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

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 µ -Dvoretzky covering : µ absolutely continuous 3 Ai-Hua FAN TPWWT 2/26

General problem Ai-Hua FAN TPWWT 3/26

General problem Setting ( X, d ) : a complete metric space ( x n ) n ≥ 1 ⊂ X : a sequence of centers ( r n ) 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 ( x n , r n ) , F := X \ J . Question 1 : J =? µ ( J = X ?, J = X ?, dim J = ?) Question 2 : F =? µ ( F = ∅ ?, F = ∅ ?, dim F = ?) NB Different from shrinking target problem (Borel-Cantelli lemma). Ai-Hua FAN TPWWT 4/26

Simple Properties of J and F Different points of view of J : � ∞ covering : { y ∈ X : 1 B ( x n ,r n ) ( y ) = ∞} n =1 � ∞ hitting : { y ∈ X : 1 B ( y,r n ) ( x n ) = ∞} . n =1 ( x n ) dense ⇒ J Baire set (so J � = ∅ ) � µ ( B ( x n , r n )) < ∞ ⇒ µ ( J ) = 0 dim µ J ≤ sup { τ > 0 : � µ ( B ( x n , r n ) τ = ∞} µ r n = r, x n = T n x, µ ergodic ⇒ J ( x ) = X a.e. Ai-Hua FAN TPWWT 5/26

Example 1 Homogeneous diophantine approximation { x n } = { p q } = Q ∩ (0 , 1) naturally ordered r n = φ ( q ) when x n = p q J { x ∈ T : � qx � < qφ ( q ) i.o. } = = T if φ ( q ) ↓ , � qφ ( q ) = ∞ Leb Khintchine : J 1 Dirichlet : J = T if φ ( q ) = q 2 Jarnik : dim J = 2 1 ν if φ ( q ) = q v with v > 2 . Ai-Hua FAN TPWWT 6/26

Example 2 Inhomogeneous diophantine approximation x n = nα (mod 1) ( α �∈ Q ), r n = 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 n τ with τ > 1 Fan-Wu (2006) : General sequence { ψ ( n ) } : • a.e. α , ∀ ψ ( n ) ↓ log n � ψ ( n ) s < ∞ � � dim J ( α ) = inf s > 0 : = lim sup (1) − log ψ ( n ) • ∃ α , ∃ ψ ( n ) ↓ s.t (1) is false. But No exceptional α if the limit exists. Ai-Hua FAN TPWWT 7/26

Example 3 (Fan-Schmeling-Troubetzkoy) Dynamical diophantine approximation Model Tx = 2 x mod 1 defined on T . µ φ , µ ψ Gibbs measures. a x n = T n x , r n = n τ ( a > 0 , τ > 0 ) � � y ∈ T : � y − 2 n x � < a J ( x ) := i.o. n τ � For µ φ -a.e. x, we have J ( x ) = T if 1 τ > e + := sup µ ( − φ ) dµ. � µ ψ = T if 1 For µ φ -a.e. x, we have J ( x ) τ > h ( µ φ | µ ψ ) := ( − ψ ) dµ ψ . The two values are optimal. NB 1. Generalization to Markov interval maps (L. M. Liao and S. Seuret). a 2. No result for ℓ n = n 1 /e + . Ai-Hua FAN TPWWT 8/26

Classical Dvoretzky covering Ai-Hua FAN TPWWT 9/26

Dvoretzky Random covering Model (1956) X = T : the unit circle ; x n = ω n : independent, identically and uniformly distributed ; ℓ n = 2 r n ↓ 0 Partial results Dvoretzky (1956) : ∃ ℓ n s. t. J ( ω ) = T a.s. Kahane (1959) : ℓ n = 1+ ǫ ⇒ J ( ω ) = T a.s. n ℓ n = 1 − ǫ ⇒ J ( ω ) � = T a.s. Billard (1963) : n Kahane(1968) : ℓ n = 1 − ǫ ⇒ dim F ( ω ) = ǫ a.s. n os, Orey, Mandelbrot : ℓ n = 1 Billard, Erd¨ n Fan-Wu (2004) / A. Durand (2008) : Assume � ℓ n < ∞ . Then � ℓ s a.s. dim J ( ω ) = inf { s > 0 : n < ∞} . (also follows from the mass transfer principle of from Beresnevich-Velani 2006 ). Ai-Hua FAN TPWWT 10/26

