Analytic Pseudo-Rotations in Area-Preserving Dynamics

• Analytic pseudo-rotations are real-analytic, area-preserving maps on surfaces that exhibit the minimal number of periodic points dictated by topology.
• They are constructed using approximation-by-conjugacy methods with entire symplectomorphisms to produce phenomena such as ergodicity, transitivity, and maximal local emergence.
• These constructions challenge classical rigidity expectations by showing that analyticity permits bounded deviation and annular behavior even beyond the toral framework.

Searching arXiv for papers on analytic pseudo-rotations to ground the article in the relevant literature. Analytic pseudo-rotations are real-analytic, area-preserving surface diffeomorphisms or symplectomorphisms with the minimal number of periodic points forced by topology, but whose global dynamics need not be analytically conjugate to a rigid rotation. In the recent literature, the term appears in several closely related settings: on the open cylinder $A=\mathbb R/\mathbb Z\times\mathbb R$, where a pseudo-rotation has zero periodic points; on the closed disk, where it has exactly one periodic point; on the sphere, where it has exactly two periodic points; and, in a different toral usage, for homeomorphisms homotopic to the identity whose rotation set is a single rational vector (Berger, 2022). The analytic theory has developed along two complementary lines: rigidity and bounded-deviation results for toral rational pseudo-rotations, and constructive Anosov–Katok-type schemes showing that ergodicity, transitivity, and maximal local emergence can occur in the analytic category on the cylinder, disk, annulus, and sphere (Koropecki et al., 2012).

1. Definitions and ambient settings

The modern analytic literature distinguishes several ambient surfaces and corresponding notions of pseudo-rotation. On the open cylinder $A=\mathbb R/\mathbb Z\times\mathbb R$, equipped with the standard symplectic form $\omega_A=d\theta\wedge dy$, an analytic symplectomorphism $F\colon A\to A$ is called an “entire symplectomorphism” when it extends to a biholomorphism $F\colon A^c\to A^c$ of the complexification $$A^c=\{(\theta,y)\in\mathbb C/\mathbb Z\times\mathbb C\}$$ preserving $d\theta\wedge dy$. In this setting, a pseudo-rotation is an analytic symplectomorphism of $A$ with zero periodic points; on $S^2$, it has exactly two periodic points, namely the two fixed points (Berger, 2022).

For compact surfaces, the relevant class is

$\Symp^\omega(M)=\bigl\{\,f\in\Diff^\omega(M):f^*\omega=\omega,\ \det Df>0\bigr\},$

where $A=\mathbb R/\mathbb Z\times\mathbb R$0 is the closed annulus $A=\mathbb R/\mathbb Z\times\mathbb R$1, the closed unit disk $A=\mathbb R/\mathbb Z\times\mathbb R$2, or the sphere $A=\mathbb R/\mathbb Z\times\mathbb R$3. In that framework, an analytic pseudo-rotation has finitely many periodic points, all fixed, with the minimal numbers dictated by the surface: $A=\mathbb R/\mathbb Z\times\mathbb R$4 on $A=\mathbb R/\mathbb Z\times\mathbb R$5, $A=\mathbb R/\mathbb Z\times\mathbb R$6 on $A=\mathbb R/\mathbb Z\times\mathbb R$7, and $A=\mathbb R/\mathbb Z\times\mathbb R$8 on $A=\mathbb R/\mathbb Z\times\mathbb R$9. Moreover, each fixed point is elliptic, with eigenvalues

$\omega_A=d\theta\wedge dy$0

When the surface has boundary, one also speaks of the irrational boundary rotation number $\omega_A=d\theta\wedge dy$1 (Berger, 2024).

A related but distinct toral notion comes from rotation theory. For $\omega_A=d\theta\wedge dy$2, a homeomorphism $\omega_A=d\theta\wedge dy$3 homotopic to the identity admits a lift $\omega_A=d\theta\wedge dy$4 with rotation set

$\omega_A=d\theta\wedge dy$5

If $\omega_A=d\theta\wedge dy$6 with $\omega_A=d\theta\wedge dy$7, then $\omega_A=d\theta\wedge dy$8 is called a rational pseudo-rotation. This usage emphasizes rotation sets and bounded deviations rather than minimal periodic-point counts (Koropecki et al., 2012).

2. Toral analytic pseudo-rotations and bounded deviation

For rational pseudo-rotations of the torus preserving a Borel probability measure of full support, the central statement is a trichotomy obtained after reduction to the irrotational case $\omega_A=d\theta\wedge dy$9. Exactly one of the following holds: $F\colon A\to A$0 contains a fully essential continuum $F\colon A\to A$1 whose complement is a disjoint union of open disks; every lifted orbit is bounded,

$F\colon A\to A$2

or there is a nonzero $F\colon A\to A$3 and $F\colon A\to A$4 such that

$F\colon A\to A$5

The last alternative is the annular case: bounded deviation in one rational direction (Koropecki et al., 2012).

