stable ergodicity beyond partial hyperbolicity1 mechanisms that activate stable ergodicity partial...
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![Page 1: Stable ergodicity beyond partial hyperbolicity1 mechanisms that activate stable ergodicity partial hyperbolicity f 2Di 1 m(M) ispartially hyperbolicif there exists a Df-invariant splitting](https://reader033.vdocuments.us/reader033/viewer/2022051916/60078303dd14bd40c41e00a7/html5/thumbnails/1.jpg)
Stable ergodicity beyond partialhyperbolicity
Jana Rodriguez HertzSUSTech - China
Dynamics on the screenAugust 5, 2020
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1 introduction
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1 setting
setting
• M compact Riemannian manifold without boundary• m volume probability measure• Diffr
m(M): Cr -diffeomorphisms preserving m
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1 stable ergodicity
stable ergodicity
• f ∈ Diff1m(M) is stably ergodic
• if there exists f ∈ U ⊂ Diff1m(M) open
• such that
g ∈ U ∩ Diff2m(M) ⇒ g ergodic
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1 question
question
• does there exist U ⊂ Diff1m open
• such that
g ∈ U ⇒ g ergodic?
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1 mechanisms that activate stable ergodicity
hyperbolicity
f ∈ Diff1m(M) is Anosov or hyperbolic if
• there exists a Df -invariant splitting TM = Es ⊕ Eu
• such that
TM = Es ⊕ Eu
↓ ↓Df − contracting Df − expanding
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1 mechanisms that activate stable ergodicity
Anosov-Sinai (1967)
• f ∈ Diff1+αm (M) hyperbolic
• ⇒ ergodic• ⇒ stably ergodic
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1 mechanisms that activate stable ergodicity
Grayson - Pugh - Shub (1995)
• ∃ a non-hyperbolic stably ergodic diffeomorphism
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1 mechanisms that activate stable ergodicity
partial hyperbolicity
f ∈ Diff1m(M) is partially hyperbolic if
• there exists a Df -invariant splittingTM = Es ⊕ Ec ⊕ Eu
• such that
TM = Es ⊕ Ec ⊕ Eu
↓ ↓ ↓contracting intermediate expanding
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1 mechanisms that activate stable ergodicity
conjecture: pugh - shub (1995)
stable ergodicity is Cr -dense among partially hyperbolicdiffeomorphisms
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1 mechanisms that activate stable ergodicity
Avila - Crovisier - Wilkinson (2016)
stable ergodicity is C1-dense among partially hyperbolicdiffeomorphisms
Hertz - H. - Ures (2008)
stable ergodicity is C∞-dense among partially hyperbolicdiffeomorphisms (c = 1)
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1 mechanisms that activate stable ergodicity
Tahzibi (2004)
• ∃ a non-partially hyperbolic stably ergodicdiffeomorphism
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1 mechanisms that activate stable ergodicity
dominated splitting
f ∈ Diff1m(M) admits a dominated splitting
• there exists a Df -invariant splitting TM = E ⊕ F• such that
TM = E ⊕ F↓ ↓
more contracting more expanding
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1 mechanisms that activate stable ergodicity
it is a necessary condition:
stable ergodicity
⇓
dominated splitting
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1 mechanisms that activate stable ergodicity
pugh - shub (1995)
a little hyperbolicity goes a long way toward guaranteeingstable ergodicity
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1 conjecture
conjecture: JRH. (2012)
generically in Diff1m(M)
dominated splitting
⇓
stable ergodicity
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2 exploring new mechanisms for SE
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2 known mechanism for PHD
accessibility
• for partially hyperbolic diffeomorphisms• Pugh - Shub proposed accessibility
•
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2 known mechanism for PHD
pugh - shub program
• for partially hyperbolic diffeomorphisms in Diff2m(M)
• stable accessibility is Cr -dense• accessibility⇒ ergodicity
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2 other mechanisms activating SE
other mechanisms
• in dimension 3• dominated splitting⇒ one hyperbolic bundle• ⇒ either E = Es or F = Eu
• suppose TM = E ⊕ Eu
• there is an invariant foliation Fu tangent to Eu
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2 other mechanisms activating SE
minimality
• a foliation is minimal• if every leaf is dense
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2 other mechanisms activating SE
program in dimension 3
• for f ∈ Diff1m(M) with a dominated splitting
• (stable) minimality of the Fu is C1-dense• C1-generically, minimality of Fu ⇒ stable ergodicity
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2 other mechanisms activating SE
theorem (G. Nuñez, JRH 2019)
generically in Diff1m(M3),
• Fu minimal
⇒ stable ergodicity
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2 question
question (2014)
generically in Diff1m(M3),
• dominated splitting• ⇒ Fu or Fs minimal?
