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Isomorphism and bi-Lipschitz equivalence between the univoque sets. (English) Zbl 1452.37028

In this paper, the main attention is given to self-similar sets (with and without the open set condition) and to the bi-Lipschitz equivalence.
Some basic notions such as orbit sets, bi-Lipschitz equivalent metric spaces, quasi-Hölder equivalent metric spaces, quasi-Lipschitz equivalent metric spaces, Pisot numbers, and configuration sets are given. Several auxiliary lemmas are given. Also, the measure-theoretic isomorphism and the bi-Lipschitz equivalence between two sets are considered.
A uniform formula of the Hausdorff dimension of \(U_1\) is given and the existence of the unique measures of maximal entropy with respect to the lazy map for the closure of \(U_1\) and \(K\), where \(K\) is a self-similar set or an attractor for a certain iterated function system, is proven.
Finally, several auxiliary statements are proven and examples are given. The proofs of the main results are explained in detail. Final remarks are devoted to a brief description of open problems.

MSC:

37C45 Dimension theory of smooth dynamical systems
28A78 Hausdorff and packing measures
28A80 Fractals
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