Continuous orbit equivalence of topological Markov shifts and dynamical zeta functions

Abstract

For continuously orbit equivalent one-sided topological Markov shifts $(X_A,{\sigma }_A)$ and $(X_B,{\sigma }_B)$, their eventually periodic points and cocycle functions are studied. As a result we directly construct an isomorphism between their ordered cohomology groups $({\bar{H}}^A,{\bar{H}}_{+}^A)$ and $({\bar{H}}^B,{\bar{H}}_{+}^B)$. We also show that the cocycle functions for the continuous orbit equivalences give rise to positive elements of the ordered cohomology, so that the the zeta functions of continuously orbit equivalent topological Markov shifts are related. The set of Borel measures is shown to be invariant under continuous orbit equivalence of one-sided topological Markov shifts.

25 pages

References in corpus (1)

• Homology and topological full groups of etale groupoids on totally disconnected spaces

Cited by in corpus (2)

• Flow equivalence and orbit equivalence for shifts of finite type and isomorphism of their groupoids
• C*-algebras, groupoids and covers of shift spaces