Quantitative Hölder Regularity, Concentration, and Spectral Applications for Lyapunov Exponents of Random $\mathrm{GL}(2,\mathbb{R})$ Cocycles, with Extensions to $\mathrm{GL}(d,\mathbb{R})$

Abstract

This paper develops a quantitative regularity theory for the Lyapunov exponents of random products of matrices in $\mathrm{GL}(2,\mathbb{R})$, with extensions to $\mathrm{GL}(d,\mathbb{R})$ for all $d\ge 2$. At every compactly supported measure $\hat{I}½$ with simple Lyapunov spectrum, we give an explicit closed-form Hölder exponent ${\hat{I}}_{\ast }^2(\hat{I}½,\hat{I}¸)$ and constant in the modulus of continuity of $\hat{I}{»}_{\pm }$ in the Wasserstein-plus-Hausdorff metric, depending only on the eccentricity of $\mathrm{supp}\hat{I}½$, the Lyapunov gap, and the Hölder index $\hat{I}¸$. At every $\hat{I}½\in {\text{M}}_c(\mathrm{GL}(2,\mathbb{R}))$ we identify the log-Hölder exponent of Tall and Viana as $\hat{I}{º}_{\ast }(\hat{I}½,\hat{I}¸)=\hat{I}¸/(2+\hat{I}¸)$ under a natural mixing hypothesis, and $\hat{I}¸/(8(1+\hat{I}¸))$ in the perpetuity regime. The same spectral-gap method yields a large deviation principle with explicit rate function, Hoeffding-Azuma concentration inequalities, an extension to Markov-chain driven cocycles with closed-form exponent, and a quantitative log-Hölder modulus of continuity for the integrated density of states of one-dimensional random Schrödinger operators with absolutely continuous disorder. The Hölder theory extends to $\mathrm{GL}(d,\mathbb{R})$ for the top exponent under spectral simplicity, and to the partial sums $\hat{I}{›}_k=\hat{I}{»}_1+\cdots +\hat{I}{»}_k$ under strong $k$-irreducibility, yielding Hölder continuity of each individual sub-top exponent. A method-optimality proposition shows that ${\hat{I}}_{\ast }^2$ is the best exponent obtainable from the linear balance of axioms (A1)-(A3) of the spectral-gap method; strict improvement requires either modifying these axioms or adopting a different proof strategy. A lower-bound proposition adapted from Duarte, Klein, and Santos rules out uniform Hölder continuity across ${\text{M}}_c(\mathrm{GL}(2,\mathbb{R}))$.

65 pages. Key words and phrases. Lyapunov exponents, random matrix products, modulus of continuity, transfer operators, Wasserstein distance, large deviations, Schrödinger cocycles, integrated density of states