partial hyperbolicity in 3-manifolds - impa · partial hyperbolicity in 3-manifolds jana rodriguez...

panorama conjectures Anosov torus

partial hyperbolicityin 3-manifolds

Jana Rodriguez Hertz

Universidad de la RepúblicaUruguay

Institut Henri PoincaréJune, 14, 2013

definition

setting

M3 closed Riemannian 3-manifoldf : M → M partially hyperbolic diffeomorphism

definition

setting

M3 closed Riemannian 3-manifold

f : M → M partially hyperbolic diffeomorphism

definition

setting

M3 closed Riemannian 3-manifoldf : M → M partially hyperbolic diffeomorphism

definition

partial hyperbolicity

f : M3 → M3 is partially hyperbolic

TM = Es ⊕ Ec ⊕ Eu

↑ ↑ ↑

definition

↑ ↑ ↑contracting intermediate expanding

definition

↑ ↑ ↑1-dim 1-dim 1-dim

examples

example

example (conservative)

f : T2 × T1 → T2 × T1

such that

2 11 1

examples

example

f : T2 × T1 → T2 × T1

such that

2 11 1

)× id

examples

example

f : T2 × T1 → T2 × T1

such that

2 11 1

)× id

examples

example

f : T2 × T1 → T2 × T1

such that

2 11 1

)× Rθ

examples

example

example (non-conservative)

f : T2 × T1 → T2 × T1

such that

2 11 1

)× NPSP

examples

open problems

problems

ergodicity

dynamical coherenceclassification

→ Anosov torus

examples

open problems

problems

ergodicitydynamical coherence

classification

→ Anosov torus

examples

open problems

problems

ergodicitydynamical coherenceclassification

→ Anosov torus

examples

open problems

problems

ergodicitydynamical coherenceclassification

→ Anosov torus

Anosov torus

Anosov torusT embedded 2-torus

∃g : M → M diffeo s.t.

1 g(T ) = T2 g|T hyperbolic

Anosov torus

Anosov torusT embedded 2-torus∃g : M → M diffeo s.t.

Anosov torus

1 g(T ) = T

2 g|T hyperbolic

Anosov torus

non-ergodicity

most ph are ergodic

conjecture (pugh-shub)partially hyperbolic diffeomorphisms

∪C1-open and Cr -dense set of ergodic diffeomorphisms

non-ergodicity

most ph are ergodic

hertz-hertz-ures08 (this setting)partially hyperbolic diffeomorphisms

∪C1-open and C∞-dense set of ergodic diffeomorphisms

non-ergodicity

open problem

open problemdescribe non-ergodic partially hyperbolic diffeomorphisms

non-ergodicity

ergodic

non-ergodic

non-ergodicity

open problem

open problemdescribe non-ergodic partially hyperbolic diffeomorphisms

non-ergodicity

ergodic

non-ergodic

non-ergodicity

open problem

open problemdescribe 3-manifolds

supporting non-ergodic partially hyperbolic diffeomorphisms

3-manifolds

ergodic

non-ergodic

ergodic

non-ergodicity

open problem

3-manifolds

ergodic

non-ergodic

ergodic

non-ergodicity

open problem

3-manifolds

ergodic

non-ergodic

ergodic

non-ergodicity

conjecture

subliminal conjecturemost 3-manifolds

do not supportnon-ergodic partially hyperbolic diffeomorphisms

non-ergodicity

conjecture

subliminal conjecturemost 3-manifolds do not support

non-ergodic partially hyperbolic diffeomorphisms

non-ergodicity

evidence

hertz-hertz-ures08N 3-nilmanifold,

then eitherN = T3,

{partially hyperbolic } ⊂ { ergodic }

nilmanifolds

ergodic

non-ergodicity

evidence

hertz-hertz-ures08N 3-nilmanifold, then either

N = T3,

{partially hyperbolic } ⊂ { ergodic }

nilmanifolds

ergodic

non-ergodicity

evidence

N = T3,

or{partially hyperbolic } ⊂ { ergodic }

nilmanifolds

ergodic

non-ergodicity

evidence

N = T3, or{partially hyperbolic } ⊂ { ergodic }

nilmanifolds

ergodic

non-ergodicity

evidence

nilmanifolds

ergodic

non-ergodicity

evidence

nilmanifolds

ergodic

non-ergodicity

non-ergodic conjecture

non-ergodic conjecture (hertz-hertz-ures)the only 3-manifolds supporting

non-ergodic PH diffeomorphisms are:

