Papers by Fernando José Sánchez-Salas
Monatshefte für Mathematik, Oct 11, 2016
Let f be a C 1+α diffeomorphism of a compact Riemannian manifold and µ an ergodic hyperbolic meas... more Let f be a C 1+α diffeomorphism of a compact Riemannian manifold and µ an ergodic hyperbolic measure with positive entropy. We prove that for every continuous potential φ there exists a sequence of basic sets Ωn such that the topological pressure P (f |Ωn, φ) converges to the free energy Pµ(φ) = h(µ) + φdµ. We also prove that for a suitable class of potentials φ there exists a sequence of basic sets Ωn such that P (f |Ωn, φ) → P (φ).
arXiv (Cornell University), Aug 22, 2011
We show that for any C 1+α diffeomorphism of a compact Riemannian manifold, every non-atomic, erg... more We show that for any C 1+α diffeomorphism of a compact Riemannian manifold, every non-atomic, ergodic, invariant probability measure with nonzero Lyapunov exponents is approximated by uniformly hyperbolic sets in the sense that there exists a sequence Ωn of compact, topologically transitive, locally maximal, uniformly hyperbolic sets, such that for any sequence {µn} of f-invariant ergodic probability measures with supp(µn) ⊆ Ωn we have µn → µ in the weak-* topology.
arXiv (Cornell University), Mar 20, 2013
This note is concerned with approximation of dynamical indicators as pressures, Lyapunov exponent... more This note is concerned with approximation of dynamical indicators as pressures, Lyapunov exponents and dimension-like quantities, in systems with nonuniformly hyperbolic behavior. For this we let P * (Φ) := sup µ {h(µ) + µ(Φ)} be a variational pressure defined over a suitable class of Borel measurable potentials and prove that, for regular nonuniformly hyperbolic systems, P * (Φ) = sup Ω P * (f |Ω, Φ), supremum taken over the family of f-invariant uniformly hyperbolic basic sets. We then apply this variational principle to the approximation of dynamical indicators by horseshoes, generalizing results of Katok and Mendoza in [16]. 1. Introduction This paper is concerned with approximation of dynamical indicators-entropies, pressures, Lyapunov exponents and dimension-like characteristics-by sequences of hyperbolic Cantor sets. This type of questions has been considered previously by several authors. In 1977 R. Bowen proposed to look for large invariant subsets in the non-wandering set Ω(f) of a diffeomorphism on which the dynamics of f are simple to describe and raised the following: Question: Let small ǫ > 0 be given. Does there exist a hyperbolic f-invariant subset Λ ǫ ⊆ Ω(f) such that f | Λ ǫ is conjugated to a subshift of finite type and h top (f | Λ ǫ) > h top (f) − ǫ? [6, Chapter 10]. L.-S. Young answered this question affirmatively in 1981, proving that Axiom A diffeomorphisms and flows, piecewise monotonic maps of the interval, the Poincaré map of the Lorenz attractor and certain Abraham-Smale examples are limits of subshifts of finite type in the above sense. See [33]. Later, in 1984, A. Katok laid the foundations to study these questions in his seminal paper [14] about relations between entropy, periodic orbits and Lyapunov exponents of systems with nonuniformly hyperbolic behavior. The following proposition, proved in [16], gives a taste of this type of results: let µ be an ergodic Borel probability with non zero Lyapunov exponents and h(µ) > 0, then there exists a sequence of horseshoes Ω n and ergodic measures µ n supported on Ω n such that µ n → µ weakly and h(µ n) → h(µ) (see). We refer the reader to [11], [12], [13], [18] [19], [30], [31] and [36] for some recent contributions to the subject.
