Integral representation of Lyapunov exponents

Abstract: We develop an abstract operator-theoretic variational principle for asymptotic growth rates arising from subadditive processes driven by Markov operators: for each invariant measure on the base, the growth rate equals the supremum of fiber integrals over invariant lifts to the bundle, and this supremum is attained on an ergodic lift. Applied to (random) linear bundle morphisms, the principle extends the classical projective formulas for sums of Lyapunov exponents, including singular cocycles, and yields new asymptotic representations in terms of conditional annealed growth along individual directions. As an application, we prove that for random linear bundle morphisms driven by Markovian place-dependent noise, the pointwise Lyapunov exponents depend only on the current noise state and initial position, not on the full noise realization.

• The paper presents an operator-theoretic variational principle that yields an integral representation for Lyapunov exponents in various dynamical systems.
• It extends classical formulas to handle singular cocycles and Markov-driven processes using measurable bundles and invariant measure lifts.
• The approach enables computational schemes by exchanging maximization and expectation and shows that pointwise exponents depend only on current state and position.

Abstract Operator-Theoretic Variational Principle for Lyapunov Exponents

Introduction and Motivation

The paper "Integral representation of Lyapunov exponents" (2604.13376) presents a comprehensive operator-theoretic framework for representing Lyapunov exponents, extending classical integral formulas to general Markov-driven processes and singular cocycles. Lyapunov exponents serve as the quantitative descriptors of asymptotic growth rates in dynamical systems and are foundational in random dynamics, smooth ergodic theory, and applied probability. Classical existence, following Kingman's subadditive ergodic theorem, is insufficient for explicit computation or optimization; the paper addresses this gap by developing operator-based variational principles that yield integral representations, facilitating both theoretical analysis and computational schemes.

Operator-Theoretic Framework

The framework is built upon a measurable bundle $\pi:\hat X \to X$ with compact fibers and associated Markov operators $P$ on $B(X)$ and $\hat P$ on $B(\hat X)$, which intertwine via $\pi$: $\hat P \circ \pi^* = \pi^* \circ P$. This structure generalizes the classical skew-product and projective approaches to Lyapunov exponent computation. The main technical result demonstrates that the fiberwise maximization of an additive process generated by an upper semicontinuous potential induces a $P$-subadditive sequence on the base space, whose asymptotic growth rate can be represented via a variational principle:

$$\Lambda(\mu) = \sup_{\hat\mu \in \mathcal{I}_\mu(\hat P)} \int \hat\phi_1\, d\hat\mu,$$

where $\mathcal{I}_\mu(\hat P)$ denotes the set of $P$0-invariant measure lifts of $P$1 (a $P$2-invariant measure), and the supremum is attained at ergodic lifts. This abstract result unifies classical Furstenberg-Kifer formulas and Ledrappier-type variational representations for sums of Lyapunov exponents and extends them to singular cocycles.

Applications to Deterministic and Random Bundle Morphisms

Deterministic Bundle Morphisms

For a linear bundle morphism $P$3 over an ergodic measure-preserving transformation $P$4 of $P$5, the induced cocycle on the projective bundle (with cemetery section for singularities) allows integral representation of Lyapunov partial sums:

$P$6

where $P$7 is $P$8-invariant and projects to $P$9. This holds for both invertible and singular cases, with explicit handling of degeneracy loci (where the potential is $B(X)$0).

Random Linear Bundle Morphisms and Markovian Noise

The framework is further extended to random bundle morphisms $B(X)$1 driven by Markovian (potentially place-dependent) noise. The construction replaces the classical path space $B(X)$2 with the more accessible state-phase space $B(X)$3, encoding random iteration via a Markov operator. The paper demonstrates that pointwise Lyapunov exponents depend only on the current noise state and position, not on the full noise path. The operator-theoretic approach yields new asymptotic representations for Lyapunov partial sums:

$B(X)$4

demonstrating a nontrivial exchange of maximization and averaging, with practical computational advantages (conditional expectations along directions rather than operator norms).

Special Cases: Bernoulli and Edge Markov Models

In the Bernoulli (i.i.d.) case, the classical $B(X)$5-independence of the Lyapunov spectrum is recovered, with explicit identification of the exponents with annealed asymptotic formulas. The operator approach generalizes variational formulas for vertex and edge Markovian matrix products and random cocycles, encompassing invertible and singular maps, finite and infinite alphabets, and a spectrum of structural and integrability conditions.

Strong Results and Claims

The paper establishes the following robust claims:

• Variational Principle for Sums of Lyapunov Exponents: For general (including singular) linear bundle morphisms and random cocycles, the sum of the first $B(X)$6 Lyapunov exponents is given by the supremum of fiber integrals over invariant lifts, attained at an ergodic lift.
• Pointwise Lyapunov Exponents Depend Only on Current State and Position: For random morphisms driven by Markov place-dependent noise, the pointwise exponents are functions of $B(X)$7, not the full noise realization.
• Unified Representation for Deterministic, Bernoulli, and Markov Cocycles: The operator-theoretic approach subsumes classical projective, Grassmannian, Bernoulli, and Markov formulas.
• Exchange of Maximization and Expectation: Asymptotic formulas are valid where the sup and expectation do not generally commute, providing new insight into the variational structure.

Theoretical and Computational Implications

The operator-theoretic framework brings coherence and generality to the representation of Lyapunov exponents, directly linking variational principles to asymptotic spectral growth. It facilitates ergodic optimization, perturbation theory, and computational schemes, particularly through conditional expectations along individual directions. The methodology is robust under singularities and degeneracy, with clear handling of bad sets, and is amenable to Monte Carlo approximations. The abstract framework also clarifies the relation between path-space and state-phase formulations, offering more transparent and accessible representations.

Future Directions

Possible avenues for development include:

• Analytical and computational exploration of ergodic optimization in Markov-driven systems, including non-compact settings.
• Extension of the variational principle to infinite-dimensional bundle morphisms and non-stationary noise models.
• Integration with statistical mechanics, stochastic control, and random operator theory for applications involving Lyapunov exponents in optimization and stability analysis.

Conclusion

The paper delivers a rigorous, operator-theoretic variational principle for Lyapunov exponents in deterministic, random, and Markov-driven settings, including singular cocycles. Its integral formulas unify and extend classical approaches, enable explicit analysis for a broad class of systems, and provide concrete computational schemes for Lyapunov exponents. The results markedly clarify the dependence structure of pointwise exponents in random dynamical systems and supply an adaptable framework for further theoretical and applied developments.