Finitely many physical measures for sectional-hyperbolic attracting sets and statistical stability

TL;DR

This paper proves that sectional-hyperbolic attracting sets have finitely many physical measures with basins covering almost all points, and these measures depend continuously on the flow, establishing statistical stability.

Contribution

It establishes the finiteness and statistical stability of physical measures for sectional-hyperbolic attracting sets in $C^1$ flows.

Findings

Finitely many physical measures cover almost all points in the basin.

Physical measures depend continuously on the flow in the $C^1$ topology.

Central-unstable disks expand to contain uniformly sized balls.

Abstract

We show that a sectional-hyperbolic attracting set for a H\"older-$C^1$ vector field admits finitely many physical/SRB measures whose ergodic basins cover Lebesgue almost all points of the basin of topological attraction. In addition, these physical measures depend continuously on the flow in the $C^1$ topology, that is, sectional-hyperbolic attracting sets are statistically stable. To prove these results we show that each central-unstable disk in a neighborhood of this class of attracting sets is eventually expanded to contain a ball whose inner radius is uniformly bounded away from zero.

Figures

Equations

$$B(\mu )=\left\{z\in M:\underset{t\to \infty }{\lim }\frac{1}{t}{\int }_0^t\psi ({\varphi }_s(z))ds=\int \psi d\mu ,\forall \psi \in C^0(M,\mathbb{R})\right\}$$

$$|\det (D{\varphi }_t(x)\mid P_x)|\ge K{e}^{\theta t}\text{for all x∈Λ, t≥0}.$$

$$\displaystyle\mu\in\Phi(G)=\big{\{}\sum_{i=1}^{k}t_{i}\mu_{i}:t_{i}\geq 0\quad\text{and}\quad\sum_{i=1}^{k}t_{i}=1\big{\}}.$$

$$W_x^s=\{y\in M:d({\varphi }_{t}y,{\varphi }_{t}x)\underset{t\to +\infty }{\to }0\}$$

$$W_x^u({\epsilon }_0)=\{y\in M:d({\varphi }_{-t}y,{\varphi }_{-t}x)\le {\epsilon }_0,\forall t\ge 0\}$$

$${\mathcal{W}}^{cs}=\{W_x^s:x\in W_{\mathcal{O}(p_i)}^u({\epsilon }_0);i=1,\dots ,l\}$$

$$\omega (x)={\omega }_G(x)=\left\{y\in M:\exists t_n\nearrow \infty \text{ s.t. }{\varphi }_{t_n}x\underset{n\to \infty }{\to }y\right\}$$

$$f(x)={\varphi }_{\tau (x)}(x)\text{where}\tau (x)=\inf \left\{t>T_1:{\varphi }_{t}x\in \bigcup _{j=1}^{\ell }\mathrm{Closure}{\Sigma }_{y_j}(a_0)\right\}.$$

$$f(y)={\varphi }_{\tau (y)}(y)\text{where}\tau (y)=\inf \left\{t>T_1:{\varphi }_{t}y\in \bigcup _{j=1}^{\ell }{\Sigma }_{y_j}\right\}.$$

$$g_k\mid {\hat{D}}_k:{\hat{D}}_k\to D_k=g_k{\hat{D}}_k\subset \Xi \text{is a diffeomorphism for each k≥1.}$$

$$\rho (\tilde{f}D)\ge {\lambda }_2^{-1}\rho (D)\text{and}(1-2a)\rho (\tilde{f}D)\le 2\mathrm{diam}(\tilde{f}D)\le (1+2a)\rho (\tilde{f}D),$$

$$\Vert {\dot{\gamma }}_1(s)\Vert$$

$$\displaystyle\geq\lambda_{1}^{-1}\big{\|}D\exp_{y}\big{(}\dot{\gamma}(s)+Dg_{1}(\gamma(s))\cdot\dot{\gamma}(s)\big{)}\big{\|}\geq\frac{\lambda_{1}^{-1}}{2}\cdot(1-2a)\|\dot{\gamma}(s)\|.$$

$$d(D_n,{\Gamma }_0)$$

$$\rho (D_n)$$

$$h_{\mu }({\varphi }_1)=\int \log |\det D{\varphi }_1\mid E^{cu}|d\mu >0$$

$$\underset{T\to \infty }{\lim }\frac{1}{T}{\int }_0^T\phi ({\varphi }_{t}x)dt=\mu (\phi )>0,\text{for each x∈B(μ)}.$$

$$h_{top}(g,K)=\underset{\epsilon \to 0}{\lim }r(K,\epsilon ),$$

$$h_{\mu }(g,\mathcal{P})=\underset{n\ge 1}{\inf }\frac{1}{n}{H}_{\mu }({\mathcal{P}}^n)\text{where}{H}_{\mu }({\mathcal{P}}^n)=\sum _{B\in {\mathcal{P}}^{n-1}}-\mu (B)\log \mu (B)$$

$$d_H(K,L)=\inf \{r>0:K\subset B(L,r)\text{and}L\subset B(K,r)\}.$$

$$\mathrm{supp}{\mu }_n\subset V\text{and}\mathrm{dist}(E_{n,x}^{\ast };E_{{\Lambda }_G(U),x}^{\ast })<\xi ,x\in V;\ast =s,cu,\forall n>N;$$

$$\displaystyle\operatorname{dist}(E,F):=\max\big{\{}\sup_{\|v\|=1,v\in E}\operatorname{dist}(v,F),\sup_{\|v\|=1,v\in F}\operatorname{dist}(v,E)\big{\}};$$

$$\partial {\mathcal{P}}^n$$

$$\displaystyle 0=\limsup_{n}\big{(}h_{\mu_{n}}(G_{n})+\int\psi_{n}\,d\mu_{n}\big{)}\leq\limsup_{n}\big{(}\frac{1}{k}H_{\mu_{n}}(\mathcal{P}_{n}^{k})+\int\psi_{n}\,d\mu_{n}\big{)}$$

$$\displaystyle\big{|}\int\psi_{n}\,d\mu_{n}-\int\psi\,d\mu\big{|}\leq\big{|}\int(\psi_{n}-\psi)\,d\mu_{n}\big{|}+\big{|}\int\psi\,d\mu_{n}-\int\psi\,d\mu\big{|}\xrightarrow[n\to\infty]{}0.$$

$$\displaystyle 0\leq\inf_{k\geq 1}\big{(}\frac{1}{k}H_{\mu}(\mathcal{P}^{k})-\int\psi\,d\mu\big{)}=h_{\mu}(G)-\int\psi\,d\mu\leq 0$$

$$|t_i-s_i|<\zeta ,t_i,s_i\in \mathbb{R},i=1,\dots ,k⟹\sum _{i=1}^k-x_i\log x_i<\omega ;$$

$$n\ge L,Q\in {\mathcal{P}}^k⟹{\mu }_n(B_{\delta }(Q))\le \mu (B_{\delta }(Q))+\frac{\zeta }{2}.$$

$$Q_n\subset B_{\delta }(Q)\text{and}{\mu }_n(Q_n)\le {\mu }_n(B_{\delta }(Q))\le \mu (B_{\delta }(Q))+\frac{\zeta }{2}\le \mu (Q)+\zeta$$

$$\displaystyle\frac{1}{k}H_{\mu_{n}}(\mathcal{P}^{k}_{n})\leq\frac{1}{k}\big{(}H_{\mu}(\mathcal{P}^{k})+\omega\big{)}\leq\frac{1}{k}H_{\mu}(\mathcal{P}^{k})+\frac{\omega}{k}$$

$$\underset{n\to \infty }{\mathrm{limsup}}\frac{1}{k}{H}_{{\mu }_n}({\mathcal{P}}_n^k)\le \frac{1}{k}{H}_{\mu }({\mathcal{P}}^k)$$

Full text

The author was partially supported by CNPq-Brazil (grant 300985/2019-3).

1. 1 Introduction and statements of the results

2. 1.1 Preliminary definitions

3. 1.2 Statement of the results

4. 1.3 Organization of the text

5. 2 Preliminary results on sectional-hyperbolic attracting sets

6. 2.1 Lorenz-like singularities

7. 2.2 Extension of the stable bundle and center-unstable cone fields

8. 2.3 Global Poincaré map on adapted cross-sections

9. 2.3.1 Construction of a global adapted cross-section

10. 2.3.2 The Poincaré map

11. 2.4 Hyperbolicity of the global Poincaré map

12. 2.4.1 Hyperbolicity at each smooth strip

13. 2.4.2 Hyperbolicity of the extensions of the Poincaré maps at smooth strips

14. 3 The basin of sectional-hyperbolic attracting sets

15. 3.1 Denseness of stable leaves of $\Lambda$ on $U$

16. 3.2 Local uniform expansion of $cu$-disks

17. 4 Finitely many ergodic physical measures for sectional hyperbolic attracting sets

18. 4.1 Uniformly center-unstable size of invariant subsets

19. 4.2 Uniform volume of ergodic basis of physical measures

20. 4.3 Full volume of stable leaves in the topological basin

21. 5 Statistical stability of sectional-hyperbolic attracting sets

22. 5.1 Entropy expansiveness

23. 5.2 Upper semicontinuity of metric entropy

24. 5.3 Equilibrium states and physical measures

25. 5.4 Statistical stability

1. Introduction and statements of the results

The term statistical properties of a dynamical system refers to the statistical behavior of typical trajectories of the system. It is well known that this relates to the properties of the evolution of measures by the dynamics. Statistical properties are often a better object to be studied than pointwise behavior. In fact, the future behavior of initial data can be unpredictable, but statistical properties are often regular and their description simpler.

Arguably one of the most influential concepts in the theory of Dynamical Systems has been the notion of physical (or $SRB$) measure. We say that an invariant probability measure $\mu$ for a flow $\phi_{t}$ is physical if the set

[TABLE]

has non-zero volume, with respect to any volume form on the ambient compact manifold $M$. The set $B(\mu)$ is by definition the basin of $\mu$. It is assumed that time averages of these orbits be observable if the flow models a physical phenomenon.

The study of the existence of these special measures and their statistical properties for uniformly hyperbolic diffeomorphisms and flows has a long and rich history, starting with the works of Sinai, Ruelle and Bowen [17, 18, 47, 48, 52]. Some classes of systems that not satisfy all the basic assumptions of uniform hyperbolicity have been shown to possess physical measures much more recently: sectional-hyperbolicity is a generalization of Smale’s notion of Axiom A [53] that allows for the inclusion of equilibria (also known as singularities or steady-states) and incorporates the classical Lorenz attractor [29] as well as the geometric Lorenz attractors of [1, 24]. For three-dimensional flows, sectional hyperbolic attractors are precisely the ones that are robustly transitive, and they reduce to Axiom A attractors when there are no equilibria [38].

For arbitrary dimensions this notion was established first in [32] and the first concrete example provided by [15]. Sectional-hyperbolic attractors are those robustly transitive attracting sets for which the flow in a star flow in the trapping region, that is, there are no bifurcations of singularities or periodic orbits for all nearby dynamics (also known as “strongly homogeneous flows”). Again these sets reduce to Axiom A attractors if there are no equilibria.

Sectional-hyperbolic attractors in $3$-manifolds were shown to have a unique physical measure in [8, 7] and sectional-hyperbolic attracting sets have finitely many ergodic physical measures whose basins cover a full volume subset of a neighborhood of the attracting set; see [51, 9]. The study of statistical properties of these measures is well developed program, we mention the recent works [30, 25, 50, 23, 10, 3, 6, 4, 5, 12] among others.

Recently it was shown the existence of a unique physical measure for sectional-hyperbolic attractors for flows in manifolds with any finite dimension in [28] using the Thermodynamical Formalism and assuming certain properties of a stable foliation in a neighborhood of the attracting set, common to the above mentioned works in the $3$-dimensional setting; see also [33] for a different proof using stochastic stability of such attractors.

Various issues regarding the existence and smoothness of the stable foliation in a neighborhood of sectional-hyperbolic attracting sets are clarified in [4]; a topological foliation always exists, and an analytic proof of smoothness of the foliation for the classical Lorenz attractor (and nearby attractors) is given in [4, 6]. In [5] sufficient conditions are provided for these foliations to have absolutely continuous holonomy maps, a crucial technical feature to obtain many statistical properties in dynamics. For higher differentiability properties of these foliations for geometric Lorenz attractors, see [54].

Here we pave the way to further study of statistical properties of sectional-hyperbolic attracting sets. We solve the basin problem for sectional-hyperbolic attracting sets, that is, we show that an open dense and full measure subset of points in a neighborhood of these sets is exponentially asymptotic to some orbit inside the set. More precisely: given a neighborhood $U$ of an invariant sectional-hyperbolic attracting set $\Lambda$ of a smooth flow $\phi_{t}$, there exists $K,\lambda>0$ and an open and dense subset $W\subset U$ with full Lebesgue measure (${\rm Leb}(U\setminus W)=0$) so that for any given $y\in W$ there exists $x\in\Lambda$ satisfying $d(\phi_{t}y,\phi_{t}x)\leq Ke^{-\lambda t}$ for all $t>0$.

Moreover, coupled with recent results from [20] on weak limits of time averages for almost all orbits in partially hyperbolic sets with applications to sectional-hyperbolic attracting sets, we complement [28] proving the existence of finitely many ergodic physical measures for sectional-hyperbolic attracting sets in any dimension. In addition, the basins of these measures cover a full Lebesgue measure subset of a neighborhood of the sectional-hyperbolic attracting set.

Having this, we use recent results from [40] on robust entropy expansiveness for sectional-hyperbolic attracting sets to prove that the physical measures depend continuously on the flow, showing that asymptotic time averages for Lebesgue almost all points in a neighborhood of such attracting sets are robust under small perturbations of the dynamics. This is known as statistical stability and our proof provides a far-reaching extension of the results already obtained for the $3$-flows having geometric Lorenz attractors in [2] and the classical Lorenz attractor in [11].

1.1. Preliminary definitions

Let $M$ be a compact Riemannian manifold with induced distance $d$ and volume form ${\rm Leb}$. Let $\mathcal{X}^{1}(M)$ be the set of $C^{1}$ vector fields on $M$ and denote by $\phi^{G}_{t}$ the flow generated by $G\in\mathcal{X}^{1}(M)$. We say that $G$ is Hölder-$C^{1}$ if on any local chart the derivative $DG$ is $\alpha$-Hölder for some fixed $0<\alpha<1$. We write $\mathcal{X}^{1+}(M)$ for the vector space of all Hölder-$C^{1}$ vector fields over $M$.

