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Thermodynamic Formalism: Ergodic theory and validated numerics - - PowerPoint PPT Presentation

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Thermodynamic Formalism: Ergodic theory and validated numerics Dvoretzky coverings Ai-Hua FAN Univ. Picardie, France CIRM, July 8-12, 2019 Ai-Hua FAN TPWWT 1/26 Outline General problem 1 Classical Dvoretzky covering 2 -Dvoretzky

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Thermodynamic Formalism:

Ergodic theory and validated numerics

Dvoretzky coverings

Ai-Hua FAN

  • Univ. Picardie, France

CIRM, July 8-12, 2019

Ai-Hua FAN TPWWT 1/26

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Outline

1

General problem

2

Classical Dvoretzky covering

3

µ-Dvoretzky covering : µ absolutely continuous

Ai-Hua FAN TPWWT 2/26

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General problem

Ai-Hua FAN TPWWT 3/26

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General problem Setting

(X, d) : a complete metric space (xn)n≥1 ⊂ X : a sequence of centers (rn)n≥1 ⊂ R+ : a sequence of radius µ : a (reference) measure on X Study subjects : (limsup set/infinitely covered set) J := lim sup

n→∞ B(xn, rn),

F := X \ J . Question 1 : J =? (J = X ?, J

µ

= X ?, dim J = ?) Question 2 : F =? (F = ∅ ?, F

µ

= ∅ ?, dim F = ?) NB Different from shrinking target problem (Borel-Cantelli lemma).

Ai-Hua FAN TPWWT 4/26

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Simple Properties of J and F

Different points of view of J : covering : {y ∈ X :

∞

  • n=1

1B(xn,rn)(y) = ∞} hitting : {y ∈ X :

∞

  • n=1

1B(y,rn)(xn) = ∞}. (xn) dense ⇒ J Baire set (so J = ∅) µ(B(xn, rn)) < ∞ ⇒ µ(J ) = 0 dimµ J ≤ sup{τ > 0 : µ(B(xn, rn)τ = ∞} rn = r, xn = T nx, µ ergodic ⇒ J (x)

µ

= X a.e.

Ai-Hua FAN TPWWT 5/26

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Example 1

Homogeneous diophantine approximation

{xn} = { p

q } = Q ∩ (0, 1) naturally ordered

rn = φ(q) when xn = p

q

J = {x ∈ T : qx < qφ(q) i.o.} Khintchine : J

Leb

= T if φ(q) ↓, qφ(q) = ∞ Dirichlet : J = T if φ(q) =

1 q2

Jarnik : dim J = 2

ν if φ(q) = 1 qv with v > 2.

Ai-Hua FAN TPWWT 6/26

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Example 2

Inhomogeneous diophantine approximation

xn = nα (mod 1) (α ∈ Q), rn = ψ(n) J (α) = {x ∈ T : x − nα < ψ(n) i.o.}

Borel-Cantelli : λ(J (α)) = 0 if ψ(q) < ∞ Bugeaud, Schemeling-Troubetzkoy (2003) : dim(J (α)) = 1 τ if ψ(q) = 1 nτ with τ > 1 Fan-Wu (2006) : General sequence {ψ(n)} :

  • a.e. α, ∀ψ(n) ↓

dim J (α) = inf

  • s > 0 :
  • ψ(n)s < ∞
  • = lim sup

log n − log ψ(n) (1)

  • ∃α , ∃ψ(n) ↓ s.t (1) is false. But No exceptional α if the limit exists.

Ai-Hua FAN TPWWT 7/26

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Example 3 (Fan-Schmeling-Troubetzkoy)

Dynamical diophantine approximation

Model Tx = 2x mod 1 defined on T. µφ, µψ Gibbs measures. xn = T nx, rn =

a nτ (a > 0, τ > 0)

J (x) :=

  • y ∈ T : y − 2nx < a

nτ i.o.

  • For µφ-a.e. x, we have J (x) = T if 1

τ > e+ := supµ

  • (−φ)dµ.

For µφ-a.e. x, we have J (x)

µψ

= T if 1

τ > h(µφ|µψ) :=

  • (−ψ)dµψ.

The two values are optimal. NB

  • 1. Generalization to Markov interval maps (L. M. Liao and S. Seuret).
  • 2. No result for ℓn =

a n1/e+ .

