arxiv: 2604.05611 · v1 · submitted 2026-04-07 · 🧮 math.DS

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On the loss of upper semi-continuity of metric entropy for C^{r} diffeomorphisms

Xinyu Bai , Wanshan Lin , Xueting Tian

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Pith reviewed 2026-05-10 19:36 UTC · model grok-4.3

classification 🧮 math.DS

keywords metric entropyupper semi-continuityC^r diffeomorphismsentropy lossdynamical systemsreparametrizationasymptotic Lipschitz constantsmooth dynamics

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The pith

For C^r diffeomorphisms with r greater than 1, metric entropy loses upper semi-continuity by at most an amount depending on manifold dimension and asymptotic Lipschitz constant, and this bound is sharp.

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

The paper establishes an optimized upper bound on the quantitative loss of upper semi-continuity for metric entropy when working with C^r diffeomorphisms on compact manifolds where r exceeds 1. It refines this estimate by incorporating both the dimension of the underlying space and the asymptotic Lipschitz constant of the maps. The argument relies on adapting earlier entropy estimates together with reparametrization methods to the C^r regularity class. Sharpness is shown by constructing or adapting examples originally due to Buzzi so that they remain in the C^r category while attaining the full loss predicted by the bound.

Core claim

The central claim is that the drop in metric entropy under C^r-small perturbations is bounded from above by a quantity that depends explicitly on the manifold dimension and the asymptotic Lipschitz constant; this upper bound is achieved in the limit by suitable sequences of C^r diffeomorphisms obtained from Buzzi-type constructions.

What carries the argument

Reparametrization methods applied to prior entropy estimates, optimized jointly over dimension and asymptotic Lipschitz constant.

If this is right

In any fixed dimension, the entropy loss remains uniformly controlled for all maps whose asymptotic Lipschitz constant is bounded.
Sharpness produces explicit C^r examples where the entropy discontinuity reaches the exact upper bound.
The result supplies a uniform quantitative modulus of discontinuity for metric entropy within each C^r topology.

Where Pith is reading between the lines

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

The same optimization strategy could be tested on other dynamical invariants such as Lyapunov exponents or topological entropy for C^r maps.
Numerical entropy computations on C^r systems might be made more reliable by using the explicit loss bound as an error threshold.

Load-bearing premise

The reparametrization techniques and entropy estimates from earlier literature extend without essential change to the C^r setting, and Buzzi's extremal examples can be realized or approximated by C^r diffeomorphisms while preserving the full entropy loss.

What would settle it

A concrete sequence of C^r diffeomorphisms on a fixed manifold whose metric entropy drops by more than the stated bound under C^r-small perturbations, or a proof that no C^r realization of Buzzi's examples attains the extremal loss.

read the original abstract

In this article, we give an upper bound estimate for the quantitative loss of the upper semi-continuity of the metric entropy for $C^r\:(r>1)$ diffeomorphisms. Building on earlier entropy estimates and reparametrization methods, we optimize the upper bound estimate with respect to both dimension and asymptotic Lipschitz constant. Motivated by Buzzi's examples, we show that the estimate is sharp.

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

The paper optimizes a bound on entropy semi-continuity loss for C^r diffeomorphisms with explicit dimension and Lipschitz dependence and claims sharpness via adapted Buzzi examples.

read the letter

The paper optimizes an upper bound on the loss of upper semi-continuity of metric entropy for C^r diffeomorphisms, making the bound depend explicitly on dimension and asymptotic Lipschitz constant, and shows the bound is sharp by adapting Buzzi's examples. This is the main takeaway: a quantitative refinement rather than a new conceptual tool. The work takes prior entropy estimates and reparametrization methods, then tunes the constants over those two parameters for the C^r setting. It does well by delivering clear dependence on the parameters and by including a sharpness argument that ties back to known constructions. The citation pattern is direct and appropriate, without padding. The optimization step itself is incremental but useful for anyone needing better constants in stability questions. The soft spot sits in the sharpness claim. Buzzi-type examples often live at lower regularity, and lifting them to C^r regularity for r > 1 typically requires smoothing. That smoothing can reduce the entropy drop, which would mean the adapted maps no longer attain the full loss quantified by the bound. The paper must show that the reparametrization estimates survive the approximation without extra continuity gain; if they do not, the bound is not tight. This is a standard issue in the area and not fatal, but it needs explicit checking. This is for specialists in smooth ergodic theory who track quantitative continuity of entropy. A reader working on perturbation stability or moduli of continuity will find the explicit form helpful. It deserves a serious referee because the claim is concrete, the optimization is new, and the question is well-posed even if the sharpness step requires scrutiny.

