Functional limit theorems for semi-dispersing billiards with cusps Fran¸ coise P` ene Univ Brest, IUF, LMBA, UMR CNRS 6205, France joint work with Paul Jung (KAIST, Daejon, south Corea) and Hong-Kun Zhang (UMASS, Amherst, USA) CIRM Thermodynamic Formalism: Ergodic Theory and Validated Numerics 12th July 2019 1/11 Fran¸ coise P` ene FCLT for Billiards with cusp
Functional limit theorems for i.i.d. random variables Let ( X k ) k be a sequence of centered R -valued i.i.d. random variables. ◮ If E [ X 2 1 ] < ∞ , then ⌊ nt ⌋ n − 1 L , J 1 � ∀ t 0 > 0 , X k n → + ∞ ( B t ) t ∈ [0 , t 0 ] , − → 2 k =1 t ∈ [0 , t 0 ] with ( B t ) t a Brownian motion. 2/11 Fran¸ coise P` ene FCLT for Billiards with cusp
Functional limit theorems for i.i.d. random variables Let ( X k ) k be a sequence of centered R -valued i.i.d. random variables. ◮ If E [ X 2 1 ] < ∞ , then ⌊ nt ⌋ n − 1 L , J 1 � ∀ t 0 > 0 , X k n → + ∞ ( B t ) t ∈ [0 , t 0 ] , − → 2 k =1 t ∈ [0 , t 0 ] with ( B t ) t a Brownian motion. ◮ If lim x →∞ x 2 P ( ± X 1 ≥ x ) = A ± , A + + A − > 0, then ⌊ nt ⌋ 1 L , J 1 � ∀ t 0 > 0 , √ n log n X k n → + ∞ ( B t ) t ∈ [0 , t 0 ] . − → k =1 t ∈ [0 , t 0 ] 2/11 Fran¸ coise P` ene FCLT for Billiards with cusp
Functional limit theorems for i.i.d. random variables Let ( X k ) k be a sequence of centered R -valued i.i.d. random variables. ◮ If E [ X 2 1 ] < ∞ , then ⌊ nt ⌋ n − 1 L , J 1 � ∀ t 0 > 0 , X k n → + ∞ ( B t ) t ∈ [0 , t 0 ] , − → 2 k =1 t ∈ [0 , t 0 ] with ( B t ) t a Brownian motion. ◮ If lim x →∞ x 2 P ( ± X 1 ≥ x ) = A ± , A + + A − > 0, then ⌊ nt ⌋ 1 L , J 1 � ∀ t 0 > 0 , √ n log n X k n → + ∞ ( B t ) t ∈ [0 , t 0 ] . − → k =1 t ∈ [0 , t 0 ] ◮ If ∃ α ∈ (1 , 2) s.t. lim x →∞ x α P ( ± X 1 ≥ x ) = A ± , A + + A − > 0, then ⌊ nt ⌋ n − 1 L , J 1 � ∀ t 0 > 0 , X k n → + ∞ ( Z t ) t ∈ [0 , t 0 ] , − → α k =1 t ∈ [0 , t 0 ] with ( Z t ) t an α -stable process (non continuous, but c` adl` ag: right continuous with left limit). 2/11 Fran¸ coise P` ene FCLT for Billiards with cusp
Billiard in a dispersing domain with cusps Q ⊂ R 2 , ∂ Q = ∪ i Γ i P i = Γ i ∩ Γ i +1 : ”corners” (distinct tangents) or ”cusps” (same tangent) Γ i curve C 3 ”convex” with non null curvature outside the cusps. Space M : set of unit reflected vectors v ) ∈ ∂ Q × S 1 : � � M := { x = ( q , � n q , � v � ≥ 0 } � n q : inward unit vector normal to ∂ Q at q Billiard map : T : M → M : T ( x )= next reflected vector 1 Invariant probability measure : µ with density ( q , � v ) �→ 2 | ∂ Q | sin( T q Q , � v ) 3/11 Fran¸ coise P` ene FCLT for Billiards with cusp
