Geodesic
Automorphisms with exotic orbit growth
Stephan Baier1 ; Sawian Jaidee2 ; Shaun Stevens1 ; Thomas Ward3
1 School of Mathematics University of East Anglia Norwich NR4 7TJ, UK
2 Department of Mathematics 123 Mittraphab Road Khon Kaen 40002, Thailand
3 Department of Mathematical Sciences Durham University Durham DH1 3LE, UK
Acta Arithmetica, Tome 158 (2013) no. 2, pp. 173-197

Voir la notice de l'article provenant de la source Institute of Mathematics Polish Academy of Sciences

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).
DOI : 10.4064/aa158-2-5
Keywords: dynamical mertens theorem describes asymptotics growth number closed orbits dynamical system construct families ergodic automorphisms fixed entropy compact connected groups continuum growth rates different growth scales shows particular space ergodic algebraic dynamical systems modulo equivalence shared orbit growth asymptotics countable contrast equivalence relation measurable isomorphism equal entropy known quotient space countable uncountable problem manifestation lehmers problem

Stephan Baier 1 ; Sawian Jaidee 2 ; Shaun Stevens 1 ; Thomas Ward 3

1 School of Mathematics University of East Anglia Norwich NR4 7TJ, UK
2 Department of Mathematics 123 Mittraphab Road Khon Kaen 40002, Thailand
3 Department of Mathematical Sciences Durham University Durham DH1 3LE, UK 
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Stephan Baier; Sawian Jaidee; Shaun Stevens; Thomas Ward. Automorphisms with exotic orbit growth. Acta Arithmetica, Tome 158 (2013) no. 2, pp. 173-197. doi: 10.4064/aa158-2-5

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