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We show the existence of physical measures for $C^{\infty}$ smooth instances of certain partially hyperbolic dynamics, both continuous and discrete, exhibiting mixed behavior (positive and negative Lyapunov exponents) along the central non-uniformly hyperbolic multidimensional invariant direction, as a consequence of assuming the existence of certain types of ''regular points'' on positive volume subsets. This includes the $C^3$ robust class of multidimensional non-hyperbolic attractors obtained by Viana, and the $C^1$ robust classes of $3$-sectionally hyperbolic wild strange attractors presented by Shilnikov and Turaev., Comment: 15 pages, 1 figure; improved references and extended comments

On a compact manifold of any dimension $d\geq 3$, we show that joint non-integrability of the stable and unstable foliation of a hyperbolic attractor with one-dimensional expanding direction, for a vector field of class $C^2$, implies exponential mixing with respect to its physical measure. Consequently, the set of Axiom A vector fields which mix exponentially with respect to the physical measure of its non-trivial attractors contains a $C^1$-open and $C^2$-dense subset of the set of all Axiom A vector fields. Moreover, for codimension one $C^2$ Anosov flows in any dimension $d\geq 3$, if the flow mixes with respect to the unique physical measure, then the flow mixes exponentially, proving the Bowen-Ruelle conjecture in this setting., Comment: A fatal flaw was found on a crucial lemma. The proof of the main statement as it stands is incomplete

We prove that a partially hyperbolic attracting set for a $C^2$ vector field supports an ergodic physical/SRB measure if, and only if, the trapping region admits non-uniform sectional expansion on a positive Lebesgue measure subset. Moreover, in this case, the attracting set supports at most finitely many ergodic physical/SRB measures, which are also Gibbs states along the central-unstable direction. This extends to continuous time systems a similar well-known result obtained for diffeomorphisms, encompassing the presence of equilibria accumulated by regular orbits within the attracting set, since it is shown that slow recurrence to hyperbolic saddle-type equilibria is automatic for Lebesgue almost every point. We present several examples of application, including the existence of physical measures for asymptotically sectional hyperbolic attracting sets., Comment: 50 pages, 9 figures; improved presentation, shortened title, included (counter-)examples, improved results showing that slow recurrence to saddle-type hyperbolic equilibria is automatic thus dropping a crucial sufficient condition from the main theorems

It is known that sectional-hyperbolic attracting sets, for a $C^2$ flow on a finite dimensional compact manifold, have at most finitely many ergodic physical invariant probability measures. We prove an upper bound for the number of distinct ergodic physical measures supported on a connected singular-hyperbolic attracting set for a $3$-flow. This bound depends only on the number of Lorenz-like equilibria contained in the attracting set. Examples of singular-hyperbolic attracting sets are provided showing that the bound is sharp., Comment: 31 pages, 13 figures

There exists a $C^2$-open and $C^1$-dense subset of vector fields exhibiting singular-hyperbolic attracting sets (with codimension-two stable bundle), in any $d$-dimensional compact manifold ($d\ge3$), which mix exponentiallu with respect to any physical/SRB invariant probability measure. More precisely, we show that given any connected singular-hyperbolic attracting set for a $C^2$-vector field $X$, there exists a $C^1$-close multiple of $X$ of class $C^2$, generating a topologically equivalent flow, which is robustly exponentially mixing with respect to any physical measure for all vector fields in a $C^2$ neighborhood. That is, every singular-hyperbolic attracting set mixes exponentially with respect to its physical measures modulo an arbitrarily small change in the speed of the flow., Comment: A crucial lemma in the argument has a fatal flaw. The proof of the main statement as it stands is incomplete

We extend results on robust exponential mixing for geometric Lorenz attractors, with a dense orbit and a unique singularity, to singular-hyperbolic attracting sets with any number of (either Lorenz- or non-Lorenz-like) singularities and finitely many ergodic physical/SRB invariant probability measures, whose basins cover a full Lebesgue measure subset of the trapping region of the attracting set. We obtain exponential mixing for any physical probability measure supported in the trapping region and also exponential convergence to equilibrium, for a $C^2$ open subset of vector fields in any $d$-dimensional compact manifold ($d\ge3$)., Comment: 56 pages; 6 figures; minor changes proposed by referees; content as accepted for publication in Journal of Dynamics and Differential Equations, 2022

