Almost all $S$-integer dynamical systems have many periodic points

Ward, T. B. (1998) Almost all $S$-integer dynamical systems have many periodic points. Ergodic Theory and Dynamical Systems, 18 (2). pp. 1-16. ISSN 1469-4417

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Abstract

We show that for almost every ergodic S-integer dynamical system the radius of convergence of the dynamical zeta function is no larger than exp(-[1/2]htop) < 1. In the arithmetic case almost every zeta function is irrational. We conjecture that for almost every ergodic S-integer dynamical system the radius of convergence of the zeta function is exactly exp(-htop) < 1 and the zeta function is irrational. In an important geometric case (the S-integer systems corresponding to isometric extensions of the full p-shift or, more generally, linear algebraic cellular automata on the full p-shift) we show that the conjecture holds with the possible exception of at most two primes p. Finally, we explicitly describe the structure of S-integer dynamical systems as isometric extensions of (quasi-)hyperbolic dynamical systems.

Item Type:	Article
Faculty \ School:	Faculty of Science > School of Mathematics
Depositing User:	Vishal Gautam
Date Deposited:	18 Mar 2011 14:48
Last Modified:	16 Dec 2022 01:02
URI:	https://ueaeprints.uea.ac.uk/id/eprint/18603
DOI:	10.1017/S0143385798113378

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