Everest, G. and Miles, R. and Stevens, S. and Ward, T. (2007) 'Orbit-counting in non-hyperbolic dynamical systems.', Journal für die reine und angewandte Mathematik = Crelles journal., 2007 (608). pp. 155-182.
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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|Publisher Web site:||http://dx.doi.org/10.1515/CRELLE.2007.056|
|Record Created:||18 Oct 2012 15:50|
|Last Modified:||14 Dec 2012 12:08|
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