Young towers and uniform statistical properties

Alexey Korepanov (University of Warwick)

Frank Adams 1,

Let \(T_a\), \(a \in A\) be a family of nonuniformly hyperbolic transformations with invariant measures \(\mu_a\). We prove statistical limit laws for sequences of the type \(\sum_{j=0}^{n-1} v \circ T_{a_n}^j\).

A key ingredient is a new martingale-coboundary decomposition, which is useful already when the family \(T_a\) is replaced by a fixed transformation \(T\), and is particularly effective when \(T_a\) varies with \(a\).

We also estimate correlations \( \int v \, w \circ T_a^n \, d\mu_a \) uniformly in \(a\).

Our results include cases where the family \(T_a\) consists of intermittent maps, unimodal maps (along the Collet-Eckmann parameters), Viana maps, and externally forced dispersing billiards.

This is a joint work with Zemer Kosloff and Ian Melbourne.

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