本文展示了如何用符号动力系统对全不连通局部紧群的任意自同构进行建模。该模型是有限符号上的全移位与两类系统之一的乘积的逆极限。一类是具有每点皆为周期的置换的可数离散空间，另一类是本质游荡的可数状态 Markov 移位。所用的部分思想来自动力学，部分来自全不连通局部紧群的研究。后者涉及尺度函数与整洁子群。文中讨论了这些思想与动力学思想之间的联系。可以看到，仅本质游荡的可数状态 Markov 移位影响尺度函数。最后，证明了相对于 Haar 测度的传递性或遍历性意味着该系统没有可数离散或本质游荡分量。
英文摘要：
It is shown how to model any automorphism of a totally disconnected, locally compact group by a symbolic dynamical system. The model is an inverse limit of a product of a full-shift, on a finite number of symbols, with one of two types of systems. One is a countable discrete space with a permutation having every point periodic and the other is an essentially wandering, countable state Markov shift. Some of the ideas used are from dynamics and some from the study of totally disconnected, locally compact groups. The later ideas concern the scale function and tidy subgroups. There is a discussion of the connections between those ideas and the dynamical ideas. It is seen that only the essentially wandering, countable state Markov shift affects the scale function. Finally, it's shown that transitivity or ergodicity with respect to Haar measure implies that the system has no countable discrete or essentially wandering component.
We show that if the complexity difference function p(n+1)-p(n) of a infinite minimal shift is bounded, then the the automorphism group of the one-sided shift is finite, and the automorphism group of the corresponding two-sided shift "modulo the shift" is finite. For the minimal Sturmian and minimal substitution shifts, the bounds can be explicitly computed, and for linearly recurrent shifts, the bound can be expressed as a function of the linear recurrence constant. We also show that any endomorphism of a linearly recurrent shift is a root of a power of the shift map.
If $α$ is an irreducible nonexpansive ergodic automorphism of a compact abelian group $X$ (such as an irreducible nonhyperbolic ergodic toral automorphism), then $α$ has no finite or infinite state Markov partitions, and there are no nontrivial continuous embeddings of Markov shifts in $X$. In spite of this we are able to construct a symbolic space $V$ and a class of shift-invariant probability measures on $V$ each of which corresponds to an $α$-invariant probability measure on $X$. Moreover, every $α$-invariant probability measure on $X$ arises essentially in this way. The last part of the pa…
A probability measure is a characteristic measure of a topological dynamical system if it is invariant to the automorphism group of the system. We show that zero entropy shifts always admit characteristic measures. We use similar techniques to show that automorphism groups of minimal zero entropy shifts are sofic.
The group of automorphisms of a symbolic dynamical system is countable, but often very large. For example, for a mixing subshift of finite type, the automorphism group contains isomorphic copies of the free group on two generators and the direct sum of countably many copies of $\mathbb{Z}$. In contrast, the group of automorphisms of a symbolic system of zero entropy seems to be highly constrained. Our main result is that the automorphism group of any minimal subshift of stretched exponential growth with exponent $<1/2$, is amenable (as a countable discrete group). For shifts of polynomial grow…
The notions of symbolic matrix system and $λ$-graph system for a subshift are generalizations of symbolic matrix and $λ$-graph (= finite symbolic matrix) for a sofic shift respectively ([Doc. Math. 4(1999), 285-340]). M. Nasu introduced the notion of textile system for a pair of graph homomorphisms to study automorphisms and endomorphisms of topological Markov shifts ([Mem. Amer. Math. Soc. 546,114(1995)]). In this paper, we formulate textile systems on $λ$-graph systems and study automorphisms on subshifts. We will prove that for a forward automorphism $ϕ$ of a subshift $(Λ,σ)$, the automorph…
We associate a homomorphism in the Heisenberg group to each hyperbolic unimodular automorphism of the free group on two generators. We show that the first return-time of some flows in "good" sections, are conjugate to niltranslations, which have the property of being self-induced.
This survey gives an account of an algebraic construction of symbolic covers and representations of ergodic automorphisms of compact abelian groups, initiated by A.M. Vershik around 1992 for hyperbolic automorphisms of finite-dimensional tori. The key ingredient in this approach, which was subsequently extended to arbitrary expansive automorphisms of compact abelian groups, is the use of homoclinic points of the automorphism. Although existence and abundance of homoclinic points is intimately connected to expansiveness of the automorphism, it is nevertheless possible to extend certain aspects …
We show that any connected locally compact group which admits an expansive automorphism is nilpotent. We also show that for any locally compact group $G$, $α\in {\rm Aut}(G)$ is expansive if and only if for any $α$-invariant closed subgroup $H$ which is either compact or normal, the restriction of $α$ to $H$ is expansive and the quotient map on $G/H$ corresponding to $α$ is expansive. We get a structure theorem for locally compact groups admitting expansive automorphisms. We prove that an automorphism on a non-discrete locally compact group can not be both distal and expansive.
