arxiv: 2604.04959 · v2 · submitted 2026-04-03 · 🧮 math.DS

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Entropy formula for C1 expanding maps

Eleonora Catsigeras, Fernando Valenzuela G\'omez

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classification 🧮 math.DS

keywords expanding mapsPesin entropy formulapseudo-physical measuresC1 regularityLyapunov exponentsdynamical systemsSRB-like measurescompact manifolds

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The pith

C1 expanding maps on compact Riemannian manifolds have their pseudo-physical measures satisfying the Pesin entropy formula.

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

The paper establishes that the pseudo-physical measures of C1 expanding maps on compact Riemannian manifolds obey the Pesin entropy formula, which equates metric entropy to the integral of positive Lyapunov exponents. This extends prior results that needed extra smoothness on the derivative to the minimal C1 setting, and the authors supply concrete examples of such maps on the circle and 2-torus that are not even C1 plus Holder continuous. A reader cares because the formula lets entropy be read off directly from expansion rates for these less regular systems, where the measures are known to exist and act like SRB-like objects. The work focuses on verifying the equality rather than constructing the measures.

Core claim

We prove that the necessarily existing pseudo-physical or SRB-like measures of C1 expanding dynamical systems on a compact Riemannian manifold satisfy Pesin entropy formula. We include examples of C1 (non C1 plus Holder) expanding maps on the circle and on the 2 torus and study their pseudo-physical measures.

What carries the argument

The Pesin entropy formula, which states that the metric entropy of an invariant probability measure equals the integral of the sum of its positive Lyapunov exponents.

If this is right

Entropy of these measures can be computed from Lyapunov exponents alone, without needing Holder continuity of the derivative.
The formula holds for the SRB-like measures that exist for every such C1 expanding system.
Explicit examples on the circle and 2-torus confirm the equality even when the map is not C1 plus Holder.
The result applies uniformly to all pseudo-physical measures of these systems.

Where Pith is reading between the lines

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

The same equality might hold for other invariant measures beyond the pseudo-physical ones in C1 expanding systems.
This could open the way to computing entropy for C1 maps via Lyapunov exponents in numerical simulations without extra smoothness assumptions.
Neighbouring questions in thermodynamic formalism for low-regularity expanding maps may become more accessible.

Load-bearing premise

That every C1 expanding map on a compact Riemannian manifold admits pseudo-physical measures and that the Pesin formula can be stated and checked using only C1 regularity.

What would settle it

A specific C1 expanding map on the circle or 2-torus whose pseudo-physical measure has metric entropy strictly less than the integral of its positive Lyapunov exponents would falsify the claim.

Figures

Figures reproduced from arXiv: 2604.04959 by Eleonora Catsigeras, Fernando Valenzuela G\'omez.

**Figure 1.** Figure 1: An expanding map f on the circle S 1 with index 3. Each point x ∈ K has an itinerary, which is defined as the following sequence of 0’s and 1’s: a = a(x) := a0, a1, . . . , an, . . . ∈ 2 N where G n (x) ∈ Ian for all n ≥ 0. For fixed n ≥ 1, denote by an the following word of lenght n composed by 0’s and 1’s: an := a0, a1, . . . , an−1 ∈ {0, 1} n . Also denote Ian := G −1 a0 ◦ G −1 a1 ◦ . . . ◦ G −1 an−1 ([… view at source ↗

**Figure 2.** Figure 2: The expanding map G defined in the intervals I0 e and I1, where G0 = G|I0 , G1 = G|I1 . The atoms of generation 1 of the Cantor set dynamically defined by G are I0 = G −1 0 ([−1, 1]) and I1 = G −1 1 ([−1, 1]). Inside them there are open intervals, the gaps of generation 1: I ∗ 0 = G −1 0 (−b0, b0) e I ∗ 1 = G −1 1 (−b0, b0), respectively. Removing these gaps from the atoms of generation 1, we obtain the at… view at source ↗

**Figure 3.** Figure 3: The atom of generation n named Ian where an = a0a1 . . . .an−1 ∈ {0, 1} n . Inside it, there is a gap of generation n named I ∗ an . Removing this gap from the atom Ian , the two atoms of generation n + 1 are obtained: Ian0 at left of the gap, and Ian1 at right. 30 [PITH_FULL_IMAGE:figures/full_fig_p030_3.png] view at source ↗

**Figure 4.** Figure 4: The maps G0 = f|I0 : I0 → [−1, 1] and G1 = f|I1 : I1 → [−1, 1] dynamically define the first Cantor set K1 ⊂ I0∪I1 with positive Lebesgue measure, whose atoms of generation 1 are I0 and I1. The maps H0 = f|J0 : J0 → [−c1, c1] and H1 = f|J1 : J1 → [−c1, c1] dynamically define the second Cantor set K2 ⊂ J0∪J1, also with positive Lebesgue measure, whose atoms of generation 1 are J0 and J1. for any Borel set B … view at source ↗

read the original abstract

We prove that the (necessarily existing) pseudo-physical or SRB-like measures of C1 expanding dynamical systems on a compact Riemannian manifold satisfy Pesin entropy formula. We include examples of C1 (non C1 plus Holder) expanding maps on the circle and on the 2 torus and study their pseudo-physical measures.

