Special Session 45: Frontiers in Topological Dynamics: Theory, Applications, and Interdisciplinary Connections

Physical measures for a class of partially hyperbolic attractors

Yongluo Cao Soochow University Peoples Rep of China Co-Author(s):    Zeya Mi

Abstract: Title: Physical measures for a class of partially hyperbolic attractors Abstract: In this talk, we consider the existence of physical measure for partially hyperbolic attractors. If the systems`s central direction can be decomposed into one dimension sub-bundles which are dominated splitting, and all u-gibbs measures are hyperbolic, then there exist finite physical measures.

Invariant measure for infinite Iterated Function Systems

Xiaopeng Chen Shantou UNiversity Peoples Rep of China Co-Author(s):

Abstract: In this talk we define some infinite iterated function systems. We study the Ruelle operator theorem for the infinite iterated function systems. We prove the existence and uniqueness of the invariant measure for these types of systems.

On Entropy-like Invariants for Dynamical Semigroup Actions

Wen-Chiao Cheng Chinese Culture University Taiwan Co-Author(s):

Abstract: During this talk, entropy-like invariants are first introduced and presented. Some questions will be shown and then talk about conditional entropy`s propositions. Variational principle, subadditivity and commutativity properties for entropy and pressure are also shown. Furthermore, some new results are also exhibited to see what we can do next step for semigroup action on entropy propositions.

A Geometric Framework for Stochastic Dynamics on Manifolds of Probability Densities

Jinqiao Duan Great Bay University Peoples Rep of China Co-Author(s):

Abstract: Traditional analysis of stochastic systems often relies on pathwise trajectories within Euclidean space. However, a more profound understanding of complex phenomena, such as critical transitions, emerges when these systems are viewed as evolutions on a manifold of probability densities. This talk explores this geometric perspective on the Onsager Machlup most probable paths, Schrodinger bridges, and information geodesics. This provides a cohesive framework that links calculus of variations and information geometry, offering new insights into the most probable behaviours of uncertain systems.

Threshold dynamics of two diffusive epidemic models with heterogeneity parameters

Lian Duan Anhui University of Science and Technology Peoples Rep of China Co-Author(s):

Abstract: In this talk, I will report the threshold dynamics of two class of diffusive epidemic models with heterogeneity parameters. One is diffusive ZIKV epidemic model, and another is diffusive SIRI epidemic model. Our analyses show that the basic reproduction number which serves as a threshold parameter that predicts whether epidemics will persist or become globally extinct. For the latter, the asymptotic profiles of the positive steady state are also discussed.

Some interactions between Dynamical systems and Harmonic analysis

Aihua Fan Central Normal University Peoples Rep of China Co-Author(s):

Abstract: Any locally compact Abelian (LCA) group acts on a torus $\mathbb{T}^n$ of arbitrary dimension. We show that this action admits a decomposition into uniquely ergodic subsystems. As application, the famous Kronecker theorem on Diophantine approximations (density) is then strengthened into a Weyl-type theorem (equidistribution). This allows us to show that the characters of an LCA group are orthogonal in the sense of Bohr and other consequences follow. Furthermore, we show that the algebra of quasi-periodic functions with spectra in a $\mathbb{Z}$-module of finite rank n is isometrically isomorphic to the algebra C($\mathbb{T}^n$) of continuous periodic functions. This is the theoretical foundation upon which the projection method in scientific computation is based. As a theoretical application of this algebra isomorphism, we can prove, in a very simple way, the Hausdorff-Young inequalities for almost periodic functions in the sense of Besicovitch. This last part is a joint work with Kai JIANG and Pingwen ZHANG.

