Special Session 50: Trends in Infinite Dimensional Topological Dynamics

Dynamics of Solenoidal Automorphisms

Sharan Gopal BITS-Pilani Hyderabad campus India Co-Author(s):    Dr. Faiz Imam

Abstract: This talk is based on a series of papers (mentioned below), which aim at studying the sets of periodic points of automorphisms on a solenoid. In these papers, we attempted to characterise the sets of periodic points of solenoidal automorphisms. In [1], we characterized the sets of periodic points of 1-dimensional solenoids and full solenoids. In the next paper [2], it was extended to some higher dimensional solenoids in terms of inverse limits of certain maps on a torus. Finally, the third paper [3] does this using the concept of adeles. The talk starts with an introduction to some basic notions of topological dynamics that are required to present our results, followed by a summary of the results mentioned above. It is then concluded with the same question for more general dynamical systems than solenoids. [1] Gopal, Sharan and Raja, C.R.E. \textit{Periodic points of solenoidal automorphisms}, Topology Proceedings, 50 (2017), 49-57. [2] Gopal, Sharan and Imam, Faiz. \textit{Periodic points of solenoidal automorphisms in terms of inverse limits}, Appl. Gen. Topology, Vol. 22 (2021), 321-330. [3] Imam, Faiz and Gopal, Sharan. \textit{Periodic points of solenoidal automorphisms in terms of adeles}, Monatsh Math 204 (2024), 501-511.

Dynamical systems and the Diophantine approximation on the Hecke group H4

Dong Han Kim Dongguk University - Seoul Korea Co-Author(s):

Abstract: Diophantine approximation is to find rational numbers that approximate irrationals. It is related to the two kinds of dynamical systems, dynamics on the parameter space and dynamics on the phase space. The Gauss map and geodesic flows are parameter space dynamics. The Diophantine approximation exponent quantifies the rate at which the geodesic approaches the cusp of the fundamental domain of the modular group. It also gives the rate of the recurrence time of irrational rotations and translations on torus which are dynamical systems of the phase space. In this talk we generalize classical results on the Diophantine approximation to the Hecke group H_q. When q=4, the Diophantine approximation on H_4 corresponds to the approximation on the unit circle and it is related with continued fraction algorithms to find best rational approximations of given parities. We also discuss the dynamical systems on the phase space associated with the Diophantine approximation on H_4.

Understanding cut-and-project sets on substitution tilings

Jeon-Yup Lee Catholic Kwandong University Korea Co-Author(s):    Boris Solomyak

Abstract: After the discovery of quasicrystals in material sciences, there has been a lot of study to understand the structure of quasicrystals. Mathematically, quasicrystals can be modeled by tilings or point sets, and the structure of quasicrystals can be described by pure discrete spectrum of the tiling dynamics. It is known that a cut-and-project set with a nice window always gives pure discrete spectrum. But the converse is not true in general settings. Here we look at substitution tilings and study the relation between the cut-and-project sets and pure discrete spectrum. We first look at the case that the expansion maps of the substitutions are diagonalizable. And then we will also talk about a recent development on non-diagonalizable case.

Spectral decomposition and skew product for group actions

Keonhee Lee Chungnam National University Korea Co-Author(s):

Abstract: Spectral decomposition which is fundamental in the qualitative theory of dynamical systems delineates that the nonwandering set can be decomposed as a finite number of disjoint compact invariant indecomposable sets. In this talk we establish various types of spectral decomposition for group actions on compact metric spaces. In particular, we use a skew-product associated with a group action to derive the spectral decomposition of the nonwandering set in a given direction. This talk is based on reference [1]. References [1] K.Lee, C. Morales and Y. Tang, Spectral decomposition and skew-product for group actions, preprint. [2] K. Lee and N. Nguyen, Spectral decomposition and $\Omega$-stability of flows with expanding measures, J. Differential Equations 269 (2020), 7574-7604.

Various Shadowing Properties in General Topological Spaces

Khundrakpam Binod Mangang Manipur University India Co-Author(s):    Khundrakpam Binod Mangang , Thiyam Thadoi Devi and Sonika Akoijam

Abstract: In this talk, we introduce various shadowing properties such as Hausdorff average shadowing property, Hausdorff ergodic shadowing property, periodic shadowing property when the phase space is a general topological space. We prove some related results. On a compact Hausdorff space, if $f$ has the Hausdorff average shadowing property, we show that $f^k$ has the Hausdorff average shadowing property for every positive integer $k$. Further, we show that a dynamical system with the Hausdorff ergodic shadowing property is Hausdorff chain transitive if $f$ is surjective. The content of the talk is from the following references. [1]POSITIVE EXPANSIVITY, CHAIN TRANSITIVITY, RIGIDITY, AND SPECIFICATION ON GENERAL TOPOLOGICAL SPACES, Bull. Korean Math. Soc. 59 (2022), No. 2. [2] ON PERIODIC SHADOWING, TRANSITIVITY, CHAIN MIXING AND EXPANSIVITY IN UNIFORM DYNAMICAL SYSTEMS, Gulf Journal of Mathematics Vol 9, Issue 2 (2020). [3] ERGODIC SHADOWING, d-SHADOWING AND EVENTUAL SHADOWING IN TOPOLOGICAL SPACES, Nonlinear Functional Analysis and Applications Vol. 27, No. 4 (2022).