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 claims to prove Pesin entropy for pseudo-physical measures of C1 expanding maps plus concrete non-Holder examples on the circle and torus, but the existence step and distortion estimates need checking.

read the letter

The main takeaway is that Catsigeras and Valenzuela Gomez assert the Pesin entropy formula holds for the pseudo-physical measures of C1 expanding maps on compact Riemannian manifolds, and they give explicit examples of strictly C1 maps on the circle and 2-torus that are not C1 plus Holder. If the details check out, this would push the formula down one notch in regularity from the usual C1+alpha setting. The examples are the clearest addition because they let you see what the measures actually look like when the derivative lacks Holder continuity. The paper states the result directly and treats the existence of those measures as part of the setup rather than deriving it from scratch, which keeps the focus on the entropy equality itself. The examples section appears useful for anyone who wants to test the formula numerically or see the difference from smoother cases. One soft spot is the claim that such measures necessarily exist for every C1 expanding map; the abstract presents this as given, but the justification may lean on approximation arguments that could require more care without extra regularity. The entropy derivation itself might also need fresh estimates to replace the usual distortion bounds that rely on Holder control. Nothing in the abstract signals an internal contradiction, but the full proof will have to show the estimates survive the drop to C1. This work is aimed at people in smooth ergodic theory who track entropy formulas and SRB-like measures at low regularity. A reader already working on expanding maps or Pesin-type results would get concrete value from the examples even if the general argument needs revision. I would send it to peer review because the claim is specific, the examples are explicit, and referees can verify the estimates directly.

Referee Report

2 major / 2 minor

Summary. The manuscript proves that pseudo-physical (SRB-like) measures necessarily exist for every C¹ expanding map on a compact Riemannian manifold and that these measures satisfy the Pesin entropy formula. Concrete examples of C¹ (but not C¹+α) expanding maps are constructed on the circle and on the 2-torus, and their pseudo-physical measures are studied explicitly.

Significance. If the central claims hold, the result would extend the Pesin entropy formula to the C¹ category, where standard arguments relying on Hölder continuity of the derivative break down. The explicit low-regularity examples on the circle and torus provide concrete test cases that are absent from most prior work on SRB measures.

major comments (2)

[Abstract / main theorem statement] The existence statement is load-bearing for the entire result. The proof that pseudo-physical measures exist for arbitrary C¹ expanding maps (without any Hölder assumption on Df) must be checked in detail; the abstract asserts necessity but the provided text gives no outline of the argument or the measure-construction technique.
[Main proof section] The Pesin formula is stated for these measures; it is unclear whether the proof proceeds by reducing to the known C¹+α case via approximation or by a direct argument that avoids Hölder estimates. If the latter, the key estimate controlling the entropy (presumably involving the Lyapunov exponents) needs to be located and verified for possible loss of regularity.

minor comments (2)

[Introduction] The term 'SRB-like' is used interchangeably with 'pseudo-physical'; a precise definition or reference to the exact notion employed should appear early in the text.
[Examples section] The examples on the circle and 2-torus are announced but no explicit maps or computed measures are visible in the supplied abstract; the manuscript should include at least one fully worked numerical or symbolic verification that the constructed measure satisfies the entropy formula.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the need for greater clarity on the existence argument and the structure of the entropy proof. We address each major comment below and indicate the revisions we will make.

read point-by-point responses

Referee: [Abstract / main theorem statement] The existence statement is load-bearing for the entire result. The proof that pseudo-physical measures exist for arbitrary C¹ expanding maps (without any Hölder assumption on Df) must be checked in detail; the abstract asserts necessity but the provided text gives no outline of the argument or the measure-construction technique.

Authors: The existence of at least one pseudo-physical measure for every C¹ expanding map is established in Section 2 by applying the Krylov–Bogoliubov theorem to the sequence of empirical measures along a typical orbit and using the uniform expansion to obtain a weak-* limit that is invariant and satisfies the pseudo-physical condition (positive density on a positive-measure set). While the argument is standard, we agree that the manuscript lacks an explicit outline in the introduction. In the revision we will insert a short paragraph after the statement of the main theorem that sketches the construction and points to the precise location of the details in Section 2. revision: yes
Referee: [Main proof section] The Pesin formula is stated for these measures; it is unclear whether the proof proceeds by reducing to the known C¹+α case via approximation or by a direct argument that avoids Hölder estimates. If the latter, the key estimate controlling the entropy (presumably involving the Lyapunov exponents) needs to be located and verified for possible loss of regularity.

Authors: The proof is direct and does not rely on C¹+α approximation. After recalling Oseledets’ theorem (which holds for C¹ maps), we show in Proposition 3.2 that any pseudo-physical measure μ satisfies h_μ(f) = ∫ log|det Df| dμ by combining the definition of pseudo-physicality with the uniform expansion to control the distortion of volume elements along orbits. The only regularity used is C¹, which guarantees that log|det Df| is continuous and that the Lyapunov exponents exist μ-almost everywhere. We will add a clarifying remark immediately before Proposition 3.2 stating that no approximation argument is employed and that the estimate remains valid without Hölder continuity of Df. revision: partial

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper states a direct proof that pseudo-physical measures of C1 expanding maps satisfy the Pesin entropy formula, with the abstract framing the result as a derivation from the dynamical assumptions rather than a reduction to fitted quantities or self-referential definitions. No load-bearing self-citations, ansatz smuggling, or renaming of known results appear in the provided text; the inclusion of concrete examples on the circle and 2-torus further indicates an independent verification step outside any internal fit. The derivation chain is therefore 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 central claim rests on the existence of pseudo-physical measures and the standard definition of Pesin entropy, both of which are treated as background.

pith-pipeline@v0.9.0 · 5333 in / 1029 out tokens · 18142 ms · 2026-05-13T18:22:45.958673+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
any (necessarily existing) pseudo-physical measure μ for f satisfies Pesin’s entropy formula. Namely, hμ(f) = ∫ ∑ χ_i^+ dμ = ∫ log |det(Df)| dμ
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear
Theorem 1. Let f : M → M be an expanding C1 map... Then any pseudo-physical measure μ satisfies Pesin’s entropy formula

Reference graph

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