Automorphisms with exotic orbit growth
Volume 158 / 2013
Acta Arithmetica 158 (2013), 173-197
MSC: Primary 37C35; Secondary 37P35, 11N12.
DOI: 10.4064/aa158-2-5
Abstract
The dynamical Mertens' theorem describes asymptotics for the growth in the number of closed orbits in a dynamical system. We construct families of ergodic automorphisms of fixed entropy on compact connected groups with a continuum of growth rates on two different growth scales. This shows in particular that the space of all ergodic algebraic dynamical systems modulo the equivalence of shared orbit-growth asymptotics is not countable. In contrast, for the equivalence relation of measurable isomorphism or equal entropy it is not known if the quotient space is countable or uncountable (this problem is a manifestation of Lehmer's problem).
