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Automorphisms with exotic orbit growth. (English) Zbl 1291.37031

This paper deals with the problem of determining the possible growth rates of the number of periodic orbits of a certain class of dynamical systems of algebraic origin: the ergodic automorphisms of compact connected groups. Unlike for continuous maps on compact metric spaces, these algebraic systems show considerable rigidity, which limits the range of dynamical properties that can vary continuously. The extent to which the growth rate of periodic orbits is determined by the topological entropy of a system, is quantified by a dynamical analogue of Mertens’ theorem in number theory. Specifically, one is interested in the large-period asymptotics of a certain function \(M_T=M_T(N)\), where \(T\) is the mapping and \(N\) is the period. The authors construct ergodic automorphisms \(T\) of one-dimensional compact metric groups, all with the same topological entropy, which exhibit a continuum of growth rates with respect to two different parametrisations, namely \(M_T(N)\sim \kappa\log(N)\) with \(\kappa\in (0,1)\), and \(M_T(N)\sim k\log(N)^\delta\), with \(\delta\in (0,1)\) and \(k>0\).

MSC:

37C35 Orbit growth in dynamical systems
37P35 Arithmetic properties of periodic points
11N13 Primes in congruence classes
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