A Parabolic System for Bacterial Colony Dynamics

Zhaosheng Feng University of Texas Rio Grande Valley USA Co-Author(s):

Abstract: In this talk, we present a nonlinear parabolic system incorporating dispersion to model bacterial aggregation dynamics. In contrast to models without dispersion, the inclusion of dispersive effects enables the propagation of bacterial clusters, suggesting that dispersion may serve as a regulatory mechanism in colony behavior. Through theoretical analysis and numerical simulations, we show that an initially random bacterial distribution evolves into a periodic wave pattern, in the absence of dispersion, this pattern ultimately transitions into a stationary solitary wave.

Symplectic blenders near whiskered tori and persistence of saddle-center homoclinics

Dongchen Li Shanghai Center for Mathematical Sciences, Fudan University Peoples Rep of China Co-Author(s):    Dmitry Turaev

Abstract: A blender is a hyperbolic basic set such that the projection of its stable/unstable set onto some center subspace has a higher topological dimension than the set itself. We prove that, for any $C^s$ symplectic diffeomorphism (where $s=2,\dots\infty,\omega$), if it has a one-dimensional whiskered torus with a homoclinic orbit, then a symplectic blender can be created by an arbitrarily $C^s$-small perturbation. Using this result, we show that the non-transverse homoclinic intersection between the invariant manifolds of a saddle-center periodic point is persistent, in the sense that the original system lies in the $C^s$-closure of a $C^1$-open set of symplectic diffeomorphisms where those having saddle-center homoclinics are dense. Our results also hold in the corresponding continuous-time settings.

The equivalence of precompactness, zero maximal pattern entropy and bounded mean complexity for finite partitions

Jian Li Shantou University Peoples Rep of China Co-Author(s):    Tao Yu, Xianliang Zhong

Abstract: In this talk, we discuss several types of low complexity of finite partitions in a standard probability space, including precompactness, zero maximal pattern entropy, bounded mean complexity and mean equicontinuity. We show that a collection of finite partitions is precompactness in the Rokhlin metric if and only if it has zero maximal pattern entropy if and only if the collection of the characteristic functions of atoms in the partitions is precompactness in $L^2$ if and only if it has bounded mean complexity with respect the Hamming distance. Then we apply this result to the complexity of a partition of countably infinite discrete amenable group actions. This talk is based on a joint work with Tao Yu and Xianliang Zhong.

Mean weak length

Bingbing Liang Soochow University Peoples Rep of China Co-Author(s):

Abstract: We introduce a weak version of the classical length function, termed the weak length function, defined on subsets of $R$-modules over a unital ring $R$, and further consider the concept of mean weak length for $R\Gamma$-modules associated with an amenable group $\Gamma$. Under an appropriate upgrading condition together with certain mild assumptions, we establish that the mean weak length function is additive with respect to short exact sequences. This result has two consequences. First, we provide a purely algebraic proof of the additivity of algebraic entropy, which is originally established via topological entropy methods. Second, within our unified framework, we give an alternative and conceptual proof of the additivity of mean length, previously obtained by Li-Liang and Virilli using different approaches.

Dynamical systems and their induced systems

Kairan Liu Chongqing University Peoples Rep of China Co-Author(s):    Hanfeng Li; Yixiao Qiao; Runju Wei

Abstract: In this report, we will discuss the topological and dynamical connections between dynamical systems and their induced systems, including openness of relatively factor maps, relative mean dimension, and local entropy theory.

Continuity of the Lyapunov exponent and measure entropy for stochastic differential equations

Zhenxin Liu Dalian University of Technology Peoples Rep of China Co-Author(s):    Lixin Zhang

Abstract: For non-autonomous linear stochastic differential equations (SDE), we establish that the top Lyapunov exponent is continuous if the coefficients almost uniformly converge. For autonomous SDEs, assuming the existence of invariant measures and the convergence of coefficients and their derivatives in pointwise sense, we get the continuity of all Lyapunov exponents. Furthermore, we talk about the Lipschitz and Holder continuity of Lyapunov exponents. For autonomous SDEs, we establish a relationship between the measure entropy of the stochastic flow and the rate of volume growth of stable submanifolds under iteration. By combining the results of Kifer and Yomdin, we obtain that for such systems with $C^\infty$ coefficients, under suitable integrability conditions, the measure entropy is upper semicontinuous with respect to the coefficients.