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2 other mechanisms activating SE
question - in any dimension
Fu minimal
⇓
stable ergodicity
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2 conjecture
conjecture JRH (2019)
generically in Diff1m(M)
Fu minimal
⇓
stable ergodicity
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2 theorem
G. Núñez - JRH (2020)
for a generic map f ∈ Diff1m(M), if
• Fu minimal
⇒ ∃ U(f ) ⊂ Diff1m(M): ∀g ∈ Diff2
m(M) ∩ U
• ∃ ergodic component Phc(q)
• Phc(q)ess
= M
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2 another mechanism for SE
G. Núñez - D. Obata - JRH (2020)
generically in Diff1m(M3)
• dominated splitting• + a little partial hyperbolicity (∗)• ⇒ stable ergodicity
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2 another mechanism for SE
G. Núñez - D. Obata - JRH (2020)
• in the isotopy class of every partially hyperbolicf ∈ Diff1
m(M3)
• there is a stably ergodic diffeomorphism• which is not (strictly) partially hyperbolic
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3 ergodic homoclinic classes
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3 Pesin invariant manifolds
Pesin stable/unstable manifolds
W−(x) =
{y ∈ M : lim sup
n→∞
1n
log d(f n(x), f n(y)) < 0}
W+(x) =
{y ∈ M : lim sup
n→∞
1n
log d(f−n(x), f−n(y)) < 0}
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3 ergodic homoclinic class
ergodic homoclinic class (HHTU11)
p ∈ PerH(f )
Phc+(p) = {x : W+(x) t W s(o(p)) 6= ∅}
W s(p)
Wu(p) W+(x)
p
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3 ergodic homoclinic class
ergodic homoclinic class (HHTU11)
p ∈ PerH(f )
Phc−(p) = {x : W−(x) t W u(o(p)) 6= ∅}
W s(p)
Wu(p) W−(x)
x
p
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3 criterion of ergodicity
criterion of ergodicity (HHTU11)
• p ∈ PerH(f ), f ∈ Diff2m(M)
• m(Phc−(p)) > 0 and m(Phc+(p)) > 0
then
1 Phc−(p)◦= Phc+(p) :
◦= Phc(p)
2 f |Phc(p) ergodic
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4 strategy
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4 strategy
strategy
prove that C1-densely, for f ∈ Diff1m(M)
• ∃ p ∈ PerH(f )
• ∃ U ⊂ Diff1m(M)
• such that g ∈ U ∩ Diff2m(M)⇒
m(Phc(p)) = 1
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4 strategy
setting
• f ∈ Diff1m(M3)
• with TM = E ⊕ Eu dominated splitting• ∃ p ∈ PerH(f ) with u(p) = dim(Eu) = 1
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4 strategy
strategy
on the one hand, use geometric approach to find• ∃ U ⊂ Diff1
m(M) such that• g ∈ U ∩ Diff2
m(M)⇒
m(Phcu(p)) = 1
• ⇒ m(Phc+(p)) = 1
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4 strategy
strategy
on the other hand, use generic approach to guarantee• ∃ U ⊂ Diff1
m(M) such that• g ∈ U ∩ Diff2
m(M)⇒
m(Phc−(p)) > 0
• thenPhc+(p)
◦= Phc−(p)
◦= Phc(p)
◦= M
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4 strategy
generic mechanism - M - B - H - ACW - AB
for a generic f ∈ Diff1m(M) with dominated splitting
• f ergodic• ∃q ∈ PerH(f ) such that Phc(q)
◦= M
• Oseledets splitting is dominated u(q) = #{LE > 0}• ∃U(f ) such that m(Phc(q)) > 0 for all g ∈ U
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4 strategy
geometric mechanism - minimality
assume f ∈ Diff1m(M3) with dominated splitting TM = E ⊕
Eu generic• ⇒ ∃p ∈ PerH(f ) with u(p) = dim Eu = 1• if Fu is minimal• ∃ U(f ) such that for all g ∈ U
Phcu(p) = M
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4 strategy
geometric mechanism - minimality
assume f ∈ Diff1m(M3) with dominated splitting TM = E ⊕
Eu generic• if u(p) = u(q)
• generically Phc(p) = Phc(q) (HHTU - AC)• on one hand Phc+(p) = M for all g ∈ U• on the other hand Phc−(p) > 0 for all g ∈ U• ⇒ f stably ergodic
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4 strategy
geometric mechanism - minimality
assume f ∈ Diff1m(M3) with dominated splitting TM = E ⊕
Eu generic• if u(q) > u(p)
• TM = Es ⊕ Ec ⊕ Eu
• ⇒ f partially hyperbolic• ⇒ f stably ergodic
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4 strategy
geometric mechanism - a little partial hyperbolicity
assume f ∈ Diff1m(M3) with dominated splitting TM = E ⊕
Eu generic• ∃ p ∈ PerH(f ) with u(p) = dim(Eu) = 1• ⇒ Phcu(p) is open• assume
Λ(f ) = M \ Phcu(p)
is partially hyperbolic (∗)• (NOH hypothesis)
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4 strategy
geometric mechanism - a little partial hyperbolicity
• g 7→ Λ(g) continuous in f• Λ(g) PH• with a generic argument,• we see m(Λ(g)) = 0 for all C2 g ∈ U• ⇒ Phcu(p)
◦= M for all g ∈ U
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4 strategy
geometric mechanism - a little partial hyperbolicity
• u(p) = u(q)
• generically Phc(p) = Phc(q) (HHTU - AC)
• on one hand Phc+(p)◦= M for all g ∈ U
• on the other hand Phc−(p) > 0 for all g ∈ U• ⇒ f stably ergodic
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4 strategy
generic argument (if there is time)
• m(Λ(fn)) > 0 with fn → f fn ∈ C2
• ⇒ Λ(fn) su-saturated (P-JRH)• ⇒ Λ(f ) su-saturated• ⇒ ∃ q1,q2 ∈ PerH(f ) such that
W ss(q1) tq W uu(q2) 6= ∅ (H)
• ⇒ non-generic situation (Kupka-Smale argument)
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4 question
question
stable ergodicity
⇓?
mixing
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thank you!
thank you!