Stability index for chaotically driven concave maps

Abstract

We study skew product systems driven by a hyperbolic base map S ^ : Θ → Θ (for example, a baker map or an Anosov surface diffeomorphism) and with simple concave fibre maps on R + like x ↦ g ^ ( 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 ↦ φ ^ ∞ ( θ ) ∈ R + . For many choices of g ^ , φ ^ ∞ has a residual set of zeros, but φ ^ ∞ > 0 μ SRB -almost surely, where μ SRB is 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 { ( θ , x ) ∈ Θ × R + : 0 ⩽ x ⩽ φ ^ ∞ ( θ ) } of φ ^ ∞ , at all regular points ( θ , 0 ) in terms of the local exponents Γ ^ ( θ ) : = lim n → ∞ ( 1 / n ) log g ^ n ( θ ) and Λ ^ ( θ ) : = lim n → ∞ ( 1 / n ) log | D u S ^ - n ( θ ) | and of the positive zero s * of a certain thermodynamic pressure function associated with S ^ and g ^ . (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.