Stable ergodicity beyond partial hyperbolicity1 mechanisms that activate stable ergodicity partial...
Transcript of Stable ergodicity beyond partial hyperbolicity1 mechanisms that activate stable ergodicity partial...
Stable ergodicity beyond partialhyperbolicity
Jana Rodriguez HertzSUSTech - China
Dynamics on the screenAugust 5, 2020
1 introduction
1 setting
setting
• M compact Riemannian manifold without boundary• m volume probability measure• Diffr
m(M): Cr -diffeomorphisms preserving m
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
1 question
question
• does there exist U ⊂ Diff1m open
• such that
g ∈ U ⇒ g ergodic?
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
1 mechanisms that activate stable ergodicity
Anosov-Sinai (1967)
• f ∈ Diff1+αm (M) hyperbolic
• ⇒ ergodic• ⇒ stably ergodic
1 mechanisms that activate stable ergodicity
Grayson - Pugh - Shub (1995)
• ∃ a non-hyperbolic stably ergodic diffeomorphism
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
1 mechanisms that activate stable ergodicity
conjecture: pugh - shub (1995)
stable ergodicity is Cr -dense among partially hyperbolicdiffeomorphisms
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)
1 mechanisms that activate stable ergodicity
Tahzibi (2004)
• ∃ a non-partially hyperbolic stably ergodicdiffeomorphism
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
1 mechanisms that activate stable ergodicity
it is a necessary condition:
stable ergodicity
⇓
dominated splitting
1 mechanisms that activate stable ergodicity
pugh - shub (1995)
a little hyperbolicity goes a long way toward guaranteeingstable ergodicity
1 conjecture
conjecture: JRH. (2012)
generically in Diff1m(M)
dominated splitting
⇓
stable ergodicity
2 exploring new mechanisms for SE
2 known mechanism for PHD
accessibility
• for partially hyperbolic diffeomorphisms• Pugh - Shub proposed accessibility
•
2 known mechanism for PHD
pugh - shub program
• for partially hyperbolic diffeomorphisms in Diff2m(M)
• stable accessibility is Cr -dense• accessibility⇒ ergodicity
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
2 other mechanisms activating SE
minimality
• a foliation is minimal• if every leaf is dense
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
2 other mechanisms activating SE
theorem (G. Nuñez, JRH 2019)
generically in Diff1m(M3),
• Fu minimal
⇒ stable ergodicity
2 question
question (2014)
generically in Diff1m(M3),
• dominated splitting• ⇒ Fu or Fs minimal?
2 other mechanisms activating SE
question - in any dimension
Fu minimal
⇓
stable ergodicity
2 conjecture
conjecture JRH (2019)
generically in Diff1m(M)
Fu minimal
⇓
stable ergodicity
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
2 another mechanism for SE
G. Núñez - D. Obata - JRH (2020)
generically in Diff1m(M3)
• dominated splitting• + a little partial hyperbolicity (∗)• ⇒ stable ergodicity
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
3 ergodic homoclinic classes
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}
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
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
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
4 strategy
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
4 strategy
setting
• f ∈ Diff1m(M3)
• with TM = E ⊕ Eu dominated splitting• ∃ p ∈ PerH(f ) with u(p) = dim(Eu) = 1
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
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
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
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
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
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
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)
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
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
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)
4 question
question
stable ergodicity
⇓?
mixing
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