Orbit Growth of Shift Spaces Induced by Bouquet Graphs and Dyck Shifts

For a discrete dynamical system, the prime orbit and Mertens’ orbit counting functions describe the growth of its closed orbits in a certain way. The asymptotic behaviours of these counting functions can be determined via Artin–Mazur zeta function of the system. Specifically, the existence of a non-vanishing meromorphic extension of the zeta function leads to certain asymptotic results. In this paper, we prove the asymptotic behaviours of the counting functions for a certain type of shift spaces induced by directed bouquet graphs and Dyck shifts. We call these shift spaces as the bouquet-Dyck shifts. Since their respective zeta function involves square roots of polynomials, the meromorphic extension is difficult to be obtained. To overcome this obstacle, we employ some theories on zeros of polynomials, including the well-known Eneström–Kakeya Theorem in complex analysis. Finally, the meromorphic extension will imply the desired asymptotic results.

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>

On generalized compensation functions for factor maps between shift spaces on countable alphabets

We show the existence of generalized compensation functions for a particular type of one-block factor maps [Formula: see text] between countable subshifts [Formula: see text] and [Formula: see text]. For factor maps between compact spaces, continuous compensation functions were studied by Walters in relation to the theory of relative pressure. Applying the thermodynamic formalism for sequences on countable subshifts, we generalize some existing results on factor maps between compact spaces to non-compact spaces. For related questions, we also study the existence of a preimage measure on [Formula: see text] of an invariant measure on [Formula: see text], and their relations.