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arxiv: 2512.13862 · v4 · submitted 2025-12-15 · 🧮 math.DS

Invariance principle in dynamical systems

Karina Marin , Mauricio Poletti This is my paper

Pith reviewed 2026-05-16 21:39 UTC · model grok-4.3

classification 🧮 math.DS
keywords invariance principledynamical systemsLyapunov exponentsholonomiesmeasure disintegrationpartially hyperbolic systems
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The pith

Disintegrations of measures with zero center Lyapunov exponents admit extra invariance under holonomies.

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

The paper surveys the Invariance Principle in dynamical systems. This principle asserts that the disintegration of measures possessing zero center Lyapunov exponents possesses additional invariance properties with respect to holonomies. Understanding this helps explain the behavior of invariant measures in systems with partial hyperbolicity. The survey covers basic definitions, key results, and some applications to dynamical systems.

Core claim

The Invariance Principle states that the disintegration of measures with zero center Lyapunov exponents admits some extra invariance by holonomies.

What carries the argument

The Invariance Principle, which establishes additional holonomy invariance for the disintegration of measures with zero center Lyapunov exponents.

If this is right

  • Measures with this property can be shown to have specific ergodic properties.
  • The principle aids in classifying invariant measures in partially hyperbolic systems.
  • Applications include rigidity results and proofs of invariance in concrete dynamical systems.

Where Pith is reading between the lines

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

  • It may extend to measures in higher-dimensional systems where center directions are present.
  • Numerical simulations of specific diffeomorphisms could test the invariance in practice.

Load-bearing premise

The measures under consideration have zero center Lyapunov exponents.

What would settle it

Finding a measure with zero center Lyapunov exponents whose disintegration does not remain invariant under holonomies would disprove the principle.

read the original abstract

In this survey we talk about what is known as Invariance Principle in dynamical systems. It states that the disintegration of measures with zero center Lyapunov exponents admits some extra invariance by holonomies. We focus on explaining the basic definitions and ideas behind a series of results about the Invariance Principle and give some basic applications on how this is used in dynamical systems.

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 / 2 minor

Summary. The manuscript is a survey on the Invariance Principle in dynamical systems. It states that the disintegration of measures with zero center Lyapunov exponents admits extra invariance by holonomies. The paper focuses on explaining the basic definitions, key ideas from existing results in the literature, and some basic applications in dynamical systems.

Significance. As a purely expository survey restating a known result without new theorems, proofs, or derivations, the paper's significance is primarily pedagogical: it may help clarify the Invariance Principle and its applications for researchers and students in dynamical systems, especially those studying partially hyperbolic diffeomorphisms or measures with zero center exponents. Credit is due for the explicit statement of the zero-center-exponent hypothesis as a prerequisite, consistent with the surveyed literature.

minor comments (2)
  1. [Abstract] The abstract could be strengthened by naming one or two key references or theorems that form the core of the surveyed results, to better guide readers to the primary literature.
  2. Ensure that all basic definitions (e.g., disintegration, holonomies, center Lyapunov exponents) are accompanied by precise citations to standard sources in the field for readers who may need to consult the original statements.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of our survey manuscript. We agree that the paper is expository in nature and focuses on clarifying the Invariance Principle, its definitions, and basic applications for the dynamical systems community. We will prepare a revised version incorporating any minor improvements suggested.

Circularity Check

0 steps flagged

No significant circularity; purely expository survey of known result

full rationale

The manuscript is an expository survey that restates the standard Invariance Principle (disintegration of measures with zero center Lyapunov exponents is invariant under holonomies) without new theorems, proofs, or parameter-dependent claims. The central statement is presented as a known result whose hypotheses (zero center exponents) are explicitly required in the literature it surveys; no internal derivation or extension is attempted that could introduce a hidden assumption or reduce any claim to fitted inputs, self-definitions, or self-citation chains. All steps are self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The survey relies on standard background from ergodic theory and smooth dynamical systems without introducing new free parameters, axioms beyond domain standards, or invented entities.

axioms (1)
  • standard math Standard definitions and properties of Lyapunov exponents and holonomies in partially hyperbolic dynamical systems.
    Invoked in the statement of the principle as background knowledge.

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Reference graph

Works this paper leans on

33 extracted references · 33 canonical work pages

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