partial hyperbolicity in 3-manifolds - impa · partial hyperbolicity in 3-manifolds jana rodriguez...
TRANSCRIPT
![Page 1: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/1.jpg)
panorama conjectures Anosov torus
partial hyperbolicityin 3-manifolds
Jana Rodriguez Hertz
Universidad de la RepúblicaUruguay
Institut Henri PoincaréJune, 14, 2013
![Page 2: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/2.jpg)
panorama conjectures Anosov torus
definition
setting
setting
M3 closed Riemannian 3-manifoldf : M → M partially hyperbolic diffeomorphism
![Page 3: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/3.jpg)
panorama conjectures Anosov torus
definition
setting
setting
M3 closed Riemannian 3-manifold
f : M → M partially hyperbolic diffeomorphism
![Page 4: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/4.jpg)
panorama conjectures Anosov torus
definition
setting
setting
M3 closed Riemannian 3-manifoldf : M → M partially hyperbolic diffeomorphism
![Page 5: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/5.jpg)
panorama conjectures Anosov torus
definition
partial hyperbolicity
partial hyperbolicity
f : M3 → M3 is partially hyperbolic
TM = Es ⊕ Ec ⊕ Eu
↑ ↑ ↑
![Page 6: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/6.jpg)
panorama conjectures Anosov torus
definition
partial hyperbolicity
partial hyperbolicity
f : M3 → M3 is partially hyperbolic
TM = Es ⊕ Ec ⊕ Eu
↑ ↑ ↑contracting intermediate expanding
![Page 7: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/7.jpg)
panorama conjectures Anosov torus
definition
partial hyperbolicity
partial hyperbolicity
f : M3 → M3 is partially hyperbolic
TM = Es ⊕ Ec ⊕ Eu
↑ ↑ ↑1-dim 1-dim 1-dim
![Page 8: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/8.jpg)
panorama conjectures Anosov torus
examples
example
example (conservative)
f : T2 × T1 → T2 × T1
such that
f =(
2 11 1
)×
×
![Page 9: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/9.jpg)
panorama conjectures Anosov torus
examples
example
example (conservative)
f : T2 × T1 → T2 × T1
such that
f =(
2 11 1
)× id
×
![Page 10: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/10.jpg)
panorama conjectures Anosov torus
examples
example
example (conservative)
f : T2 × T1 → T2 × T1
such that
f =(
2 11 1
)× id
×
![Page 11: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/11.jpg)
panorama conjectures Anosov torus
examples
example
example (conservative)
f : T2 × T1 → T2 × T1
such that
f =(
2 11 1
)× Rθ
×
![Page 12: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/12.jpg)
panorama conjectures Anosov torus
examples
example
example (non-conservative)
f : T2 × T1 → T2 × T1
such that
f =(
2 11 1
)× NPSP
×
![Page 13: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/13.jpg)
panorama conjectures Anosov torus
examples
open problems
problems
ergodicity
dynamical coherenceclassification
→ Anosov torus
![Page 14: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/14.jpg)
panorama conjectures Anosov torus
examples
open problems
problems
ergodicitydynamical coherence
classification
→ Anosov torus
![Page 15: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/15.jpg)
panorama conjectures Anosov torus
examples
open problems
problems
ergodicitydynamical coherenceclassification
→ Anosov torus
![Page 16: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/16.jpg)
panorama conjectures Anosov torus
examples
open problems
problems
ergodicitydynamical coherenceclassification
→ Anosov torus
![Page 17: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/17.jpg)
panorama conjectures Anosov torus
Anosov torus
Anosov torus
Anosov torusT embedded 2-torus
∃g : M → M diffeo s.t.
1 g(T ) = T2 g|T hyperbolic
![Page 18: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/18.jpg)
panorama conjectures Anosov torus
Anosov torus
Anosov torus
Anosov torusT embedded 2-torus∃g : M → M diffeo s.t.
