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arxiv: 2604.13376 · v1 · submitted 2026-04-15 · 🧮 math.DS · math.PR

Integral representation of Lyapunov exponents

Pablo G. Barrientos , Isaia Nisoli This is my paper

Pith reviewed 2026-05-10 12:54 UTC · model grok-4.3

classification 🧮 math.DS math.PR
keywords Lyapunov exponentssubadditive processesMarkov operatorsrandom dynamical systemsvariational principleergodic liftsbundle morphismsplace-dependent noise
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The pith

A variational principle shows that pointwise Lyapunov exponents for random linear bundle morphisms depend only on the current noise state and initial position.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes an operator-theoretic variational principle for the asymptotic growth rates of subadditive processes driven by Markov operators. For any invariant measure on the base space, the growth rate equals the supremum of certain fiber integrals taken over all invariant lifts to the bundle, and this supremum is achieved by an ergodic lift. When specialized to random linear bundle morphisms, the principle recovers and extends the classical projective formulas that express sums of Lyapunov exponents, including cases with singular cocycles, and produces new representations that use conditional annealed growth along individual directions. The main application proves that, under Markovian place-dependent noise, the pointwise Lyapunov exponents at a given position become independent of the entire past noise realization and are determined solely by the present noise value and the starting point in the bundle.

Core claim

The central claim is that an abstract variational principle equates the growth rate of a subadditive process driven by a Markov operator to the supremum of fiber integrals over invariant lifts of any base invariant measure, with the supremum attained on an ergodic lift. Applied to random linear bundle morphisms, this yields integral representations for sums of Lyapunov exponents that hold even for singular cocycles and express the exponents via conditional annealed growth along directions. The resulting simplification shows that, for Markovian place-dependent driving noise, the pointwise Lyapunov exponents are functions only of the current noise state and the initial position, independent of

What carries the argument

The operator-theoretic variational principle for subadditive processes driven by Markov operators, which equates growth rate to the supremum of fiber integrals over invariant lifts of a base measure and shows attainment on ergodic lifts.

If this is right

  • Classical projective formulas for sums of Lyapunov exponents extend directly to singular cocycles.
  • New asymptotic representations appear that express growth via conditional annealed integrals along individual directions.
  • Pointwise Lyapunov exponents reduce to functions of the current noise state and position alone under place-dependent Markov noise.
  • The supremum in the variational equality is attained by an ergodic lift for every base invariant measure.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The reduction to local state dependence may allow computation of Lyapunov exponents from finite Markov chain data on the base without simulating long noise histories.
  • The same variational structure could supply integral formulas for other subadditive quantities arising in random dynamical systems.
  • Stability criteria in noisy linear systems may now be checked using only the stationary distribution of the driving Markov chain rather than the full skew-product dynamics.

Load-bearing premise

Subadditive processes admit invariant measures on the base together with corresponding ergodic lifts to the bundle such that the variational equality for the growth rates holds.

What would settle it

An explicit example of a Markovian place-dependent noise system on a linear bundle in which two different full noise realizations produce different pointwise Lyapunov exponents at the same fixed noise state and initial position.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript develops an abstract operator-theoretic variational principle for asymptotic growth rates of 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, attained on an ergodic lift. Applied to random linear bundle morphisms, the principle extends classical projective formulas for sums of Lyapunov exponents (including singular cocycles) and yields representations via conditional annealed growth along directions. As an application, it proves that for random linear bundle morphisms driven by Markovian place-dependent noise, pointwise Lyapunov exponents depend only on the current noise state and initial position, not on the full noise realization.

Significance. If the variational principle holds with the required invariant measures and ergodic lifts, the work offers a new integral representation framework for Lyapunov exponents in random bundle systems, extending beyond classical Oseledets-type formulas to place-dependent driving. This could enable localized analysis of growth rates in applications like stochastic stability or non-autonomous dynamics, with the pointwise dependence result providing a reduction in complexity for computing exponents.

