×

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

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.

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.
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] Chothi V., Math. 489 pp 99– (1997)
[2] Denef J., Progr. Math. 59 pp 25– (1985)
[3] Everest G. R., Cont. Math. 385 pp 293– (2005)
[4] DOI: 10.1017/S0143385700009573 · Zbl 0507.58034
[5] Noorani Salmi Md., Bull. Malays. Math. Soc. 22 pp 127– (2)
[6] DOI: 10.1007/BF02760669 · Zbl 0552.28020
[7] Parry W., Ann. Math. 118 pp 573– (2) · Zbl 0537.58038
[8] DOI: 10.2307/2159634 · Zbl 0756.58040
[9] Sharp R., S.) 21 pp 205– (1991)
[10] DOI: 10.1007/BF01297343 · Zbl 0737.28008
[11] DOI: 10.1112/S0024609397003330 · Zbl 0903.22001
[12] DOI: 10.1017/S0143385798113378 · Zbl 0915.58081
[13] DOI: 10.1090/S0002-9939-04-07626-9 · Zbl 1052.37017
[14] DOI: 10.1007/BF01692444 · JFM 24.0176.02
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.