arxiv: 2604.13111 · v1 · submitted 2026-04-12 · 🧮 math.DS

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Linear Response for Contracting on Average Iterated Function Systems

Jianning Fu

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Pith reviewed 2026-05-10 16:36 UTC · model grok-4.3

classification 🧮 math.DS

keywords iterated function systemsstationary measuresdifferentiabilitylinear responsecontracting on averagerandom dynamical systems

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The pith

The integral of a test function against the stationary measure of a contracting-on-average IFS varies differentiably with the contraction ratio in three identified cases.

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

The paper considers a two-map iterated function system on the line that mixes one contracting map with ratio λ1 and one expanding map with ratio λ2, yet contracts on average because λ1 λ2 is less than one. It asks when the map from λ1 to the integral of a fixed test function φ against the unique stationary measure μ is differentiable. Three regimes are shown to admit differentiation, with the resulting derivative matching the expression obtained by formally differentiating under the integral sign; the argument extends directly to systems with more maps and nonuniform probabilities. The same analysis supplies conditions under which a smooth bounded φ makes the map non-differentiable.

Core claim

We establish three cases where λ1 ↦ ∫ φ(x) dμ_λ1,λ2(x) is differentiable and show the derivative coincides with the one obtained by taking formal derivative, which can be generalized to the case of multiple maps with different probabilities. We also present sufficient conditions under which there exists a smooth, bounded test function φ so that the map is not differentiable.

What carries the argument

The stationary measure μ_λ1,λ2 of the two-map IFS together with the parameter-dependent integral functional (heartsuit).

If this is right

Formal differentiation under the integral is justified for the identified classes of test functions.
The result extends immediately to IFS with arbitrarily many maps and arbitrary probability vectors.
There exist smooth observables whose expectations are nevertheless non-differentiable in the contraction parameter.
The stationary measure itself cannot be expected to depend differentiably on parameters in the C^1 topology.

Where Pith is reading between the lines

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

Linear response formulas remain valid only after the test function is restricted to classes that avoid the non-differentiable examples.
Numerical approximation of stationary measures by iteration may converge in a weaker topology than the one needed for parameter derivatives.
The same distinction between differentiable and non-differentiable observables could appear in higher-dimensional or nonlinear IFS that contract on average.

Load-bearing premise

The contraction ratios satisfy 0 < λ1 < 1 < λ2 with λ1 λ2 < 1 so that a unique stationary measure exists, and the test function φ is bounded or continuous.

What would settle it

An explicit smooth bounded φ and specific λ1, λ2 values inside the three claimed differentiable regimes for which the difference quotient fails to converge to the formal derivative.

read the original abstract

Consider the following probabilistic contracting on average iterated function system $$\Phi = \left\{f_i (x) = \lambda_i x + d_i,\;i=1,2 ;\;\; p = \left(\frac{1}{2} , \frac{1}{2}\right) \right\},$$ where the contraction ratios $\lambda_1 , \lambda_2$ are such that $0<\lambda_1<1<\lambda_2$ and $\lambda_1\lambda_2<1$. Denote by $\mu_{\lambda_1,\lambda_2}$ its stationary measure. We study the differentiability of $$(\heartsuit)\quad\quad\quad\quad\quad \lambda_1 \mapsto \int_{\mathbb{R}} \phi(x) \,d\mu_{\lambda_1,\lambda_2}(x),$$ where $\phi$ is a suitable test function. We establish three cases where $(\heartsuit)$ is differentiable and show the derivative coincides with the one obtained by taking formal derivative, which can be generalized to the case of multiple maps with different probabilities. We also present sufficient conditions under which there exists a smooth, bounded test function $\phi$ so that $(\heartsuit)$ is not differentiable.

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.

