The paper is concerned with maximal subgroups of the ample (better known as topological full) groups of homeomorphisms of totally disconnected compact metrizable topological spaces. We describe all maximal subgroups that are stabilizers of finite sets. Under certain assumptions on the ample group (including minimality), we describe all maximal subgroups that are stabilizers of closed sets or stabilizers of partitions into clopen sets. In particular, our results apply to the ample groups associated with Cantor minimal systems and to some Higman-Thompson groups.

We consider the substitution subshift generated by the Thue-Morse substitution 0→01, 1→10. We prove that the topological full group of the subshift contains a subgroup of intermediate growth. Namely, one group from the family known as the Grigorchuk groups embeds into this group. To obtain our main result, we prove an embedding theorem for topological full groups, and also develop a technique to prove isomorphism of groups using the Schreier graphs.

A finite, connected, (d+1)-regular graph G is called Ramanujan if every its eigenvalue λ satisfies either λ=± (d+1) or |λ|≤ 2√(d). The Ramanujan condition corresponds to the optimal rate of decay of correlations for the associated non-backtracking edge subshift. We consider a higher-dimensional generalization of this observation. We introduce the notion of a d-regular ℤ^δ-subshift of finite type, and we define a Ramanujan subshift as a d-regular ℤ^δ-subshift with an optimal rate of decay of correlations. We show that for every odd prime power q≥ 3 and dimension δ<q, there exists a q-regular Ramanujan ℤ^δ-subshift. The construction is based on the quaternionic lattices over 𝔽_q(t) introduced by Rungtanapirom-Stix-Vdovina (2019). Each of our q-regular Ramanujan subshifts gives rise to a family of non-bipartite (q+1)-regular Ramanujan graphs. These graphs are very explicit and local in the strong sense: the neighbors of any vertex can be computed by an explicit Mealy automaton associated with the subshift. As a byproduct, for every odd prime power q, we get a single lifting rule that can be iterated to produce an infinite family of (q+1)-regular Ramanujan graphs.

We explain how certain tools from convex analysis and probability theory may be used in order to obtain counting results for the number of words with prescribed frequencies of letters in regular languages.

We canonically identify the groups of isometries and dilations of local fields and their rings of integers with subgroups of the automorphism group of the $(d+1)$-regular tree $\widetilde T_{d+1}$, where $d$ is the residual degree. Then we introduce the class of liftable self-similar groups acting on a $d$-regular rooted tree whose ascending HNN extensions act faithfully and vertex transitively on $\widetilde T_{d+1}$ fixing one of the ends. The closures of these extensions in $\mathrm{Aut}(\widetilde T_{d+1})$ are totally disconnected locally compact group that belong to the class of scale groups. We give numerous examples of liftable groups coming from self-similar groups acting essentially freely or groups admitting finite $L$-presentations. In particular, we show that the finitely presented group constructed by the first author and the finitely presented HNN extension of the Basilica group embed into the group $\mathcal D(\mathbb Q_2)$ of dilations of the field $\mathbb Q_2$ of $2$-adic numbers. These actions, translated to $\widetilde T_3$, are 2-transitive on the punctured boundary of $\widetilde T_3$. Also we explore scale-invariant groups with the purpose of getting new examples of scale groups.

We study symbolic dynamical representations of actions of the first Grigorchuk group G, namely its action on the boundary of the infinite rooted binary tree, its representation in the topological full group of a minimal substitutive ℤ-shift, and its representation as a minimal system of Schreier graphs. We show that the first system admits an SFT cover, and the latter two systems are conjugate to sofic subshifts on G, but are not of finite type.

