Related papers: Ergodicity of iterated function systems via minima…

In this paper, we give a precise meaning to the following fact, and we prove it: $C^1$-open and densely, all the non-hyperbolic ergodic measures generated by a robust cycle are approximated by periodic measures. We apply our technique to…

For a dynamical system satisfying the approximate product property and asymptotically entropy expansiveness, we characterize a delicate structrue of the space of invariant measures: The ergodic measures of intermediate entropies and…

We study random dynamical systems of certain continuous functions on the unit interval. We use bounded variation to provide sufficient conditions for unique ergodicity of these systems. Several classes of examples are provided.

Let T be an ergodic automorphism of the d-dimensional torus T^d, and f be a continuous function from T^d to R^l. On the probability space T^d equipped with the Lebesgue-Haar measure, we prove the weak convergence of the sequential empirical…

We continue the investigation of which non-dierentiable maps can occur in the framework of ergodic theory started in [2]. We construct a Besicovitch-Morse function map which preserves the Lebesgue measure. We also show that the set of…

We consider finitely generated groups of real-analytic circle diffeomorphisms. We show that if such a group admits an exceptional minimal set (i.e., a minimal invariant Cantor set), then its Lebesgue measure is zero; moreover, there are…

We show that $C^\infty$ surface diffeomorphisms with positive topological entropy have at most finitely many ergodic measures of maximal entropy in general, and at most one in the topologically transitive case. This answers a question of…

Let $f$ be a $C^2$ diffeomorphism on compact Riemannian manifold $M$ with partially hyperbolic splitting $$ TM=E^u\oplus E_1^c\oplus\cdots\oplus E_k^c \oplus E^s, $$ where $E^u$ is uniformly expanding, $E^s$ is uniformly contracting, and…

This paper introduces a theory of Thermodynamic Formalism for Iterated Function Systems with Measures (IFSm). We study the spectral properties of the Transfer and Markov operators associated to a IFSm. We introduce variational formulations…

In 1989 Kusuoka started the study of probability measures on the shift space that are defined with the help of products of matrices. In particular, he derived a sufficient condition for the ergodicity of such measures, which have since been…

For dynamical systems satisfying the approximate $\mathbb{Z}^{d}$ or $\mathbb{Z}_+^{d}$-product property and asymptotically entropy expansiveness, we establish a precise description of the structure of their space of invariant measures. In…

Given sequence of measure preserving transformations $\{U_k:\,k=1,2,\ldots, n\}$ on a measurable space $(X,\mu)$. We prove a.e. convergence of the ergodic means \begin{equation} \frac{1}{s_1\cdots…

In this paper we develop a new approach for studying overlapping iterated function systems. This approach is inspired by a famous result due to Khintchine from Diophantine approximation. This result shows that for a family of limsup sets,…

In this paper, we study physical measures for partially hyperbolic diffeomorphisms with multi one-dimensional centers under the condition that all Gibbs $u$-states are hyperbolic. We prove the finiteness of ergodic physical measures. Then…

In this paper, we show that for several interesting systems beyond uniform hyperbolicity, any generic continuous function has a unique maximizing measure with zero entropy. In some cases, we also know that the maximizing measure has full…

We construct examples of minimal and uniquely ergodic systems realizing all possible behaviors in the interplay of measurable and topological nilfactors. To build such examples, we adapt an idea that stems from Furstenberg's construction of…

We show that for a minimal system $(X,T)$, the set of saturated points along cubes with respect to its maximal $\infty$-step pro-nilfactor $X_\infty$ has a full measure. As an application, it is shown that if a minimal system $(X,T)$ has no…

We define a class of dynamical maps on the quasi-local algebra of a quantum spin system, which are quantum analogues of probabilistic cellular automata. We develop criteria for such a system to be ergodic, i.e., to possess a unique…

In this paper, we prove that ergodic measures with large entropy give uniformly large measure to the set of points with simultaneously long unstable and long stable manifolds. As a consequence, for $C^{\infty}$ surface diffeomorphisms, we…

In this paper, for iterated function systems, we define the classic concept of the dynamical systems: topological conjugacy of diffeomorphisms. We generalize the Hartman-Grobman theorem for one dimensional iterated function systems on R.…