Dynamics of primitive elements under group actions

Pratyush Mishra Wake Forest University USA Co-Author(s):    Pratyush Mishra

Abstract: There has been some substantial work on studying the structure of a group by analyzing the behavior of primitive elements, sometimes under strong assumptions. We formulate and study a conjecture of Platonov and Potapchik for general group actions via analyzing the dynamics of primitive elements for a given action. Such studies led us to further afield to produce results that combines computational, dynamical, geometric, and purely algebraic viewpoints. As an application, we introduce the notion of Nielsen and Schreier girth and obtain a class of groups with finite Nielsen girth but having infinite girth. Reference: 1) P. Mishra, Dynamics of primitive elements under group actions, arxiv.org/pdf/2403.16769

Spectral Decomposition and Topological Stability for Dynamical Systems on Non-metrizable Spaces

Jumi Oh Sungkyunkwan University Korea Co-Author(s):    Jumi Oh

Abstract: In this talk, we introduce the notions of symbolic expansivity and symbolic shadowing for homeomorphisms on non-metrizable spaces which are generalizations of expansivity and shadowing for metric spaces, respectively. The main result is to generalize the Smales spectral decomposition theorem to symbolically expansive homeomorphisms with symbolic shadowing on non-metrizable compact Hausdorff totally disconnected spaces. Furthermore, we consider the topological stability for homeomorphisms on the spaces.

Topological Lipschitz Shadowing Property

Nisarg Purohit BITS Pilani (Hyderabad Campus) India Co-Author(s):    Sharan Gopal

Abstract: In the theory of shadowing properties, there is a recent trend of generalizing various notions of these properties to topological spaces which are not necessarily metrizable. In this talk, we will show a generalization of Lipschitz Shadowing property (LpSp) to uniform spaces, which will be called as Topological Lipschitz Shadowing property (TLpSp). Several results that hold for LpSp are shown to be true for TLpSp. Also, we prove that if a compact metric space possesses TLpSp then it also possesses LpSp.

Expansive Minimal Flows

Elias Rego AGH university of Science and Technology Poland Co-Author(s):

Abstract: In this talk we shall discuss the relation between minimality and expansiveness for regular flows. Our main goal is to extend a famous result due Ma\~n\`e which states that if an expansive homeomorphism is minimal, then it must be defined on a zero dimensional space. An attempt of extending this result to flows was made by Keynes and Sears in 1981 with the extra assumption of no spiral points or the flows being Axiom A. Here we will see how remove those extra conditions and obtain a result analogous to Ma\~n\`e`s in full generality. Precisely, we show that a expansive flow is minimal if and only if it is a suspension of a minimal subshift of finite type. We further apply our findings to study the minimal subsets of expansive flows. This is a joint work with Alfonso Artigue.

Measurable spectral decomposition for homeomorphisms

Bomi Shin Sungkyunkwan University Korea Co-Author(s):    Bomi Shin

Abstract: The Spectral Decomposition Property (SDP) plays a central role in understanding the structure of nonwandering sets in dynamical systems. In this talk, we extend the classical SDP to a measure-theoretic setting for homeomorphisms on compact metric spaces. We demonstrate that a homeomorphism has the SDP if and only if each Borel probability measure satisfies this property. Additionally, we have the relationship between shadowing properties and spectral decomposition by proving that shadowable measures for expansive homeomorphisms exhibit the SDP. This talk is based on reference \cite{s}. \begin{thebibliography}{20} \bibitem{s} Shin, Bomi A measurable spectral decomposition. Monatsh. Math. 204 (2024), no. 2, 311--322. \end{thebibliography}

Joint ergodicity of piecewise monotone maps

Younghwan Son POSTECH Korea Co-Author(s):

Abstract: Joint ergodicity is a generalization of the notion of ergodicity to a finite number of measure preserving transformations. Berend and Bergelson provided a characterization of joint ergodicity for commuting invertible measure preserving systems. In this talk we present a generalization of their characterization and provide some examples of joint ergodicity of piecewise monotone maps. This result demonstrates that the phenomena of joint ergodicity takes place even when the involved measure preserving transformations are neither commuting nor invertible, and have different invariant measures. This is a joint work of Vitaly Bergelson.