Orbit-counting in Non-hyperbolic Dynamical Systems
نویسندگان
چکیده
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 orbitcounting 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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