Orbit-counting in non-hyperbolic dynamical systems

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

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
• Public URL: https://durham-repository.worktribe.com/output/1502557