Ai-Hua FAN TPWWT 8/26

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Classical Dvoretzky covering

Ai-Hua FAN TPWWT 9/26

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Dvoretzky Random covering

Model (1956) X = T : the unit circle ; xn = ωn : independent, identically and uniformly distributed ; ℓn = 2rn ↓0 Partial results Dvoretzky (1956) : ∃ℓn s. t. J (ω) = T a.s. Kahane (1959) : ℓn = 1+ǫ

n

⇒ J (ω) = T a.s. Billard (1963) : ℓn = 1−ǫ

n

⇒ J (ω) = T a.s. Kahane(1968) :ℓn = 1−ǫ

n

⇒ dim F(ω) = ǫ a.s. Billard, Erd¨

  • s, Orey, Mandelbrot : ℓn = 1

n

Fan-Wu (2004) / A. Durand (2008) : Assume ℓn < ∞. Then a.s. dim J (ω) = inf{s > 0 :

  • ℓs

n < ∞}.

(also follows from the mass transfer principle of from Beresnevich-Velani2006).

Ai-Hua FAN TPWWT 10/26

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Complete solutions : Shepp condition/Kahane condition

Theorem (L. Shepp, 1972) The circle is a.s. covered (i.e. J (ω) = T a.s.) iff

∞

  • n=1

1 n2 eℓ1+···+ℓn = ∞. Theorem (J. P. Kahane, 1987) A compact set F is a.s. covered (i.e. J (ω) ⊃ F a.s.) iff CapΦF = 0, where Φ(t) = exp

  • (ℓn − |t|)+

Φ-energy : Iµ

Φ :=

Φ(t − s)dµ(t)dµ(s). CapΦF = 0 means Iµ

Φ = ∞ for all probability measures µ supported

by F. Shepp’s condition means

  • Φ(t)dt = ∞.

Ai-Hua FAN TPWWT 11/26

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Proof of the necessity : main lines Billard martingale method/Multiplicative chaos

Consider the (positive) martingales ∀t ∈ T, Qn(t) :=

n

  • k=1

1 − 1(0,ℓk)(t − ωk) 1 − ℓk . Mn :=

  • T

Qn(t)dt. Qn(t) = 0 iff t ∈ ωk + (0, ℓk) for some 1 ≤ k ≤ n. lim Mn > 0 = ⇒ T is not covered. EM 2

n = O(1) =

⇒ lim Mn > 0 a.s. EM 2

n = O(1)⇐

⇒

  • Φ(t)dt = ∞ (Shepp’s condition).

For the necessity of Kahane’s condition, we need the equilibrium measure σF instead of the Lebesgue measure and consider Mn :=

  • F

Qn(t)dσF (t).

Ai-Hua FAN TPWWT 12/26

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Potential theory/Equilibrium measure

Define the potential U µ

Φ(t) := Φ ∗ µ(t) =

  • Φ(t − s)dµ(s).

and the capacity CapΦ(F) := 1/IΦ(E) where IΦ(E) := infµ Iµ

Φ.

Theorem (Kahane-Salem, Ensembles parfaits et s´ eries trigonometriques)

  • Φ(n) ≥ 0.

Iµ

Φ =

Φ(n)| µ(n)|2. If IΦ(F) < ∞, there exists a unique probability σF such that IσF

Φ

= IΦ(F). {t ∈ T : U σF

Φ (t) < IΦ(F)} is of zero measure for any measure of

finite energy. NB 1. σF is called the equilibrium measure of F ; the last property is useful in the proof of sufficiency of Kahane’s condition.

  • 2. Results hold for all convex kernel Φ defined in (0, 1).

Ai-Hua FAN TPWWT 13/26

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Proof of the sufficiency : ideas

Dvoretzky covering is equivalent to Poisson covering (JPK). Poisson process (Xn, Yn) on R × R+ associated to dt ⊗ δℓn. Possion covering problem : R = (Xn, Xn + Yn) a.s. ? (B.M.) Consider a cloded set F ⊂ T and the martingale Mǫ := ∞ e−t1t∈Gǫd σǫ(t). where Gǫ := ∪Yn≥ǫ(Xn, Xn + Yn) σǫ : equilibrium measure of F associated to Φǫ Φǫ(t) := exp

ℓn≥ǫ(ℓn − |t|)+

  • σǫ : periodization of σǫ.

