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Variational description of uniform Lyapunov exponents via adapted metrics on exterior products

Mikhail Anikushin

TL;DR

This work develops a variational framework for uniform Lyapunov exponents of linear cocycles in infinite-dimensional spaces by leveraging adapted metrics on exterior powers. It introduces a Growth Formula for infinitesimal growth exponents in Banach spaces, establishes a Lyapunov-metric variational principle, and extends Liouville-type trace formulas to mean representations and exterior powers, enabling effective computation of Lyapunov exponents and Lyapunov dimensions. Special attention is given to Hilbert-space cocycles, where sums of exponents correspond to wedge-power top exponents and to a Schur-type variational description that yields computable Lyapunov dimensions, including a robust Ergodic Variational Principle for subadditive families. The theory is then applied to delay equations in $\mathbb{R}^n$, introducing adapted-delay metrics, a decomposition/symmetrization framework for delay operators, and concrete, asymptotically sharp dimension estimates for Mackey--Glass and Suarez--Schopf delayed oscillators, with scaling yielding sharper bounds as delay grows. The results provide practical tools for bounding attractor dimensions and understanding stabilization/bifurcation phenomena in infinite-dimensional delay systems.

Abstract

In this work, we present a comprehensive study of the relationship among uniform Lyapunov exponents, the Liouville trace formula, and adapted metrics for cocycles in Hilbert spaces. First, we prove that uniform Lyapunov exponents can be approximated by constructing adapted metrics on exterior products. Next, we develop a general computational theory in an abstract setting, establish a generalized Liouville trace formula, and pose and discuss the symmetrization problem related to computations. Third, we discuss ergodic properties and upper semicontinuity in the context of subadditive families over a noncompact base. Furthermore, we use adapted metrics and the trace formula to obtain, for the first time, effective dimension estimates for a general class of delay equations. In particular, we illustrate this approach by deriving upper estimates for the Lyapunov dimension of global attractors in the Mackey--Glass equations and the periodically forced Suarez--Schopf delayed oscillator. As the delay value tends to infinity, the estimates appear to be asymptotically sharp.

Variational description of uniform Lyapunov exponents via adapted metrics on exterior products

TL;DR

This work develops a variational framework for uniform Lyapunov exponents of linear cocycles in infinite-dimensional spaces by leveraging adapted metrics on exterior powers. It introduces a Growth Formula for infinitesimal growth exponents in Banach spaces, establishes a Lyapunov-metric variational principle, and extends Liouville-type trace formulas to mean representations and exterior powers, enabling effective computation of Lyapunov exponents and Lyapunov dimensions. Special attention is given to Hilbert-space cocycles, where sums of exponents correspond to wedge-power top exponents and to a Schur-type variational description that yields computable Lyapunov dimensions, including a robust Ergodic Variational Principle for subadditive families. The theory is then applied to delay equations in , introducing adapted-delay metrics, a decomposition/symmetrization framework for delay operators, and concrete, asymptotically sharp dimension estimates for Mackey--Glass and Suarez--Schopf delayed oscillators, with scaling yielding sharper bounds as delay grows. The results provide practical tools for bounding attractor dimensions and understanding stabilization/bifurcation phenomena in infinite-dimensional delay systems.

Abstract

In this work, we present a comprehensive study of the relationship among uniform Lyapunov exponents, the Liouville trace formula, and adapted metrics for cocycles in Hilbert spaces. First, we prove that uniform Lyapunov exponents can be approximated by constructing adapted metrics on exterior products. Next, we develop a general computational theory in an abstract setting, establish a generalized Liouville trace formula, and pose and discuss the symmetrization problem related to computations. Third, we discuss ergodic properties and upper semicontinuity in the context of subadditive families over a noncompact base. Furthermore, we use adapted metrics and the trace formula to obtain, for the first time, effective dimension estimates for a general class of delay equations. In particular, we illustrate this approach by deriving upper estimates for the Lyapunov dimension of global attractors in the Mackey--Glass equations and the periodically forced Suarez--Schopf delayed oscillator. As the delay value tends to infinity, the estimates appear to be asymptotically sharp.
Paper Structure (13 sections, 46 theorems, 245 equations)

This paper contains 13 sections, 46 theorems, 245 equations.

Table of Contents

  1. Introduction
  2. Infinitesimal growth exponents and Lyapunov metrics for cocycles in Banach spaces
  3. Uniform Lyapunov exponents and Lyapunov dimension for cocycles in Hilbert spaces
  4. Infinitesimal growth exponents of compound cocycles in associated metrics
  5. Infinitesimal growth exponents in the conformal class of a constant metric
  6. Adapted metrics for delay equations in $\mathbb{R}^{n}$
  7. Linearization of delay equations
  8. Additive symmetrization of delay operators
  9. Mackey--Glass equations
  10. Periodically forced Suarez--Schopf delayed oscillator
  11. Ergodic Variational Principle for subadditive families
  12. Upper semicontinuity of the uniform growth exponent for subadditive families
  13. Additive symmetrization of operators

Key Result

Lemma 2.1

Let $\Xi$ be a linear cocycle in $\mathbb{E}$ over a semiflow $(\mathcal{Q},\vartheta)$. Then for all $t \geq 0$ and $q \in \mathcal{Q}$, we have In other words, the function is right lower semicontinuous. In particular, it is Borel measurable.

Theorems & Definitions (114)

  • Remark 2.1
  • Lemma 2.1
  • proof
  • Theorem 2.1: Growth Formula
  • proof
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Theorem 2.2
  • proof
  • ...and 104 more