1 g(T ) = T2 g|T hyperbolic
![Page 19: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/19.jpg)
panorama conjectures Anosov torus
Anosov torus
Anosov torus
Anosov torusT embedded 2-torus∃g : M → M diffeo s.t.
1 g(T ) = T
2 g|T hyperbolic
![Page 20: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/20.jpg)
panorama conjectures Anosov torus
Anosov torus
Anosov torus
Anosov torusT embedded 2-torus∃g : M → M diffeo s.t.
1 g(T ) = T2 g|T hyperbolic
![Page 21: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/21.jpg)
panorama conjectures Anosov torus
non-ergodicity
most ph are ergodic
conjecture (pugh-shub)partially hyperbolic diffeomorphisms
∪C1-open and Cr -dense set of ergodic diffeomorphisms
![Page 22: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/22.jpg)
panorama conjectures Anosov torus
non-ergodicity
most ph are ergodic
hertz-hertz-ures08 (this setting)partially hyperbolic diffeomorphisms
∪C1-open and C∞-dense set of ergodic diffeomorphisms
![Page 23: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/23.jpg)
panorama conjectures Anosov torus
non-ergodicity
open problem
open problemdescribe non-ergodic partially hyperbolic diffeomorphisms
non-ergodicity
ergodic
non-ergodic
![Page 24: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/24.jpg)
panorama conjectures Anosov torus
non-ergodicity
open problem
open problemdescribe non-ergodic partially hyperbolic diffeomorphisms
non-ergodicity
ergodic
non-ergodic
![Page 25: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/25.jpg)
panorama conjectures Anosov torus
non-ergodicity
open problem
open problemdescribe 3-manifolds
supporting non-ergodic partially hyperbolic diffeomorphisms
3-manifolds
ergodic
non-ergodic
ergodic
![Page 26: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/26.jpg)
panorama conjectures Anosov torus
non-ergodicity
open problem
open problemdescribe 3-manifolds
supporting non-ergodic partially hyperbolic diffeomorphisms
3-manifolds
ergodic
non-ergodic
ergodic
![Page 27: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/27.jpg)
panorama conjectures Anosov torus
non-ergodicity
open problem
open problemdescribe 3-manifolds
supporting non-ergodic partially hyperbolic diffeomorphisms
3-manifolds
ergodic
non-ergodic
ergodic
![Page 28: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/28.jpg)
panorama conjectures Anosov torus
non-ergodicity
conjecture
subliminal conjecturemost 3-manifolds
do not supportnon-ergodic partially hyperbolic diffeomorphisms
![Page 29: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/29.jpg)
panorama conjectures Anosov torus
non-ergodicity
conjecture
subliminal conjecturemost 3-manifolds do not support
non-ergodic partially hyperbolic diffeomorphisms
![Page 30: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/30.jpg)
panorama conjectures Anosov torus
non-ergodicity
evidence
hertz-hertz-ures08N 3-nilmanifold,
then eitherN = T3,
or
{partially hyperbolic } ⊂ { ergodic }
nilmanifolds
ergodic
![Page 31: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/31.jpg)
panorama conjectures Anosov torus
non-ergodicity
evidence
hertz-hertz-ures08N 3-nilmanifold, then either
N = T3,
or
{partially hyperbolic } ⊂ { ergodic }
nilmanifolds
ergodic
![Page 32: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/32.jpg)
panorama conjectures Anosov torus
non-ergodicity
evidence
hertz-hertz-ures08N 3-nilmanifold, then either
N = T3,
or{partially hyperbolic } ⊂ { ergodic }
nilmanifolds
ergodic
![Page 33: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/33.jpg)
panorama conjectures Anosov torus
non-ergodicity
evidence
hertz-hertz-ures08N 3-nilmanifold, then either
N = T3, or{partially hyperbolic } ⊂ { ergodic }
nilmanifolds
ergodic
![Page 34: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/34.jpg)
panorama conjectures Anosov torus
non-ergodicity
evidence
hertz-hertz-ures08N 3-nilmanifold, then either
N = T3, or{partially hyperbolic } ⊂ { ergodic }
nilmanifolds
ergodic
![Page 35: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/35.jpg)
panorama conjectures Anosov torus
non-ergodicity
evidence
hertz-hertz-ures08N 3-nilmanifold, then either
N = T3, or{partially hyperbolic } ⊂ { ergodic }
nilmanifolds
ergodic
![Page 36: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/36.jpg)
panorama conjectures Anosov torus
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
![Page 37: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/37.jpg)
panorama conjectures Anosov torus
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
![Page 38: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/38.jpg)
panorama conjectures Anosov torus
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
![Page 39: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/39.jpg)
panorama conjectures Anosov torus
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