Periodic and Horseshoe-like Structures in Partially Hyperbolic Systems

Xiao Ma University of Science and Technology of China Peoples Rep of China Co-Author(s):    Xinyu Bai, Wen Huang, Zeng Lian, Leiye Xu, Hang Zhao

Abstract: This talk explores periodic and horseshoe-like structures in partially hyperbolic systems, with an emphasis on how these structures relate to the complexity of the system, including entropy and expansion rates. The presentation is based on joint works with Xinyu Bai, Wen Huang, Zeng Lian, Leiye Xu, and Hang Zhao.

Uniform rokhlin property and double variational principle for mean dimension

Ruxi Shi Fudan University Peoples Rep of China Co-Author(s):    Alessandro Codenotti and Petr Naryshkin

Abstract: As a new topological invariant, the notion of mean topological dimension was introduced by Gromov (1999). Then it was developed systematically by Lindenstrauss and Weiss (2000). Under marker property, Lindenstrauss and Tsukamoto (2019) developed a variational principle between mean dimension theory and rate distortion theory. In this talk, we extend Lindenstrauss-Tsukamoto double variational principle to dynamics of amenable group action with uniform rokhlin property. This talk is based on the ongoing work with Alessandro Codenotti and Petr Naryshkin.

Development of Bowen`s specification legacy on thermodynamic formalism and applications to stochastic properties

Tianyu Wang School of Mathematical Sciences Shanghai Jiao Tong University Peoples Rep of China Co-Author(s):

Abstract: In 1970`s, Rufus Bowen proposed a general criterion on establishing thermodynamic formalism of a dynamical system by extracting critical features of a uniformly hyperbolic system equipped with a H\{o}lder continuous potential. Precisely, he showed that on an expansive homeomorphism satisfying a strong orbital concatenation property called specification, any boundedly distorted continuous potential must possess a unique equilibrium state. In this talk, we briefly survey the development of Bowen`s argument in the past two decades by illustrating how Bowen`s idea is extended to a variety of systems beyond compact uniform setting. This includes the introduction of a recent variant of Bowen`s approach, as well as its applications on detecting a few types of strong stochastic behaviors

Preimage Mean Dimension

Xinsheng Wang Shantou University Peoples Rep of China Co-Author(s):    Weisheng Wu, Yujun Zhu

Abstract: In this talk, the mean dimension theory via the preimage structure for noninvertible infinite dimensional dynamical systems with infinite topological entropy is considered. Several invariants, such as the topological preimage mean dimensions, the metric preimage mean dimensions and the rate distortion preimage mean dimensions involving two variables (metrics and measures) are introduced and the relations among these quantities are considered. Particularly, for a non-invertible system with the backward marker property, a double variational principle relating rate distortion preimage mean dimensions and topological preimage mean dimension is established. This is a joint work with Weisheng Wu and Yujun Zhu.

Product saturation theorem in polynomials

Qinqi Wu Shanghai University of Finance and Economics Peoples Rep of China Co-Author(s):    Jiaqi Yu

Abstract: In this talk, we introduce the saturation theorem in product systems for polynomials. It is a joint work with Jiaqi Yu.

Symmetry vs Ergodicity

Disheng Xu Great Bay University Peoples Rep of China Co-Author(s):

Abstract: In this talk, I will present my joint work with Damjanovic, Wilkinson and Wu. More precisely I will show for smooth dynamics, at least for affine maps on homogeneous spaces, ergodic theoretic property of a dynamical system is closely related to its symmetries.