Given a compact invariant set $\Lambda$ for $G\in\mathcal{X}^{1}(M)$, we say that $\Lambda$ is isolated if there exists an open set $U\supset\Lambda$ such that $\Lambda=\bigcap_{t\in\mathcal{R}}\phi_{t}(U)$. If $U$ can be chosen so that $\phi_{t}(U)\subset U$ for all $t>0$, then we say that $\Lambda$ is an attracting set.

A compact invariant set $\Lambda$ is partially hyperbolic if the tangent bundle over $\Lambda$ can be written as a continuous $D\phi_{t}$-invariant sum $T_{\Lambda}M=E^{s}\oplus E^{cu},$ where $d_{s}=\dim E^{s}_{x}\geq 1$ and $d_{cu}=\dim E^{cu}_{x}\geq 2$ for $x\in\Lambda$, and there exist constants $C>0$, $\lambda\in(0,1)$ such that for all $x\in\Lambda$, $t\geq 0$, we have

• •

uniform contraction along $E^{s}$: $\|D\phi_{t}|E^{s}_{x}\|\leq C\lambda^{t};$ and

• •

domination of the splitting: $\|D\phi_{t}|E^{s}_{x}\|\cdot\|D\phi_{-t}|E^{cu}_{\phi_{t}x}\|\leq C\lambda^{t}.$

We say that $E^{s}$ is the stable bundle and $E^{cu}$ the center-unstable bundle. A partially hyperbolic attracting set is a partially hyperbolic set that is also an attracting set.

We say that the center-unstable bundle $E^{cu}$ is sectional expanding if for every two-dimensional subspace $P_{x}\subset E^{cu}_{x}$,

[TABLE]

If $\sigma\in M$ and $G(\sigma)=0$, then $\sigma$ is called an equilibrium or singularity in what follows and we denote by $\operatorname{Sing}(G)$ the family of all such points. An invariant set is nontrivial if it is neither a periodic orbit nor an equilibrium.

We say that a compact invariant set $\Lambda$ is a sectional hyperbolic set if $\Lambda$ is partially hyperbolic with sectional expanding center-unstable bundle and all equilibria in $\Lambda$ are hyperbolic. A sectional hyperbolic set which is also an attracting set is called a sectional hyperbolic attracting set.

A singular hyperbolic set is a compact invariant set $\Lambda$ which is partially hyperbolic with volume expanding center-unstable subbundle and all equilibria within the set are hyperbolic. A sectional hyperbolic set is singular hyperbolic and both notions coincide if, and only if, $d_{cu}=2$.

Remark 1.1**.**

1. (1)

A sectional hyperbolic set with no equilibria is necessarily a hyperbolic set, that is, the central unstable subbundle admits a splitting $E^{cu}_{x}={\mathbb{R}}\{G(x)\}\oplus E^{u}_{x}$ for all $x\in\Lambda$ where $E^{u}_{x}$ is uniformly contracting under the time reversed flow; see e.g. **[7]**. 2. (2)

A sectional hyperbolic attracting set cannot contain isolated periodic orbits. For otherwise such orbit must be a periodic sink, contradicting volume expansion.

We recall that a subset $\Lambda\subset M$ is transitive if it has a full dense orbit, that is, there exists $x\in\Lambda$ such that $\operatorname{Closure}{\{\phi_{t}x:t\geq 0\}}=\Lambda=\operatorname{Closure}{\{\phi_{t}x:t\leq 0\}}$.

A nontrivial transitive sectional hyperbolic attracting set is a sectional hyperbolic attractor. For more details on these notions, see e.g. [7] and references therein.

1.2. Statement of the results

The definition of singular-hyperbolicity ensures that every invariant probability measure supported in a singular-hyperbolic set is a hyperbolic measure. Moreover, if the vector field is smooth (at least Hölder-$C^{1}$) from the proof of [8, Theorem B, Section 4] or explicitly from [51, Theorem 1.5], we get that every singular-hyperbolic attracting set admits finitely many $\mu_{1},\dots,\mu_{k}$ ergodic physical/SRB invariant measures; and the union of the ergodic basins of these measures covers a full Lebesgue measure subset of the topological basin of attraction of $\Lambda$, i.e. ${\rm Leb}(U\setminus\cup_{i=1}^{k}B(\mu_{i}))=0$.

We show here that the same result is true in higher dimensions for sectional-hyperbolic attracting sets.

Theorem A**.**

*Every sectional-hyperbolic attracting set for a Hölder-$C^{1}$ vector field admits finitely many $\mu_{1},\dots,\mu_{k}$ ergodic physical/SRB invariant probability measures. Moreover, the union of the ergodic basins of these measures covers a full Lebesgue measure subset of the topological basin of attraction of $\Lambda$. *

In [28] existence a uniqueness of the physical measure was obtained for sectional-hyperbolic attractors of $C^{2}$ vector fields. We extend the argument from [28] avoiding the use of a dense orbit taking advantage of the recent results from [20] which hold in the $C^{1+}$ topology.

By robustness of partial hyperbolicity and sectional expansion, given a sectional-hyperbolic attracting set $\Lambda_{G}(U)=\cap_{t>0}\phi_{t}(U)$ with trapping region $U$, then there exists a neighborhood ${\mathcal{U}}\subset\mathcal{X}^{1+}(M)$ of $G$ so that $U$ is a trapping region and $\Lambda_{Y}(U)$ is sectional-hyperbolic for all $Y\in{\mathcal{U}}$. It is then natural to study the stability of the physical measures under small perturbation of the vector field $G$.

Theorem B**.**

Let $G\in\mathcal{X}^{1+}(M)$ be a vector field having a trapping region $U$ whose attracting set $\Lambda_{G}(U)=\cap_{t>0}\phi_{t}(U)$ is sectional-hyperbolic. Then there exists a neighborhood ${\mathcal{U}}\subset\mathcal{X}^{1+}(M)$ of $G$ so that, for each choice of $G_{n}\in{\mathcal{U}}$ and $\mu_{n}$ a physical measures for $G_{n}$ supported in $U$ such that $\|G_{n}-G\|_{C^{1}}\to 0$ when $n\nearrow\infty$, each weak∗ accumulation point $\mu$ of $(\mu_{n})_{n\geq 1}$ is a linear convex combination of the ergodic physical measures of $\Lambda_{G}$ provided in Theorem A:

[TABLE]

In other words, the convex hull $\Phi(G)$ of the ergodic physical measures of a sectional-hyperbolic attracting set depends continuously on the vector field, with respect to the $C^{1}$ topology of vector fields and weak∗ topology of probability measures on a manifold.

Statistical stability means that time averages $\widetilde{\psi}^{G}=\lim\frac{1}{t}\int_{0}^{t}\psi\circ\phi^{G}_{s}\,ds$ of continuous observables ${\varphi}:U\to{\mathbb{R}}$ in a neighborhood of the sectional-hyperbolic attracting sets, well-defined Lebesgue almost everywhere in $U$, depend continuously on the vector field $G$ generating the flow $\phi^{G}_{t}$, so that we can assure that $|\widetilde{\psi}^{G}-\widetilde{\psi}^{G^{\prime}}|$ is small as long has $\|G-G^{\prime}\|_{C^{1}}$ is small enough.

Theorem B improves both [2] and [11] since, although not dealing with the density of the invariant probability of the quotient map along stable leaves on global cross-section of the geometric Lorenz attractor, its statement and proof applies to a much larger family of sectional-hyperbolic attracting sets.

In particular, the attracting sets appearing as small perturbations of singular-hyperbolic attractors as in Morales [36], which must have a singular component, are statistical stable whatever the number of singularities involved.

We note that there are many examples of singular-hyperbolic attracting sets, non-transitive and containing non-Lorenz-like singularities; see Figure 1 for an example obtained by conveniently modifying the geometric Lorenz construction, and many others in [37]. statistical stability follows for all these examples.

Moreover Theorem B applies to the multidimensional Lorenz attractor described in [15] without further ado.

In addition, the open families of Lorenz-like attractors obtained after bifurcating saddle-connections by many authors [44, 49, 27, 21, 35, 34, 45, 46, 39] are automatically endowed with statistical stability after Theorem B, that is, in the (generic) unfolding of double (resonant) homoclinic cycle or saddle-connections, the physical measure for the ensuing Lorenz-like attractors depends continuously on the parameters.

We mention that in the preprint [33] the authors prove stochastic stabilty for $C^{2}$ transitive sectional-hyperbolic attracting sets which, in particular, provides an alternative proof of the existence of a unique SRB measure for $C^{2}$ sectional-hyperbolic attractors. This is a different kind of stability which is in general unrelated to statistical stability. Whereas statistical stability compares the SRB measures of close vector fields, stochastic stability considers random perturbations of a given vector field, usually though a diffusion, and checks whether the stationary measure for the randomly perturbed vector field converges to some special invariant measure for the orinal unperturbed vector field when the size of the diffusion vanishes. The results strongly depend on the type of random perturbation chosen, see e.g. [14, Appendix D].

The proofs of Theorems A and B use a construction of adapted cross-sections, generalizing the one presented in the $3$-flow setting in [8] and in the codimension $2$ setting in [5], which has been used to prove many delicate statistical properties of these flows; a similar construction (but built in a different way) of higher-dimensional adapted cross-sections was recently proposed in the work [19]. This enables us to solve the basin problem, as follows; see e.g. [13] for a similar but more delicate instance in a highly non-uniformly hyperbolic setting.

For a periodic point $p$ of $G$ we write $\mathcal{O}(p)$ for its compact orbit $\{\phi_{t}p:t\in{\mathbb{R}}\}=\{\phi_{t}p:t\in[0,T_{p}]\}$ where the minimal $T_{p}>0$ satisfying this is the period of any point $q\in\mathcal{O}(p)$. Moreover, we write

[TABLE]

for the stable manifold of $x\in M$ and for a given $\varepsilon_{0}>0$ we write

[TABLE]

for the local unstable manifold of $x$ of size $\varepsilon_{0}$. There are analogous and dual notions of local stable manifolds and unstable manifolds.

We say that a periodic point $p$ is hyperbolic if $D\phi_{T_{p}}(p):T_{p}M\circlearrowleft$ admits three $D\phi_{T_{p}}$-invariant subspaces forming a splitting $T_{p}M=E^{s}_{p}\oplus\langle G\rangle\oplus E^{u}_{p}$, where $\langle G\rangle={\mathbb{R}}\cdot G$ is an eigenspace with eigenvalue $1$; $E^{s}_{p}$ is contracted and $E^{u}_{p}$ expanded. The (Un)Stable Manifold Theorem ensures that $W^{u}_{q}(\varepsilon_{0})$ is an embedded manifold with $T_{q}W^{u}_{q}(\varepsilon_{0})=E^{u}_{q}$ and $W^{s}_{q}$ is an immersed submanifold with $T_{q}W^{s}_{q}=E^{s}_{q}$, for each $q\in\mathcal{O}(p)$. We write $W^{u}_{\mathcal{O}(p_{i})}(\varepsilon_{0})$ for the union $\cup\{W^{u}_{q}(\varepsilon_{0}):q\in\mathcal{O}(p)\}$ and analogously $W^{s}_{\mathcal{O}(p_{i})}=\cup\{W^{s}_{q}(\varepsilon_{0}):q\in\mathcal{O}(p)\}$. For more details on these notions from Hyperbolic Dynamics, see e.g. [41].

Theorem C**.**

Let $G\in\mathcal{X}^{1}(M)$ be a vector field having a trapping region $U$ whose attracting set $\Lambda=\cap_{t>0}\phi_{t}(U)$ is sectional-hyperbolic. Then there are $\varepsilon_{0}>0$ and finitely many (hyperbolic) periodic points $p_{1},\dots,p_{l}$ of $\Lambda$ so that

[TABLE]

is open and dense in {\mathcal{U}}=\{y\in M:d\big{(}\phi_{t}y,\Lambda\big{)}\xrightarrow[t\to+\infty]{}0\}\supset U. In particular, $\cup_{i}W^{s}_{\mathcal{O}(p_{i})}$ is dense in ${\mathcal{U}}$.

*In addition, if $G\in\mathcal{X}^{1+}(M)$, then ${\mathcal{W}}^{cs}$ contains the basin of any physical probability measure supported in $\Lambda$ and has full volume: ${\rm Leb}({\mathcal{U}}\setminus{\mathcal{W}}^{cs})=0$. *

Recently [56] obtains a similar structure for $C^{2}$ sectional-hyperbolic attractors, showing that they are homoclinic classes.

We conjecture that in this setting ${\mathcal{W}}^{sc}$ is the entire basin, as follows.

Conjecture 1**.**

For any sectional-hyperbolic attracting set $\Lambda$ the topological basin of attraction coincides with the family of stable manifolds through the points of local unstable leaves of finitely many periodic orbits, that is, ${\mathcal{U}}={\mathcal{W}}^{cs}$.

The following example shows that partially hyperbolic attracting sets which are not sectional-hyperbolic do not necessarily satisfy the conclusions of Theorem C.

Example 1**.**

Consider Bowen’s example flow (see [55] for the not very clear reason for the name) is a folklore example showing that Birkhoff averages need not exist almost everywhere. Indeed, in the system pictured in Figure 2 time averages only exist for the sources $s_{3}$, $s_{4}$ and for the set of separatrixes and saddle equilibria $\Lambda=W_{1}\cup W_{2}\cup W_{3}\cup W_{4}\cup\{s_{1},s_{2}\}$, which is an attracting set.

The orbit under this flow $\phi_{t}$ of every point $z\in{\mathbb{S}}^{1}\times[-1,1]=M$ not in $\Lambda\cup\{s_{3},s_{4}\}$ accumulates on either side of the separatrixes, as suggested in the figure, if we impose the condition $\lambda_{1}^{-}\lambda_{2}^{-}>\lambda_{1}^{+}\lambda_{2}^{+}$ on the eigenvalues of the saddle fixed points $s_{1}$ and $s_{2}$; for more specifics on this see e.g. [55] and references therein.

Due to the very long sojourn times around $s_{1}$ and $s_{2}$, future time averages of continuous functions ${\varphi}:M\to{\mathbb{R}}$ with ${\varphi}(s_{1})\neq{\varphi}(s_{2})$ do not exist; see e.g. [26]. However, time averages do exist for points in $\Lambda\cup\{s_{3},s_{4}\}$.

