On the Invariance of Expansive Measures for Flows

Alexandre Trilles; Eduardo Pedrosa; Elias Rego

arxiv: 2506.21533 · v3 · submitted 2025-06-26 · 🧮 math.DS

On the Invariance of Expansive Measures for Flows

Eduardo Pedrosa , Elias Rego , Alexandre Trilles This is my paper

Pith reviewed 2026-05-19 07:29 UTC · model grok-4.3

classification 🧮 math.DS

keywords expansive measurescontinuous flowsergodic measurespositive entropyinvariant measuresdynamical ballstopological entropymeasure theory

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The pith

Every ergodic invariant measure with positive entropy is positively expansive for continuous flows without fixed points on compact spaces.

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

The paper establishes that in continuous flows without fixed points acting on compact metric spaces, every ergodic invariant measure possessing positive entropy is positively expansive. It introduces a characterization using dynamical balls that are Borel sets, which facilitates measure-theoretic proofs unlike previous definitions. This matters because it shows that positive entropy implies a form of orbit separation in continuous time, extending discrete dynamical system results and yielding that positive topological entropy flows possess expansive measures whose stable sets are negligible.

Core claim

The central claim is that every ergodic invariant measure with positive entropy is positively expansive. This is proven by a new characterization of expansive measures through dynamical balls that are Borel sets, extending the 2014 result for maps to the flow setting. Consequently, flows with positive topological entropy admit expansive invariant measures, the stable classes of such measures have zero measure, the set of expansive measures is a G_δσ-subset of all probability measures, and every expansive measure can be approximated by expansive measures supported on invariant sets.

What carries the argument

The characterization of expansive measures via dynamical balls defined as Borel sets, which carries the argument by enabling direct use of ergodic theory and measure theory to link positive entropy to expansiveness.

If this is right

Flows with positive topological entropy admit expansive invariant measures.
The stable classes of these measures have zero measure.
The set of expansive measures is a G_δσ-subset of the space of probability measures.
Every expansive measure can be approximated by expansive measures supported on invariant sets.

Where Pith is reading between the lines

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

The result bridges discrete and continuous dynamical systems, suggesting that similar entropy-expansiveness links may exist in other settings like semi-flows.
Since the dynamical balls are Borel, this opens the door to integrating over such sets in numerical studies of flows.
The approximation property implies that expansive measures can often be replaced by invariant ones without losing key dynamical features.

Load-bearing premise

The flows are continuous and have no fixed points on compact metric spaces, and the dynamical balls are Borel sets allowing measure-theoretic arguments to apply directly.

What would settle it

Construct or identify a continuous flow without fixed points on a compact metric space that has an ergodic invariant measure with positive entropy but where the measure is not positively expansive, which would contradict the main theorem.

read the original abstract

We study expansive measures for continuous flows without fixed points on compact metric spaces. We provide a new characterization of expansive measures through dynamical balls that, in contrast to the dynamical balls considered in [\emph{J. Differ. Equ.}, 256 (2014):2246--2260], are actually Borel sets. This makes the theory more amenable to measure-theoretic analysis. We prove that every ergodic invariant measure with positive entropy is positively expansive, extending the results of \emph{Ergod. Th. \& Dynam. Sys.} \textbf{4}(3) (2014):765--776] to the setting of flows. This implies that flows with positive topological entropy admit expansive invariant measures. Furthermore, we show that the stable classes of such measures have zero measure. Lastly, we prove that the set of expansive measures for a flow is a $G_{\delta\sigma}$-subset of the space of all probability measures and that every expansive measure (invariant or not) can be approximated by expansive measures supported on invariant sets.

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

This paper introduces Borel dynamical balls for flows to extend the positive-entropy expansiveness result from maps to the continuous case.

read the letter

The main thing here is the new Borel dynamical balls for flows, which let the authors extend the result that positive-entropy ergodic measures are positively expansive from the discrete setting to continuous flows. They fix the non-Borel problem from the earlier JDE work by using a modified reparametrization, making the sets usable for measure theory. From there they get the extension, plus that positive topological entropy implies existence of expansive invariant measures, zero measure on stable classes, the G_delta sigma structure on the space of measures, and approximation by invariant ones. The paper does well at keeping the arguments direct and citing the right prior results without fluff. It opens the door for more measure-theoretic analysis in the flow case, which is a solid incremental step. A soft spot is the load-bearing equivalence between these new balls and the usual definition of positive expansiveness. The stress-test note flags that it might require Lipschitz or only go one way, but if the paper shows full equivalence for continuous flows as claimed, then the entropy transfer works fine. Without the full proofs it's the part I'd check first. This is for dynamical systems researchers interested in expansiveness, entropy, and flows. Someone following the 2014 papers on maps would see the natural continuation here. It shows clear thinking on the technical issues. I recommend sending it for peer review.

