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Francisco Arana-Herrera

Rice University

Title: Polygonal Billiards: Classic and Quantum

Abstract: The classical dynamics of billiard flows on rational polygons has been successfully studied by reduction to straight line flows on translation surfaces. Nevertheless, the quantum counterpart of this picture remains much less understood. We will discuss joint work with Athreya and Forni studying quantizations of straight line flows on translation surfaces and applications to some classical problems of Furstenberg in ergodic theory.

Vitaly Bergelson

The Ohio State University

Title: Uniform distribution of subpolynomial functions along primes and applications.

Abstract: We will review some old and new results which deal with the uniform distribution of Hardy field functions along primes and discuss various applications to ergodic theory, additive combinatorics and Benford law. The talk is based on joint work with Grigori Kolesnik and Younghwan Son.

Valérie Berthé

Université Paris Cité

Title: Spectral properties and symbolic discrepancy

Abstract: Discrepancy is a measure of equidistribution for sequences of points. In its symbolic version, discrepancy measures the difference between the number of occurrences of a factor in a prefix of an infinite word in a given shift and its expected value, stated in terms of frequency of occurrence of this factor. In other words, it is a particular instance of an ergodic deviation in a Birkhoff sum. The aim of this lecture is to stress the relationships that exist between characterizations of bounded discrepancy and of continuous eigenvalues. This is a joint work with Paulina Cecchi Bernales and Bastian Espinoza.

Nicolás Bitar

Université de Picardie Jules Verne

Title: Substitutive and S-adic systems beyond abelian groups

Abstract:  Substitutive and S-adic systems have a rich theory in the one-dimensional and multidimensional settings, providing powerful tools for the study of topological dynamical systems. In the hope of recovering these tools for group actions, in this talk I will show how we can extend the notion of substitutions to countable groups. I will explore how different classes of groups admit different types of hierarchical decompositions that allow us to define S-adic and substitutive subshifts. I will then show how to recover the classical results on minimality, zero entropy, unique ergodicity, and realization results, provided that the underlying group has good combinatorial properties. This work is done in collaboration with Christopher Cabezas and Pierre Guillon.

Julien Boulanger

Universidad de Chile

Title: The Khinchin constant for interval exchange transformations

Abstract: In this talk we discuss an ongoing project with R.Gutierrez-Romo, E.Lanneau, A.Pardo and V.Maturana in which we are interested in the generalization of the Khinchin constant from continued fractions (the almost sure limit of the geometric means of the digits of the continued fraction of a real number) to the setting of Interval Exchange Transformations. In this case, the existence of a Khinchin constant is guaranteed by the existence of an ergodic measure for the (accelerated) Rauzy induction, as discovered by A.Zorich, but the computation of this constant is not at all obvious. The aim of the talk is to introduce the notions discussed in this abstract, discuss the case with 3 intervals as well as some questions in the case where there are more intervals.

Christopher Cabezas

Universidad de Chile

Title: Notions of openness in semigroup actions

Abstract: We study the notion of openness of shifts transformations given by semigroup actions. In 1966 Parry proved that the only subshifts where the shift is a non-invertible open map are SFTs. We showed a similar result for any semigroup actions, introducing the notion of synchronizing sets and study some applications, such as the existence of periodic points. We also  discuss some interesting examples on N^d, Q+, and free semigroups. This is a joint work with Sebastian Donoso and Matheiu Sablik.

Haritha Cheriyath

Universidad de Chile

Title: Forbidding words from subshifts

Abstract: We consider perturbations of subshifts by forbidding additional words. Unlike in smooth dynamics, such perturbations are unusual in the discrete setting of symbolic dynamics. However, the seemingly simple operation of deleting a word from a subshift has a diverse range of unexpected connections, for instance, in coding theory, ergodic theory, or non-transitive games. In this talk, we explore some of these connections and we also study the dynamical behaviour under such perturbations. We focus on how the structure or the topological entropy may vary and we ask when two perturbations are conjugate.

Tomasz Downarowicz

Wrocław University of Science and Technology

Title: Universality of G-subshifts with specification

Abstract: In a joint work with Benjy Weiss, Matuesz Więcek and Guohua Zhang, we show that $G$-subshifts $(X,G)$ ($G$ – countable amenable group, $X$ – subset of $A^G$, $A$- finite alphabet) with appropriately defined specification property, are universal in the sense that for any free ergodic measure-preserving G-action $(Y,\Sigma,\mu,G)$ of entropy less than $h_{top}(X)$, there exists an invariant measure on X isomorphic to $\mu$. In fact, this is not exactly true; a sufficient and necessary condition for the above to be true is the existence of a free element $x \in X$, i.e., such that $g(x) = x$ only for $g=e$ (the unity of G).

