Classification complexity of chaotic systems — Benjamin Vejnar

The aim of this talk is first to briefly describe a natural way of measuring simplicity/complexity of classification problems by using Invariant Descriptive Set Theory and then to discuss recent applications in the context of topological dynamics. We mainly deal with the classification of transitive systems on the interval, on the Cantor set and on the Hilbert cube with respect to the topological conjugacy relation. At the end, we provide some attempts to identify the complexity of classification of minimal systems.

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Completely invariant sets and Lorenz maps — Piotr Oprocha

In this talk we will discuss relations between completely invariant sets and renormalizations of expanding Lorenz maps, that is maps $f:[0,1]\to [0,1]$ satisfying the following three conditions: 1. there is a critical point $c\in (0,1)$ such that $f$ is continuous and strictly increasing on $[0,c)$ and $(c,1]$; 2. $\underset{x\to c^{-}}{lim}f(x)=1$ and $\underset{x\to c^{+}}{lim}f(x)=0$; 3. $f$ is differentiable for all points not belonging to a finite set $F\subseteq [0,1]$ and $\underset{x\notin F}{inf}{f}^{\prime }(x)>1$; with special emphasis on piecewise linear case. The talk is based on joint works with L. Cholewa.

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Cubic polynomials and laminations — Nikita Selinger

I will review the notion of laminations as introduced by W. Thurston and explain how laminations can be used to study parameter spaces of polynomials.

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Dimension of Lyapunov spectrum for non uniformly hyperbolic settings — Emma Dinowitz

We study the Hausdorff dimension of the set of points with a fixed lyapunov exponent inside a family of subsets of a 3 dimensional flow with non uniform hyperbolicity properties. Recent work of Sarig, Lima, and others have constructed countable state markov partitions modeling these sets. Using their framework we prove upper bounds analogous to the uniformly hyperbolic situation.

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Ergodic Averages along Sequences of Slow Growth — Kaitlyn Loyd

Given Birkhoff's pointwise ergodic theorem, it is natural to consider whether convergence still holds along subsequences of the integers. In this talk, we investigate convergence of ergodic averages along the number theoretic sequence $\mathrm{\Omega }(n)$, where $\mathrm{\Omega }(n)$ denotes the number of prime factors of $n$ counted with multiplicities. In particular, we demonstrate that, although a pointwise ergodic theorem does not hold along $\mathrm{\Omega }(n)$, there are multiple instances in which we can recover convergence. We also present a more general criterion for identifying slow-growing sequences possessing a certain divergence property exhibited by $\mathrm{\Omega }(n)$. This talk is based on joint work with Sovanlal Mondal (Ohio State).

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Ergodic optimization with linear constraints — Kevin McGoff

Let $T:X\to X$ be a continuous map on a compact metrizable space, let $f:X\to \mathbb{R}$ be continuous, and let $W\subset C(X)$ be a closed subspace of continuous functions from $X$ to $\mathbb{R}$. We consider the set $M_W(X,T)$ of all $T$-invariant Borel probability measures $\mu$ such that $\int gd\mu =0$ for all $g$ in $W$. Then we consider optimization problems of the form $$max\int fd\mu +\tau h(\mu ),$$ where $\mu$ ranges over $M_W(X,T)$, $h(\mu )$ denotes the entropy of $\mu$ with respect to $T$, and $\tau$ is either $0$ or $1$. Our main results concern the basic properties of such optimization problems, including feasibility, geometry of the solution set, uniqueness of solutions, and realizability. This talk is based on ongoing joint work with Shengwen Guo (UNC Charlotte).

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Flow equivalence and PSL_2(Q)-equivalence — Scott Schmieding

A real number gives rise to a Sturmian system encoding a rotation of the circle, and there are several beautiful connections between these systems and arithmetic properties of the associated parameters. One is a result of Fokkink, which shows that two Sturmian subshifts with parameters \alpha and \beta are flow equivalent if and only if \alpha and \beta lie in the same orbit of the action of PSL_2(Z) on the set of reals via Mobius transformations, a condition which is itself characterized by the tails of their continued fraction expansions. I'll describe some recent work, joint Christopher-Lloyd Simon, describing the action of PSL_2(Q) in terms of a certain relation on systems called isogeny.

