Mathematics > Dynamical Systems
[Submitted on 23 Jul 2022 (v1), last revised 13 Mar 2024 (this version, v2)]
Title:Assouad type dimensions of infinitely generated self-conformal sets
View PDF HTML (experimental)Abstract:We study the dimension theory of limit sets of iterated function systems consisting of a countably infinite number of conformal contractions. Our focus is on the Assouad type dimensions, which give information about the local structure of sets. Under natural separation conditions, we prove a formula for the Assouad dimension and prove sharp bounds for the Assouad spectrum in terms of the Hausdorff dimension of the limit set and dimensions of the set of fixed points of the contractions. The Assouad spectra of the family of examples which we use to show that the bounds are sharp display interesting behaviour, such as having two phase transitions. Our results apply in particular to sets of real or complex numbers which have continued fraction expansions with restricted entries, and to certain parabolic attractors.
Submission history
From: Amlan Banaji [view email][v1] Sat, 23 Jul 2022 21:43:24 UTC (33 KB)
[v2] Wed, 13 Mar 2024 16:39:52 UTC (59 KB)
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