Bastián Espinoza

Université de Liège

Title: Dynamics of dendric subshifts

Abstract:  Dendric subshifts are a recently introduced generalization of both interval exchange transformations and Arnoux-Rauzy subshifts that preserve some of their main properties. In particular, it has shown to be the adequate framework for the design of low complexity multidimensional continued fraction algorithms, thus representing an exciting class of objects from dynamical, geometric, and combinatorial perspectives. In this talk, I will present new results concerning their dynamical factors and S-adic structure. This is a joint work with Julien Leroy and Pierre Stas.

Felipe García-Ramos

Jagiellonian University and Universidad Autónoma de San Luis Potosí

Title: Local entropy theory and applications

Abstract: In this talk we will provide a survey on local entropy theory, a framework that began with the notions of topological completely positive topological entropy and the development of entropy pairs. Beyond clarifying the behavior of these properties, local entropy theory has grown into a perspective with applications across topological dynamics, connecting to the Ellis semigroup, maximal equicontinuous factors, sequence entropy, homoclinic points, mean dimension, spectral properties, and nilsystems. The talk follows a survey paper co-written with Hanfeng Li.

Till Hauser

Pontificia Universidad Católica de Chile

Title: Mean Diameter, Regularity and Diam-Mean Equicontinuity

Abstract: The notion of a regular factor map plays a central role in the interplay between topological dynamics and ergodic theory. In this talk, we study mean diameters and use them to obtain a characterization of regular factor maps in the context of actions of countable amenable groups. Building on this characterization, we show that an action is diam-mean equicontinuous if and only if it is a regular extension of its maximal equicontinuous factor. This extends earlier results that were previously known only for certain minimal actions. In addition, we establish the existence of a maximal diam-mean equicontinuous factor.

Wen Huang

University of Science and Technology of China

Title: On Systems Disjoint from All Minimal Systems

Abstract: In 1967, H. Furstenberg introduced the notion of disjointness in both ergodic theory and topological dynamics as a tool for comparing distinct systems. In this talk we present several characterizations of systems that are disjoint from every minimal system. We show that a topological dynamical system (X,T) is disjoint from all minimal systems if and only if there exist minimal subsets (M_i)_{i∈ℕ} of X whose union is dense in X and each of which is disjoint from X. For a semi-simple system (X,T), it is disjoint from all minimal systems if and only if there exists a dense G_δ set Ω⊂X×X such that for every pair (x₁,x₂)∈Ω the subsystems O(x₁,T) and O(x₂,T) are disjoint. This is joint work with Song Shao, Hui Xu, and Xiangdong Ye.

Jennifer Jones

University of Denver

Title: The stabilized automorphism group of minimal systems

Abstract: The stabilized automorphism group of a dynamical system (X,T) is the group of all self-homeomorphisms of X that commute with some power of T. While this is an algebraic object, we show that it captures rich dynamical information. We begin by characterizing the stabilized automorphism groups of odometers and Toeplitz subshifts, establishing an invariance property in these settings. We then extend our results to a broader class of minimal systems, proving that if two such systems have isomorphic stabilized automorphism groups and each has a non-trivial rational eigenvalue, then they must share the same set of rational eigenvalues. We further identify a class of systems for which the assumption of having a non-trivial rational eigenvalue can be removed. Finally, we generalize a known result for mixing shifts of finite type to include all irreducible shifts of finite type.

Bryna Kra

Northwestern University

Title: Infinite patterns in large sets of integers

Abstract: Since Szemeredi’s Theorem and Furstenberg’s proof thereof using ergodic theory, ergodic methods have been used to show the existence of numerous patterns in sets of integers with positive upper density. These tools have led to uncovering new patterns that occur in any sufficiently large set of integers, but until recently all such patterns have been finite. This talk will be an overview of our resolution of questions and conjectures of Erdos on infinite configurations in sets with positive upper density, along with a review of further directions. This is joint work with Joel Moreira, Florian Richter, and Donald Robertson.

Borys Kuca

Jagiellonian University

Title: Recent advances on multiple ergodic averages

Abstract: In 1977, Furstenberg gave a dynamical proof of the theorem of Szemerédi on the existence of arithmetic progressions in dense subsets of integers. In doing so, he initiated the use of ergodic methods to solve problems originating from additive combinatorics and number theory. The main tool in this field are multiple ergodic averages, a class of multilinear operators that generalize classical Birkhoff averages and can be used to count the number of arithmetic patterns in dense sets of integers. This talk will survey recent developments on multiple ergodic averages and ensuing applications to additive combinatorics.

