Schedule for: 19w5093 - Symbolic Dynamical Systems

Many of the classical models of statistical physics, such as the Ising and Potts models, can be defined over any underlying finite graph. The case of a sparse, randomly-generated underlying graph has received considerable recent attention from probabilists, largely guided by several far-reaching predictions about its behaviour from the physics literature.

When the underlying graph is chosen uniformly at random from all $(2d)$-regular graphs on $n$ vertices, and then $n$ is sent to infinity, the local neighbourhoods around most vertices look like larger and larger trees with high probability. This observation allows one to extract weak limit processes over an infinite $(2d)$-regular tree from sequences of models built over the finite graphs. That infinite tree can be viewed as the Cayley graph of a free group, and the limit process becomes a probability-preserving action of that group on a shift-space.

This point of view is the basis for various asymptotic analyses of probabilistic features of the finite models, and also for the definition of sofic entropy for free-group actions in ergodic theory. This talk will be a gentle introduction to these two fields and the connections between them.

Given a countable amenable group $G$ one can ask which are the real numbers that can be realized as the topological entropy of a subshift of finite type (SFT). A famous result by Hochman and Meyerovitch completely characterizes these numbers for $\mathbb{Z}^2$. I will show that the same characterization is valid for any amenable group with decidable word problem which admits an action of $\mathbb{Z}^2$ which is free and bounded. Using this result we can give a full characterization of the entropies of SFTs for polycyclic groups. Furthermore, the same result holds for any countable group with decidable word problem which contains the direct product of any pair of infinite, finitely generated and amenable groups. In particular, it holds for many branch groups such as the Grigorchuk group.

Consider a translation-invariant process $X$ indexed by $\mathbb{Z}^d$. Suppose that $X$ is finitely-dependent in the sense that its restrictions to sets which are sufficiently separated (at least some fixed distance apart) are independent. Block factors of i.i.d. provide natural examples of such processes, and the question of whether all such $X$ are of this form was raised by Ibragimov and Linnik over 50 years ago. It took roughly 30 years until Burton, Goulet and Meester constructed an example which showed that this is not the case, that is, such an $X$ may not be a block factor of an i.i.d. process. On the other hand, we show that $X$ is a finitary factor of an i.i.d. process. This means that $X=F(Y)$ for some i.i.d. process $Y$ and some measurable map F which commutes with translations, and moreover, that in order to determine the value of $X_v$ for a given $v$, one only needs to look at a finite (but random) region of $Y$. The result extends to finitely-dependent processes indexed by the vertex set of any transitive amenable graph.

This talk is about topological and combinatorial properties of finite rank minimal systems. We establish a clear connection with the $S$-adic subshifts and provide necessary and sufficient conditions for a subshift to be of finite rank. Using these conditions we study the number of asympototic components of a finite rank subshift and show that there is a rank two subshift with non superlinear complexity. I will also mention results concerning the automorphism group of a finite rank subshift.

This is work in progress with Fabien Durand, Alejandro Maass and Samuel Petite.

A dynamical system is asymptotically nilpotent if every trajectory tends to some special zero point. When such convergence is necessarily uniform for some family of dynamical systems, we speak of nilrigidity. We discuss nilrigidity in particular in the context of (cold) dynamics of cellular automata and subactions of subshifts. Joint work with Ilkka Törmä.

It is well known that if $f$ is a Holder continuous function from a mixing shift of finite type $X$ to $\mathbb R$, then there exists a unique equilibrium state which is an invariant Gibbs measure having $f$ as a potential function. This result has been generalized to wider classes, such as when $X$ is a subshift with the specification property and $f$ is a function in the Bowen class. Recently Baker and Ghenciu showed that there exists a (non-invariant) Gibbs measures for the zero potential if and only if $X$ is (right-)balanced. We extend this result and show that a necessary and sufficient condition for the existence of invariant Gibbs measures on $X$ for the potential $0$ is the bi-balanced condition for $X$. We define a new condition, called $f$-balanced condition for the pair $(X,f)$ and present a similar result for the existence of Gibbs measure with respect to $f$. Using this result, we construct a class of shift spaces which have a Gibbs measure but do not have invariant Gibbs measures for the potential $0$, or equivalently, which are one-sided balanced but not bi-balanced, answering a question raised by Baker and Ghenciu.

The automorphism group $\text{Aut}(\sigma)$ of a subshift $(X,\sigma)$ consists of all homeomorphisms $\phi \colon X \to X$ such that $\phi \sigma = \sigma \phi$. When $(X,\sigma)$ is a shift of finite type, $\text{Aut}(\sigma)$ is known to have a rich group structure, and we'll discuss some background and problems related to the study of $\text{Aut}(\sigma)$. Finally, we'll discuss recent work with Yair Hartman and Bryna Kra in which we introduce a certain stabilized automorphism group, and outline results which, among other things, allow us to distinguish (up to isomorphism) the stabilized automorphism groups of various full shifts.​

Let $G$ be an amenable and virtually orderable finitely generated group—e.g., any group of polynomial growth—and let $X$ be a $G$-SFT having a safe symbol. In this talk I will give a simple condition sufficient for having a special representation of the topological entropy of such an $X$ and for having an efficient $\epsilon$-additive approximation of it. In addition, I will also explain the relation between these results and the existence of computational phase transitions.