Hence, no point in $M\setminus(\Lambda\cup\{s_{3},s_{4}\})$ belongs to the stable manifold of any point of $\Lambda$. That is, we have that $W^{s}(\Lambda)=M\setminus(\Lambda\cup\{s_{3},s_{4}\})$ but the union of the stable manifolds of the points of $\Lambda$ is simply $\Lambda$.

To obtain an example with Lorenz-like singularities, just multiply this system by the North-South flow on ${\mathbb{S}}^{1}$ so that the derivative of the flow at the sink in the south pole $S$ has an eigenvalue $\lambda<0$ so that $\lambda_{i}^{+}+\lambda>0,i=1,2$; and then take the attracting set $\Lambda\times\{S\}$ in $M\times{\mathbb{S}}^{1}$.

1.3. Organization of the text

In Section 2 we present precise statements of the main properties of sectional-hyperbolic attracting sets together with a precise description of the construction of a family of adapted cross-sections $\Xi$ and a corresponding piecewise smooth and uniformly hyperbolic global Poincaré return map (with singularities) on a subset $\Xi^{\prime\prime}$ of $\Xi$, which might be of independent interest for further work on statistical properties of these systems.

In Section 3 we consider the basin of a $C^{1}$ sectional-hyperbolic attracting set $\Lambda$ proving the topological part of the statement of Theorem C as a consequence of showing that every center-unstable disk contains subdisks which are sent by arbitrarily large iterates of the Poincaré map to center-unstable disks with inner radius uniformly bounded away from zero accumulating the local unstable manifold of a hyperbolic periodic orbit from a finite subset of such orbits in the attracting set. In Subsection 4.1 we obtain as a consequence of the previous result that every positively invariant subset of $\Lambda$ containing ${\rm Leb}$-a.e. point of a central-unstable disk must contain a central-unstable disk with uniform inner radius.

This enables us to complete the proof of Theorem A in Subsection 4.2 by using and completing the relevant steps presented in [28] together with the results from Subsection 4.1 and the more recent results from [20], under the assumption that the vector field is Hölder-$C^{1}$. Following this, a proof of the measure theoretic part of the statement of Theorem C is presented in Subsection 4.3.

Finally, we present a proof of Theorem B on statistical stability in Section 5, coupling the previous results with robust entropy expansiveness of sectional hyperbolic attracting sets obtained in [40].

Acknowledgement

I thank the Mathematics Department at UFBA and CAPES-Brazil for the basic support of research activities; and also the anonymous referee for many suggestions that helped to improve the text.

2. Preliminary results on

sectional-hyperbolic attracting sets

Let $G$ be a $C^{1}$ vector field admitting a singular-hyperbolic attracting set $\Lambda$ with isolating neighborhood $U$. Given $x\in M$ we denote the omega-limit set

[TABLE]

and the alpha-limit set $\alpha(x)=\omega_{-G}(x)$ which are non-empty on a compact ambient space $M$.

2.1. Lorenz-like singularities

We first recall some properties of sectional-hyperbolic attracting sets extending some results from [4, 5] which hold for $d_{cu}\geq 2$.

Proposition 2.1**.**

Let $\Lambda$ be a sectional hyperbolic attracting set and let $\sigma\in\Lambda$ be an equilibrium. If there exists $x\in\Lambda\setminus\{\sigma\}$ so that $\sigma\in\omega(x)\cup\alpha(x)$, then $\sigma$ is generalized Lorenz-like: that is, $DG(\sigma)|E^{cu}_{\sigma}$ has a real eigenvalue $\lambda^{s}$ and $\lambda^{u}=\inf\{\Re(\lambda):\lambda\in\operatorname{sp}(DG(\sigma)),\Re(\lambda)\geq 0\}$ satisfies $-\lambda^{u}<\lambda^{s}<0<\lambda^{u}$ and so the index of $\sigma$ is $\dim E^{s}_{\sigma}=d_{s}+1$.

Remark 2.2**.**

1. (1)

If $\sigma\in\operatorname{Sing}(G)\cap\Lambda$ is a generalized Lorenz-like singularity and $\gamma_{\sigma}^{s}$ is its local stable manifold, then at $w\in\gamma_{\sigma}^{s}\setminus\{\sigma\}$ we have $T_{w}\gamma_{\sigma}^{s}=E^{cs}_{w}=E^{s}_{w}\oplus{\mathbb{R}}\cdot\{G(w)\}$ since $T\gamma_{\sigma}^{s}$ is $D\phi_{t}$-invariant and contains $G(w)$ (because $\gamma_{\sigma}^{s}$ is $\phi_{t}$-invariant) and the dimensions coincide. 2. (2)

If an equilibrium $\sigma\in\operatorname{Sing}(G)\cap\Lambda$ is not generalized Lorenz-like, then $\sigma$ is not in the limit set of $\Lambda\setminus\{\sigma\}$, i.e. there is no $x\in\Lambda\setminus\{\sigma\}$ so that $\sigma\in\alpha(x)\cup\omega(x)$. An example is provided by the pair of equilibria of the Lorenz system of equations away from the origin: these are saddles with an expanding complex eigenvalue which belong to the attracting set of the trapping ellipsoid already known to E. Lorenz; see e.g. **[7, Section 3.3]** and references therein.

Proof of Proposition 2.1.

It follows from sectional-hyperbolicity that $\sigma$ is a hyperbolic saddle and that at most $d_{cu}$ eigenvalues have positive real part. If there are only $d_{cu}-1$ such eigenvalues, then the constraints on $\lambda^{s}$ and $\lambda^{u}$ follow from sectional expansion.

Let $\gamma$ be the local stable manifold for $\sigma$. If $\sigma\in\omega(x)\cap\alpha(x)$ for some $x\in\Lambda\setminus\{x\}$, it remains to rule out the case $\dim\gamma=d-d_{cu}=d_{s}$.

In this case, $T_{p}\gamma=E^{s}_{p}$ for all $p\in\gamma\cap\Lambda$ and in particular $G(p)\in E^{s}_{p}$. On the one hand, $G(p)\in E^{cu}_{p}$ (see e.g. [7, Lemma 6.1]), so we deduce that $G(p)=0$ for all $p\in\gamma\cap\Lambda$ and so $\gamma\cap\Lambda=\{\sigma\}$.

On the other hand, if $\sigma\in\omega(x)$ (the case $\sigma\in\alpha(x)$ is analogous), then by the local behavior of orbits near hyperbolic saddles, there exists $p\in(\gamma\setminus\{\sigma\})\cap\omega(x)\subset(\gamma\setminus\{\sigma\})\cap\Lambda$ which, as we have seen, is impossible. ∎

2.2. Extension of the stable bundle and

center-unstable cone fields

Let $\mathcal{D}^{k}$ denote the $k$-dimensional open unit disk and let $\mathrm{Emb}^{r}(\mathcal{D}^{k},M)$ denote the set of $C^{r}$ embeddings $\psi:\mathcal{D}^{k}\to M$ endowed with the $C^{r}$ distance. We say that the image of any such embedding is a $C^{r}$ $k$-dimensional disk.

Proposition 2.3**.**

[4*, Proposition 3.2, Theorem 4.2 and Lemma 4.8]**

Let $\Lambda$ be a partially hyperbolic attracting set.*

1. (1)

The stable bundle $E^{s}$ over $\Lambda$ extends to a continuous uniformly contracting $D\phi_{t}$-invariant bundle $E^{s}$ on an open positively invariant neighborhood $U_{0}$ of $\Lambda$. 2. (2)

There exists a constant $\lambda\in(0,1)$, such that

1. (a)

for every point $x\in U_{0}$ there is a $C^{r}$ embedded $d_{s}$-dimensional disk $W^{s}_{x}\subset M$, with $x\in W^{s}_{x}$, such that $T_{x}W^{s}_{x}=E^{s}_{x}$; $\phi_{t}(W^{s}_{x})\subset W^{s}_{\phi_{t}x}$ and $d(\phi_{t}x,\phi_{t}y)\leq\lambda^{t}d(x,y)$ for all $y\in W^{s}_{x}$, $t\geq 0$ and $n\geq 1$. 2. (b)

the disks $W^{s}_{x}$ depend continuously on $x$ in the $C^{0}$ topology: there is a continuous map $\gamma:U_{0}\to{\rm Emb}^{0}(\mathcal{D}^{d_{s}},M)$ such that $\gamma(x)(0)=x$ and $\gamma(x)(\mathcal{D}^{d_{s}})=W^{s}_{x}$. Moreover, there exists $L>0$ such that $\operatorname{Lip}\gamma(x)\leq L$ for all $x\in U_{0}$. 3. (c)

the family of disks ${\mathcal{F}}^{s}=\{W^{s}_{x}:x\in U_{0}\}$ defines a topological foliation ${\mathcal{W}}^{s}$ of $U_{0}$: every $x_{0}\in U_{0}$ admits a neighborhood $V\subset U_{0}$ and a homeomorphism $\psi:V\to{\mathbb{R}}^{d_{s}}\times{\mathbb{R}}^{d_{cu}}$ so that $\psi(W^{s}_{x})=\pi_{s}^{-1}\{\pi_{s}(\psi(x))\}$ where $\pi^{s}:{\mathbb{R}}^{d_{s}}\times{\mathbb{R}}^{d_{cu}}\to{\mathbb{R}}^{d_{s}}$ is the canonical projection.

Remark 2.4**.**

For any two close enough $d_{cu}$-disks $D_{1},D_{2}$ contained in $U_{0}$ and transverse to ${\mathcal{F}}^{s}$ there exists an open subset $\hat{D}_{1}$ of $D_{1}$ so that $W^{s}_{x}\cap D_{2}$ is a singleton. This defines the holonomy map $h:\hat{D}_{1}\to D_{2},\hat{D}_{1}\ni x\mapsto W^{s}_{x}\cap D_{2}$ and Proposition 2.3 ensures that $h$ is continuous.

The splitting $T_{\Lambda}M=E^{s}\oplus E^{cu}$ extends continuously to a splitting $T_{U_{0}}M=E^{s}\oplus E^{cu}$ where $E^{s}$ is the invariant uniformly contracting bundle in Proposition 2.3 (however $E^{cu}$ is not invariant in general). Given $a>0$ and $x\in U_{0}$, we define the center-unstable cone field as $\mathcal{C}^{cu}_{x}(a)=\{v=v^{s}+v^{cu}\in E^{s}_{x}\oplus E^{cu}_{x}:\|v^{s}\|\leq a\|v^{cu}\|\}$.

Proposition 2.5**.**

Let $\Lambda$ be a partially hyperbolic attracting set.

1. (1)

There exists $T_{0}>0$ such that for any $a>0$, after possibly shrinking $U_{0}$, $D\phi_{t}\cdot\mathcal{C}^{cu}_{x}(a)\subset\mathcal{C}^{cu}_{\phi_{t}x}(a)$ for all $t\geq T_{0}$, $x\in U_{0}$. 2. (2)

Let $\lambda_{1}\in(0,1)$ be given. After possibly increasing $T_{0}$ and shrinking $U_{0}$, there exist constants $K,\theta>0$ such that $|\det(D\phi_{t}|P_{x})|\geq K\,e^{\theta t}$ for each $2$-dimensional subspace $P_{x}\subset E^{cu}_{x}$ and all $x\in U_{0}$, $t\geq 0$.

Proof.

For item (1) see [4, Proposition 3.1]. Item (2) follows from the robustness of sectional expansion; see [5, Proposition 2.10] with straightforward adaptation to area expansion along any two-dimensional subspace of $E^{cu}_{x}$. ∎

2.3. Global Poincaré map on adapted

cross-sections

We assume that $\Lambda$ is a partially hyperbolic attracting set and recall how to construct a piecewise smooth Poincaré map $f:\Xi\to\Xi$ preserving a contracting stable foliation ${\mathcal{W}}^{s}(\Xi)$. This largely follows [8] (see also [7, Chapter 6]) and [5, Section 3] with slight modifications to account for the higher dimensional set up.

We write $\rho_{0}>0$ for the injectivity radius of the exponential map $\exp_{z}:T_{z}M\to M$ for all $z\in U$, so that $\exp_{z}\mid B_{z}(0,\rho_{0}):B_{z}(0,\rho_{0})\to M,v\mapsto\exp_{z}v$ is a diffeomorphism with $B_{z}(0,\rho_{0})=\{v\in T_{z}M:\|v\|\leq\rho_{0}\}$ and $D\exp_{z}(0)=Id$ and also $d(z,\exp_{z}(v))=\|v\|$ for all $v\in B_{z}(0,\rho_{0})$.

2.3.1. Construction of a global adapted

cross-section

Let $y\in\Lambda$ be a regular point ($G(y)\neq\vec{0}$). Then there exists an open flow box $V_{y}\subset U_{0}$ containing $y$. That is, if we fix $\varepsilon_{0}\in(0,1)$ small, then we can find a diffeomorphism $\chi:\mathcal{D}^{d-1}\times(-\varepsilon_{0},\varepsilon_{0})\to V_{y}$ with $\chi(0,0)=y$ such that $\chi^{-1}\circ\phi_{t}\circ\chi(z,s)=(z,s+t)$. Define the cross-section $\Sigma_{y}=\chi(\mathcal{D}^{d-1}\times\{0\})$.

Remark 2.6**.**

We assume that $\Sigma_{y}\subset\exp_{y}(B_{y}(0,\rho_{0}/3)\cap G(z)^{\perp})$ and $\|D(\exp_{y})^{-1}_{x}\|\leq 2$ for all $x\in\Sigma_{y}$ without loss of generality.

For each $x\in\Sigma_{y}$, let $W^{s}_{x}(\Sigma_{y})=\bigcup_{|t|<\varepsilon_{0}}\phi_{t}(W^{s}_{x})\cap\Sigma_{y}$. This defines a topological foliation ${\mathcal{W}}^{s}(\Sigma_{y})$ of $\Sigma_{y}$. We can also assume that $\Sigma_{y}$ is diffeomorphic to $\mathcal{D}^{d_{cu}-1}\times\mathcal{D}^{d_{s}}$ by reducing the size of the $\Sigma_{y}$ if needed. The stable boundary $\partial^{s}\Sigma_{y}\cong\partial\mathcal{D}^{d_{cu}-1}\times\mathcal{D}^{d_{s}}\cong{\mathbb{S}}^{d_{cu}-2}\times\mathcal{D}^{d_{s}}$ is a regular topological manifold homeomorphic to a cylinder of stable leaves, since ${\mathcal{W}}^{s}$ is a topological foliation; i.e. $\cong$ denotes only the existence of a homeomorphism and the subspace topology of $\partial^{s}\Sigma_{y}$ induced by $M$ coincides with the manifold topology.

