Hölder regularity and exponential decay of correlations for a class of piecewise partially hyperbolic maps

TL;DR

The paper studies a class of piecewise partially hyperbolic endomorphisms $F$ on $\Sigma = M \times K$ with a non-uniformly expanding base and uniformly contracting fibers. It constructs anisotropic Banach spaces built from leafwise disintegration and proves a spectral gap for the transfer operator $\mathop{\mathrm{F}}_{*}$ on those spaces, yielding exponential convergence to equilibrium. It further shows that the unique $F$-invariant measure has a $\zeta$-Hölder disintegration along the stable leaves and derives exponential decay of correlations for $\zeta$-Hölder observables. The framework accommodates potential discontinuities parallel to the contracting direction and provides explicit rates linked to contraction constants and Lasota–Yorke bounds, advancing the understanding of statistical properties for a broad class of skew-product dynamics.

Abstract

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 $ζ$-Hölder. We use this fact to obtain exponential decay of correlations on the set of $ζ$-Hölder functions.

Theorems & Definitions (63)

• Theorem A
• Theorem B: Spectral gap on $S^{\infty}$
• Theorem C
• Theorem D
• Theorem E
• Theorem 1.1
• Remark 1
• Lemma 1.2
• proof
• Example 1.1