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Armando Castro

Publications and source records attributed to Armando Castro.

10 recordsLinked to original sources
arXiv2024

Regularity, linear response formula and differentiability of the free energy for non-uniformly expanding local homeomorphisms

We study equilibrium states for an open class of non-uniformly expanding local homeomorphisms defined by a mild condition such that for some iterate each point admits at least one contracting inverse branch. We prove the existence and uniqueness of equilibrium states and the differentiability of statistical quantities (such as the equilibrium states and the free energy function) with respect to the dynamical system.

math.DS↗
arXiv2017

Linear response, and consequences for differentiability of statistical quantities and Multifractal Analysis

In this article we initially fix ourselves to smooth (C^r) expanding dynamical systems. We prove the C^{r-1} differentiability of the topological pressure, equilibrium states and their densities with respect to smooth expanding dynamical systems and any smooth potential (C^{r-1}- linear response formula wiyh respect to the dynamics, and analytical response formula with respect to the potential). This is done by proving the regularity of the dominant eigenvalue of the transfer operator with respect to dynamics and potential. From that, we obtain strong consequences on the regularity of the dynamical system statistical properties, that apply in more general contexts. Indeed, we prove that the average and variance obtained from the central limit theorem vary $C^{r-1}$ with respect to the $C^{r}-$expanding dynamics and $C^{r}-$potential, and also, there is a large deviations principle with its rate $C^{r-1}$ with respect to the dynamics and potential. An application for multifractal analysis is given.

math.DS↗
arXiv2016

Statistical properties of the maximal entropy measure for partially hyperbolic attractors

We show the existence and uniqueness of the maximal entropy probability measure for partially hyperbolic diffeomorphisms which are semi-conjugate to nonuniformly expanding maps. Using the theory of projective metric on cones we then prove exponential decay of correlations for H\"older continuous observables and the central limit theorem for the maximal entropy probability measure. Moreover, for systems derived from solenoid we also prove the statistical stability for the maximal entropy probability measure. Finally, we use such techniques to obtain similar results in a context containing partially hyperbolic systems derived from Anosov. Paper published in ETDS in Jan. 2016.

math.DS↗
arXiv2012

New Criteria of Generic Hyperbolicity based on Periodic Points

We prove a criteria for uniform hyperbolicity based on the periodic points of the transformation. More precisely, if a mild (non uniform) hyperbolicity condition holds for the periodic points of any diffeomorphism in a residual subset of a $C^1$-open set $\SU$ then there exists an open and dense subset $\SA\subset \SU$ of Axiom A diffeomorphisms. Moreover, we also prove a noninvertible version of Ergodic Closing Lemma which we use to prove a counterpart of this result for local diffeomorphisms. As a simple corollary of our techniques, we have that an arbitrary $\mathrm{C}^1$-class local diffeomorphism $f$ of a closed manifold $M^n$ is uniformly expanding on the closure $\mathrm{Cl}_{M^n}(\mathrm{Per}(f))$ of its periodic point set $\mathrm{Per}(f)$, if it is nonuniformly expanding on $\mathrm{Per}(f)$.

math.DS↗
arXiv2012

Differentiability of thermodynamical quantities in non-uniformly expanding dynamics

In this paper we study the ergodic theory of a robust non-uniformly expanding maps where no Markov assumption is required. We prove that the topological pressure is differentiable as a function of the dynamics and analytic with respect to the potential. Moreover we not only prove the continuity of the equilibrium states and their metric entropy as well as the differentiability of the maximal entropy measure and extremal Lyapunov exponents with respect to the dynamics. We also prove a local large deviations principle and central limit theorem and show that the rate function, mean and variance vary continuously with respect to observables, potentials and dynamics. Finally, we show that the correlation function associated to the maximal entropy measure is differentiable with respect to the dynamics and it is $C^1$-convergent to zero. In addition, precise formulas for the derivatives of thermodynamical quantities are given.

math.DS↗
arXiv2012

Equilibrium states for non-uniformly expanding maps: decay of correlations and strong stability

We study the rate of decay of correlations for equilibrium states associated to a robust class of non-uniformly expanding maps where no Markov assumption is required. We show that the Ruelle-Perron-Frobenius operator acting on the space of Holder continuous observables has a spectral gap and deduce the exponential decay of correlations and the central limit theorem. In particular, we obtain an alternative proof for the existence and uniqueness of the equilibrium states and we prove that the topological pressure varies continuously. Finally, we use the spectral properties of the transfer operators in space of differentiable observables to obtain strong stability results under deterministic and random perturbations.

math.DS↗
arXiv2006

Shadowing by non uniformly hyperbolic periodic points and uniform hyperbolicity

We prove that, under a mild condition on the hyperbolicity of its periodic points, a map $g$ which is topologically conjugated to a hyperbolic map (respectively, an expanding map) is also a hyperbolic map (respectively, an expanding map). In particular, this result gives a partial positive answer for a question done by A. Katok, in a related context.

math.DS↗