# Automorphisms with exotic orbit growth

Stephan Baier; Sawian Jaidee; Shaun Stevens; Thomas Ward

Acta Arithmetica (2013)

- Volume: 158, Issue: 2, page 173-197
- ISSN: 0065-1036

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topStephan 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 -

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