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arxiv: 2604.20402 · v2 · submitted 2026-04-22 · 🧮 math.DS

Quenched and annealed linear response for some partially hyperbolic skew products

Davor Dragicevic , Yeor Hafouta This is my paper

Pith reviewed 2026-05-09 23:23 UTC · model grok-4.3

classification 🧮 math.DS
keywords partially hyperbolic skew productslinear responsestatistical stabilityquenched and annealedrandom hyperbolic mapsparametric familiesasymptotic momentsdynamical systems
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The pith

For partially hyperbolic skew products with parameter-dependent base maps and random hyperbolic fiber maps, quenched and annealed statistical stability and linear response hold, along with differentiability of asymptotic moments.

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

The paper shows that these skew-product systems remain statistically stable under small parameter changes, both when the base is fixed almost surely (quenched) and when averaged over the base (annealed). It derives explicit linear response formulas that include an extra term arising from the base map's own dependence on the parameter. This setup also makes the long-run average moments differentiable with respect to the parameter. The results cover partially hyperbolic maps outside the scope of earlier annealed-only arguments.

Core claim

We prove quenched and annealed statistical stability, linear response, and differentiability of asymptotic moments for parametric families of partially hyperbolic skew products whose fiber maps are random and hyperbolic while the base maps themselves vary with the parameter. The resulting response formulas and moment derivatives contain additional contributions from the base's parameter derivative that are absent when the base is held fixed.

What carries the argument

The skew-product decomposition separating a parameter-dependent base from random hyperbolic fiber maps, which permits fiberwise transfer-operator estimates that survive the base variation.

If this is right

  • Both quenched and annealed invariant measures vary continuously with the parameter.
  • The linear-response formula gains an extra integral term coming from the base map's derivative.
  • Asymptotic moments such as integrated observables or Lyapunov exponents become differentiable functions of the parameter.
  • The annealed results apply to a strictly larger class of partially hyperbolic maps than those treated in prior work.

Where Pith is reading between the lines

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

  • The same separation of base and fiber could be used to obtain response formulas in systems where the base evolves slowly compared with the fiber.
  • Numerical checks on low-dimensional examples with explicit hyperbolic fibers would directly test the new base-dependent terms.
  • The differentiability of moments may allow parameter optimization in models whose statistics are governed by such skew products.

Load-bearing premise

The fiber maps stay uniformly hyperbolic and the overall partial hyperbolicity is preserved under small parameter shifts so that the required operator bounds remain valid.

What would settle it

A concrete parametric family of skew products where the numerically measured change in the invariant measure for a small parameter increment fails to match the explicit linear-response formula derived in the paper.

read the original abstract

We prove quenched and annealed statistical stability, linear response, and differentiability of asymptotic moments for parametric families of partially hyperbolic skew products, with random hyperbolic maps on the fibers. The main novelty is that the base maps also depend on the parameter, which leads to different formulas in the linear response and the derivative of the asymptotic moments with respect to the parameter. Our annealed results apply to partially hyperbolic maps that are not covered in \cite{BashCastro26,Dol,DS}.

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

0 major / 3 minor

Summary. The manuscript proves quenched and annealed statistical stability, linear response, and differentiability of asymptotic moments for parametric families of partially hyperbolic skew products with random hyperbolic maps on the fibers. The central novelty is that the base maps depend on the parameter, which produces modified formulas for the linear response and for the derivative of the asymptotic moments; the annealed results are claimed to cover partially hyperbolic maps outside the scope of BashCastro26, Dol, and DS.

Significance. If the proofs are complete, the work supplies a technically useful extension of linear-response theory to a broader class of partially hyperbolic skew products in which the base dynamics itself varies with the parameter. The provision of explicit (and different) formulas for the response and for the derivatives of asymptotic moments, together with the annealed results that reach beyond the cited prior literature, strengthens the toolkit available for analyzing parameter sensitivity in random dynamical systems.

minor comments (3)
  1. [§2.1] §2.1: the precise regularity assumptions on the parameter dependence of the base map (e.g., C^1 or Hölder) are stated only implicitly through the standing hypotheses; an explicit list would clarify the scope of the main theorems.
  2. [Theorem 3.2] Theorem 3.2 and the subsequent linear-response formula: the term arising from the base-map derivative is written without an explicit reference to the corresponding term in the earlier literature; adding a short comparison sentence would help readers track the modification.
  3. [annealed linear response proof] The proof of the annealed linear response (around p. 18) invokes a uniform bound on the fiber derivatives that is not restated in the statement of the theorem; moving this bound into the hypotheses would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation for minor revision. The summary accurately reflects the main results on quenched and annealed statistical stability, linear response, and differentiability of asymptotic moments for the parametric families of partially hyperbolic skew products, with emphasis on the novelty arising from parameter-dependent base maps and the extended scope of the annealed results beyond the cited literature.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper establishes quenched and annealed statistical stability, linear response, and differentiability of asymptotic moments for parametric families of partially hyperbolic skew products where base maps depend on the parameter. The abstract positions the work as an extension of prior results in the literature, with the novelty consisting of modified formulas arising from the parameter dependence in the base. No self-definitional reductions, fitted inputs renamed as predictions, or load-bearing self-citations appear in the stated claims or abstract. The derivation relies on standard techniques for such systems and is presented as independent of the target results by construction, consistent with external mathematical benchmarks in dynamical systems.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based solely on the abstract, the central claims rest on standard domain assumptions of partial hyperbolicity and random hyperbolicity in skew products; no free parameters or invented entities are mentioned.

axioms (2)
  • domain assumption The maps form partially hyperbolic skew products with random hyperbolic maps on the fibers.
    Explicitly stated as the class of systems for which the results hold.
  • domain assumption The base maps depend on the parameter in a way that permits the quenched and annealed analysis.
    Identified as the source of the new formulas in the abstract.

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