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

  • NIGEL D. BURKE (a1) and IAN F. PUTNAM (a2)

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.

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