We present criteria for statistical stability of attracting sets for vector fields using dynamical conditions on the corresponding generated flows. These conditions are easily verified for all singular-hyperbolic attracting sets of $C^2$ vector fields using known results, providing robust examples of statistically stable singular attracting sets (encompassing in particular the Lorenz and geometrical Lorenz attractors). These conditions are shown to hold also on the persistent but non-robust family of contracting Lorenz flows (also known as Rovella attractors), providing examples of statistical stability among members of non-open families of dynamical systems. In both instances, our conditions void the use of detailed information about perturbations of the one-dimensional induced dynamics on specially chosen Poincar\'e sections., Comment: 15 pages; 01 figure; online version https://rdcu.be/cf6tH

We prove that sectional-hyperbolic attracting sets for $C^1$ vector fields are robustly expansive (under an open technical condition of strong dissipative for higher codimensional cases). This extends known results of expansiveness for singular-hyperbolic attractors in $3$-flows even in this low dimensional setting. We deduce some converse results taking advantage of recent progress in the study of star vector fields: a robustly transitive attractor is sectional-hyperbolic if, and only if, it is robustly expansive. In a low dimensional setting, we show that an attracting set of a $3$-flow is singular-hyperbolic if, and only if, it is robustly chaotic (robustly sensitive to initial conditions)., Comment: 33 pages; 07 figures; keywords: sectional-hyperbolicity, robust expansiveness, strong dissipativity, star flow, robust transitivity, robust chaotic, attracting sets. Refocused statements of main theorems and proof of more important results. Corrected some typos and improved some definitions

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., Comment: 27 pages; 2 figures. Improved statements and proofs after referee comments; version accepted to appear in Ergodic Theory and Dynamical Systems

Let $f:M\to M$ be a $C^1$ map of a compact manifold $M$, with dimension at least $2$, admitting some point whose future trajectory has only negative Lyapunov exponents. Then this trajectory converges to a periodic sink. We need only assume that $Df$ is never the null map at any point (in particular, we need no extra smoothness assumption on $Df$), encompassing a wide class of possible critical behavior. Similarly, a trajectory having only positive Lyapunov exponents for a $C^1$ diffeomorphism is itself a periodic repeller (source). Analogously for a $C^1$ open and dense subset of vector field on finite dimensional manifolds: for a flow $\phi_t$ generated by such a vector field, if a trajectory admits weak asymptotic sectional contraction (the extreme rates of expansion of the Linear Poincar\'e Flow are all negative), then this trajectory belongs either to the basin of attraction of a periodic hyperbolic attracting orbit (a periodic sink or an attracting equilibrium); or the trajectory accumulates a codimension one saddle singularity. Similar results hold for weak sectional expanding trajectories. Both results extend part of the non-uniform hyperbolic theory (Pesin's Theory) from the $C^{1+}$ diffeomorphism setting to $C^1$ endomorphisms and $C^1$ flows. Some ergodic theoretical consequences are discussed. The proofs use versions of Pliss' Lemma for maps and flows translated as (reverse) hyperbolic times, and a result ensuring that certain subadditive cocycles over vector fields are in fact additive., Comment: 37 pages; 10 figures; presentation improved after referee's report and correction of imprecise statements of lemmas 3.3, 3.4 and 3.5, and small adjustements in the proofs. To appear in Osaka J Math 2020

Singular and sectional hyperbolic sets are the objects of the extension of the classical Smale Hyperbolic Theory to flows having invariant sets with singularities accumulated by regular orbits within the set. It is by now well-known that (partially) hyperbolic sets admit adapted metrics. We show the existence of singular adapted metrics for any singular hyperbolic set with respect to a $C^{1}$ vector field on finite dimensional compact manifolds. Moreover, we obtain 2-sectional adapted metrics for certain open classes of 2-sectional hyperbolic sets and also for any hyperbolic set., Comment: 23 pages, 1 figure. Improved the statements of the results and corrected the proof of the main theorem. To appear in Bull. Brazilian Mathematical Society

