Assouad type dimensions of infinitely generated self-conformal sets
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.
Citation
Banaji , A & Fraser , J 2024 , ' Assouad type dimensions of infinitely generated self-conformal sets ' , Nonlinearity , vol. 37 , no. 4 , 045004 . https://doi.org/10.1088/1361-6544/ad2864
Publication
Nonlinearity
Status
Peer reviewed
ISSN
0951-7715Type
Journal article
Description
Funding: Both authors were financially supported by a Leverhulme Trust Research Project Grant (RPG-2019-034). JMF was also supported by an EPSRC Standard Grant (EP/R015104/1) and an RSE Sabbatical Research Grant (70249). Most of this work was completed while AB was JMF’s PhD student at the University of St Andrews, but AB was also supported by an EPSRC New Investigators Award (EP/W003880/1) while a postdoc at Loughborough University.Collections
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