Physical Measures

Stefano LuzzattoAbdus Salam International Centre for Theoretical Physics

Trieste, Italy.

International conference on Dynamical SystemsHammamet, TunisiaSeptember 5-7, 2017

Let f : M →M . For x ∈M let O+(x) := {fn(x)}n∈N be the orbit of x.Define a sequence of probability measures

µn(x) :=1

n

n−1∑

i=0

δf i(x)

Question: Does µn(x) converge?

Example (Convergence)

1 fn(x)→ p implies µn(x)→ δp2 µ ergodic invariant implies µn(x)→ µ for µ a.e. x (Birkhoff).

Example (Non-Convergence - Bowen’s eye)

∃ subsequences ni, nj such that µni(x)→ δp1 and µnj (x)→ δp2

Stefano Luzzatto (ICTP) 2 / 14

Let µ be a probability measure. The basin of µ is

Bµ := {x ∈ X : µn(x)→ µ}.

Definition

µ is a physical measure if Leb(Bµ) > 0.

Example (Physical measures)

1 δp where p is an attracting fixed point;

2 µ invariant, ergodic, and µ� Leb (µ(Bµ) = 1⇒ Leb(Bµ) > 0);

3 Sinai-Ruelle-Bowen (SRB) measures.

Example (No physical measures)

1 The identity map;

2 Bowen’s eye.

Conjecture (Palis conjecture)

Typical systems have (finitely many) physical measures.

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Conjecture (Palis conjecture)

Typical systems have (finitely many) physical measures.

Definition (Hyperbolicity)

A measurable invariant set Λ ⊂M is (nonuniformly) hyperbolic if:

1 there exists an invariant measurable splitting TxM = Esx ⊕Eux s.t.

limn→±∞

](Esfn(x), E

ufn(x)

)= 0

2 there exists χ > 0 such that for every vs ∈ Esx, vu ∈ Eux we have

limn→±∞

1

nln ‖Dfnx (vs)‖ < χ and lim

n→±∞

1

nln ‖Dfnx (vu)‖ > χ.

Λ is uniformly hyperbolic if the splitting is continuous, ](Esx, Eux) is

uniformly bounded, and 1/n ln ‖Dfnx (vs/u)‖ converge uniformly in x.

Conjecture (Viana conjecture)

Hyperbolic systems have (finitely many) physical measures.

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Existence of SRB physical measures in higher dimensions:

1 Eu uniform, Es uniform, splitting continous.Sinai-Ruelle-Bowen, 1960’s-1970’s

2 Eu uniform, Es non-uniform, splitting continous.Pesin-Sinai, Erg. Th. & Dyn. Syst. 1982Bonatti-Viana, Israel J. Math. 2000

3 Eu non-uniform, Es uniform, splitting continous.Alves-Bonatti-Viana, Inv. Math. 2000Alves-Dias-L-Pinheiro, J. Eur. Math. Soc. 2015

4 Eu non-uniform, Es trivial (non-uniformly expanding)Alves-L.-Pinheiro, Ann. IHP, 2005Alves-Dias-L. Ann. IHP, 2013

5 Es, Eu non-uniform, splitting measurable.Climenhaga-Dolgopyat-Pesin, Comm. Math. Phys. 2017Climenhaga-L.-Pesin, In progress, 2017

Remark: most results assume implicitly or explicitly additionaltechnical conditions including recurrence assumptions.

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Three methods for construction of SRB physical measures:

1 Symbolic coding and thermodynamical formalism;

2 Pushing forward Lebesgue measure;

3 Inducing (+ combination of previous two).

Inducing: Let Γ ⊂M , τ : Γ→ N a return time s.t. f τ(x)(x) ∈ Γ. Then

f := f τ : Γ→ Γ

is the corresponding induced map. The induced map f may havebetter properties than the original map f and may be easier to study.In particular we may be able to show that f admits an SRB measure.

Proposition

Suppose f : Γ→ Γ admits an SRB probability µ and τ :=∫τdµ <∞.

Then f admits an SRB probability measure.