1 the 3-torus,2 the mapping torus of −id : T2 → T2

3 the mapping tori of hyperbolic automorphisms of T2

non-ergodicity

1 the 3-torus,

2 the mapping torus of −id : T2 → T2

non-ergodicity

1 the 3-torus

non-ergodicity

2 the mapping torus of −id

non-ergodicity

3 the mapping torus of a hyperbolic automorphism

non-ergodicity

stronger non-ergodic conjecture

stronger non-ergodic conjecturef : M → M non-ergodic partially hyperbolic diffeomorphism,then

∃ torus tangent to Es ⊕ Eu

non-ergodicity

stronger non-ergodic conjecture

stronger non-ergodic conjecturef : M → M non-ergodic partially hyperbolic diffeomorphism,then

∃ torus tangent to Es ⊕ Eu

dynamical coherence

integrability

↑ ↑ ↑Fs

dynamical coherence

integrability

dynamical coherence

integrability

↑ ↑ ↑Fs ? Fu

dynamical coherence

1 ∃ invariant Fcs tangent to Es ⊕ Ec

2 ∃ invariant Fcu tangent to Ec ⊕ Eu

remark⇒ ∃ invariant Fc tangent to Ec

dynamical coherence

dynamical coherence1 ∃ invariant Fcs tangent to Es ⊕ Ec

dynamical coherence

open question

longstanding open question

f : M3 → M3 partially hyperbolic ?⇒ f dynamically coherent

hertz-hertz-ures10

dynamical coherence

open question

longstanding open question

f : M3 → M3 partially hyperbolic ?⇒ f dynamically coherent

hertz-hertz-ures10

dynamical coherence

counterexample

hertz-hertz-ures10∃f : T3 → T3 partially hyperbolic

non-dynamically coherentnon-conservative“robust”

dynamical coherence

counterexample

non-dynamically coherent

non-conservative“robust”

dynamical coherence

counterexample

non-dynamically coherentnon-conservative

“robust”

dynamical coherence

counterexample

non-dynamically coherentnon-conservative“robust”

inspiring result

theorem

theorem (HHU10)

If f : M3 → M3 is partially hyperbolic, then

any invariant Fcu tangent to Ec ⊕ Eu

cannot have compact leaves

inspiring result

theorem

theorem (HHU10)

If f : M3 → M3 is partially hyperbolic, thenany invariant Fcu tangent to Ec ⊕ Eu

inspiring result

theorem

theorem (HHU10)

If f : M3 → M3 is partially hyperbolic, thenany invariant Fcu tangent to Ec ⊕ Eu

structure of the example

let us build an example f : T3 → T3 such that:

there is an invariant torus T cu tangent to Ec ⊕ Eu

Ec ⊕ Eu is uniquely integrable in T3 \ T cu

Ec ⊕ Eu is NOT integrable at T cu

let us build an example f : T3 → T3 such that:there is an invariant torus T cu tangent to Ec ⊕ Eu

non-integrability at T cu

view of a center-stable leaf

W cs(x)F cu

non-integrability at T cu

view of a center-stable leaf

W cs(x)F cu

construction

how it is built

f : Anosov× NP-SP

f (x , θ) = (Ax , ψ(θ))

g ∼ Anosov× NP-SP

g(x , θ) = (Ax+v(θ)es, ψ(θ))

v(θs) = 0

construction

how it is built

f : Anosov× NP-SP

f (x , θ) = (Ax , ψ(θ))

g(x , θ) = (Ax+v(θ)es, ψ(θ))

v(θs) = 0

construction

how it is built

f : Anosov× NP-SP

f (x , θ) = (Ax , ψ(θ))

g(x , θ) = (Ax+v(θ)es, ψ(θ))

v(θs) = 0

construction

how it is built

f : Anosov× NP-SP

f (x , θ) = (Ax , ψ(θ))

