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On a generalized self-similarity in the \(p\)-adic field. (English) Zbl 1357.28012

Summary: In the present paper, we introduce a new set which defines a generalized self-similar set for contractive functions \(\{fi\}^N_{i=1}\) on the unit ball \(\mathbb{Z}_p\) of \(p\)-adic numbers. This set is called unconventional limit set. We prove that the unconventional limit set is compact, perfect and uniformly disconnected. Moreover, we provide an example of two contractions for which the corresponding unconventional limiting set is quasi-symmetrically equivalent to the symbolic Cantor set.

MSC:

28A80 Fractals
11S82 Non-Archimedean dynamical systems
37P25 Dynamical systems over finite ground fields
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