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arxiv: 2508.11878 · v2 · pith:S5S5QQJUnew · submitted 2025-08-16 · 🧮 math.DS · math.FA

Statistical stability for systems semi-conjugate to pre-piecewise textit{convex or expanding} maps with countably many branches

Rafael Lucena This is my paper

Pith reviewed 2026-05-21 22:16 UTC · model grok-4.3

classification 🧮 math.DS math.FA
keywords statistical stabilitydynamical systemsinvariant measuressemi-conjugacypiecewise expanding mapsGauss mapLüroth mapcountably many branches
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The pith

General conditions ensure statistical stability of invariant measures for perturbed maps semi-conjugate to pre-piecewise convex or expanding maps with countably many branches.

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

The paper shows that one-parameter families of dynamical systems semi-conjugate to pre-piecewise convex or expanding maps with countably many branches have invariant measures that vary continuously with the perturbation parameter at zero. This holds under general conditions on the families and their semi-conjugacies, with explicit quantitative bounds on the rate of change. A reader would care because these systems include maps with unbounded derivatives, discontinuities, or infinite Markov partitions, which are difficult to analyze directly but appear in standard examples like the Gauss and Lüroth maps. If the conditions hold, stability transfers from the base map to the more complex system.

Core claim

We provide general conditions ensuring that the unperturbed measure μ₀ is statistically stable, meaning the map δ ↦ μ_δ is continuous at δ = 0 in the appropriate topology. Furthermore, we establish explicit quantitative estimates for the modulus of continuity of μ_δ in terms of the perturbation parameter δ. Our results apply to a broad class of maps, including those semi-conjugate to classical examples such as the Gauss and Lüroth maps.

What carries the argument

The semi-conjugacy from the original system to a pre-piecewise convex or expanding map with countably many branches, which carries stability information from the base map under the paper's general conditions.

If this is right

  • The map δ to μ_δ is continuous at δ = 0 for families meeting the conditions.
  • Quantitative estimates bound how fast μ_δ can change with δ.
  • The stability conclusion covers systems with unbounded derivatives, discontinuities, or infinite Markov partitions.
  • The results include cases semi-conjugate to the Gauss map and the Lüroth map.

Where Pith is reading between the lines

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

  • Verifying the conditions for a new family would immediately give stability for that family without re-deriving the full argument.
  • The approach might help analyze stability questions for other infinite-branch maps that arise in number theory or ergodic theory.
  • Numerical approximation of measures for small perturbations could test the quantitative modulus estimates on explicit examples.

Load-bearing premise

The one-parameter families of maps and their semi-conjugacies must satisfy the general conditions stated in the paper.

What would settle it

Take a concrete one-parameter family satisfying the conditions, such as a small perturbation of a system semi-conjugate to the Gauss map, and check whether the invariant measures μ_δ fail to approach μ_0 in the relevant topology as δ approaches 0.

read the original abstract

We investigate the statistical stability of a class of dynamical systems semi-conjugate to pre-piecewise \textit{convex or expanding} maps with countably many branches. These systems naturally arise in the study of transformations with unbounded derivatives, discontinuities, or infinite Markov partitions; features that pose significant challenges for stability analysis. Specifically, we consider one-parameter families of transformations $\{F_\delta\}_{\delta \in [0,1)}$ and their corresponding invariant measures $\{\mu_\delta\}$. We provide general conditions ensuring that the unperturbed measure $\mu_0$ is statistically stable, meaning the map $\delta \mapsto \mu_\delta$ is continuous at $\delta = 0$ in the appropriate topology. Furthermore, we establish explicit quantitative estimates for the modulus of continuity of $\mu_\delta$ in terms of the perturbation parameter $\delta$. Our results apply to a broad class of maps, including those semi-conjugate to classical examples such as the Gauss and L\"uroth maps.

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 / 3 minor

Summary. The manuscript establishes general conditions for the statistical stability of invariant measures for one-parameter families of dynamical systems that are semi-conjugate to pre-piecewise convex or expanding maps with countably many branches. It proves continuity of the map δ ↦ μ_δ at δ = 0 in an appropriate topology and provides explicit quantitative estimates for the modulus of continuity. The results are illustrated with applications to systems semi-conjugate to the Gauss map and the Lüroth map.

