Assouad type dimensions of infinitely generated self-conformal sets

TL;DR

This paper investigates the local geometric complexity of limit sets from infinite conformal iterated function systems, providing formulas and bounds for Assouad dimensions and spectra, with applications to continued fractions and parabolic attractors.

Contribution

It establishes a formula for Assouad dimension and sharp bounds for the Assouad spectrum for infinitely generated conformal sets under natural separation conditions.

Findings

Derived a formula for Assouad dimension of infinite conformal sets.

Established sharp bounds for the Assouad spectrum in terms of Hausdorff dimension.

Identified phase transitions in the Assouad spectrum behavior.

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.