Gibbs-Markov structures and limit laws for partially hyperbolic attractors with mostly expanding central direction

TL;DR

This paper establishes Gibbs-Markov structures for certain partially hyperbolic attractors with mostly expanding central directions and derives statistical properties like decay of correlations and the Central Limit Theorem.

Contribution

It introduces conditions under which partially hyperbolic sets admit Gibbs-Markov structures and links decay rates to the expansion behavior in the central-unstable direction.

Findings

Gibbs-Markov structures are induced under specified hyperbolicity conditions.

Decay of correlations is established with rates depending on expansion times.

Large deviations and CLT are proven for the dynamical system.

Abstract

We consider a partially hyperbolic set $K$ on a Riemannian manifold $M$ whose tangent space splits as $T_K M=E^{cu}\oplus E^{s}$, for which the centre-unstable direction $E^{cu}$ expands non-uniformly on some local unstable disk. We show that under these assumptions $f$ induces a Gibbs-Markov structure. Moreover, the decay of the return time function can be controlled in terms of the time typical points need to achieve some uniform expanding behavior in the centre-unstable direction. As an application of the main result we obtain certain rates for decay of correlations, large deviations, an almost sure invariance principle and the validity of the Central Limit Theorem.