Regularity and irregularity of fiber dimension of non-autonomous dynamical systems

TL;DR

This paper investigates the fractal properties and stability of Julia sets in non-autonomous rational dynamical systems, establishing conditions for stability and exploring the irregularity of dimension functions.

Contribution

It provides a necessary and sufficient condition for holomorphic stability and analyzes the regularity of Julia set dimensions under perturbations.

Findings

Holomorphic stability characterized by a specific condition.

Hölder continuity of hyperbolic Julia set dimensions.

Dimension functions are not differentiable at any point in certain families.

Abstract

This note concerns non-autonomous dynamics of rational functions and, more precisely, the fractal behavior of the Julia sets under perturbation of non-autonomous systems. We provide a necessary and sufficient condition for holomorphic stability which leads to H\"older continuity of dimensions of hyperbolic non-autonomous Julia sets with respect to the $l^\infty$-topology on the parameter space. On the other hand we show that, for some particular family, the Hausdorff and packing dimension functions are not differentiable at any point and that these dimensions are not equal on an open dense set of the parameter space still with respect to the $l^\infty$-topology.