g(x , θ) = (Ax+v(θ)es, ψ(θ))

v(θs) = 0

construction

how it is built

f : Anosov× NP-SP

f (x , θ) = (Ax , ψ(θ))

g(x , θ) = (Ax+v(θ)es, ψ(θ))

v(θs) = 0

construction

how it is built

f : Anosov× NP-SP

f (x , θ) = (Ax , ψ(θ))

g(x , θ) = (Ax+v(θ)es, ψ(θ))

v(θs) = 0

construction

how it is built

f : Anosov× NP-SP

f (x , θ) = (Ax , ψ(θ))

g(x , θ) = (Ax+v(θ)es, ψ(θ))

v(θs) = 0

W cs(x)F cu

construction

derivative of the perturbation

f (x , θ) = (Ax , ψ(θ))

Df (x , θ) =

λ 0 00 1/λ 00 0 ψ′(θ)

g(x , θ) = (Ax + v(θ)es, ψ(θ))

Dg(x , θ) =

λ 0 v ′(θ)0 1/λ 00 0 ψ′(θ)

construction

f (x , θ) = (Ax , ψ(θ))

Df (x , θ) =

λ 0 00 1/λ 00 0 ψ′(θ)

g(x , θ) = (Ax + v(θ)es, ψ(θ))

Dg(x , θ) =

λ 0 v ′(θ)0 1/λ 00 0 ψ′(θ)

construction

f (x , θ) = (Ax , ψ(θ))

Df (x , θ) =

λ 0 00 1/λ 00 0 ψ′(θ)

g(x , θ) = (Ax + v(θ)es, ψ(θ))

Dg(x , θ) =

λ 0 v ′(θ)0 1/λ 00 0 ψ′(θ)

construction

f (x , θ) = (Ax , ψ(θ))

Df (x , θ) =

λ 0 00 1/λ 00 0 ψ′(θ)

g(x , θ) = (Ax + v(θ)es, ψ(θ))

Dg(x , θ) =

λ 0 v ′(θ)0 1/λ 00 0 ψ′(θ)

calculations

goal 1: g(x , θ) = (Ax + v(θ)es, ψ(θ))→ v

g is semiconjugated to A : T2 → T2

semiconjugacy: h : T3 → T2

goal 2: h preserves Fcsf

h(x , θ) = x − u(θ)es

calculations

goal 1: g(x , θ) = (Ax + v(θ)es, ψ(θ))→ vg is semiconjugated to A : T2 → T2

h(x , θ) = x − u(θ)es

calculations

h(x , θ) = x − u(θ)es

calculations

h(x , θ) = x − u(θ)es

calculations

h(x , θ) = x − u(θ)es

calculations

semiconjugacy

T3 g→ T3

h ↓ ↓ h

T2 A→ T2

h(x , θ) = x − u(θ)es

h(Ax + v(θ)es, ψ(θ)) = Ah(x , θ)

Ax + v(θ)es − u(ψ(θ))es = Ax − λu(θ)es

v(θ)es − u(ψ(θ))es = −λu(θ)es

twisted cohomological equation:

u ◦ ψ − λu = v

calculations

semiconjugacy

T3 g→ T3

h ↓ ↓ h

T2 A→ T2

h(x , θ) = x − u(θ)es

u ◦ ψ − λu = v

calculations

semiconjugacy

T3 g→ T3

h ↓ ↓ h

T2 A→ T2

h(x , θ) = x − u(θ)es

u ◦ ψ − λu = v

calculations

semiconjugacy

T3 g→ T3

h ↓ ↓ h

T2 A→ T2

h(x , θ) = x − u(θ)es

u ◦ ψ − λu = v

calculations

semiconjugacy

T3 g→ T3

h ↓ ↓ h

T2 A→ T2

h(x , θ) = x − u(θ)es

u ◦ ψ − λu = v

calculations

cohomological equation

twisted cohomological equation

u ◦ ψ − λu = v

has solution

u(θ) =1λ

∞∑k=1

λkv(ψ−k (θ))

u(0) = 0well defined and continuous

derivative of u

u′(θ) =1λ

∞∑k=1

λkv ′(ψ−k (θ))(ψ−k

)′(θ)

