Analytic pseudo-rotations Pierre Berger October 10 2022

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Analytic pseudo-rotations

Pierre Berger∗

October 10, 2022

Abstract

We construct analytic symplectomorphisms of the cylinder or the sphere with zero or exactly

two periodic points and which are not conjugated to a rotation. In the case of the cylinder, we

show that these symplectomorphisms can be chosen ergodic or to the contrary with local emer-

gence of maximal order. In particular, this disproves a conjecture of Birkhoﬀ (1941) and solve

a problem of Herman (1998). One aspect of the proof provides a new approximation theorem,

it enables in particular to implement the Anosov-Katok scheme in new analytic settings.

Contents

1 Statement of the main theorems 2

1.1 Ergodic analytic and symplectic pseudo-rotation . . . . . . . . . . . . . . . . . . . . 2

1.2 Analytic and symplectic pseudo-rotation with maximal local emergence . . . . . . . 3

1.3 Sketch of proof and main approximation theorem . . . . . . . . . . . . . . . . . . . . 6

1.4 A complex analytic consequence of the main results . . . . . . . . . . . . . . . . . . 9

2 Proof of the main theorems 10

2.1 Notations for probability measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 Proof of Theorem Aon ergodic pseudo-rotations . . . . . . . . . . . . . . . . . . . . 10

2.2.1 Existence of ergodic smooth pseudo-rotations . . . . . . . . . . . . . . . . . . 10

2.2.2 Existence of ergodic analytic pseudo-rotations . . . . . . . . . . . . . . . . . . 12

2.3 Proof of Theorem Con pseudo-rotation with maximal local emergence . . . . . . . . 14

2.3.1 Existence of smooth pseudo-rotations with maximal local emergence . . . . . 14

2.3.2 Existence of analytic pseudo-rotations with maximal local emergence . . . . . 17

2.4 Proofofthecorollaries................................... 18

3 Approximation Theorems 19

3.1 GeneratorsofHam..................................... 19

3.2 The smooth case: proof of Theorem 3.3 ......................... 21

3.3 The analytic case: proof of Theorem 3.4 ......................... 23

3.4 Proof that Theorem 3.4 implies Theorem 1.8 ...................... 24

4 Smooth Lemmas 27

4.1 A consequence of Moser’s trick . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.2 Lemma for Anosov-Katok’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.3 Lemmaforemergence ................................... 29

∗IMJ-PRG, CNRS, Sorbonne University, Paris University, partially supported by the ERC project 818737 Emer-

gence of wild diﬀerentiable dynamical systems.

arXiv:2210.03438v1 [math.DS] 7 Oct 2022

1 Statement of the main theorems

In low dimensional analytic dynamics, a fundamental question is wether far from periodic points,

the dynamics is rigid. In real or complex dimension 1, dynamics are there rigid [Yoc84,Sul85]: the

dynamics restricted to an invariant domain without periodic point is either a rotation or in the

basin of periodic points. In dimension 2, for real analytic volume preserving diﬀeomorphisms of

the cylinder or the real sphere, the Birkhoﬀ rigidity conjecture for pseudo-rotations [Bir41] states

that such are also conjugate to rotations. We are going to disprove this conjecture. More precisely

we are going to give two examples of entire real symplectomorphisms of the annulus A=R/Z×R,

without periodic points and such that outside a region A0

0of Abounded by two disjoint analytic

curves, the dynamics is analytically conjugate to a rotation, while inside A0

0the dynamics is not

rigid. In the ﬁrst example, the restriction f|A0

0will be ergodic – see Theorem A– while in the

second example it will be extremely far from being ergodic – see Theorem C. More precisely the

ergodic decomposition of the latter map will be inﬁnite dimensional and even of maximal local

order, i.e. with local emergence of maximal order 2. This conﬁrms a conjecture on typicality of

high emergence in many categories [Ber17].

From this we will deduce a corollary disproving the Birkhoﬀ rigidity conjecture. The proof of

the main theorems are shown by developing the Anosov-Katok method [AK70], together with a

new approximation theorem for entire symplectomorphisms developing a recent work with Turaev

[BT22]. The corollary regarding the sphere is obtained by using a blow-down techniques introduced

in [Ber22].

Finally we will remark that these analytic constructions give examples of entires symplectomor-

phisms of C/Z×Cwithout periodic points and with a non-empty instability region J.

1.1 Ergodic analytic and symplectic pseudo-rotation

Let Sbe an orientable analytic surface. We recall a diﬀeomorphism of Sis symplectic if it

preserves the orientation and the volume.

Conjecture (Birkhoﬀ [Bir41, Pb 14-15]).An analytic symplectomorphism of the sphere with only

two ﬁxed points and no other periodic point is necessarily topologically conjugate to a rotation.

Analogously, an analytic symplectomorphism of a compact cylinder without periodic points is

topologically conjugate to a rotation.