Let $\mathcal{D}_{a}^{d_{s}}$ denote the open disk of radius $a\in(0,1]$ in $\mathcal{R}^{d_{s}}$. Define the sub-cross-section $\Sigma_{y}(a)\cong\mathcal{D}_{a}^{d_{cu}-1}\times\mathcal{D}_{a}^{d_{s}}$, and the corresponding sub-flow box $V_{y}(a)\cong\Sigma_{y}(a)\times(-\varepsilon_{0},\varepsilon_{0})$ consisting of trajectories in $V_{y}$ which pass through $\Sigma_{y}(a)$. In what follows we fix $a_{0}=3/4$.

For each equilibrium $\sigma\in\Lambda$, we let $V_{\sigma}$ be an open neighborhood of $\sigma$ on which the flow is locally conjugated by a homeomorphism to a linear flow (Hartman-Grobman Theorem). Let $\gamma^{s}_{\sigma}$ and $\gamma^{u}_{\sigma}$ denote the local stable and unstable manifolds of $\sigma$ within $V_{\sigma}$; trajectories starting in $V_{\sigma}$ remain in $V_{\sigma}$ for all future time if and only if they lie in $\gamma^{s}_{\sigma}$.

Define $V_{0}=\bigcup_{\sigma\in\operatorname{Sing}(G)\cap U}V_{\sigma}$. We shrink the neighborhoods $V_{\sigma}$ so that they are disjoint; $\Lambda\not\subset V_{0}$; and $\gamma^{u}_{\sigma}\cap\partial V_{\sigma}\subset V_{y}(a_{0})$ for some regular point $y=y(\sigma)$.

By compactness of $\Lambda$, there exists $\ell\in{\mathbb{Z}}^{+}$ and regular points $y_{1},\dots,y_{\ell}\in\Lambda$ such that $\Lambda\setminus V_{0}\subset\bigcup_{j=1}^{\ell}V_{y_{j}}(a_{0})$. We enlarge the set $\{y_{j}\}$ to include the points $y(\sigma)$ mentioned above; adjust the positions of the cross-sections $\Sigma_{y_{j}}$ if necessary to ensure that they are disjoint; and define the global cross-section $\Xi=\bigcup_{j=1}^{\ell}\Sigma_{y_{j}}$ and its smaller version $\Xi(a)=\bigcup_{j=1}^{\ell}\Sigma_{y_{j}}(a)$ for each $a\in(0,1)$.

In what follows we modify the choices of $U_{0}$ and $T_{0}$. However, $V_{y_{j}}$, $\Sigma_{y_{j}}$ and $\Xi$ remain unchanged from now on and correspond to our current choice of $U_{0}$ and $T_{0}$. All subsequent choices will be labeled $U_{1}\subset U_{0}$ and $T_{1}\geq T_{0}$. In particular $U_{1}\subset V_{0}\cup\bigcup_{j=1}^{\ell}V_{y_{j}}(a_{0})$. We set $\delta_{0}=d(\partial\Xi,\partial\Xi(a_{0}))>0$ where $\partial\Xi(a)$ is the boundary of the submanifold $\Xi(a)$ of $M$, $a\in(0,1]$, and $\Xi=\Xi(1)$.

2.3.2. The Poincaré map

By Proposition 2.3, for any $\delta>0$ we can choose $T_{1}\geq T_{0}$ such that $\operatorname{diam}\phi_{t}(W^{s}_{x}(\Sigma_{y_{j}}))<\delta$, for all $x\in\Sigma_{y_{j}}$, $j=1,\dots,\ell$ and $t>T_{1}$. We fix $T_{1}=T_{1}(\delta_{0})$ for $\delta=\delta_{0}=d(\partial\Xi,\partial\Xi(a_{0}))$ in what follows and define $\Gamma_{0}=\{x\in\Xi:\phi_{T_{1}+1}(x)\in\bigcup_{\sigma\in\operatorname{Sing}(G)\cap U_{0}}(\gamma^{s}_{\sigma}\setminus\{\sigma\})\}$ and $\Xi^{\prime}=\Xi\setminus\Gamma_{0}$. If $x\in\Xi^{\prime}$, then $\phi_{T_{1}+1}(x)$ cannot remain inside $V_{0}$ so there exists $t>T_{1}+1$ and $j=1,\dots,\ell$ such that $\phi_{t}x\in V_{y_{j}}(a_{0})$. Since $\varepsilon_{0}<1$, there exists $t>T_{1}$ such that $\phi_{t}x\in\Sigma_{y_{j}}(a_{0})$.

For each $\Sigma=\Sigma_{y_{j}}\in\Xi$, we choose a center-unstable disk $W_{\Sigma}$ which crosses $\Sigma$ and is transversal to ${\mathcal{W}}^{s}(\Sigma)$, that is, every stable leaf $W^{s}_{x}(\Sigma)$ intersects $W_{\Sigma}$ transversely at only one point, for each $x\in\Sigma$. Then, for every given $x\in W_{\Sigma}\cap\Xi^{\prime}$, we define

[TABLE]

We note that by the choice of $T_{1}=T_{1}(\delta_{0})$ we have $\operatorname{diam}\phi_{\tau(x)}(W^{s}_{x}(\Sigma))<\delta_{0}$ and so the disk $\phi_{\tau(x)}(W^{s}_{x}(\Sigma))$, although not necessarily contained in any $\Sigma_{y_{j}}$, is certainly contained in some $V_{y_{j}}$ by construction. Thus we can define

[TABLE]

for each $y\in{\mathcal{W}}^{s}_{x}(\Sigma)$. This defines $\tau$ and $f$ on $\Xi^{\prime}$.

We define the topological foliation ${\mathcal{W}}^{s}(\Xi)=\bigcup_{j=1}^{\ell}{\mathcal{W}}^{s}(\Sigma_{y_{j}})$ of $\Xi$ with leaves $W^{s}_{x}(\Xi)$ passing through each $x\in\Xi$. From the uniform contraction of stable leaves together with the choice of $\delta_{0}$, $T_{1}(\delta_{0})$ and the definition of ${\mathcal{W}}^{s}(\Xi)$ and flow invariance of ${\mathcal{W}}^{s}$ we obtain

Proposition 2.7**.**

[5*, Proposition 3.4]**

For big enough $T_{1}>T_{0}$, $f(W^{s}_{x}(\Xi))\subset W^{s}_{fx}(\Xi)$ for all $x\in\Xi^{\prime}$.*

Remark 2.8**.**

The definition of $\tau$ in [5] is pointwise, viz. $\tau(x)=\inf\{t>T_{1}:\phi_{t}x\in\bigcup_{j=1}^{\ell}\operatorname{Closure}{\Sigma_{y_{j}}(a_{0})}\}$ and this allowed discontinuities of the return time and return map along stable leaves, i.e., a pair of indexes $i\neq j$ and of close points $w,z\in W^{s}_{x}$ so that $\phi_{\tau(w)}(w)\in\operatorname{Closure}{\Sigma_{y_{j}}(a_{0})}$ and $\phi_{\tau(z)}(z)\in\operatorname{Closure}{\Sigma_{y_{i}}(a_{0})}$. So the statement of [5, Proposition 3.4] (which is the analogous to Proposition 2.7 with $d_{cu}=2$) does not make sense in this set up, since we would have $f(W^{s}_{x})$ with elements in distinct cross-sections.

The definition of $\tau$ first on the points of a fixed collection of center-unstable leaves (2.1), and then the smooth extension to the stable leaves through each of these points (2.2), provides for the crucial property stated in Proposition 2.7.

The rest of the features of the global Poincaré map $f$ stated and used in [4, 9, 5] remain valid with the same proofs.

In this way we obtain a piecewise $C^{r}$ global Poincaré map $f:\Xi^{\prime}\to\Xi=\bigcup_{j=1}^{\ell}\Sigma_{j}(a_{0})$ with piecewise $C^{r}$ roof function $\tau:\Xi^{\prime}\to[T_{1},\infty)$, and deduce the following standard result.

Lemma 2.9**.**

[5, Lemma 3.2]** If $\Sigma_{y}$ contains no equilibria (i.e. $\Gamma_{0}\cap\Sigma_{y}=\emptyset$), then $\tau\mid\Sigma_{y}\leq T_{1}+2$. In general, there is $C>0$ so that $\tau(x)\leq-C\log\operatorname{dist}(x,\Gamma_{0})$ for all $x\in\Xi^{\prime}$; in particular, $\tau(x)\to\infty$ as $\operatorname{dist}(x,\Gamma_{0})\to 0$.

We define $\partial^{s}\Xi(a_{0})=\bigcup_{j=1}^{\ell}\partial^{s}\Sigma_{y_{j}}(a_{0})$ and $\Gamma_{1}=\{x\in\Xi^{\prime}:fx\in\partial^{s}\Xi(a_{0})\}$ and then set $\Gamma=\Gamma_{0}\cup\Gamma_{1}$. Clearly $\Gamma_{0}\cap\Gamma_{1}=\emptyset$.

Lemma 2.10**.**

1. (1)

$\Gamma_{0}$* is a $d_{s}$-submanifold of $\Xi$ given by a finite union of stable leaves $W_{s_{i}}(\Xi),i=1,\dots,k$; and* 2. (2)

$\Gamma_{1}$* is a regular embedded $(d-2)$-topological submanifold foliated by stable leaves from ${\mathcal{W}}^{s}(\Xi)$ with finitely many connected components.*

Remark 2.11**.**

Note that $\Gamma_{0}$ is a (smooth) submanifold of $\Xi$ with codimension $d_{cu}-1$, so it separates $\Xi$ only if $d_{cu}=2$; while $\Gamma_{1}$ is a regular topological codimension $1$ submanifold of $\Xi$ and so it separates $\Xi$.

Proof.

It is clear that $W^{s}_{x}(\Xi)\subset\Gamma$ for all $x\in\Gamma$, so $\Gamma$ is foliated by stable leaves. We claim that $\Gamma$ is precisely the set of those points of $\Xi$ which are sent to the boundary of $\Xi$ or never visit $\Xi$ in the future.

Indeed, if $x_{0}\in\Xi^{\prime}\setminus\Gamma_{1}$, then $fx_{0}=\phi_{\tau(x_{0})}(x_{0})\in\Sigma^{\prime}$ for some $\Sigma^{\prime}\in\Xi(a_{0})=\{\Sigma_{y_{j}}(a_{0})\}$. For $x$ close to $x_{0}$, it follows from continuity of the flow that $fx\in\Sigma^{\prime}$ (with $\tau(x)$ close to $\tau(x_{0})$). Hence $x\in\Xi^{\prime}\setminus\Gamma_{1}$ and since $\Xi^{\prime}=\Xi\setminus\Gamma_{0}$, then the claim is proved and, moreover, $\Gamma$ is closed.

For item (1), we note that \Gamma_{0}\subset\Xi\cap\phi^{-1}_{[0,T_{1}+1]}\big{(}\bigcup_{\sigma}\gamma^{s}_{\sigma}\big{)} and we may assume without loss of generality that the above union comprises only generalized Lorenz-like equilibria; cf. Remark 2.2(2). Hence $T_{w}\gamma_{\sigma}^{s}=E^{cs}_{w}$ for $w\in\gamma_{\sigma}^{s}\setminus\{\sigma\}$; see Remark 2.2(1). Thus $\Gamma_{0}$ is contained in the transversal intersection between a compact $(d_{s}+1)$-submanifold and a compact $(d-1)$-manifold, so $\Gamma_{0}$ is a compact differentiable $d_{s}$-submanifold of $M$ and $\Xi$. In addition, since $\Gamma_{0}$ is foliated by stable leaves which are $d_{s}$-dimensional, then $\Gamma_{0}$ has only finitely many connected components in $\Xi$.

For item (2), note that for each $x\in\Gamma_{1}$ we have that $fx\in\partial\Sigma_{j}(a_{0})\subset\Sigma_{j}$. Thus there exists a neighborhood $W_{x}$ of $x$ in $\Xi$ and $V_{fx}$ of $fx$ in $\Sigma_{j}$ so that $f\mid W_{x}:W_{x}\to V_{fx}$ is a diffeomorphism. Hence $\Gamma_{1}\cap W_{x}=(f\mid W_{x})^{-1}(V_{x}\cap\partial^{s}\Sigma(a_{0}))$ is homeomorphic to a $(d_{cu}-2+d_{s})$-dimensional disk. Moreover, this shows that the topology of $\Gamma_{1}$ is the same as the subspace topology induced by the topology of $\Xi$. We conclude that $\Gamma_{1}$ is a regular topological $(d-2)$-dimensional submanifold.

It remains to rule out the possibility of existence of infinitely many connected components $\Gamma_{1}^{m},m\in{\mathbb{Z}}^{+}$ of $\Gamma_{1}$ in $\Xi$. Since $\Xi$ contains finitely many sections only, then there exists cross-sections $\Sigma_{j},\Sigma_{i}$ in $\Xi$ and, taking a subsequence if necessary, an accumulation set $\widetilde{\Gamma}=\lim_{m}\Gamma_{1}^{m}$ within $\operatorname{Closure}(\Sigma_{j})$ so that $f(\Gamma_{1}^{m})\subset\partial^{s}\Sigma_{i}(a_{0})$ for all $m\geq 1$. By the continuity of the stable foliation, $\widetilde{\Gamma}$ is an union of stable leaves.

We claim that the Poincaré times $\tau(x_{m})$ for $x_{m}\in\Gamma_{1}^{m},m\geq 1$ are uniformly bounded from above. For otherwise the trajectory $\phi_{[0,\tau(x_{m})]}(x_{m})$ intersects $V_{\sigma}$ for some $\sigma\in\operatorname{Sing}(G)\cap U$ and accumulates $\sigma$. Hence, by the local behavior of trajectories near saddles and the choice of the cross-sections near $V_{\sigma}$, we get that $\widetilde{\Gamma}\subset\Sigma_{i}(a_{0})$ is not contained in the boundary of the cross-section. This contradiction proves the claim. Let $T$ be an upper bound for $\tau(x_{m})$.

