Constant slope, entropy, and horseshoes for a map on a tame graph

Abstract

We study continuous countably (strictly) monotone maps defined on a tame graph, i.e., a special Peano continuum for which the set containing branchpoints and endpoints has a countable closure. In our investigation we confine ourselves to the countable Markov case. We show a necessary and sufficient condition under which a locally eventually onto, countably Markov map f of a tame graph G is conjugate to a constant slope map g of a countably affine tame graph. In particular, we show that in the case of a Markov map f that corresponds to recurrent transition matrix, the condition is satisfied for constant slope e htop(f ) , where h top (f ) is the topological entropy of f . Moreover, we show that in our class the topological entropy h top (f ) is achievable through horseshoes of the map f . Classification: Primary 37E25; Secondary 37B40, 37B45.

Dendrites and measures with discrete spectrum

“…In this paper, we especially focus on dendrites in which E(X) has countable closure. These dendrites are a special case of a tame graph, as introduced in [5]. Note that when E(X) is countable, so also is B(X) ∪ E(X), because B(X) is countable in any dendrite and has accumulation points only in E(X).…”

“…Using Lemma 3.2, we know that there are uncountably many singleton sets J θ . Now a dendrite whose endpoint set has countable closure is always the union of a countable sequence of free arcs and a countable set, see [5,Theorem 2.2]. It follows that we can find θ with the singleton J θ in the interior of a free arc A in X.…”

“…Throughout the proof, we will use freely the following well-known property of -limit sets (e.g. [6]): if for fixed we write for , then and . In particular, if is uncountable, then so is each , and if contains a given fixed point, then each contains it as well.…”

Dendrites and measures with discrete spectrum

“…In this paper we especially focus on dendrites in which E(X) has countable closure. These dendrites are a special case of a tame graph, as introduced in [6]. Note that when E(X) is countable, so also is B(X) ∪ E(X), since B(X) is countable in any dendrite and has accumulation points only in E(X).…”

“…Using Lemma 3.2 we know that there are uncountably many singleton sets J θ . Now a dendrite whose endpoint set has countable closure is always the union of a countable sequence of free arcs and a countable set, see [6,Theorem 2.2]. It follows that we can find θ with the singleton J θ in the interior of a free arc A in X.…”