Complete solutions : Shepp condition/Kahane condition Theorem (L. Shepp, 1972) The circle is a.s. covered (i.e. J ( ω ) = T a.s. ) iff � ∞ 1 n 2 e ℓ 1 + ··· + ℓ n = ∞ . n =1 Theorem (J. P. Kahane, 1987) A compact set F is a.s. covered (i.e. J ( ω ) ⊃ F a.s. ) iff � ( ℓ n − | t | ) + Cap Φ F = 0 , where Φ( t ) = exp � � Φ -energy : I µ Φ := Φ( t − s ) dµ ( t ) dµ ( s ) . Cap Φ F = 0 means I µ Φ = ∞ for all probability measures µ supported by F . � Shepp’s condition means Φ( t ) dt = ∞ . Ai-Hua FAN TPWWT 11/26

Proof of the necessity : main lines Billard martingale method/Multiplicative chaos Consider the (positive) martingales n � 1 − 1 (0 ,ℓ k ) ( t − ω k ) ∀ t ∈ T , Q n ( t ) := . 1 − ℓ k k =1 � M n := Q n ( t ) dt. T Q n ( t ) = 0 iff t ∈ ω k + (0 , ℓ k ) for some 1 ≤ k ≤ n . ⇒ T is not covered. lim M n > 0 = E M 2 n = O (1) = ⇒ lim M n > 0 a.s. � E M 2 n = O (1) ⇐ ⇒ Φ( t ) dt = ∞ (Shepp’s condition). For the necessity of Kahane’s condition, we need the equilibrium measure σ F instead of the Lebesgue measure and consider � M n := Q n ( t ) dσ F ( t ) . F Ai-Hua FAN TPWWT 12/26

Potential theory/Equilibrium measure Define the potential � U µ Φ ( t ) := Φ ∗ µ ( t ) = Φ( t − s ) dµ ( s ) . and the capacity Cap Φ ( F ) := 1 /I Φ ( E ) where I Φ ( E ) := inf µ I µ Φ . Theorem (Kahane-Salem, Ensembles parfaits et s´ eries trigonometriques) � Φ( n ) ≥ 0 . Φ = � � I µ µ ( n ) | 2 . Φ( n ) | � If I Φ ( F ) < ∞ , there exists a unique probability σ F such that I σ F = I Φ ( F ) . Φ { t ∈ T : U σ F Φ ( t ) < I Φ ( F ) } is of zero measure for any measure of finite energy. NB 1. σ F 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

Proof of the sufficiency : ideas Dvoretzky covering is equivalent to Poisson covering (JPK). Poisson process ( X n , Y n ) on R × R + associated to dt ⊗ � δ ℓ n . Possion covering problem : R = � ( X n , X n + Y n ) a.s. ? (B.M.) Consider a cloded set F ⊂ T and the martingale � ∞ e − t 1 t �∈ G ǫ d � M ǫ := σ ǫ ( t ) . 0 where G ǫ := ∪ Y n ≥ ǫ ( X n , X n + Y n ) σ ǫ : equilibrium measure of F associated to Φ ǫ Φ ǫ ( t ) := exp � ℓ n ≥ ǫ ( ℓ n − | t | ) + σ ǫ : periodization of σ ǫ . � ℓn ≥ ǫ ℓ n � ∞ First way to compute I ǫ := E M ǫ : I ǫ = e − � e − t d � σ ǫ ( t ) . 0 Second way to compute I ǫ : involving the stopping time (S. Janson) τ ǫ = inf { t > 0 : t �∈ G ǫ } . a.s. lim ǫ → 0 τ ǫ = + ∞ . Ai-Hua FAN TPWWT 14/26

