\} $ and $D^2=\{(z,t); \|z\|= 1/4, \,\,t \in (-\varepsilon /2,\varepsilon /2)\}$ .

Define X in such a way that it crosses inward transversally $\Sigma $ and, for each point $p\in \Sigma ^1$ (respectively $\Sigma ^2$ ), there is a first positive time $t_p$ such that $X^{t_p}(p) \in D^1$ (respectively $D^2$ ). We set, to keep the previous notation, $X^{t_p}(p)\stackrel {\scriptscriptstyle \mathrm {def}}{=} S(p)$ .

Moreover, for each point $\tilde {q}=S(p) \in D^1$ (respectively $D^2$ ), there is a first positive time $t_{\tilde {q}}$ such that $X^{t_{\tilde {q}}}(q)$ crosses inward transversally $\Sigma $ . Again, to keep the previous notation, we set $X^{t_{\tilde {q}}}(q) \stackrel {\scriptscriptstyle \mathrm {def}}{=} R(\tilde {q})$ . This completes the definition of X on the cross-section $\Sigma $ . Clearly, $\Sigma $ is a global cross-section to X and the first return map $P: \Sigma \to \Sigma $ is equal to $R\circ S$ . Then we easily complete the definition of X on $\bigcup _{t\geq 0}\Sigma $ satisfying all the properties described above.

By construction, the region ${\mathbb U}$ defined in equation (2) is homeomorphic to the attracting region for our flow $X^t$ , and it is contained in ${\mathbb R}^3$ . Figure 9 represents the geometric realization of the union $\bigcup _{p\in D^i}X^{[0,t(p)]}(p)$ , where $1\leq i\leq 2$ . This illustration demonstrates that this construction can be realized within a ball contained in $\mathbb {R}^3$ and, consequently, within any three-dimensional compact manifold.

Figure 9 (a) Trajectories of X from $D^1$ to $\Sigma $ , (b) the cross-section $\Sigma =\Sigma ^1\cup \Sigma ^2$ , and the images $S(\Sigma ^1) \subset D^1$ and $S(\Sigma ^2) \subset D^2$ , (c) trajectories of X from $D^2$ to $\Sigma $ .

4.1 Quasi-attractor

We start by noticing that there is non-ambiguity on what can be the attractor.

As U is an attracting region, Conley’s theory implies that U contains at least one quasi-attractor. Now Proposition 4.2 is a direct consequence of the following lemma.

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

4.2 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 is 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 $\bar P^n(\Sigma )$ ,

$$ \begin{align*}\Lambda_P=\bigcap_{n\geq 0} \bar P^n(\Sigma).\end{align*} $$

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, and the boundary is tangent to the unstable cone. The intersection of two annuli is contained in this finite set of pinched points. Finally, the thickness of these annuli is exponentially decreasing with n. The intersection is an essential circle for any nested sequence of such pinched annuli in $\bar P^n(\Sigma )$ , $n>0$ . In particular, the topological dimension of $\Lambda _P$ is $1$ . Moreover, the boundary of any annulus is tangent to the unstable cone and 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 $E^u$ .

One easily checks that is enough to prove the following lemma.

We now present tools to ensure that the orbit of an unstable segment (that is, a segment in $\Sigma $ tangent to the unstable cone) cuts almost every stable leaf.

4.3 Fixing the expansion rate bigger than the golden number $\varphi = ({1+\sqrt 5})/2$

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 send them into $\Sigma $ as a pinched essential annulus. Therefore, the expansion rate $\unicode{x3bb} $ 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 10 displays the main features of the return map P.

Figure 10 Another view of the cross-section $\Sigma $ and the image of $\Sigma $ under the Poincaré map.

The proof of the main topological properties consists of iterating unstable segments (tangent to the unstable cone) and estimating the length of these iterated components. In this section, we will choose a rate $\unicode{x3bb} $ which will allow us to estimate these lengths. 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 lengths $\unicode{x3bb} ^2\ell ([a,b])$ or $\unicode{x3bb} ^2\ell ([b, c])$ is strictly larger than $\ell ([a,c]$ if $\unicode{x3bb}>\sqrt {2}$ . The choice of $\unicode{x3bb}>\sqrt 2$ ensures the increase of a segment split into two components after two iterations if those components are not split again. This is the standard rate of expansion to guarantee the transitivity of a Lorenz attractor. We note that there are examples of one-dimensional Lorenz maps with a rate of expansion smaller than $\sqrt {2}$ that are not transitive, see [Reference Araújo and Pacifico9, p. 126]. Consider the maximum of the lengths $\unicode{x3bb} \ell ([a,b])$ and $\unicode{x3bb} ^2\ell ([b,c])$ . Notice that the golden ratio $\varphi = ({1+\sqrt 5})/2\in\ ]1,2[$ .