First way to compute Iǫ := EMǫ : Iǫ = e−

ℓn≥ǫ ℓn ∞

e−td σǫ(t). Second way to compute Iǫ : involving the stopping time (S. Janson) τǫ = inf{t > 0 : t ∈ Gǫ}. a.s. limǫ→0 τǫ = +∞.

Ai-Hua FAN TPWWT 14/26

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Multiplicative chaos operators

(JPK, 1987, Chin. Ann. Math.)

Recall the martingales ∀t ∈ T, Qn(t) :=

n

  • k=1

1 − 1(0,ℓk)(t − ωk) 1 − ℓk . For any finite measure σ ∈ M(T), define the random measure Qσ Qσ(A) := lim

n

  • A

Qn(t)dσ(t) (∀A ∈ B(T)). The multiplicative chaos operator EQ : M(T) → M(T) is defined by EQσ(A) := E[Qσ(A)] (∀A ∈ B(T)).

  • NB. (Fan2001/Barral-Fan2005) Similar operators Qa, EQa are defined for

the martingales (producing ”Gibbs measures”) Qa

n(t) = n

  • k=1

a1(0,ℓk)(t−ωk) 1 + (a − 1)ℓk (a > 0 a parameter).

Ai-Hua FAN TPWWT 15/26

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Multiplicative chaos operators (continued)

Theorem (Kahane/Fan) EQ is a projection ; M(T) = Im EQ ⊕ Ker EQ. σ ∈ Ker EQ iff σ is supported by a set of Φ-capacity zero. σ ∈ Im EQ iff σ =

k σk with Iσk Φ < ∞.

Assume that Q′, Q′′ come from two sequences {ℓ′

n} and {ℓ′′ n}. Let

σ′, σ′′ ∈ M(T), σ′′ ∈ Im EQ′′ and σ′ ≪ σ′′. Then (1) |ℓ′

n − ℓ′′ n| = ∞ =

⇒ Q′σ′ ⊥ Q′′σ′′. (2) |ℓ′

n − ℓ′′ n| < ∞ =

⇒ Q′σ′ ≪ Q′′σ′′. Assume Q′, Q′′ comes from two independent models, Q is the ”mixture”. Then (a)Qσ = Q′′Q′σ a.s. for any measure σ ∈ M(T). (b) EQ = EQ′′EQ′ = EQ′EQ′′. (c) σ ∈ Im EQ ⇒ Q′σ ∈ Im EQ′′ for almost all ω′ ∈ Ω′. (d) σ ∈ Ker EQ ⇒ Q′σ ∈ Ker EQ′′ for almost all ω′ ∈ Ω′.

dim Qλ = inf{τ > 0 : n2−τeℓ1+···ℓn = ∞} = 1 − lim sup ℓ1+···+ℓn

log n

.

NB Similar results for percolation on trees (Fan).

Ai-Hua FAN TPWWT 16/26

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  • L. Carleson problem

Question When T is infinitely covered, how to describe the infinity ?

  • A. H. Fan (1989) : two ways

Nn(t) :=

n

  • k=1

1(0,ℓk)(t − ωk) =? ∀(an) ⊂ R+,

∞

  • n=1

an = ∞, S(t) :=

∞

  • k=1

an1(0,ℓk)(t − ωk) = ∞?

Fan-Kahane (1993) for ℓn = 1+ǫ

n

: a.s. ∀t, Nn(t) ≈ log n;

∞

  • n=1

an n = ∞ = ⇒ a.s. ∀t, S(t) = ∞;

∞

  • n=1

an n < ∞ = ⇒ a.s. ∀t, S(t) < ∞.

Fan (2001), Barral-Fan (2005) : a.s. Nn(t) multifractally behaves.

