On one-sided topological conjugacy of topological Markov shifts and gauge actions on Cuntz–Krieger algebras

We characterize topological conjugacy classes of one-sided topological Markov shifts in terms of the associated Cuntz–Krieger algebras and their gauge actions with potentials.

Cohomology groups, continuous full groups and continuous orbit equivalence of topological Markov shifts

<p style='text-indent:20px;'>We will study several subgroups of continuous full groups of one-sided topological Markov shifts from the view points of cohomology groups of full group actions on the shift spaces. We also study continuous orbit equivalence and strongly continuous orbit equivalence in terms of these subgroups of the continuous full groups and the cohomology groups.</p>

Fair measures for countable-to-one maps

In this paper, we generalize the recently introduced concept of fair measure [M. Misiurewicz and A. Rodrigues, Counting preimages, Ergod. Theor. Dyn. Syst. 38 (2018) 1837–1856]. We study fair measures for Markov and mixing interval maps with countably many branches. We investigate them in terms of the recurrence properties of some underlying countable Markov shifts, both from the stochastic viewpoint and from the viewpoint of thermodynamical formalism. Finally, we move beyond the interval and look for fair measures for graph maps.