Stability index for chaotically driven concave maps

Gerhard Keller

Published 2012-09-11Version 1

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.