![Page 40: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/40.jpg)
panorama conjectures Anosov torus
non-ergodicity
non-ergodic conjecture
1 the 3-torus
![Page 41: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/41.jpg)
panorama conjectures Anosov torus
non-ergodicity
non-ergodic conjecture
2 the mapping torus of −id
![Page 42: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/42.jpg)
panorama conjectures Anosov torus
non-ergodicity
non-ergodic conjecture
3 the mapping torus of a hyperbolic automorphism
![Page 43: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/43.jpg)
panorama conjectures Anosov torus
non-ergodicity
stronger non-ergodic conjecture
stronger non-ergodic conjecturef : M → M non-ergodic partially hyperbolic diffeomorphism,then
∃ torus tangent to Es ⊕ Eu
![Page 44: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/44.jpg)
panorama conjectures Anosov torus
non-ergodicity
stronger non-ergodic conjecture
stronger non-ergodic conjecturef : M → M non-ergodic partially hyperbolic diffeomorphism,then
∃ torus tangent to Es ⊕ Eu
![Page 45: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/45.jpg)
panorama conjectures Anosov torus
dynamical coherence
integrability
integrability
f : M3 → M3 is partially hyperbolic
TM = Es ⊕ Ec ⊕ Eu
↑ ↑ ↑Fs
?
Fu
![Page 46: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/46.jpg)
panorama conjectures Anosov torus
dynamical coherence
integrability
integrability
f : M3 → M3 is partially hyperbolic
TM = Es ⊕ Ec ⊕ Eu
↑
↑
↑Fs
?
Fu
![Page 47: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/47.jpg)
panorama conjectures Anosov torus
dynamical coherence
integrability
integrability
f : M3 → M3 is partially hyperbolic
TM = Es ⊕ Ec ⊕ Eu
↑ ↑ ↑Fs ? Fu
![Page 48: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/48.jpg)
panorama conjectures Anosov torus
dynamical coherence
dynamical coherence
dynamical coherence
1 ∃ invariant Fcs tangent to Es ⊕ Ec
2 ∃ invariant Fcu tangent to Ec ⊕ Eu
remark⇒ ∃ invariant Fc tangent to Ec
![Page 49: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/49.jpg)
panorama conjectures Anosov torus
dynamical coherence
dynamical coherence
dynamical coherence1 ∃ invariant Fcs tangent to Es ⊕ Ec
2 ∃ invariant Fcu tangent to Ec ⊕ Eu
remark⇒ ∃ invariant Fc tangent to Ec
![Page 50: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/50.jpg)
panorama conjectures Anosov torus
dynamical coherence
dynamical coherence
dynamical coherence1 ∃ invariant Fcs tangent to Es ⊕ Ec
2 ∃ invariant Fcu tangent to Ec ⊕ Eu
remark⇒ ∃ invariant Fc tangent to Ec
![Page 51: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/51.jpg)
panorama conjectures Anosov torus
dynamical coherence
dynamical coherence
dynamical coherence1 ∃ invariant Fcs tangent to Es ⊕ Ec
2 ∃ invariant Fcu tangent to Ec ⊕ Eu
remark⇒ ∃ invariant Fc tangent to Ec
![Page 52: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/52.jpg)
panorama conjectures Anosov torus
dynamical coherence
open question
longstanding open question
f : M3 → M3 partially hyperbolic ?⇒ f dynamically coherent
hertz-hertz-ures10
NO
![Page 53: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/53.jpg)
panorama conjectures Anosov torus
dynamical coherence
open question
longstanding open question
f : M3 → M3 partially hyperbolic ?⇒ f dynamically coherent
hertz-hertz-ures10
NO
![Page 54: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/54.jpg)
panorama conjectures Anosov torus
dynamical coherence
counterexample
hertz-hertz-ures10∃f : T3 → T3 partially hyperbolic
non-dynamically coherentnon-conservative“robust”
![Page 55: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/55.jpg)
panorama conjectures Anosov torus
dynamical coherence
counterexample
hertz-hertz-ures10∃f : T3 → T3 partially hyperbolic
non-dynamically coherent
non-conservative“robust”
![Page 56: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/56.jpg)
panorama conjectures Anosov torus
dynamical coherence
counterexample
hertz-hertz-ures10∃f : T3 → T3 partially hyperbolic
non-dynamically coherentnon-conservative
“robust”
![Page 57: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/57.jpg)
panorama conjectures Anosov torus
dynamical coherence
counterexample
hertz-hertz-ures10∃f : T3 → T3 partially hyperbolic
non-dynamically coherentnon-conservative“robust”
![Page 58: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/58.jpg)
panorama conjectures Anosov torus
inspiring result
theorem
theorem (HHU10)
If f : M3 → M3 is partially hyperbolic, then
any invariant Fcu tangent to Ec ⊕ Eu
cannot have compact leaves
![Page 59: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/59.jpg)
panorama conjectures Anosov torus
inspiring result
theorem
theorem (HHU10)
If f : M3 → M3 is partially hyperbolic, thenany invariant Fcu tangent to Ec ⊕ Eu
