Stability index for chaotically driven concave maps

TL;DR

This paper analyzes the stability index of attractors in skew product systems driven by hyperbolic maps with concave fiber maps, linking it to local exponents and thermodynamic pressure functions.

Contribution

It introduces a method to evaluate the stability index for systems with hyperbolic base maps and concave fiber maps, connecting it to local exponents and thermodynamic concepts.

Findings

The stability index can be expressed in terms of local exponents and the zero of a pressure function.

Residual zeros of the fiber graph occur for many functions g, with positive measure almost everywhere.

The approach generalizes the understanding of basin scaling in complex dynamical systems.

Abstract

We study skew product systems driven by a hyperbolic base map S (e.g. a baker map or an Anosov surface diffeomorphism) and with simple concave fibre maps on an interval [0,a] like h(x)=g(\theta) tanh(x) where g(\theta) is a factor driven by the base map. The fibre-wise attractor is the graph of an upper semicontinuous function \phi(\theta). For many choices of the function g, \phi has a residual set of zeros but \phi>0 almost everywhere w.r.t. the Sinai-Ruelle-Bowen measure of S^(-1). In such situations we evaluate the stability index of the global attractor of the system, which is the subgraph of \phi, at all regular points (\theta,0) in terms of the local exponents \Gamma(\theta):=\lim_{n\to\infty} 1/n log g_n(\theta) and \Lambda(\theta):=\lim_{n\to\infty} 1/n\log|D_u S^{-n}(\theta)| and of the positive zero s_* of a certain thermodynamic pressure function associated with S^(-1) and g. (In queuing theory, an analogon of s_* is known as Loyne's exponent.) The stability index was introduced by Podvigina and Ashwin in 2011 to quantify the local scaling of basins of attraction.