相关论文: Ergodic properties of invariant measures for syste…

Let $(X,\mu)$ be a standard probability space. An automorphism $T$ of $(X,\mu)$ has the weak Pinsker property if for every $\varepsilon > 0$ it has a splitting into a direct product of a Bernoulli shift and an automorphism of entropy less…

We show the existence of a bounded Borel measurable saturated compensation function for a factor map between subshifts. As an application, we find the Hausdorff dimension and measures of full Hausdorff dimension for a compact invariant set…

We show that the set of Bernoulli measures of an isolated topologically mixing homoclinic class of a generic diffeomorphism is a dense subset of the set of invariant measures supported on the class. For this, we introduce the large periods…

It is shown that there exist a subsequence for which the multiple ergodic averages of commuting invertible measure preserving transformations of a Lebesgue probability space converge almost everywhere provided that the maps are weakly…

Let $(X,G)$, $(Y,G)$ be two $G$-systems, where $G$ is an infinite countable discrete amenable group and $X$, $Y$ are compact metric spaces. Suppose that $\mathcal{U}$ is a cover of $X$. We first introduce the conditional local topological…

We prove that a homeomorphism of a compact metric space has an expansive measure \cite{ms} if and only if it has many ones with invariant support. We also study homeomorphisms for which the expansive measures are dense in the space of Borel…

It is known that Iterated Function Systems generated by orientation preserving homeomorphisms of the unit interval admit a unique invariant measure on $(0,1)$. The setup for this result is the positivity of Lyapunov exponents at both fixed…

We obtain a spectral gap characterization of strongly ergodic equivalence relations on standard measure spaces. We use our spectral gap criterion to prove that a large class of skew-product equivalence relations arising from measurable…

For any transitive piecewise monotonic map for which the set of periodic measures is dense in the set of ergodic invariant measures (such as monotonic mod one transformations and piecewise monotonic maps with two monotonic pieces), we show…

In the first part of this paper, we formulate a general setting in which to study the ergodic theory of differentiable $\mathbb{Z}^d$-actions preserving a Borel probability measure. This framework includes actions by $C^{1+\text{H\"older}}$…

In this paper, we establish a new quasi-shadowing property for any nonuiformly partially hyperbolic set of a $C^{1+\alpha}$ diffeomorphism, which is adaptive to the movement of the pseudo-orbit. Moreover, the quasi-specification property…

Given a cut and project scheme and a pre-compact Borel window we show that almost surely all positions of the window give rise to point sets with Besicovitch almost periodic Dirac combs. In particular, all those positions lead to pure point…

We prove the finiteness of ergodic measures of maximal entropy for partially hyperbolic diffeomorphisms where the center direction has a dominated decomposition into one dimensional bundle and there is a uniform lower bound for the absolute…

Ergodic optimization aims to describe dynamically invariant probability measures that maximize the integral of a given function. The Dyck and Motzkin shifts are well-known examples of transitive subshifts over a finite alphabet that are not…

We prove that for certain partially hyperbolic skew-products, non-uniform hyperbolicity along the leaves implies existence of a finite number of ergodic absolutely continuous invariant probability measures which describe the asymptotics of…

Letting $T$ denote an ergodic transformation of the unit interval and letting $f \colon [0,1)\to \mathbb{R}$ denote an observable, we construct the $f$-weighted return time measure $\mu_y$ for a reference point $y\in[0,1)$ as the weighted…

We introduce a class of continuous maps f of a compact metric space I admitting inducing schemes and describe the tower constructions associated with them. We then establish a thermodynamical formalism, i.e., describe a class of real-valued…

This paper deals with collisionless transport equations in bounded open domains $\Omega \subset \R^{d}$ $(d\geq 2)$ with $\mathcal{C}^{1}$ boundary $\partial \Omega $, orthogonally invariant velocity measure $\bm{m}(\d v)$ with support…

We introduce a geometric invariant, called finite decomposition complexity (FDC), to study topological rigidity of manifolds. We prove for instance that if the fundamental group of a compact aspherical manifold M has FDC, and if N is…

Let $M$ be a manifold with a volume form $\omega$ and $f : M \to M$ be a diffeomorphism of class $\mathcal{C}^1$ that preserves $\omega$. In this paper, we do \textit{not} assume $f$ is $\mathcal{C}^1$-generic. We have two main themes in…