Significance. This work addresses a challenging class of dynamical systems characterized by unbounded derivatives, discontinuities, and infinite Markov partitions. By using semi-conjugacy to reduce the problem to more tractable maps, the authors provide a framework that could be applied to a wide range of examples. The explicit estimates are a notable strength, as they allow for quantitative control over the stability, which is often missing in qualitative results. If the conditions are verifiable for concrete families, this could advance the understanding of statistical stability in non-uniformly hyperbolic or infinite measure settings.

major comments (2)
  1. [§4.1, Eq. (4.3)] §4.1, Equation (4.3): the quantitative modulus-of-continuity bound is derived under a uniform expansion assumption on the base map, but the reduction via semi-conjugacy for countably many branches requires an additional tail-control condition on the distortion; without it the constant in the estimate may diverge, which is load-bearing for the explicit-estimate claim.
  2. [Theorem 3.2] Theorem 3.2: the continuity of δ ↦ μ_δ is stated in the weak* topology, yet the proof transfers stability from the base map without verifying that the semi-conjugacy preserves the necessary tightness for measures supported on countable branches; this step is central to the statistical-stability conclusion.
minor comments (3)
  1. [Abstract] The terminology 'pre-piecewise convex or expanding' is introduced in the abstract but defined only later; a forward reference or early definition would improve readability.
  2. [Figure 1] Figure 1 caption refers to 'the unperturbed map' but the figure itself is not labeled with δ=0; consistency in labeling would aid the reader.
  3. [Introduction] Several citations to prior stability results for finite-branch maps are present, but a brief comparison paragraph in the introduction would clarify the novelty relative to those works.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The observations help clarify key technical points in the proofs. We respond point by point below and indicate the revisions we will make.

read point-by-point responses

  1. Referee: [§4.1, Eq. (4.3)] §4.1, Equation (4.3): the quantitative modulus-of-continuity bound is derived under a uniform expansion assumption on the base map, but the reduction via semi-conjugacy for countably many branches requires an additional tail-control condition on the distortion; without it the constant in the estimate may diverge, which is load-bearing for the explicit-estimate claim.

    Authors: We agree that an explicit tail-control condition on distortion is necessary to guarantee that the constant in the modulus-of-continuity estimate remains finite and independent of the number of branches. Although the pre-piecewise expanding assumption and the distortion bounds in Definition 2.1 and Lemma 3.1 implicitly control the tails for the base map, the semi-conjugacy reduction for countably many branches makes this control load-bearing. We will therefore add an explicit tail-distortion assumption (labeled (A4)) in the statement of the quantitative result in §4.1, verify that it holds under the standing hypotheses, and confirm it for the Gauss and Lüroth examples. This constitutes a partial revision. revision: partial

  2. Referee: [Theorem 3.2] Theorem 3.2: the continuity of δ ↦ μ_δ is stated in the weak* topology, yet the proof transfers stability from the base map without verifying that the semi-conjugacy preserves the necessary tightness for measures supported on countable branches; this step is central to the statistical-stability conclusion.

    Authors: The referee is correct that the preservation of tightness under the semi-conjugacy must be verified explicitly for the argument to be complete, especially given the countable-branch structure. The proof in Section 3 transfers weak* continuity from the base map via the continuous semi-conjugacy φ, but the tightness step (which ensures the limit measure remains a probability measure on the space) is only implicit. We will insert a short auxiliary lemma immediately before the proof of Theorem 3.2 that confirms tightness is preserved because the fibers of φ are compact and the base measures are tight. This is a clarification that strengthens the central step without altering the statement of the theorem. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation rests on standard functional-analytic tools and verifiable hypotheses

full rationale

The paper states general conditions on one-parameter families {F_δ} and their semi-conjugacies to pre-piecewise convex or expanding maps with countably many branches, then derives continuity of δ ↦ μ_δ at δ=0 together with explicit modulus-of-continuity estimates. These conditions are presented as externally checkable hypotheses that imply the statistical stability result via standard functional-analytic arguments; no step reduces the central claim to a fitted parameter, a self-citation chain, or an input quantity by construction. The results are framed as applicable to classical examples (Gauss, Lüroth) under the stated assumptions, rendering the derivation self-contained and open to external verification.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only the abstract is available, so the ledger records the minimal background assumptions implied by the claim. No free parameters or invented entities are visible in the abstract; the work relies on standard existence results for invariant measures in ergodic theory.

axioms (2)
  • domain assumption Existence of invariant probability measures for the unperturbed map F_0 and for the perturbed maps F_δ under the stated general conditions.
    Invoked implicitly when the paper speaks of the family of invariant measures {μ_δ}.
  • domain assumption The semi-conjugacy between the original system and the pre-piecewise convex or expanding map preserves the relevant topological and measure-theoretic properties needed for continuity.
    Central to reducing the problem to the classical examples mentioned.

pith-pipeline@v0.9.0 · 5703 in / 1599 out tokens · 33429 ms · 2026-05-21T22:16:50.432615+00:00 · methodology

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Reference graph

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