The analytic setting sharpens this classification. If $F\colon A\to A$6 is real analytic and area-preserving, then the fully essential fixed-point alternative is ruled out. The stated reason is that $F\colon A\to A$7 is a real-analytic subset of a surface and hence locally connected, whereas in the fully essential case the essential component of $F\colon A\to A$8 is non-locally-connected. Consequently, a real-analytic area-preserving rational pseudo-rotation on $F\colon A\to A$9 satisfies only the bounded-orbit alternative or the annular alternative (Koropecki et al., 2012).

One corollary given explicitly is that any real-analytic area-preserving rational pseudo-rotation on $F\colon A^c\to A^c$0 must satisfy bounded deviations in some rational direction. Equivalently, some finite power of $F\colon A^c\to A^c$1 leaves invariant an essential topological annulus on which it acts like an annular homeomorphism. By contrast, there are $F\colon A^c\to A^c$2 non-analytic examples in which the excluded fixed-point pathology does occur: the fixed-point set is a large non-locally-connected continuum bounding a disk in which orbits wander unboundedly in all directions (Koropecki et al., 2012).

This toral theory is not the same as the disk–annulus–sphere theory developed through approximation by conjugacy. Nevertheless, it gives an analytic obstruction principle: on $F\colon A^c\to A^c$3, analyticity constrains the topology of the fixed locus strongly enough to exclude one branch of the general trichotomy. A plausible implication is that, in the toral context, analytic pseudo-rotation phenomena are organized more by bounded deviation and annularity than by the extreme recurrence patterns that remain possible in lower regularity.

3. Entire symplectomorphisms on the cylinder and the first analytic counterexamples

A major constructive breakthrough produced analytic symplectomorphisms of the cylinder and sphere with the minimal number of periodic points and dynamics not conjugated to a rotation. On the open cylinder, there exists an entire symplectomorphism $F\colon A^c\to A^c$4 leaving invariant an analytic subcylinder $F\colon A^c\to A^c$5 such that $F\colon A^c\to A^c$6 is ergodic, $F\colon A^c\to A^c$7 is analytically conjugate to the rigid rotation $F\colon A^c\to A^c$8 for some $F\colon A^c\to A^c$9, and $$A^c=\{(\theta,y)\in\mathbb C/\mathbb Z\times\mathbb C\}$$0 has no periodic point in the whole complex cylinder $$A^c=\{(\theta,y)\in\mathbb C/\mathbb Z\times\mathbb C\}$$1 (Berger, 2022).

The same construction yields, by analytic blow-down of the two boundary circles, an analytic symplectomorphism of $$A^c=\{(\theta,y)\in\mathbb C/\mathbb Z\times\mathbb C\}$$2 with exactly two fixed points and no other periodic points, whose restriction to the open equatorial cylinder is ergodic. A parallel construction gives a pseudo-rotation on $$A^c=\{(\theta,y)\in\mathbb C/\mathbb Z\times\mathbb C\}$$3 with zero periodic points whose ergodic decomposition has local emergence of maximal order $$A^c=\{(\theta,y)\in\mathbb C/\mathbb Z\times\mathbb C\}$$4: at Lebesgue-almost every point $$A^c=\{(\theta,y)\in\mathbb C/\mathbb Z\times\mathbb C\}$$5, the empirical measures

$$A^c=\{(\theta,y)\in\mathbb C/\mathbb Z\times\mathbb C\}$$6

vary in an infinite-dimensional family of local dimension $$A^c=\{(\theta,y)\in\mathbb C/\mathbb Z\times\mathbb C\}$$7 (Berger, 2022).

These results explicitly disprove Birkhoff’s 1941 conjecture that any real-analytic pseudo-rotation of the disk, annulus, or sphere must be analytically conjugate to a rigid rotation. They also solve Herman’s 1998 problem asking for an analytic symplectic diffeomorphism of $$A^c=\{(\theta,y)\in\mathbb C/\mathbb Z\times\mathbb C\}$$8 or $$A^c=\{(\theta,y)\in\mathbb C/\mathbb Z\times\mathbb C\}$$9 with a finite number of periodic points and a dense orbit. The same work records an application to complex dynamics: when $d\theta\wedge dy$0 is viewed as an entire automorphism of $d\theta\wedge dy$1, its Julia set $d\theta\wedge dy$2 is nonempty, contains the analytic core $d\theta\wedge dy$3, but has no hyperbolic periodic point; in particular $d\theta\wedge dy$4 (Berger, 2022).