Dominik Kwietniak

Jagiellonian University

Title: Realising Choquet simplices as spaces of invariant measures of minimal homeomorphisms on manifolds

Abstract: The realisation problem asks, given a property $P$ and a class $C$ of dynamical systems, whether there is a member of $C$ exhibiting $P$. For example, the smooth realisation problem, inspired by a remark of von Neumann (1932), asks: is every measure-preserving system isomorphic to one given by a smooth diffeomorphism of a compact manifold preserving a measure equivalent to the volume element? I will discuss the following variant: which Choquet simplices arise as spaces of invariant measures of minimal homeomorphisms on manifolds? It turns out that, at least for some manifolds, there are no restrictions on the structure of possible spaces of invariant measures: if a manifold $M$ admits a strictly ergodic homeomorphism and $K$ is any Choquet simplex, then $K$ is affinely homeomorphic to the simplex of invariant probability measures of some minimal homeomorphism on $M$. This is joint work with Sejal Babel, Jernej Činč, Till Hauser, and Piotr Oprocha.

Andrés Navas

Universidad de Santiago de Chile

Title: Growth of derivatives, distortion, and connexion for 1-dimensional diffeomorphisms

Abstract: For diffeomorphisms of either the circle or the compact interval, the growth of the derivative along iterates is very particular: in class C^1, subexponential growth arises only in absence of hyperbolic periodic points (Herman), and in class C^2, this forces the growth to be at most quadratic (Polterovich & Sodin, Watanabe, Dinamarca & Navas). For other metrics, as for instance an L^1 metric for the affine derivative of C^2  interval diffeomorphisms, there is still a dichotomy, closely related to the so-called Mather invariant (Eynard-Bontemps & Navas). We will see that this has an impact in the study of distorted diffeomorphisms (in the sense of Gromov), and for the important problem of the path-connectedness for the space of  commuting diffeomorphisms (Eynard-Bontemps & Navas). If the time permits, we will also discuss similar problems in higher regularity, in particular the recent remarkable work of Eynard-Bontemps & Militon for the C^{infty} category.

Ronnie Pavlov

University of Denver

Title: Maximal pattern complexity and pattern Sturmian sequences

Abstract: Maximal pattern complexity (MPC) of a subshift X was defined by Kamae and Zamboni in 2002 as a generalization of the usual word complexity function. It has nice connections to notions of independence in dynamics studied by Kerr and Li; for instance, subexponential growth of MPC implies that X is an almost 1-1 extension of a (compact abelian) group rotation. At the opposite end of the spectrum, it is known that 2n is the minimum possible growth rate for MPC of an aperiodic sequence, and sequences achieving this are called pattern Sturmian. Simple Toeplitz sequences, codings of irrational rotations by 2 intervals, and some characteristic functions of sparse sets are known to be pattern Sturmian, and it has been an open question whether these are the only examples. I’ll present some recent joint work with Anh Le and Casey Schlortt where we positively resolve this question, and describe some ideas of the proof and some open questions.

Samuel Petite

Université de Picardie Jules Verne

Title: Multiple Topological Minimal Self Joining over Groups

Abstract: In a joint work with Nicolás Bitar and Sebastián Donoso, we study dynamical system $G \overset{T}{\curvearrowright} X$ that are topologically self disjoint for some power $n$, in the sense that any tuple off the diagonal has a dense orbit in $X^n$ for the diagonal action. We generalize a result of King providing a bound on the interger $n$ for some nilpotent groups (abelian, Heisenberg group, …). This result follows from a general one on continuous group action on an infinite compact metric space: When the acting group is countable and admits a proper left and right invariant distance satisfying a mild asumption, we show the intersection of all its non-expansive horoballs is trivial.

Mathieu Sablik

Université de Toulouse

Title: Study of phase diagrams for perturbed cellular automata

Abstract: Alejandro’s early work focused on the iteration of measures by cellular automata, including the study of $\mu$-limit sets, the randomisation of linear cellular automata and rigidity. In this talk, we consider the perturbed counterpart of a cellular automaton $F$, obtained by independently modifying each cell with probability $\epsilon$ and selecting a new value uniformly at random after each iteration of $F$. We denote the perturbed cellular automaton with noise parameter $\epsilon$ by $F_\epsilon$. We consider two natural questions: – For which set of parameters does $F_\epsilon$ admit a unique invariant measure? – Which set of invariant measures are selected when $\epsilon$ goes to 0? We will address these questions in the context of specific examples, aiming to describe the possible sets that can be reached.

