arXiv subjects
Publications and source records attributed to Fernando José Sánchez-Salas.
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 $C^r$, $r > 1$ piecewise expanding maps on a $d$-dimensional smooth manifold $M$.
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 $Ω_n$ such that the topological pressure $P(f|Ω_n,ϕ)$ converges to the free energy $P_μ(ϕ) = h(μ) + \intϕ{dμ}$. Then we introduce a class of potentials $ϕ$ for which there exists sequence of basic sets $Ω_n$ such that $P(f|Ω_n,ϕ) \to P(ϕ)$.
We introduce a class of $C^{1+α}$ isolated nonuniformly hyperbolic sets $Λ$ for which $\sup_{μ\in {\mathcal M}_f}\{h(μ) - χ^+(μ)\}$ equals the rate of escape from $Λ$, where $χ^+(μ)$ is the average of the sum of positive Lyapunov exponents counted with their multiplicity.
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: $P^*(Φ) = \sup_{Ω\in {\mathcal H}}P^*(f|Ω,Φ)$ 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 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, $P^*(Φ) = \sup_ΩP^*(f|Ω,Φ)$, 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.