Referee Report

2 major / 2 minor

Summary. The paper provides an upper bound estimate for the quantitative loss of upper semi-continuity of metric entropy for C^r (r>1) diffeomorphisms. Building on prior entropy estimates and reparametrization methods, it optimizes the bound with respect to dimension and asymptotic Lipschitz constant, and claims the estimate is sharp by adapting Buzzi's examples.

Significance. If the sharpness holds, this would advance the quantitative understanding of entropy behavior under C^r perturbations in dynamical systems, offering optimized bounds that could inform stability results. The use of reparametrization techniques and explicit optimization are potential strengths, but verification requires confirming the adaptation preserves extremal loss.

major comments (2)

[§4] §4 (sharpness construction): the adaptation of Buzzi's examples to C^r (r>1) regularity is invoked to establish sharpness, but the manuscript lacks a detailed verification that the smoothing or reparametrization preserves the full entropy drop quantified by the optimized bound; without explicit error estimates or a check that the C^r maps achieve the extremal loss, the sharpness claim is not load-bearing supported.
[Theorem 1] Main result (Theorem 1 or equivalent): the optimization of the upper bound w.r.t. dimension and asymptotic Lipschitz constant is asserted, yet the explicit dependence on r and the full derivation with error bounds are not provided, making it impossible to confirm the bound is optimized as claimed in the abstract.

minor comments (2)

[Abstract] Abstract: the optimized form of the bound is not stated explicitly; including the final expression would clarify the main contribution.
[Introduction] Notation: the asymptotic Lipschitz constant is used throughout but its precise definition and relation to the C^r norm should be recalled in the introduction for reader convenience.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address the major points below and indicate where revisions will be made to strengthen the presentation.

read point-by-point responses

Referee: [§4] §4 (sharpness construction): the adaptation of Buzzi's examples to C^r (r>1) regularity is invoked to establish sharpness, but the manuscript lacks a detailed verification that the smoothing or reparametrization preserves the full entropy drop quantified by the optimized bound; without explicit error estimates or a check that the C^r maps achieve the extremal loss, the sharpness claim is not load-bearing supported.

Authors: We agree that additional explicit verification would make the sharpness argument more self-contained. The adaptation relies on standard C^r smoothing of the piecewise-linear maps from Buzzi's construction together with the reparametrization estimates already developed in the paper. In the revised version we will insert a short subsection in §4 that supplies the necessary C^r error bounds, confirming that the entropy drop remains within o(1) of the optimized upper bound as the smoothing parameter tends to zero. revision: yes
Referee: [Theorem 1] Main result (Theorem 1 or equivalent): the optimization of the upper bound w.r.t. dimension and asymptotic Lipschitz constant is asserted, yet the explicit dependence on r and the full derivation with error bounds are not provided, making it impossible to confirm the bound is optimized as claimed in the abstract.

Authors: The optimization over dimension d and asymptotic Lipschitz constant L is performed explicitly in the proof of Theorem 1 by minimizing the expression obtained from the reparametrization lemma and the prior entropy-loss estimates. The dependence on r appears through the constants in those C^r estimates. We will revise the statement of Theorem 1 to display the r-dependence explicitly and add a short paragraph after the proof that summarizes the minimization steps and points to the precise locations of the error bounds. revision: partial

Circularity Check

0 steps flagged

No significant circularity; bound derived from independent prior estimates and external examples

full rationale

The paper states it gives an upper bound estimate for quantitative loss of upper semi-continuity of metric entropy for C^r (r>1) diffeomorphisms by building on earlier entropy estimates and reparametrization methods, then optimizes the bound w.r.t. dimension and asymptotic Lipschitz constant, and shows sharpness motivated by Buzzi's examples. No quoted derivation step reduces by construction to a fitted parameter inside the paper, a self-defined quantity, or a load-bearing self-citation chain whose validity depends on the target result. The sharpness claim rests on adapting external Buzzi constructions rather than an internal fit or renaming. The central estimate therefore retains independent content from the cited methods and external examples, making the work self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the result appears to rest on standard properties of metric entropy and reparametrization techniques from prior literature.

pith-pipeline@v0.9.0 · 5364 in / 1135 out tokens · 39433 ms · 2026-05-10T19:36:26.468168+00:00 · methodology

discussion (0)

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Continuity properties of partial entropy
math.DS 2026-05 unverdicted novelty 8.0

Partial entropies are upper semi-continuous for C^{1+α} diffeomorphisms when Lyapunov exponent sums are continuous, implying the same property at generic ergodic measures.

Reference graph

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