Mixing with zero entropy and rigidity of pseudo-rotations: the open questions of Avila-Fayad-Le Calvez-Xu-Zhang (2020) on disc diffeomorphisms

In a 2020 Inventiones mathematicae paper, Avila, Fayad, Le Calvez, Xu and Zhang proved that area-preserving real analytic diffeomorphisms of the disc or the sphere that are topologically mixing must have positive topological entropy, established rigidity of pseudo-rotations with non-Brjuno rotation number, and posed nine questions on whether mixing can coexist with vanishing entropy and on rigidity of pseudo-rotations. Question 2 (existence of real analytic zero-metric-entropy transitive or ergodic disc/sphere diffeomorphisms) has been answered positively after publication, by Berger (transitive, 2024) and Delaporte (ergodic, 2025), building on Berger's disproof of Birkhoff's 1941 conjecture (published in the Annals of Mathematics, 2026). The remaining core is open: a smooth mixing disc diffeomorphism with zero entropy (Q1), $C^0$-rigidity of general pseudo-rotations (Q3), polynomial derivative bounds (Q4), uniform local rigidity (Q5), mixing $C^0$ pseudo-rotations (Q6), annulus rigidity including the analytic non-mixing gap for rotation numbers failing Yoccoz's condition $\mathcal{H}$ (Q7), and non-mixing of non-conservative zero-topological-entropy disc diffeomorphisms (Q8, Q9).

Background

An area-preserving (conservative) diffeomorphism $f$ of the closed unit disc $\mathbb{D}\subset\mathbb{R}^2$ is a smooth invertible map that preserves two-dimensional Lebesgue measure (area). It is mixing in the measure-theoretic sense if for all measurable sets $A,B$ one has $\mu(f^n(A)\cap B)\to\mu(A)\mu(B)$ as $n\to\infty$; mixing implies ergodicity (every invariant measurable set has measure $0$ or $1$) but is strictly stronger, and it implies topological mixing (for any two nonempty open sets $U,V$ there exists $N$ such that $f^n(U)\cap V\neq\emptyset$ for all $n\geq N$), which in turn implies transitivity (existence of a dense orbit). Weak mixing, an intermediate condition, asks that the time-averaged correlations $\frac{1}{N}\sum_{n<N}|\mu(f^n(A)\cap B)-\mu(A)\mu(B)|$ tend to $0$. Two invariants grade dynamical complexity: the metric (Kolmogorov-Sinai) entropy $h_\mu(f)$, the asymptotic exponential rate at which the dynamics distinguishes initial conditions from the viewpoint of the invariant measure, and the topological entropy $h_{\mathrm{top}}(f)$, the exponential growth rate of the number of distinguishable orbit segments; always $h_\mu(f)\leq h_{\mathrm{top}}(f)$. A theorem of Katok (whose effective form is relevant below) asserts that a $C^{1+\alpha}$ surface diffeomorphism carrying an invariant probability measure of positive metric entropy must possess hyperbolic periodic points (periodic points at which the derivative along the orbit expands one direction and contracts another) - indeed horseshoes (dynamics containing a copy of the full shift on two symbols) - and hence infinitely many periodic points.

The classical device for producing zero-entropy conservative dynamics is the approximation-by-conjugation method of Anosov and Katok: one writes $f_n=h_n^{-1}\circ R_{\alpha_n}\circ h_n$, where $R_{\alpha_n}$ is the rigid rotation by a rational $\alpha_n$, $h_n$ is an area-preserving diffeomorphism, and $\alpha_n\to\alpha$ irrational, arranged so that $f_n$ converges to a diffeomorphism with rotation number $\alpha$ and exotic ergodic properties. The method yields ergodic and weakly mixing area-preserving disc diffeomorphisms with zero entropy, but no scheme has ever produced a mixing limit on a surface: known limits are rigid or at best weakly mixing. Rigidity is the obstruction: $f$ is $C^k$-rigid if $f^{q_j}\to\mathrm{Id}$ in the $C^k$ topology along some sequence $q_j\to\infty$, and rigidity plainly precludes topological mixing. Whether mixing conservative dynamics with zero entropy exists in the smooth surface category is an old problem, raised by Fayad and Katok in connection with the smooth realization problem. In the continuous category the metric-entropy version is not obstructed: Katok's construction of Bernoulli surface diffeomorphisms, combined with the Oxtoby-Ulam theorem on conjugating full-support invariant measures to area, yields mixing homeomorphisms of the sphere with zero metric entropy; the obstruction is smoothness and, for zero topological entropy, even continuity.

