Orbit-counting in non-hyperbolic dynamical systems

Details Orbit-counting in non-hyperbolic dynamical systems Orbit-counting in non-hyperbolic dynamical systems

2005-11-22

arxiv.org/abs/math/0511569v1

Journal f\~A1/4r die reine und angewandte Mathematik, 608, 155-182, 2007

Mathematics

Dynamical Systems

Scientific paper

10.1515/CRELLE.2007.056

There are well-known analogs of the prime number theorem and Mertens' theorem for dynamical systems with hyperbolic behaviour. Here we consider the same question for the simplest non-hyperbolic algebraic systems. The asymptotic behaviour of the orbit-counting function is governed by a rotation on an associated compact group, and in simple examples we exhibit uncountably many different asymptotic growth rates for the orbit-counting function. Mertens' Theorem also holds in this setting, with an explicit rational leading coefficient obtained from arithmetic properties of the non-hyperbolic eigendirections.

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