Equidistribution for nonuniformly expanding dynamical systems, and application to the almost sure invariance principle

Let $T \colon M \to M$ be a nonuniformly expanding dynamical system, such as logistic or intermittent map. Let $v \colon M \to \mathbb{R}^d$ be an observable and $v_n = \sum_{k=0}^{n-1} v \circ T^k$ denote the Birkhoff sums. Given a probability measure $\mu$ on $M$, we consider $v_n$ as a discrete time random process on the probability space $(M, \mu)$. In smooth ergodic theory there are various natural choices of $\mu$, such as the Lebesgue measure, or the absolutely continuous $T$-invariant measure. They give rise to different random processes. We investigate relation between such processes. We show that in a large class of measures, it is possible to couple (redefine on a new probability space) every two processes so that they are almost surely close to each other, with explicit estimates of "closeness". The purpose of this work is to close a gap in the proof of the almost sure invariance principle for nonuniformly hyperbolic transformations by Melbourne and Nicol

Linear response for intermittent maps with summable and nonsummable decay of correlations

We consider a family of Pomeau-Manneville type interval maps $T_\alpha$, parametrized by $\alpha \in (0,1)$, with the unique absolutely continuous invariant probability measures $\nu_\alpha$, and rate of correlations decay $n^{1-1/\alpha}$. We show that despite the absence of a spectral gap for all $\alpha \in (0,1)$ and despite nonsummable correlations for $\alpha \geq 1/2$, the map $\alpha \mapsto \int \varphi \, d\nu_\alpha$ is continuously differentiable for $\varphi \in L^{q}[0,1]$ for $q$ sufficiently large.Comment: Corrected and improved technical estimates; minor general correction

Linear response for intermittent maps with summable and nonsummable decay of correlations

We consider a family of Pomeau–Manneville type interval maps ${{T}_{\alpha}}$ , parametrized by $\alpha \in (0,1)$ , with the unique absolutely continuous invariant probability measures ${{\nu}_{\alpha}}$ , and rate of correlations decay ${{n}^{1-1/\alpha}}$ . We show that despite the absence of a spectral gap for all $\alpha \in (0,1)$ and despite nonsummable correlations for $\alpha \geqslant 1/2$ , the map $\alpha \mapsto {\int}^{}\varphi \,\text{d}{{\nu}_{\alpha}}$ is continuously differentiable for $\varphi \in {{L}^{q}}\left[0,1\right]$ for q sufficiently large

Rates of mixing for the measure of maximal entropy of dispersing billiard maps

In a recent work, Baladi and Demers constructed a measure of maximal entropy for finite horizon dispersing billiard maps and proved that it is unique, mixing and moreover Bernoulli. We show that this measure enjoys natural probabilistic properties for H\"older continuous observables, such as at least polynomial decay of correlations and the Central Limit Theorem. The results of Baladi and Demers are subject to a condition of sparse recurrence to singularities. We use a similar and slightly stronger condition, and it has a direct effect on our rate of decay of correlations. For billiard tables with bounded complexity (a property conjectured to be generic), we show that the sparse recurrence condition is always satisfied and the correlations decay at a super-polynomial rate

Deterministic homogenization under optimal moment assumptions for fast-slow systems. Part 1

We consider deterministic homogenization (convergence to a stochastic differential equation) for multiscale systems of the form $$x_{k+1} = x_k + n^{-1} a_n(x_k,y_k) + n^{-1/2} b_n(x_k,y_k), \quad y_{k+1} = T_n y_k,$$ where the fast dynamics is given by a family $T_n$ of nonuniformly expanding maps. Part 1 builds on our recent work on martingale approximations for families of nonuniformly expanding maps. We prove an iterated weak invariance principle and establish optimal iterated moment bounds for such maps. (The iterated moment bounds are new even for a fixed nonuniformly expanding map T.) The homogenization results are a consequence of this together with parallel developments on rough path theory in Part 2 by Chevyrev, Friz, Korepanov, Melbourne & Zhang