Scott Schmieding

Penn State University

Title: Invariant random compacts

Abstract: I’ll discuss the notion of an invariant random compact for a group action. The focus will be on the setting where the action satisfies either minimality or an almost minimality property: that there are no proper infinite compact invariant sets. The talk will outline a number of questions, and a few results, which are all joint with Bryna Kra.

Anne Siegel

IRISA / CNRS

Title: Symbiosis, Systems and Environmental Biology, and Discretization of Dynamic Systems

Abstract: Molecular biology often pushes researchers to revisit established concepts to address questions that seem mathematically straightforward but quickly become intractable due to the complexity and limitations of available data and knowledge. Here, we explore how predicting symbiotic mechanisms between marine bacteria and their hosts or between Atacama desert bacteria – though theoretically “simple” – requires leveraging discrete abstractions of dynamic systems and innovative combinatorial optimization methods to adapt to real-world constraints. This approach highlights the intersection of systems biology, computational modeling, and the challenges of translating biological complexity into actionable insights.

Víctor Sirvent

Universidad Católica del Norte

Title: Homological data on the periodic structure of self-maps on wedge sums

Abstract: The Lefschetz fixed point theory and its generalizations are  important tools in the study of the periodic structure of self-maps on spaces with “nice structure”, e.g. compact polyhedra. In this talk we use this approach to study the set of periods  for continuous self-maps on a wedge sum of topological manifolds, which  exhibits a particular combinatorial structure.
We  explicitly compute the Lefschetz numbers, the Dold coefficients and consider the corresponding  set of algebraic periods.
Moreover, we study the special case of maps on a wedge sum of tori, and highlight  some of  the homological obstructions that arise in defining these  maps.

Marcelo Sobottka

Universidade Federal de Santa Catarina

Title: Fantastic Shift Spaces and the Graphs Where to Find Them

 Abstract: Given a non-empty countable set $\mathcal{A}$ (an alphabet), we define the full shift $\mathcal{A}^\mathbb{S}$, where $\mathbb{S}$ stands for the set of the integers ($\mathbb{Z}$) or for the set of non-negative integers ($\mathbb{N}$). We endow $\mathcal{A}^\mathbb{S}$ with the prodiscret topology, and on $\mathcal{A}^\mathbb{S}$ we consider the shift map $\sigma$.

A shift space (or subshift) is any closed set $\Lambda\subset \mathcal{A}^\mathbb{S}$ such that $\sigma(\Lambda)\subset\Lambda$.

It is well known that if $\mathcal{A}$ is finite, then only sofic shifts can be presented by finite directed labeled graphs (in fact, to be presented by such a graph is an alternative definition for finite-alphabet sofic shifts). In contrast, it is easy to check that every countable-alphabet one-sided shift space can be presented by a countable directed labeled graph.

In this talk I will present definitions for {\em shifts of finite type} and for {\em sofic shifts} that can be used in the general context, and that coincide with the classical ones in the context of finite-alphabet shifts. Following from these definitions, I will define two new classes of shift spaces that can properly exist only in the context of infinite-alphabet shift spaces: {\em weakly sofic shifts}; and {\em shifts of variable length}. I then investigate when a two-sided shift space can be presented by a countable directed labeled graph, and present some results that characterize graphs that are presenting one- or two-sided shifts of finite type and (weakly) sofic shifts.

Wenbo Sun

Virginia Tech

Title: Partition regularity for quadratic equations

Abstract: Let p(x,y,z) be a polynomial. The partition regularity problem asks the following question: for any finite coloring of integers, is there any x,y,z of the same color such that p(x,y,z)=0. In this talk, we will go over the history of the partition regularity problem for quadratic polynomials. We will also talk about a recent joint work with S. Donoso, A. Koutsogiannis and A. Ferre Moragues on the partition regularity problem over imaginary quadratic number fields.

Reem Yassawi

Queen Mary University of London

Title: Some of Alejandro’s work on invariant measures for cellular automata

Abstract: In this talk I will give an overview of Alejandro’s work on asymptotic randomisation of linear cellular automata, as well as resulting literature.

Xiangdong Ye

University of Science and Technology of China

Title: Scattering and its connection to density problems

Abstract: The notion of scattering was introduced by Blanchard, Host, and Maass in 2000, defined via the complexity of open covers. Their striking result shows that a topological dynamical system is scattering if and only if it is weakly disjoint from all minimal systems. This concept was subsequently studied in depth by Huang and Ye in 2004. In a recent joint work with Guo, Qiu and Xu, we uncovered a link between scattering and certain density-type problems. In this talk, I will present this connection and explore several related questions.