partial hyperbolicity and topology of 3-manifolds Jana Rodriguez - PowerPoint PPT Presentation

panorama conjectures Anosov torus dynamical coherence non-dynamically coherent conjecture non-dynamically coherent conjecture (hertz-hertz-ures) f : M 3 → M 3 non-dynamically coherent,

panorama conjectures Anosov torus dynamical coherence non-dynamically coherent conjecture non-dynamically coherent conjecture (hertz-hertz-ures) f : M 3 → M 3 non-dynamically coherent, then M is either:

panorama conjectures Anosov torus dynamical coherence non-dynamically coherent conjecture non-dynamically coherent conjecture (hertz-hertz-ures) f : M 3 → M 3 non-dynamically coherent, then M is either: T 3 1 3-manifolds

panorama conjectures Anosov torus dynamical coherence non-dynamically coherent conjecture non-dynamically coherent conjecture (hertz-hertz-ures) f : M 3 → M 3 non-dynamically coherent, then M is either: T 3 1 the mapping torus of − id 2 3-manifolds

panorama conjectures Anosov torus dynamical coherence non-dynamically coherent conjecture non-dynamically coherent conjecture (hertz-hertz-ures) f : M 3 → M 3 non-dynamically coherent, then M is either: T 3 1 the mapping torus of − id 2 the mapping torus of a hyperbolic automorphism 3 3-manifolds

panorama conjectures Anosov torus dynamical coherence stronger conjecture stronger non-dynamically coherent conjecture f : M 3 → M 3 non-dynamically coherent, then either

panorama conjectures Anosov torus dynamical coherence stronger conjecture stronger non-dynamically coherent conjecture f : M 3 → M 3 non-dynamically coherent, then either ∃ torus tangent to E c ⊕ E u , or

panorama conjectures Anosov torus dynamical coherence stronger conjecture stronger non-dynamically coherent conjecture f : M 3 → M 3 non-dynamically coherent, then either ∃ torus tangent to E c ⊕ E u , or ∃ torus tangent to E s ⊕ E c

panorama conjectures Anosov torus dynamical coherence intermediate conjecture intermediate conjecture f volume preserving ⇒ f dynamically coherent

panorama conjectures Anosov torus dynamical coherence evidence potrie11 f : T 3 → T 3 non-dynamically coherent, then

panorama conjectures Anosov torus dynamical coherence evidence potrie11 f : T 3 → T 3 non-dynamically coherent, then ∃ torus tangent to E c ⊕ E u , or

panorama conjectures Anosov torus dynamical coherence evidence potrie11 f : T 3 → T 3 non-dynamically coherent, then ∃ torus tangent to E c ⊕ E u , or ∃ torus tangent to E s ⊕ E c

panorama conjectures Anosov torus classification examples of ph dynamics known ph dynamics in dimension 3

panorama conjectures Anosov torus classification examples of ph dynamics known ph dynamics in dimension 3 perturbations of time-one maps of Anosov flows

panorama conjectures Anosov torus classification examples of ph dynamics known ph dynamics in dimension 3 perturbations of time-one maps of Anosov flows certain skew-products

panorama conjectures Anosov torus classification examples of ph dynamics known ph dynamics in dimension 3 perturbations of time-one maps of Anosov flows certain skew-products certain DA-maps

panorama conjectures Anosov torus classification examples of ph dynamics known ph dynamics in dimension 3 perturbations of time-one maps of Anosov flows certain skew-products certain DA-maps new example non-dynamically coherent example

panorama conjectures Anosov torus classification question question are there more examples?

panorama conjectures Anosov torus classification conjecture pujals classification conjecture (pujals01) If f : M 3 → M 3 is a transitive partially hyperbolic diffeomorphism, then f is (finitely covered by) either

panorama conjectures Anosov torus classification conjecture pujals classification conjecture (pujals01) If f : M 3 → M 3 is a transitive partially hyperbolic diffeomorphism, then f is (finitely covered by) either a perturbation of a time-one map of an Anosov flow 1

panorama conjectures Anosov torus classification conjecture pujals classification conjecture (pujals01) If f : M 3 → M 3 is a transitive partially hyperbolic diffeomorphism, then f is (finitely covered by) either a perturbation of a time-one map of an Anosov flow 1 a skew-product 2

