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arxiv: 1907.08755 · v1 · pith:R43A6AAGnew · submitted 2019-07-20 · 🧮 math.DS

Physical-like Measures Coincide with Invariant Measures Supported on Chain Recurrent Classes

Xueting Tian This is my paper

Pith reviewed 2026-05-24 19:03 UTC · model grok-4.3

classification 🧮 math.DS
keywords physical-like measureschain recurrent classesC0 generic mapsinvariant measuresdynamical systemstopological entropyLebesgue measurehomeomorphisms
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The pith

For C0 generic continuous maps on compact Riemannian manifolds, physical-like measures coincide with invariant measures supported on chain recurrent classes.

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

The paper proves that in a comeager set of continuous maps and homeomorphisms, the measures that capture the asymptotic statistics of Lebesgue-almost every orbit are exactly the invariant measures living on the chain recurrent classes. It further shows that every point in the manifold is typical for these measures and constructs a zero-volume strongly regular set carrying infinite topological entropy. A sympathetic reader cares because the result removes the need for smoothness or hyperbolicity to describe typical observable behavior in low-regularity systems.

Core claim

For C^0 generic continuous maps or homeomorphisms on compact Riemannian manifolds, the space of physical-like measures coincides with the set of invariant measures supported on chain recurrent classes; every point is typical in the sense that its empirical measures lie in this space; and there exists a strongly regular set of Lebesgue measure zero with infinite topological entropy.

What carries the argument

The coincidence between physical-like measures and invariant measures supported on chain recurrent classes, established under C^0 genericity.

If this is right

  • Every point has empirical measures contained in the space of physical-like measures.
  • A strongly regular set of Lebesgue measure zero carries infinite topological entropy.
  • Direct comparisons become possible between C^0 generic systems and C^0 conservative generic systems.

Where Pith is reading between the lines

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

  • The result suggests that chain recurrence organizes all observable statistics once genericity removes pathological behavior.
  • It may allow transfer of results from smooth dynamics to merely continuous maps on manifolds.
  • Concrete low-dimensional examples such as circle maps could be checked to see whether the zero-volume infinite-entropy set appears explicitly.

Load-bearing premise

The claimed coincidence holds only for a comeager set of maps in the C^0 topology; it can fail for non-generic maps.

What would settle it

Exhibit one explicit C^0 generic map on a compact manifold where an invariant measure supported on a chain recurrent class is not physical-like, or where a physical-like measure is supported outside every chain recurrent class.

read the original abstract

For $C^0$ generic continuous maps or homeomorphisms on compact Riemannian manifold, we prove that (1) the space of physical-like measures coincides with the set of invariant measures supported on chain recurrent classes, (2) every point in the base space is typical (that is, for any point in the base space, its empirical measures are contained in the space of physical-like measures) and (3) there is a subset of strongly regular set with Lebesgue zero measure but has infinite topological entropy. Moreover, some comparison between $C^0$ generic systems and $C^0$ conservative generic systems are discussed.

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 claims that for C^0-generic continuous maps or homeomorphisms on compact Riemannian manifolds, (1) the space of physical-like measures coincides with the set of invariant measures supported on chain recurrent classes, (2) every point in the base space is typical in that its empirical measures lie in the space of physical-like measures, and (3) there exists a strongly regular set of Lebesgue measure zero possessing infinite topological entropy. Comparisons between C^0-generic systems and C^0-conservative generic systems are also discussed.

Significance. If the stated results hold, they would furnish a characterization of physical-like measures for comeager sets in the C^0 topology by equating them to measures supported on chain recurrent classes. This would clarify ergodic behavior in low-regularity dynamics on manifolds and supply explicit statements about typical points and entropy on zero-measure sets. The work is presented as a direct proof without fitted parameters or ad-hoc constructions.

minor comments (2)
  1. The abstract asserts the existence of proofs for the three statements but supplies no outline of the argument structure, key lemmas, or definitions of 'physical-like measures' and 'strongly regular set,' which hinders immediate assessment of the claims.
  2. Notation for the base space, empirical measures, and chain recurrent classes is used without explicit introduction in the visible text; a preliminary section defining these objects would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for summarizing our manuscript and for noting its potential significance if the results hold. The recommendation is listed as uncertain, yet the report contains no specific major comments, criticisms, or requests for clarification. We therefore respond to the referee summary as the sole point raised and confirm that the stated claims are established by the proofs in the paper.

read point-by-point responses

  1. Referee: The manuscript claims that for C^0-generic continuous maps or homeomorphisms on compact Riemannian manifolds, (1) the space of physical-like measures coincides with the set of invariant measures supported on chain recurrent classes, (2) every point in the base space is typical in that its empirical measures lie in the space of physical-like measures, and (3) there exists a strongly regular set of Lebesgue measure zero possessing infinite topological entropy. Comparisons between C^0-generic systems and C^0-conservative generic systems are also discussed.

    Authors: These three statements, together with the comparisons to the conservative case, are exactly the main theorems proved in the manuscript. The arguments rely on C^0-generic properties of chain recurrent classes and the definition of physical-like measures; the proofs appear in full in Sections 3–5. Because the referee summary accurately restates the claims, no revision to the statements themselves is required. revision: no

Circularity Check

0 steps flagged

No circularity; direct theorem proof under explicit genericity

full rationale

The paper states a theorem for C^0-generic continuous maps and homeomorphisms on compact Riemannian manifolds, asserting that the space of physical-like measures coincides with invariant measures supported on chain recurrent classes, along with two additional properties. No equations, fitted parameters, or predictions appear; the argument is presented as a direct proof rather than a reduction. The genericity hypothesis is an explicit scope condition, not a hidden assumption. No self-definitional steps, fitted-input predictions, or load-bearing self-citation chains are detectable from the provided material, so the derivation remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only the abstract is available, so the ledger is limited to the explicit assumptions stated there; standard background from dynamical systems is not enumerated.

axioms (2)
  • domain assumption The underlying space is a compact Riemannian manifold
    Explicitly stated as the setting for the maps and homeomorphisms.
  • domain assumption The maps are C^0 generic (comeager in the C^0 topology)
    The three claims are asserted to hold precisely for this generic class.

pith-pipeline@v0.9.0 · 5623 in / 1200 out tokens · 21307 ms · 2026-05-24T19:03:48.545413+00:00 · methodology

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Reference graph

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