While following Birkhoﬀ “considerable evidence was adducted for this conjecture”, Anosov-

Katok [AK70] gave examples of smooth symplectomorphisms of the sphere or the annulus with

resp. 2 or 0 periodic points which are ergodic. Anosov and Katok proved their theorems by

introducing the approximation by conjugacy method, that we will recall in the sequel. Also,

in his famous list of open problems, Herman wrote that a positive answer to the following question

would disprove the Birkhoﬀ rigidity conjecture:

Question 1.1 (Herman [Her98][Q.3.1]).Does there exist an analytic symplectomorphism of the

cylinder or the sphere with a ﬁnite number of periodic points and a dense orbit?

Corollary Bwill disprove both Birkhoﬀ’s conjectures and answer positively to Herman’s question

in the cylinder case. It will also bring a new analytic case of application of the approximation by

conjugacy method, as wondered by Fayad-Katok in [FK04,§7]. To state the main theorems, we set:

T:= R/Z,I:= (−1,1),A:= T×R,and A0=A×I.

We recall that a map of Ais entire if it extends to an analytic map on C/Z×C. An entire

symplectomorphism is a symplectomorphism which is entire and whose inverse is entire.

An analytic cylinder of A0

0of Ais a subset of the form {(θ, y)∈A:γ−(θ)< y < γ+(θ)}, for

two analytic functions γ−< γ+.

Theorem A. There is an entire symplectomorphism Fof Awhich leaves invariant an analytic

cylinder A0

0⊂A, whose restriction to A0

0is ergodic and whose restriction to A\cl(A0

0)is analytically

conjugated to a rotation. Moreover the complex extension of Fhas no periodic point in C/Z×C.

A consequence of Theorem Ais a counter example of both Birkhoﬀ conjectures and a positive

answer to Herman’s question in the case of the cylinder:

Corollary B. 1. There is an analytic symplectomorphism of cl(A0)which is ergodic and has

no periodic point.

2. There is an analytic symplectomorphism of the sphere, whose restriction to a sub-cylinder is

ergodic and which displays only two periodic points.

This corollary is proved in Section 2.4 using the blow up techniques introduced in [Ber22]. It

seems that the proof of Theorem Amight be pushed forward to obtain a positive answer to the

following:

Problem 1.2. In Theorem A, show that we can replace A0

0by a cylinder with whose boundary is

formed by two pseudo-circles.

Theorem Awill be proved by proving a new approximation Theorem 1.8 which enables to

implement the approximation by conjugacy method to analytic maps of the cylinder. Analytic

symplectomorphisms of the 2-torus which are ergodic and without periodic points are known since

Furstenberg [Fur61, Thm 2.1] (one of the explicit examples is (θ1, θ2)∈T27→ (θ1+α, θ1+θ2) for

any αirrational). Actually the approximation by conjugacy method is known to provide examples

of analytic symplectomorphisms of T2which are isotopic to the identity, see [BK19] for stronger

results. Up to now, the only other known analytic realization of approximation by conjugacy

method was Fayad-Katok’s theorem [FK14] showing the existence of analytic uniquely ergodic

volume preserving maps on odd spheres. As a matter of fact, we also answer a question of Fayad-

Katok [FK04,§7.1] on wether analytic realization of the approximation by conjugacy method may

be done on other manifolds than tori or odd spheres .

A way to repair the Birkhoﬀ conjecture might be to ask rigidity for dynamics without pe-

riodic point1whose rotation number satisﬁes a diophantine condition, as wondered Herman in

[Her98][Q.3.2]. The approximation by conjugacy method has been been useful to construct many

other interesting examples of dynamical systems with special property, see for instance the survey

[FH78,Cro06]. Certainly the new approximation Theorem 1.8 enables to adapt these examples to

the case of analytic maps of the cylinder. In the next subsection, we will study a new property.

We show how to use this scheme to construct diﬀeomorphisms with high local emergence. The

readers only interested by the Birkhoﬀ conjecture can skip the next subsection and go directly to

Section 1.3.

1.2 Analytic and symplectic pseudo-rotation with maximal local emergence

While an ergodic dynamics might sound complicated, the description of the statistical behavior of

its orbits is by deﬁnition trivial. We recall that by Birkhoﬀ’s ergodic theorem, given a symplectic

1These are called pseudo-rotation in [BCLR06].

map fof a compact surface S, for Leb .a.e. point xthe following limit is a well deﬁned probability

measure called the empirical measure of x.

e(x) := 1

k=1

δfk(x).

The measure e(x) describes the statistical behavior of the orbit of x. The empirical function

e:x∈S7→ e(x) is a function with value in the space M(S) of probability measures on S. Note

that eis a measurable function.

A natural question is how complex is the diversity of statistical behaviors of the orbits of points

in a Lebesgue full set. To study this, we shall look at the size of the push forward e∗Leb of

the Lebesgue probability measure Leb of Sby e. The measure is e∗Leb is called the ergodic

decomposition; it is a probability measure on the space M(S) of probability measures of S. The

ergodic decomposition describes the distribution of the statistical behaviors of the orbits.