panorama conjectures Anosov torus classification conjecture pujals classification conjecture (pujals01) If f : M 3 → M 3 is a transitive partially hyperbolic diffeomorphism, then f is (finitely covered by) either a perturbation of a time-one map of an Anosov flow 1 a skew-product 2 a DA-map 3

panorama conjectures Anosov torus classification conjecture classification conjecture (hhu) If f : M 3 → M 3 is partially hyperbolic and dynamically coherent, then f is

panorama conjectures Anosov torus classification conjecture classification conjecture (hhu) If f : M 3 → M 3 is partially hyperbolic and dynamically coherent, then f is a perturbation of a time-one map of an Anosov flow, 1

panorama conjectures Anosov torus classification conjecture classification conjecture (hhu) If f : M 3 → M 3 is partially hyperbolic and dynamically coherent, then f is a perturbation of a time-one map of an Anosov flow, 1 a skew-product, or 2

panorama conjectures Anosov torus classification conjecture classification conjecture (hhu) If f : M 3 → M 3 is partially hyperbolic and dynamically coherent, then f is a perturbation of a time-one map of an Anosov flow, 1 a skew-product, or 2 a DA-map. 3

panorama conjectures Anosov torus classification conjecture classification conjecture (hhu) If f : M 3 → M 3 is partially hyperbolic and dynamically coherent, then f is leafwise conjugate to an Anosov flow 1 a skew-product, or 2 a DA-map. 3

panorama conjectures Anosov torus classification conjecture classification conjecture (hhu) If f : M 3 → M 3 is partially hyperbolic and dynamically coherent, then f is leafwise conjugate to an Anosov flow 1 leafwise conjugate to a skew-product with linear base 2 a DA-map. 3

panorama conjectures Anosov torus classification conjecture classification conjecture (hhu) If f : M 3 → M 3 is partially hyperbolic and dynamically coherent, then f is leafwise conjugate to an Anosov flow 1 leafwise conjugate to a skew-product with linear base 2 leafwise conjugate to an Anosov map in T 3 . 3

panorama conjectures Anosov torus Anosov torus problems ergodicity

panorama conjectures Anosov torus Anosov torus problems ergodicity dynamical coherence

panorama conjectures Anosov torus Anosov torus problems ergodicity dynamical coherence classification

panorama conjectures Anosov torus Anosov torus problems          ergodicity    → Anosov torus dynamical coherence   classification         

panorama conjectures Anosov torus Anosov torus Anosov torus Anosov torus T embedded 2-torus

panorama conjectures Anosov torus Anosov torus Anosov torus Anosov torus T embedded 2-torus ∃ f : M → M s.t.

panorama conjectures Anosov torus Anosov torus Anosov torus Anosov torus T embedded 2-torus ∃ f : M → M s.t. f ( T ) = T 1

panorama conjectures Anosov torus Anosov torus Anosov torus Anosov torus T embedded 2-torus ∃ f : M → M s.t. f ( T ) = T 1 f | T isotopic to Anosov 2

panorama conjectures Anosov torus Anosov torus invariant tori in ph dynamics invariant tori in PH dynamics T invariant torus tangent to

panorama conjectures Anosov torus Anosov torus invariant tori in ph dynamics invariant tori in PH dynamics T invariant torus tangent to E s ⊕ E u

panorama conjectures Anosov torus Anosov torus invariant tori in ph dynamics invariant tori in PH dynamics T invariant torus tangent to E s ⊕ E u E c ⊕ E u

panorama conjectures Anosov torus Anosov torus invariant tori in ph dynamics invariant tori in PH dynamics T invariant torus tangent to E s ⊕ E u E c ⊕ E u E s ⊕ E u

panorama conjectures Anosov torus Anosov torus invariant tori in ph dynamics invariant tori in PH dynamics T invariant torus tangent to E s ⊕ E u ⇒ E c ⊕ E u T Anosov torus E s ⊕ E u

panorama conjectures Anosov torus Anosov torus conjectures

panorama conjectures Anosov torus Anosov torus conjectures non-ergodic conjecture f : M → M non-ergodic partially hyperbolic