We consider now the open subset ${\mathcal O}_\varphi \subset {\mathcal O}_1$ consisting of flows X so that the expansion rate $\unicode{x3bb} $ of the return map in the unstable cone is larger than $\varphi $ .

4.4 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 ${\mathcal H}^i$ into 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 11 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 in the following.

Figure 11 (a) $ q_1 \in \gamma _+^s$ implying that $X \in {\mathcal H}^1_+ $ and (b) $ q_1 \in \gamma _-^s$ implying that $X \in {\mathcal H}^1_- $ .

The submanifolds ${\mathcal H}^1$ and ${\mathcal H}^2$ cut ${\mathcal O}_\varphi $ into $4$ 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 we have the following lemma.

Figure 12 $ p_1$ and $p_2$ are fixed points of $X \in {\mathcal O}_{\varphi }\setminus {\mathcal H}^{1}\cup {\mathcal H}^{2}$ .

Consider now the region ${\mathcal O}_\varphi ^{+,+}$ : every vector field $X\in {\mathcal O}_\varphi ^{+,+}$ has exactly one fixed point $p_1$ in $\Sigma ^1$ and one 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 ${\mathcal H}{\mathcal E}_i$ as 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 13. As in Lemma 4.11, one has the following lemma.

Figure 13 (a) $X \in {\mathcal H}{\mathcal E}_{1}$ and (b) $X \in {\mathcal H}{\mathcal E}_{2}$ .

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 two connected components: one, denoted by $\Sigma _+$ , contains $\gamma ^s_+$ , and the other, denoted by $\Sigma _-$ , contains $\gamma ^s_-$ . We denote 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 components $\Sigma _\pm $ and $\Sigma _\mp $ . We will denote ${\mathcal L}^{+,-}$ as the union

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

Recall that ${\mathcal H}^i$ corresponds to a homoclinic loop and ${\mathcal H}^i_\pm $ distinguishes the up or down connected components of $W^{s}(\sigma )\setminus W^{ss}(\sigma )$ involved in that loop.

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

This section aims to prove the following result.

Proof. The component $\Sigma _+$ is a rectangle. The stable leaf $\gamma ^s_+$ cuts $\Sigma _+$ into two components. 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 14. Then $L_1\cup L_2$ cuts $\Sigma ^+$ into three 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 $R_{L_2}\subset \Sigma _2$ .

Figure 14 The leaves $L_1$ and $L_2$ , and the region $\Sigma _+$ .

Consider the restriction of P to $R_L\setminus \gamma ^+_s$ . The images of $R_{L_1}$ and $R_{L_2}$ are cuspidal triangles with cusps at $q_1$ and $q_2$ , and are contained in $R_L$ . Recall that P is hyperbolic. Thus, the restriction of P to $R_L$ satisfies all the properties of the return map in the geometric model of the Lorenz attractor with a rate expansion larger than $\varphi>\sqrt 2$ . The rectangle $R_L$ is an attracting region for P. One deduces that U contains an attracting sub-region, in which X is a geometric model of the Lorenz attractor.

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 ${\mathcal O}_\varphi ^{-,-}$

Proof. According to Lemma 4.9, for proving the transitivity of the maximal invariant, it is enough to verify that the iterates of an unstable segment S in $\Sigma $ cut all the stable leaves with the possible exception of a set with an empty interior, see Figure 15.

Figure 15 $q_1 \in \Sigma ^1$ and $q_2 \in \Sigma ^2$ .

So consider an 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, 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 $S'$ in any iterate $P^n(S)$ has a length bounded by $\delta \ell (S)$ . We now fix $\delta \in ]1,{\unicode{x3bb} ^2}/2[$ .

If S is disjoint from $\gamma ^s_+\cup \gamma ^s_-$ , then $\ell (P(S))>\unicode{x3bb} \ell (S)>\delta \ell (S)$ contradicts 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 one, 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 a length larger or equal to $\frac 12\ell (S)$ . The length of $P(S_1)$ is at least $(\unicode{x3bb} /2) \ell (S)$ .

If $P(S_1)$ is disjoint from $\gamma ^s_+\cup \gamma ^s_-$ , then

$$ \begin{align*} \ell(P(S_1)>\unicode{x3bb} \ell(S_1)> \frac{\unicode{x3bb}^2}2\ell(S)> \delta\ell(S), \end{align*} $$

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 endpoints 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 of a union of essential circles in $\Sigma $ and therefore always cuts $\gamma ^s_+$ and $\gamma ^s_-$ , so that the maximal invariant set $\Lambda _X$ intersects non-trivially both stable sets $W^s_+(\sigma )$ and $W^s_-(\sigma )$ . Thus, $\Lambda _X$ is a two-sided Lorenz attractor, ending the proof. See Figure 15.

