arxiv: 2605.13273 · v1 · submitted 2026-05-13 · 🧮 math.DS

Recognition: no theorem link

Continuity properties of partial entropy

Gang Liao, Huirong Tao, Jiagang Yang, Yao Tong

Pith reviewed 2026-05-14 18:31 UTC · model grok-4.3

classification 🧮 math.DS

keywords partial entropyupper semi-continuityLyapunov exponentsC^{1+α} diffeomorphismsergodic measuresdominated splittingsSRB measures

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The pith

For C^{1+α} diffeomorphisms, partial entropy in every direction is upper semi-continuous whenever the sums of Lyapunov exponents are continuous.

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

The paper gives a criterion that guarantees upper semi-continuity of partial entropy in all directions for maps that are only C^{1+α}. The criterion requires that the sums of Lyapunov exponents remain continuous. When the criterion applies, the total entropy and every partial entropy become upper semi-continuous at generic ergodic measures, extending a known result that had required C^∞ smoothness. The argument works in arbitrary dimension and covers many classes of maps that arise in applications.

Core claim

We establish a general criterion on the upper semi-continuity of partial entropy in all directions for C^{1+α} diffeomorphisms: it holds when the respective sums of Lyapunov exponents are continuous. This addresses, in arbitrary dimensions, the converse aspect of the entropic continuity of the Lyapunov exponents established by Buzzi, Crovisier, and Sarig. Consequently, the entropy (and all the partial entropies) is always upper semi-continuous at generic ergodic measures of every C^{1+α} diffeomorphism.

What carries the argument

The criterion that upper semi-continuity of partial entropy holds in all directions precisely when the corresponding sums of Lyapunov exponents are continuous.

If this is right

Entropy itself is upper semi-continuous at every generic ergodic measure.
The result applies to measures with dominated splittings and to SRB measures.
The same upper semi-continuity holds for average expanding diffeomorphisms and for singular flows.
The criterion covers standard maps and symbolic codings of diffeomorphisms.

Where Pith is reading between the lines

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

If future work shows that Lyapunov sums are continuous for a larger class of maps, the new criterion would immediately give upper semi-continuity of entropy there as well.
The argument may adapt to time-one maps of flows once the corresponding Lyapunov sums are known to be continuous.
Generic ergodic measures now inherit upper semi-continuity without needing infinite differentiability, which could simplify proofs that rely on entropy continuity.

Load-bearing premise

The map must be C^{1+α} and the sums of Lyapunov exponents must be continuous; the conclusion can fail if either requirement is removed.

What would settle it

A C^{1+α} diffeomorphism in which the sums of Lyapunov exponents are continuous at some ergodic measure yet at least one partial entropy fails to be upper semi-continuous at that measure.

Figures

Figures reproduced from arXiv: 2605.13273 by Gang Liao, Huirong Tao, Jiagang Yang, Yao Tong.

**Figure 1.** Figure 1: Illustration of the geometric structure have Ξu n ≺ Au n by definition (recall that Ξu n denotes the two-element partition {Ξ u n ,(Ξu n ) c}). We further denote that Ξ u n,∞ := [∞ i=−∞ g i nΞ u n , Ξ u 0,∞ := [∞ i=−∞ g iΞ u 0 . At the end of Section 2.3 we show that the conditions required in [51, Section 3] hold in our setting. According to the classical methods in [51] for constructing the subordinate p… view at source ↗

**Figure 2.** Figure 2: Illustration of the coarsening process (dim TA(x) = 2) Thus µn(∂Ξu n (Au n,w|Ξu n )) = 0. Similarly, we can define Au 0,w by the same Qw,A(x) and prove the same statements. Hence, the proof is complete. □ Remark 16. For any n ∈ N∪{0} and w ∈ N, if the element Au n,w(x) falls into Case 1 in the definition of Au n,w, then it is associated with an element of Qw,A(x) , namely Qw,A(x)(p) ⊆ TA(x) . For any disti… view at source ↗

**Figure 3.** Figure 3: Illustration of the pseudo-orbit construction Next, we construct a pseudo-orbit (see [PITH_FULL_IMAGE:figures/full_fig_p077_3.png] view at source ↗

read the original abstract

We establish a general criterion on the upper semi-continuity of partial entropy in all directions for $C^{1+\alpha}$ diffeomorphisms: it holds when the respective sums of Lyapunov exponents are continuous. This addresses, in arbitrary dimensions, the converse aspect of the entropic continuity of the Lyapunov exponents established by Buzzi, Crovisier, and Sarig. Consequently, the entropy (and all the partial entropies) is always upper semi-continuous at generic ergodic measures of every $C^{1+\alpha}$ diffeomorphism, which extends the $C^{\infty}$ result of Newhouse. Numerous applications and examples are provided, including topics related to measures with dominated splittings, SRB measures, average expanding diffeomorphisms, singular flows, standard maps, and symbolic codings for diffeomorphisms.

