Parabolic carpets

TL;DR

This paper introduces parabolic carpets, a new class of fractals generated by non-conformal, non-uniformly contracting iterated function systems, and analyzes their measure spectra and dimensions.

Contribution

It defines and studies parabolic carpets, extending self-affine carpets to include systems with parabolic fixed points, and computes their measure spectra and box dimensions.

Findings

Computed the $L^q$-spectrum of weak Gibbs measures on parabolic carpets.

Determined the box dimensions of the parabolic carpets.

Extended the theory of self-affine carpets to non-conformal, parabolic cases.

Abstract

We introduce and study a family of non-conformal and non-uniformly contracting iterated function systems. We refer to the attractors of such systems as parabolic carpets. Roughly speaking they may be thought of as nonlinear analogues of self-affine carpets which are allowed to have parabolic fixed points. We compute the $L^q$-spectrum of a class of weak Gibbs measures supported on parabolic carpets as well as the box dimensions of the carpet itself.