Quantitative statistical stability for equilibrium states of piecewise partially hyperbolic maps

Rafael Bilbao; Ricardo Bioni; Rafael Lucena

doi:10.3934/dcds.2023129

Article Contents

2024, Volume 44, Issue 3: 855-881. Doi: 10.3934/dcds.2023129

This issue Previous Article Decay of correlations for some non-uniformly hyperbolic attractors Next Article

Quantitative statistical stability for equilibrium states of piecewise partially hyperbolic maps

1.
Universidad Pedagógica y Tecnológica de Colombia, Colombia
2.
Federal University of Alagoas, Brazil

^*Corresponding author: Rafael Lucena
^*Corresponding author: Rafael Lucena

Received: August 2021

Revised: July 2023

Early access: October 2023

Published: March 2024

The 3rd author is supported by [Alagoas Research Foundation-FAPEAL (Brazil) Grants 60030.0000000161/2022 and CNPq (Brazil) Grants 409198/2021-8].

Abstract / Introduction Full Text(HTML) Figure(1) Related Papers Cited by

Abstract

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 $-Hölder 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. Furthermore, we establish that certain interesting classes of perturbations exhibit a modulus of continuity estimated by $ D_2\delta^\zeta \log \delta $, where $ D_2 $ is a constant.

Keywords:

Mathematics Subject Classification: Primary: 37A25, 37A10; Secondary: 37C30, 37D50.

Citation:

Full Text(HTML)

Figure 1. The graph of the perturbed map g

Download: Full-size image PowerPoint slide

Related Papers

Cited by

References

[1]	V. Araujo and M. J. Pacifico, Three-Dimensional Flows, Springer-Verlag, New York, 2010. doi: 10.1007/978-3-642-11414-4.
[2]	R. A. Bilbao, R. Bioni and R. Lucena, Hölder regularity and exponential decay of correlations for a class of piecewise partially hyperbolic maps, Nonlinearity, 33 (2020), 6790-6818. doi: 10.1088/1361-6544/aba888.
[3]	A. Castro and T. Nascimento, Statistical Properties of the maximal entropy measure for partially hyperbolic attractors, Ergodic Theory and Dynamical Systems, 37 (2017), 1060-1101. doi: 10.1017/etds.2015.86.
[4]	A. Castro and P. Varandas, Equilibrium states for non-uniformly expanding maps: Decay of correlations and strong stability, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 30 (2013), 225-249. doi: 10.1016/j.anihpc.2012.07.004.
[5]	S. Galatolo, Quantitative statistical stability, speed of convergence to equilibrium for partially hyperbolic skew products, J. Éc. polytech. Math., 5 (2018), 377-405. doi: 10.5802/jep.73.
[6]	S. Galatolo and R. Lucena, Spectral Gap and quantitative statistical stability for systems with contracting fibers and Lorenz like maps, Discrete & Continuous Dynamical Systems - A, 40 (2020), 1309-1360. doi: 10.3934/dcds.2020079.
[7]	S. Galatolo and M. J. Pacifico, Lorenz-like flows: Exponential decay of correlations for the Poincaré map, logarithm law, quantitative recurrence, Ergodic Theory and Dynamical Systems, 30 (2010), 1703-1737. doi: 10.1017/S0143385709000856.
[8]	F. Ledrappier and P. Walters, A relativised variational principle for continuous transformations, J. London Math. Soc., 16 (1977), 568-576. doi: 10.1112/jlms/s2-16.3.568.
[9]	D. Lima and R. Lucena, Lipschitz regularity of the invariant measure and statistical properties for a class of random dynamical systems, preprint, 2020, arXiv: 2001.08265.
[10]	R. Lucena, Spectral Gap for Contracting Fiber Systems and Applications, Ph.D thesis, Federal University of Rio de Janeiro in Brazil, 2015.
[11]	K. Oliveira and M. Viana, Fundamentos da Teoria Ergódica, Colecão Fronteiras da Matematica - SBM, Rio de Janeiro, 2014.
[12]	P. Varandas and M. Viana, Existence, uniqueness and stability of equilibrium states for non-uniformly expanding maps, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 27 (2010), 555-593. doi: 10.1016/j.anihpc.2009.10.002.