We obtain a exponential large deviation upper bound for continuous observables on suspension semiflows over a non-uniformly expanding base transformation with non-flat singularities and/or discontinuities, where the roof function defining the suspension behaves like the logarithm of the distance to the singular/discontinuous set of the base map. To obtain this upper bound, we show that the base transformation exhibits exponential slow recurrence to the singular set. The results are applied to semiflows modeling singular-hyperbolic attracting sets of $C^2$ vector fields. As corollary of the methods we obtain result on the existence of physical measure for classes of piecewise $C^{1+}$ expanding maps of the interval with singularities and discontinuities. We are also able to obtain exponentially fast escape rates from subsets without full measure., Comment: 55 pages; 6 figures; accepted version JDDE. Major changes after referee report: Theorem C weakened, new Corollary D included. Deep changes in Section 4 due to issue with bounded distortion estimates. To the best of out knowledge, this is the first time non-exponential initial partitions are used in this setting and still provide upper exponential decay for large deviations

Over the last 10 years or so, advanced statistical properties, including exponential decay of correlations, have been established for certain classes of singular hyperbolic flows in three dimensions. The results apply in particular to the classical Lorenz attractor. However, many of the proofs rely heavily on the smoothness of the stable foliation for the flow. In this paper, we show that many statistical properties hold for singular hyperbolic flows with no smoothness assumption on the stable foliation. These properties include existence of SRB measures, central limit theorems and associated invariance principles, as well as results on mixing and rates of mixing. The properties hold equally for singular hyperbolic flows in higher dimensions provided the center-unstable subspaces are two-dimensional., Comment: Accepted version. To appear in Advances in Math

We prove existence of equilibrium states with special properties for a class of distance expanding local homeomorphisms on compact metric spaces and continuous potentials. Moreover, we formulate a C$^1$ generalization of Pesin's Entropy Formula: all ergodic weak-SRB-like measures satisfy Pesin's Entropy Formula for $C^1$ non-uniformly expanding maps. We show that for weak-expanding maps such that $\Leb$-a.e $x$ has positive frequency of hyperbolic times, then all the necessarily existing ergodic weak-SRB-like measures satisfy Pesin's Entropy Formula and are equilibrium states for the potential $\psi=-\log|\det Df|$. In particular, this holds for any $C^1$-expanding map and in this case the set of invariant probability measures that satisfy Pesin's Entropy Formula is the weak$^*$-closed convex hull of the ergodic weak-SRB-like measures., Comment: 47 pages, 5 figures. Content updated to last accepted version to appear in Math. Z

We show that a uniformly continuous random perturbation of a transitive map defines an aperiodic Harris chain which also satisfies Doeblin's condition. As a result, we get exponential decay of correlations for suitable random perturbations of such systems. We also prove that, for transitive maps, the limiting distribution for Extreme Value Laws (EVLs) and Hitting/Return Time Statistics (HTS/RTS) is standard exponential. Moreover, we show that the Rare Event Point Process (REPP) converges in distribution to a standard Poisson process., Comment: 1 figure; 21 pages. Added affiliation of both authors and more keywords. Now the contents match the final version published in Nonlinearity

Using Caratheodory measures, we associate to each positive orbit ${\mathcal O}_{f}^{+}(x)$ of a measurable map $f$, a Borel measure $\eta_{x}$. We show that $\eta_{x}$ is $f$-invariant whenever $f$ is continuous or $\eta_{x}$ is a probability. These measures are used to study the \emph{historic} points of the system, that is, \emph{points with no Birkhoff averages}, and we construct topologically generic subset of \emph{wild historic points} for wide classes of dynamical models. We use properties of the measure $\eta_x$ to deduce some features of the dynamical system involved, like the \emph{existence of heteroclinic connections from the existence of open sets of historic points}., Comment: 34 pages; 4 figures; this version focuses on abundance of wild historic points and leaves the ergodic decomposition and generalized physical measures part for another publication; updated with referees comments and suggestions. To appear in Bull Braz Math Soc

We prove exponential decay of correlations for a class of $C^{1+\alpha}$ uniformly hyperbolic skew product flows, subject to a uniform nonintegrability condition. In particular, this establishes exponential decay of correlations for an open set of geometric Lorenz attractors. As a special case, we show that the classical Lorenz attractor is robustly exponentially mixing., Comment: Final version. Appeared online, Annales Henri Poincare

