## Young towers and uniform statistical properties

#### Alexey Korepanov (University of Warwick)

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.