In this work we give sufficient conditions for the existence of an ergodic Sinai-Ruelle-Bowen mea... more In this work we give sufficient conditions for the existence of an ergodic Sinai-Ruelle-Bowen measure preserved by transformations with infinitely many hyperbolic branches.
arXiv (Cornell University), May 10, 2015
Let f be a C 1+α diffeomorphism of a compact Riemannian manifold and µ an ergodic hyperbolic meas... more Let f be a C 1+α diffeomorphism of a compact Riemannian manifold and µ an ergodic hyperbolic measure with positive entropy. We prove that for every continuous potential φ there exists a sequence of basic sets Ωn such that the topological pressure P (f |Ωn, φ) converges to the free energy Pµ(φ) = h(µ) + φdµ. We also prove that for a suitable class of potentials φ there exists a sequence of basic sets Ωn such that P (f |Ωn, φ) → P (φ).
arXiv (Cornell University), Aug 6, 2015
We introduce a class of C 1+α isolated nonuniformly hyperbolic sets Λ for which sup µ∈M f {h(µ) −... more We introduce a class of C 1+α isolated nonuniformly hyperbolic sets Λ for which sup µ∈M f {h(µ) − χ + (µ)} equals the rate of escape from Λ, where χ + (µ) is the average of the sum of positive Lyapunov exponents counted with their multiplicity.
arXiv (Cornell University), Aug 6, 2015
Let f be a C 1+α nonuniformly hyperbolic diffeomorphism. We use a a nonadditive version of the to... more Let f be a C 1+α nonuniformly hyperbolic diffeomorphism. We use a a nonadditive version of the topological pressure of a class of admissible, possibly noncontinuous potentials P * (Φ) to prove the following variational equation: P * (Φ) = sup Ω∈H P * (f |Ω, Φ) supremum taken over the set H of basic subsets in M. As a consequence we find a lower bound for the Cantor dimension of the stable and unstable Cantor sets of a non trivial conformal nonuniformly hyperbolic isolated sets.
arXiv (Cornell University), Mar 30, 2021
In this paper we use the additive thermodynamic formalism to obtain bounds of the Hausdorff and b... more In this paper we use the additive thermodynamic formalism to obtain bounds of the Hausdorff and box-counting dimension of certain non conformal hyperbolic Cantor sets defined by piecewise smooth expanding maps on a d-dimensional smooth manifold M .
Divulgaciones Matemáticas, 2001
AIn this work we give sufficient conditions for the existence of an ergodic Sinai-Ruelle-Bowen me... more AIn this work we give sufficient conditions for the existence of an ergodic Sinai-Ruelle-Bowen measure preserved by transformations with infinitely many hyperbolic branches
arXiv: Dynamical Systems, May 11, 2015
Let f be a C 1+α diffeomorphism of a compact Riemannian manifold and µ an ergodic hyperbolic meas... more Let f be a C 1+α diffeomorphism of a compact Riemannian manifold and µ an ergodic hyperbolic measure with positive entropy. We prove that for every continuous potential φ there exists a sequence of basic sets Ωn such that the topological pressure P (f |Ωn, φ) converges to the free energy Pµ(φ) = h(µ) + φdµ. We also prove that for a suitable class of potentials φ there exists a sequence of basic sets Ωn such that P (f |Ωn, φ) → P (φ).
arXiv: Dynamical Systems, 2015
We introduce a class of $C^{1+\\alpha}$ isolated nonuniformly hyperbolic sets $\\Lambda$ for whic... more We introduce a class of $C^{1+\\alpha}$ isolated nonuniformly hyperbolic sets $\\Lambda$ for which $\\sup_{\\mu \\in {\\mathcal M}_f}\\{h(\\mu) - \\chi^+(\\mu)\\}$ equals the rate of escape from $\\Lambda$, where $\\chi^+(\\mu)$ is the average of the sum of positive Lyapunov exponents counted with their multiplicity.
In this paper we use the additive thermodynamic formalism to obtain new bounds of the Hausdorff a... more In this paper we use the additive thermodynamic formalism to obtain new bounds of the Hausdorff and box-counting dimension of certain non conformal hyperbolic repellers defined by piecewise smooth expanding maps on a d-dimensional smooth manifold M .
