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Almost all \(S\)-integer dynamical systems have many periodic points. (English) Zbl 0915.58081

Summary: We show that for almost every ergodic \(S\)-integer dynamical system the radius of convergence of the dynamical zeta function is no longer than \(\exp(-{1\over 2} h_{\text{top}})< 1\). 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 \(\exp(-h_{\text{top}})< 1\) 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.

MSC:

37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics
37D99 Dynamical systems with hyperbolic behavior
37C30 Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc.
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