cannot have compact leaves
![Page 60: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/60.jpg)
panorama conjectures Anosov torus
inspiring result
theorem
theorem (HHU10)
If f : M3 → M3 is partially hyperbolic, thenany invariant Fcu tangent to Ec ⊕ Eu
cannot have compact leaves
![Page 61: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/61.jpg)
panorama conjectures Anosov torus
structure of the example
structure of the example
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
![Page 62: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/62.jpg)
panorama conjectures Anosov torus
structure of the example
structure of the example
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
![Page 63: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/63.jpg)
panorama conjectures Anosov torus
structure of the example
structure of the example
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
![Page 64: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/64.jpg)
panorama conjectures Anosov torus
structure of the example
structure of the example
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
![Page 65: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/65.jpg)
panorama conjectures Anosov torus
structure of the example
non-integrability at T cu
view of a center-stable leaf
>>
<<>>
>
<>
><
<<
>
<<
θn
TORUS
θn
θs
W cs(x)F cu
F s
![Page 66: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/66.jpg)
panorama conjectures Anosov torus
structure of the example
non-integrability at T cu
view of a center-stable leaf
>>
<<>>
>
<>
><
<<
>
<<
θn
TORUS
θn
θs
W cs(x)F cu
F s
![Page 67: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/67.jpg)
panorama conjectures Anosov torus
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
![Page 68: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/68.jpg)
panorama conjectures Anosov torus
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
![Page 69: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/69.jpg)
panorama conjectures Anosov torus
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
xn
xs
![Page 70: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/70.jpg)
panorama conjectures Anosov torus
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
xn
xs
![Page 71: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/71.jpg)
panorama conjectures Anosov torus
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
xn
xs
![Page 72: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/72.jpg)
panorama conjectures Anosov torus
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
xn
xs
![Page 73: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/73.jpg)
panorama conjectures Anosov torus
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
>>
<<>>
>
<>
><
<<
>
<<
θn
TORUS
θn
θs
W cs(x)F cu
F s
![Page 74: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/74.jpg)
panorama conjectures Anosov torus
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)
![Page 75: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/75.jpg)
panorama conjectures Anosov torus
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)
![Page 76: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/76.jpg)
panorama conjectures Anosov torus
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)
![Page 77: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/77.jpg)
panorama conjectures Anosov torus
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)
![Page 78: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/78.jpg)
panorama conjectures Anosov torus
calculations
goals
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
![Page 79: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/79.jpg)
panorama conjectures Anosov torus
calculations
goals
goal 1: g(x , θ) = (Ax + v(θ)es, ψ(θ))→ vg is semiconjugated to A : T2 → T2
semiconjugacy: h : T3 → T2
goal 2: h preserves Fcsf
h(x , θ) = x − u(θ)es
![Page 80: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/80.jpg)
panorama conjectures Anosov torus
calculations
goals
goal 1: g(x , θ) = (Ax + v(θ)es, ψ(θ))→ vg is semiconjugated to A : T2 → T2
semiconjugacy: h : T3 → T2
goal 2: h preserves Fcsf
h(x , θ) = x − u(θ)es
![Page 81: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/81.jpg)
panorama conjectures Anosov torus
calculations
goals
goal 1: g(x , θ) = (Ax + v(θ)es, ψ(θ))→ vg is semiconjugated to A : T2 → T2
semiconjugacy: h : T3 → T2
goal 2: h preserves Fcsf
h(x , θ) = x − u(θ)es
![Page 82: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/82.jpg)
panorama conjectures Anosov torus
calculations
goals
goal 1: g(x , θ) = (Ax + v(θ)es, ψ(θ))→ vg is semiconjugated to A : T2 → T2
semiconjugacy: h : T3 → T2
goal 2: h preserves Fcsf
h(x , θ) = x − u(θ)es
![Page 83: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/83.jpg)
panorama conjectures Anosov torus