We deal with a multivariate growth series $\Gamma_L(\mathbf{z})$, $\mathbf{z} \in \mathbb{C}^d$, associated with a regular language $L$ over an alphabet of cardinality $d \geq 2$. Our focus is on languages coming from subgroups of the free group $F_m$ of finite rank $m$ and from the subshifts of finite type. We suggest a tool for computing the rate of growth $\varphi_L(\mathbf{r})$ of $L$ in the direction $\mathbf{r} \in \mathbb{R}^d$. Using the concave growth condition introduced by the second author in [Comment. Math. Helv. 2002, 77 (3), 563-608] and the results of Convex Analysis we represent $\psi_L(\mathbf{r}) = \log\left(\varphi_L(\mathbf{r})\right)$ as a support function of a convex set that is the closure of the $\textrm{Relog}$ image of the domain of absolute convergence of $\Gamma_L(\mathbf{z})$. This allows us to compute $\psi_L(\mathbf{r})$ in some cases, including a Fibonacci language or a language of freely reduced words representing elements of a free group $F_2$. Also we show that the methods of the Large Deviation Theory can be used as an alternative approach.

We show that the second rational map $G$ associated with the group $\mathcal{G}$ of intermediate growth constructed by the first author in 1980 is semiconjugate with the antiholomorphic map $z\rightarrow\bar{z}^2$. For doing this we use a family of $G$--invariant curves found by the third author and for each invariant curve, create a Markov partition of it.

Motivated by the study of profinite topology in branch groups, we prove a structural result about their finitely generated subgroups. More precisely, we show that finitely generated subgroups of a branch group with the subgroup induction property have a block structure, which roughly means that, up to a finite index, they are products of finite index subgroups, embedded in the group in a way that is coherent with its branch action on the rooted tree.

J. S. Wilson proved in 1971 an isomorphism between the structural lattice associated to a group belonging to his second class of groups with every proper quotient finite and the Boolean algebra of clopen subsets of Cantor's ternary set. In this paper we generalize this isomorphism to the class of branch groups. Moreover, we show that for every faithful branch action of a group $G$ on a spherically homogeneous rooted tree $T$ there is a canonical $G$-equivariant isomorphism between the Boolean algebra associated with the structure lattice of $G$ and the Boolean algebra of clopen subsets of the boundary of $T$.

A one-by-one exhaustion is a combinatorial/geometric condition which excludes eigenvalues from the spectra of Laplace and Schr\"odinger operators on graphs. Isoperimetric inequalities in graphs with a cocompact automorphism group provide an upper bound on the von Neumann dimension of the space of eigenfunctions. Any finitely generated indicable amenable group has a Cayley graph without eigenvalues. There exists a finitely generated group G with finite generating sets S and S' such that the adjacency operator of the Cayley graph of (G,S) has no eigenvalue while the adjacency operator of the Cayley graph of (G,S') has pure point spectrum.

Using the construction by Bencs and Tóth of invariant random subgroups on weakly branch groups acting on regular rooted trees we produce uncountably many indecomposable characters on these groups. In fact, we study three types of characters coming from the action of a weakly branch group on a regular tree, paying attention to their similarities and differences. We use obtained results to show that each countable amenable branch group has uncountably many pairwise not quasi-equivalent embeddings into Murray-von Neumann hyperfinite factor. For the canonical character associated with a self-similar group and studied by the second author as a self-similar trace we provide a number of examples when it is explicitly computed.

AbstractThe ring$\mathbb Z_{d}$ofd-adic integers has a natural interpretation as the boundary of a rootedd-ary tree$T_{d}$. Endomorphisms of this tree (that is, solenoidal maps) are in one-to-one correspondence with 1-Lipschitz mappings from$\mathbb Z_{d}$to itself. In the case when$d=p$is prime, Anashin [‘Automata finiteness criterion in terms of van der Put series of automata functions’,p-Adic Numbers Ultrametric Anal. Appl.4(2) (2012), 151–160] showed that$f\in \mathrm {Lip}^{1}(\mathbb Z_{p})$is defined by a finite Mealy automaton if and only if the reduced coefficients of its van der Put series constitute ap-automatic sequence over a finite subset of$\mathbb Z_{p}\cap \mathbb Q$. We generalize this result to arbitrary integers$d\geq 2$and describe the explicit connection between the Moore automaton producing such a sequence and the Mealy automaton inducing the corresponding endomorphism of a rooted tree. We also produce two algorithms converting one automaton to the other andvice versa. As a demonstration, we apply our algorithms to the Thue–Morse sequence and to one of the generators of the lamplighter group acting on the binary rooted tree.

