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Orbit-counting in non-hyperbolic dynamical systems

Everest, G.; Miles, R.; Stevens, S.; Ward, T.

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Authors

G. Everest

R. Miles

S. Stevens

T. Ward



Abstract

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.

Citation

Everest, G., Miles, R., Stevens, S., & Ward, T. (2007). Orbit-counting in non-hyperbolic dynamical systems. Journal für die reine und angewandte Mathematik, 2007(608), 155-182. https://doi.org/10.1515/crelle.2007.056

Journal Article Type Article
Publication Date Jan 1, 2007
Deposit Date Oct 12, 2012
Publicly Available Date Dec 14, 2012
Journal Journal für die reine und angewandte Mathematik
Print ISSN 0075-4102
Electronic ISSN 1435-5345
Publisher De Gruyter
Peer Reviewed Peer Reviewed
Volume 2007
Issue 608
Pages 155-182
DOI https://doi.org/10.1515/crelle.2007.056

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