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Full Groups of Cantor Dynamical Systems: Characters and Invariant Measures — Constantine Medynets

Given a Cantor minimal dynamical system $(X,T)$, the topological full group $[[T]]$ consists of all homeomorphisms of $X$ that locally act as powers of $T$. These groups can be viewed as generalized symmetric groups on the continuous orbit equivalence relation of $(X,T)$. A series of works by Giordano–Putnam–Skau, Matui, Medynets, Nekrashevych, and others have demonstrated that the algebraic structure of topological full groups completely determines the orbit structure of the underlying systems. This naturally leads to the question of whether the structure of invariant measures, an invariant of orbit equivalence, is similarly reflected in the full group's algebraic properties. In this talk, we present joint work with Artem Dudko (IMPAN) on the classification of characters of topological full groups of Cantor minimal systems. We establish that every extreme character of the commutator subgroup of $[[T]]$ is of the form $\mu (Fix(g))$, where $\mu$ is an ergodic product measure on $X^n$, thereby confirming Vershik’s conjecture for the class of full groups. As a consequence, we show that prime indecomposable characters are in one-to-one correspondence with ergodic measures.

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Homomorphisms from aperiodic subshifts to subshifts with the finite extension property — Robert Bland

We are inspired by recent efforts to generalize the classical embedding theorem of Krieger for $\mathbb{Z}$ subshifts, which states that if $X$ is an SFT and $Y$ is a mixing SFT, then $X$ embeds into $Y$ if certain necessary conditions on the periodic points and entropy are satisfied. Moving to subshifts over groups $G$ beyond $\mathbb{Z}$, an extra essential hypothesis emerges: that there is a homomorphism (a continuous and shift-commuting map, not necessarily injective) from $X$ to $Y$ at all. This is trivially satisfied if, e.g., $Y$ contains a fixed point, but necessary and sufficient conditions for the existence of a homomorphism are not known in general. In this talk, we present joint work with K. McGoff that constructs a homomorphism $\varphi :X\to Y$ in the case that $X$ is aperiodic, $Y$ has the finite extension property, and the underlying group $G$ has the property that every finitely generated subgroup of $G$ has polynomial growth (i.e., $G$ is locally virtually nilpotent by Gromov's theorem). The finite extension property (FEP) can be seen as a very strong mixing-like condition which has been considered before for subshifts over ${\mathbb{Z}}^d$ [Briceño, McGoff, Pavlov 2016].

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Maximal Pattern Complexity for General Alphabets — Casey Schlortt

Maximal pattern complexity was introduced by Teturo Kamae and Luca Zamboni in 2002 as a way to link word complexity and sequence entropy. In this same paper, they introduced the idea of a pattern Sturmian over two letters, an aperiodic sequence with the lowest possible maximal pattern complexity on a two letter alphabet. In this talk, we will introduce some established results about sequences with low maximal pattern complexity and some new results extending the understanding of sequences of low maximal pattern complexity on larger alphabets.

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Measures of maximal entropy on coded shift spaces — Christian Wolf

In this talk, we present results about the uniqueness of measures of maximal entropy on coded shift spaces. A coded shift space is defined as the closure of all bi-infinite concatenations of words from a fixed countable generating set. We derive sufficient conditions for the uniqueness of measures of maximal entropy and equilibrium states of Hoelder continuous potentials based on the partition of the coded shift into its concatenation set (sequences that are concatenations of generating words) and its residual set (sequences added under the closure). We also discuss flexibility results for the entropy on the concatenation and residual sets. Finally, we present a local structure theorem for intrinsically ergodic coded shift spaces. This shows that our results apply to a larger class of coded shift spaces compared to previous works by Climenhaga, Climenhaga and Thompson, and Pavlov. The results presented in this talk are joint work with Tamara Kucherenko and Martin Schmoll.

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Neighborhood N-Shadowing — Elyssa Stephens

We define neighborhood $N$-shadowing property and discuss the relationship of this property to mixing sofic shifts. Specifically, we show all mixing sofic shifts over a finite alphabet have neighborhood 2-shadowing.

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On the Equivalence of Equilibrium and Freezing States — Evans Hedges

This talk is concerned with freezing phase transitions in general dynamical systems. A freezing phase transition is one in which, for a given potential $\varphi$, there exists some inverse temperature ${\beta }_0>0$ such that for all $\alpha ,\beta >{\beta }_0$, the collection of equilibrium states for $\alpha \varphi$ and $\beta \varphi$ coincide. In this sense, below the temperature $1/{\beta }_0$, the system "freezes" on a fixed collection of equilibrium states. We will provide an overview of this direction of study, and conclude with some novel results related to the obtainability of a given measure as a freezing state, as well as the fact that the collection of potentials that freeze is dense in $C(X)$ under certain conditions on the dynamical system.