We present a strategy for a geometric construction of cross sections for the geodesic flow on locally symmetric orbifolds of rank one. We work it out in detail for $Γ\backslash H$, where $H$ is the upper half plane and $Γ=\PGamma_0(p)$, $p$ prime. Its associated discrete dynamical system naturally induces a symbolic dynamics on $\R$. The transfer operator produced from this symbolic dynamics has a particularly simple structure.
The decimal expansion real numbers, familiar to us all, has a dramatic generalization to representation of dynamical system orbits by symbolic sequences. The natural way to associate a symbolic sequence with an orbit is to track its history through a partition. But in order to get a useful symbolism, one needs to construct a partition with special properties. In this work we develop a general theory of representing dynamical systems by symbolic systems by means of so-called Markov partitions. We apply the results to one of the more tractable examples: namely hyperbolic automorphisms of the two…
We study automorphisms $α$ of a totally disconnected, locally compact group $G$ which are expansive in the sense that, for some identity neighbourhood $U$, the sets $α^n(U)$ (for integers $n$) intersect in the trivial group. Notably, we prove that the automorphism induced by $α$ on $G/N$ for an $α$-stable closed normal subgroup $N$ of $G$ is always expansive. Further results involve the associated contraction groups $U_α$ consisting of all $x$ in $G$ such that $α^n(x) \to e$ as $n$ tends to infinity. If $α$ is expansive, then $W := U_αU_{α^{-1}}$ is an open identity neighbourhood in $G$. We gi…
We provide special cross sections for the Weyl chamber flow on a sample class of Riemannian locally symmetric spaces of higher rank, namely the direct product spaces of Schottky surfaces. We further present multi-parameter transfer operator families for the discrete dynamical systems on Furstenberg boundary that are related to these cross sections.
We study the arithmetic codings of hyperbolic automorphisms of the 2-torus, i.e. the continuous mappings acting from a certain symbolic space of sequences with a finite alphabet endowed with an appropriate structure of additive group onto the torus which preserve this structure and turn the two-sided shift into a given automorphism of the torus. This group is uniquely defined by an automorphism, and such an arithmetic coding is a homomorphism of that group onto the 2-torus. The necessary and sufficient condition of the existence of a bijective arithmetic coding is obtained; it is formulated in…
For a locally compact metrizable group $G$, we consider the action of ${\rm Aut}(G)$ on ${\rm Sub}_G$, the space of all closed subgroups of $G$ endowed with the Chabauty topology. We study the structure of groups $G$ admitting automorphisms $T$ which act expansively on ${\rm Sub}_G$. We show that such a group $G$ is necessarily totally disconnected, $T$ is expansive and that the contraction groups of $T$ and $T^{-1}$ are closed and their product is open in $G$; moreover, if $G$ is compact, then $G$ is finite. We also obtain the structure of the contraction group of such $T$. For the class of g…
We prove that for any transitive subshift $X$ with word complexity function $c_n(X)$, if $\liminf \frac{\log (c_n(X)/n)}{\log \log \log n} = 0$, then the quotient group $\textrm{Aut}(X,σ) / \langle σ\rangle$ of the automorphism group of $X$ by the subgroup generated by the shift $σ$ is locally finite. We prove that significantly weaker upper bounds on $c_n(X)$ imply the same conclusion if the Gap Conjecture from geometric group theory is true. Our proofs rely on a general upper bound for the number of automorphisms of $X$ of range $n$ in terms of word complexity, which may be of independent in…
To a given Pisot unit $β$ we associate a finite abelian group whose size appears to be equal to the discriminant of $β$. We call it the Pisot group and find its representation in the two-sided $β$-compactum in the case of $β$ satisfying the relation $Fin(β)=\Bbb Z[β]\cap[0,1)$. As a motivation for the definition, we show that the Pisot group is the kernel of some important arithmetic coding of the toral automorphism given by the companion matrix naturally associated with $β$.
