Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-03T00:11:36.716Z Has data issue: false hasContentIssue false

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

Published online by Cambridge University Press:  01 April 1998

T. B. WARD
Affiliation:
School of Mathematics, University of East Anglia, Norwich NR4 7TJ, UK (e-mail: [email protected])

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 $\exp(-\frac{1}{2}h_{\rm top})<1$. 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 $\exp(-h_{\rm top})<1$ 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.

Type
Research Article
Copyright
1998 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)