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Automorphisms with exotic orbit growth

Volume 158 / 2013

Stephan Baier, Sawian Jaidee, Shaun Stevens, Thomas Ward 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).

Authors

  • Stephan BaierSchool of Mathematics
    University of East Anglia
    Norwich NR4 7TJ, UK
    e-mail
  • Sawian JaideeDepartment of Mathematics
    123 Mittraphab Road
    Khon Kaen 40002, Thailand
    e-mail
  • Shaun StevensSchool of Mathematics
    University of East Anglia
    Norwich NR4 7TJ, UK
    e-mail
  • Thomas WardDepartment of Mathematical Sciences
    Durham University
    Durham DH1 3LE, UK
    e-mail

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