We consider a family of isometric extensions of the full shift on p symbols (for p a prime) parametrized by a probability space. Using Heath-Brown's work on the Artin conjecture, it is shown that for all but two primes p the set of limit points of the growth rate of periodic points is infinite almost surely. This shows in particular that the dynamical zeta function is not algebraic almost surely.
We introduce a class of coded systems that we construct from sofic systems and Dyck shifts and we study a class of subshifts that we obtain by excluding words of length two from Dyck shifts. We derive expressions for zeta functions and topological entropy. We derive an expression for the zeta function of certain subshifts that we obtain by excluding words from Dyck shifts and of certain subshifts that we obtain by excluding words from the subshifts that are constructed from full shifts and Dyck shifts.
The Markov-Dyck shifts arise from finite directed graphs. An expression for the zeta function of a Markov-Dyck shift is given. The derivation of this expression is based on a formula in Keller (G. Keller, {\it Circular codes, loop counting, and zeta-functions}, J. Combinatorial Theory {\bf 56} (1991), pp. 75--83). For a class of examples that includes the Fibonacci-Dyck shift the zeta functions and topological entropy ae determined.
We discuss a method of calculating the zeta function of subshifts which have a presentation by a finite directed graph labeled by elements of the associated inverse semigroup. This class of subshifts is introduced as a class of property A subshifts(T.Hamachi, K.Inoue and W.Krieger/Subsystems of finite type and semigroup invariants of subshifts, preprint), and the Dyck shift and the Motzkin shift are representative subshifts in this class. The exact number of the periodic points and entropies are also given by this method, and these values are used in the embedding condition for an irreducible …
Given a real number $ β> 1$, we study the associated $ (-β)$-shift introduced by S. Ito and T. Sadahiro. We compares some aspects of the $(-β)$-shift to the $β$-shift. When the expansion in base $ -β$ of $ -\fracβ{β+1} $ is periodic with odd period or when $ β$ is strictly less than the golden ratio, the $ (-β)$-shift, as defined by S. Ito and T. Sadahiro cannot be coded because its language is not transitive. This intransitivity of words explains the existence of gaps in the interval. We observe that an intransitive word appears in the $(-β)$-expansion of a real number taken in the gap. Furth…
This article introduces and investigates the basic features of a dynamical zeta function for group actions, motivated by the classical dynamical zeta function of a single transformation. A product formula for the dynamical zeta function is established that highlights a crucial link between this function and the zeta function of the acting group. A variety of examples are explored, with a particular focus on full shifts and closely related variants. Amongst the examples, it is shown that there are infinitely many non-isomorphic virtually cyclic groups for which the full shift has a rational zet…
A decomposition theorem for the Lind zeta function of a reversal system $(X, T, R)$ of finite order is established. A reversal system can be regarded as an action of a certain group $G$ on $X$. To establish an explicit formula for the Lind zeta function of $(X, T, R)$, we need to consider finite index subgroups $H$ of $G$ with induced actions given by automorphisms or by flips. When the underlying dynamical system $(X, T)$ is either a shift of finite type or a sofic shift, we express the Lind zeta function of $(X, T, R)$ in terms of matrices.
We introduce notions of suspension and flow equivalence on one-sided topological Markov shifts, which we call one-sided suspension and one-sided flow equivalence, respectively. We prove that one-sided flow equivalence is equivalent to continuous orbit equivalence on one-sided topological Markov shifts. We also show that the zeta function of the flow on a one-sided suspension is a dynamical zeta function with some potential function and that the set of certain dynamical zeta functions is invariant under one-sided flow equivalence of topological Markov shifts.
Transfer operators and Ruelle zeta functions for super-continuous functions on one-sided topological Markov shifts are considered. For every super-continuous function, we construct a Banach space on which the associated transfer operator is compact. Using this Banach space, we establish the trace formula and spectral representation of Ruelle zeta functions for a certain class of super-continuous functions. Our results include, as a special case, the classical trace formula and spectral representation for the class of locally constant functions.
