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Assouad type dimensions of infinitely generated self-conformal sets

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posted on 2024-03-08, 15:31 authored by Amlan BanajiAmlan Banaji, Jonathan M Fraser

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.

Funding

Leverhulme Trust Research Project Grant (RPG-2019-034)

Fourier analytic techniques in geometry and analysis

Engineering and Physical Sciences Research Council

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RSE Sabbatical Research Grant (70249)

Overlapping iterated function systems: New approaches and breaking the super-exponential barrier

Engineering and Physical Sciences Research Council

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History

School

  • Science

Department

  • Mathematical Sciences

Published in

Nonlinearity

Volume

37

Issue

4

Publisher

IOP Publishing

Version

  • VoR (Version of Record)

Rights holder

© IOP Publishing Ltd & London Mathematical Society

Publisher statement

Original Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence (https://creativecommons.org/licenses/by/3.0/). Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.

Acceptance date

2024-02-12

Publication date

2024-02-29

Copyright date

2024

ISSN

0951-7715

eISSN

1361-6544

Language

  • en

Depositor

Amlan Banaji. Deposit date: 12 February 2024

Article number

045004

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