ACTA MATHEMATICA UNIVERSITATIS COMENIANAE

Vol. 65,   1   (1996)
pp.   149-157

RESCALING OF MARKOV SHIFTS
T. WARD

Abstract.  Given a $\Bbb Z^d$ topological Markov shift $\Sigma$ and a $d\times d$ integer matrix $M$ with $\det(M)\neq 0$, we introduce the $M$-rescaling of $\Sigma$, denoted $\Sigma^(M)$. We show that some (internal) power of the $\Bbb Z^d$-action on $\Sigma^(M)$ is isomorphic to some (Cartesian, or external) power of $\Sigma$, and deduce that the two Markov shifts have the same topological entropy. Several examples from the theory of group automorphisms are discussed. Full shifts in any dimension are shown to be invariant under rescaling, and the problem of whether the reverse is true is interpreted as a higher-dimensional analogue of William's problem.

AMS subject classification.  58F03
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