Stability index for chaotically driven concave maps

We study skew product systems driven by a hyperbolic base map Ŝ : Θ → Θ (for example, a baker map or an Anosov surface diffeomorphism) and with simple concave fibre maps on R+ like x → ĝ(θ) arctan(x), where θ ∈ Θ is a parameter driven by the base map. The fibrewise attractor is the graph of an upper semicontinuous function θ → φ̂∞(θ) ∈ R+. For many choices of ĝ, φ̂∞ has a residual set of zeros, but φ̂∞ > 0 μSRB-almost surely, where μSRB is the Sinai–Ruelle–Bowen measure of Ŝ−1. In such situations, we evaluate the stability index of the global attractor of the system, which is the subgraph {(θ, x) ∈ Θ× R+ : 0 x φ̂∞(θ)} of φ̂∞, at all regular points (θ, 0) in terms of the local exponents Γ̂(θ) := limn→∞(1/n) log ĝn(θ) and Λ̂(θ) := limn→∞(1/n) log |DuŜ−n(θ)| and of the positive zero s∗ of a certain thermodynamic pressure function associated with Ŝ and ĝ. (In queuing theory, an analogue of s∗ is known as Loynes’ exponent [R. M. Loynes, ‘The stability of a queue with non-independent inter-arrival and service times’, Proc. Cambridge Philos. Soc. 58 (1962) 497–520].) The stability index was introduced by Podvigina and Ashwin [‘On local attraction properties and a stability index for heteroclinic connections’, Nonlinearity 24 (2011) 887–929.] to quantify the local scaling of basins of attraction.