Then, for an accumulation point $x\in\widetilde{\Gamma}$ of $(x_{m})_{m\geq 1}$ we have that the trajectories $\phi_{[0,T]}(x_{m})$ converge in the $C^{1}$ topology (taking a subsequence if necessary) to a limit curve $\phi_{[0,T]}(x)$ and so $fx=\phi_{\tau(x)}(x)\in\partial^{s}\Sigma_{i}(a_{0})$. Thus we can find neighborhoods $W_{x}$ of $x$ and $V_{fx}$ of $fx$ in $\Xi$ so that for arbitrarily large $m$ we have that $f\mid W_{x}:W_{x}\to V_{fx}$ is a diffeomorphism and $\Gamma_{1}\cap W_{x}=(f\mid W_{x})^{-1}(V_{x}\cap\partial^{s}\Sigma_{i}(a_{0}))$, which contradicts the regularity of $\Gamma_{1}$ as topological submanifold.

This concludes the proof of item (2) and the lemma. ∎

Let us set $\Xi^{\prime\prime}=\Xi(a_{0})\setminus\Gamma$ from now on. Then $\Xi^{\prime\prime}=S_{1}\cup\dots\cup S_{m}$ for some $m\geq 1$, where each $S_{i}$ is a connected smooth strip, homeomorphic to either (i) $\mathcal{D}^{d_{cu}}\times\mathcal{D}^{d_{s}}$ if $\Gamma_{0}\cap\operatorname{Closure}{S_{i}}=\emptyset$; or (ii) $\mathcal{D}^{d_{cu}}\times(\mathcal{D}^{d_{s}}\setminus\{0\})$ otherwise. The latter are singular (smooth) strips.

We note that $f\mid S_{i}:S_{i}\to\Xi(a_{0})$ is a diffeomorphism onto its image, $\tau\mid S_{i}:S_{i}\to[T_{1},\infty)$ is smooth for each $i$, $\tau\mid S_{i}\leq T_{1}+2$ on non-singular strips $S_{i}$ and also on a neighborhood of $\partial^{s}(S_{i}\cup\Gamma_{0})$ for singular strips $S_{i}$. The foliation ${\mathcal{W}}^{s}(\Xi)$ restricts to a foliation ${\mathcal{W}}^{s}(S_{i})$ on each $S_{i}$.

Remark 2.12**.**

In what follows it may be necessary to increase $T_{1}$ leading to changes to $f$, $\tau$, $\Gamma$ and $\{S_{i}\}$ (and the constant $C$ in Lemma 2.9). However, the global cross-section $\Xi=\bigcup\Sigma_{y_{j}}$ is fixed throughout the argument.

Remark 2.13**.**

Since $f$ sends $\Xi^{\prime\prime}$ into $\Xi=\Xi(1)$, there are smooth extensions $\widetilde{f_{i}}:\widetilde{S_{i}}\to\Xi$ of $f\mid S_{i}:S_{i}\to\Xi$, where $\tilde{S}_{i}\supset\operatorname{Closure}(S_{i})\setminus\Gamma_{0}$.

2.4. Hyperbolicity of the global Poincaré map

We assume from now on that $\Lambda$ is a sectional hyperbolic attracting set with $d_{cu}>2$ and proceed to show that, for large enough $T_{1}>1$, the global Poincaré map $f:\Xi^{\prime\prime}\to\Xi$ is piecewise uniformly hyperbolic (with discontinuities and singularities).

2.4.1. Hyperbolicity at each smooth strip

Let $S\in\{S_{i}\}$ be one of the smooth strips. Then there are cross-sections $\Sigma$, $\widetilde{\Sigma}\in\Xi$ so that $S\subset\Sigma$ and $f(\Sigma)\subset\widetilde{\Sigma}$. The splitting $T_{U_{0}}M=E^{s}\oplus E^{cu}$ induces the continuous splitting $T\Sigma=E^{s}(\Sigma)\oplus E^{u}(\Sigma)$, where $E^{s}_{x}(\Sigma)=(E^{s}_{x}\oplus{\mathbb{R}}\{G(x)\})\cap T_{x}{\Sigma}$ and $E^{u}_{x}(\Sigma)=E^{cu}_{x}\cap T_{x}{\Sigma}$ for $x\in\Sigma$; and analogous definitions apply to $\widetilde{\Sigma}$.

Proposition 2.14**.**

The splitting $T\Sigma=E^{s}(\Sigma)\oplus E^{u}(\Sigma)$ is

**invariant: **

$Df\cdot E^{s}_{x}(\Sigma)=E^{s}_{fx}(\widetilde{\Sigma})$* for all $x\in S$, and $Df\cdot E^{u}_{x}(\Sigma)=E^{u}_{fx}(\widetilde{\Sigma})$ for all $x\in\Lambda\cap S$.*

**uniformly hyperbolic: **

for each given $\lambda_{1}\in(0,1)$ there exists $T_{1}>0$ so that if $\inf\tau>T_{1}$, then $\|Df\mid E^{s}_{x}(\Sigma)\|\leq\lambda_{1}$ for each $x\in S$; and \|\big{(}Df\mid E^{u}_{x}(\Sigma)\big{)}^{-1}\|\geq\lambda_{1}^{-1} for all $x\in S\cap\Lambda$.

Moreover, there exists $0<\widetilde{\lambda_{1}}<\lambda_{1}$ so that, for all $x$ on a non-singular strip $S$, or for $x$ on a neighborhood of $\partial^{s}(S\cup\Gamma_{0})$ of a singular strip $S$ we have \widetilde{\lambda_{1}}<\|\big{(}Df\mid E^{s}_{x}(\Sigma)\big{)}^{-1}\| and $\|Df\mid E^{u}_{x}(\Sigma)\|<\widetilde{\lambda_{1}}^{-1}$.

Proof.

See [5, Proposition 4.1] with straightforward adaptation to use area expansion along each two-dimensional subspaces within $E^{u}_{x}(\Sigma)$ in order to obtain uniform expansion; cf. [7, Lemma 8.25]. The last statement follows from the boundedness of $\tau$ on the designated domains; cf. Lemma 2.9. ∎

2.4.2. Hyperbolicity of the extensions of the

Poincaré maps at smooth strips

For a given $a>0$, $x\in\Sigma$ and $\Sigma\in\Xi$ we define the unstable cone field at $x$ as $\mathcal{C}^{u}_{x}(\Sigma,a)=\{w=w^{s}+w^{u}\in E^{s}_{x}(\Sigma)\oplus E^{u}_{x}(\Sigma):\|w^{s}\|\leq a\|w^{u}\|\}.$

Remark 2.15**.**

We assume that $\mathcal{C}^{u}_{x}(\Sigma_{y},a)\subset D(\exp_{y})_{\exp_{y}^{-1}x}\cdot\mathcal{C}^{u}_{y}(\Sigma_{y},2a)$ for all $x\in\Sigma_{y}$ and each $\Sigma_{y}\in\Xi$ without loss of generality; recall Remark 2.6. Consequently, letting $\pi^{u}:E^{s}_{y}(\Sigma_{y})\oplus E^{u}_{y}(\Sigma_{y})\to E^{u}_{y}(\Sigma_{y})$ be the canonical projection, we get $\|\pi^{u}w\|/\|w\|\in(1-2a,1+2a)$ for all $w\in\mathcal{C}^{u}_{x}(\Sigma_{y},a)$, where we implicitly identify $\mathcal{C}^{u}_{x}(\Sigma_{y},a)$ with a subcone of $\mathcal{C}^{u}_{y}(\Sigma_{y},2a)$, for $x\in\Sigma_{y}$ and $\Sigma_{y}\in\Xi$.

Proposition 2.16**.**

For any $a>0$, $\lambda_{1}\in(0,1)$, we can increase $T_{1}$ and shrink $U_{1}$ such that, if $\inf\tau>T_{1}$ and $x\in S$ and $S,S^{\prime}\in\{S_{i}\}$ so that $fx\in S^{\prime}$, then

• •

$Df(x)\cdot\mathcal{C}^{u}_{x}(S,a)\subset\mathcal{C}^{u}_{fx}(S^{\prime},a)$; and

• •

$\|Df(x)w\|\geq\|\pi^{u}Df(x)w\|\geq\lambda_{1}^{-1}\|w\|$* for all $w\in\mathcal{C}^{u}_{x}(S,a)$.*

Moreover $\|Df(x)w\|\leq\widetilde{\lambda_{1}}^{-1}\|w\|$ for $x$ in a non-singular $S$ or $x$ in a neighborhood of $\partial^{s}(S\cup\Gamma_{0})$ for a singular $S$.

Proof.

See [5, Proposition 4.2], use $\widetilde{\lambda_{1}}$ from Proposition 2.14 and the estimate on $\|\pi^{u}\|$ from Remark 2.15. ∎

Considering the union of the smooth strips $S$, the previous results shows that we obtain a global continuous uniformly hyperbolic splitting $T\Xi^{\prime\prime}=E^{s}(\Xi)\oplus E^{u}(\Xi)$ in the following sense.

Theorem 2.17**.**

For given $a>0$ and $\lambda_{1}\in(0,1)$ we obtain a global Poincaré map $f$ so that the stable bundle $E^{s}(\Xi)$ and the restricted splitting $T_{\Lambda}\Xi^{\prime\prime}=E^{s}_{\Lambda}(\Xi)\oplus E^{u}_{\Lambda}(\Xi)$ are $Df$-invariant; and $Df\cdot\mathcal{C}^{u}_{x}(\Xi,a)\subset\mathcal{C}^{u}_{fx}(\Xi,a)$ and $\|\pi_{u}Df(x)w\|\geq\lambda_{1}^{-1}\|w\|$ for all $x\in\Xi^{\prime\prime}$ and $w\in\mathcal{C}^{u}_{x}(\Xi,a)$.

Remark 2.18**.**

The extensions $\widetilde{f_{i}}:\widetilde{S_{i}}\to\Xi$ of $f\mid S_{i}:S_{i}\to\Xi(a_{0})$ mentioned in Remark 2.13 are such that on $\tilde{S}_{i}\supset\operatorname{Closure}(S_{i})\setminus\Gamma_{0}$ the map $\widetilde{f_{i}}$ behaves as $f$ in Propositions 2.14 and 2.16. In particular, $\delta_{1}=d(S_{i},\partial\widetilde{S_{i}})\geq\widetilde{\lambda_{1}}\cdot d(\Xi(a_{0}),\Xi)=\widetilde{\lambda_{1}}\delta_{0}$.

3. The basin of sectional-hyperbolic

attracting sets

Here we prove the topological part of the statement of Theorem C using first the following technical result. The measure theoretic part is dealt with in the next section; see Subsection 4.3.

Theorem 3.1**.**

There are finitely many (hyperbolic) periodic points $p_{1},\dots,p_{l}$ of $\Lambda$ in $\Xi$ so that ${\mathcal{W}}^{s}=\{W^{s}_{x}(\Xi):x\in W^{u}_{p_{i}}(\Xi);i=1,\dots,l\}$ is open and dense in $\Xi$ and, in particular, $\cup_{i}W^{u}_{p_{i}}(\Xi)$ is dense in $\Xi$.

This is enough to deduce the topological part of Theorem C, since this implies that ${\mathcal{W}}^{cs}$ is open and dense in ${\mathcal{U}}$. In the rest of the section we prove Theorem 3.1.

3.1. Denseness of stable leaves of $\Lambda$ on $U$

Proof of Theorem 3.1.

In what follows we say that a $C^{1}$ $(d_{cu}-1)$-dimensional disk $D\subset\Sigma$ such that $T_{x}D\subset\mathcal{C}^{u}_{x}(\Sigma,a)$ for all $x\in D$ is a center-unstable disk, or just a $cu$-disk. A $cu$-disk $D$ is an unstable disk, or just a $u$-disk, if for any given $x,y\in D$ there exists a sequence $\widetilde{f_{i}}:\tilde{S_{i}}\to\Xi$ of smooth extensions of $f_{i}=f\mid S_{i}$ together with a subsequence $i_{k}$ and $x_{k},y_{k}\in\Xi$ so that $g_{k}=\widetilde{f_{i_{k}}}\circ\widetilde{f_{i_{k}-1}}\circ\cdots\circ\widetilde{f_{2}}\circ\widetilde{f_{1}}$ satisfies $g_{k}x_{k}=x,g_{k}y_{k}=y$ and $d(x_{k},y_{k})\leq\lambda_{1}^{i_{k}}d(x,y)$ for all $k\geq 1$111Note that an unstable disk is necessarily contained in the attracting set $\Lambda$..

From Remark 2.15, if $S\subset\Sigma_{y}$ for some $\Sigma_{y}\in\Xi$, then $\widetilde{D}=\exp_{y}^{-1}(D)=\operatorname{Graph}(g:D_{u}\to E^{s}_{y}(\Sigma))$ where $D_{u}=\pi_{u}\widetilde{D}\subset E^{u}_{y}(\Sigma)$ is a open subset of $E^{u}_{y}(\Sigma_{y})$ and $g$ is a $C^{1}$ map such that $\|Dg\|\leq 2a$. Indeed, $D$ is transverse to ${\mathcal{W}}^{s}(\Sigma_{y})$ and each $W^{s}_{x}(\Sigma_{y})$ is the graph of ${\varphi}_{x}:B(0,\rho)\cap E^{s}_{y}(\Sigma_{y})\to E^{u}_{y}(\Sigma_{y})$ which is $C^{1}$ and depends continuously on $x$ in the $C^{1}$ topology; and the tangent space at any point of $\widetilde{D}$ is contained in $\mathcal{C}^{u}_{y}(\Sigma_{y},2a)$.

We define $\rho(D)=\sup\{r>0:B(x,r)\subset D_{u},x\in E^{u}_{y}(\Sigma_{y})\}$ as the inner radius of any given $cu$-disk $D$.