Ai-Hua FAN TPWWT 17/26

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Multifractality of Nn(t)

Fβ :=

  • t ∈ T : lim

n→∞

Nn(t) ℓ1 + · · · + ℓn = β

  • ,

α := lim sup

n→∞

n

j=1 ℓj

− log ℓn . dα(β) := 1 + α(β − 1 − β log β)

Theorem (Barral-Fan 2005) (Slow like ℓn =

a n log n) If lim supn→∞ nℓn < ∞ and α = 0, then

a.s ∀β ≥ 0 , dim(Fβ) = 1. (2) (Normal like ℓn = a

n) If lim supn→∞ nℓn < ∞ and 0 < α < ∞, then

a.s ∀β ≥ 0 (dα(β) > 0), dim(Fβ) = dα(β). (3) (Rapide like ℓn = a log n

n

) If lim supn→∞ nℓn = ∞, then a.s ∀t ∈ T, lim Nn(t) ℓ1 + · · · + ℓn = 1. (4)

Ai-Hua FAN TPWWT 18/26

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µ-Dvoretzky covering I

Ai-Hua FAN TPWWT 19/26

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µ-Dvoretzky Random covering

Model X = T : the unit circle ; xn = ωn : independent, identically and µ-distributed ; ℓn = 2rn ↓0 We restricted to the case µ = λf i.e. dµ(x) = f(x)dx.

Ai-Hua FAN TPWWT 20/26

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Essential infimum points

Assume f : T → R : Borel measurable function, I ⊂ T : an interval. Essential infimum of f on I : essinfIf := sup{a ∈ R : a ≤ f(x) for almost all x ∈ I}. Essential infimum at x0 ∈ T of f : Ef(x0) := lim

n→∞ essinfB(x0, 1

n )f.

Set of essential infimum points of f : Kf := {x ∈ T : Ef(x) = mf}. Notation : mf := essinfTf. Proposition Kf is a non empty and compact set.

Ai-Hua FAN TPWWT 21/26

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Flat points

A point x ∈ T is flat for the measure µf and the sequence (ℓn) if

∞

  • n=1

|µf(B(x, rn)) − mfℓn| < ∞. Notation : Ff := {x ∈ T : x is flat}.

Ai-Hua FAN TPWWT 22/26

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Results (Fan and Karagulyan, 2018)

Theorem Assume |Kf ∩ Ff| > 0. Then the circle is covered for the µf-Dvoretzky covering if and only if

∞

  • n=1

1 n2 emf (ℓ1+···+ℓn) = ∞. (5)

Theorem Assume |Kf| = 0 and Kf is covered by Ff and a countable translates

  • f Ff, except a countable set. Under some conditions, the NSC for µf-

Dvoretzky covering is

∀x ∈ Kf,

∞

  • n=1

µf(B(x, rn)) = ∞; (6) ∀a > mf,

∞

  • n=1

1 n2 ea(ℓ1+···+ℓn) = ∞; CapΦ(mf )(Ff ∩ Kf) = 0, (7)

Theorem Assume ℓn = c/n. Then a NSC is cmf ≥ 1.

Ai-Hua FAN TPWWT 23/26

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comparison principle I

Consider two Dvoretzky covering respectively : µ and ν : two Borel probability measures on T. U ⊂ T : a non-empty open set. K ⊂ U : a compact set. Theorem Suppose µ|U ≤ ν|U. If K is covered for the µ-Dvoretzky covering, then it is covered for the ν-Dvoretzky covering. NB (localization) No need to know what happens outside U.

Ai-Hua FAN TPWWT 24/26

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comparison principle II

Compare µf and µg where g is locally constant : µf : Borel probability measures with density f. U ⊂ T : an open set. K ⊂ U : a compact set. Theorem Suppose f = a on U (a > 0 being a constant). If K is covered for the µf-Dvoretzky covering iff CapΦ(a)(K) = 0 where Φ(a)(t) = exp

  • a

∞

  • n=1

(ℓn − |t|)+

  • .

Idea of proof (Compare with Kahane’s condition).

  • 1. Let M be the inverse of ω. Then (M −1ωn) are i.i.d. and uniform.
  • 2. M : I → M(I) (I ⊂ U interval) is affine.
  • 3. For ny interval J ⊂ I, |M(J)| = a|J|.

Ai-Hua FAN TPWWT 25/26

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Questions

Non complete answer to any singular measure. (Conjecture) The non-covered set is a Salem set. Supporting result a.s. EIQλ

α

≈

  • n

| Φ(n)| n1−α Multifractal analysis of Qλ, Qσ. High dimensional Dvoretzky covering. How about xn = 2nx mod 1, ℓn = a

n with respect to Lebesgue

measure ?

Ai-Hua FAN TPWWT 26/26