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A zero-dimensional (resp. symbolic) flow is a suspension flow over a zero-dimensional system (resp. a subshift). We show that any topological flow admits a principal extension by a zero-dimensional flow. Following [Bur19] we deduce that any topological flow admits an extension by a symbolic flow if and only if its time-$t$ map admits an extension by a subshift for any $t eq 0$. Moreover the existence of such an extension is preserved under orbit equivalence for regular topological flows, but this property does not hold more true for singular flows. Finally we investigate symbolic extensions for singular suspension flows. In particular, the suspension flow over the full shift on ${0,1}^{mathbb Z}$ with a roof function $f$ vanishing at the zero sequence $0^infty$ admits a principal symbolic extension or not depending on the smoothness of $f$ at $0^infty$.
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We construct symbolic dynamics on sets of full measure (w.r.t. an ergodic measure of positive entropy) for $C^{1+epsilon}$ flows on compact smooth three-dimensional manifolds. One consequence is that the geodesic flow on the unit tangent bundle of a compact $C^infty$ surface has at least const $times(e^{hT}/T)$ simple closed orbits of period less than $T$, whenever the topological entropy $h$ is positive -- and without further assumptions on the curvature.
Symbolic Extension Entropy Theorem (SEET) describes the possibility of a lossless digitalization of a dynamical system by extending it to a subshift. It gives an estimate on the entropy of symbolic extensions (and the necessary number of symbols). Unlike in the measure-theoretic case, where Kolmogorov--Sinai entropy is the estimate, in the topological setup the task reaches beyond the classical theory of entropy. Tools from an extended theory of entropy structures are needed. The main goal of this paper is to prove the SEET for actions of countable amenable groups: Let a countable amenable group $G$ act by homeomorphisms on a compact metric space $X$ and let $mathcal M_G(X)$ denote the simplex of $G$-invariant probability measures on $X$. A function $E $ on $mathcal M_G(X)$ equals the extension entropy function $h^pi$ of a symbolic extension $pi:(Y,G)to (X,G)$, where $h^pi(mu)=sup{h_ u(Y,G): uinpi^{-1}(mu)}$ ($muinmathcal M_G(X)$), if and only if $E $ is an affine superenvelope of the entropy structure of $(X,G)$. The statement is preceded by presentation of the concepts of an entropy structure and superenvelopes, adapted from $mathbb Z$-actions. In full generality we prove a slightly weaker version of SEET, in which symbolic extensions are replaced by quasi-symbolic extensions, i.e., extensions in form of a joining of a subshift with a zero-entropy tiling system. The notion of a tiling system is a subject of earlier works and in this paper we review and complement the theory developed there. The full version of the SEET is proved for groups which are either residually finite or enjoy the comparison property. In order to describe the range of our theorem, we devote a large portion of the paper to the comparison property. Our main result in this aspect shows that all subexponential groups have the comparison property (and thus satisfy the SEET).
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.
In this survey we will present the symbolic extension theory in topological dynamics, which was built over the past twenty years.
Given a piecewise $C^{1+beta}$ map of the interval, possibly with critical points and discontinuities, we construct a symbolic model for invariant probability measures with nonuniform expansion that do not approach the critical points and discontinuities exponentially fast almost surely. More specifically, we code the lift of these measures in the natural extension of the map.
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