Rotational properties of nilpotent groups of diffeomorphisms of surfaces

Rotational properties of nilpotent groups of diffeomorphisms of surfaces

Javier Ribón javier@mat.uff.br

April 2014

Flows

Theorem (Lima)

LetX₁, . . . ,X_nbe vector fields inS²such that the flows ofX_i andX_j commute for all 1≤i,j ≤n. ThenSing(X₁)∩. . .∩Sing(X_n)6=∅.

Close to Id diffeomorphisms

Theorem (Bonatti)

LetSbe a compact surface of non-vanishing Euler characteristic. Let f₁, . . . ,f_nbe pairwise commutingC¹-diffeomorphisms close toId. Then Fix(f₁)∩. . .∩Fix(fn)6=∅.

hf₁, . . . ,fniis an abelian group

The result holds for nilpotent groups on the sphere (Druck-Fang-Firmo).

Close to Id diffeomorphisms

Theorem (Bonatti)

LetSbe a compact surface of non-vanishing Euler characteristic. Let f₁, . . . ,f_nbe pairwise commutingC¹-diffeomorphisms close toId. Then Fix(f₁)∩. . .∩Fix(fn)6=∅.

hf₁, . . . ,fniis an abelian group

The result holds for nilpotent groups on the sphere (Druck-Fang-Firmo).

Close to Id diffeomorphisms

Theorem (Bonatti)

LetSbe a compact surface of non-vanishing Euler characteristic. Let f₁, . . . ,f_nbe pairwise commutingC¹-diffeomorphisms close toId. Then Fix(f₁)∩. . .∩Fix(fn)6=∅.

hf₁, . . . ,fniis an abelian group

The result holds for nilpotent groups on the sphere (Druck-Fang-Firmo).

Isotopy theory

Idea

Finding models of the group up to isotopy with irreducible elements and then transferring the properties of the model to the initial group.

Irreducible elements Finite order elements Pseudo-Anosov elements

Thurston classification

Given an orientation-preserving homeomorphismf :S→S, there exists a homeomorphismgisotopic tof such that:

g is a finite order element or g is pseudo-Anosov or

g preserves a finite union of disjoint simple essential closed curves (=reducing curves).

There exists a common Thurston decomposition for abelian groups.

Irreducible elements Finite order elements Pseudo-Anosov elements Thurston classification

Given an orientation-preserving homeomorphismf :S→S, there exists a homeomorphismgisotopic tof such that:

g is a finite order element or g is pseudo-Anosov or

g preserves a finite union of disjoint simple essential closed curves (=reducing curves).

There exists a common Thurston decomposition for abelian groups.

Irreducible elements Finite order elements Pseudo-Anosov elements Thurston classification

Given an orientation-preserving homeomorphismf :S→S, there exists a homeomorphismgisotopic tof such that:

g is a finite order element or g is pseudo-Anosov or

g preserves a finite union of disjoint simple essential closed curves (=reducing curves).

There exists a common Thurston decomposition for abelian groups.

Theorem (Franks-Handel-Parwani ’07)

LetGbe an abelian subgroup ofDiff¹₊(S²). Then there exists either a global fixed point or a 2-orbit. MoreoverGhas a global fixed point if w(f,g) =0 for allf,g ∈G.

w :G×G→Z/2Zis a morphism of groups

Theorem (Franks-Handel-Parwani ’07)

LetGbe an abelian subgroup ofDiff¹₊(S²). Then there exists either a global fixed point or a 2-orbit. MoreoverGhas a global fixed point if w(f,g) =0 for allf,g ∈G.

w :G×G→Z/2Zis a morphism of groups

Theorem (Franks-Handel-Parwani ’07)

LetGbe an abelian subgroup ofDiff¹₊(R²). Suppose thatGhas a non-empty compact invariant set. Then there exists a global fixed point.

Theorem (R)

LetGbe a nilpotent subgroup ofDiff¹₊(S²). Then there exists either a global fixed point or a 2-orbit.

Theorem (R)

LetGbe a nilpotent subgroup ofDiff¹₊(R²). Suppose thatGhas a non-empty compact invariant set. Then there exists a global fixed point.

Theorem (R)

LetGbe a nilpotent subgroup ofDiff¹₊(S²). Then there exists either a global fixed point or a 2-orbit.

Theorem (R)

LetGbe a nilpotent subgroup ofDiff¹₊(R²). Suppose thatGhas a non-empty compact invariant set. Then there exists a global fixed point.

Applications

Theorem (R)

LetGbe a nilpotent subgroup ofDiff¹₊(S²). Suppose thatGhas an odd finite invariant set. Then there exists either a global fixed point or a 2-orbit.

Applications

Consider a fixed point free nilpotent subgroupGofDiff¹₊(S²).

Definition

We say that two 2-orbitsO₁andO₂have the same class if {f ∈G:f_|O1 ≡Id}={f ∈G:f_|O2 ≡Id}

Definition

We say that 2-orbitsO₁, . . . ,O_nare independent if the action ofGon O₁∪. . .∪ O_nis(Z/2Z)ⁿ.