For Hoelder cocycles over a Lipschitz base transformation, possibly non-invertible, we show that the subbundles given by the Oseledets Theorem are Hoelder-continuous on compact sets of measure arbitrarily close to 1. The results extend to vector bundle automorphisms, as well as to the Kontsevich-Zorich cocycle over the Teichmueller flow on the moduli space of abelian differentials. Following a recent result of Chaika-Eskin, our results also extend to any given Teichmueller disk., Comment: 29 pages; corrected a few typos and added several references and comments following referee suggestions (this is the final version content to appear in JLMS)

It is proved that an endomorphism $\varphi$ of a hyperbolic group $G$ satisfies a H\"older condition with respect to a visual metric if and only if $\varphi$ is virtually injective and $G\varphi$ is a quasiconvex subgroup of $G$. If $G$ is virtually free or torsion-free co-hopfian, then $\varphi$ is uniformly continuous if and only if it it satisfies a H\"older condition if and only if it is virtually injective. Lipschitz conditions are discussed for free group automorphisms., Comment: 27 pages

Four geometric conditions on a geodesic metric space, which are stronger variants of classical conditions characterizing hyperbolicity, are proved to be equivalent. In the particular case of the Cayley graph of a finitely generated group, it is shown that they characterize virtually free groups., Comment: 12 pages

For any dimension $d\geq 3$ we construct $C^{1}$-open subsets of the space of $C^{3}$ vector fields such that the flow associated to each vector field is Axiom A and exhibits a non-trivial attractor which mixes exponentially with respect to the unique SRB measure., Comment: 15 pages, no figures; content as published in journal; includes text of the Erratum as an appendix

We discuss in the context of finite extensions two classical theorems of Takahasi and Howson on subgroups of free groups. We provide bounds for the rank of the intersection of subgroups within classes of groups such as virtually free groups, virtually nilpotent groups or fundamental groups of finite graphs of groups with virtually polycyclic vertex groups and finite edge groups. As an application of our generalization of Takahasi's Theorem, we provide an uniform bound for the rank of the periodic subgroup of any endomorphism of the fundamental group of a given finite graph of groups with finitely generated virtually nilpotent vertex groups and finite edge groups., Comment: 20 pages; no figures. Keywords: finite extensions, Howson's Theorem, Hanna Neumann Conjecture, Takahasi's Theorem, periodic subgroups

We obtain stochastic stability of C2 non-uniformly expanding one-dimensional endomorphisms, requiring only that the first hyperbolic time map be L^{p}-integrable for p>3. We show that, under this condition (which depends only on the unperturbed dynamics), we can construct a random perturbation that preserves the original hyperbolic times of the unperturbed map and, therefore, to obtain non-uniform expansion for random orbits. This ensures that the first hyperbolic time map is uniformly integrable for all small enough noise levels, which is known to imply stochastic stability. The method enables us to obtain stochastic stability for a class of maps with infinitely many critical points. For higher dimensional endomorphisms, a similar result is obtained, but under stronger assumptions., Comment: 25 pages; 1 figure; many typos corrected. Keywords: stochastic stability, additive perturbation, non-uniform expansion, slow recurrence, adapted random perturbation, stationary measures, physical measures, first hyperbolic time map. To be published in Stochastics and Dynamics

We relate dominated splitting for a linear multiplicative cocyle with dominated splitting for the exterior powers of this cocycle. For a C1 vector field X on a 3-manifold, we can obtain singular-hyperbolicity using only the tangent map DX of X and a family of indefinite and non-degenerate quadratic forms without using the associated flow X_t and its derivative DX_t. As a consequence, we show the existence of adapted metrics for singular-hyperbolic sets for three-dimensional C1 vector fields., Comment: 19 pages; major changes to the introduction and main statements with more precise results. 1 figure. Extra examples and counter-examples to certain higher-dimensional generalizations. Streamlined presentation to motivate mais results. arXiv admin note: text overlap with arXiv:1201.2550

