Collapse of the Lyapunov spectrum for Perron-Frobenius operator cocycles

Cecilia González-Tokman, Anthony Quas

Published 2018-06-22Version 1

In this paper, we study random Blaschke products, acting on the unit circle and consider the cocycle of Perron-Frobenius operators acting on Banach spaces of analytic functions on an annulus. We completely describe the Lyapunov spectrum of these cocycles. We then consider a perturbation in which the Perron-Frobenius operators are composed with a convolution with a normal distribution with small variance (this corresponds to replacing a map by an annealed version where the image point is moved by a random normal amount). Combining the above, we obtain a very natural cocycle, where the unperturbed cocycle has infinitely many distinct Lyapunov exponents, but arbitrarily small natural perturbations cause a complete collapse of the Lyapunov spectrum except for the exponent 0 associated with the absolutely continuous invariant measure. The phenomenon is superficially similar to the finite-dimensional phenomenon, discovered by Bochi, where a generic area-preserving diffeomorphism is either hyperbolic, or has all zero Lyapunov exponents. In this paper, however, the cocycle and its perturbation are explicitly described; and further, the mechanism for collapse is quite different.