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
![Page 84: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/84.jpg)
panorama conjectures Anosov torus
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
![Page 85: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/85.jpg)
panorama conjectures Anosov torus
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
![Page 86: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/86.jpg)
panorama conjectures Anosov torus
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
![Page 87: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/87.jpg)
panorama conjectures Anosov torus
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
![Page 88: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/88.jpg)
panorama conjectures Anosov torus
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
![Page 89: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/89.jpg)
panorama conjectures Anosov torus
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
![Page 90: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/90.jpg)
panorama conjectures Anosov torus
calculations
cohomological equation
twisted cohomological equation
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
λkv ′(ψ−k (θ))(ψ−k
)′(θ)
∑converges uniformly in any compact 63 θs
⇒ u′ continuous for θ 6= θs
![Page 91: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/91.jpg)
panorama conjectures Anosov torus
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
![Page 92: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/92.jpg)
panorama conjectures Anosov torus
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
![Page 93: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/93.jpg)
panorama conjectures Anosov torus
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
![Page 94: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/94.jpg)
panorama conjectures Anosov torus
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
![Page 95: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/95.jpg)
panorama conjectures Anosov torus
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
![Page 96: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/96.jpg)
panorama conjectures Anosov torus
calculations
calculation of v
u(θ) =1λ
∞∑k=1
λkv(ψ−k (θ))
u′(θ) =1λ
∞∑k=1
λkv ′(ψ−k (θ))(ψ−k
)′(θ)
if v has a unique minimum at θs, then limθ→θs u′(θ) =∞curves θ 7→ (x + u(θ)es, θ) invariant partition
![Page 97: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/97.jpg)
panorama conjectures Anosov torus
calculations
calculation of v
u(θ) =1λ
∞∑k=1
λkv(ψ−k (θ))
u′(θ) =1λ
∞∑k=1
λkv ′(ψ−k (θ))(ψ−k
)′(θ)
if v has a unique minimum at θs, then limθ→θs u′(θ) =∞curves θ 7→ (x + u(θ)es, θ) invariant partition
![Page 98: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/98.jpg)
panorama conjectures Anosov torus
calculations
calculation of v
u(θ) =1λ
∞∑k=1
λkv(ψ−k (θ))
u′(θ) =1λ
∞∑k=1
λkv ′(ψ−k (θ))(ψ−k
)′(θ)
if v has a unique minimum at θs, then limθ→θs u′(θ) =∞
curves θ 7→ (x + u(θ)es, θ) invariant partition
![Page 99: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/99.jpg)
panorama conjectures Anosov torus
calculations
calculation of v
u(θ) =1λ
∞∑k=1
λkv(ψ−k (θ))
u′(θ) =1λ
∞∑k=1
λkv ′(ψ−k (θ))(ψ−k
)′(θ)
if v has a unique minimum at θs, then limθ→θs u′(θ) =∞curves θ 7→ (x + u(θ)es, θ) invariant partition
![Page 100: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/100.jpg)
panorama conjectures Anosov torus
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
λK
σK
![Page 101: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/101.jpg)
panorama conjectures Anosov torus
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
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
λK
σK
![Page 102: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/102.jpg)
panorama conjectures Anosov torus
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
λK
σK
![Page 103: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/103.jpg)
panorama conjectures Anosov torus
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) ≤ ε
note d(ψ−K (θ), θs) ≥ 1Ld(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
λK
σK
![Page 104: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/104.jpg)
panorama conjectures Anosov torus
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
λK
σK
![Page 105: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/105.jpg)
panorama conjectures Anosov torus
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
λK
σK
![Page 106: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/106.jpg)
panorama conjectures Anosov torus
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
λK
σK
![Page 107: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/107.jpg)
panorama conjectures Anosov torus
