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Almost all S-integer dynamical systems have many periodic points

Ward, T.

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Authors

T. Ward



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

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