Limit Theorems and Quantitative Statistical Stability for the Equilibrium States of Piecewise Partially Hyperbolic Maps
Universidad Pedagógica y Tecnológica de Colombia（哥伦比亚教学技术大学）；Universidade Federal de Alagoas（阿拉戈阿斯联邦大学）
Rafael A. Bilbao, Rafael Lucena
备注：
In this version, we have substantially refined Example 2.2. Additionally, we have improved the overall readability of the manuscript. arXiv admin note: text overlap with arXiv:2311.05577
This paper establishes limit theorems and quantitative statistical stability for a class of piecewise partially hyperbolic maps that are not necessarily continuous nor locally invertible. By employing a flexible functional-analytic framework that bypasses the classical requirement of compact embeddings between Banach spaces, we obtain explicit rates of convergence for the variation of equilibrium states under perturbations. Furthermore, we prove the exponential decay of correlations and the Central Limit Theorem for Hölder observables. A key feature of our approach is its applicability to systems where traditional spectral gap techniques fail due to the presence of singularities and the lack of invertibility. We provide several examples illustrating the scope of our results, including partially hyperbolic attractors over horseshoes, non-invertible dynamics semi-conjugated to Manneville--Pomeau maps, and fat solenoidal attractors.
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A class of piecewise $C^2$ Lozi-like maps in three-dimensional Euclidean spaces is introduced, and the existence of Sinai-Ruelle-Bowen measures is studied, where the dimension of the instability is equal to two. Further, an example with computer simulations is provided to illustrate the theoretical results.
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Katok's entropy formula is an important formula in entropy theory. This paper is devoted to establishing Katok's entropy formula of unstable metric entropy which is the entropy caused by the unstable part of partially hyperbolic systems.
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It was shown that in robustly transitive, partially hyperbolic diffeomorphisms on three dimensional closed manifolds, the strong stable or unstable foliation is minimal. In this article, we prove ``almost all'' leaves of both stable and unstable foliations are dense in the whole manifold.
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We show stable ergodicity of a class of conservative diffeomorphisms which do not have any hyperbolic invariant subbundle. Moreover the uniqueness of SRB measures for non-conservative $C^1$ perturbations of such diffeomorphisms. This class contains strictly non-partially hyperbolic robustly transitive diffeomorphisms by Bonatti-Viana and so we answer their question about the stable ergodicity of such systems.
