Transition space for the continuity of the Lyapunov exponent of quasiperiodic Schrödinger cocycles

Abstract

We construct discontinuous point of the Lyapunov exponent of quasiperiodic Schrödinger cocycles in the Gevrey space G s with s > 2. In contrast, the Lyapunov exponent has been proved to be continuous in the Gevrey space G s with s < 2 [17,37]. This shows that G 2 is the transition space for the continuity of the Lyapunov exponent.

Quantitative reducibility of Gevrey quasi-periodic cocycles and its applications

“…Bourgain-Jitomirskaya [17] showed that the Lyapunov exponent L(α, A) : T × C ω (T, SL(2, C)) → R is jointly continuous for irrational α, and this result can be partially extended to ν-Gevrey topology with 1/2 < ν < 1. Recently, Ge-Wang-You-Zhao [27] showed that there exists a continuous-discontinuous transition for the Lyapunov exponent in the ν-Gevrey space at ν = 1/2.…”