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arxiv: 2411.16628 · v3 · submitted 2024-11-25 · 🧮 math.DS

On linear response for discontinuous perturbations of smooth endomorphisms

Giovanni Canestrari This is my paper

Pith reviewed 2026-05-23 17:14 UTC · model grok-4.3

classification 🧮 math.DS
keywords linear responsephysical measuresdiscontinuous perturbationssmooth endomorphismsuniform mixingstandard pairsdynamical systemsstatistical stability
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The pith

Uniform mixing on standard pairs makes the physical measure Lipschitz in the perturbation parameter for discontinuous perturbations of smooth endomorphisms.

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

The paper studies families of maps formed by applying discontinuous perturbations to smooth endomorphisms. When these families obey uniform mixing conditions expressed via standard pairs, the associated physical measures vary Lipschitz continuously with the size of the perturbation. The analysis also addresses linear response, which tracks the first-order change in the measure, and confirms that the abstract mixing conditions hold in at least one concrete example. This supplies a route to statistical stability results in the presence of discontinuities that would otherwise obstruct direct application of smooth perturbation techniques.

Core claim

If a family of discontinuous perturbations of smooth endomorphisms satisfies uniform mixing assumptions on standard pairs, then the physical measure is Lipschitz continuous in the perturbation parameter, and linear response holds for the family.

What carries the argument

Uniform mixing assumptions on standard pairs, which provide quantitative decay of correlations sufficient to bound the variation of the physical measure with the parameter.

If this is right

  • The physical measure changes at most linearly under small changes to the perturbation parameter.
  • A linear response formula for the derivative of the measure with respect to the parameter can be obtained from the mixing assumptions.
  • The result applies whenever the abstract mixing conditions can be verified directly, including in the concrete example treated in the paper.
  • Statistical stability of the physical measure extends to this class of discontinuous perturbations.

Where Pith is reading between the lines

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

  • The same mixing framework could be checked in other discontinuous settings, such as billiards with moving obstacles, to obtain Lipschitz continuity there as well.
  • If the mixing rate can be made explicit, quantitative bounds on the Lipschitz constant become available for numerical checks.
  • The approach may connect to random perturbations by treating noise as a special case of the discontinuous family.

Load-bearing premise

The perturbed family satisfies uniform mixing assumptions on standard pairs.

What would settle it

A concrete family of discontinuous perturbations that obeys the uniform mixing conditions on standard pairs yet has a physical measure whose variation with the parameter is not Lipschitz would falsify the main claim.

Figures

Figures reproduced from arXiv: 2411.16628 by Giovanni Canestrari.

Figure 1
Figure 1. Figure 1: On the left M+ 1,t in red, M+ 2,t in blue, M+ 3,t in green and M+ 4,t in brown, for t = 1/8. On the right, there are the corresponding images. We denote by L A t the transfer operator associated with (5.3). For r0 > 0 that will be specified shortly, we define C u =  (ξ, η) ∈ R 2 : ξη > 0 [PITH_FULL_IMAGE:figures/full_fig_p021_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: S − 0 (left), S − t (center), S − 0 ∪ S− t (right), t = 1 8 . In red and blue, the sets Rt and Bt respectively. Lemma 5.3. For any ϕ ∈ C0 (M), limt→0 R M(ϕ ◦ FA,t − ϕ)dm t = m(ϕ) − Z 1 0 ϕ(s, s)ds. Proof. By a direct computation (see also (Fig.1)), for all t ∈ (0, 1/8], (5.13) L A t 1 − 1 t = 1 1 − t 1M\Ht − 1 t 1Ht , where Ht is the white portion of M in the right side of (Fig.1), i.e., Ht = Dt ∪ Rt, Dt =… view at source ↗
read the original abstract

We consider discontinuous perturbations of smooth endomorphisms and show that if the perturbed family satisfies uniform mixing assumptions on standard pairs the physical measure is Lipschitz in the parameter defying the perturbation. We also study the problem of linear response for this class of perturbations. Finally we discuss the applicability of the abstract assumptions proving linear response for a concrete example.