For any group $G$ and set $A$, let $\text{CA}(G;A)$ be the monoid of all cellular automata over the configuration space $A^G$. In this talk, we present some algebraic results on $\text{CA}(G;A)$ when $G$ and $A$ are both finite. First, we show that any generating set of $\text{CA}(G;A)$ must have a cellular automaton with minimal memory set equal to $G$ itself. Second, we describe the structure of the group of units of $\text{CA}(G;A)$ in terms of a set of representatives of the conjugacy classes of subgroups of $G$. Third, we discuss the minimal cardinality of a generating set of $\text{CA}(G;A)$: in some cases we give it precisely, while in others we give some bounds. We apply this to provide a simple proof that $\text{CA}(G;A)$ is not finitely generated for various kinds of infinite groups $G$.

In this talk I'll discuss the notion of "invariant random orders", and explain how it can be useful in studying actions of countable groups. In particular, we'll formulate a unified "Kieffer-Pinsker formula" for the Kolmogorov-Sinai entropy of measure preserving actions of amenable groups, and show how it can be used to prove that a topologically predictable action of a countable amenable group has zero topological entropy, as conjectured by Hochman, and mention some related open problems. Based on joint work with Andrei Alpeev and Sieye Ryu.

In joint work with Ibrahim Salama, we study the complexity function $p_\tau(n)$ of a labeled tree or tree shift, which counts as a function of $n$ the number of different labelings of a shape of size $n$. We give a definition of entropy, prove that the limit in the definition exists, and that the limit is the infimum. For tree shifts determined by adjacency constraints a version of Pavlov's strip technique proves strict inequality with dimension and provides an efficient approximation method. Attractive questions concern equilibrium measures and relations with other kinds of entropy.

In this talk we will step back from the generality of multidimensional shifts of finite type and focus on two specific classes: The space of graph homomorphisms from $\mathbb Z^d$ to a fixed connected undirected graph and the space of tilings of $\mathbb Z^d$ by rectangular shapes. While these classes cover fairly well-studied examples like the space of proper 3-colourings and domino tilings, a lot still remains to be understood. We will discuss some recent results about these shift spaces and try to pinpoint the difficulties which arise in their investigation.

In 1999 R. Exel and M. Laca extended the construction of the Cuntz-Krieger algebras for infinite countable symbols and transitive matrices, they introduced a class of algebras which today are known as Exel-Laca algebras $O_A$, which are related to its respective countable Markov shift $\Sigma_A$, similarly to what happens to the Cuntz-Krieger algebras. Such construction gave birth a locally compact version of the space $\Sigma_A$, the space $X_A$, which in general contains the standard space $\Sigma_A$ as a dense subset, when $\Sigma_A$ is locally compact these two spaces coincide. On another hand, M. Denker, M. Urbański, O. Sarig, and many others developed the Thermodynamic Formalism on the standard countable Markov shifts $\Sigma_A$, this space, in general, is not locally compact. Despite a big success of Exel-Laca algebras in the Operator Algebra community, the measure-theoretic aspects of this generalization of the symbolic space and interaction with the dynamical system community were essentially zero until now. We obtained the first results about thermodynamic formalism (conformal and DLR measures, pressure, phase transitions, etc.) on the space $X_A$. We will give some geometric interpretation of $X_A$, results about the existence of conformal and DLR states on this new setting and we will answer the first natural question: If $\Sigma_A$ is a subset of $X_A$, what is the connection between the standard thermodynamic formalism on $\Sigma_A$ and the results on $X_A$? We will see that not only new phenomena appear (phase transitions) as well we can recover conformal measures living on $\Sigma_A$ as a limit of new conformal measures which are detected only in the new space $X_A$. The results are part of a project which is still in developing with T. Razseja (IME-USP), Ruy Exel (UFSC/University of Nebraska–Lincoln) and Rodrigo Frausino (IME-USP).

We discuss the automorphism group, i.e. the centralizer of the shift action inside the group of self-homeomorphisms of a subshift, together with the extended symmetry group (the corresponding normalizer) of certain $\mathbb Z^d$-subshifts with a hierarchical structure, like bijective substitutive subshifts and the Robinson tiling. This group has been previously studied in e.g. Michael Baake, John Roberts and Reem Yassawi's previous works, among others.

Treating those subshifts as geometrical objects, we introduce techniques to identify allowed extended symmetries from large-scale structures present in certain special points of the subshift, leading to strong restrictions on the group of extended symmetries. We prove that in the aforementioned cases, $\text{Sym}(X, \mathbb Z^d)$ (and thus $\text{Aut}(X, \mathbb Z^d)$) is virtually-$\mathbb Z^d$ and we explicitly represent the nontrivial extended symmetries, associated with the quotient $\text{Sym}(X, \mathbb Z^d)/\text{Aut}(X, \mathbb Z^d)$, as a subset of rigid transformations of the coordinate axes. We also show how our techniques carry over to the study of the Robinson tiling, both in its minimal and non-minimal version. We emphasize the geometric nature of these techniques and how they reflect the capability of extended symmetries to capture such properties in a subshift.