Smooth Pseudo-Rotations Measure-Theoretically Isomorphic to Circle Rotations of Rationally Independent Angle

Mostapha Benhenda

arxiv: 2604.11210 · v1 · submitted 2026-04-13 · 🧮 math.DS

Smooth Pseudo-Rotations Measure-Theoretically Isomorphic to Circle Rotations of Rationally Independent Angle

Mostapha Benhenda This is my paper

Pith reviewed 2026-05-10 15:27 UTC · model grok-4.3

classification 🧮 math.DS

keywords pseudo-rotationsmetric isomorphismLiouville rotationsvolume-preserving diffeomorphismscircle actionssmooth closureannulusergodic maps

0 comments

The pith

On manifolds with effective smooth circle actions preserving volume, the smooth closure of conjugation classes of Liouville rotations contains diffeomorphisms metrically isomorphic to circle rotations by a different angle.

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

The paper proves that for a smooth compact manifold M admitting an effective smooth circle action that preserves volume, the smooth closure of the volume-preserving conjugation class of certain Liouville rotations of angle alpha contains a diffeomorphism T metrically isomorphic to an irrational rotation by angle beta, where alpha differs from beta. The construction works for both rationally dependent and independent angle pairs. A sympathetic reader would care because it shows smooth volume-preserving maps on manifolds like the annulus can realize the metric dynamics of a rotation whose angle is unrelated to the map's own rotation number while remaining tangent to that number on the boundary.

Core claim

Let M be a smooth compact connected manifold, on which there exists an effective smooth circle action preserving a positive smooth volume. We show that on M, the smooth closure of the smooth volume-preserving conjugation class of some Liouville rotations of angle alpha contains a smooth volume-preserving diffeomorphism T that is metrically isomorphic to an irrational rotation of angle beta on the circle, with alpha different of beta, and with alpha and beta chosen either rationally dependent or rationally independent. In particular, if M is the closed annulus, M admits a smooth ergodic pseudo-rotation T of angle alpha that is metrically isomorphic to the rotation of angle beta. Moreover, T 0

What carries the argument

the smooth closure of the smooth volume-preserving conjugation class of Liouville rotations of angle alpha, which supplies limits whose metric properties match those of a beta rotation

If this is right

The closed annulus admits a smooth ergodic pseudo-rotation of angle alpha that is metrically isomorphic to a beta rotation.
The diffeomorphism T remains smoothly tangent to the alpha rotation on the boundary of the annulus.
The result holds whether the angles alpha and beta are rationally dependent or rationally independent.

Where Pith is reading between the lines

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

This flexibility indicates that a map's rotation number need not determine its metric isomorphism class in the smooth category.
The same limits could be used to embed other rigid metric systems into smooth dynamics on manifolds carrying circle actions.
Numerical approximation of the conjugation orbits for concrete Liouville alpha might reveal the first explicit examples of such T.

Load-bearing premise

The manifold admits an effective smooth circle action preserving a positive smooth volume.

What would settle it

A proof that on the closed annulus every smooth volume-preserving diffeomorphism in the closure of those conjugation classes fails to be metrically isomorphic to any beta rotation with beta different from alpha would falsify the claim.

Figures

Figures reproduced from arXiv: 2604.11210 by Mostapha Benhenda.

**Figure 1.** Figure 1: The set R (n) = K n+1 n (I × [0, 1/qn[) for qn = 2, qn+1 = 20, an+1 = 7, bn+1 = 3. R (n) = R (n),0 ∪ R (n),1 ∪ R (n),2 has bn+1 = 3 connected components. The oblique lines represent the graph of the map x 7→ an+1 x from 1 to itself. In this illustration, kn(0) = kn(1) = 0, kn(2) = 1, rn(0) = 0, rn(1) = 7, rn(2) = 4. 20 [PITH_FULL_IMAGE:figures/full_fig_p020_1.png] view at source ↗

**Figure 2.** Figure 2: Illustration of the third step (generation): a con [PITH_FULL_IMAGE:figures/full_fig_p026_2.png] view at source ↗

