Yi SHI Peking University Institute de Math ematiques de ......"invariant", and in the...

Hyperbolic Dynamics on Heisenberg Nilmanifold

Yi SHI

Peking University

Institute de Mathematiques de Bourgogne

IHP, June 13, 2013

Y. SHI, Peking University and IMB Hyperbolic Dynamics on Heisenberg Nilmanifold, 2013

Heisenberg Group

Consider the 3-dimensional Heisenberg group

: x, y, z ∈ R

with the matrix operation

(x, y, z) · (a, b, c) = (x+ a, y + b, z + c+ xb).

Heisenberg Nilmanifold

Heisenberg Nilmanifold H = H/Γ, where

Γ = (x, y, z) ∈ H : x, y, z ∈ Z

is the integer lattice.

Heisenberg Group

Consider the 3-dimensional Heisenberg group

: x, y, z ∈ R

with the matrix operation

(x, y, z) · (a, b, c) = (x+ a, y + b, z + c+ xb).

Heisenberg Nilmanifold H = H/Γ, where

Γ = (x, y, z) ∈ H : x, y, z ∈ Z

is the integer lattice.

Look at a fundamental domain of H, then we can see that H is

an S1-bundle over T2 with Euler number 1.

Partially Hyperbolic Automorphisms

Automorphism f : H −→ H:

f(g1) · f(g2) = f(g1 · g2), ∀g1, g2 ∈ H

If f(Γ) = Γ, then it induce an automorphism:

f : H −→ H.

Remark

If f is a partially hyperbolic automorphism on H, then it is a

lift of an linear Anosov diffeomorphism on T2 and the

S1-bundle tangent to the central space.

Partially Hyperbolic Automorphisms

Automorphism f : H −→ H:

f(g1) · f(g2) = f(g1 · g2), ∀g1, g2 ∈ H

If f(Γ) = Γ, then it induce an automorphism:

f : H −→ H.

Remark

If f is a partially hyperbolic automorphism on H, then it is a

lift of an linear Anosov diffeomorphism on T2 and the

S1-bundle tangent to the central space.

Example

Consider the cat map on T2

one of its lift on H is:

fA : (x, y, z) 7→ (2x+ y, x+ y, z + f(x, y))

f(x, y) = x2 + xy +1

2y2 − x− 1

Remark

Ec is tangent to the S1-fibers, and Es ⊕ Eu is a contact plane

field, which is not integrable everywhere.

Example

Consider the cat map on T2

one of its lift on H is:

fA : (x, y, z) 7→ (2x+ y, x+ y, z + f(x, y))

f(x, y) = x2 + xy +1

2y2 − x− 1

Remark

Ec is tangent to the S1-fibers, and Es ⊕ Eu is a contact plane

field, which is not integrable everywhere.

Contact Plane Fields

Denote the canonical vector fields

X =∂

∂x, Y =

∂y+ x

∂z, Z =

Es ⊕ Eu = 〈X − 1

2Z , Y − 1

2Z〉.

is a contact plane field.

Partially Hyperbolic Diffeomorphisms

All the C2 partially hyperbolic volume preserving

diffeomorphisms on H is ergodic. [HHU], 2008.

Every partially hyperbolic diffeomorphisms on H is leaf

conjugate to some automorphism. [H],[HP], 2013.

Main Result

Theorem. For any partially hyperbolic automorphism

fA : H −→ H,

there exists a sequence of C∞ diffeomorphisms fn, s.t.

fn −→ fA in C1 topology.

fn is structural stable.

Ω(fn) consists of one attractor and one repeller.

Remark

All these partially hyperbolic automorphisms on H is stably

ergodic, but not robustly transitive.

Main Result

Theorem. For any partially hyperbolic automorphism

fA : H −→ H,

there exists a sequence of C∞ diffeomorphisms fn, s.t.

fn −→ fA in C1 topology.

fn is structural stable.

Ω(fn) consists of one attractor and one repeller.

Remark

All these partially hyperbolic automorphisms on H is stably

ergodic, but not robustly transitive.

Naive Example

Consider A× Id : T2 × S1 −→ T2 × S1.

Es ⊕ Eu integrable =⇒ A× fMS |S1C∞−→ A× Id.

Contact Structure

Milnor-Wood Inequality implies that there does not exists any

closed surfaces or foliations transverse to the S1-fibers in H.

Birkhoff Sections

Definition of Birkhoff Sections

An imbedded surface Σ → H is a Birkhoff section if:

∂Σ = Sp1 ∪ · · · ∪ Spk consists of finite many S1-fibers.

Int(Σ) is transverse to the S1-fibers of H.

Remark. Int(Σ) is an l-cover of T2 \ p1, · · · , pk, where l ≤ k.

Birkhoff Sections

Break Transivity

Choose two parallel Birkhoff sections to be the candidates of

attractor and repeller.

That is if fA(Σ) ≈ Σ and fA(Σ′) ≈ Σ′, then

far from boundary, Σ central contracting, Σ′ central

expanding, just like T3;

close to boundary, apply the central DA-construction from

C.Bonatti and N.Guelman.

Difficulty:

How could a Birkhoff section Σ is ”invariant” by fA?

When its image fA(Σ) is isotopic to Σ along the fibers?

Break Transivity

Choose two parallel Birkhoff sections to be the candidates of

attractor and repeller.

That is if fA(Σ) ≈ Σ and fA(Σ′) ≈ Σ′, then

far from boundary, Σ central contracting, Σ′ central

expanding, just like T3;

close to boundary, apply the central DA-construction from

C.Bonatti and N.Guelman.

Difficulty:

How could a Birkhoff section Σ is ”invariant” by fA?

When its image fA(Σ) is isotopic to Σ along the fibers?

Boundary Property

For a Birkhoff section Σn with boundary

∂Σn = π−1(( k

m2n) : 0 ≤ k, l ≤ m2n − 1).

Here m = |det(A− I)|, and the twist number of each fiber is

positive.

∂fA(Σn) = ∂Σn.

Close to each boundary fiber, fA(Σn) locally isotopic to Σ.

Boundary Property

For a Birkhoff section Σn with boundary

∂Σn = π−1(( k

m2n) : 0 ≤ k, l ≤ m2n − 1).

Here m = |det(A− I)|, and the twist number of each fiber is

positive.

∂fA(Σn) = ∂Σn.

Close to each boundary fiber, fA(Σn) locally isotopic to Σ.

Isotopic Class of Birkhoff Sections

Theorem

With the boundary property as above, ∃ Σn isotopic to

fA(Σn) along the S1-fibers, and the isotopy class is

unique.

Isotopic Class of Birkhoff Sections

Moreover, we can choose Σn, such that ”far” from the

boundary, fA(Σn) is C1 close to Σn.

Contact Structure and Birkhoff Section

Central DA-Construction

Close to the boundary fiber, we applying the central

DA-construction due to C.Bonatti and N.Guelman, which tear

the Birkhoff section to be branched:

the candidate of attractor along unstable direction;

the candidate of repeller along stable direction.

Central DA-Construction

Perturb fA to get fn preserve these two branched surfaces

”invariant”, and in the central direction one is contracting to

get the attractor, the other is expanding to get the repeller.

Thank You !

Yi SHI Peking University Institute de Math ematiques de ......"invariant", and in the...

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