∑converges uniformly in any compact 63 θs

⇒ u′ continuous for θ 6= θs

calculations

u ◦ ψ − λu = v

has solution

u(θ) =1λ

∞∑k=1

λkv(ψ−k (θ))

derivative of u

u′(θ) =1λ

∞∑k=1

)′(θ)

calculations

u ◦ ψ − λu = v

has solution

u(θ) =1λ

∞∑k=1

λkv(ψ−k (θ))

u(0) = 0

well defined and continuousderivative of u

u′(θ) =1λ

∞∑k=1

)′(θ)

calculations

u ◦ ψ − λu = v

has solution

u(θ) =1λ

∞∑k=1

λkv(ψ−k (θ))

derivative of u

u′(θ) =1λ

∞∑k=1

)′(θ)

calculations

u ◦ ψ − λu = v

has solution

u(θ) =1λ

∞∑k=1

λkv(ψ−k (θ))

derivative of u

u′(θ) =1λ

∞∑k=1

)′(θ)

calculations

u ◦ ψ − λu = v

has solution

u(θ) =1λ

∞∑k=1

λkv(ψ−k (θ))

derivative of u

u′(θ) =1λ

∞∑k=1

)′(θ)

calculations

u ◦ ψ − λu = v

has solution

u(θ) =1λ

∞∑k=1

λkv(ψ−k (θ))

derivative of u

u′(θ) =1λ

∞∑k=1

)′(θ)

calculations

u ◦ ψ − λu = v

has solution

u(θ) =1λ

∞∑k=1

λkv(ψ−k (θ))

derivative of u

u′(θ) =1λ

∞∑k=1

)′(θ)

calculations

calculation of v

u(θ) =1λ

∞∑k=1

λkv(ψ−k (θ))

u′(θ) =1λ

∞∑k=1

)′(θ)

if v has a unique minimum at θs, then limθ→θs u′(θ) =∞curves θ 7→ (x + u(θ)es, θ) invariant partition

calculations

calculation of v

u(θ) =1λ

∞∑k=1

λkv(ψ−k (θ))

u′(θ) =1λ

∞∑k=1

)′(θ)

calculations

calculation of v

u(θ) =1λ

∞∑k=1

λkv(ψ−k (θ))

u′(θ) =1λ

∞∑k=1

)′(θ)

if v has a unique minimum at θs, then limθ→θs u′(θ) =∞

curves θ 7→ (x + u(θ)es, θ) invariant partition

calculations

calculation of v

u(θ) =1λ

∞∑k=1

λkv(ψ−k (θ))

u′(θ) =1λ

∞∑k=1

)′(θ)

calculations

u′ →∞

let v ′(θs) = 0, v ′′(θs) > 0, ψ′(θs) = σ ∈ (λ,1)

let ε > 0 small, choose L > 1 so that for d(θ, θs) ≤ ε1L ≤

v ′(θ)d(θ,θs)

≤ L and 1L ≤

ψ′(θ)σ ≤ L

for θ ∼ θs let K be the last d(ψ−K (θ), θs) ≤ εnote d(ψ−K (θ), θs) ≥ 1

Ld(ψ−K−1(θ), θs) >εL

then u′(θ) ≥ λK−1v ′(ψ−K (θ))(ψ−K )′(θ)

u′(θ) ≥ λK−1 1Ld(ψ−K (θ), θs)

1Lσ−K

u′(θ) ≥ ελL3

calculations

u′ →∞

let v ′(θs) = 0, v ′′(θs) > 0, ψ′(θs) = σ ∈ (λ,1)let ε > 0 small, choose L > 1 so that for d(θ, θs) ≤ ε

1L ≤

v ′(θ)d(θ,θs)

≤ L and 1L ≤

ψ′(θ)σ ≤ L

u′(θ) ≥ λK−1 1Ld(ψ−K (θ), θs)

1Lσ−K

u′(θ) ≥ ελL3

calculations

u′ →∞

let v ′(θs) = 0, v ′′(θs) > 0, ψ′(θs) = σ ∈ (λ,1)let ε > 0 small, choose L > 1 so that for d(θ, θs) ≤ ε1L ≤

v ′(θ)d(θ,θs)

