Conjugacy of local homeomorphisms via groupoids and C*-algebras

Abstract

We investigate dynamical systems consisting of a locally compact Hausdorff space equipped with a partially defined local homeomorphism. Important examples of such systems include self-covering maps, one-sided shifts of finite type and, more generally, the boundary-path spaces of directed and topological graphs. We characterize the topological conjugacy of these systems in terms of isomorphisms of their associated groupoids and C*-algebras. This significantly generalizes recent work of Matsumoto and of the second- and third-named authors.

Entropy of local homeomorphisms with applications to infinite alphabet shift spaces

“…}, which we denote by N. Deaconu-Renault systems, or singly generated dynamical systems, were introduced independently by Deaconu in [12] and by Renault in [28]. In this subsection, we recall some of its concepts and terminology, as introduced in [1], which in turn follows [9].…”

“…Let ε > 0 and choose J such that 1 2 J < ε. Let k > 0 be such that, for all 0 ≤ i ≤ J, there exists l i < k such that p ′ l i = p ′′ i .…”

“…We show that d X is uniformly continuous with respect to d 1 . Given ε > 0, let J > 0 be such that 1 2 J < ε. Use Lemma 7.11 and write each r(p ′ i ), 0 ≤ i ≤ J as r(p ′ i ) = ∪A i n i , with A i k i the distinguished set with ε(A i k i ) < ∞.…”

“…We show that Id X : (X, d X ) → (X, d) is uniformly continuous. Given ε > 0, choose k ∈ N such that 1 2 k < ε. Consider the set A = {p i : |p i | ≤ k}, which is finite.…”

“…Deaconu-Renault systems routinely appear as the key ingredient in the description of C*algebras as groupoid C*-algebras (and of algebras as Steinberg algebras) and their dynamical properties are often linked with properties of the associated algebras. For example, conjugacy of Deaconu-Renault systems is described in terms of isomorphisms of the associated C*-algebras in [1]. In our work, we describe, among other things, when our proposed metric entropy is an invariant for conjugacy of Deaconu-Renault systems.…”