The stability index for dynamically defined Weierstrass functions

TL;DR

This paper investigates the stability index of invariant graphs in skew-product dynamical systems with affine fiber maps, analyzing their local structure and multifractal properties.

Contribution

It introduces a method to compute the stability index for typical points and performs a multifractal analysis of its level sets in such systems.

Findings

Explicit calculation of the stability index at typical points

Identification of complex local structures of basins of attraction

Multifractal analysis of the stability index level sets

Abstract

Let $\hat{T} : X \times \mathbb{R} \to X \times \mathbb{R}$ given by $\hat{T}(x,t) = (Tx, g_x(t))$ be a skew-product dynamical system where $T : X \to X$ is a mixing conformal expanding map and, for each $x \in X$, $g_x : \mathbb{R} \to \mathbb{R}$ is an affine map of the form $g_x(t)=-f(x)+\lambda(x)^{-1}t$. Under a suitable contraction hypotheses on $\lambda$ there exists a measurable function $u: X \to \mathbb{R}$ such that $\text{graph}(u) = \{(x,u(x)) \mid x\in X\}$ is $\hat{T}$-invariant and divides $X \times \mathbb{R}$ into two regions, $\mathbb{B}^+$ and $\mathbb{B}^-$, consisting of points that are repelled under iteration by $\hat{T}$ to $\pm\infty$. These two regions act as basins of attraction to $\pm \infty$ in the sense of Milnor. The two basins have a complicated local structure: a neighbourhood of a point $(x,t) \in ^+$ will typically intersect $\mathbb{B}^-$ in a set of positive measure. The stability index (as introduced by Podvigina and Ashwin \cite{podviginaashwin:11} for general Milnor attractors) is the rate of polynomial decay of the measure of this intersection. We calculate the stability index at typical points in $X \times \mathbb{R}$. We also perform a multifractal analysis of the level sets of the stability index.