In this paper, we explore the spectral measures of the Laplacian on Schreier graphs for several self-similar groups (the Grigorchuk, Lamplighter, and Hanoi groups) from the dynamical and algebro-geometric viewpoints. For these graphs, classical Schur renormalization transformations act on appropriate spectral parameters as rational maps in two variables. We show that the spectra in question can be interpreted as asymptotic distributions of slices by a line of iterated pullbacks of certain algebraic curves under the corresponding rational maps (leading us to a notion of a spectral current ). We follow up with a dynamical criterion for discreteness of the spectrum. In case of atomic spectrum, the precise rate of convergence of finite-scale approximands to the limiting spectral measure is given. For the three groups under consideration, the corresponding rational maps happen to be fibered over polynomials in one variable. We reveal the algebro-geometric nature of this integrability phenomenon.

In this article we study spectral properties of the family of Schreier graphs associated to the action of the Thompson group F F on the interval [ 0 , 1 ] [0,1] . In particular, we describe spectra of Laplace type operators associated to these Schreier graphs and calculate certain spectral measures associated to the Schreier graph Υ \Upsilon of the orbit of 1 / 2 1/2 . As a byproduct we calculate the asymptotics of the return probabilities of the simple random walk on Υ \Upsilon starting at 1 / 2 1/2 . In addition, given a Laplace type operator L L on a tree-like graph we study relations between the spectral measures of L L associated to delta functions of different vertices and the spectrum of L L .

We introduce a class of subshifts governed by finitely many two-sided infinite words. We call these words leading sequences. We show that any locally constant cocycle over such a subshift is uniform. From this we obtain Cantor spectrum of Lebesgue measure zero for associated Jacobi operators if the subshift is aperiodic. Our class covers all simple Toeplitz subshifts as well as all Sturmian subshifts. We apply our results to the spectral theory of Schreier graphs for uncountable families of groups acting on rooted trees.

Anatolii Mikhailovich Stepin, a prominent scientist and educator, expert in dynamical systems and ergodic theory, passed away on 7 November 2020. His death was a bitter loss for his family, students, colleagues, and mathematicians at large. He was born in Moscow on 20 July 1940. During the war he was evacuated to Chelyabinsk together with his mother, while his father, a chemical engineer, worked at a defense factory. After the war their family returned to Moscow. Anatolii graduated from Moscow School no. 434 with a gold medal and, intending to follow in his father’s footsteps, enrolled at the Moscow Power Engineering Institute, the Division of Thermal Physics. At that time M. I. Vishik, who was well known for his work on partial differential equations, read mathematics courses at the Institute. The talented student became interested in these lectures, and after his 3rd year at Power Engineering Institute, he decided to transfer to the Faculty of Mechanics and Mathematics at Moscow State University. After solving the tricky problem of being transferred to another university, Stepin became a student at the Department of Theory of Functions and Functional Analysis, and chose F.A. Berezin to be his scientific advisor. In addition to Berezin’s research seminar, Stepin was also

We introduce the notion of the automatic logarithm LogA(B) of a finite initial Mealy automaton B, with another automaton A as the base. It allows one to find for any input word w a power n such that B(w)=An(w). The purpose is to study the expanding properties of graphs describing the action of the group generated by A and B on input words of a fixed length interpreted as levels of a regular d-ary rooted tree T. Formally, the automatic logarithm is a single map LogA(B):∂T→Zd from the boundary of the tree to the d-adic integers. Under the assumption that theaction of the automaton A on the tree T is level-transitive andof bounded activity, we show that LogA(B) can be computed bya Moore machine. The distribution of values of the automatic logarithm yields a probabilistic measure μ on ∂T, which in some cases can be computed by a Mealy-type machine (we then say that μ is finite-state). We provide a criterion to determine whether μ is finite-state. A number of examples with A being the adding machine are considered.