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Finitely represented closed-orbit subdynamics for commuting automorphisms

Published online by Cambridge University Press:  13 October 2009

RICHARD MILES
RICHARD MILES*
Affiliation:
Department of Mathematics, KTH, SE-100 44 Stockholm, Sweden (email: ricmiles@math.kth.se)
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Abstract

The purpose of this paper is to exhibit highly structured subdynamics for a class of non-expansive algebraic ℤd-actions based on the closed orbits of elements of an action. This is done using dynamical Dirichlet series to encode orbit counts. It is shown that there is a distinguished group homomorphism from ℤd onto a finite abelian group that controls the form of the Dirichlet series of elements of an action and that these series have common analytic properties. Corresponding orbit growth asymptotics are subsequently investigated.

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Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 30 , Issue 6 , December 2010 , pp. 1787 - 1802
Copyright
Copyright © Cambridge University Press 2009

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References

[1]Artin, M. and Mazur, B.. On periodic points. Ann. of Math. (2) 81 (1965), 82–99.CrossRefGoogle Scholar
[2]Boyle, M. and Lind, D.. Expansive subdynamics. Trans. Amer. Math. Soc. 349(1) (1997), 55–102.CrossRefGoogle Scholar
[3]Cohn, P. M.. Algebraic Numbers and Algebraic Functions (Chapman and Hall Mathematics Series). Chapman & Hall, London, 1991.CrossRefGoogle Scholar
[4]Einsiedler, M. and Lind, D.. Algebraic ℤd-actions of entropy rank one. Trans. Amer. Math. Soc. 356(5) (2004), 1799–1831 (electronic).Google Scholar
[5]Einsiedler, M., Lind, D., Miles, R. and Ward, T.. Expansive subdynamics for algebraic ℤd-actions. Ergod. Th. & Dynam. Sys. 21(6) (2001), 1695–1729.CrossRefGoogle Scholar
[6]Everest, G., Miles, R., Stevens, S. and Ward, T.. Dirichlet series for finite combinatorial rank dynamics. Trans. Amer. Math. Soc. Published electronically, July 30, 2009, DOI: 10.1090/S0002-9947-09-04962-9.CrossRefGoogle Scholar
[7]Everest, G., Miles, R., Stevens, S. and Ward, T.. Orbit-counting in non-hyperbolic dynamical systems. J. Reine Angew. Math. 608 (2007), 155–182.Google Scholar
[8]Everest, G., Stangoe, V. and Ward, T.. Orbit counting with an isometric direction. Algebraic and Topological Dynamics (Contemporary Mathematics, 385). Eds. Kolyada, S., Manin, Y. and Ward, T.. American Mathematical Society, Providence, RI, 2005, pp. 293–302.Google Scholar
[9]Kechris, A. S.. On the classification problem for rank 2 torsion-free abelian groups. J. London Math. Soc. (2) 62(2) (2000), 437–450.Google Scholar
[10]Margulis, G. A.. On Some Aspects of the Theory of Anosov Systems (Springer Monographs in Mathematics. With a survey by R. Sharp: Periodic orbits of hyperbolic flows. Translated from the Russian by Valentina Vladimirovna Szulikowska). Springer, Berlin, 2004.CrossRefGoogle Scholar
[11]Miles, R.. Zeta functions for elements of entropy rank-one actions. Ergod. Th. & Dynam. Sys. 27(2) (2007), 567–582.Google Scholar
[12]Miles, R.. Periodic points of endomorphisms on solenoids and related groups. Bull. Lond. Math. Soc. 40(4) (2008), 696–704.CrossRefGoogle Scholar
[13]Parry, W.. An analogue of the prime number theorem for closed orbits of shifts of finite type and their suspensions. Israel J. Math. 45(1) (1983), 41–52.Google Scholar
[14]Parry, W. and Pollicott, M.. An analogue of the prime number theorem for closed orbits of Axiom A flows. Ann. of Math. (2) 118(3) (1983), 573–591.CrossRefGoogle Scholar
[15]Ruelle, D.. Dynamical zeta functions and transfer operators. Notices Amer. Math. Soc. 49(8) (2002), 887–895.Google Scholar
[16]Schmidt, K.. Dynamical Systems of Algebraic Origin (Progress in Mathematics, 128). Birkhäuser, Basel, 1995.Google Scholar
[17]Schmidt, K.. Algebra, arithmetic and multi-parameter ergodic theory. Internat. Math. News 200 (2005), 1–21.Google Scholar
[18]Stanley, R. P.. Enumerative Combinatorics. Vol. 1 (Cambridge Studies in Advanced Mathematics. With a foreword by Gian-Carlo Rota. Corrected reprint of the 1986 original, 49). Cambridge University Press, Cambridge, 1997.CrossRefGoogle Scholar
[19]Yu, K. R.. Linear forms in p-adic logarithms. II. Compositio Math. 74(1) (1990), 15–113.Google Scholar