pith. sign in

arxiv: 2604.02810 · v1 · submitted 2026-04-03 · 🧮 math.DS

Invariant measures with full support and approximation by zero-entropy systems in the C⁰-Gromov--Hausdorff topology

Richard Javier Cubas Becerra , Jorge Cris\'ostomo Parejas This is my paper

Pith reviewed 2026-05-13 18:44 UTC · model grok-4.3

classification 🧮 math.DS
keywords homeomorphismsinvariant measuresfull supporttopological entropyC0-Gromov-Hausdorff topologyergodic decompositionzero entropyapproximation
0
0 comments X

The pith

Every homeomorphism admitting a full-support invariant probability measure can be approximated in the C^0-Gromov-Hausdorff topology by zero-entropy homeomorphisms.

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

The paper shows that any homeomorphism of a compact metric space preserving a probability measure with support equal to the whole space can be approximated arbitrarily closely by zero-entropy homeomorphisms when distance is measured in the C^0-Gromov-Hausdorff topology. The result follows from decomposing the invariant measure into ergodic components and using the existence of dense positive orbits within those supports. This immediately yields that topological entropy is not stable under such perturbations. Additional statements establish that topologically GH-stable members of the class have dense periodic points and that transitive GH-stable homeomorphisms must themselves carry a full-support invariant measure.

Core claim

Every homeomorphism of a compact metric space admitting an invariant probability measure with full support can be approximated in the C^0-Gromov-Hausdorff topology by homeomorphisms with zero topological entropy. The argument relies on the ergodic decomposition theorem and on the existence of points with dense positive orbit in the supports of suitable ergodic components. As a consequence, topological entropy is not stable under C^0-Gromov-Hausdorff perturbations within this class. If the homeomorphism is topologically GH-stable, its periodic points are dense in the ambient space. By combining this framework with a previous result on transitive and topologically GH-stable homeomorphisms, the

What carries the argument

Ergodic decomposition of the invariant measure, combined with the existence of dense positive orbits in the supports of its ergodic components, to produce C^0-Gromov-Hausdorff approximations by zero-entropy homeomorphisms.

If this is right

  • Topological entropy fails to be stable under C^0-Gromov-Hausdorff perturbations for all homeomorphisms that admit a full-support invariant measure.
  • Any topologically GH-stable homeomorphism in the class must have dense periodic points in the ambient space.
  • Every topologically transitive and GH-stable homeomorphism admits an invariant measure with full support.
  • The set of homeomorphisms approximable by zero-entropy systems includes the entire class of systems possessing a full-support invariant measure.

Where Pith is reading between the lines

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

  • The same approximation strategy may extend to other dynamical invariants that are not stable under C^0-Gromov-Hausdorff limits.
  • Zero-entropy homeomorphisms are dense, in the C^0-Gromov-Hausdorff metric, inside the subclass of homeomorphisms that carry full-support invariant measures.
  • Stability results for transitive systems can now be transferred directly to questions about the existence of full-support measures.

Load-bearing premise

The existence of points with dense positive orbits in the supports of suitable ergodic components of the invariant measure.

What would settle it

A concrete homeomorphism on a compact metric space that carries a full-support invariant probability measure yet cannot be approximated arbitrarily closely in the C^0-Gromov-Hausdorff topology by any zero-entropy homeomorphism.

read the original abstract

In this paper we prove that every homeomorphism of a compact metric space admitting an invariant probability measure with full support can be approximated in the $C^0$-Gromov--Hausdorff topology by homeomorphisms with zero topological entropy. The argument relies on the ergodic decomposition theorem and on the existence of points with dense positive orbit in the supports of suitable ergodic components. As a consequence, topological entropy is not stable under $C^0$-Gromov--Hausdorff perturbations within this class. We also show that if, in addition, the homeomorphism is topologically $GH$-stable, then its periodic points are dense in the ambient space. Finally, by combining this framework with a previous result on transitive and topologically $GH$-stable homeomorphisms, we deduce that every dynamics in this class admits an invariant measure with full support and therefore falls within the scope of the general approximation theorem by zero-entropy 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 paper proves that every homeomorphism of a compact metric space admitting an invariant probability measure with full support can be approximated in the C^0-Gromov-Hausdorff topology by homeomorphisms with zero topological entropy. The argument applies the ergodic decomposition theorem to the given measure and invokes the existence of points with dense positive orbits in the supports of suitable ergodic components to construct the approximating sequence. Consequences include the non-stability of topological entropy under such perturbations and, for topologically GH-stable maps, the density of periodic points. The framework is also combined with a prior result on transitive GH-stable homeomorphisms to deduce that every such system admits a full-support invariant measure.

