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Markov partitions and homology for $n/m$-solenoids

Published online by Cambridge University Press:  27 November 2015

NIGEL D. BURKE  and
IAN F. PUTNAM
NIGEL D. BURKE
Affiliation:
Department of Pure Mathematics, Cambridge University, Cambridge, CB3 OWB, UK email ndb35@cam.ac.uk
IAN F. PUTNAM
Affiliation:
Department of Mathematics and Statistics, University of Victoria, Victoria, BC, CanadaV8W 2Y2 email ifputnam@uvic.ca
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Abstract

Given a relatively prime pair of integers, $n\geq m>1$, there is associated a topological dynamical system which we refer to as an $n/m$-solenoid. It is also a Smale space, as defined by David Ruelle, meaning that it has local coordinates of contracting and expanding directions. In this case, these are locally products of the real and various $p$-adic numbers. In the special case, $m=2,n=3$ and for $n>3m$, we construct Markov partitions for such systems. The second author has developed a homology theory for Smale spaces and we compute this in these examples, using the given Markov partitions, for all values of $n\geq m>1$ and relatively prime.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 37 , Issue 3 , May 2017 , pp. 716 - 738
Copyright
© Cambridge University Press, 2015 

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