To measure the size of the ergodic decomposition, for every compact metric space X, we endow

the space M(X) of probability measures on Xwith the Kantorovitch-Wasserstein metric d:

∀µ1, µ2∈ M(X),d(µ1, µ2) := inf ZX2

d(x1, x2)dµ :µ∈ M(X2) s.t. pi∗µ=µi∀i∈ {1,2},

where pi: (x1, x2)7→ X2→xi∈Xfor i∈ {1,2}. This distance induces the weak ?topology on

M(X) which is compact. Also it holds:

Proposition 1.3. For any compact metric spaces X, Y , any µ∈ M(X)and f, g ∈C0(X, Y )it

holds:

d(f∗µ, g∗µ)≤max

x∈Xd(f(x), g(x)).

Proof. Let ˆµbe the measure on the diagonal of X×Xwhich is pushed forward by the 1st and 2sd

coordinate projection to µ. Then observe that the pushforward νof ˆµby the product (f, g) is a

transport from f∗µto g∗µ; its cost RY×Yd(y, y0)dν is at most maxx∈Xd(f(x), g(x)).

The following has been proved several times (see [BB21, Thm 1.3] for references):

Theorem 1.4. The box order of (M(S),d)is 2:

lim

→0

log log N()

|log |= 2 with N()the -covering number of (M(S),d).

In contrast, up to now, all the bounds on the emergence of symplectic and analytic dynamics

are ﬁnite dimensional:

Example 1.5 (Case study).1. If the measure Leb is ergodic, then the ergodic decomposition is

a Dirac measure at the Lebesgue measure: e∗Leb = δLeb .

2. For a straight irrational rotation of the annulus A0=T×I, we have e∗Leb = RIδLeb T×{y}dLeb (y)

with Leb T×{y}the one-dimensional Lebesgue measure on T×{y}. Hence the ergodic rotation

of a straight irrational rotation of the annulus is one dimensional. The same occurs for an

integrable twist map of A0.

A natural problem (see [Ber17,Ber20,BB21]) is to ﬁnd dynamics for which e∗Leb is inﬁnite

dimensional in many categories. The notion of emergence has been introduced to precise this

problem . In the present conservative setting2, the emergence describes the size of the ergodic

decomposition. More precisely, the emergence of fat scale  > 0 is the minimum number

E()≥1 of probability measures (µi)1≤i≤E()such that:

Zmin

id(e(x), µi)dLeb <.

The order of the emergence is

OEf= lim sup

→0

log log E()

|log |.

In [BB21, ineq. (2.2) and Prop. 3.14], it has been shown that OEf≤dim S= 2. Also an

example of C∞-ﬂows on the disk with maximal emergence order: OEf= 2. We do not know how

to extend these examples to the analytic setting3. Also this example has ergodic decomposition of

local dimension 1. Actually, in view of Theorem 1.4, on can hope for the existence of an ergodic

decomposition of inﬁnite local dimension and even of positive local order.

Let us denote ˆ

e:= e∗Leb the ergodic decomposition of f. The order of the local emergence

of fis4:

OEloc(f) = lim sup

→0Zlog |log ˆ

e(B(µ, ))|

|log |dˆ

In [Hel22], Helfter showed that for ˆ

ea.e. µ∈ M(S) it holds:

Olocˆ

e(µ) := lim sup

→0

log |log ˆ

e(B(µ, ))|

|log |≤ OEf.

As OEf≤2 by Theorem 1.4, it comes that Olocˆ

e(µ)≤2 a.e. Thus if OEloc(f) = 2, then the local

order Olocˆ

e(µ) of ˆ

eis 2 for ˆ

e-a.e. µ∈ M(S). In this work we give the ﬁrst example of smooth

symplectomorphism with inﬁnite dimensional local emergence for smooth dynamics. Moreover our

example is entire and of maximal order of local emergence:

Theorem C. There is an entire symplectomorphism Fof Awhich leaves invariant an analytic

cylinder A0

0⊂A, whose restriction to A0

0has local emergence 2 and whose restriction to A\cl(A0

is analytically conjugated to a rotation. Moreover the complex extension of Fhas no periodic point

in C/Z×C.

A more restrictive version of local emergence could be done by replacing the lim inf instead of

lim sup. Then a natural open problem is:

Question 1.6. Does there exist a smooth conservative map such that the following limit is positive:

OEloc(f) = lim inf

→0Zlog |log ˆ

e(B(µ, ))|

|log |dˆ

2The notion of emergence is also deﬁned for dissipative system.

3Yet in the dissipative setting, a locally dense set of area contracting polynomial automorphism of R2has been

show to have emergence of order 2 in [BB22].

4This deﬁnition of order of the local emergence is at most the one of [Ber20] were the limsup is inside the integral.

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Analyticpseudo-rotationsPierreBerger*October10,2022AbstractWeconstructanalyticsymplectomorphismsofthecylinderorthespherewithzeroorexactlytwoperiodicpointsandwhicharenotconjugatedtoarotation.Inthecaseofthecylinder,weshowthatthesesymplectomorphismscanbechosenergodicortothecontrarywithlocalemer-genceof...

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