Automorphisms with exotic orbit growth

Stephan Baier, et al. "Automorphisms with exotic orbit growth." Acta Arithmetica 158.2 (2013): 173-197. <http://eudml.org/doc/279638>.

@article{StephanBaier2013,
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).},
author = {Stephan Baier, Sawian Jaidee, Shaun Stevens, Thomas Ward},
journal = {Acta Arithmetica},
keywords = {orbit growth; Mertens theorem; algebraic dynamics; thin set of primes},
language = {eng},
number = {2},
pages = {173-197},
title = {Automorphisms with exotic orbit growth},
url = {http://eudml.org/doc/279638},
volume = {158},
year = {2013},
}

TY - JOUR
AU - Stephan Baier
AU - Sawian Jaidee
AU - Shaun Stevens
AU - Thomas Ward
TI - Automorphisms with exotic orbit growth
JO - Acta Arithmetica
PY - 2013
VL - 158
IS - 2
SP - 173
EP - 197
AB - 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).
LA - eng
KW - orbit growth; Mertens theorem; algebraic dynamics; thin set of primes
UR - http://eudml.org/doc/279638
ER -