[PPT] - Functional limit theorems for semi-dispersing billiards with cusps PowerPoint Presentation

SLIDE 1

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

SLIDE 2

Functional limit theorems for i.i.d. random variables

Let (Xk)k be a sequence of centered R-valued i.i.d. random variables. ◮ If E[X 2

1 ] < ∞, then

∀t0 > 0,  n− 1

2

⌊nt⌋

k=1

Xk  

t∈[0,t0] L, J1

− →

n→+∞ (Bt)t∈[0,t0] ,

with (Bt)t a Brownian motion.

2/11 Fran¸ coise P` ene FCLT for Billiards with cusp

SLIDE 3

Functional limit theorems for i.i.d. random variables

Let (Xk)k be a sequence of centered R-valued i.i.d. random variables. ◮ If E[X 2

1 ] < ∞, then

∀t0 > 0,  n− 1

2

⌊nt⌋

k=1

Xk  

t∈[0,t0] L, J1

− →

n→+∞ (Bt)t∈[0,t0] ,

with (Bt)t a Brownian motion. ◮ If limx→∞ x2P(±X1 ≥ x) = A±, A+ + A− > 0, then ∀t0 > 0,   1 √n log n

⌊nt⌋

k=1

Xk  

t∈[0,t0] L, J1

− →

n→+∞ (Bt)t∈[0,t0] .

2/11 Fran¸ coise P` ene FCLT for Billiards with cusp

SLIDE 4

Functional limit theorems for i.i.d. random variables

Let (Xk)k be a sequence of centered R-valued i.i.d. random variables. ◮ If E[X 2

1 ] < ∞, then

∀t0 > 0,  n− 1

2

⌊nt⌋

k=1

Xk  

t∈[0,t0] L, J1

− →

n→+∞ (Bt)t∈[0,t0] ,

with (Bt)t a Brownian motion. ◮ If limx→∞ x2P(±X1 ≥ x) = A±, A+ + A− > 0, then ∀t0 > 0,   1 √n log n

⌊nt⌋

k=1

Xk  

t∈[0,t0] L, J1

− →

n→+∞ (Bt)t∈[0,t0] .

◮ If ∃α ∈ (1, 2) s.t. limx→∞ xαP(±X1 ≥ x) = A±, A+ + A− > 0, then ∀t0 > 0,  n− 1

α

⌊nt⌋

k=1

Xk  

t∈[0,t0] L, J1

− →

n→+∞ (Zt)t∈[0,t0] ,

with (Zt)t an α-stable process (non continuous, but c` adl` ag: right continuous with left limit).

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SLIDE 5

Billiard in a dispersing domain with cusps

Q ⊂ R2, ∂Q = ∪iΓi Pi = Γi ∩ Γi+1: ”corners” (distinct tangents) or ”cusps” (same tangent) Γi curve C 3 ”convex” with non null curvature outside the cusps. Space M : set of unit reflected vectors M := {x = (q, v) ∈ ∂Q × S1 : nq, v ≥ 0}

nq: inward unit vector normal to ∂Q at q

Billiard map : T : M → M : T(x)= next reflected vector Invariant probability measure : µ with density (q, v) →

1 2|∂Q| sin(TqQ,

v)

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SLIDE 6

Billiard in a dispersive domain with/without cusp

Let Hη be the set of functions f : M → R η-H¨

lder inside each Γi.

Let f ∈ Hη s.t.

M f dµ = 0.

◮ If there is no cusp : ∀t0 > 0, ⌊nt⌋−1

k=0

f ◦ T k √n

t∈[0,t0]

L, J1

− →

n→+∞ (Σ(f )Bt)t∈[0,t0] ,

B = (Bt)t BM (brownian motion), Σ2(f ) :=

n∈Z Cov(f , f ◦ T n).

[Sinai,70], [Young,98], [Chernov,99], [De Simoi & T´

th,14]

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SLIDE 7

Billiard in a dispersive domain with/without cusp

Let Hη be the set of functions f : M → R η-H¨

lder inside each Γi.

