Related papers: Exponential multiple mixing for commuting automorp…

We study dynamical properties of automorphisms of compact nilmanifolds and prove that every ergodic automorphism is exponentially mixing and exponentially mixing of higher orders. This allows to establish probabilistic limit theorems and…

We study mixing properties of commutative groups of automorphisms acting on compact nilmanifolds. Assuming that every nontrivial element acts ergodically, we prove that such actions are mixing of all orders. We further show exponential…

Consider actions of $\Z ^r$ by ergodic automorphisms on a compact nilmanifolds for $r \geq 2$. We show that small $C^k$ perturbations of such higher rank partially hyperbolic actions are smoothly conjugate to the original action, using a…

Let f be a holomorphic automorphism of positive entropy on a compact Kaehler surface. We show that the equilibrium measure of f is exponentially mixing. The proof uses some recent development on the pluripotential theory. The result also…

Given a $\mathbb Z^r$-action $\alpha$ on a nilmanifold $X$ by automorphisms and an ergodic $\alpha$-invariant probability measure $\mu$, we show that $\mu$ is the uniform measure on $X$, unless modulo finite index modification, one of the…

Let $f$ be a holomorphic automorphism of a compact K\"ahler manifold. Assume moreover that $f$ admits a unique maximal dynamic degree $d_p$ with only one eigenvalue of maximal modulus. Let $\mu$ be its equilibrium measure. In this paper, we…

An exponential automorphism of $\mathbf{C}$ is a function $\alpha: \mathbf{C} \rightarrow \mathbf{C}$ such that $\alpha(z_1 + z_2) = \alpha(z_1) + \alpha(z_2)$ and $\alpha\left( e^z \right) = e^{\alpha(z)}$ for all $z, z_1, z_2 \in…

Let $f$ be a $C^{1+\alpha}$ diffeomorphism of a compact manifold $M$ preserving a smooth measure $\mu$. We show that if $f:(M,\mu)\to (M,\mu)$ is exponentially mixing then it is Bernoulli.

We consider the unique measure of maximal entropy of an automorphism of a compact K{\"a}hler manifold with simple action on cohomology. We show that it is exponentially mixing of all orders with respect to H{\"o}lder observables. It follows…

We present: 1) a mixing $Z ^ 2$-action with the following asymmetry of multiple mixing property: for some commuting measure-preserving transformations $S$, $T$ and a sequence $n_j$ $$ \lim_{j\to \infty}\mu(A\bigcap S^{-n_j}A\bigcap…

An algebraic $\Gamma$-action is an action of a countable group $\Gamma$ on a compact abelian group $X$ by continuous automorphisms of $X$. We prove that any expansive algebraic action of a finitely generated nilpotent group $\Gamma$ on a…

We show that genuinely higher rank expanding actions of abelian semi-groups on compact manifolds are $C^{\infty}$-conjugate to affine actions on infra-nilmanifolds. This is based on the classification of expanding diffeomorphisms up to…

Let S be a non-exceptional oriented surface of finite type. We give a new proof based on symbolic coding of the following result of Avila and Gouezel. The Teichmueller flow is exponentially mixing with respect to any ergodic…

We study mixing properties of epimorphisms of a compact connected finite-dimensional abelian group $X$. In particular, we show that a set $F$, $|F|>\dim X$, of epimorphisms of $X$ is mixing iff every subset of $F$ of cardinality $(\dim…

Let $(X,\mathcal{B},\mu)$ be a probability space and let $T_1,..., T_l$ be $l$ commuting invertible measure preserving transformations \linebreak of $X$. We show that if $T_1^{c_1} ... T_l^{c_l}$ is ergodic for each $(c_1,...,c_l)\neq…

Let $\{T^z\}$ be an ergodic action of the group $Z^n$ by automorphisms of the probability space $(X,m)$, $\sum_{i}^\infty a_i<\infty$, $a_i>0$. For any sequence $M_k\to +\infty$ there exist $N_k>M_k$ and a function $ f\in L_1(X,m)$ such…

We develop a new analytic method for quantitative mixing of automorphisms on nilmanifolds. The method is based on the introduction and solvability of \emph{multiple fractional cohomological equations of Type~$I$} (sum type). We prove that…

We prove the $L^{2}$ convergence for the linear multiple ergodic averages of commuting transformations $T_{1}, ..., T_{l}$, assuming that each map $T_i$ and each pair $T_iT_j^{-1}$ is ergodic for $i\neq j$. The limiting behavior of such…

Let $f$ be a holomorphic automorphism of a compact K\"ahler manifold with simple action on cohomology and $\mu$ its unique measure of maximal entropy. We prove that $\mu$ is exponentially mixing of all orders for all d.s.h.\ observables,…

We show that a residual set of non-degenerate IETs on more than 3 letters is topologically mixing. This shows that there exists a uniquely ergodic topologically mixing IET. This is then applied to show that some billiard flows in a fixed…