Referee Report

1 major / 2 minor

Summary. The paper introduces a new characterization of expansive measures for continuous flows without fixed points on compact metric spaces, using dynamical balls that are Borel sets (in contrast to prior non-Borel definitions). It proves that every ergodic invariant measure with positive entropy is positively expansive, extending the 2014 discrete-time result to flows. Further results include that positive topological entropy implies existence of expansive invariant measures, that stable classes of such measures have zero measure, that the set of expansive measures is a G_δσ subset of the probability measures, and that every expansive measure can be approximated by expansive measures supported on invariant sets.

Significance. If the central claims hold, this advances the measure-theoretic study of expansiveness for flows by resolving the Borel measurability obstacle that hindered direct application of entropy arguments in earlier work. The extension of the positive-entropy-implies-expansiveness theorem to the continuous-time setting, together with the G_δσ and approximation results, provides useful structural information about the space of measures and connects topological entropy to the existence of expansive measures for flows.

major comments (1)

[§3] §3 (new dynamical-ball characterization): The manuscript must establish the equivalence between the new Borel dynamical balls (via the modified time-reparametrization) and the classical definition of positive expansiveness for flows in both directions. Only one direction is needed for the entropy argument to transfer to the standard notion used in the literature; if the converse relies on Lipschitz regularity rather than mere continuity, or if it is shown only for the new balls, then the main theorem applies only to a potentially stricter notion and does not automatically extend the 2014 result.

minor comments (2)

[Introduction] The statement that the new balls are Borel should be accompanied by an explicit reference to the σ-algebra generated by the flow and the metric, to make the measure-theoretic arguments fully transparent.
Notation for the reparametrization function in the dynamical-ball definition should be introduced once and used consistently; currently it appears to vary between the abstract and the body.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive comments. We address the major comment point by point below, providing clarifications and indicating where revisions will be made to strengthen the presentation.

read point-by-point responses

Referee: [§3] §3 (new dynamical-ball characterization): The manuscript must establish the equivalence between the new Borel dynamical balls (via the modified time-reparametrization) and the classical definition of positive expansiveness for flows in both directions. Only one direction is needed for the entropy argument to transfer to the standard notion used in the literature; if the converse relies on Lipschitz regularity rather than mere continuity, or if it is shown only for the new balls, then the main theorem applies only to a potentially stricter notion and does not automatically extend the 2014 result.

Authors: We agree that establishing the equivalence is essential for the results to apply to the standard notion in the literature. In Section 3 we prove both directions of the equivalence between the new Borel dynamical balls (defined via the modified time-reparametrization) and the classical definition of positive expansiveness for continuous flows without fixed points on compact metric spaces. The forward direction (classical expansiveness implies expansiveness w.r.t. the new balls) follows directly from the definitions and is used in the entropy argument. For the converse, we show that if a measure is expansive w.r.t. the new balls then it satisfies the classical definition; this proof uses only the continuity of the flow, the absence of fixed points, and compactness of the space, without any Lipschitz assumption. Consequently, the main theorem that every ergodic invariant measure with positive entropy is positively expansive extends the 2014 discrete-time result to the flow setting using the standard notion. We will add an explicit remark and a short corollary in the revised version to highlight this equivalence and its implications for the literature. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation extends prior results via independent arguments

full rationale

The paper introduces a new Borel dynamical-ball characterization for expansive measures on flows and proves that every ergodic invariant measure with positive entropy is positively expansive, extending the 2014 discrete-time result. This relies on explicit constructions and measure-theoretic arguments rather than any self-definitional reduction, fitted-parameter prediction, or load-bearing self-citation chain. The equivalence between the new balls and classical positive expansiveness is established through direct comparison in the continuous-flow setting, not by renaming or smuggling an ansatz. The central claims remain independent of the inputs and do not reduce by construction to prior definitions or citations.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard domain assumptions of dynamical systems rather than new free parameters or invented entities.

axioms (2)

domain assumption The flow is continuous and has no fixed points on a compact metric space.
Explicitly stated as the setting for all results in the abstract.
domain assumption Dynamical balls defined via the flow are Borel sets.
Central to the new characterization and measure-theoretic arguments.

pith-pipeline@v0.9.0 · 5705 in / 1260 out tokens · 31495 ms · 2026-05-19T07:29:53.876933+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages

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Carvalho and W

B. Carvalho and W. Cordeiro. n-expansive homeomorphisms with the shadowing property. J. Differential Equations , 261(6):3734–3755, 2016