A homeomorphism of $\mathbb{D}$ fixing the origin has a rotation number $\rho(f)\in\mathbb{R}/\mathbb{Z}$, the average angular displacement $\lim_{n\to\infty}\arg(f^n(z))/n$ of orbits when this limit exists and is independent of the point $z\neq 0$; one says that $f$ has a unique rotation number when this holds. An area-preserving disc homeomorphism whose only periodic point is the fixed origin is called a pseudo-rotation; its rotation number is irrational. Pseudo-rotations are precisely the transitive objects of the zero-topological-entropy conservative world: Franks and Handel proved that an area-preserving diffeomorphism of a genus-zero surface with zero topological entropy and at least three fixed points decomposes into countably many invariant annuli, so a transitive area-preserving disc diffeomorphism with zero topological entropy is necessarily a pseudo-rotation. Hence the search for a smooth mixing zero-entropy disc diffeomorphism (Q1 below) is exactly the search for a mixing pseudo-rotation, and rigidity of pseudo-rotations is the corresponding obstruction. The arithmetic of the rotation number governs rigidity: writing $p_n/q_n$ for the continued-fraction convergents of $\alpha$, a rotation number is Diophantine if $|\alpha-p/q|\geq c/q^{\tau}$ for some $c,\tau>0$ and all rationals $p/q$, of Brjuno type if $\sum_n \log q_{n+1}/q_n<\infty$ (a strictly larger class), Liouville if $\limsup_n (\log q_{n+1})/(\log q_n)=\infty$ (equivalently, approximable by rationals to arbitrarily high order), and super-Liouville if $\limsup_n (\log q_{n+1})/q_n=\infty$, in increasingly extreme opposite cases; every super-Liouville number is Liouville, but Liouville numbers need not be super-Liouville, and Liouville numbers of Brjuno type exist, so the non-Brjuno rigidity theorems do not exhaust the Liouville class. Small-divisor methods from KAM (Kolmogorov-Arnold-Moser) theory - specifically a theorem of Rüssmann that an analytic area-preserving map with Brjuno rotation number, tangent to a rotation, is analytically conjugate to it near an elliptic fixed point (one whose derivative is conjugate to a rotation) - provide local control in the analytic category.

In this landscape the paper of Avila, Fayad, Le Calvez, Xu and Zhang (AFLXZ, 2020) proved: (Theorem A) an area-preserving real analytic diffeomorphism of the disc or the sphere that is topologically mixing has positive topological entropy, so analytic mixing with zero topological entropy is impossible - the proof combines Rüssmann's theorem with the Franks-Handel structure theorem; (Theorem 1) a $C^k$ ($k\geq 2$) pseudo-rotation whose rotation number is not of Brjuno type is $C^{k-1}$-rigid; (Theorem 2) a quantitative rigidity criterion for Hölder pseudo-rotations whose rotation number is super-Liouville; (Theorem B) real analytic pseudo-rotations of the disc are $C^\infty$ locally rigid at the center; (Theorem C, whose method was published separately by Fayad and Zhang as an effective version of Katok's horseshoe theorem) a growth gap for conservative $C^2$ surface diffeomorphisms: a suitable lacunary quantitative lower bound on the derivative growth $\|Df^{q_n}\|$ forces a hyperbolic periodic point, so between exponential derivative growth with hyperbolic periodic points and subexponential growth there is a gap; and (Corollary A) a local rigidity statement: for each irrational $\alpha$ and $\delta>0$ there is $\epsilon(\delta,\alpha)>0$ such that a real analytic pseudo-rotation with rotation number $\alpha$ that is $\epsilon$-close to the rigid rotation inside an analyticity band of width $\delta$ is $C^\infty$-rigid. Pseudo-rotations of the closed annulus $\mathbb{T}\times[0,1]$ are area-preserving maps preserving each boundary circle with a unique irrational rotation number in the angular coordinate; for analytic annulus pseudo-rotations, the analytic non-mixing argument of Theorem A requires analytic conjugacy to the rotation on the boundary circles, which is guaranteed when the rotation number satisfies Yoccoz's condition $\mathcal{H}$ - the arithmetic condition from Yoccoz's theorem on the analytic linearization of circle diffeomorphisms, under which an analytic circle diffeomorphism with rotation number $\alpha$ is analytically conjugate to the rotation; it is a proper strengthening of the Brjuno condition, so Brjuno numbers failing $\mathcal{H}$ exist. At the time of writing no real analytic pseudo-rotation was known to exist - Birkhoff had conjectured in 1941 that none does - and the paper poses nine questions, stated in the problem statement.

