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).
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-aa158-2-5,
author = {Stephan Baier and Sawian Jaidee and Shaun Stevens and Thomas Ward},
title = {Automorphisms with exotic orbit growth},
journal = {Acta Arithmetica},
volume = {161},
year = {2013},
pages = {173-197},
zbl = {1291.37031},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa158-2-5}
}
Stephan Baier; Sawian Jaidee; Shaun Stevens; Thomas Ward. Automorphisms with exotic orbit growth. Acta Arithmetica, Tome 161 (2013) pp. 173-197. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa158-2-5/