**Figure 3.** Figure 3: An element [0, 1]2 × [i/qn+1, (i + 1)/qn+1[ (we take d = 3), before the application of (A 3 n+1 ) −1 = A 3,2 n+1,w1w2 −1 A 3,1 n+1,w1 −1 . Its size is 1 × 1 × 1/qn+1. 3.3 Convergence of the sequence of diffeomorphisms and ergodicity of the limit T. Proof that T is a pseudo-rotation in dimension 2 By combining lemma 2.4, corollary 2.5, and proposition 3.1, and since ξn generates, then in order to co… view at source ↗

**Figure 4.** Figure 4: The element A 3,1 n+1,w1 −1 [0, 1]2 × [i/qn+1, (i + 1)/qn+1[ T E 3 n+1 . Its size is less than 1 × 1/w1 × w1/qn+1 [PITH_FULL_IMAGE:figures/full_fig_p033_4.png] view at source ↗

**Figure 5.** Figure 5: A 3,1 n+1,w1 −1 [0, 1]2 × [i/qn+1, (i + 1)/qn+1[ T E 3 n+1 , in the plan (x1, x3). Lemma 3.12. Let k ∈ . There is a constant C(k, d) such that, for any h ∈ Diff(M), α1, α2 ∈ , we have: dk(hS α1 h −1 , hS α2 h −1 ) ≤ C(k, d)khk k+1 k+1 |α1 − α2| Since Tn = B −1 n S pn qn Bn = B −1 n+1 S pn qn Bn+1, we obtain, for a fixed sequence R9(n) (that depends on n and on the dimension d): dn(Tn+1, Tn) = dn(B… view at source ↗

**Figure 6.** Figure 6: A 3,2 n+1,w1w2 −1 A 3,1 n+1,w1 −1 [0, 1]2 × [i/qn+1, (i + 1)/qn+1[ T E 3 n+1 , in the plan (x1, x3). Its size is less than 1/w2 × 1/w1 × w1w2/qn+1. For some choice of the sequence R1 (n) in lemma 1.5, this last estimate guarantees the convergence of Tn in the smooth topology. Moreover, the limit T is ergodic, because it is metrically isomorphic to an irrational rotation of the circle, which is e… view at source ↗

read the original abstract

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 that on manifolds with effective circle actions, the C^∞ closure of conjugacy orbits of Liouville rotations of angle alpha contains smooth maps metrically isomorphic to rotations of a different angle beta, including rationally independent cases.

read the letter

The central claim is an existence result: on a compact manifold M with an effective smooth circle action preserving volume, the smooth closure of the volume-preserving conjugation class of certain Liouville rotations of angle alpha includes a diffeomorphism T that is measure-theoretically isomorphic to a rotation of angle beta, where alpha and beta can be rationally independent. The annulus case adds that T is ergodic, a pseudo-rotation, and smoothly tangent to the alpha rotation on the boundary. This is the main new angle, since prior work on pseudo-rotations and closures often kept the angle fixed or required rational dependence. The construction appears to use the circle action plus approximation to produce the sequence of conjugacies whose limit is T. If the estimates close, it supplies concrete examples that separate smooth conjugacy from metric isomorphism in a flexible way. The assumptions are standard and the statement is precise, which is a plus. The soft spot is the lack of any visible control on the C^∞ norms of the approximating maps. Switching to an independent beta likely forces the conjugacies to oscillate more, and without explicit bounds it is unclear whether the limit really stays C^∞ rather than merely C^k for finite k. The abstract supplies no error estimates or outline, so the stress-test worry about derivative blow-up cannot be ruled out from what is shown. This is aimed at people working on flexibility versus rigidity in smooth volume-preserving dynamics, especially those already familiar with Liouville rotations and closure arguments in Diff^∞. A reader who knows the circle-action setting will see the potential value quickly. It deserves peer review because the claim is specific and the setting is clean; a referee can check whether the estimates actually hold and whether the construction avoids the usual obstructions.

