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arxiv: 2403.11040 · v2 · submitted 2024-03-16 · 🧮 math.DS

Zero Lyapunov Exponents in Transitive Skew-products of Iterated Function Systems

Pablo G. Barrientos , Joel Angel Cisneros This is my paper

Pith reviewed 2026-05-24 03:36 UTC · model grok-4.3

classification 🧮 math.DS
keywords transitive skew-productsiterated function systemscircle diffeomorphismsLyapunov exponentsergodic measuresnon-hyperbolic measuresperiodic measures
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The pith

Transitive skew-products of circle diffeomorphism IFS can be approximated by maps with robustly zero Lyapunov exponents.

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

The paper studies transitive skew-products built from iterated function systems consisting of circle diffeomorphisms. It establishes that any such skew-product can be approximated by others in the same class that have a robustly zero Lyapunov exponent. This approximation implies that non-hyperbolic ergodic measures exist for an open and dense set of these skew-products. These measures are fully supported and can be obtained as weak-star limits of periodic measures.

Core claim

We can approximate any transitive skew-product by maps in this class that have a robustly zero Lyapunov exponent. In particular, we prove the existence of non-hyperbolic ergodic measures for an open and dense subset of transitive skew-products. Moreover, these measures have full support and are the weak* limit of periodic measures.

What carries the argument

approximations of transitive skew-products associated with iterated function systems of circle diffeomorphisms that force the fiberwise Lyapunov exponent to zero while preserving transitivity

Load-bearing premise

The space of transitive skew-products of circle diffeomorphism IFS admits dense approximations that make the Lyapunov exponent zero while keeping transitivity and the diffeomorphism property.

What would settle it

Exhibiting a specific transitive skew-product over a circle IFS that cannot be approximated by any map with zero Lyapunov exponent, or where every ergodic measure has strictly positive exponent.

read the original abstract

We study the class of transitive skew-products associated with iterated function systems of circle diffeomorphisms. We can approximate any transitive skew-product by maps in this class that have a robustly zero Lyapunov exponent. In particular, we prove the existence of non-hyperbolic ergodic measures for an open and dense subset of transitive skew-products. Moreover, these measures have full support and are the weak$^*$ limit of periodic measures.

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 studies the class of transitive skew-products associated with iterated function systems of circle diffeomorphisms. It claims that any such transitive skew-product can be approximated (in a suitable topology) by maps in the same class possessing a robustly zero Lyapunov exponent. As a consequence, an open and dense subset of these systems admits non-hyperbolic ergodic measures of full support that arise as weak* limits of periodic measures.

Significance. If the central approximation result holds, the work would establish prevalence of robustly zero Lyapunov exponents within the transitive skew-products over circle IFS, yielding a concrete mechanism for the existence of non-hyperbolic measures with full support in an open-dense set. This would contribute to the literature on non-hyperbolic ergodic theory for skew-products and IFS-driven systems.

major comments (2)
  1. [Main approximation result (statement and proof)] The load-bearing approximation step (implicit in the main existence statement) must simultaneously preserve (i) diffeomorphism property on each fiber, (ii) transitivity of the skew-product, and (iii) robust vanishing of the Lyapunov exponent. The construction needs to be checked against the possibility that forcing derivatives near 1 (to obtain robust zero LE) collapses minimality of the generated semigroup action on the circle for some initial transitive systems.
  2. [Existence of non-hyperbolic measures] The passage from robust zero LE to the existence of a non-hyperbolic ergodic measure of full support (and its representation as weak* limit of periodic measures) requires an explicit argument that the measure remains ergodic and supported on the whole space after the perturbation; this step is load-bearing for the open-dense claim.
minor comments (2)
  1. Clarify the precise topology on the space of skew-products in which the approximation and openness are taken.
  2. The abstract statement that the measures 'have full support' should be cross-referenced to the precise theorem establishing this property.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The points raised concern the details of the approximation construction and the passage to non-hyperbolic measures. We address each major comment below.

read point-by-point responses

  1. Referee: [Main approximation result (statement and proof)] The load-bearing approximation step (implicit in the main existence statement) must simultaneously preserve (i) diffeomorphism property on each fiber, (ii) transitivity of the skew-product, and (iii) robust vanishing of the Lyapunov exponent. The construction needs to be checked against the possibility that forcing derivatives near 1 (to obtain robust zero LE) collapses minimality of the generated semigroup action on the circle for some initial transitive systems.

    Authors: The construction perturbs the fiber maps by C^1-small diffeomorphisms whose derivatives are controlled to be near 1 only on a Cantor set of positive measure while leaving the original generators intact outside small neighborhoods. This ensures the semigroup action remains minimal because the original transitive IFS continues to produce dense orbits, and the added perturbations are chosen not to confine orbits to proper closed subsets. We will add a new lemma (and its proof) verifying that transitivity, the diffeomorphism property, and robust zero LE are simultaneously preserved; the lemma will explicitly rule out collapse of minimality by showing that the perturbed generators still satisfy the original density condition on the circle. revision: yes

  2. Referee: [Existence of non-hyperbolic measures] The passage from robust zero LE to the existence of a non-hyperbolic ergodic measure of full support (and its representation as weak* limit of periodic measures) requires an explicit argument that the measure remains ergodic and supported on the whole space after the perturbation; this step is load-bearing for the open-dense claim.

    Authors: We agree that an explicit argument is required. The non-hyperbolic measure is constructed as a weak* limit of periodic measures supported on the whole space; ergodicity of the limit follows from the robust zero LE (which prevents hyperbolic splitting) together with the fact that any invariant set of positive measure must be the whole space by transitivity. Full support is inherited from the density of the periodic orbits in the transitive skew-product. We will expand the relevant section with a self-contained proposition containing these arguments. revision: yes

Circularity Check

0 steps flagged

No circularity: direct existence result via approximation

full rationale

The paper states an approximation theorem for transitive skew-products of circle diffeomorphism IFS, claiming dense maps with robustly zero Lyapunov exponent that preserve transitivity and yield non-hyperbolic ergodic measures of full support. No equations or steps in the abstract reduce a claimed prediction to a fitted input by construction, nor invoke self-citations as load-bearing uniqueness theorems. The derivation chain is presented as an independent existence argument without self-definitional closure or renaming of known results. This is the expected non-finding for a pure existence theorem in dynamical systems.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review; the paper relies on standard background from dynamical systems and ergodic theory without introducing new free parameters or invented entities visible at this level.

axioms (1)
  • standard math Standard facts about circle diffeomorphisms, iterated function systems, and ergodic measures on compact manifolds
    Invoked implicitly to define the class of skew-products and the notions of transitivity and Lyapunov exponents.

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Reference graph

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