We use uniform expansion along center-unstable cones by the extension of $f$ to obtain

Proposition 3.2**.**

There exist $\rho_{0}>0$ and finitely many periodic points $p_{1},\dots,p_{l}$ of $\Lambda$ in $\Xi$ so that any given center-unstable disk $D_{0}$ contains a nested sequence $D_{0}\supset\hat{D}_{1}\supset\hat{D}_{2}\supset\dots$ of disks admitting

• •

*a sequence $\widetilde{f_{i}}:\tilde{S_{i}}\to\Xi$ of smooth extensions and; *

• •

a subsequence $i_{k}$ so that $g_{k}=\widetilde{f_{i_{k}}}\circ\widetilde{f_{i_{k}-1}}\circ\cdots\circ\widetilde{f_{2}}\circ\widetilde{f_{1}}$ satisfies:

[TABLE]

Moreover, $(D_{k})_{k\geq 1}$ accumulates a $u$-disk $D$ in the $C^{1}$ topology which

• •

contains the local unstable manifold $W^{u}_{q}(\Xi)$ with respect to $f$ of a point of $q$ of the orbit of $p_{i}$ for some $i\in\{1,\ldots,l\}$; and

• •

whose inner radius is uniformly bounded away from zero: $\rho(D)\geq\rho_{0}$.

We prove Proposition 3.2 in the next subsection. Since ${\mathcal{W}}^{s}(\Xi)$ is transversal to any $cu$-disk and the nested disks $\hat{D}_{k}$ with vanishing diameter intersect in a unique point $r\in D_{0}\cap\Lambda$, then this shows that every center-unstable disk in any smooth strip contains the transversal intersection of the stable manifold of all points in the local unstable manifold of a periodic point of $\Lambda$, completing the proof of Theorem 3.1. ∎

3.2. Local uniform expansion of $cu$-disks

Here we fix a $cu$-disk $D_{0}$ in $S\in\{S_{i}\}$ and prove Proposition 3.2.

We obtain by induction a sequence of disks $D_{n},n\geq 0$ in $\Xi$ as follows. First, the inner radius of any $cu$-disk contained in a smooth strip $\widetilde{S}$ is uniformly expanded by the global Poincaré map.

Lemma 3.3**.**

If $\lambda_{2}\in(0,1)$ satisfies $2\lambda_{1}<\lambda_{2}(1-2a)$ and $D$ is a $cu$-disk contained in some extension $\widetilde{S}$ of a smooth strip $S\in\{S_{i}\}$, then

[TABLE]

where $\widetilde{f}:\tilde{S}\to\Xi$ is the extension of $f\mid S:S\to\Xi(a_{0})$.

Proof.

Let $S\subset\Sigma_{y}\in\Xi$. From Remark 2.18, $\widetilde{f}D$ is a $cu$-disk contained in some $\Sigma_{y^{\prime}}\in\Xi$ and we can write $\exp_{y^{\prime}}^{-1}(\widetilde{f}D)=\operatorname{Graph}(g:D_{u}^{1}\to E^{s}_{y^{\prime}}(\Sigma_{y^{\prime}}))$ where $D_{u}^{1}\subset E_{y^{\prime}}^{u}(\Sigma_{y^{\prime}})$ is an open subset. Then for a ball $B(x^{\prime},r)\subset D_{u}^{1}$ and $C^{1}$ curve $\gamma_{1}:(I,0,1)\to(\operatorname{Closure}{B(x^{\prime},r)},x^{\prime},\partial B(x^{\prime},r))$ there exists a unique curve $\gamma:I\to D_{u}=\pi_{u}\exp_{y}^{-1}D$ such that $\gamma_{1}(s)=\pi_{u}\widetilde{f}\exp_{y}(\gamma(s)+g_{1}\gamma(s))$, where $s\in I=[0,1]$. By Theorem 2.17 and Remark 2.18 together with the choice of $\Sigma_{y},\Sigma_{y^{\prime}}$ in Remark 2.6

[TABLE]

Then the bound on the inner radius follows by the choice of $\lambda_{2}$, since $\gamma_{1}$ is any curve joining $\gamma_{1}(0)=x^{\prime}$ to the boundary $\gamma_{1}(1)\in\partial B(x^{\prime},r)$ inside $D_{u}^{1}$. For the diameter, note that $\|u-v\|(1-2a)\leq\|u+g_{1}u-(v+g_{1}v)\|\leq(1+2a)\|u-v\|$ for all $u,v\in D_{u}^{1}$ and account the effect of $\exp_{y^{\prime}}$ on distances, cf. Remark 2.6. ∎

We let $\lambda_{2}$ be as in the statement of Lemma 3.3 in what follows; fix $\lambda_{2}<a_{1}<1$ and assume without loss of generality that $a_{1}\lambda_{2}^{-1}>5$. We assume that $cu$-disks $D_{0},\dots,D_{n}$ have already been obtained so that there are smooth strips $S_{0},\dots,S_{n}$ satisfying $D_{i}\subset\widetilde{S_{i}}\subset\Sigma_{y_{i}}$ and $D_{i+1}\subset\widetilde{f_{i}}D_{i}$, $i=0,\dots,n-1$.

Letting $D_{n}=\exp_{y_{n}}\operatorname{Graph}(g_{n})$ we consider the balls $\mathcal{B}=\{B(x,a_{1}\rho(D_{n}))\subset\pi_{u}\exp_{y_{n}}^{-1}D_{n}\}$ and corresponding disks $\mathcal{D}=\{D=\exp_{y_{n}}\operatorname{Graph}(g_{n}\mid B),B\in\mathcal{B}\}$. We set $\hat{\mathcal{D}}=\{D\in\mathcal{D}:\exists S,D\cap\partial^{s}S\neq\emptyset\}$ and $\hat{\mathcal{D}}_{\sigma}=\{D\in\mathcal{D}:D\cap\Gamma_{0}\neq\emptyset\}$. Then we have the following cases.

1. (1)

If $\mathcal{D}\not\subseteq\hat{\mathcal{D}}\cup\hat{\mathcal{D}}_{\sigma}$, then we choose some $D\in\mathcal{D}\setminus(\hat{\mathcal{D}}\cup\hat{\mathcal{D}}_{\sigma})$. There exists a smooth section $S$ so that $D\subset S$ and we reset $D_{n}=D$ and define $D_{n+1}=fD_{n}=(f\mid S)(D_{n})\subset\Xi(a_{0})$. 2. (2)

Otherwise: either $\hat{\mathcal{D}}\neq\emptyset$ or $\hat{\mathcal{D}}_{\sigma}\neq\emptyset$.

1. (a)

If $\hat{\mathcal{D}}_{\sigma}\neq\emptyset$, then we choose $D\in\hat{\mathcal{D}}_{\sigma}$ and $B\subset\pi_{u}\exp_{y_{n}}^{-1}D$ a ball of radius $a_{1}\rho(D_{n})/4$ so that, resetting $D_{n}=\exp_{y_{n}}B$, we have

[TABLE]

We then define $D_{n+1}=fD_{n}$. 2. (b)

Otherwise, we have $\hat{\mathcal{D}}\neq\emptyset=\hat{\mathcal{D}}_{\sigma}$ and consider the subfamily $\widetilde{\mathcal{D}}=\{D\in\hat{\mathcal{D}}:\exists S,D\cap\partial^{s}S\neq\emptyset\neq D\cap\partial^{s}\widetilde{S}\}$ of those disks which intersect both $\partial^{s}S$ and $\partial^{s}\widetilde{S}$ for some $S$.

1. (i)

If $\widetilde{\mathcal{D}}=\emptyset$, then we choose some $D\in\hat{\mathcal{D}}$ and $S$ so that $D\cap\partial^{s}S\neq\emptyset$; reset $D_{n}=D$; and define $D_{n+1}=\widetilde{f}D_{n}$, where $\widetilde{f}$ is the extension of $f\mid S$ to $\widetilde{S}$. 2. (ii)

Otherwise, we choose $D\in\widetilde{\mathcal{D}}$. There exists a subdisk $\hat{D}\subset D$ such that $\hat{D}\subset\widetilde{S}$ and $\rho(\hat{D})\geq a_{0}\delta_{1}/2$ by Remark 2.18 and definition of $\delta_{1}$ and $\hat{D}$. We reset $D_{n}=\hat{D}$ and define $D_{n+1}=\widetilde{f}D_{n}$, with $\widetilde{f}$ denoting the extension of $f\mid S$ to $\widetilde{S}$.

This complete the inductive step of the construction of a sequence $(D_{n})_{n\geq 0}$ of $cu$-disks in $\Xi$. Lemma 3.3 ensures that $\rho(D_{n+1})\geq(a_{2}\lambda_{2}^{-1}/4)\rho(D_{n})$ and $a_{1}\lambda_{2}^{-1}/4>5/4>1$ by the choice of $a_{1}$.

Since $\operatorname{diam}S<\operatorname{diam}\widetilde{S}$ is bounded by a uniform constant for all smooth strips $S\in\{S_{i}\}$, the expansion of the inner radius implies that the induction cannot go through cases (1), (2a) or (2b-i) above consecutively infinitely many times. Each time we are in case (2b-ii) we restart the algorithm choosing a subdisk of $D_{n+1}$ with half the inner radius.

We conclude that, starting with any disk $D_{0}$ as above, we obtain a subsequence $n_{k}\nearrow\infty$ so that $D_{n_{k}}$ is in case (2b-ii) and $\rho(D_{n_{k}})>a_{0}\delta_{1}/2$ for all $k\geq 1$. Moreover, there exists a subdisk $D^{\prime}_{n_{k-1}}$ with $2\rho(D^{\prime}_{n_{k-1}})=\rho(D_{n_{k}})$ and an iterate $f^{m_{k}}$ so that $f^{m_{k}}\mid D^{\prime}_{n_{k-1}}:D^{\prime}_{n_{k-1}}\to D_{n_{k}}$ is a diffeomorphism. In addition, the uniform bound on the diameter also ensures that $m_{k}=n_{k}-n_{k-1}$ is bounded: $m_{k}\leq\bar{m}$.

Finally, since $\Xi$ contains finitely many cross-sections, we can assume without loss of generality that $D_{n_{k}}\subset\Sigma_{y}\in\Xi$ for (possibly a subsequence of) all $k$. This is a sequence of graphs of $C^{1}$ functions with uniformly bounded derivative and domains given by balls with radius uniformly bounded away from zero. It follows that there exists a subsequence of such disks uniformly converging to a $cu$-disk $D$ in the $C^{1}$-topology.

In particular, for big enough $k$ we have that the stable holonomy map $h:D^{\prime}_{n_{k-1}}\to D_{n_{k}}$ within $\Sigma$ is well-defined by the choice of $D^{\prime}_{n_{k-1}}$ with half of the inner radius of $D_{n_{k}}$ and, moreover, is a continuous map; see Remark 2.4. Hence $g=h\circ(f^{m_{k}}\mid D_{n_{k}})^{-1}:D_{n_{k}}\to D_{n_{k}}$ has a fixed point by Brower’s Fixed Point Theorem, that is, there exists a stable leaf satisfying $f^{m_{k}}W^{s}_{x}(\Xi)\subset W^{s}_{x}(\Xi)$. The contraction of stable leaves implies the existence of a fixed point $p$ of $f^{m_{k}}\mid\Sigma$ which is a periodic point for the flow whose stable manifold crosses $D_{n_{k}}$. Since we can take $k$ as big as needed, we obtain that $W^{s}_{p}(\Sigma)$ transversely crosses $D$ also.

To complete the proof, since $D_{n+1}\subset\widetilde{f_{n}}D_{n}$ by construction, if we set $g_{n}=\widetilde{f_{n}}\circ\cdots\circ\widetilde{f_{1}},n\geq 1$, then we can find $\hat{D}_{n+1}\subset D_{0}$ so that $g_{n}\hat{D}_{n}=D_{n}$ and $D_{n+1}\subset D_{n}$, $n\geq 1$. Since $D_{n_{k}}\to D$ uniformly as graphs, for $x,y\in D$ there are $\widetilde{x_{k}},\widetilde{y_{k}}\in\hat{D}_{n_{k}}$ such that $(g_{n_{k}}\widetilde{x_{k}},g_{n_{k}}\widetilde{y_{k}})\to(x,y)$. By uniform expansion on $cu$-disks, for any given $i\geq 1$ we get $d(g_{n_{k}-i}\widetilde{x_{k}},g_{n_{k}-i}\widetilde{y_{k}})\leq\lambda_{1}^{i}d(g_{n_{k}}\widetilde{x_{k}},g_{n_{k}}\widetilde{y_{k}})$ for all $k\geq 1$. Thus, for an accumulation pair $(x_{i},y_{i})$ of $(g_{n_{k}-i}\widetilde{x_{k}},g_{n_{k}-i}\widetilde{y_{k}})$ and sequence $g^{i}=\widetilde{f_{i}}\circ\cdots\circ\widetilde{f_{0}}$ of $\widetilde{f_{n_{k}-i}}\circ\dots\circ\widetilde{f_{n_{k}}}$ as $k\nearrow\infty$, we get $(g^{i}x_{i},g^{i}y_{i})=(x,y)$ and $d(x_{i},y_{i})\leq\lambda_{1}^{i}d(x,y)$. Hence $D$ is a $u$-disk as claimed.

In addition, by the Inclination Lemma (“$\lambda$-lemma” [41, Chap. 2, §7]), we can assume without loss of generality that $n_{k}$ are multiples of $m_{k}$ for all large enough $k$ and that $D\subset W^{u}_{p}$ is a piece of the local unstable manifold of the hyperbolic periodic orbit $p$ of $f$, whose period is a divisor of $m_{k}$. In particular, $p$ is a hyperbolic periodic orbit for the flow whose period $\tau_{p}$ is bounded: $\tau_{p}\leq T=T(\bar{m})$.

This ensures the uniform size of the unstable manifold of the periodic orbits obtained by the previous algorithm and, moreover, since their period is bounded, that the possible periodic orbits are finite in number, by hyperbolicity and compactness.

This completes the proof of Proposition 3.2.

4. Finitely many ergodic physical measures for

sectional hyperbolic attracting sets

Here we prove Theorem A. We first obtain an auxiliary result consequence of the previous arguments on $cu$-disks contained in adapted cross-sections.

4.1. Uniformly center-unstable size of invariant

subsets

We prepare the proof of Theorem A obtaining a result on uniform size of positively flow-invariant subsets along the center-unstable direction.

We say that a $d_{cu}$-dimensional $C^{1}$ disk $D_{0}\subset U$ is a $cu$-disk if $T_{x}D_{0}\subset\mathcal{C}^{cu}_{x}(a)$ for all $x\in D$ (observe that such $D$ is not contained in any cross-section $\Sigma\in\Xi$).