We consider two dimensional maps preserving a foliation which is uniformly contracting and a one dimensional associated quotient map having exponential convergence to equilibrium (iterates of Lebesgue measure converge exponentially fast to physical measure). We prove that these maps have exponential decay of correlations over a large class of observables. We use this result to deduce exponential decay of correlations for the Poincare maps of a large class of singular hyperbolic flows. From this we deduce logarithm laws for these flows., Comment: 39 pages; 03 figures; proof of Theorem 1 corrected; many typos corrected; improvements on the statements and comments suggested by a referee. Keywords: singular flows, singular-hyperbolic attractor, exponential decay of correlations, exact dimensionality, logarithm law

We present an extension of the notion of infinitesimal Lyapunov function to singular flows, and from this technique we deduce a characterization of partial/sectional hyperbolic sets. In absence of singularities, we can also characterize uniform hyperbolicity. These conditions can be expressed using the space derivative DX of the vector field X together with a field of infinitesimal Lyapunov functions only, and are reduced to checking that a certain symmetric operator is positive definite at the tangent space of every point of the trapping region., Comment: 37 pages, 1 figure; corrected the statement of Lemma 2.2 and item (2) of Theorem 2.7; removed item (5) of Theorem 2.7 and its wrong proof since the statement of this item was false; corrected items (1) and (2) of Theorem 2.23 and their proofs. Included Example 6 on smooth reduction of families of quadratic forms. The published version in Math Z journal needs an errata

We prove existence of (at most denumerable many) absolutely continuous invariant probability measures for random one-dimensional dynamical systems with asymptotic expansion. If the rate of expansion (Lyapunov exponents) is bounded away from zero, we obtain finitely many ergodic absolutely continuous invariant probability measures, describing the asymptotics of almost every point. We also prove a similar result for higher-dimensional random non-uniformly expanding dynamical systems. The results are consequences of the construction of such measures for skew-products with essentially arbitrary base dynamics and asymptotic expansion along the fibers. In both cases our method deals with either critical or singular points for the random maps., Comment: 30 pages; 2 figures. Keywords: non-uniform expansion, random dynamics, slow recurrence, singular and critical set, absolutely continuous invariant measures, skew-product. To appear in Math Z, 2014

We obtain sufficient conditions for an invariant splitting over a compact invariant subset of a $C^1$ flow $X_t$ to be dominated. In particular, we reduce the requirements to obtain sectional hyperbolicity and hyperbolicity., Comment: 15 pages, 4 figures; streamlined presentation; statement and proof of main theorem corrected; included new application to obtain hyperbolicity for three-dimensional flows; some corrections suggested by the referee

We construct open sets of Ck (k bigger or equal to 2) vector fields with singularities that have robust exponential decay of correlations with respect to the unique physical measure. In particular we prove that the geometric Lorenz attractor has exponential decay of correlations with respect to the unique physical measure., Comment: Final version accepted for publication with added corrections (not in official published version) after O. Butterley pointed out to the authors that the last estimate in the argument in Subsection 4.2.3 of the previous version is not enough to guarantee the uniform non-integrability condition claimed. We have modified the argument and present it here in the same Subsection. 3 figures, 34 pages

We show that the time-1 map of an Anosov flow, whose strong-unstable foliation is $C^2$ smooth and minimal, is $C^2$ close to a diffeomorphism having positive central Lyapunov exponent Lebesgue almost everywhere and a unique physical measure with full basin, which is $C^r$ stably ergodic. Our method is perturbative and does not rely on preservation of a smooth measure., Comment: 27 pages, 5 figures; main theorem with strong hypothesis; proofs corrected

We prove that, under a mild summability condition on the growth of the derivative on critical orbits any piecewise monotone interval map possibly containing discontinuities and singularities with infinite derivative (cusp map) admits an ergodic invariant probability measures which is absolutely continuous with respect to Lebesgue measure., Comment: 15 pages, 1 Figure

We describe some recent results on the dynamics of singular-hyperbolic (or Lorenz-like) attractors: attractors in this class are expansive and so sensitive with respect to initial data; they admit a unique physical measure whose support is the whole attractor, which is hyperbolic and the equilibrium state with respect to the center-unstable Jacobian; the hitting time associated to a geometric Lorenz attractor satisfies a logarithm law; the rate of large deviations for the physical measure on the ergodic basin of a geometric Lorenz attractor is exponential., Comment: 19 pages, 6 figures, short survey paper for the Springer Proceedings associated to the DYNA2008 Conference: Dynamics & Applications in honour of Mauricio Peixoto and David Rand