Problem:find GOOD induced map with INTEGRABLE return times

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Example: Pomeau-Manneville maps: Let γ ≥ 0 and

f(x) = x+ x1+γ mod 1.

For γ = 0, f(x) = 2x mod 1 and Lebesgue is ergodic and invariant.For γ > 0, f ′(x) = 1 + (1 + γ)xγ and so f ′(0) = 1.Since 0 is a fixed point, f is not uniformly expanding and we cannotapply any existing methods to construct physical measures.

The inducing time is integrable if and only if γ ∈ [0, 1).Stefano Luzzatto (ICTP) 7 / 14

Example: Anosov DiffeomorphismΓ = element of the Markov partition; τ = first return time

fk fk

fk

fn

fk

fn

F = f τ : Γ→ Γ is a Young Tower and therefore ∃ µ SRB for F .Moreover, τ integrable and therefore ⇒ µ SRB for f .

Similar construction without a priori Markov partitions (Young ’98)

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Definition (Rectangle)

Let Λ be a hyperbolic set. A set Γ ⊂ Λ is a rectangle if for everyx, y ∈ Γ, the intersection V s

x ∩ V uy is a single point which belongs to Γ.

P

Q

Wu(P)

Ws(Q) Ws(P)

Wu(Q)

R

Γ = CS ∩ CU where CS =⋃

x∈Γ

V s(x), CU =⋃

x∈Γ

V u(x)

In general Γ is a Cantor set so the Young Tower is harder to get.

Stefano Luzzatto (ICTP) 9 / 14

P

Q

Wu(P)

Ws(Q) Ws(P)

Wu(Q)

R

We let Γ denote the region bounded by V sp , V

up , V

sq , V

uq .

Definition

A rectangle Γ is

1 nice if f i(V sp/q) ∩ intΓ = ∅ and f−i(V u

p/q) ∩ intΓ = ∅ ∀i > 0.

2 recurrent if every x ∈ Γ returns to Γ;

3 fat if Leb(∪x∈Γ0Vsx ) > 0.

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Definition

A rectangle Γ is

1 nice if f i(V sp/q) ∩ intΓ = ∅ and f−i(V u

p/q) ∩ intΓ = ∅ ∀i > 0.

2 recurrent if every x ∈ Γ returns to Γ;

3 fat if Leb(∪x∈Γ0Vsx ) > 0.

Theorem (Climenhaga-L-Pesin)

Let f : M →M be a C2 surface diffeomorphism with a

1 hyperbolic set Λ ⊆M , and a

2 nice recurrent rectangle Γ ⊆ Λ.

Then there exists a Topological First Return Young Tower.

If Γ is fat then this Topological Young Tower satisfies hyperbolicity anddistortion estimates and f admits an SRB probability measure.

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Definition (Hyperbolic branch)

A hyperbolic branch f i : Cs → Cu is a stable strip that gets mappedto an unstable strip with uniformly hyperbolic estimates.

p

bCS

bCU

q

b�0

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Definition (Hyperbolic recurrence)

A rectangle Γ is hyperbolically recurrent if every return to Γ givesrise to a hyperbolic branch.

Theorem

Let Γ0 ⊆ Λ be a nice recurrent rectangle.Then Γ0 is a nice hyperbolically recurrent rectangle.

Definition (Saturation)

A hyperbolically recurrent rectangle Γ = Cs ∩ Cu is saturated if forevery hyperbolic branch f i : Cs → Cu we have

f−i(Cu ∩ Cs) ⊂ Cs and f i(Cs ∩ Cu) ⊂ Cu.

Theorem

Let Γ0 be a nice hyperbolically recurrent rectangle.Then there is a nice hyperbolically recurrent saturated rectangle Γ.

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Theorem

Let Γ be a nice hyperbolically recurrent saturated rectangle.Then Γ admits a First Return Topological Young Tower.

Theorem

Let Γ be a fat nice hyperbolically recurrent saturated rectangle.Then Γ admits a First Return Young Tower.

Theorem

Let Λ be a hyperbolic set.Let Γ ⊂ Λ be a fat nice hyperbolically recurrent saturated rectangle.Then Γ admits a Young Tower with Integrable Return Times.

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