Lyapunov Exponents of Free Operators

TL;DR

This paper extends the concept of Lyapunov exponents to free operators using the extended Fuglede-Kadison determinant, establishing their properties, calculation formulas, and connections to free probability and random matrix theory.

Contribution

It introduces a new definition of Lyapunov exponents for free operators, proves their existence, and explores their additive property and relation to free probability tools.

Findings

Lyapunov exponents for free operators are well-defined and additive.

Derived formulas for calculating Lyapunov exponents of free operators.

Connected Lyapunov exponents distribution to Marchenko-Pastur law and Voiculescu's S-transform.

Abstract

Lyapunov exponents of a dynamical system are a useful tool to gauge the stability and complexity of the system. This paper offers a definition of Lyapunov exponents for a sequence of free linear operators. The definition is based on the concept of the extended Fuglede-Kadison determinant. We establish the existence of Lyapunov exponents, derive formulas for their calculation, and show that Lyapunov exponents of free variables are additive with respect to operator product. We illustrate these results using an example of free operators whose singular values are distributed by the Marchenko-Pastur law, and relate this example to C. M. Newman's "triangle" law for the distribution of Lyapunov exponents of large random matrices with independent Gaussian entries. As an interesting by-product of our results, we derive a relation between the extended Fuglede-Kadison determinant and Voiculescu's S-transform.