Orbit-counting in non-hyperbolic dynamical systems

There are well-known analogues 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. The proof of the dynamical analogue of Mertens' Theorem uses transcendence theory and Dirichlet characters.