Abstract
In the study of aperiodic order via dynamical methods, topological entropy is an important concept. In this paper, parts of the theory, like Bowen’s formula for fibre wise entropy or the independence of the definition from the choice of a Van Hove sequence, are extended to actions of several non-discrete groups. To establish these results, we will show that the Ornstein–Weiss lemma is valid for all considered groups which appear in the study of cut and project schemes.
Highlights
Aperiodic order, an intermediate concept between order and disorder, has attracted a lot of attention over the last three decades in the fields of physics, geometry, number theory and harmonic analysis [1,2,4,7,8,9,44,49]
The construction of aperiodic point sets via cut and project schemes was pioneered by Yves Meyer in his famous monograph on ”Algebraic numbers and harmonic analysis”
We present that in a similar way one obtains the relative topological entropy of an action as the scaled entropy of the restricted action to certain model sets and in particular any uniform lattice
Summary
An intermediate concept between order and disorder, has attracted a lot of attention over the last three decades in the fields of physics, geometry, number theory and harmonic analysis [1,2,4,7,8,9,44,49]. We present that in a similar way one obtains the relative topological entropy of an action as the scaled entropy of the restricted action to certain model sets and in particular any uniform lattice This allows to transfer several Theorems proven for discrete amenable groups to our context. As the mentioned topology in the construction of Delone dynamical systems is naturally defined via the notion of a uniformity [7,50] and the corresponding arguments are well known [33,41,43,56,57,58,60,61], we took the freedom to follow an idea from [19,30,41,59] and use the language of uniformities in order to prove the results for actions on compact Hausdorff spaces.
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