≤ L and 1L ≤

ψ′(θ)σ ≤ L

u′(θ) ≥ λK−1 1Ld(ψ−K (θ), θs)

1Lσ−K

u′(θ) ≥ ελL3

calculations

u′ →∞

v ′(θ)d(θ,θs)

≤ L and 1L ≤

ψ′(θ)σ ≤ L

for θ ∼ θs let K be the last d(ψ−K (θ), θs) ≤ ε

note d(ψ−K (θ), θs) ≥ 1Ld(ψ−K−1(θ), θs) >

u′(θ) ≥ λK−1 1Ld(ψ−K (θ), θs)

1Lσ−K

u′(θ) ≥ ελL3

calculations

u′ →∞

v ′(θ)d(θ,θs)

≤ L and 1L ≤

ψ′(θ)σ ≤ L

u′(θ) ≥ λK−1 1Ld(ψ−K (θ), θs)

1Lσ−K

u′(θ) ≥ ελL3

calculations

u′ →∞

v ′(θ)d(θ,θs)

≤ L and 1L ≤

ψ′(θ)σ ≤ L

u′(θ) ≥ λK−1 1Ld(ψ−K (θ), θs)

1Lσ−K

u′(θ) ≥ ελL3

calculations

u′ →∞

v ′(θ)d(θ,θs)

≤ L and 1L ≤

ψ′(θ)σ ≤ L

u′(θ) ≥ λK−1 1Ld(ψ−K (θ), θs)

1Lσ−K

u′(θ) ≥ ελL3

calculations

u′ →∞

v ′(θ)d(θ,θs)

≤ L and 1L ≤

ψ′(θ)σ ≤ L

u′(θ) ≥ λK−1 1Ld(ψ−K (θ), θs)

1Lσ−K

u′(θ) ≥ ελL3

calculations

open problem

open problemdescribe 3-manifolds supporting non-dynamically coherentexamples

3-manifolds

non DC

calculations

open problem

3-manifolds

non DC

calculations

open problem

3-manifolds

non DC

calculations

non-dynamically coherent conjecture

non-dynamically coherent conjecture (hertz-hertz-ures)

f : M3 → M3 non-dynamically coherent,

then M is either:1 T3

2 the mapping torus of −id3 the mapping torus of a hyperbolic automorphism

3-manifolds

calculations

f : M3 → M3 non-dynamically coherent,then M is either:

3-manifolds

calculations

3-manifolds

calculations

2 the mapping torus of −id

3 the mapping torus of a hyperbolic automorphism

3-manifolds

calculations

3-manifolds

calculations

stronger conjecture

stronger non-dynamically coherent conjecture

f : M3 → M3 non-dynamically coherent, then either

∃ torus tangent to Ec ⊕ Eu, or∃ torus tangent to Es ⊕ Ec

calculations

stronger conjecture

f : M3 → M3 non-dynamically coherent, then either∃ torus tangent to Ec ⊕ Eu, or

∃ torus tangent to Es ⊕ Ec

calculations

stronger conjecture

f : M3 → M3 non-dynamically coherent, then either∃ torus tangent to Ec ⊕ Eu, or∃ torus tangent to Es ⊕ Ec

calculations

intermediate conjecture

intermediate conjecturef volume preserving⇒ f dynamically coherent

calculations

evidence

hammerlind-potrie13f partially hyperbolic & non-dynamically coherent on a3-manifold with solvable fundamental group, then

calculations

evidence

∃ torus tangent to Ec ⊕ Eu, or

∃ torus tangent to Es ⊕ Ec

calculations

evidence

classification

examples of ph dynamics

known ph dynamics in dimension 3

perturbations of time-one maps of Anosov flowscertain skew-productscertain DA-maps

new example

non-dynamically coherent example

classification

known ph dynamics in dimension 3perturbations of time-one maps of Anosov flows

certain skew-productscertain DA-maps

new example

classification

known ph dynamics in dimension 3perturbations of time-one maps of Anosov flowscertain skew-products

certain DA-maps

new example

classification

known ph dynamics in dimension 3perturbations of time-one maps of Anosov flowscertain skew-productscertain DA-maps

new example

classification

known ph dynamics in dimension 3perturbations of time-one maps of Anosov flowscertain skew-productscertain DA-maps

new examplenon-dynamically coherent example

classification

question

questionare there more examples?

