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Automorphisms with exotic orbit growth

Baier, Stephan; Jaidee, Sawian; Stevens, Shaun; Ward, Thomas

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Authors

Stephan Baier

Sawian Jaidee

Shaun Stevens

Thomas Ward



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).

Citation

Baier, S., Jaidee, S., Stevens, S., & Ward, T. (2013). Automorphisms with exotic orbit growth. Acta Arithmetica, 158, 173-197. https://doi.org/10.4064/aa158-2-5

Journal Article Type Article
Publication Date Jan 1, 2013
Deposit Date Feb 28, 2013
Publicly Available Date Apr 17, 2013
Journal Acta Arithmetica
Print ISSN 0065-1036
Electronic ISSN 1730-6264
Publisher Instytut Matematyczny
Peer Reviewed Peer Reviewed
Volume 158
Pages 173-197
DOI https://doi.org/10.4064/aa158-2-5

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Copyright © 2013 IMPAN. All rights reserved.





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