Physical Measures for Partially Hyperbolic Diffeomorphisms Stefano Luzzatto Fifth International Conference and School Geometry, Dynamics, Integrable Systems June 2014
Physical Measuresfor Partially Hyperbolic Diffeomorphisms
Stefano Luzzatto
Fifth International Conference and School Geometry, Dynamics,Integrable Systems
June 2014
Physical Measures
f : M →M C1+ diffeomorphism, µ probability measure,
The basin (of attraction) of µ is
Bµ :=
x :1
n
n−1∑j=0
ϕ(f j(x))→∫ϕ dµ for any ϕ ∈ C0(M,R).
=
x :1
n
n−1∑j=0
δf ix → µ
Definition
µ is a physical measure if Leb(Bµ) > 0.
Stefano Luzzatto (ICTP) 2 / 11
Example
f contraction. fn(x)→ p ∀ x. δp is a physical measure.
Example
f(x) = 2x mod 1. Lebesgue measure is a physical measure.
Birkhoff’s Ergodic Theorem: µ ergodic and invariant ⇒ µ(Bµ) = 1.
Example
If µ is ergodic, invariant and µ << Leb, then µ is a physical measure.
Not all dynamical systems have physical measures.
Counterexample
The identity map f(x) = x has no physical measure.
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Counterexample
Question
Which systems have physical measures? How many do they have?
Conjecture (Palis)
Typical systems have (finitely many) physical measures.
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If f has an attractor Λ with Leb(Λ) = 0 then any invariant measure issingular w.r.t Lebesgue. Then µ(Bµ) = 1 ; Leb(Bµ) > 0.
Example
1) µ has conditional measures on unstable manifolds which areabsolutely continuous w.r.t Leb (Sinai-Ruelle-Bowen or SRB property);2) absolutely continuous stable foliation.Then µ is a physical measure.
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Hyperbolicityf is uniformly hyperbolic (or Anosov) if
TM = Es ⊕ Eu s.t. m(Df |Eux) > 1 > ‖Df |Es
x‖ ∀x ∈M.
f is (absolutely) partially hyperbolic if ∃ λ > 0 s.t.
TM = Es(c) ⊕ Eu(c) s.t. m(Df |Eux) > λ > ‖Df |Es
x‖ ∀x ∈M.
f is (pointwise) partially hyperbolic if
TM = Es ⊕ Euc s.t. min{1,m(Df |Eucx
)} > ‖Df |Esx‖ ∀x ∈M.
or
TM = Ecs ⊕ Eu s.t. m(Df |Eucx
) > max{1, ‖Df |Esx‖} ∀x ∈M.
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Uniformly Expanding case: Ecs ⊕ Eu
Theorem ((Sinai, Ruelle, Bowen, 1970’s))
Es ⊕ Eu ⇒ a finite number of physical (SRB) measures.
Theorem (Pesin-Sinai, 1982)
Ecs ⊕ Eu (absolute) ⇒ SRB measures (not physical).
Theorem ((Bonatti-Viana, 00))
Ecs ⊕ Eu (pointwise) and negative Lyapunov exponents:
lim supn→∞
ln ‖Dfn|Ecsx‖1/n < 0,
⇒ a finite number of physical (SRB) measures .
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Uniformly Expanding case: Ecs ⊕ Eu
Proof: Let γ = W uloc(x) for some x ∈M and consider the sequence
µn :=1
n
n−1∑i=0
f i∗ Lebγ .
where
f i∗ Lebγ(A) := Leb(f−i(A) ∩ γ) = Leb({x ∈ γ : f i(x) ∈ A}).
Letµ = weak-star limit point of {µn}.
Then:
µ has conditional measures µΓ on local unstable manifolds Γ withµΓ � LebΓ. (SRB property)
Absolute continuity of the stable foliation. (⇒ physical)
Uniform size of local unstable manifolds. (⇒ finiteness).
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Uniformly contracting case Es ⊕ Euc
Theorem (Alves-Bonatti-Viana ’00)
Es ⊕ Ecu and positive (lower) Lyapunov exponents:
lim infn→∞
1
n
n∑i=1
lnm(Df |Ecufi(x)
) > ε > 0
⇒ there exist a finite number of physical SRB measures.
Theorem (Alves-Dias-L.-Pinheiro ’13)
Es ⊕ Ecu and positive (upper) Lyapunov exponents:
lim supn→∞
1
n
n∑i=1
lnm(Df |Ecufi(x)
) > ε > 0
⇒ there exist a finite number of physical SRB measures.
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Proposition (Hyperbolic times. Es ⊕ Ecu)
There exists δ > 0 such that if
lim supn→∞
1
n
n∑i=1
logm(Df |Ecufi(x)
) > ε
there exists a sequence {ni(x)} of hyperbolic times for x, and cu-disks
V cun1
(x) ⊃ V cun2
(x) ⊃ V cun3s
(x) ⊃ · · ·such that
fni : Vni(x)→ Bδ(fni(x))
is uniformly expanding and has bounded distortion. If
lim infn→∞
1
n
n∑i=1
logm(Df |Ecufi(x)
) > ε
then the sequence {ni} has positive density at infinity.
Using the positive density, the construction of physical SRB measurescan be carried in the Es ⊕Ecu setting almost in the same way as in theEcs ⊕ Eu or Es ⊕ Eu setting by taking the limit of µni along asubsequence of positive density.
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Gibbs-Markov-Young structuresA Gibbs-Markov-Young structure is set Λ = Γs ∩ Γu with productstructure s.t.:
1
1) Positivemeasure: Lebγ(Λ ∩ γ) > 0for all γ ∈ Γu and the holonomymap map along Γs is absolutelycontinuous with densitybounded above and below.
2) Markov returns: There exists a partition of Λ into s-subsetsΛs1,Λ
s2, ... and a sequence of integers {Ri} such that Λui := fRi(Λsi ) is a
u-subset and fRi : Λsi → Λuiis a hyperbolic branch.3) Integrable returns: ∞∑
i=1
Ri <∞.
Gibbs-Markov-Young structure ⇒ physical SRB measure.
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