Let f ∈ Hη s.t.

M f dµ = 0.

◮ If there is no cusp : ∀t0 > 0, ⌊nt⌋−1

k=0

f ◦ T k √n

t∈[0,t0]

L, J1

− →

n→+∞ (Σ(f )Bt)t∈[0,t0] ,

B = (Bt)t BM (brownian motion), Σ2(f ) :=

n∈Z Cov(f , f ◦ T n).

[Sinai,70], [Young,98], [Chernov,99], [De Simoi & T´

th,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

k=0

f ◦T k √n log n

t

L,J1

− →

n→+∞ (σf Bt)t BM

σf := c.

S1 f (P,

v) |sin(TPQ, v)|

1 2 d

v ; f (P, v) := limx→(P,

v), x∈M f (x).

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Billards with higher order cusps

f ∈ Hη s.t.

M f dµ = 0, with constant sign around each optimal cusp.

◮ [Jung & Zhang,18] (1 cusp P : z±(s) = ±c0sβ, β > 2, α :=

β β−1) :

n− 1

α n−1

k=0 f ◦ T k L

− →

n→+∞ σf Z : E[eiuZ] = e−|u|α−i sign(u) tan πα

2

σf ,P := c.

S1 f (P,

v) |sin(TPQ, v)|

1 α d

v

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Billards with higher order cusps

f ∈ Hη s.t.

M f dµ = 0, with constant sign around each optimal cusp.

◮ [Jung & Zhang,18] (1 cusp P : z±(s) = ±c0sβ, β > 2, α :=

β β−1) :

n− 1

α n−1

k=0 f ◦ T k L

− →

n→+∞ σf Z : E[eiuZ] = e−|u|α−i sign(u) tan πα

2

σf ,P := c.

S1 f (P,

v) |sin(TPQ, v)|

1 α d

v ◮ [Jung, P. & Zhang,19+] If

◮ βi-Cusp in Pi: zi,±(s) = ±ci,±sβi /βi + O

s2βi −1

, z′

i,±(s) = ±ci,±sβi + O

s2βi −2

, with ci,± ≥ 0 not both 0, ◮ T(Pi) = Pj, ◮ β∗ := max βi > 2, α :=

β∗ β∗−1.

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Billards with higher order cusps

f ∈ Hη s.t.

M f dµ = 0, with constant sign around each optimal cusp.

◮ [Jung & Zhang,18] (1 cusp P : z±(s) = ±c0sβ, β > 2, α :=

β β−1) :

n− 1

α n−1

k=0 f ◦ T k L

− →

n→+∞ σf Z : E[eiuZ] = e−|u|α−i sign(u) tan πα

2

σf ,P := c.

S1 f (P,

v) |sin(TPQ, v)|

1 α d

v ◮ [Jung, P. & Zhang,19+] If

◮ βi-Cusp in Pi: zi,±(s) = ±ci,±sβi /βi + O

s2βi −1

, z′

i,±(s) = ±ci,±sβi + O

s2βi −2

, with ci,± ≥ 0 not both 0, ◮ T(Pi) = Pj, ◮ β∗ := max βi > 2, α :=

β∗ β∗−1.

Then

n− 1

α ⌊nt⌋−1

k=0

f ◦ T k

t L, M1

− →

n→+∞

Zt =

i : βi=β∗ σf ,PiZ(i) t

t

Z(i) independent α-stable processes with independent and stationary increments s.t. : Z(i)

t L

= t

1 α Z, so that:

E[eiuZt] = exp

−t|u|α
i

|σf ,Pi|α − i sign(u)

i

sign(σf ,i)|σf ,i|α tan πα 2

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Fran¸ coise P` ene FCLT for Billiards with cusp

SLIDE 11

Convergence for M1 but not for J1

◮ dJ1(f , g)(resp. dM1(f , g)): infimum of ℓ s.t. two ”ants” can travel

ne the graph of f and the other the one of g, stayingℓ-close one

from the other, without turning back, jumping (resp. walking vertically) when they meet a discontinuity.