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
λK
σK
![Page 108: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/108.jpg)
panorama conjectures Anosov torus
calculations
open problem
open problemdescribe 3-manifolds supporting non-dynamically coherentexamples
3-manifolds
DC
non DC
DC
![Page 109: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/109.jpg)
panorama conjectures Anosov torus
calculations
open problem
open problemdescribe 3-manifolds supporting non-dynamically coherentexamples
3-manifolds
DC
non DC
DC
![Page 110: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/110.jpg)
panorama conjectures Anosov torus
calculations
open problem
open problemdescribe 3-manifolds supporting non-dynamically coherentexamples
3-manifolds
DC
non DC
DC
![Page 111: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/111.jpg)
panorama conjectures Anosov torus
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
![Page 112: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/112.jpg)
panorama conjectures Anosov torus
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
![Page 113: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/113.jpg)
panorama conjectures Anosov torus
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
![Page 114: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/114.jpg)
panorama conjectures Anosov torus
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 −id
3 the mapping torus of a hyperbolic automorphism
3-manifolds
![Page 115: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/115.jpg)
panorama conjectures Anosov torus
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
![Page 116: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/116.jpg)
panorama conjectures Anosov torus
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
![Page 117: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/117.jpg)
panorama conjectures Anosov torus
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
![Page 118: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/118.jpg)
panorama conjectures Anosov torus
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
![Page 119: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/119.jpg)
panorama conjectures Anosov torus
calculations
intermediate conjecture
intermediate conjecturef volume preserving⇒ f dynamically coherent
![Page 120: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/120.jpg)
panorama conjectures Anosov torus
calculations
evidence
hammerlind-potrie13f partially hyperbolic & non-dynamically coherent on a3-manifold with solvable fundamental group, then
∃ torus tangent to Ec ⊕ Eu, or∃ torus tangent to Es ⊕ Ec
![Page 121: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/121.jpg)
panorama conjectures Anosov torus
calculations
evidence
hammerlind-potrie13f partially hyperbolic & non-dynamically coherent on a3-manifold with solvable fundamental group, then
∃ torus tangent to Ec ⊕ Eu, or
∃ torus tangent to Es ⊕ Ec
![Page 122: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/122.jpg)
panorama conjectures Anosov torus
calculations
evidence
hammerlind-potrie13f partially hyperbolic & non-dynamically coherent on a3-manifold with solvable fundamental group, then
∃ torus tangent to Ec ⊕ Eu, or∃ torus tangent to Es ⊕ Ec
![Page 123: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/123.jpg)
panorama conjectures Anosov torus
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
![Page 124: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/124.jpg)
panorama conjectures Anosov torus
classification
examples of ph dynamics
known ph dynamics in dimension 3perturbations of time-one maps of Anosov flows
certain skew-productscertain DA-maps
new example
non-dynamically coherent example
![Page 125: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/125.jpg)
panorama conjectures Anosov torus
classification
examples of ph dynamics
known ph dynamics in dimension 3perturbations of time-one maps of Anosov flowscertain skew-products
certain DA-maps
new example
non-dynamically coherent example
![Page 126: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/126.jpg)
panorama conjectures Anosov torus
classification
examples of ph dynamics
known ph dynamics in dimension 3perturbations of time-one maps of Anosov flowscertain skew-productscertain DA-maps
new example
non-dynamically coherent example
![Page 127: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/127.jpg)
panorama conjectures Anosov torus
classification
examples of ph dynamics
known ph dynamics in dimension 3perturbations of time-one maps of Anosov flowscertain skew-productscertain DA-maps
new examplenon-dynamically coherent example
![Page 128: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/128.jpg)
panorama conjectures Anosov torus
classification
question
questionare there more examples?
