Strong Fast Invertibility and Lyapunov Exponents for Linear Systems

TL;DR

This paper characterizes strongly fast invertible linear systems with bounded coefficients, linking their properties to Lyapunov exponents and providing convergence results for Lyapunov exponent computation methods.

Contribution

It offers a full characterization of regular strongly fast invertible systems and establishes conditions for Lyapunov exponent convergence even without regularity.

Findings

Strongly fast invertibility relates to Lyapunov exponent multiplicities.

Stability of Lyapunov exponents implies strong fast invertibility.

Convergence conditions for Benettin's algorithm stepsizes are derived.

Abstract

In 2019 Anthony Quas, Philippe Thieullen and Mohamed Zarrabi introduced the concept of strong fast invertibility for linear cocycles. It relates the growth of volumes between different initial times and, together with a condition on singular value gaps, yields the existence of a dominated splitting of the dynamics. The properties of this splitting largely coincide with those for systems with stable Lyapunov exponents. In this article, we take a closer look at strongly fast invertible systems with bounded coefficients. By linking the dimensions at which a system admits strong fast invertibility to the multiplicities of Lyapunov exponents, we are able to give a full characterization of regular strongly fast invertible systems similar to that of systems with stable Lyapunov exponents. In particular, we show that the stability of Lyapunov exponents implies strong fast invertibility (even in the absence of regularity). Central to our arguments are certain induced systems on spaces of exterior products that represent the evolution of volumes. Finally, we derive convergence results for the computation of Lyapunov exponents via Benettin's algorithm using perturbation theory. While the stronger assumption of stable Lyapunov exponents clearly leaves more freedom on how to choose stepsizes, we derive conditions for the stepsizes with which convergence can be ensured even if a system is only strongly fast invertible.