The significance of these constructions is twofold. First, they replace a rigidity expectation by explicit analytic counterexamples. Second, they do so without relaxing the symplectic or pseudo-rotation constraints: zero periodic points on the cylinder, exactly two on the sphere. This suggests that analyticity alone does not force rotation-like dynamics once one leaves the torus and works in the cylinder–disk–sphere framework.

4. Approximation by conjugacy in the analytic category

The constructive theory is based on the Anosov–Katok approximation-by-conjugacy method. One builds maps of the form

$d\theta\wedge dy$5

on the cylinder, with $d\theta\wedge dy$6, where $d\theta\wedge dy$7. Outside a small core $d\theta\wedge dy$8, the limit of the conjugacies exists and the resulting map is conjugate to $d\theta\wedge dy$9; inside $A$0, the conjugacies diverge and the limit is not globally conjugate to a rigid rotation (Berger, 2022).

At each stage one uses horizontal and vertical twists,

$A$1

with $A$2 entire and small on complex strips. The analytic control is expressed on

$A$3

using the norm

$A$4

Choosing $A$5 and parameters $A$6 with $A$7, $A$8, one arranges

$A$9

with $S^2$0 (Berger, 2022).

The keystone approximation result states that if $S^2$1 and $S^2$2, then for every $S^2$3 and every $S^2$4-neighborhood $S^2$5 of $S^2$6, there exists $S^2$7 whose restriction lies in $S^2$8 and such that

$S^2$9

where $\Symp^\omega(M)=\bigl\{\,f\in\Diff^\omega(M):f^*\omega=\omega,\ \det Df>0\bigr\},$0. The proof proceeds by expressing compactly supported Hamiltonian maps as finite products of commutators $\Symp^\omega(M)=\bigl\{\,f\in\Diff^\omega(M):f^*\omega=\omega,\ \det Df>0\bigr\},$1 of vertical and horizontal twists, then using Runge’s theorem to replace local factors by entire ones while keeping control on large complex sets (Berger, 2022).

A complementary and more general formulation appears as an AbC principle on $\Symp^\omega(M)=\bigl\{\,f\in\Diff^\omega(M):f^*\omega=\omega,\ \det Df>0\bigr\},$2, $\Symp^\omega(M)=\bigl\{\,f\in\Diff^\omega(M):f^*\omega=\omega,\ \det Df>0\bigr\},$3, and $\Symp^\omega(M)=\bigl\{\,f\in\Diff^\omega(M):f^*\omega=\omega,\ \det Df>0\bigr\},$4: any dynamical property $\Symp^\omega(M)=\bigl\{\,f\in\Diff^\omega(M):f^*\omega=\omega,\ \det Df>0\bigr\},$5 that is $\Symp^\omega(M)=\bigl\{\,f\in\Diff^\omega(M):f^*\omega=\omega,\ \det Df>0\bigr\},$6-AbC-realizable can be realized by a real-analytic symplectomorphism. In the paper’s formulation, for $\Symp^\omega(M)=\bigl\{\,f\in\Diff^\omega(M):f^*\omega=\omega,\ \det Df>0\bigr\},$7, any CT-AbC-realizable $\Symp^\omega(M)=\bigl\{\,f\in\Diff^\omega(M):f^*\omega=\omega,\ \det Df>0\bigr\},$8-property on $\Symp^\omega(M)=\bigl\{\,f\in\Diff^\omega(M):f^*\omega=\omega,\ \det Df>0\bigr\},$9 is satisfied by some analytic symplectomorphism $A=\mathbb R/\mathbb Z\times\mathbb R$00 (Berger, 2024).

5. Sphere and disk: transitivity, ergodicity, and emergence

The analytic AbC principle has been used to transfer smooth $A=\mathbb R/\mathbb Z\times\mathbb R$01- or $A=\mathbb R/\mathbb Z\times\mathbb R$02-realizable phenomena to genuine analytic pseudo-rotations on the sphere and disk. One stated consequence is the existence of $A=\mathbb R/\mathbb Z\times\mathbb R$03 that is transitive and has exactly two periodic points, both elliptic fixed points, and of $A=\mathbb R/\mathbb Z\times\mathbb R$04 that is transitive and has exactly one periodic point, an elliptic fixed point. Since transitivity implies the existence of an elliptic fixed point that is not Lyapunov-stable, this also yields an analytic counterexample to Birkhoff’s 1927 question on stability of analytic elliptic points (Berger, 2024).

A later development sharpened the dynamical conclusions from transitivity to ergodicity and from orbit-density phenomena to maximal local emergence. For $A=\mathbb R/\mathbb Z\times\mathbb R$05 equal to the closed disk or the sphere, $A=\mathbb R/\mathbb Z\times\mathbb R$06 is taken to be the group of real-analytic, area-preserving diffeomorphisms whose holomorphic continuation to a fixed complex neighborhood remains uniformly bounded, equipped with the analytic Fréchet topology

$A=\mathbb R/\mathbb Z\times\mathbb R$07

In this framework, a pseudo-rotation on the disk has exactly one periodic point, necessarily an elliptic fixed point at the center, while on the sphere it has exactly two periodic points, necessarily the two poles, and no other periodic orbits in the interior (Delaporte, 17 Jul 2025).