Significance. If the central claim holds, the result is a useful addition to topological dynamics: it shows that zero-entropy homeomorphisms are dense (in the C^0-GH topology) within the class of systems possessing a full-support invariant measure. The proof strategy correctly combines the ergodic decomposition theorem with standard facts about dense orbits and handles the metric requirements of the approximation without circularity. The derived statements on entropy non-stability and periodic-point density for GH-stable maps are natural corollaries. The linkage to earlier work on transitive GH-stable maps broadens the scope and is a clear strength.

minor comments (2)
  1. The statement of the main approximation theorem would benefit from an explicit sentence clarifying that the approximating homeomorphisms are required to be defined on the same space (or on spaces that converge in the GH sense) to avoid any ambiguity in the topology.
  2. A brief remark on the dependence of the approximation on the choice of ergodic component (or on how the sequence is extracted from the decomposition) would improve readability in the proof section.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for the positive assessment of its contributions. We are grateful for the recommendation to accept the paper.

Circularity Check

0 steps flagged

Minor self-citation to prior result on transitive GH-stable maps; central approximation independent

full rationale

The paper's core argument applies the ergodic decomposition theorem (external standard result) to an invariant measure with full support, then invokes the existence of dense positive orbits in suitable ergodic components to construct approximating zero-entropy homeomorphisms in the C^0-Gromov-Hausdorff topology. This chain is self-contained and does not reduce any prediction or claim to a fitted parameter or self-referential definition. The only self-reference appears in the final deduction, where a prior result on transitive GH-stable maps is combined to show that such systems admit full-support measures and thus fall under the main theorem; this corollary is not required to establish the primary approximation result.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on two standard background results from ergodic theory and one prior theorem on transitive GH-stable maps; no free parameters, ad-hoc constants, or new postulated entities are introduced.

axioms (1)
  • standard math Ergodic decomposition theorem for invariant probability measures on compact metric spaces
    Invoked to decompose the given fully supported measure into ergodic components whose supports admit dense positive orbits.

pith-pipeline@v0.9.0 · 5480 in / 1300 out tokens · 50805 ms · 2026-05-13T18:44:02.806223+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

  1. [1]

    Arbieto and C

    A. Arbieto and C. A. Morales Rojas. Topological stability from Gromov–Hausdorff viewpoint. Discrete and Continuous Dynamical Systems, 37(7):3531–3544, 2017

    work page 2017

  2. [2]

    Burago, Y

    D. Burago, Y. Burago, and S. Ivanov.A Course in Metric Geometry, volume 33 ofGraduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2001

    work page 2001

  3. [3]

    R. J. Cubas Becerra. Propriedades de um homeomorfismo GH estável. Master’s thesis, Uni- versidade Federal de Uberlândia, Uberlândia, 2018

    work page 2018

  4. [4]

    Gromov.Metric Structures for Riemannian and Non-Riemannian Spaces

    M. Gromov.Metric Structures for Riemannian and Non-Riemannian Spaces. Modern Birkhäuser Classics. Birkhäuser, Boston, 2007

    work page 2007

  5. [5]

    W. Jung. The closure of periodic orbits in the Gromov–Hausdorff space.Topology and its Applications, 264:493–497, 2019

    work page 2019

  6. [6]

    Katok and B

    A. Katok and B. Hasselblatt.Introduction to the Modern Theory of Dynamical Systems, volume 54 ofEncyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 1995

    work page 1995

  7. [7]

    Lee and C

    J. Lee and C. Morales.Gromov–Hausdorff Stability of Dynamical Systems and Applications to PDEs. SpringerBriefs in Mathematics. Springer, Cham, 2022

    work page 2022

  8. [8]

    Robinson.Dynamical Systems: Stability, Symbolic Dynamics, and Chaos

    C. Robinson.Dynamical Systems: Stability, Symbolic Dynamics, and Chaos. CRC Press, Boca Raton, FL, 2 edition, 1999

    work page 1999

  9. [9]

    Viana and K

    M. Viana and K. Oliveira.Foundations of Ergodic Theory, volume 151 ofCambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2016

    work page 2016

  10. [10]

    P. Walters. Anosov diffeomorphisms are topologically stable.Topology, 9(1):71–78, 1970

    work page 1970

  11. [11]

    P. Walters. On the pseudo-orbit tracing property and its relationship to stability. InThe Structure of Attractors in Dynamical Systems, volume 668 ofLecture Notes in Mathematics, pages 231–244. Springer, Berlin, 1978. APPROXIMATION BY ZERO-ENTROPY SYSTEMS 15

    work page 1978

  12. [12]

    Walters.An Introduction to Ergodic Theory, volume 79 ofGraduate Texts in Mathematics

    P. Walters.An Introduction to Ergodic Theory, volume 79 ofGraduate Texts in Mathematics. Springer-Verlag, New York, 1982. F acultad de Ciencias Matemáticas, UNMSM, Lima, Perú Email address:jcrisostomop@unmsm.edu.pe Universidad Científica del Sur, Lima, Perú Email address:cubas.mat@usp.br

    work page 1982