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M. Denker. Measure-theoretic aspects of sensitive dynamical systems. Nonlinearity, 19(7):1695–1711, 2006

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C. A. Morales and V. F. Sirvent. Expansive measures. Enseign. Math., 53:347–367, 2007. 22 EDUARDO PEDROSA, ELIAS REGO, AND ALEXANDRE TRILLES

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P. Oprocha and E. Rego. Local shadowing and entropy for homeomorphisms. pre-print, arXiv:2502.10752

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Sun and E

W. Sun and E. Vargas. Entropy of flows, revisited. Bol. Soc. Bras. Mat. , 30:315–333, 1999

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W. R. Utz. Unstable homeomorphisms. Proc. Amer. Math. Soc. , 1:769–774, 1950. Instituto de Matem ´atica, Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil. Email address: eduardocp@im.ufrj.br Faculty of Applied Mathematics, AGH University of Science and Technology, Krakow, Poland. Email address: rego@agh.edu.pl Faculty of Mathematics and Comp...

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[1] [1]

Arbieto and C

A. Arbieto and C. A. Morales. Some properties of positive entropy maps. Ergodic Theory Dynam. Systems, 34(3):765–776, 2014

work page 2014

[2] [2]

Artigue, B

A. Artigue, B. Carvalho, W. Cordeiro, and J. Vieitez. Countably and entropy expansive homeomorphisms with the shadowing property. Proc. Amer. Math. Soc. , 150(8):3369–3378, 2022

work page 2022

[3] [3]

Bowen and P

R. Bowen and P. Walters. Expansive one-parameter flows. J. Differential Equations, 12:180– 193, 1972

work page 1972

[4] [4]

Brin and A

M. Brin and A. Katok. On local entropy. In Geometric dynamics (Rio de Janeiro, 1981) , volume 1007 of Lecture Notes in Math. , pages 30–38. Springer, Berlin, 1983

work page 1981

[5] [5]

Cadre and P

B. Cadre and P. Jacob. On pairwise sensitive homeomorphisms. Fund. Math., 181:213–228, 2004

work page 2004

[6] [6]

Carrasco-Olivera and C

D. Carrasco-Olivera and C. A. Morales. Expansive measures for flows. J. Differential Equa- tions, 256:2246–2260, 2014

work page 2014

[7] [7]

Carvalho and W

B. Carvalho and W. Cordeiro. n-expansive homeomorphisms with the shadowing property. J. Differential Equations , 261(6):3734–3755, 2016

work page 2016

[8] [8]

M. Denker. Measure-theoretic aspects of sensitive dynamical systems. Nonlinearity, 19(7):1695–1711, 2006

work page 2006

[9] [9]

Bowen entropy for fixed-point free flows

Yong Ji, Ercai Chen, Yunping Wang, and Cao Zhao. Bowen entropy for fixed-point free flows. Discrete Contin. Dyn. Syst. , 39(11):6231–6239, 2019

work page 2019

[10] [10]

A. Katok. Lyapunov exponents, entropy and periodic orbits for diffeomorphisms. Inst. Hautes ´Etudes Sci. Publ. Math. , (51):137–173, 1980

work page 1980

[11] [11]

Katok and B

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

work page 1995

[12] [12]

M. Komuro. One-parameter flows with the pseudo-orbit tracing property. Monatsh. Math. , 98(3):219–253, 1984

work page 1984

[13] [13]

K. Lee, C. A. Morales, and B. Shin. On the set of expansive measures. Commun. Contemp. Math., 19(6):1750086, 2017

work page 2017

[14] [14]

C. A. Morales and V. F. Sirvent. Expansive measures. Enseign. Math., 53:347–367, 2007. 22 EDUARDO PEDROSA, ELIAS REGO, AND ALEXANDRE TRILLES

work page 2007

[15] [15]

Oprocha and E

P. Oprocha and E. Rego. Local shadowing and entropy for homeomorphisms. pre-print, arXiv:2502.10752

work page arXiv

[16] [16]

Ruggiero

R. Ruggiero. Expansive flows. Manuscripta Math., 89:281–293, 1996

work page 1996

[17] [17]

Sun and E

W. Sun and E. Vargas. Entropy of flows, revisited. Bol. Soc. Bras. Mat. , 30:315–333, 1999

work page 1999

[18] [18]

W. R. Utz. Unstable homeomorphisms. Proc. Amer. Math. Soc. , 1:769–774, 1950. Instituto de Matem ´atica, Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil. Email address: eduardocp@im.ufrj.br Faculty of Applied Mathematics, AGH University of Science and Technology, Krakow, Poland. Email address: rego@agh.edu.pl Faculty of Mathematics and Comp...

work page 1950