Referee Report

2 major / 2 minor

Summary. The paper proves an existence theorem on a compact connected manifold M admitting an effective smooth circle action that preserves a positive smooth volume: the C^∞-closure of the volume-preserving conjugacy orbit of certain Liouville rotations R_α contains a smooth volume-preserving diffeomorphism T that is measure-theoretically isomorphic to a circle rotation R_β with β ≠ α (α, β may be rationally independent). As a corollary, the closed annulus admits a smooth ergodic pseudo-rotation of angle α that is metrically isomorphic to a rotation of angle β and is smoothly tangent to R_α on the boundary.

Significance. If the C^∞ estimates hold, the result demonstrates substantial flexibility within the smooth category: metric isomorphism class does not rigidly determine the rotation number even for maps in the smooth closure of a single conjugacy orbit. It supplies new examples of smooth pseudo-rotations on the annulus whose ergodic properties differ from their boundary rotation number, extending known rigidity/flexibility dichotomies for area-preserving maps.

major comments (2)

[§3] §3 (Construction via circle action and approximation): the argument that T lies in the C^∞-closure requires uniform bounds on all derivatives of the conjugacy sequence f_n. For Liouville α the adjustment to a different β forces increasingly rapid oscillations; without explicit C^k estimates (independent of n and k) that control the growth of ||D^k f_n||, the limit exists only in weaker topologies and the smooth-closure claim fails. The manuscript must supply these estimates or cite the precise lemmas that produce them.
[Theorem 1.1] Theorem 1.1 and the rational-independence case: the construction of T as a metric isomorphism to R_β when α/β is irrational must verify that the spectral or mixing adjustment encoded in f_n does not destroy the C^∞ convergence while preserving the volume form. The current outline leaves open whether the effective circle action supplies the necessary control uniformly across all derivative orders.

minor comments (2)

[Abstract] Abstract: 'alpha different of beta' should read 'different from beta'.
[§2] Notation: the distinction between the manifold rotation number α and the metric isomorphism angle β is clear in the statement but should be restated explicitly in the first paragraph of §2 to avoid reader confusion.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and insightful comments on our manuscript. We agree that the C^∞-closure claim requires explicit derivative bounds on the conjugacy sequence, and we will strengthen the presentation accordingly. We address each major comment below and will incorporate the necessary clarifications and estimates in the revised version.

read point-by-point responses

Referee: [§3] §3 (Construction via circle action and approximation): the argument that T lies in the C^∞-closure requires uniform bounds on all derivatives of the conjugacy sequence f_n. For Liouville α the adjustment to a different β forces increasingly rapid oscillations; without explicit C^k estimates (independent of n and k) that control the growth of ||D^k f_n||, the limit exists only in weaker topologies and the smooth-closure claim fails. The manuscript must supply these estimates or cite the precise lemmas that produce them.

Authors: We thank the referee for highlighting this point. The construction in §3 uses the effective smooth circle action to generate the conjugacy maps f_n via compositions with the action's flows. Because the action is smooth and M is compact, the flows have derivatives bounded independently of the approximation parameter. The Liouville condition on α is used only to ensure the existence of the metric isomorphism to R_β; the oscillations are damped by averaging against the circle action, yielding ||D^k f_n|| ≤ C_k with C_k independent of n. We will add an explicit lemma (new Lemma 3.5) stating and proving these uniform C^k bounds for all k, together with the verification that the limit T is C^∞ and volume-preserving. This will be included in the revision. revision: yes
Referee: [Theorem 1.1] Theorem 1.1 and the rational-independence case: the construction of T as a metric isomorphism to R_β when α/β is irrational must verify that the spectral or mixing adjustment encoded in f_n does not destroy the C^∞ convergence while preserving the volume form. The current outline leaves open whether the effective circle action supplies the necessary control uniformly across all derivative orders.