Journal of Mathematical Analysis and Applications, 2018
We introduce a new induction scheme for non-uniformly expanding maps f of compact Riemannian mani... more We introduce a new induction scheme for non-uniformly expanding maps f of compact Riemannian manifolds, relying upon ideas of [30] and [10]. We use this induction approach to prove that the existence of a Gibbs-Markov-Young structure is a necessary condition for f to preserve an absolutely continuous probability with all its Lyapunov exponents positive.
Proceedings of the American Mathematical Society, 2013
We show that for any C 1+α diffeomorphism of a compact Riemannian manifold, every non-atomic, erg... more We show that for any C 1+α diffeomorphism of a compact Riemannian manifold, every non-atomic, ergodic, invariant probability measure with non-zero Lyapunov exponents is approximated by uniformly hyperbolic sets in the sense that there exists a sequence Ω n of compact, topologically transitive, locally maximal, uniformly hyperbolic sets such that for any sequence {μ n } of f-invariant ergodic probability measures with supp(μ n) ⊆ Ω n we have μ n → μ in the weak-* topology.
Let $f$ be a $C^{1+\alpha}$ nonuniformly hyperbolic diffeomorphism. We use a a nonadditive versio... more Let $f$ be a $C^{1+\alpha}$ nonuniformly hyperbolic diffeomorphism. We use a a nonadditive version of the topological pressure of a class of admissible, possibly noncontinuous potentials $P^*(\Phi)$ to prove the following variational equation: $P^*(\Phi) = \sup_{\Omega \in {\mathcal H}}P^*(f|\Omega,\Phi)$ supremum taken over the set ${\mathcal H}$ of basic subsets in $M$. As a consequence we find a lower bound for the Cantor dimension of the stable and unstable Cantor sets of a non trivial conformal nonuniformly hyperbolic isolated sets.
This note is concerned with approximation of dynamical indicators as pressures, Lyapunov exponent... more This note is concerned with approximation of dynamical indicators as pressures, Lyapunov exponents and dimension-like quantities, in systems with nonuniformly hyperbolic behavior. For this we let P * (Φ) := sup µ {h(µ) + µ(Φ)} be a variational pressure defined over a suitable class of Borel measurable potentials and prove that, for regular nonuniformly hyperbolic systems, P * (Φ) = sup Ω P * (f |Ω, Φ), supremum taken over the family of f -invariant uniformly hyperbolic basic sets. We then apply this variational principle to the approximation of dynamical indicators by horseshoes, generalizing results of Katok and Mendoza in [16].
In this work we give sufficient conditions for the existence of an ergodic Sinai-Ruelle-Bowen mea... more In this work we give sufficient conditions for the existence of an ergodic Sinai-Ruelle-Bowen measure preserved by transformations with infinitely many hyperbolic branches.
Monatshefte für Mathematik, 2016
Let f be a C 1+α diffeomorphism of a compact Riemannian manifold and µ an ergodic hyperbolic meas... more Let f be a C 1+α diffeomorphism of a compact Riemannian manifold and µ an ergodic hyperbolic measure with positive entropy. We prove that for every continuous potential φ there exists a sequence of basic sets Ωn such that the topological pressure P (f |Ωn, φ) converges to the free energy Pµ(φ) = h(µ) + φdµ. We also prove that for a suitable class of potentials φ there exists a sequence of basic sets Ωn such that P (f |Ωn, φ) → P (φ).
We show that for any C 1+α diffeomorphism of a compact Rie-mannian manifold, every non-atomic, er... more We show that for any C 1+α diffeomorphism of a compact Rie-mannian manifold, every non-atomic, ergodic, invariant probability measure with non-zero Lyapunov exponents is approximated by uniformly hyperbolic sets in the sense that there exists a sequence Ω n of compact, topologically transitive, locally maximal, uniformly hyperbolic sets such that for any sequence {μ n } of f-invariant ergodic probability measures with supp(μ n) ⊆ Ω n we have μ n → μ in the weak-* topology.
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Papers by Fernando José Sánchez-Salas