![Page 129: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/129.jpg)
panorama conjectures Anosov torus
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
![Page 130: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/130.jpg)
panorama conjectures Anosov torus
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 flow
2 a skew-product3 a DA-map
![Page 131: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/131.jpg)
panorama conjectures Anosov torus
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-product
3 a DA-map
![Page 132: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/132.jpg)
panorama conjectures Anosov torus
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
![Page 133: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/133.jpg)
panorama conjectures Anosov torus
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.
![Page 134: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/134.jpg)
panorama conjectures Anosov torus
classification
conjecture
classification conjecture (hhu)
If f : M3 → M3 is partially hyperbolic and dynamically coherent,then f is
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.
![Page 135: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/135.jpg)
panorama conjectures Anosov torus
classification
conjecture
classification conjecture (hhu)
If f : M3 → M3 is partially hyperbolic and dynamically coherent,then f is
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.
![Page 136: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/136.jpg)
panorama conjectures Anosov torus
classification
conjecture
classification conjecture (hhu)
If f : M3 → M3 is partially hyperbolic and dynamically coherent,then f is
1 a perturbation of a time-one map of an Anosov flow,2 a skew-product, or3 a DA-map.
![Page 137: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/137.jpg)
panorama conjectures Anosov torus
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 a skew-product, or3 a DA-map.
![Page 138: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/138.jpg)
panorama conjectures Anosov torus
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 a DA-map.
![Page 139: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/139.jpg)
panorama conjectures Anosov torus
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.
![Page 140: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/140.jpg)
panorama conjectures Anosov torus
classification
invariant tori in ph dynamics
invariant tori in PH dynamicsT invariant torus tangent to
Es ⊕ Eu
Ec ⊕ Eu
Es ⊕ Eu
⇒ T Anosov torus
![Page 141: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/141.jpg)
panorama conjectures Anosov torus
classification
invariant tori in ph dynamics
invariant tori in PH dynamicsT invariant torus tangent to
Es ⊕ Eu
Ec ⊕ Eu
Es ⊕ Eu
⇒ T Anosov torus
![Page 142: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/142.jpg)
panorama conjectures Anosov torus
classification
invariant tori in ph dynamics
invariant tori in PH dynamicsT invariant torus tangent to
Es ⊕ Eu
Ec ⊕ Eu
Es ⊕ Eu
⇒ T Anosov torus
![Page 143: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/143.jpg)
panorama conjectures Anosov torus
classification
invariant tori in ph dynamics
invariant tori in PH dynamicsT invariant torus tangent to
Es ⊕ Eu
Ec ⊕ Eu
Es ⊕ Eu
⇒ T Anosov torus
![Page 144: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/144.jpg)
panorama conjectures Anosov torus
classification
invariant tori in ph dynamics
invariant tori in PH dynamicsT invariant torus tangent to
Es ⊕ Eu
Ec ⊕ Eu
Es ⊕ Eu
⇒ T Anosov torus
![Page 145: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/145.jpg)
panorama conjectures 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
1 T3
2 the mapping torus of −id : T2 → T2
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
![Page 146: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/146.jpg)
panorama conjectures 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
1 T3
2 the mapping torus of −id : T2 → T2
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
![Page 147: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/147.jpg)