We describe a pair of invariants for actions of finite groups on shifts of finite type, the left-reduced and right-reduced shifts. The left-reduced shift was first constructed by U. Fiebig, who showed that its zeta function is an invariant, and in fact equal to the zeta function of the quotient dynamical system. We also give conditions for expansivity of the quotient, and applications to combinatorial group theory, knot theory and topological quantum field theory.
We introduce "puzzles of quasi-finite type" which are the counterparts of our subshifts of quasi-finite type (Invent. Math. 159 (2005)) in the setting of combinatorial puzzles as defined in complex dynamics. We are able to analyze these dynamics defined by entropy conditions rather completely, obtaining a complete classification with respect to large entropy measures and a description of their measures with maximum entropy and periodic orbits. These results can in particular be applied to entropy-expanding maps like (x,y)-->(1.8-x^2+sy,1.9-y^2+sx) for small s. We prove in particular the meromo…
We address the analysis of the following problem: given a real Hölder potential $f$ defined on the Bernoulli space and $μ_f$ its equilibrium state, it is known that this shift-invariant probability can be weakly approximated by probabilities in periodic orbits associated to certain zeta functions. Given a Hölder function $f>0$ and a value $s$ such that $0<s<1$, we can associate a shift-invariant probability $ν_{s}$ such that for each continuous function $k$ we have \[\int k dν_{s}=\frac{\sum_{n=1}^{\infty}\sum_{x\in Fix_{n}}e^{sf^{n}(x)-nP(f)}\frac{k^{n}(x)}{n}}{\sum_{n=1}^{\infty}\sum_{x\in F…
This paper extends and applies algebraic invariants and constructions for mixing finite group extensions of shifts of finite type. For a finite abelian group G, Parry showed how to define a G-extension S_A from a square matrix A over Z_+G, and classified the extensions up to topological conjugacy by the strong shift equivalence class of A over Z_+G. Parry asked in this case if the det(I-tA) (which captures the "periodic data" of the extension) would classify up to finitely many topological conjugacy classes the extensions by G of a fixed mixing shift of finite type. When the algebraic K-theory…
We study sets of nontypical points under the map $f_β\mapsto βx $ mod 1, for non-integer $β$ and extend our results from [Färm, Persson, Schmeling, 2010] in several directions. In particular we prove that sets of points whose forward orbit avoid certain Cantor sets, and set of points for which ergodic averages diverge, have large intersection properties. We observe that the technical condition $β>1.541$ found in [Färm, Persson, Schmeling, 2010] can be removed.
Let $S=\{s_{n}\}$ be an increasing finite or infinite subset of $\mathbb N \bigcup \{0\}$ and $X(S)$ the $S$-gap shift associated to $S$. Let $f_{S}(x)=1-\sum\frac{1}{x^{s_{n}+1}}$ be the entropy function which will be vanished at $2^{h(X(S))}$ where $h(X(S))$ is the entropy of the system. Suppose $X(S)$ is sofic with adjacency matrix $A$ and the characteristic polynomial $χ_{A}$. Then for some rational function $ Q_{S} $, $χ_{A}(x)=Q_{S}(x)f_{S}(x)$. This $ Q_{S} $ will be explicitly determined. We will show that $ζ(t)=\frac{1}{f_{S}(t^{-1})}$ or $ζ(t)=\frac{1}{(1-t)f_{S}(t^{-1})}$ when $|S|<…
In this paper we show that, for topological dynamical systems with a dense set (in the weak topology) of periodic measures, a typical (in Baire's sense) invariant measure has, for each $q>0$, zero lower $q$-generalized fractal dimension. This implies, in particular, that a typical invariant measure has zero upper Hausdorff dimension and zero lower rate of recurrence. Of special interest is the full-shift system $(X,T)$ (where $X= M^{\Z}$ is endowed with a sub-exponential metric and the alphabet $M$ is a perfect and compact metric space), for which we show that a typical invariant measure has, …