major comments (2)
  1. [§4] §4 (Application to place-dependent noise): The claim that pointwise Lyapunov exponents depend only on the current noise state and initial position relies on applying the abstract principle to Markovian place-dependent noise. This requires existence of invariant measures on the base and ergodic lifts to the bundle such that the variational equality holds, but the manuscript provides no verification or sufficient conditions (e.g., Feller property, tightness, or continuity assumptions on the Markov operator) to guarantee these for place-dependent noise; without this, the representation and the dependence conclusion do not follow.
  2. [§2] §2 (Abstract variational principle), proof of main theorem: The transition from the variational equality (growth rate = sup fiber integrals over lifts) to the pointwise dependence on current state alone requires explicit justification of how the sup is attained and how it localizes; the abstract states proofs are given but lacks error estimates or step-by-step verification from the equality to the pointwise claim, which is load-bearing for the application result.
minor comments (2)
  1. [§2] Notation for the Markov operator and bundle morphisms should be clarified with explicit definitions early in §2 to avoid ambiguity when transitioning to the linear case.
  2. Add references to classical results on projective formulas for Lyapunov exponents (e.g., works by Arnold or Oseledets extensions) to better contextualize the extension claimed in the abstract.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and valuable comments on the manuscript. We address each major comment below, indicating the revisions we plan to incorporate to improve clarity and rigor.

read point-by-point responses

  1. Referee: [§4] §4 (Application to place-dependent noise): The claim that pointwise Lyapunov exponents depend only on the current noise state and initial position relies on applying the abstract principle to Markovian place-dependent noise. This requires existence of invariant measures on the base and ergodic lifts to the bundle such that the variational equality holds, but the manuscript provides no verification or sufficient conditions (e.g., Feller property, tightness, or continuity assumptions on the Markov operator) to guarantee these for place-dependent noise; without this, the representation and the dependence conclusion do not follow.

    Authors: We acknowledge that the application in §4 would benefit from explicit sufficient conditions. The abstract variational principle holds whenever invariant measures on the base and corresponding ergodic lifts to the bundle exist. For place-dependent Markov noise on a compact metric space with a continuous (Feller) transition kernel, existence of invariant measures follows from the Krylov-Bogoliubov theorem, and ergodic lifts are obtained via the ergodic decomposition theorem applied to the skew-product. In the revised manuscript, we will add a remark in §4 stating these assumptions and briefly verifying that they ensure the variational equality, thereby supporting the pointwise dependence result. revision: yes

  2. Referee: [§2] §2 (Abstract variational principle), proof of main theorem: The transition from the variational equality (growth rate = sup fiber integrals over lifts) to the pointwise dependence on current state alone requires explicit justification of how the sup is attained and how it localizes; the abstract states proofs are given but lacks error estimates or step-by-step verification from the equality to the pointwise claim, which is load-bearing for the application result.

    Authors: We agree that the localization argument in the proof of the main theorem in §2 can be made more explicit. The attainment of the supremum on an ergodic lift, combined with the invariance of the lift, ensures that the fiber integral depends only on the marginal measure on the base (i.e., the current noise state). For the subadditive case, we will expand the proof with a detailed chain of inequalities, including a quantitative error bound derived from the subadditive ergodic theorem to control the approximation between the growth rate and the integrals. This step-by-step verification will clarify the transition to state-only dependence. revision: yes

Circularity Check

0 steps flagged

No circularity: abstract variational principle derived from ergodic theory without self-referential reduction

full rationale

The paper develops an operator-theoretic variational principle stating that for invariant measures on the base, growth rates equal the supremum of fiber integrals over invariant lifts to the bundle, attained at ergodic lifts. This is presented as a general result for subadditive processes driven by Markov operators, then specialized to random linear bundle morphisms to extend projective formulas. The application to place-dependent noise follows directly from the general principle without equations that define the target Lyapunov exponents in terms of themselves or rename fitted quantities as predictions. No self-citation chains, ansatzes smuggled via prior work, or uniqueness theorems imported from the authors are load-bearing in the derivation chain. The framework remains self-contained against external benchmarks in ergodic theory.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claims rest on standard domain assumptions from ergodic theory and operator theory; no free parameters, invented entities, or ad-hoc axioms are indicated in the abstract.

axioms (2)
  • domain assumption Existence of invariant measures for the Markov operators on the base space
    Invoked to state the variational principle for each invariant measure.
  • domain assumption Existence of invariant lifts to the bundle that are ergodic
    Required for the supremum to be attained and for the growth-rate equality.

pith-pipeline@v0.9.0 · 5410 in / 1360 out tokens · 28897 ms · 2026-05-10T12:54:44.879055+00:00 · methodology

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