Proposition 4.1**.**

For a sectional hyperbolic attracting set $\Lambda$ of a $C^{1}$ vector field $G$, there exists $\delta>0$ so that, given a positively $G$-invariant subset $E\subset\Lambda$ having a $cu$-disk $D$ such that $D\cap E$ has full Lebesgue induced measure in $D$, then there exists a $cu$-disk $\widetilde{D}$ whose inner radius is larger than $\delta$ and such that $\widetilde{D}\cap E$ has full Lebesgue induced measure in $\widetilde{D}$. Moreover, there exists $i\in\{1,\dots,l\}$ so that for any $\varepsilon>0$ we can find $\widetilde{D}$ as above $\varepsilon$-$C^{1}$-close to the local unstable manifold of $\mathcal{O}(p_{i})$, where the periodic point $p_{i}$ is given by Proposition 3.2.

Proof.

This is a consequence of Proposition 3.2. Indeed, if $E\subset\Lambda$ and $D$ are as stated, then we project $D$ into $D_{0}$ through the flow to the nearest cross-section, that is, for any $x\in D$ we consider $t(x)=\inf\{t>0:\phi_{t}x\in\Xi(a_{0})\}$ and $p(x)=\phi_{t(x)}x,x\in D$.

We claim that $p(D)$ contains a $cu$-disk $D_{0}$ inside some $\Sigma\in\Xi$ and moreover $E\cap D_{0}$ has full Lebesgue induced measure in $D_{0}$.

Assuming this claim, then $\hat{D}_{k}\cap E$ also has full Lebesgue induced measure in $\hat{D}_{k}$ for each of the disks $\hat{D}_{k}\subset D_{0}$ provided by Proposition 3.2. Moreover, since the Poincaré map $f$ is a piecewise $C^{1}$ diffeomorphism as well as its extensions, then $D_{k}=g_{k}\hat{D}_{k}$ is such that $D_{k}\cap E$ also has full Lebesgue induced measure in $D_{k}$ by invariance of $E$ under all transformations $\phi_{t},t\in{\mathbb{R}}$. The statement of Proposition 4.1 follows since, by construction, (i) the $cu$-disks $D_{k}$ have inner radius larger than some $\delta>0$ inside $\Sigma$; (ii) fixing some $k\geq 1$ we have that $\widetilde{D}=\phi_{[-\delta,\delta]}(D_{k})$ is a $d_{cu}$-dimensional center-unstable disk for the flow of $G$ with inner radius bounded away from zero; and (iii) by smoothness of the flow and invariance of $E$ we have that $\widetilde{D}\cap E=\phi_{[-\delta,\delta]}(D_{k}\cap E)$ also has full Lebesgue induced measure inside $\phi_{[-\delta,\delta]}(D_{k})$. Moreover, the $C^{1}$-closeness to the local unstable manifold of one of the hyperbolic periodic orbits provided by Proposition 3.2 follows, using the transversal intersection of the stable manifold of this orbit with $\widetilde{D}$ together with the Inclination Lemma.

We are left to prove the claim. Since $D\subset U$ we have $t(x)<\infty$ for all $x\in D$ and we fix $x_{0}\in D$ and $y_{0}=p(x_{0})\in\Sigma$, for some adapted cross-section $\Sigma\in\Xi(a_{0})$ in what follows, where we assume without loss of generality that $x_{0}$ is not a singularity.

We take a cross-section $S$ to $G$ at $x_{0}$ and note that since $D$ is a $cu$-disk for the flow, then there exists a neighborhood $V$ of $x_{0}$ in $M$ such that (i) $p(V)\subset\Sigma$ and (ii) $S$ is transversal to $D\cap V$. So $D_{S}=S\cap D\cap V$ is a submanifold of $M$ of codimension $1+d_{s}$. Hence, $D_{S}$ is a submanifold of $S$ of dimension $d_{cu}-1$ and a $cu$-disk inside $S$, that is, $T_{x}D_{S}\subset\mathcal{C}^{cu}_{x}(a,S)$ according to the definition of the induced center-unstable cone fields on a cross-section $S$. Consequently, $p(D_{S})$ is a $cu$-disk inside $\Sigma$ and contained in $p(D)$. We are left to show that $E$ has full Lebesgue induced measure in $p(D_{S})$.

We now conveniently choose coordinates on a local chart of $M$ at $V$ so that $S={\mathbb{R}}^{d-1}\times\{0\}$, $G(x_{0})=(0,\dots,0,1)$ and $E^{s}={\mathbb{R}}^{d_{s}}\times\{0^{d_{cu}}\},E^{cu}_{x_{0}}=\{0^{d_{s}}\}\times{\mathbb{R}}^{d_{cu}}$, and also $D\cap V$ is the graph of a $C^{1}$ map ${\varphi}:{\mathbb{R}}^{d_{cu}}\to{\mathbb{R}}^{d_{s}}$. Since $\Phi:\{0^{d_{s}}\}\times{\mathbb{R}}^{d_{cu}}\to D\cap V,u\mapsto({\varphi}u,u)$ is a $C^{1}$ diffeomorphism and $E\cap D\cap V$ has full Lebesgue induced measure in $D\cap V$, then $\widetilde{E}=\Phi^{-1}(E\cap D\cap V)$ has full Lebesgue measure in $\{0^{d_{s}}\}\times{\mathbb{R}}^{d_{cu}}$.

However $D_{S}=\Phi(\{0^{d_{s}}\}\times\{{\mathbb{R}}^{d_{cu}-1}\times\{0\})$ does not necessarily intersect $E$ in a full Lebesgue induced measure subset. But Fubbini’s Theorem ensures that $\widetilde{E}\cap\{0^{d_{s}}\}\times{\mathbb{R}}^{d_{cu}-1}\times\{t\}$ has full Lebesgue measure for Lebesgue almost every $t\in{\mathbb{R}}$.

Thus we can choose $t$ as close to [math] as needed so that $S_{t}={\mathbb{R}}^{d-1}\times\{t\}$ is a cross-section to $G$; $D_{t}=S_{t}\cap D\cap V$ is a $cu$-disk inside $S_{t}$ and $E\cap D_{t}$ has full Lebesgue induced measure in $D_{t}$. Moreover, we also have that $p(D_{t})\subset p(D)\subset\Sigma$ is a $cu$-disk inside $\Sigma$ and $p(D_{t}\cap E)$ has full Lebesgue induced measure in $p(D_{t})$, since $p\mid D_{t}:D_{t}\to p(D_{t})$ is a diffeomorphism as smooth as $G$.

This completes the proof of the claim with $D_{0}=p(D_{t})$ and Proposition 4.1 follows. ∎

4.2. Uniform volume of ergodic basis of physical

measures

We now extend the steps presented in [28] together with Proposition 4.1 and the following result.

Theorem 4.2**.**

[20, Appendix: Corollary B.1& Theorem I]** A $C^{1}$ vector field having a sectional hyperbolic attracting set $\Lambda$ supports an SRB measure. More precisely, for Lebesgue almost every point $x$ in the trapping region of $\Lambda$, any weak∗ limit measure of the family $\left(T^{-1}\int_{0}^{T}\delta_{\phi_{t}x}\,dt\right)_{T>0}$ is an SRB measure. Moreover, if the vector field is Hölder-$C^{1}$, then each limit measure is a physical measure.

The above result states that any weak∗ accumulation point $\mu$ of the empirical measures along the orbit of a Lebesgue generic point in $U$ is an equilibrium state for the logarithm of the center-unstable Jacobian, that is

[TABLE]

the positiveness being a consequence of sectional-hyperbolicity.

Moreover, if the flow is Hölder-$C^{1}$, then this SRB measure is also a physical measure since its support contains the (Pesin) unstable manifold through $\mu$-a.e. point and the stable foliation is absolutely continuous222This is a consequence of the partial hyperbolicity of the attracting set, that the vector field is Hölder-$C^{1}$ and Proposition 2.3; this can be seen adapting known arguments from [43, 42]., following standard geometric and ergodic arguments; see e.g. [28, Sections 2&3] and the proof of [20, Theorem I]. In particular, the center unstable manifold $W^{cu}_{x}$ through $\mu$-a.e. $x$ is a $cu$-disk contained in the attracting set $\Lambda$.

Remark 4.3**.**

From Proposition 4.1, since the support of any SRB measure $\mu$ is a forward invariant closed subset and contains a $cu$-disk, then there exists a periodic orbit $\mathcal{O}(p_{i})$ contained in $\operatorname{supp}\mu$ for some $i\in\{1,\dots,l\}$. The stable leaves through the points of the local unstable manifold $W^{cu}_{\mathcal{O}(p_{i})}(\varepsilon_{0})$, for all small enough $\varepsilon_{0}>0$, intersect each center-unstable manifold through $\mu$-a.e. point in an open subset, which contains $\mu$-generic points by the absolute continuity property of the stable foliation. In particular, this shows that no such periodic orbit can be shared by distinct SRB measures of a sectional-hyperbolic attracting subset.

Proof of Theorem A.

From Theorem 4.2 we have that any sectional hyperbolic attracing set for a $C^{1}$ flow admits some physical/SRB probability measure $\mu$ which we can assume, without loss of generality, to be ergodic. Indeed, using ergodic decomposition, by Ruelle’s Inequality [31] we have $h_{\mu}(\phi_{1})\leq\int\log|\det D\phi_{1}\mid E^{cu}|\,d\mu$ and so if $\mu$ satisfies (4.1), then each ergodic component of $\mu$ also satisfies (4.1)

Now we use that the ergodic basin $B(\mu)$ of $\mu$ contains a full Lebesgue measure subset of some center-unstable disk $D_{0}$ inside the sectional-hyperbolic attracting set together with Proposition 4.1.

Corollary 4.4**.**

Every sectional hyperbolic attracting set $\Lambda$ for a Hölder-$C^{1}$ vector field admits $\varepsilon_{0}>0$ so that the volume of the ergodic basin $B(\mu)$ of any ergodic $SRB$ measure $\mu$ supported in $\Lambda$ is uniformly bounded away from zero: ${\rm Leb}(B(\mu))\geq\varepsilon_{0}$.

Proof.

By assumption, $\mu$ is an ergodic $SRB$-measure and, as explained above, in our setting the stable holonomies are absolutely continuous. Then by [28, Lemma 3.2] we have that there exists a open subset $V$ of the basin of attraction of $\Lambda$ so that ${\rm Leb}$-a.e. $x\in V$ is $\mu$-generic, that is, ${\rm Leb}(V\setminus B(\mu))=0$.

Hence there exists a $cu$-disk $D_{0}\subset V$ such that $D_{0}\cap B(\mu)$ has full Lebesgue induced measure in $D_{0}$. Proposition 4.1 implies that the positively invariant subset $B(\mu)$ contains a $cu$-disk $D$ with $\rho(D)\geq\delta$ for some uniform $\delta>0$ depending only on $\Lambda$. The same proof of [28, Lemma 3.2], using the uniform size of local stable leaves of ${\mathcal{W}}^{s}$ and the angle between $E^{s}_{x}$ and $E^{cu}_{x}$ at $x\in D$ uniformly bounded away from zero (due to domination), implies that the set $W=\bigcup\{W^{s}_{x}:x\in D\}$ is open, diffeomorphic to a cylinder $D\times\mathcal{D}^{d_{s}}$ of uniform height. So ${\rm Leb}(W)\geq\varepsilon_{0}$ for some uniform $\varepsilon_{0}>0$. In addition, ${\rm Leb}$-a.e $x\in W$ belongs to $B(\mu)$ by the absolute continuity of the stable foliation. ∎

We are now ready to complete the proof of Theorem A: let $U$ be a trapping region for $\Lambda$. If ${\rm Leb}(U\setminus B(\mu))=0$, then $\mu$ is the unique physical/SRB measure supported in $\Lambda$. Otherwise, let $\mu_{1}=\mu$ and since $U_{1}=U\setminus B(\mu_{1})$ is such that ${\rm Leb}(U_{1})>0$ we can use [20, Theorem I] to ensure that ${\rm Leb}$-a.e. $x\in U_{1}$ belongs to the ergodic basin of some SRB measure $\mu_{2}\neq\mu_{1}$. This measure $\mu_{2}$ is a physical measure, satisfies ${\rm Leb}(B(\mu_{2}))>\delta>0$ by Corollary 4.4 and $B(\mu_{1})\cap B(\mu_{2})=\emptyset$ and $B(\mu_{1})\cup B(\mu_{2})\subset U$.

Again, if {\rm Leb}\big{(}U\setminus(B(\mu_{1})\cup B(\mu_{2}))\big{)}=0, then $\Lambda$ supports exactly the pair $\mu_{1},\mu_{2}$ of ergodic physical measures whose ergodic basins cover the topological basin of $\Lambda$ except perhaps a Lebesgue zero subset. Otherwise $U_{2}=U\setminus(B(\mu_{1})\cup B(\mu_{2}))$ is such that ${\rm Leb}(U_{2})>0$ and we can repeat the argument.

Since the ergodic basins of distinct ergodic physical probability measures are disjoint subsets of the trapping region $U$ which has finite volume, and each ergodic basin has a minimum volume bounded away from zero, this inductive process stops with finitely many $\mu_{1},\dots,\mu_{k}$ ergodic physical/SRB measures supported on $\Lambda$, whose basis cover the trapping region $U$, ${\rm Leb}\bmod 0$. This completes the proof of Theorem A. ∎

4.3. Full volume of stable leaves in the topological

basin

Here we prove the measure-theoretic part of the statement of Theorem C.

Let $\mu$ be an ergodic SRB measure supported in $\Lambda$ for a Hölder-$C^{1}$ vector field, that is, a physical measure. Let $\mathcal{O}(p_{i})$ be the hyperbolic periodic orbit contained in $\operatorname{supp}\mu$; see Remark 4.3. Let $V$ be an open neighborhood of $\mathcal{O}(p)$ and ${\varphi}:M\to{\mathbb{R}}$ a non-negative continuous function supported in $V$, so that $\mu({\varphi})=\int{\varphi}\,d\mu>0$. Hence

[TABLE]

Thus ${\varphi}(\phi_{t}x)>0$ for some $t>0$ and so $\phi_{t}x\in V$. This ensures that $x\in W^{s}_{y}$ for some $y\in W^{cu}_{\mathcal{O}(p_{i})}$, that is, $x\in{\mathcal{W}}^{cs}$ for each $x\in B(\mu)$.