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On the existence of freezing phase transitions for lattice systems — Tamara Kucherenko

We establish the existence of freezing phase transitions in the settings of multi-dimensional shift spaces. Precisely, given an arbitrary proper subshift $X$ of a d-dimensional shift space we explicitly construct a continuous potential $\varphi$ such that for all $\beta$ above some critical value ${\beta }_c$ the equilibrium states of $\beta \varphi$ are the measures of maximal entropy of $X$, whereas for $\beta$ below ${\beta }_c$ no equilibrium state of $\beta \varphi$ is supported on $X$. This phenomenon is referred to as a freezing phase transition for potential $\varphi$ with the motivation stemming from quasicrystal models in statistical physics. To contrast this result we establish sufficient conditions on the potential which guaranty that the system never freezes. This is a joint work with J.-R. Chazottes and A. Quas.

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Percolation Theory and the Diversity of Cellular Automata on Groups — Felipe García-Ramos

We will explain a connection between the diversity of cellular automata observable on a given countable group and the percolation threshold associated with the Cayley graphs of such groups. As a consequence, we show that Gilman's dichotomy holds for the endomorphism semigroup of a countable group if and only if the group is locally virtually cyclic.

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Phase transitions in the Potts model on Cayley tree. — Diyath Pannipitiya

The Ising model is one of the most important theoretical models in statistical physics, which was originally developed to describe ferromagnetism. A system of magnetic particles, for example, can be modeled as a linear chain in one dimension or a lattice in two dimensions, with one particle at each lattice point. Then each particle is assigned a spin ${\sigma }_i\in \{\pm 1\}$. The $q$-state Potts model is a generalization of the Ising model where each spin ${\sigma }_i$ may take on $q\ge 3$ number of states $\{0,\cdots ,q-1\}$. Both models have temperature $T$ and an externally applied magnetic field $h$ as parameters. Many statistical and physical properties of the $q$-state Potts model can be derived by studying its partition function. This includes phase transitions as $T$ and/or $h$ are varied. The celebrated Lee-Yang Theorem characterizes such phase transitions of the $2$-state Potts model (the Ising model). This theorem does not hold for $q>2$. Thus, phase transitions for the Potts model as $h$ is varied are more complicated and mysterious. We give some results that characterize the phase transitions of the $3$-state Potts model as $h$ is varied for constant $T$ on the binary rooted Cayley tree. Similarly to the Ising model, we show that for fixed $T>0$ the $3$-state Potts model for the ferromagnetic case exhibits a phase transition at one critical value of $h$ or not at all, depending on $T$. However, an interesting new phenomenon occurs for the $3$-state Potts model because the critical value of $h$ can be non-zero for some range of temperatures. The $3$-state Potts model for the antiferromagnetic case exhibits phase transition at up to two critical values of $h$. The recursive constructions of the $(n+1{)}^{st}$ level Cayley tree from two copies of the $n^{th}$ level Cayley tree allow one to write a relatively simple rational function relating the Lee-Yang zeros at one level to the next. This allows us to use techniques from dynamical systems.

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Sets of pointwise recurrence and answers to some questions of Host, Kra, and Maass — Anh Le

A subset of the positive integers is dynamically central syndetic if it contains the times of return of a point to a neighborhood of itself in a minimal dynamical system. This class of syndetic sets forms an important bridge between dynamics and combinatorics. We show that a set is dynamically central syndetic if and only if it is a member of a syndetic, idempotent filter. We elaborate on the consequences of this characterization for the dual family: sets of pointwise recurrence. For example, we provide several combinatorial characterizations of sets of pointwise recurrence, show that these sets do not have the Ramsey property, and they are sets of multiple recurrence. These results answer several questions asked by Host, Kra, and Maass. This talk is based on an joint work with Daniel Glasscock (University of Massachusetts Lowell).

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Specification and $\omega$-chaos in non-compact systems — Jonathan Meddaugh

We demonstrate conditions under which a dynamical system on a Lindelöf space exhibits $\omega$-chaos. In particular, we show that a system which satisfies a generalized version of the specification property and which contains at least three mutually separated orbit closures exhibits dense $\omega$-chaos.

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The Variational Principle for Entropy of Countable State Shift Spaces With Specification — Alex Paschal

We define and discuss specification properties for countable state shift spaces, which are special cases of definitions from an upcoming paper by Climenhaga, Thompson, and Wang and generalize the well-studied compact specification property to non-compact shift spaces. We present an infinite class of examples of such shift spaces and prove the variational principle for these spaces.

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The stabilized automorphism group of minimal systems — Jennifer N. Jones-Baro

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. In this talk, we will describe the stabilized automorphism group of minimal systems. The main result we will prove is that if two minimal systems have isomorphic stabilized automorphism groups and each has at least one non-trivial rational eigenvalue, then the systems have the same rational eigenvalues.

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