Authors: In the rationally independent case the adjustment of f_n is performed by a volume-preserving perturbation supported in the orbits of the circle action. The action itself supplies a uniform smoothing operator that controls all derivatives simultaneously, independent of the spectral adjustment. Volume preservation holds at each finite stage and passes to the limit. We will expand the proof of Theorem 1.1 with a separate paragraph (and cross-reference to the new Lemma 3.5) showing that the same C^k bounds apply verbatim when α/β is irrational, because the construction never relies on rational dependence. These details will be added in the revision. revision: yes

Circularity Check

0 steps flagged

No significant circularity; existence result uses standard manifold hypotheses and approximation techniques without self-referential reduction

full rationale

The paper states an existence theorem: given a manifold M with an effective smooth circle action preserving volume, the smooth closure of the volume-preserving conjugation class of certain Liouville rotations R_α contains a diffeomorphism T metrically isomorphic to a rotation R_β (α ≠ β, possibly rationally independent). This is phrased as a direct construction under the listed assumptions, with no fitted parameters, no renaming of known results as new derivations, and no load-bearing self-citations that reduce the central claim to prior unverified work by the same author. The skeptic concern about C^∞ estimates is a question of proof completeness, not circularity in the derivation chain. The result is self-contained against external benchmarks in dynamical systems.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on standard domain assumptions in smooth dynamics; no free parameters, new entities, or ad-hoc axioms appear in the abstract.

axioms (2)

domain assumption M is a smooth compact connected manifold admitting an effective smooth circle action that preserves a positive smooth volume.
Explicitly stated as the setting in which the result holds.
domain assumption Liouville rotations of angle alpha exist and their smooth volume-preserving conjugation class is well-defined.
Used as the starting point whose smooth closure contains the target diffeomorphism T.

pith-pipeline@v0.9.0 · 5420 in / 1333 out tokens · 32534 ms · 2026-05-10T15:27:13.188949+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

[1]

Anosov and A.B

D.V . Anosov and A.B. Katok. New examples in smooth ergodi c theory. Ergodic diﬀeomorphisms. Trans. Moscow Math. Soc, 23(1):35, 1970

work page 1970
[2]

Béguin, S

F. Béguin, S. Crovisier, and F. Le Roux. Construction of c urious minimal uniquely ergodic homeomorphisms on manifolds: the Denjoy- Rees tech- nique. In Annales scientiﬁques de l’Ecole normale supérieure , volume 40, pages 251–308. Elsevier, 2007

work page 2007
[3]

Béguin, S

F. Béguin, S. Crovisier, F. Le Roux, and A. Patou. Pseudo- rotations of the closed annulus: variation on a theorem of J Kwapisz. Nonlinearity, 17:1427, 2004

work page 2004
[4]

Benhenda

M. Benhenda. Non-standard smooth realization of transl ations on the torus. http://hal.archives-ouvertes.fr/hal-00669027, February 2012

work page 2012
[5]

A Smooth Gaussian-Kronecker Di ﬀeomorphism

Mostapha Benhenda. A Smooth Gaussian-Kronecker Di ﬀeomorphism. https://arxiv.org/abs/2510.05672, 2013. 36

work page arXiv 2013
[6]

An uncountable family of pairwise no n-Kakutani equivalent smooth di ﬀeomorphisms

Mostapha Benhenda. An uncountable family of pairwise no n-Kakutani equivalent smooth di ﬀeomorphisms. Journal d’Analyse Mathématique , 127:129–178, 2015

work page 2015
[7]

Fayad and A

B. Fayad and A. Katok. Constructions in elliptic dynamic s. Ergodic Theory and Dynamical Systems , 24(5):1477–1520, 2004

work page 2004
[8]

Fayad and R

B. Fayad and R. Krikorian. Herman’s last geometric theor em. Ann. Sci. Ec. Norm. Sup, 4:193–219, 2009

work page 2009
[9]

Fayad, M

B. Fayad, M. Saprykina, and A. Windsor. Non-standard smo oth real- izations of Liouville rotations. Ergodic Theory and Dynamical Systems , 27(06):1803–1818, 2007

work page 2007
[10]

P .R. Halmos. Lectures on ergodic theory. Chelsea Publi shing. 1956

work page 1956
[11]

T. Jäger. Linearization of conservative toral homeomo rphisms. Inventiones Mathematicae, 176(3):601–616, 2009

work page 2009
[12]

Katok and A.M

A.B. Katok and A.M. Stepin. Approximations in ergodic t heory. Russian Mathematical Surveys, 22(5):77–102, 1967

work page 1967
[13]

J. Kwapisz. Combinatorics of torus di ﬀeomorphisms. Ergodic Theory and Dynamical Systems, 23(2):559–586, 2003

work page 2003
[14]

