Lyapunov-maximising measures for pairs of weighted shift operators

TL;DR

This paper investigates the measures that maximize the top Lyapunov exponent for pairs of weighted shift operators, establishing generic uniqueness, prescribed entropy values, and differences from the matrix case through explicit constructions and metric analysis.

Contribution

It proves generic uniqueness of Lyapunov-maximising measures, constructs pairs with prescribed entropy, and shows measures are not solely characterized by support, unlike in the matrix case.

Findings

Generic pairs have unique Lyapunov-maximising measures.

Existence of pairs with prescribed entropy less than log 2.

Lyapunov-maximising measures are not determined solely by their supports.

Abstract

Motivated by recent investigations of ergodic optimisation for matrix cocycles, we study the measures of maximum top Lyapunov exponent for pairs of bounded weighted shift operators on a separable Hilbert space. We prove that for generic pairs of weighted shift operators the Lyapunov-maximising measure is unique, and show that there exist pairs of operators whose unique Lyapunov-maximising measure takes any prescribed value less than $\log 2$ for its metric entropy. We also show that in contrast to the matrix case, the Lyapunov-maximising measures of pairs of bounded operators are in general not characterised by their supports: we construct explicitly a pair of operators, and a pair of ergodic measures on the 2-shift with identical supports, such that one of the two measures is Lyapunov-maximising for the pair of operators and the other measure is not. Our proofs make use of the Ornstein $\overline{d}$-metric to estimate differences in the top Lyapunov exponent of a pair of weighted shift operators as the underlying measure is varied.