Orbit-counting in non-hyperbolic dynamical systems. (English) Zbl 1137.37006

Everest, Graham; Miles, Richard; Stevens, Shaun; Ward, Thomas B.

J. Reine Angew. Math. 608, 155-182 (2007).

This paper is concerned with analogues of the prime number theorem and Mertens’ theorem for dynamical systems. Let \(T: X\to X\) be a continuous map on a compact metric space. Define a closed orbit \(\tau\) of length \(|\tau|= n\) as the set \(\{x,T(x), T^2(x),\dots, T^n(x)= x\}\). Then one is interested in the asymptotic behaviour of expressions of the type \[ \pi_T(N)= |\{\tau: |\tau|\leq N\}|,\quad M_T(N)= \sum_{|\tau|\leq N} e^{-h(T)|\tau|}, \] where \(h(T)\) denotes the topological entropy of the map. The first problem can be seen as an analogous problem to the prime number theorem and the second to Mertens’ problem.
Some results are known, provided the dynamical system has hyperbolic behaviour. The purpose of this paper is to consider dynamical systems with non-hyperbolic behaviour. Some asymptotic representations for \(\pi_T(N)\) and \(M_T(N)\) are given.

Reviewer: Ekkehard Krätzel (Jena)

MSC:

37B99	Topological dynamics
37A45	Relations of ergodic theory with number theory and harmonic analysis (MSC2010)
11N05	Distribution of primes
37C30	Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc.

Keywords:

dynamical systems; dynamical zeta function; prime number theorem; Mertens’ theorem

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