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

The proof of the proposition for $X\in {\mathcal O}_{\varphi }^{-,+}$ is identical to the proof for $X\in {\mathcal O}_{\varphi }^{+,-} $ interchanging the components $\Sigma ^1$ and $\Sigma ^2$ , so we will write the proof only for $X\in {\mathcal O}_{\varphi }^{+,-}$ . See Figure 16.

Figure 16 $q_1, q_2 \in \Sigma ^2$ for $X\in {\mathcal O}^{-,+}$ .

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

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 symmetrical in these two open sets. We provide here the proof where $q_1\in \Sigma _+$ . See Figure 17.

Figure 17 $q_1 \in \Sigma _+$ for $X\in \widetilde {{\mathcal O}_\varphi }^{+,+}$ .

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

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 admit an identical proof, mutatis mutandis). See Figure 18.

Figure 18 $X \in {\mathcal H}^1 \cup {\mathcal H}^2$ .

Similarly to the proof of Propositions 5.1, 5.2, and 5.3, the proof consists in considering an unstable segment S which does not admit any segment $\tilde S$ of length larger than $\delta \ell (S)$ , $1<\delta <\unicode{x3bb} /\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 one 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 $\unicode{x3bb} \ell (S)$ 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 $\unicode{x3bb} ^2\ell (S)$ 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 $q_2$ . However, $P(S_2)$ is a segment starting at $q_2$ . Now $P^2(S_1)\cup \{q_2\}\cup P(S_2)$ is a segment of length at least $\unicode{x3bb} \ell (S)$ . This contradicts the choice of S, ending the proof.

The statement of Theorem B does not announce a two-sided Lorenz attractor, because the ${\mathcal H}^i$ are not open subsets, and then the robustness of the transitivity is 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 ${\mathcal H}^{1,2}_-$ be 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 19. Thus, both unstable separatrices of $\sigma $ are homoclinic connections and are included in the same connected component of $W^s(\sigma )\setminus W^{ss}(\sigma )$ .

Figure 19 $ X \in {\mathcal H}^1\cup {\mathcal H}^2$ , and $q_1,q_2\in \gamma ^s_+$ and $q_1,q_2\in \gamma ^s_-$ .

The 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.

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

Note that Propositions 5.1, 5.2, 5.3, and B and Lemma C together prove the following theorem.

The region ${\mathcal L}^{+,-}$ has been defined as a union of many regions, and the propositions listed above prove the conclusion of Proposition D in each of these regions. Finally, some of these regions are not open, so Lemma C checks that the vector fields in these non-open regions admit neighborhoods contained in 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 has two fixed points $p_i\in \Sigma ^i$ , $i=1,2$ , a heteroclinic connection 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.

Proof. The first return map is illustrated in Figure 20. The study of the first return map in each rectangle $\overline \Sigma _i$ is classical for the expansion rate larger than $\sqrt 2<\varphi $ .

Figure 20 $X\in {\mathcal H}{\mathcal E}_1\cap {\mathcal H}{\mathcal E}_2$ .

The submanifold ${\mathcal H}{\mathcal E}_1\cap {\mathcal H}{\mathcal E}_2$ cuts ${\mathcal H}{\mathcal E}_1$ into two (relative) open subsets, as $q_2$ belongs either to $\Sigma _+$ or $\Sigma _-$ (connected components of $\Sigma \setminus (W^s_1\cup W^s_2)$ . In the same way, the submanifold ${\mathcal H}{\mathcal E}_2\cap {\mathcal H}{\mathcal E}_2$ cuts ${\mathcal H}{\mathcal E}_2$ into two (relative) open subsets, as $q_1$ belongs either to $\Sigma _+$ or $\Sigma _-$ (connected components of $\Sigma \setminus (W^s_1\cup W^s_2)$ .

We will consider $X\in {\mathcal H}{\mathcal E}_1\setminus ({\mathcal H}{\mathcal E}_1\cap {\mathcal H}{\mathcal E}_2)$ , and $q_2\in \Sigma _+$ , the other cases are similar.

Then the stable leaf through $q_2$ cuts $\Sigma _+$ into two components, one, denoted as $\Sigma ^+_1$ , bounded by the stable leaf $W^s_1$ through $p_1$ 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 $q_2\in \Sigma ^1$ . Note that if $X\in {\mathcal H}{\mathcal E}_1\setminus ({\mathcal H}{\mathcal E}_1\cap {\mathcal H}{\mathcal E}_2)$ , then $q_1\in W^s(p_2)$ and ${q_2 \notin W^s(p_1)}$ .