Finally, from Theorem A, there are finitely many ergodic SRB measures whose basis cover ${\rm Leb}$-a.e. point of ${\mathcal{U}}$. This ensures that {\rm Leb}\big{(}{\mathcal{U}}\setminus{\mathcal{W}}^{cs}\big{)}=0, and completes the proof of the measure-theoretic part of the statement of Theorem C.

5. Statistical stability of sectional-hyperbolic

attracting sets

Statistical stability is essentially a consequence of the existence of finitely many physical measures whose basins cover ${\rm Leb}$-a.e points of the trapping region together with recent results from [40] on robust entropy expansiveness of sectional hyperbolic attractors on their trapping regions. We recall some relevant notions in what follows to be able to present a proof of Theorem B in Subsection 5.4.

5.1. Entropy expansiveness

Let $g:M\to M$ be a continuous map and $K$ a not necessarily invariant subset of $M$. For $\varepsilon>0$ and $n\geq 1$, the $(\varepsilon,n)$-dynamical ball around $x\in M$ is $B(x,\varepsilon,n)=\{y\in M:d(g^{j}x,g^{j}y)<\varepsilon,\quad\forall 0\leq j<n\}.$ A subset $E\subset M$ is a $(n,\varepsilon)$-generator for $K$ if, given $x\in K$, there exists $y\in E$ so that $d(g^{i}x,g^{i}y)<\varepsilon$ for each $0\leq i<n$. Equivalently, the dynamical ball $\{B(y,\varepsilon,n):y\in E\}$ are an open cover of $K$.

Let $r_{n}(K,\varepsilon)$ be the cardinality of the smallest $(n,\varepsilon)$-generator for $K$ and $r(K,\varepsilon)=\limsup_{n\to\infty}\frac{1}{n}\log r_{n}(K,\varepsilon).$ The topological entropy of $g$ on $K$ is given by

[TABLE]

and the topological entropy of $g$ is defined by $h_{top}(g)=h_{top}(g,M)$.

For $x\in M$ and $\varepsilon>0$ we define the two-sided $\varepsilon$-dynamical ball at $x$ as $B(x,\varepsilon,\infty)=\{y:d(g^{n}x,g^{n}y)<\varepsilon\,\forall n\in{\mathbb{Z}}\}$ and say that $g$ is $\varepsilon$-entropy expansive if all these infinite dynamical balls have zero topological entropy, that is, \sup_{x\in M}h_{top}\big{(}g,B(x,\varepsilon,\infty)\big{)}=0.

5.2. Upper semicontinuity of metric entropy

Let $\mu$ be a $g$-invariant measure and $\mathcal{P}$ a finite $\mu\bmod 0$ measurable partition. The metric entropy of $\mu$ with respect to the partition $\mathcal{P}$ is given by

[TABLE]

and $\mathcal{P}^{n}$ is the $n$th dynamical refinement of $\mathcal{P}$: $\mathcal{P}^{n}=\mathcal{P}\vee g^{-1}\mathcal{P}\vee\cdots\vee g^{-(n-1)}\mathcal{P}$. The metric entropy of $\mu$ is $h_{\mu}(g)=\sup_{\mathcal{P}}h_{\mu}(g,\mathcal{P})$ and the supremum is taken over all finite measurable partitions.

If $g$ is $\varepsilon$-entropy expansive, then every finite partition $\mathcal{P}$ with $\operatorname{diam}\mathcal{P}<\varepsilon$ is generating, that is, it satisfies $h_{\mu}(g)=h_{\mu}(g,\mathcal{P})$ for all $\mu\mu\in\mathcal{M}_{1}^{g}$, where $\mathcal{M}_{1}^{g}$ is the family of all $g$-invariant probability; measures see e.g. [16].

The metric entropy of a vector field is the metric entropy of the time-one map of its induced flow. A vector field is $\varepsilon$-entropy expansive if the time-one map of its induced flow is $\varepsilon$-entropy expansive.

Entropy expansiveness is a sufficient condition to ensure upper semicontinuity of the entropy map $\mu\in\mathcal{M}_{1}^{g}\mapsto h_{\mu}(g)$, as follows.

Lemma 5.1**.**

[16]** If $G$ is entropy expansive, then the metric entropy function is upper semicontinuous.

5.3. Equilibrium states and physical measures

Since the family $\mathcal{M}_{1}^{G}$ of $G$-invariant probability measures is compact in the weak∗ topology, then for entropy expansive vector fields there exist some measure which maximizes the function $\mu\in\mathcal{M}_{1}^{G}\mapsto h_{\mu}(G)+\int\psi\,d\mu$ for any given continuous function $\psi:M\to{\mathbb{R}}$, known as an equilibrium state for $\psi,G$.

In order to use equilibrium states to obtain statistical stability, we relate equilibrium states for the potencial $\psi=\log|\det D\phi_{1}\mid E^{cu}|$ with physical measures in the same way as for hyperbolic attracting sets; see e.g. [18].

Theorem 5.2**.**

Let $\Lambda$ be a sectional-hyperbolic attracting set for a Hölder-$C^{1}$ vector field $G$ with the open subset $U$ as trapping region. Then

1. (1)

Each $G$-invariant ergodic probability measure $\mu$ supported in $\Lambda$ the following are equivalent

1. (a)

$h_{\mu}(\phi_{1})=\int\psi\,d\mu>0$; 2. (b)

$\mu$* is a $SRB$ measure, that is, admits an absolutely continuous disintegration along unstable manifolds;* 3. (c)

$\mu$* is a physical measure, i.e., its basin $B(\mu)$ has positive Lebesgue measure.* 2. (2)

In addition, the family ${\mathbb{E}}$ of all $G$-invariant probability measures which satisfy item (a) above is the convex hull ${\mathbb{E}}=\{\sum_{i=1}^{k}t_{i}\mu_{i}:\sum_{i}t_{i}=1;0\leq t_{i}\leq 1,i=1,\dots,k\}.$

We recall that from sectional hyperbolicity together with Ruelle’s Inequality [48] we have $h_{\nu}(\phi_{1})\leq\int\psi\,d\nu$ for all $\nu\in\mathcal{M}_{1}^{G}$. Hence, the set ${\mathbb{E}}$ defined above is formed by equilibrium states for $-\psi,G$. The proof of Theorem 5.2 can be found in [9, Section 2.3] where the same properties were stated and proved in the $d_{cu}=2$ setting (singular-hyperbolic attracting sets). However, the proof presented there also holds in the present setting without change.

5.4. Statistical stability

Here we prove Theorem B.

We consider vector fields $G$ on a subset ${\mathcal{U}}$ of $\mathcal{X}^{1+}(M)$ with a trapping region $U$ of a sectional hyperbolic attracting set $\Lambda_{G}=\Lambda_{G}(U)=\cap_{t>0}\phi_{t}^{G}(U)$ so that each $G\in{\mathcal{U}}$ is $\varepsilon$-entropy expansive. Then the map ${\mathcal{U}}\to\mathcal{K}(U),G\mapsto\Lambda_{G}(U)$ is continuous, where $\mathcal{K}(U)$ is the family of compact subsets of $U$ with the Hausdorff distance between compact subsets $K,L\subset U$ of a metric space given by (see e.g. [22])

[TABLE]

Lemma 5.3**.**

[7*, Lemma 2.3]**

For every $\varepsilon>0$ there is a neighborhood ${\mathcal{V}}$ of $G$ in $\mathcal{X}^{1}(M)$ such that $\Lambda_{Y}(U)\subset B(\Lambda_{G}(U),\varepsilon)$ and $\Lambda_{G}(U)\subset B(\Lambda_{Y}(U),\varepsilon)$ for all $Y\in{\mathcal{V}}$.*

Moreover the map $\nu\in\mathcal{M}\mapsto\operatorname{supp}\nu\in\mathcal{K}(M)$ is also continuous, where $\mathcal{M}$ is the family of probability measures in $M$ with the weak∗ topology. In addition, the domination of the splitting $E_{\Lambda}^{s}\oplus E_{\Lambda}^{cu}$ implies its continuity for nearby vector fields; see e.g. [14, Appendix B.1].

For any fixed $G\in{\mathcal{U}}$ and any sequence $G_{n}\in{\mathcal{U}}$ such that $\|G_{n}-G\|_{C^{1}}\to 0$ when $n\nearrow\infty$, we let $\mu_{n}\in\mathcal{M}_{1}^{G_{n}}$ be equilibrium states for $\psi_{n},G_{n},n\geq 1$, where \psi_{n}=\psi_{G_{n}}=\log\big{|}\det D\phi_{1}^{G_{n}}\mid E^{cu}_{\Lambda_{G_{n}}(U)}\big{|}, and $\mu$ be a weak∗ limit point of $(\mu_{n})_{n\geq 1}$. We assume that $\mu=\lim\mu_{n}$ restricting to a subsequence if necessary. Since the splitting $E^{s}_{\Lambda_{G_{n}}(U)}\oplus E^{cu}_{\Lambda_{G_{n}}(U)}$ is continuous, we can deal with its continuous extension $E^{s}_{n}\oplus E^{cu}_{n}$ to define $\psi_{n}$ on the whole of $M$.

The continuity of dominated splittings for nearby vector fields means that for each $\xi>0$ there exists $N\geq 1$ and a neighborhood $V$ of $\operatorname{supp}\mu$ so that

[TABLE]

where the distance $\operatorname{dist}(E,F)$ between two subspaces $E,F$ of $T_{x}M$ is defined to be

[TABLE]

and $\operatorname{dist}(v,H):=\min_{w\in E}\|v-w\|$ for each subspace $H$ of $T_{x}M$ and any $x\in M$.

Moreover, since $D\phi^{G_{n}}_{1}(x)$ converges to $D\phi_{1}^{G}(x)$ uniformly in $x$ when $n\nearrow\infty$, then $\psi_{n}\to\psi=\psi_{G}$ uniformly by definition of the $C^{1}$ topology, in the following sense: for any given $\xi>0$ there is $N\geq 1$ and a neighborhood $V$ of $\operatorname{supp}\mu$ so that $|\psi_{n}(x)-\psi(x)|<\xi$ for all $x\in V$ and each $n>N$.

Proof of Theorem B.

Using the compactness of the manifold $M$, we construct a finite open cover $\{B(x_{i},\delta):i=1,\dots,k\}$ for some $2\delta<\varepsilon$ such that $\mu(B(x,\delta))=0$, $i=1,\dots,k$; and obtain the partition $\mathcal{P}=\bigvee_{i=1}^{k}B(x_{i},\varepsilon/2)$ with diameter smaller than $\varepsilon$ and the boundaries of each atom with zero $\mu$-measure. Hence, for each $k\geq 1$, we have that $\mu(\partial\mathcal{P}^{k})=0$ since by continuity we have

[TABLE]

Now for each fixed $k\geq 1$ we find

[TABLE]

where $\mathcal{P}_{n}^{k}=\bigvee_{i=0}^{k-1}\phi_{-i}^{G_{n}}\mathcal{P}$ and $(\phi_{t}^{G_{n}})_{t}$ is the flow induced by $G_{n}$.

Lemma 5.4**.**

For each fixed $k\geq 1$ we have $\limsup_{n}\frac{1}{k}H_{\mu_{n}}(\mathcal{P}_{n}^{k})\leq\frac{1}{k}H_{\mu}(\mathcal{P}^{k})$ where $\mathcal{P}^{k}=\bigvee_{i=0}^{k-1}\phi_{-i}^{G}\mathcal{P}$.

Assuming the lemma, since $k\geq 1$ is arbitrary and (possibly taking a subsequence) we have $\mu_{n}\to\mu$ in the weak∗ topology, we have

[TABLE]

Consequently, we deduce that

[TABLE]

and so $\mu$ achieves the maximum of $\mu\in\mathcal{M}_{1}^{G}\mapsto h_{\mu}(G)-\int\psi\,d\mu$. From Theorem 5.2 we have that $\mu$ is a convex linear combination of the finitely many ergodic physical measures supported in $\Lambda_{G}(U)$ provided by Theorem A. ∎

To complete the proof of Theorem B we present the proof of the lemma.

Proof of Lemma 5.4.

Observe that \sup_{|t|<k}d\big{(}\phi_{t}^{G_{n}}(x),\phi_{t}^{G}(x))\xrightarrow[n\to\infty]{u}0 for all fixed $k\geq 1$ and uniformly in $x\in M$. Moreover, we may assume without loss of generality that each $P\in\mathcal{P}$ has non-empty interior by construction.

Thus for each $\delta>0$ and atom $Q\in\mathcal{P}$ there exists $N=N(\delta,Q)\in{\mathbb{Z}}^{+}$ such that for all $n\geq N$ and $0\leq t\leq k$

• •

$\phi^{G}_{-t}(Q)\cap\phi_{-t}^{G_{n}}(Q)\neq\emptyset$ and $\phi_{-t}^{G_{n}}(Q)\subset B_{\delta}(\phi_{-t}^{G}(Q))$; and

• •

$\mu(\partial B_{\delta}(Q))=0$;

where $B_{\delta}(Q)=\cup_{x\in Q}B(x,\delta)$ is the $\delta$-neighborhood of the set $Q$. Let $N(\delta,\mathcal{P}^{k})=\max_{Q\in\mathcal{P}^{k}}N(\delta,Q)$ be chosen to satisfy the previous relations for all $Q\in\mathcal{P}^{k}$ simultaneously.

For $\omega>0$ let $\zeta>0$ be such that

[TABLE]

and, for each $\delta>0$, let $L=L(\zeta,\delta,\mathcal{P}^{k})$ be such that $\mu(\partial B(Q,\delta))=0,\forall Q\in\mathcal{P}^{k}$ and

[TABLE]

Since $\mu(\partial\mathcal{P}^{k})=0$, let $\delta_{0}$ be such that $\mu(B_{\delta}(Q))\leq\mu(Q)+\zeta/2$ for all $Q\in\mathcal{P}^{k}$.

Now, we take $0<\delta<\delta_{0}$ in the previous choices, and for $n\geq L(\zeta,\delta,\mathcal{P}^{k})+N(\delta,\mathcal{P}^{k})$ we have for each $Q_{n}\in\mathcal{P}^{k}_{n}$ that there exists $Q\in\mathcal{P}^{k}$ so that

[TABLE]

which ensures by the choice of the pair $(\zeta,\omega)$ that

[TABLE]

for all big enough $n$ depending on $\omega$. Since $\omega>0$ is arbitrary, this shows that

[TABLE]

and completes the proof of the lemma ∎