Multiplicative chaos operators (JPK, 1987, Chin. Ann. Math.) Recall the martingales � n 1 − 1 (0 ,ℓ k ) ( t − ω k ) ∀ t ∈ T , Q n ( t ) := . 1 − ℓ k k =1 For any finite measure σ ∈ M ( T ) , define the random measure Qσ � Qσ ( A ) := lim Q n ( t ) dσ ( t ) ( ∀ A ∈ B ( T )) . n A The multiplicative chaos operator E Q : M ( T ) → M ( T ) is defined by E Qσ ( A ) := E [ Qσ ( A )] ( ∀ A ∈ B ( T )) . NB . (Fan 2001 /Barral-Fan 2005 ) Similar operators Q a , E Q a are defined for the martingales (producing ”Gibbs measures”) n � a 1 (0 ,ℓk ) ( t − ω k ) Q a n ( t ) = ( a > 0 a parameter) . 1 + ( a − 1) ℓ k k =1 Ai-Hua FAN TPWWT 15/26

Multiplicative chaos operators (continued) Theorem (Kahane/Fan) E Q is a projection ; M ( T ) = Im E Q ⊕ Ker E Q . σ ∈ Ker E Q iff σ is supported by a set of Φ -capacity zero. σ ∈ Im E Q iff σ = � k σ k with I σ k Φ < ∞ . Assume that Q ′ , Q ′′ come from two sequences { ℓ ′ n } and { ℓ ′′ n } . Let σ ′ , σ ′′ ∈ M ( T ) , σ ′′ ∈ Im E Q ′′ and σ ′ ≪ σ ′′ . Then (1) � | ℓ ′ ⇒ Q ′ σ ′ ⊥ Q ′′ σ ′′ . n − ℓ ′′ n | = ∞ = (2) � | ℓ ′ ⇒ Q ′ σ ′ ≪ Q ′′ σ ′′ . n − ℓ ′′ n | < ∞ = 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) E Q = E Q ′′ E Q ′ = E Q ′ E Q ′′ . (c) σ ∈ Im E Q ⇒ Q ′ σ ∈ Im E Q ′′ for almost all ω ′ ∈ Ω ′ . (d) σ ∈ Ker E Q ⇒ Q ′ σ ∈ Ker E Q ′′ for almost all ω ′ ∈ Ω ′ . dim Qλ = inf { τ > 0 : � n 2 − τ e ℓ 1 + ··· ℓ n = ∞} = 1 − lim sup ℓ 1 + ··· + ℓ n . log n NB Similar results for percolation on trees (Fan). Ai-Hua FAN TPWWT 16/26

L. Carleson problem Question When T is infinitely covered, how to describe the infinity ? A. H. Fan (1989) : two ways n � N n ( t ) := 1 (0 ,ℓ k ) ( t − ω k ) =? k =1 ∞ ∞ � � ∀ ( a n ) ⊂ R + , a n = ∞ , S ( t ) := a n 1 (0 ,ℓ k ) ( t − ω k ) = ∞ ? n =1 k =1 Fan-Kahane (1993) for ℓ n = 1+ ǫ : a.s. ∀ t, N n ( t ) ≈ log n ; n ∞ ∞ a n a n � � n = ∞ = ⇒ a.s. ∀ t, S ( t ) = ∞ ; n < ∞ = ⇒ a.s. ∀ t, S ( t ) < ∞ . n =1 n =1 Fan (2001), Barral-Fan (2005) : a.s. N n ( t ) multifractally behaves. Ai-Hua FAN TPWWT 17/26

Multifractality of N n ( t ) � � N n ( t ) F β := t ∈ T : lim ℓ 1 + · · · + ℓ n = β , n →∞ � n j =1 ℓ j α := lim sup − log ℓ n . n →∞ d α ( β ) := 1 + α ( β − 1 − β log β ) Theorem (Barral-Fan 2005) a (Slow like ℓ n = n log n ) If lim sup n →∞ nℓ n < ∞ and α = 0 , then a.s ∀ β ≥ 0 , dim( F β ) = 1 . (2) (Normal like ℓ n = a n ) If lim sup n →∞ nℓ n < ∞ and 0 < α < ∞ , then a.s ∀ β ≥ 0 ( d α ( β ) > 0) , dim( F β ) = d α ( β ) . (3) (Rapide like ℓ n = a log n ) If lim sup n →∞ nℓ n = ∞ , then n N n ( t ) a.s ∀ t ∈ T , lim = 1 . (4) ℓ 1 + · · · + ℓ n Ai-Hua FAN TPWWT 18/26