Dynamical zeta functions for typical extensions of full shifts. (English) Zbl 0956.37016

The author considers the growth rate of the number of periodic points in a class of hyperbolic dynamical systems, which generalizes the shift on \(p\) symbols (\(p\) is a prime) in an arithmetical setting. It is shown that with respect to a probabilistic parametrization of valuation-theoretic significance, for all but two primes \(p\), the set of limit points of the growth rate is almost surely infinite, and the dynamical zeta function almost surely non-algebraic. The exclusion of two primes results from a connection with Artin’s conjecture on primitive roots. The minimalist presentation requires familiarity with the author’s previous work.


37C30 Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc.
37D99 Dynamical systems with hyperbolic behavior
11R45 Density theorems
37B99 Topological dynamics
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