Almost all S-integer dynamical systems have many periodic points

Ward, T.

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.

Citation

Ward, T. (1998). Almost all S-integer dynamical systems have many periodic points. Ergodic Theory and Dynamical Systems, 18(2), 471-486. https://doi.org/10.1017/s0143385798113378

• Journal Article Type: Article
• Publication Date: Sep 1, 1998
• Deposit Date: Oct 12, 2012
• Publicly Available Date: Oct 24, 2012
• Journal: Ergodic Theory and Dynamical Systems
• Print ISSN: 0143-3857
• Electronic ISSN: 1469-4417
• Publisher: Cambridge University Press
• Peer Reviewed: Peer Reviewed
• Volume: 18
• Issue: 2
• Pages: 471-486
• DOI: https://doi.org/10.1017/s0143385798113378
• Public URL: https://durham-repository.worktribe.com/output/1502425