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

Ward, T. (1998) 'Almost all S-integer dynamical systems have many periodic points.', Ergodic theory and dynamical systems., 18 (2). pp. 471-486.

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
Full text:PDF - Accepted Version (396Kb)
Full text:PDF - Published Version (111Kb)
Status:Peer-reviewed
Publisher Web site:http://dx.doi.org/10.1017/S0143385798113378
Publisher statement:© Copyright Cambridge University Press 1998. This paper has been published in a revised form subsequent to editorial input by Cambridge University Press in "Ergodic theory and dynamical systems" (18: 2 (2007) 471-486) http://journals.cambridge.org/action/displayJournal?jid=ETS
Record Created:12 Oct 2012 12:50
Last Modified:25 Oct 2012 11:55

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