panorama conjectures 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
1 T3
2 the mapping torus of −id : T2 → T2
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
![Page 148: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/148.jpg)
panorama conjectures 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 either1 T3
2 the mapping torus of −id : T2 → T2
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
![Page 149: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/149.jpg)
panorama conjectures 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 either1 T3
2 the mapping torus of −id : T2 → T2
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
![Page 150: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/150.jpg)
panorama conjectures 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 either1 T3
2 the mapping torus of −id : T2 → T2
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
![Page 151: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/151.jpg)
panorama conjectures 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 either1 T3
2 the mapping torus of −id : T2 → T2
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
![Page 152: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/152.jpg)
panorama conjectures 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 either1 T3
2 the mapping torus of −id : T2 → T2
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
![Page 153: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/153.jpg)
panorama conjectures Anosov torus
classification
why stronger conjectures?
hertz-hertz-ures11M irreducible contains an Anosov torus,
then M is either1 T3
2 the mapping torus of −id : T2 → T2
3 a mapping torus of a hyperbolic automorphism of T2
3-manifolds
![Page 154: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/154.jpg)
panorama conjectures Anosov torus
classification
why stronger conjectures?
hertz-hertz-ures11M irreducible contains an Anosov torus, then M is either
1 T3
2 the mapping torus of −id : T2 → T2
3 a mapping torus of a hyperbolic automorphism of T2
3-manifolds
![Page 155: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/155.jpg)
panorama conjectures Anosov torus
classification
why stronger conjectures?
hertz-hertz-ures11M irreducible contains an Anosov torus, then M is either
1 T3
2 the mapping torus of −id : T2 → T2
3 a mapping torus of a hyperbolic automorphism of T2
3-manifolds
![Page 156: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/156.jpg)
panorama conjectures Anosov torus
classification
why stronger conjectures?
hertz-hertz-ures11M irreducible contains an Anosov torus, then M is either
1 T3
2 the mapping torus of −id : T2 → T2
3 a mapping torus of a hyperbolic automorphism of T2
3-manifolds
![Page 157: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/157.jpg)
panorama conjectures Anosov torus
classification
why stronger conjectures?
hertz-hertz-ures11M irreducible contains an Anosov torus, then M is either
1 T3
2 the mapping torus of −id : T2 → T2
3 a mapping torus of a hyperbolic automorphism of T2
3-manifolds
![Page 158: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/158.jpg)
panorama conjectures Anosov torus
classification
remark
remarkf : M3 → M3 partially hyperbolic⇒ M irreducible
why
Rosenberg68Burago-Ivanov08
![Page 159: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/159.jpg)
panorama conjectures Anosov torus
classification
remark
remarkf : M3 → M3 partially hyperbolic⇒ M irreducible
why
Rosenberg68Burago-Ivanov08
![Page 160: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/160.jpg)
panorama conjectures Anosov torus
classification
remark
remarkf : M3 → M3 partially hyperbolic⇒ M irreducible
why
Rosenberg68
Burago-Ivanov08
![Page 161: partial hyperbolicity in 3-manifolds - IMPA · partial hyperbolicity in 3-manifolds Jana Rodriguez Hertz Universidad de la República Uruguay Institut Henri Poincaré June, 14, 2013](https://reader035.vdocuments.us/reader035/viewer/2022081611/5f06b4187e708231d41950c8/html5/thumbnails/161.jpg)
panorama conjectures Anosov torus
classification
remark
remarkf : M3 → M3 partially hyperbolic⇒ M irreducible
why
Rosenberg68Burago-Ivanov08