Applications

Consider a fixed point free nilpotent subgroupGofDiff¹₊(S²).

Definition

We say that two 2-orbitsO₁andO₂have the same class if {f ∈G:f_|O1 ≡Id}={f ∈G:f_|O2 ≡Id}

Definition

We say that 2-orbitsO₁, . . . ,O_nare independent if the action ofGon O₁∪. . .∪ O_nis(Z/2Z)ⁿ.

Applications

Consider a fixed point free nilpotent subgroupGofDiff¹₊(S²).

Definition

We say that two 2-orbitsO₁andO₂have the same class if {f ∈G:f_|O1 ≡Id}={f ∈G:f_|O2 ≡Id}

Definition

We say that 2-orbitsO₁, . . . ,O_nare independent if the action ofGon

Consider a fixed point free nilpotent G of Diff

( S

).

Definition

We say that two 2-orbitsO₁andO₂have the same class if {f ∈G:f_|O1 ≡Id}={f ∈G:f_|O2 ≡Id}

Definition

We say that 2-orbitsO₁, . . . ,O_nare independent if the action ofGon O₁∪. . .∪ O_nis(Z/2Z)ⁿ.

Exercise

If there are 4 classes of 2-orbits then there are 3 independent 2-orbits.

Abelian case

{f ∈G:f_|O1 ≡Id} ∩ {f ∈G:f_|O2 ≡Id}={f ∈G:w(f,g) =0∀g ∈G}

Consider a fixed point free nilpotent G of Diff

( S

).

Definition

We say that two 2-orbitsO₁andO₂have the same class if {f ∈G:f_|O1 ≡Id}={f ∈G:f_|O2 ≡Id}

Definition

We say that 2-orbitsO₁, . . . ,O_nare independent if the action ofGon O₁∪. . .∪ O_nis(Z/2Z)ⁿ.

Exercise

If there are 4 classes of 2-orbits then there are 3 independent 2-orbits.

Consider a fixed point free nilpotent G of Diff

( S

).

Definition

We say that two 2-orbitsO₁andO₂have the same class if {f ∈G:f_|O1 ≡Id}={f ∈G:f_|O2 ≡Id}

We say that 2-orbitsO₁, . . . ,O_nare independent if the action ofGon O₁∪. . .∪ O_nis(Z/2Z)ⁿ.

Exercise

If there are 4 classes of 2-orbits then there are 3 independent 2-orbits.

Abelian case

There are no 4 different classes of 2-orbits. More precisely there are 3 classes of 2-orbits.

Consider a fixed point free nilpotent G of Diff

( S

).

Definition

We say that two 2-orbitsO₁andO₂have the same class if {f ∈G:f_|O1 ≡Id}={f ∈G:f_|O2 ≡Id}

We say that 2-orbitsO₁, . . . ,O_nare independent if the action ofGon O₁∪. . .∪ O_nis(Z/2Z)ⁿ.

Exercise

If there are 4 classes of 2-orbits then there are 3 independent 2-orbits.

Consider a fixed point free nilpotent G of Diff

( S

).

Nilpotent case

F =union of two 2-orbits or three 2-orbits whose classes are pairwise different.

G→[G]⊂Mod(S²,F)

[G]is abelian and irreducible.

It is possible to definew :G×G→Z/2Zas in the abelian case. Theorem (R)

LetGbe a fixed-point-free nilpotent subgroup ofDiff¹₊(S²). Then there are either 1 or 3 classes of 2-orbits.

Consider a fixed point free nilpotent G of Diff

( S

).

Nilpotent case

F =union of two 2-orbits or three 2-orbits whose classes are pairwise different.

G→[G]⊂Mod(S²,F) [G]is abelian and irreducible.

It is possible to definew :G×G→Z/2Zas in the abelian case. Theorem (R)

LetGbe a fixed-point-free nilpotent subgroup ofDiff¹₊(S²). Then there are either 1 or 3 classes of 2-orbits.

Consider a fixed point free nilpotent G of Diff

( S

).

Nilpotent case

F =union of two 2-orbits or three 2-orbits whose classes are pairwise different.

G→[G]⊂Mod(S²,F) [G]is abelian and irreducible.

It is possible to definew :G×G→Z/2Zas in the abelian case.

Theorem (R)

LetGbe a fixed-point-free nilpotent subgroup ofDiff¹₊(S²). Then there are either 1 or 3 classes of 2-orbits.

Consider a fixed point free nilpotent G of Diff

( S

).

Nilpotent case

F =union of two 2-orbits or three 2-orbits whose classes are pairwise different.

G→[G]⊂Mod(S²,F) [G]is abelian and irreducible.

It is possible to definew :G×G→Z/2Zas in the abelian case.

Theorem (R)

LetGbe a fixed-point-free nilpotent subgroup ofDiff¹₊(S²). Then there

Theorem (Firmo-R)

LetG=hH, φibe a nilpotent subgroup ofDiff¹₀(T²)whereH is a normal subgroup ofG. Suppose that there exists aφ-invariant ergodic measureµsuch that the support ofµis contained inFix(H)and ρ_µ(φ) = (0,0). ThenGhas a global fixed point.

