Analytic quasi-perodic cocycles with singularities and the Lyapunov Exponent of Extended Harper's Model

TL;DR

This paper extends Avila's global theory of analytic SL(2,C) cocycles to include singularities, enabling the calculation of Lyapunov exponents for the extended Harper's model across all parameters and irrational frequencies, including the self-dual regime.

Contribution

It introduces a method to analyze cocycles with singularities, expanding the applicability of Avila's global theory to new regimes and ensuring continuity of the Lyapunov exponent.

Findings

Lyapunov exponent can be determined for all parameter values.

Extension of theory includes the self-dual regime.

Lyapunov exponent varies continuously with cocycle parameters.

Abstract

We show how to extend (and with what limitations) Avila's global theory of analytic SL(2,C) cocycles to families of cocycles with singularities. This allows us to develop a strategy to determine the Lyapunov exponent for extended Harper's model, for all values of parameters and all irrational frequencies. In particular, this includes the self-dual regime for which even heuristic results did not previously exist in physics literature. The extension of Avila's global theory is also shown to imply continuous behavior of the LE on the space of analytic $M_2(\mathbb{C})$-cocycles. This includes rational approximation of the frequency, which so far has not been available.