Search

In this article, we develop a novel functional analytic approach and demonstrate its effectiveness in establishing the existence, uniqueness, regularity of disintegration, and statistical properties of equilibrium states associated with a class of discontinuous dynamics. We apply this approach to a group of piecewise partially hyperbolic maps and a set of potentials. Furthermore, we prove several limit theorems, such as exponential decay of correlations, and propose certain conjectures. Additionally, we provide a set of examples where these results can be applied, including discontinuous dynamics semi-conjugated to the Manneville-Pomeau map., Comment: In this version, we have corrected the proof of Proposition 2.20, which stated that the operator \overline{\func F}_\Phi was a weak contraction in \mathcal{L}^{\infty}. Additionally, we have implemented other typographical improvements

We consider a class of endomorphisms that contains a set of piecewise partially hyperbolic dynamics semi-conjugated to non-uniformly expanding maps. Our goal is to study a class of endomorphisms that preserve a foliation that is almost everywhere uniformly contracted, with possible discontinuity sets parallel to the contracting direction. We apply the spectral gap property and the $\zeta$-H\"older regularity of the disintegration of its equilibrium states to prove a quantitative statistical stability statement. More precisely, under deterministic perturbations of the system of size $\delta$, we show that the $F$-invariant measure varies continuously with respect to a suitable anisotropic norm. Moreover, we prove that for certain interesting classes of perturbations, its modulus of continuity is $O(\delta^\zeta \log \delta)$. This article has been accepted for publication in the Discrete and Continuous Dynamical Systems journal., Comment: Many improvements in the text have been implemented. Lemmas and corollaries that help to understand the proofs have been added. The text has been restructured, and an important example has been included, where all the required hypotheses were explicitly verified. arXiv admin note: text overlap with arXiv:1911.06457

We consider a class of Random Dynamical Systems and prove that its invariant measure has a Lipschitz disintegration. As a consequence, we prove exponential decay of correlations over regular observables. Both results are obtained as an application of spectral gap for its transfers operator which is also proved in the paper., Comment: The overall presentation and readability was improved

We consider a class of endomorphisms which contains a set of piecewise partially hyperbolic skew-products with a non-uniformly expanding base map. The aimed transformation preserves a foliation which is almost everywhere uniformly contracted with possible discontinuity sets, which are parallel to the contracting direction. We prove that the associated transfer operator, acting on suitable anisotropic normed spaces, has a spectral gap (on which we have quantitative estimation) and the disintegration of the unique invariant physical measure, along the stable leaves, is $\zeta$-H\"older. We use this fact to obtain exponential decay of correlations on the set of $\zeta$-H\"older functions., Comment: Typos corrected, example added and the overall presentation and readability was improved. arXiv admin note: substantial text overlap with arXiv:1507.08191

We consider transformations preserving a contracting foliation, such that the associated quotient map satisfies a Lasota-Yorke inequality. We prove that the associated transfer operator, acting on suitable normed spaces, has a spectral gap (on which we have quantitative estimation). As an application we consider Lorenz-like two dimensional maps (piecewise hyperbolic with unbounded contraction and expansion rate): we prove that those systems have a spectral gap and we show a quantitative estimate for their statistical stability. Under deterministic perturbations of the system of size $\delta$, the physical measure varies continuously, with a modulus of continuity $O(\delta \log \delta )$, which is asymptotically optimal for this kind of piecewise smooth maps., Comment: 1) Some consequences of Spectral Gap and of the stability result were included in the text. 2) An appendix on disintegration of measures was included. 3) Typos were corrected. 4) Other comments and explanations were included