The stability index for dynamically defined Weierstrass functions

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.