Problem Statement

The problem is the cluster of nine questions posed by Avila-Fayad-Le Calvez-Xu-Zhang in On mixing diffeomorphisms of the disc (Inventiones mathematicae 220, 2020, 673-714): Questions 1 and 2 are stated in the introduction, and Questions 3 through 9 in Section 2.4, Some questions around the rigidity of pseudo-rotations. The common objective is to determine to what extent topological mixing can coexist with vanishing entropy, and to what extent pseudo-rotations are necessarily rigid, for diffeomorphisms of the closed disc, the sphere, and the closed annulus, across the regularity classes $C^0$, $C^k$, $C^\infty$, and real analytic ($C^\omega$). The questions, as the source states them, are:

Q1. Does there exist a smooth area-preserving diffeomorphism of the disc that is mixing with zero metric entropy? With zero topological entropy? (Open. Mixing homeomorphisms of the sphere with zero metric entropy exist by Katok's construction and Oxtoby-Ulam conjugacy, so the question is about smoothness, and about topological entropy.)

Q2. Does there exist a real analytic area-preserving diffeomorphism of the disc or the sphere with zero metric entropy that is transitive? Ergodic? (Resolved positively after publication.)

Q3. Is every $C^k$ pseudo-rotation $C^0$-rigid? The question can be asked for any $k\geq 1$, $k=\infty$, or $k=\omega$ (real analytic). (Open in the Brjuno and Diophantine regimes; for $k=\omega$, or for Diophantine rotation number and $k=\infty$, it becomes an intermediate step towards the Birkhoff-Herman problem on conjugacy to the rigid rotation.)

Q4. Is there any polynomial bound on the growth of the derivatives of a pseudo-rotation? Is every $C^\infty$ pseudo-rotation with Liouville rotation number $C^0$-rigid, or even $C^\infty$-rigid? (Open. AFLXZ Theorem 1 covers only non-Brjuno rotation numbers and Bramham's earlier result only super-Liouville ones; since Liouville numbers of Brjuno type exist, and the source itself notes that its direct rigidity approach cannot go much beyond non-Brjuno type numbers to include all Liouville numbers, the $C^0$-rigidity part for Liouville rotation number, like the $C^\infty$-rigidity part and the polynomial bound, remains open. As the source notes, a positive answer to the polynomial-bound part would yield $C^\infty$-rigidity for any Liouville rotation number by the same proof as that of Theorem 1, and, combined with Herman's smooth version of Rüssmann's theorem, would imply that smooth pseudo-rotations - hence smooth area-preserving disc diffeomorphisms with zero topological entropy - are never topologically mixing.)

Q5. Does the local rigidity result of AFLXZ Corollary A hold for an $\epsilon$ that is independent of the rotation number of the pseudo-rotation? (Open.)

Q6. Does there exist a mixing $C^0$ pseudo-rotation of the disc? (Open.)

Q7. Are pseudo-rotations of the annulus rigid? In particular, is it true that a real analytic annulus pseudo-rotation whose rotation number is of Brjuno type but fails Yoccoz's condition $\mathcal{H}$ is not topologically mixing? (Here $\mathcal{H}$ is the arithmetic condition from Yoccoz's theorem on the analytic linearization of circle diffeomorphisms, under which an analytic circle diffeomorphism with rotation number $\alpha$ is analytically conjugate to the rotation; it is a proper strengthening of the Brjuno condition, so Brjuno numbers failing $\mathcal{H}$ exist. Open; the KAM argument of Theorem A extends to the annulus only under $\mathcal{H}$.)

Q8. Is it true that a diffeomorphism of the disc with zero topological entropy that has a unique rotation number is never topologically mixing? (Open.)

Q9. Is it true that a diffeomorphism of the disc with zero topological entropy that preserves a measure strictly positive on every open set is never topologically mixing? Is it true that a diffeomorphism of the disc with zero topological entropy is never topologically mixing? (Open.)

The sub-questions are independent: each is separately answerable, and a complete resolution of any one of them - a proof of the corresponding statement or a counterexample refuting it - constitutes a complete answer to that sub-question, so resolving the whole cluster is not required. With Q2 resolved, the surviving open core is Q1, Q3, Q4, Q5, Q6, Q7 (including the analytic Brjuno-but-not-$\mathcal{H}$ annulus case), Q8, and Q9.

References

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