We show that the the shift on the reduced C*--algebras of RD--groups, including the free group on infinitely many generators, and the amalgamated free product C*--algebras, enjoys the very strong ergodic property of the convergence to the equilibrium. Namely, the free shift converges, pointwise in the weak topology, to the conditional expectation onto the fixed--point subalgebra. Provided the invariant state is unique, we also show that such an ergodic property cannot be fulfilled by any classical dynamical system, unless it is conjugate to the trivial one--point dynamical system.
We extend the theory of ergodic optimization and maximizing measures to the non-commutative field of C*-dynamical systems. We then employ this ergodic optimization machinery to provide an alternate characterization of unique erogdicity of C*-dynamical systems when the resident group action satisfies certain Choquet-theoretic assumptions.
Starting from a discrete $C^*$-dynamical system $(\mathfrak{A}, θ, ω_o)$, we define and study most of the main ergodic properties of the crossed product $C^*$-dynamical system $(\mathfrak{A}\rtimes_α\mathbb{Z}, Φ_{θ, u},\om_o\circ E)$, $E:\mathfrak{A}\rtimes_α\mathbb{Z}\rightarrow\ga$ being the canonical conditional expectation of $\mathfrak{A}\rtimes_α\mathbb{Z}$ onto $\mathfrak{A}$, provided $\a\in\aut(\ga)$ commute with the $*$-automorphism $þ$ up tu a unitary $u\in\ga$. Here, $Φ_{θ, u}\in\aut(\mathfrak{A}\rtimes_α\mathbb{Z})$ can be considered as the fully noncommutative generalisation of …
We show that for every topological dynamical system with the approximate product property, zero topological entropy is equivalent to unique ergodicity. Equivalence of minimality is also proved under a slightly stronger condition. Moreover, we show that unique ergodicity implies the approximate product property if the system has periodic points.
We extend the theory of ergodic optimization and maximizing measures to the non-commutative field of C*-dynamical systems. We then provide a result linking the ergodic optimizations of elements of a C*-dynamical system to the convergence of certain ergodic averages in a suitable seminorm. We also provide alternate proofs of several results in this article using the tools of nonstandard analysis.
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 intermediate pressures are generic in certain subspaces. This proves a conjecture of Katok for a broad class of systems and extends a sequence of known results.
Invariant ergodic measures for generalized Boole type transformations are studied using an invariant quasi-measure generating function approach based on special solutions to the Frobenius--Perron operator. New two-dimensional Boole type transformations are introduced, and their invariant measures and ergodicity properties are analyzed.
A dynamical theory for singular stochastic delay differential equations I: Linear equations and a Multiplicative Ergodic Theorem on fields of Banach spaces
For any 1-lipschitz ergodic map $F:\; \mathbb{Z}^{k}_{p} \mapsto \mathbb{Z}^{k}_{p},\;k>1\in\mathbb{N},$ there are 1-lipschitz ergodic map $G:\; \mathbb{Z}_{p} \mapsto \mathbb{Z}_{p}$ and two bijection $H_k$, $T_{k,\;P}$ that $$G = H_{k} \circ T_{k,\;P}\circ F\circ H^{-1}_{k} \text{ and } F = H^{-1}_{k} \circ T_{k,\;P^{-1}}\circ G\circ H_{k}.$$
Ergodic Properties of weak Asymptotic Pseudotrajectories for Set-valued Dynamical Systems
描述诸如渐近自治微分方程或随机逼近过程等各种确定性和随机过程的渐近行为的一个成功方法是将其与适当选择的极限半流相关联。Benaïm和Schreiber（2000）定义了一类这样的随机过程，他们称之为弱渐近伪轨线，并研究了它们的遍历行为。特别是，他们证明了弱 * 极限点的经验措施与这些过程几乎肯定是不变的相关的确定性半流。Benaïm，Hofbauer and Sorin（2005）将这种方法推广到集值动力系统。我们追求的类比延伸到这些设置弱渐近伪轨线的遍历性。
First and higher order uniform dual ergodic theorems for dynamical systems with infinite measure
本文将Karamata定理的多项式逼近证明推广到算子情形。作为结果，我们提供了一个简单的证明\emph{uniform dual ergodicity}的一类非常大的动力系统无限测度，我们得到的收敛速度的界限。在许多感兴趣的情况下，包括Pomeau-Manneville族的不确定性映射，通过实际Tauberian剩余理论获得的估计是非常弱的。基于复Tauberian剩余理论的技术，我们开发了一种方法，提供了\emph{second (and higher) order asymptotics}。在这个过程中，我们推导出一个\emph{higher order Tauberian theorem}的标量幂级数，据我们所知，以前没有被覆盖。
We consider $(1,2)$-rational functions given on the field of $p$-adic numbers $\mathbb Q_p$. In general, such a function has four parameters. We study the case when such a function has two fixed points and show that when there are two fixed points then $(1,2)$-rational function is conjugate to a two-parametric $(1,2)$-rational function. Depending on these two parameters we determine the type of the fixed points, find Siegel disks and the basin of attraction of the fixed points. Moreover, we classify invariant sets and study the ergodicity properties of the function on each invariant set. We de…
In this article, we pay attention to transitive dynamical systems having the shadowing property and the entropy functions are upper semicontinuous. As for these dynamical systems, when we consider ergodic optimization restricted on the subset of invariant measures whose metric entropy are equal or greater than a given constant, we prove that for generic real continuous functions the ergodic optimization measure is unique, ergodic, full support and have metric entropy equal to the given constant. Similar results also hold for suspension flows over transitive subshift of finite type, Cr (r\geq 2…
We show the existence, over an arbitrary infinite ergodic $\mathbb{Z}$-dynamical system, of a generic ergodic relatively distal extension of arbitrary countable rank and arbitrary infinite compact extending groups (or more generally, infinite quotients of compact groups) in its canonical distal tower.
The ergodic decomposition theorem is a cornerstone result of dynamical systems and ergodic theory. It states that every invariant measure on a dynamical system is a mixture of ergodic ones. Here we formulate and prove the theorem in terms of string diagrams, using the formalism of Markov categories. We recover the usual measure-theoretic statement by instantiating our result in the category of stochastic kernels. Along the way we give a conceptual treatment of several concepts in the theory of deterministic and stochastic dynamical systems. In particular, - ergodic measures appear very natural…
We establish connections between several properties of topological dynamical systems, such as: - every point is generic for an ergodic measure, - the map sending points to the measures they generate is continuous, - the system splits into uniquely (alternatively, strictly) ergodic subsystems, - the map sending ergodic measures to their topological supports is continuous, - the Cesaro means of every continuous function converge uniformly.
At the beggining of the 80's, H.Masur and W.Veech started the study of generic properties of interval exchange transformations proving that almost every such transformation is uniquely ergodic. About the same time, S.Novikov's school and French mathematicians independently discovered very intriguing phenomena for classes of measured foliations on surfaces and respective IETs. For instance, minimality is exceptional in these families. A precise version of this statement is a conjecture by Novikov. The French and Russian constructions are very different ones. Nevertheless, in the most simple sit…
