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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Synchronization points and associated dynamical invariants


Author: Richard Miles
Journal: Trans. Amer. Math. Soc. 365 (2013), 5503-5524
MSC (2010): Primary 37A35, 37B05, 37C25, 37C85, 37P30, 11G50, 11Z05
Published electronically: April 2, 2013
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Abstract: This paper introduces new invariants for multiparameter dynamical systems. This is done by counting the number of points whose orbits intersect at time $ n$ under simultaneous iteration of finitely many endomorphisms. We call these points synchronization points. The resulting sequences of counts together with generating functions and growth rates are subsequently investigated for homeomorphisms of compact metric spaces, toral automorphisms and compact abelian group epimorphisms. Synchronization points are also used to generate invariant measures and the distribution properties of these are analysed for the algebraic systems considered. Furthermore, these systems reveal strong connections between the new invariants and problems of active interest in number theory, relating to heights and greatest common divisors.


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Additional Information

Richard Miles
Affiliation: School of Mathematics, University of East Anglia, Norwich, NR4 7TJ, United Kingdom
Email: r.miles@uea.ac.uk

DOI: http://dx.doi.org/10.1090/S0002-9947-2013-05829-1
PII: S 0002-9947(2013)05829-1
Received by editor(s): March 13, 2011
Received by editor(s) in revised form: March 4, 2012
Published electronically: April 2, 2013
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.