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.

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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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