Billiard in a dispersive domain with/without cusp Let H η be the set of functions f : M → R η -H¨ older inside each Γ i . � Let f ∈ H η s.t. M f d µ = 0. ◮ If there is no cusp : ∀ t 0 > 0, �� ⌊ nt ⌋− 1 � f ◦ T k L , J 1 k =0 √ n n → + ∞ (Σ( f ) B t ) t ∈ [0 , t 0 ] , − → t ∈ [0 , t 0 ] B = ( B t ) t BM (brownian motion), Σ 2 ( f ) := � n ∈ Z Cov ( f , f ◦ T n ). [Sinai,70], [Young,98], [Chernov,99], [De Simoi & T´ oth,14] 4/11 Fran¸ coise P` ene FCLT for Billiards with cusp
Billiard in a dispersive domain with/without cusp Let H η be the set of functions f : M → R η -H¨ older inside each Γ i . � Let f ∈ H η s.t. M f d µ = 0. ◮ If there is no cusp : ∀ t 0 > 0, �� ⌊ nt ⌋− 1 � f ◦ T k L , J 1 k =0 √ n n → + ∞ (Σ( f ) B t ) t ∈ [0 , t 0 ] , − → t ∈ [0 , t 0 ] B = ( B t ) t BM (brownian motion), Σ 2 ( f ) := � n ∈ Z Cov ( f , f ◦ T n ). [Sinai,70], [Young,98], [Chernov,99], [De Simoi & T´ oth,14] ◮ Machta model : 3 pairwise tangent circles. [Machta,83], [Chernov & Markarian,07], [Chernov&Zhang,08]: mixing rate [B´ alint, Chernov & Dolgopyat,11] (1 cusp P ) : � � ⌊ nt ⌋− 1 � f ◦ T k L , J 1 k =0 n → + ∞ ( σ f B t ) t BM − → √ n log n t 1 2 d � � σ f := c . S 1 f ( P , � v ) | sin( T P Q , � v ) | v ; f ( P , � v ) := lim x → ( P ,� v ) , x ∈M f ( x ). 4/11 Fran¸ coise P` ene FCLT for Billiards with cusp
Billards with higher order cusps � f ∈ H η s.t. M f d µ = 0, with constant sign around each optimal cusp. ◮ [Jung & Zhang,18] (1 cusp P : z ± ( s ) = ± c 0 s β , β > 2, α := β β − 1 ) : α � n − 1 n → + ∞ σ f Z : E[e iuZ ] = e −| u | α − i sign( u ) tan πα L n − 1 k =0 f ◦ T k − → 2 1 α d � � σ f , P := c . S 1 f ( P , � v ) | sin( T P Q , � v ) | v 5/11 Fran¸ coise P` ene FCLT for Billiards with cusp
Billards with higher order cusps � f ∈ H η s.t. M f d µ = 0, with constant sign around each optimal cusp. ◮ [Jung & Zhang,18] (1 cusp P : z ± ( s ) = ± c 0 s β , β > 2, α := β β − 1 ) : α � n − 1 n → + ∞ σ f Z : E[e iuZ ] = e −| u | α − i sign( u ) tan πα L n − 1 k =0 f ◦ T k − → 2 1 α d � � σ f , P := c . S 1 f ( P , � v ) | sin( T P Q , � v ) | v ◮ [Jung, P. & Zhang,19+] If ◮ β i -Cusp in P i : z i , ± ( s ) = ± c i , ± s β i /β i + O � s 2 β i − 1 � , i , ± ( s ) = ± c i , ± s β i + O z ′ s 2 β i − 2 � � , with c i , ± ≥ 0 not both 0, ◮ T ( P i ) � = P j , ◮ β ∗ := max β i > 2, α := β ∗ β ∗ − 1 . 5/11 Fran¸ coise P` ene FCLT for Billiards with cusp
Billards with higher order cusps � f ∈ H η s.t. M f d µ = 0, with constant sign around each optimal cusp. β ◮ [Jung & Zhang,18] (1 cusp P : z ± ( s ) = ± c 0 s β , β > 2, α := β − 1 ) : L n − 1 α � n − 1 n → + ∞ σ f Z : E[e iuZ ] = e −| u | α − i sign( u ) tan πα k =0 f ◦ T k − → 2 1 α d � � σ f , P := c . S 1 f ( P , � v ) | sin( T P Q , � v ) | v ◮ [Jung, P. & Zhang,19+] If ◮ β i -Cusp in P i : z i , ± ( s ) = ± c i , ± s β i /β i + O s 2 β i − 1 � � , i , ± ( s ) = ± c i , ± s β i + O z ′ s 2 β i − 2 � � , with c i , ± ≥ 0 not both 0, ◮ T ( P i ) � = P j , ◮ β ∗ := max β i > 2, α := β ∗ β ∗ − 1 . � f ◦ T k � L , M 1 � � α � ⌊ nt ⌋− 1 i : β i = β ∗ σ f , P i Z ( i ) n − 1 Then − → Z t = � k =0 t n → + ∞ t t Z ( i ) independent α -stable processes with independent and stationary increments s.t. : Z ( i ) L 1 α Z , so that: = t t � �� �� sign( σ f , i ) | σ f , i | α tan πα | σ f , P i | α − i sign( u ) � E [ e iu Z t ] = exp − t | u | α 2 i i 5/11 Fran¸ coise P` ene FCLT for Billiards with cusp
Convergence for M 1 but not for J 1 ◮ d J 1 ( f , g ) (resp. d M 1 ( f , g ) ) : infimum of ℓ s.t. two ”ants” can travel one the graph of f and the other the one of g , staying ℓ -close one from the other, without turning back, jumping (resp. walking vertically) when they meet a discontinuity. d =1/n d =1/2 d =1/2 d =1/2 J1 J1 J1 J 1 d =1/n d =1/n d =1/n d =1/4 M M M1 M1 1 1 1 1-(1/n) 1/2 1/n 0 0 1/2 1 6/11 Fran¸ coise P` ene FCLT for Billiards with cusp
Convergence for M 1 but not for J 1 ◮ d J 1 ( f , g ) (resp. d M 1 ( f , g ) ) : infimum of ℓ s.t. two ”ants” can travel one the graph of f and the other the one of g , staying ℓ -close one from the other, without turning back, jumping (resp. walking vertically) when they meet a discontinuity. d =1/n d =1/2 d =1/2 d =1/2 J1 J1 J1 J 1 d =1/n d =1/n d =1/n d =1/4 M M M1 M1 1 1 1 1-(1/n) 1/2 1/n 0 0 1/2 1 k =0 f ◦ T k and � f � ∞ < ∞ : ◮ Since S n f = � n − 1 � � S ⌊ nt ⌋ f + ( nt − ⌊ nt ⌋ ) f ◦ T ⌈ nt ⌉ � S ⌊ nt ⌋ f � L , J 1 L , J 1 n → + ∞ ( Z t ) t ⇒ − → n → + ∞ ( Z t ) t − → 1 1 n n α α t t which would imply that a sequence of continuous process converges for J 1 to a discontinuous process. Impossible! 6/11 Fran¸ coise P` ene FCLT for Billiards with cusp
Proof of the functional limit theorem � Let f ∈ H η s.t. M f d µ = 0. ◮ Induced system ( M , ˜ µ, F ) M := { x = ( q , v ) ∈ M : d ( q , cusps ) ≥ ǫ } , µ := µ ( ·| M ), F ( x ) = T R ( x ) ( x ), R ( x ) := min { n ≥ 1 : T n ( x ) ∈ M } ˜ f ◦ T k and Set ˜ f := � R ( · ) − 1 k =0 N n ( x ) = # { k = 1 , ..., n − 1 : T k ( x ) ∈ M } ≈ n µ ( M ). 7/11 Fran¸ coise P` ene FCLT for Billiards with cusp
Proof of the functional limit theorem � Let f ∈ H η s.t. M f d µ = 0. ◮ Induced system ( M , ˜ µ, F ) M := { x = ( q , v ) ∈ M : d ( q , cusps ) ≥ ǫ } , µ := µ ( ·| M ), F ( x ) = T R ( x ) ( x ), R ( x ) := min { n ≥ 1 : T n ( x ) ∈ M } ˜ f ◦ T k and Set ˜ f := � R ( · ) − 1 k =0 N n ( x ) = # { k = 1 , ..., n − 1 : T k ( x ) ∈ M } ≈ n µ ( M ). N ⌊ nt ⌋ ( x ) − 1 ⌊ nt ⌋− 1 ⌊ n µ ( M ) t ⌋− 1 1 µ ( M ) α n − 1 � f ◦ T k ( x ) ≈ n − 1 � ˜ � ˜ f ◦ F k ( x ) ≈ f ◦ F k ( x ) α α 1 ( n µ ( M )) α k =0 k =1 k =1 7/11 Fran¸ coise P` ene FCLT for Billiards with cusp
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