Almost Countable Spectrum and Logarithmic Sarnak Conjecture

Leiye Xu University of Science and Technology of China Peoples Rep of China Co-Author(s):    Wen Huang and Maoru Tan

Abstract: We introduce topological dynamical systems with almost countable spectrum. We prove that the Logarithmic Sarnak Conjecture holds for zero-entropy topological dynamical systems whose spectrum is almost countable. This class includes Anzai skew product on $\mathbb{T}^2$ over a rotation of $\mathbb{T}^1$, time-one maps of continuous suspension flows over rotations, systems with finite maximal pattern entropy, and bounded tame systems.

Metric mean dimension of amenable group actions via entropy of subsets

Ruifeng Zhang Hefei University of Technology Peoples Rep of China Co-Author(s):    Xinyao He and Guohua Zhang

Abstract: We investigates the theory of metric mean dimension for actions of countable discrete amenable groups on compact metric spaces. We introduce the metric mean dimension, packing metric mean dimension, and Bowen metric mean dimension, respectively, using the topological entropy, packing topological entropy and Bowen`s dimensional entropy, and establish the equivalence among these three metric mean dimensions. Our main results demonstrate that the global metric mean dimension of the system can be completely characterized by the asymptotic behavior of various entropies of local sets, particularly the $\epsilon$-stable sets at individual points. These results generalize previous work on single transformations, and provide new tools for studying dynamical systems with infinite topological entropy. This is a joint work with Xinyao He and Guohua Zhang.

On fast Lyapunov spectra for Markov-R\`{e}nyi maps

Yiwei Zhang Anhui University of Science and Technology Peoples Rep of China Co-Author(s):    Lulu Fang, Carlos Gustavo Moreira, Zhichao Wang

Abstract: In this talk, we study the multifractal analysis for Markov-R\`{e}nyi maps, which form a canonical class of piecewise differentiable interval maps, with countably many branches and may contain a parabolic fixed point simultaneously, and do not assume any distortion hypotheses. We develop a geometric approach, independent of thermodynamic formalism, to study the fast Lyapunov spectrum for Markov-R\`{e}nyi maps. Our study can be regarded as a refinement of the Lyapunov spectrum at infinity. We demonstrate that the fast Lyapunov spectrum is a piecewise constant function, possibly exhibiting a discontinuity at infinity. Our results extend the works in \cite[Theorem 1.1]{FLWW13}, \cite[Theorem 1.2]{LR}, and \cite[Theorem 1.2]{FSW} from the Gauss map to arbitrary Markov-R\`{e}nyi maps, and highlight several intrinsic differences between the fast Lyapunov spectrum and the classical Lyapunov spectrum. Moreover, we establish the upper and lower fast Lyapunov spectra for Markov-R\`{e}nyi maps.

Discrete spectrum of probability measures for locally compact group actions

Xiaomin Zhou Huazhong University of Science and Technology Peoples Rep of China Co-Author(s):    Zongrui Hu, Xiao Ma, and Leiye Xu

Abstract: In this talk we study discrete spectrum of probability measures for locally compact group actions. It turns out that a probability measure has discrete spectrum if and only if it has bounded measure-max-mean-complexity. As an application, an invariant measure of a locally compact amenable group action has discrete spectrum if and only if it has bounded mean-complexity along F\olner sequences. This is a joint work with Zongrui Hu, Xiao Ma, and Leiye Xu.

Variational principles for (metric) mean dimension

Xiaoyao Zhou Nanjing Normal University Peoples Rep of China Co-Author(s):

Abstract: Mean dimension is a topological invariant of topological dynamical systems, first introduced by Gromov. Subsequently, Lindenstrauss and Weiss conducted further investigations on this quantity, aiming to address an embedding problem proposed by Auslander in the 1970s. Furthermore, from the perspective of entropy theory, they introduced the concept of metric mean dimension. In 2018--2019, Lindenstrauss and Tsukamoto established the variational principles for (metric) mean dimension. In this talk, we present our recent progress regarding the variational principles for (metric) mean dimension.