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

1 major / 0 minor

Summary. The paper proves an abstract result: if a family of discontinuous perturbations of smooth endomorphisms satisfies uniform mixing assumptions on standard pairs (with rates independent of the perturbation parameter), then the physical measures vary Lipschitz continuously with the parameter and linear response holds. It then discusses applicability of the assumptions to a concrete example, claiming linear response for that case.

Significance. The conditional abstract theorem extends linear response theory to a class of discontinuous perturbations under verifiable mixing hypotheses on standard pairs. This is a useful technical contribution in dynamical systems if the mixing conditions can be checked with parameter-independent constants, as the paper attempts for its example.

major comments (1)
  1. [Section discussing applicability to the concrete example] The section discussing the concrete example: the claim that the example satisfies the uniform mixing assumptions on standard pairs with rates independent of the perturbation parameter is asserted but not supported by explicit bounds or a self-contained verification. This verification is load-bearing for transferring the abstract Lipschitz and linear-response conclusions to the example; without it the applicability statement remains conditional on an unconfirmed hypothesis.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the constructive comment on the manuscript. We respond to the major comment below.

read point-by-point responses

  1. Referee: [Section discussing applicability to the concrete example] The section discussing the concrete example: the claim that the example satisfies the uniform mixing assumptions on standard pairs with rates independent of the perturbation parameter is asserted but not supported by explicit bounds or a self-contained verification. This verification is load-bearing for transferring the abstract Lipschitz and linear-response conclusions to the example; without it the applicability statement remains conditional on an unconfirmed hypothesis.

    Authors: We agree that the discussion of the concrete example in the manuscript asserts the applicability of the uniform mixing assumptions without providing fully explicit, self-contained bounds on the mixing rates that are independent of the perturbation parameter. While the section offers heuristic arguments and references to related estimates, it does not constitute a complete verification. In the revised manuscript we will expand this section with additional detailed estimates and calculations to establish the parameter-independent rates explicitly, thereby making the transfer of the abstract conclusions to the example rigorous and self-contained. revision: yes

Circularity Check

0 steps flagged

No circularity detected; central result is conditional on external mixing assumptions with no reduction to fitted inputs or self-citation chains

full rationale

The paper's main theorem states that if the perturbed family satisfies uniform mixing assumptions on standard pairs, then the physical measure is Lipschitz in the parameter and linear response holds. This is an implication derived from the assumptions rather than a self-definitional or fitted-input construction. The discussion of applicability to a concrete example is presented as a separate verification step. No quotes or equations in the provided material show any prediction reducing by construction to its inputs, any load-bearing self-citation, or an ansatz smuggled via prior work. The derivation chain remains self-contained against the stated assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on the uniform mixing assumption on standard pairs, which is treated as an external hypothesis rather than derived; no free parameters or invented entities are introduced in the abstract statement.

axioms (1)
  • domain assumption The perturbed family satisfies uniform mixing assumptions on standard pairs.
    This is the load-bearing hypothesis invoked to obtain Lipschitz continuity of the physical measure and linear response.

pith-pipeline@v0.9.0 · 5560 in / 1207 out tokens · 19042 ms · 2026-05-23T17:14:23.637436+00:00 · methodology

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Forward citations

Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Quenched and annealed linear response for some partially hyperbolic skew products

    math.DS 2026-04 unverdicted novelty 7.0

    Establishes quenched and annealed linear response and statistical stability for partially hyperbolic skew products with parameter-dependent base maps, yielding new formulas for the response and moment derivatives.

  2. Linear response for Sinai billiards with small holes

    math.DS 2026-04 unverdicted novelty 7.0

    The conditional survival probability measure for a Sinai billiard with a small hole is differentiable with respect to hole size at zero, and the derivative is computed.

  3. Linear response for Sinai billiards with small holes

    math.DS 2026-04 unverdicted novelty 6.0

    The conditional survival probability measure for Sinai billiards with small holes is differentiable at t=0 and its derivative is computed.

Reference graph

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