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Numeration systems as dynamical systems – introduction. (English) Zbl 1122.37011

Denteneer, Dee (ed.) et al., Dynamics and stochastics. Festschrift in honor of M. S. Keane. Selected papers based on the presentations at the conference ‘Dynamical systems, probability theory, and statistical mechanics’, Eindhoven, The Netherlands, January 3–7, 2005, on the occasion of the 65th birthday of Mike S. Keane. Beachwood, OH: IMS, Institute of Mathematical Statistics (ISBN 0-940600-64-1/pbk). Institute of Mathematical Statistics Lecture Notes - Monograph Series 48, 198-211 (2006).
This is a short introduction to a longer treatment by the author [available from the author’s website http://www.14.plala.or.jp/kamae/] of a class of numeration systems arising from substitutions (which includes the important family of \(\beta\)-expansions for algebraic \(\beta>1\)). The emphasis is on the compactification of the reals corresponding to a numeration system rather than properties of the numeration system as a way to expand real numbers. Thus a ‘numeration system with \(G\)’ (where \(G\) is a closed subgroup of \(\mathbb R_{>0}\)) is a compact metric space \(\Omega\) carrying an action \((\lambda,t):\omega\mapsto\lambda\omega+t\) of \(G\times\mathbb R\) with the property that the translation action of \(t\) is uniquely ergodic, with unique measure attaining the topological entropy \(|\log\lambda|\) of the action \(\omega\mapsto\lambda\omega\) for any \(\lambda\neq1\). Such numeration systems are constructed for weighted substitutions, and an associated zeta function is determined explicitly.
For the entire collection see [Zbl 1113.60008].

MSC:

37B10 Symbolic dynamics
11A63 Radix representation; digital problems
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