蒋云平 (纽约城市大学女王学院)

Ergodic Theory Motivated by Conjectures in Number

Abstract: The Birkhoff ergodic theorem, a fundamental result in ergodic theory, asserts that, in an ergodic measure-preserving dynamical system, the time average equates to the space average for almost every point. In the realm of uniquely ergodic systems on compact metric spaces, this equivalence extends to every point, extending the theorem’s applicability. Our current research, inspired by Sarnak’s conjecture and Chowla’s conjecture in number theory, delves into the study of weighted time averages and time averages along a sequence of natural numbers for continuous functions within certain dynamical systems on compact metrics. We aim to exploit the oscillatory nature of weights and the uniform behavior of sequences of natural numbers as tools for categorizing zero entropy dynamical systems. Two arithmetic functions, the Möbius function as a weight and the big prime omega function as a sequence of natural numbers, exhibit these properties, respectively. We will introduce additional weights and sequences of natural numbers with similar properties. This presentation will offer an overview of recent developments in this field. Sarnak’s conjecture, intimately linked with Chowla’s conjecture in number theory, provides a crucial motivation for our research. Further investigation into this connection reveals intriguing relationships between invariant measures, particularly in Möbius and square-free flows. I will also discuss recent advancements in this area.

牛壮 (怀俄明大学)

Mean dimension, comparison, and the classification of C*-algebras

Abstract: The recent classification theorem asserts that the well-behaved C*-algebras are classified by the ordered K-groups together with the pairing of the trace simplex. Consider the crossed product C*-algebra of a free and minimal dynamical system. It may or may not be well behaved, but having zero mean dimension (or small boundary property) will guarantee the C*-algebra to be well behaved (at least in the case of Zd-actions). In the talk, I will discuss the mean dimension and the (Cuntz) comparison of positive elements, for crossed product C*-algebras and also for AH algebras with diagonal maps if time permits.

陈二才 (南京师范大学)

Group extensions for random shifts of finite type

Abstract: In this talk, I will present a study on the group extensions for topologically mixing random shifts of finite type and consider the natural comparison of relative Gurevič entropies between random group G and Gab = G/[G, G] extensions. We show that the relative Gurevič entropy of random group G extensions is equal to the relative Gurevič entropy of random group Gab extensions if and only if G is amenable. This is a joint work with K. Yang, Z. Lin and X. Zhou.

张国华 (复旦大学)

Further Characterizations for Weak Expansiveness of Amenable

Group Actions

Abstract: In this talk I discuss further characterizations for weak expan- siveness of amenable group actions, by interpret equivalently topological conditional entropy of amenable group actions via topological entropy of stable subsets and dimensional entropy of subsets, respectively. This is a joint work with Dou Dou and Ying Wang.

David Burguet(法国索邦大学)

Topological mean dimension of induced systems

Abstract: For a topological system (X,T), we consider the induced map T∗ on the set M(X) of Borel probability measures. It is well known that T∗ has infinite topological entropy, if T has positive topological entropy. We show that the topological mean dimension of T∗ is also infinite. Moreover we give precise rates of divergence of hW (T∗, ϵ) when ϵ goes to zero, where hW (T∗, ϵ) denotes the Bowen metric entropy with respect to the Wasserstein distance W . The proof uses the independence theory developed by Glasner-Weiss, Kerr-Li and others. Joint work with Ruxi Shi.

Yonatan Gutman (波兰科学院数学研究所)

The delay dimension threshold for prediction of dynamical systems

Abstract: We study the problem of reconstructing and predicting the future of a dynamical system by the use of time-delay measurements of typical observables. Considering the case of too few measurements, we prove that for Lipschitz systems on compact sets in Euclidean spaces, equipped with a Borel probability measure µ of Hausdorff dimension d, one needs at least d measurements of a typical (prevalent) Lipschitz observable for reliable µ-almost sure reconstruction and prediction. Consequently, the Hausdorff dimension of µ is the precise threshold for the minimal delay (embedding) dimension for such systems in a probabilistic setting. This allows us to establish a variant of a lower bound in the prediction error conjecture of Schroer–Sauer–Ott–Yorke from 1998. Based on joint works with Krzysztof Barański and Adam Śpiewak.

周效尧 (南京师范大学)

Metric mean dimension of some subsets in dynamical systems

Abstract: In this talk, we will introduce some recent results about the metric mean dimension of some subsets (generic points, non-dense orbit set etc) . This is a joint work with Yang Rui, Yang Jiao and Ercai Chen. 

史汝西 (法国索邦大学)

Finite mean dimension and marker property

Abstract: The mean dimension was introduced by Michel Gromov (1999) as a new topological invariant. It was developed and studied systematically by Lindenstrauss and Weiss. In this talk, I will discuss about the relation between the marker property and dynamical systems of finite mean dimension.

金磊 (中山大学)

Embedding possibility and mean dimension for discrete and

continuous time flows

Abstract: The aim of this talk is to overview some embedding results in relation to mean dimension theory, within the framework of continuous and discrete time flows. With a view towards the similarity and difference between these behaviours, we may further pose natural questions asking if there is a reasonable perspective from which it is worth looking for analogues with those universal properties.

刘春麟 (中国科学技术大学)

Metric Mean Dimension via Preimage Structures

Abstract: The preimage entropy provides a quantitative estimate of how “invertible” a system is. Once there are several examples where the preimage entropy is infinite, it cannot provide more information. Thus, we introduce the concept of preimage metric mean dimension, and study many properties of it. Meanwhile, we provide many examples to compute their preimage mean metric dimension. This work joint with Fagner B. Rodrigues.

王旭磊 (复旦大学)

MEASURE OF MAXIMAL ENTROPY OF GENERAL SUBSETS

Abstract: Based on the Variational principles for topological entropies of subsets written by Feng and Huang in 2012, in this talk, we mainly discuss the measure of maximal entropy with respect to the Variational principles above. We will show that under the assumption of h-expansiveness, if the set cannot be written as a countable union of sets with lower entropy, then there exists a measure of maximal entropy, and in fact the condition is also necessary. Additionally, we will discuss several examples and properties of it.