classification

conjecture pujals

classification conjecture (pujals01)

If f : M3 → M3 is a transitive partially hyperbolicdiffeomorphism, then f is (finitely covered by) either

1 a perturbation of a time-one map of an Anosov flow2 a skew-product3 a DA-map

classification

conjecture pujals

1 a perturbation of a time-one map of an Anosov flow

2 a skew-product3 a DA-map

classification

conjecture pujals

1 a perturbation of a time-one map of an Anosov flow2 a skew-product

3 a DA-map

classification

conjecture pujals

1 a perturbation of a time-one map of an Anosov flow2 a skew-product3 a DA-map

classification

conjecture

classification conjecture (hhu)

If f : M3 → M3 is partially hyperbolic and dynamically coherent,then f is

1 leafwise conjugate to an Anosov flow2 leafwise conjugate to a skew-product with linear base3 leafwise conjugate to an Anosov map in T3.

classification

conjecture

1 a perturbation of a time-one map of an Anosov flow,

2 leafwise conjugate to a skew-product with linear base3 leafwise conjugate to an Anosov map in T3.

classification

conjecture

1 a perturbation of a time-one map of an Anosov flow,2 a skew-product, or

3 leafwise conjugate to an Anosov map in T3.

classification

conjecture

1 a perturbation of a time-one map of an Anosov flow,2 a skew-product, or3 a DA-map.

classification

conjecture

1 leafwise conjugate to an Anosov flow2 a skew-product, or3 a DA-map.

classification

conjecture

1 leafwise conjugate to an Anosov flow2 leafwise conjugate to a skew-product with linear base3 a DA-map.

classification

conjecture

1 leafwise conjugate to an Anosov flow2 leafwise conjugate to a skew-product with linear base3 leafwise conjugate to an Anosov map in T3.

classification

invariant tori in ph dynamics

invariant tori in PH dynamicsT invariant torus tangent to

Es ⊕ Eu

Ec ⊕ Eu

Es ⊕ Eu

⇒ T Anosov torus

classification

Es ⊕ Eu

Ec ⊕ Eu

Es ⊕ Eu

⇒ T Anosov torus

classification

Es ⊕ Eu

Ec ⊕ Eu

Es ⊕ Eu

⇒ T Anosov torus

classification

Es ⊕ Eu

Ec ⊕ Eu

Es ⊕ Eu

⇒ T Anosov torus

classification

Es ⊕ Eu

Ec ⊕ Eu

Es ⊕ Eu

⇒ T Anosov torus

classification

conjectures

non-ergodic conjecturef : M → M non-ergodic partiallyhyperbolic

non-dyn. coh. conjecturef : M → M non-dyn. coherentpartially hyperbolic

then M is either

3 the mapping torus of a hyperbolic map of T2

stronger conjecture∃ Anosov torus tangent toEs ⊕ Eu

stronger conjecture∃ Anosov torus tangent toEc ⊕ Eu or Es ⊕ Ec

classification

conjectures

then M is either

classification

conjectures

then M is either

classification

conjectures

then M is either1 T3

classification

conjectures

classification

conjectures

classification

conjectures

classification

conjectures

classification

why stronger conjectures?

hertz-hertz-ures11M irreducible contains an Anosov torus,

3 a mapping torus of a hyperbolic automorphism of T2

3-manifolds

classification

hertz-hertz-ures11M irreducible contains an Anosov torus, then M is either

3-manifolds

classification

3-manifolds

classification

3-manifolds

classification

3-manifolds

classification

remark

remarkf : M3 → M3 partially hyperbolic⇒ M irreducible

Rosenberg68Burago-Ivanov08

classification

remark

classification

remark

Rosenberg68

Burago-Ivanov08

classification

remark

partial hyperbolicity in 3-manifolds - impa · partial hyperbolicity in 3-manifolds jana rodriguez...

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