Using Berger’s analytic-lifting principle, one obtains real-analytic pseudo-rotations of the disk and sphere that preserve area and are metric ergodic. The same principle also yields real-analytic pseudo-rotations on the disk and sphere with maximal local emergence. Emergence is described through empirical measures

$A=\mathbb R/\mathbb Z\times\mathbb R$08

their limit set $A=\mathbb R/\mathbb Z\times\mathbb R$09, and the ergodic decomposition measure $A=\mathbb R/\mathbb Z\times\mathbb R$10. On a surface, the growth exponent $A=\mathbb R/\mathbb Z\times\mathbb R$11 is bounded above by $A=\mathbb R/\mathbb Z\times\mathbb R$12, and maximal emergence means equality $A=\mathbb R/\mathbb Z\times\mathbb R$13 (Delaporte, 17 Jul 2025).

The constructions use two dual AbC schemes: a $A=\mathbb R/\mathbb Z\times\mathbb R$14-scheme for ergodicity and a $A=\mathbb R/\mathbb Z\times\mathbb R$15-scheme for maximal local emergence. In the ergodic scheme on the cylinder $A=\mathbb R/\mathbb Z\times\mathbb R$16, one chooses $A=\mathbb R/\mathbb Z\times\mathbb R$17 commuting with $A=\mathbb R/\mathbb Z\times\mathbb R$18, supported in $A=\mathbb R/\mathbb Z\times\mathbb R$19, so that for every horizontal circle the push-forward of Lebesgue is $A=\mathbb R/\mathbb Z\times\mathbb R$20-close in Kantorovich–Wasserstein distance to full Lebesgue measure on $A=\mathbb R/\mathbb Z\times\mathbb R$21; after iteration and passage to the analytic category, this yields ergodic pseudo-rotations on $A=\mathbb R/\mathbb Z\times\mathbb R$22 and $A=\mathbb R/\mathbb Z\times\mathbb R$23. In the emergence scheme, one forces conditional measures on horizontal lines to separate at scales governed by $A=\mathbb R/\mathbb Z\times\mathbb R$24, thereby obtaining $A=\mathbb R/\mathbb Z\times\mathbb R$25 in the limit (Delaporte, 17 Jul 2025).

6. Conceptual consequences and open problems

Several long-standing problems are explicitly recorded as resolved. The cylinder and sphere constructions disprove Birkhoff’s 1941 conjecture that any real-analytic pseudo-rotation of the disk, annulus, or sphere must be analytically conjugate to a rigid rotation, and they solve Herman’s 1998 problem on analytic symplectic diffeomorphisms with finitely many periodic points and a dense orbit (Berger, 2022). The AbC principle on spheres, disks, and annuli is stated to solve problems of Birkhoff (1927), Herman (1998), Fayad–Katok (2004), and Fayad–Krikorian (2018), including the existence of transitive analytic pseudo-rotations of the closed disk and unstable analytic elliptic points on $A=\mathbb R/\mathbb Z\times\mathbb R$26 (Berger, 2024).

The constructive viewpoint also changes how rigidity questions are posed. Rather than asking whether analyticity enforces conjugacy to a rigid rotation, the current literature isolates arithmetic and geometric conditions under which rigidity may re-emerge. One open question asks whether an analytic pseudo-rotation with Diophantine rotation number regains rigidity; another asks whether analytic boundary curves can be replaced by analytic pseudo-circles. In the more recent sphere–disk framework, further questions include whether analytic pseudo-rotations can exhibit mixing of all orders or even weak mixing, whether positive metric entropy can occur without destroying the pseudo-rotation property, and which Bryuno or Brjuno-type conditions on the rotation number are compatible with ergodicity or maximal emergence (Berger, 2022).

Across these works, two complementary pictures coexist. On the torus, analyticity excludes one topological pathology and leads to bounded-orbit or annular behavior for area-preserving rational pseudo-rotations (Koropecki et al., 2012). On the cylinder, annulus, disk, and sphere, analytic approximation-by-conjugacy produces pseudo-rotations that are ergodic, transitive, or display maximal local emergence, while still having the topology-forced minimal number of periodic points (Delaporte, 17 Jul 2025). This suggests that “analytic pseudo-rotation” is not a single rigidity class but a family of analytically constrained dynamical regimes whose precise form depends strongly on the underlying surface and on whether the organizing invariant is a toral rotation set or a minimal periodic-point count.