Theorem (Firmo-R)

LetGbe an irrotational nilpotent subgroup ofDiff¹₀(T²). ThenGhas a global fixed point.

Theorem (Firmo-R)

LetGbe a nilpotent subgroup ofHomeo(T²)(resp. Homeo₊(T²), Homeo0(T²)). ThenG⁰⁰⁰(resp. G⁰⁰,G⁰) is irrotational.

Theorem (Firmo-R)

LetG=hH, φibe a nilpotent subgroup ofDiff¹₀(T²)whereH is a normal subgroup ofG. Suppose that there exists aφ-invariant ergodic measureµsuch that the support ofµis contained inFix(H)and ρ_µ(φ) = (0,0). ThenGhas a global fixed point.

Theorem (Firmo-R)

LetGbe an irrotational nilpotent subgroup ofDiff¹₀(T²). ThenGhas a global fixed point.

Theorem (Firmo-R)

LetGbe a nilpotent subgroup ofHomeo(T²)(resp. Homeo₊(T²), Homeo0(T²)). ThenG⁰⁰⁰(resp. G⁰⁰,G⁰) is irrotational.

Theorem (Firmo-R)

LetG=hH, φibe a nilpotent subgroup ofDiff¹₀(T²)whereH is a normal subgroup ofG. Suppose that there exists aφ-invariant ergodic measureµsuch that the support ofµis contained inFix(H)and ρ_µ(φ) = (0,0). ThenGhas a global fixed point.

Theorem (Firmo-R)

LetGbe an irrotational nilpotent subgroup ofDiff¹₀(T²). ThenGhas a global fixed point.

Theorem (Firmo-R)

LetGbe a nilpotent subgroup ofHomeo(T²)(resp. Homeo₊(T²), Homeo0(T²)). ThenG⁰⁰⁰(resp. G⁰⁰,G⁰) is irrotational.

Theorem (Firmo-Le Calvez-R)

LetGbe a nilpotent subgroup ofDiff¹(S²)such thatFix(G) ={∞}.

Then:

i) Ghas a fixed point onS∞;

ii) iff ∈G, then∞is not isolated inFix(f)ifFix(f)6={∞};

iii) for everyf ∈Gand everyz ∈Fix(f)\ {∞}one hasR_f(z) ={0};

iv) every finite invariant measure off ∈Gis supported onFix(f);

v) every periodic point off ∈Gis fixed;

S∞=circle of half-directions at∞, it is obtained by blowing-up∞. R_f(z) =rotation set of the annulus homeomorphism induced byf in R²\ {z}

The result was already known for abelian groups (Beguin - Le Calvez - Firmo - Miernowski)

Theorem (Firmo-Le Calvez-R)

LetGbe a nilpotent subgroup ofDiff¹(S²)such thatFix(G) ={∞}.

Then:

i) Ghas a fixed point onS∞;

ii) iff ∈G, then∞is not isolated inFix(f)ifFix(f)6={∞};

iii) for everyf ∈Gand everyz ∈Fix(f)\ {∞}one hasR_f(z) ={0};

iv) every finite invariant measure off ∈Gis supported onFix(f);

v) every periodic point off ∈Gis fixed;

S∞=circle of half-directions at∞, it is obtained by blowing-up∞.

R_f(z) =rotation set of the annulus homeomorphism induced byf in R²\ {z}

The result was already known for abelian groups (Beguin - Le Calvez - Firmo - Miernowski)

Theorem (Firmo-Le Calvez-R)

LetGbe a nilpotent subgroup ofDiff¹(S²)such thatFix(G) ={∞}.

Then:

i) Ghas a fixed point onS∞;

ii) iff ∈G, then∞is not isolated inFix(f)ifFix(f)6={∞};

iii) for everyf ∈Gand everyz ∈Fix(f)\ {∞}one hasR_f(z) ={0};

iv) every finite invariant measure off ∈Gis supported onFix(f);

v) every periodic point off ∈Gis fixed;

S∞=circle of half-directions at∞, it is obtained by blowing-up∞.

R_f(z) =rotation set of the annulus homeomorphism induced byf in R²\ {z}

The result was already known for abelian groups (Beguin - Le Calvez - Firmo - Miernowski)

Theorem (Firmo-Le Calvez-R)

LetGbe a nilpotent subgroup ofDiff¹(S²)such thatFix(G) ={∞}.

Then:

i) Ghas a fixed point onS∞;

ii) iff ∈G, then∞is not isolated inFix(f)ifFix(f)6={∞};

iii) for everyf ∈Gand everyz ∈Fix(f)\ {∞}one hasR_f(z) ={0};

iv) every finite invariant measure off ∈Gis supported onFix(f);

v) every periodic point off ∈Gis fixed;

S∞=circle of half-directions at∞, it is obtained by blowing-up∞.

R_f(z) =rotation set of the annulus homeomorphism induced byf in R²\ {z}

The result was already known for abelian groups (Beguin - Le Calvez -