On the Regularity of the Dimension of Cookie-Cutter-Like Sets

TL;DR

This paper investigates the Hausdorff and box-counting dimensions of cookie-cutter-like fractal sets generated by multiple expanding maps, revealing conditions under which their dimensions are extremal and exploring non-differentiability in parameter-dependent systems.

Contribution

It establishes that the dimensions of cookie-cutter-like sets are extremal under certain conditions and demonstrates non-differentiability in one-parameter families, supporting conjectures about spectral dimensions.

Findings

Dimensions are the min and max of individual maps' dimensions.

Non-differentiability of dimensions as functions of parameters.

Supports conjecture on spectral dimension non-differentiability.

Abstract

We study the Hausdorff and box-counting dimensions of cookie-cutter-like sets formed by sequential dynamics of a finite number of expanding maps. Under some natural conditions, these dimensions turn out to be the minimum and maximum of the corresponding dimensions of the cookie-cutter sets generated by the individual expanding maps. In the case of one-parameter families of such systems, this provides a simple mechanism for producing non-differentiable fractal dimensions as functions of the parameter. This supports a conjecture that the Hausdorff dimension of the spectrum of a Sturmian Hamiltonian, in general, does not have to be differentiable as a function of the coupling constant. This is in drastic contrast to the analytic dependence of the dimensions of such spectra with quadratic irrational frequencies, e.g. the Fibonacci Hamiltonian, previously shown by M. Pollicott.