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

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.

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• The Lyapunov dimension and its estimation via the Leonov method
• The Lyapunov dimension formula for the global attractor of the Lorenz system
• Mather sets for sequences of matrices and applications to the study of joint spectral radii
• Tensor Products, Positive Linear Operators, and Delay-Differential Equations
• Finite Dimensionality of the attractor for the hyperbolic Cahn-Hilliard-Oono Equation in R^3
• Hidden and unstable periodic orbits as a result of homoclinic bifurcations in the Suarez-Schopf delayed oscillator and the irregularity of ENSO
• Frequency theorem and inertial manifolds for neutral delay equations