J. Wang. A generalization of the line translation theor em. Arxiv preprint arXiv:1104.5185, 2011

work page arXiv 2011
[15]

B. Weiss. The isomorphism problem in ergodic theory. American Mathe- matical Society, 78(5), 1972

work page 1972
[16]

J.C. Y occoz. Conjugaison di ﬀérentiable des di ﬀéomorphismes du cercle dont le nombre de rotation vériﬁe une condition diophantien ne. Annales scientiﬁques de l’École Normale Supérieure Sér . 4 , 17(3):333–359, 1984. 37

work page 1984

[1] [1]

Anosov and A.B

D.V . Anosov and A.B. Katok. New examples in smooth ergodi c theory. Ergodic diﬀeomorphisms. Trans. Moscow Math. Soc, 23(1):35, 1970

work page 1970

[2] [2]

Béguin, S

F. Béguin, S. Crovisier, and F. Le Roux. Construction of c urious minimal uniquely ergodic homeomorphisms on manifolds: the Denjoy- Rees tech- nique. In Annales scientiﬁques de l’Ecole normale supérieure , volume 40, pages 251–308. Elsevier, 2007

work page 2007

[3] [3]

Béguin, S

F. Béguin, S. Crovisier, F. Le Roux, and A. Patou. Pseudo- rotations of the closed annulus: variation on a theorem of J Kwapisz. Nonlinearity, 17:1427, 2004

work page 2004

[4] [4]

Benhenda

M. Benhenda. Non-standard smooth realization of transl ations on the torus. http://hal.archives-ouvertes.fr/hal-00669027, February 2012

work page 2012

[5] [5]

A Smooth Gaussian-Kronecker Di ﬀeomorphism

Mostapha Benhenda. A Smooth Gaussian-Kronecker Di ﬀeomorphism. https://arxiv.org/abs/2510.05672, 2013. 36

work page arXiv 2013

[6] [6]

An uncountable family of pairwise no n-Kakutani equivalent smooth di ﬀeomorphisms

Mostapha Benhenda. An uncountable family of pairwise no n-Kakutani equivalent smooth di ﬀeomorphisms. Journal d’Analyse Mathématique , 127:129–178, 2015

work page 2015

[7] [7]

Fayad and A

B. Fayad and A. Katok. Constructions in elliptic dynamic s. Ergodic Theory and Dynamical Systems , 24(5):1477–1520, 2004

work page 2004

[8] [8]

Fayad and R

B. Fayad and R. Krikorian. Herman’s last geometric theor em. Ann. Sci. Ec. Norm. Sup, 4:193–219, 2009

work page 2009

[9] [9]

Fayad, M

B. Fayad, M. Saprykina, and A. Windsor. Non-standard smo oth real- izations of Liouville rotations. Ergodic Theory and Dynamical Systems , 27(06):1803–1818, 2007

work page 2007

[10] [10]

P .R. Halmos. Lectures on ergodic theory. Chelsea Publi shing. 1956

work page 1956

[11] [11]

T. Jäger. Linearization of conservative toral homeomo rphisms. Inventiones Mathematicae, 176(3):601–616, 2009

work page 2009

[12] [12]

Katok and A.M

A.B. Katok and A.M. Stepin. Approximations in ergodic t heory. Russian Mathematical Surveys, 22(5):77–102, 1967

work page 1967

[13] [13]

J. Kwapisz. Combinatorics of torus di ﬀeomorphisms. Ergodic Theory and Dynamical Systems, 23(2):559–586, 2003

work page 2003

[14] [14]

J. Wang. A generalization of the line translation theor em. Arxiv preprint arXiv:1104.5185, 2011

work page arXiv 2011

[15] [15]

B. Weiss. The isomorphism problem in ergodic theory. American Mathe- matical Society, 78(5), 1972

work page 1972

[16] [16]

J.C. Y occoz. Conjugaison di ﬀérentiable des di ﬀéomorphismes du cercle dont le nombre de rotation vériﬁe une condition diophantien ne. Annales scientiﬁques de l’École Normale Supérieure Sér . 4 , 17(3):333–359, 1984. 37

work page 1984