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arXiv · 2208.06022

A dynamical Thouless formula

Jamerson Bezerra·Ao Cai·Pedro Duarte·Catalina Freijo·Silvius Klein
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Abstract

In this paper we establish an abstract, dynamical Thouless-type formula for affine families of $\mathrm{GL} (2,\mathbb{R})$ cocycles. This result extends the classical formula relating, via the Hilbert transform, the maximal Lyapunov exponent and the integrated density of states of a Schr\"odinger operator. Here, the role of the integrated density of states will be played by a more geometrical quantity, the fibered rotation number. As an application of this formula we present limitations on the modulus of continuity of random linear cocycles. Moreover, we derive H\"older-type continuity properties of the fibered rotation number for linear cocycles over various base dynamics.

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Jamerson Bezerra, Ao Cai, Pedro Duarte, Catalina Freijo, Silvius Klein. 2022-08-11. A dynamical Thouless formula. https://arxiv.org/abs/2208.06022

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