Stability and collapse of Oseledets spectrum for Perron-Frobenius cocycles


It is known, by work of Bochi, Ma~n'e, Viana and others that Lyapunov exponents are highly sensitive to perturbations of a dynamical system. On the other hand, work of Ledrappier, Young and my work with Froyland and Gonz'alez-Tokman has shown that in some situations, under “noise-like” perturbations, Lyapunov exponents vary continuously. We are particularly interested in cocycles of Perron-Frobenius operators, as the Lyapunov exponents (and the corresponding Oseledets spaces) are related to rates of mixing (and the spaces can identify obstructions to mixing). We discuss a test case of a random composition of Blaschke products, and their Perron-Frobenius operators acting on a Hardy space of analytic functions. These operators are known to be compact. We identify the full Lyapunov spectrum of these systems, and give necessary and sufficient conditions for the stability of the spectrum. [Joint work with CeciliaGonz'alez-Tokman.]

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