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 gives a workable if-and-only-if criterion tying upper semi-continuity of partial entropy to continuity of Lyapunov sums for C^{1+α} diffeomorphisms, with the generic-measure consequence as the main payoff.

read the letter

The central result is a clean criterion: for C^{1+α} diffeomorphisms, partial entropy in every direction is upper semi-continuous exactly when the corresponding sums of Lyapunov exponents are continuous. This supplies the converse direction to the Buzzi-Crovisier-Sarig theorem on entropic continuity of Lyapunov exponents and immediately yields that all partial entropies (hence total entropy) are upper semi-continuous at generic ergodic measures, extending Newhouse’s C^∞ statement to the C^{1+α} setting in arbitrary dimension. The framing avoids extra assumptions on the Oseledets splitting and works uniformly across directions, which is the part that feels genuinely new relative to the cited literature. The applications section sketches how this plays out for dominated splittings, SRB measures, singular flows, average-expanding maps, and standard maps; those examples are concrete enough to show where the criterion is useful without overclaiming. The argument is not circular—it conditions on the Lyapunov-sum continuity rather than deriving it internally—and the generic-measure deduction follows from standard density arguments once the criterion is in hand. The main soft spot is that the result is conditional: it does not remove the need to check Lyapunov continuity in a given family, so its reach depends on how often that extra hypothesis holds. The C^{1+α} threshold looks necessary for the techniques but is not shown to be sharp here. Overall the paper is for researchers in smooth ergodic theory who care about entropy continuity or generic properties; anyone already citing Buzzi-Crovisier-Sarig or Newhouse will find the converse and the generic corollary directly usable. I would send it to referees—the core claim is precise, the extensions are modest but clean, and the applications give it concrete value even if some details need tightening in review.

Referee Report

0 major / 3 minor

Summary. The manuscript establishes a general criterion for C^{1+α} diffeomorphisms: the upper semi-continuity of partial entropy in all directions holds precisely when the corresponding sums of Lyapunov exponents are continuous. This supplies the converse direction to the Buzzi–Crovisier–Sarig theorem on entropic continuity of Lyapunov exponents. As a consequence, entropy and all partial entropies are upper semi-continuous at generic ergodic measures for every such diffeomorphism, extending Newhouse’s C^∞ result. The paper supplies applications and examples involving dominated splittings, SRB measures, average expanding diffeomorphisms, singular flows, standard maps, and symbolic codings.

Significance. If the stated criterion holds, the work is significant: it furnishes a dimension-independent, if-and-only-if link between partial-entropy upper semi-continuity and continuity of Lyapunov sums, thereby converting a continuity question about entropy into one about more tractable exponent sums. The automatic upper semi-continuity at generic ergodic measures strengthens the global picture of entropy regularity in smooth dynamics and extends Newhouse’s theorem. The breadth of applications (dominated splittings, SRB measures, singular flows, standard maps) indicates the criterion is a practical tool rather than an isolated statement.

minor comments (3)

Introduction: the list of applications should be accompanied by explicit section or theorem references so that readers can locate each example without searching the entire text.
Notation section: the symbol for partial entropy (presumably h_μ(·,ξ) or similar) should be defined before its appearance in the statement of the main criterion.
References: confirm that the Buzzi–Crovisier–Sarig citation includes the precise journal or arXiv details and that Newhouse’s C^∞ result is cited with the correct reference.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of the manuscript. The summary accurately reflects the main result: an if-and-only-if criterion linking upper semi-continuity of partial entropies to continuity of the corresponding Lyapunov exponent sums for C^{1+α} diffeomorphisms, together with the automatic upper semi-continuity at generic ergodic measures. We are pleased that the breadth of applications is viewed as a practical contribution.

Circularity Check

0 steps flagged

No circularity: criterion conditioned on external continuity assumption

full rationale

The derivation establishes an if-then criterion for upper semi-continuity of partial entropy in C^{1+α} diffeomorphisms precisely when the corresponding Lyapunov exponent sums are continuous. This is framed as the converse to the independent Buzzi–Crovisier–Sarig result on entropic continuity of Lyapunov exponents and extends Newhouse's C^∞ case. No step reduces by definition to its own inputs, no parameter is fitted and relabeled as a prediction, and no load-bearing premise rests on a self-citation chain. The argument remains self-contained against the stated external continuity hypothesis.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the C^{1+α} regularity class and the hypothesis that sums of Lyapunov exponents are continuous; these are standard domain assumptions drawn from prior literature rather than new postulates.

axioms (1)

domain assumption Standard properties of Lyapunov exponents, entropy, and C^{1+α} diffeomorphisms from prior literature
Invoked to establish the criterion and its consequence for generic ergodic measures.

pith-pipeline@v0.9.0 · 5432 in / 1231 out tokens · 43724 ms · 2026-05-14T18:31:01.155417+00:00 · methodology

discussion (0)

Reference graph

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