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.
Stefano Luzzatto (ICTP) 3 / 11
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.
Stefano Luzzatto (ICTP) 5 / 11
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.
Stefano Luzzatto (ICTP) 6 / 11
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 .
Stefano Luzzatto (ICTP) 7 / 11
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).
Stefano Luzzatto (ICTP) 8 / 11
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.
Stefano Luzzatto (ICTP) 9 / 11
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.
Stefano Luzzatto (ICTP) 10 / 11
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.
Stefano Luzzatto (ICTP) 11 / 11
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