Let S be an ergodic measure-preserving automorphism on a non-atomic probability space, and let T be the time-one map of a topologically weak mixing suspension flow over an irreducible subshift of finite type under a Holder ceiling function. We show that if the measure-theoretic entropy of S is strictly less than the topological entropy of T, then there exists an embedding from the measure-preserving automorphism into the suspension flow. As a corollary of this result and the symbolic dynamics for geodesic flows on compact surfaces of negative curvature developed by Bowen and Ratner, we also ob…
We consider the actions of (semi)groups on a locally compact group by automorphisms. We show the equivalence of distality and pointwise distality for the actions of a certain class of groups. We also show that a compactly generated locally compact group of polynomial growth has a compact normal subgroup $K$ such that $G/K$ is distal and the conjugacy action of $G$ on $K$ is ergodic; moreover, if $G$ itself is (pointwise) distal then $G$ is Lie projective. We prove a decomposition theorem for contraction groups of an automorphism under certain conditions. We give a necessary and sufficient cond…
We consider 1-dimensional, unimodular Pisot substitution tilings with three intervals, and discuss conditions under which pairs of such tilings are locally isomorhphic (LI), or mutually locally derivable (MDL). For this purpose, we regard the substitutions as homomorphisms of the underlying free group with three generators. Then, if two substitutions are conjugated by an inner automorphism of the free group, the two tilings are LI, and a conjugating outer automorphism between two substitutions can often be used to prove that the two tilings are MLD. We present several examples illustrating the…
We show that the automorphism group of a one-dimensional full shift (the group of reversible cellular automata) does not satisfy the Tits alternative. That is, we construct a finitely-generated subgroup which is not virtually solvable yet does not contain a free group on two generators. We give constructions both in the two-sided case (spatially acting group $\Z$) and the one-sided case (spatially acting monoid $\N$, alphabet size at least eight). Lack of Tits alternative follows for several groups of symbolic (dynamical) origin: automorphism groups of two-sided one-dimensional uncountable sof…
This survey paper is aimed to describe a relatively new branch of symbolic dynamics which we call Arithmetic Dynamics. It deals with explicit arithmetic expansions of reals and vectors that have a "dynamical" sense. This means precisely that they (semi-) conjugate a given continuous (or measure-preserving) dynamical system and a symbolic one. The classes of dynamical systems and their codings considered in the paper involve: (1) Beta-expansions, i.e., the radix expansions in non-integer bases; (2) "Rotational" expansions which arise in the problem of encoding of irrational rotations of the cir…
Classical ergodic theory deals with measure (or measure class) preserving actions of locally compact groups on Lebesgue spaces. An important tool in this setting is a theorem of Mackey which provides spatial models for Boolean G-actions. We show that in full generality this theorem does not hold for actions of Polish groups. In particular there is no Borel model for the Polish automorphism group of a Gaussian measure. In fact, we show that this group as well as many other Polish groups do not admit any nontrivial Borel measure preserving actions.
Let $J$ be a semisimple Lie group with all simple factors of real rank at least two. Let $Γ<J$ be a lattice. We prove a very general local rigidity result about actions of $J$ or $Γ$. This shows that almost all so-called "standard actions" are locally rigid. As a special case, we see that any action of $Γ$ by toral automorphisms is locally rigid. More generally, given a manifold $M$ on which $Γ$ acts isometrically and a torus $\Ta^n$ on which it acts by automorphisms, we show that the diagonal action on $\Ta^n{\times}M$ is locally rigid. This paper is the culmination of a series of papers and …
In this paper we give a number of explicit constructions for II$_1$ factors and II$_1$ equivalence relations that have prescribed fundamental group and outer automorphism group. We construct factors and relations that have uncountable fundamental group different from $\IRpos$. In fact, given any II$_1$ equivalence relation, we construct a II$_1$ factor with the same fundamental group. Given any locally compact unimodular second countable group $G$, our construction gives a II$_1$ equivalence relation $\RelR$ whose outer automorphism group is $G$. The same construction does not give a II$_1$ fa…
Let $G$ be a group and let $X$ be an algebraic variety over an algebraically closed field $k$ of characteristic zero. Denote $A=X(k)$ the set of rational points of $X$. We investigate invertible algebraic cellular automata $τ\colon A^G \to A^G$ whose local defining map is induced by some morphism of algebraic varieties $X^M \to X$ where $M$ is a finite memory. When $G$ is locally embeddable into finite groups (LEF), then we show that the inverses of reversible algebraic cellular automata are automatically algebraic cellular automata. Generalizations are also obtained for finite product Hopfian…