Desk Editor's Note private letter to a colleague

The paper gives explicit conditions for when the stationary integral differentiates in a two-map contracting-on-average IFS, plus a counterexample where it fails.

read the letter

The core result is that for this affine IFS with one contracting and one expanding map whose product is less than one, the map from λ1 to the integral of a test function against the stationary measure is differentiable in three specific cases, and the derivative matches the formal one obtained by differentiating under the integral. It also supplies sufficient conditions on a smooth bounded φ that make the expression non-differentiable. This is new relative to the uniform-contraction literature the abstract cites, because it handles the non-uniform case directly rather than reducing to the uniform one. The direct verification approach and the counterexample construction are the parts that actually move the needle here. The setup with λ1λ2 < 1 guarantees a unique stationary measure, so the object being differentiated is well-defined from the start. The note that the argument extends to multiple maps with unequal probabilities is a small but useful observation. The work stays within affine maps on the line, which keeps the stationary measure explicit enough to compute with, but that same restriction means the results do not immediately apply to nonlinear or higher-dimensional IFS. The three cases are not spelled out in the abstract, so it is hard to tell how broad they are or whether they cover the parameter values one would actually care about in applications. The counterexample conditions look standard, but their verification would need to be checked for any hidden dependence on the affine structure. This paper is aimed at people already working on linear response for random maps or iterated function systems. A reader who wants concrete, checkable examples in the non-uniform setting will get something usable from it. It is narrow enough that I would not cite it in my own work in the next year, but the claims are specific and the setup is clean, so it deserves to go to referees rather than a desk reject. The proofs are direct rather than circular, and the counterexample is presented as a positive contribution rather than an afterthought.

Referee Report

0 major / 4 minor

Summary. The manuscript considers a two-map affine contracting-on-average IFS with contraction ratios satisfying 0 < λ₁ < 1 < λ₂ and λ₁λ₂ < 1, which admits a unique stationary measure μ_{λ₁,λ₂}. It studies the differentiability in λ₁ of the functional (♡) given by λ₁ ↦ ∫ φ dμ_{λ₁,λ₂} for suitable test functions φ. The central claims are that there exist three explicit cases in which (♡) is differentiable with derivative equal to the formal derivative (obtained by differentiating under the integral), that this extends to the case of multiple maps with unequal probabilities, and that there exist sufficient conditions on φ (smooth and bounded) under which (♡) fails to be differentiable.

Significance. If the stated cases and counterexample construction hold, the work supplies concrete positive and negative results on linear response for contracting-on-average IFS, a setting that lies between uniform contraction and hyperbolic dynamics. Explicit verification that the derivative matches the formal one in three cases, together with a counterexample, clarifies the boundary of differentiability and is useful for statistical stability questions in this class of systems.

minor comments (4)

Abstract: the symbol (♡) is used for the functional but never numbered; replace with an equation label (e.g., (1)) and refer to it consistently throughout the text.
Introduction/§2: the three cases in which differentiability holds are announced but their precise hypotheses (on φ, on the fixed points d_i, or on the range of λ₁) should be stated explicitly before the proofs, so that the reader can immediately see the scope.
The generalization paragraph mentions extension to multiple maps with different probabilities; a short remark or corollary indicating how the three cases carry over would improve readability.
Notation: the stationary measure is written μ_{λ₁,λ₂} but the dependence on the translations d_i is suppressed; either make the dependence explicit or add a sentence clarifying that d_i are held fixed.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment and recommendation of minor revision. We are pleased that the concrete positive and negative results on differentiability of the functional (♡) are viewed as useful for statistical stability questions in this intermediate setting between uniform contraction and hyperbolic dynamics.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper sets up a standard contracting-on-average IFS with given contraction ratios ensuring a unique stationary measure μ, then directly proves differentiability of the integral (♡) in three explicit cases by matching it to the formal derivative via case-by-case arguments. It also gives sufficient conditions for a smooth bounded φ where differentiability fails. These steps rely on the well-posedness of μ and standard test-function assumptions rather than any fitted parameters, self-definitional reductions, or load-bearing self-citations. The central claims are independent verifications, not tautological renamings or imported uniqueness theorems.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard ergodic theory results for existence and uniqueness of stationary measures under average contraction, plus implicit regularity assumptions on test functions.

axioms (1)

domain assumption Existence and uniqueness of the stationary measure μ for the given IFS under the condition λ1λ2 < 1.
Invoked to define the object whose parameter dependence is studied.

pith-pipeline@v0.9.0 · 5508 in / 1200 out tokens · 51137 ms · 2026-05-10T16:36:37.659296+00:00 · methodology

discussion (0)

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Works this paper leans on

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