1 1/2 1 1-(1/n) 1/n J1

d =1/2 d =1/n

1/2 1 J

d =1/2

J1

d =1/n

M 1

d =1/n

M

d =1/4

M1 1

d =1/2

J1

d =1/n

M1

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SLIDE 12

Convergence for M1 but not for J1

◮ dJ1(f , g)(resp. dM1(f , g)): infimum of ℓ s.t. two ”ants” can travel

ne the graph of f and the other the one of g, stayingℓ-close one

from the other, without turning back, jumping (resp. walking vertically) when they meet a discontinuity.

1 1/2 1 1-(1/n) 1/n J1

d =1/2 d =1/n

1/2 1 J

d =1/2

J1

d =1/n

M 1

d =1/n

M

d =1/4

M1 1

d =1/2

J1

d =1/n

M1

◮ Since Snf = n−1

k=0 f ◦ T k and f ∞ < ∞ :

S⌊nt⌋f n

1 α

t

L, J1

− →

n→+∞ (Zt)t ⇒

S⌊nt⌋f + (nt − ⌊nt⌋)f ◦ T ⌈nt⌉

n

1 α

t

L, J1

− →

n→+∞ (Zt)t

which would imply that a sequence of continuous process converges for J1 to a discontinuous process. Impossible!

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Proof of the functional limit theorem

Let f ∈ Hη s.t.

M f dµ = 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} Set ˜ f := R(·)−1

k=0

f ◦ T k and Nn(x) = #{k = 1, ..., n − 1 : T k(x) ∈ M} ≈ nµ(M).

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SLIDE 14

Proof of the functional limit theorem

Let f ∈ Hη s.t.

M f dµ = 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} Set ˜ f := R(·)−1

k=0

f ◦ T k and Nn(x) = #{k = 1, ..., n − 1 : T k(x) ∈ M} ≈ nµ(M). n− 1

α

⌊nt⌋−1

k=0

f ◦T k(x) ≈ n− 1

α

N⌊nt⌋(x)−1

k=1

˜ f ◦F k(x) ≈ µ(M)

1 α

(nµ(M))

1 α

⌊nµ(M)t⌋−1

k=1

˜ f ◦F k(x)

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SLIDE 15

Proof of the functional limit theorem

Let f ∈ Hη s.t.

M f dµ = 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} Set ˜ f := R(·)−1

k=0

f ◦ T k and Nn(x) = #{k = 1, ..., n − 1 : T k(x) ∈ M} ≈ nµ(M). n− 1

α

⌊nt⌋−1

k=0

f ◦T k(x) ≈ n− 1

α

N⌊nt⌋(x)−1

k=1

˜ f ◦F k(x) ≈ µ(M)

1 α

(nµ(M))

1 α

⌊nµ(M)t⌋−1

k=1

˜ f ◦F k(x) ◮ We prove that  n− 1

α

⌊nt⌋−1

k=0

˜ f ◦ F k  

t L, J1

− →

n→+∞

Zt/µ(M)

1 α

t

7/11 Fran¸ coise P` ene FCLT for Billiards with cusp

SLIDE 16

Proof of the functional limit theorem

Let f ∈ Hη s.t.

M f dµ = 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} Set ˜ f := R(·)−1

k=0

f ◦ T k and Nn(x) = #{k = 1, ..., n − 1 : T k(x) ∈ M} ≈ nµ(M). n− 1

α

⌊nt⌋−1

k=0

f ◦T k(x) ≈ n− 1

α

N⌊nt⌋(x)−1

k=1

˜ f ◦F k(x) ≈ µ(M)

1 α

(nµ(M))

1 α

⌊nµ(M)t⌋−1

k=1

˜ f ◦F k(x) ◮ We prove that  n− 1

α

⌊nt⌋−1

k=0

˜ f ◦ F k  

t L, J1

− →

n→+∞

Zt/µ(M)

1 α

t

◮ If the sign of f is constant around each cusp, we conclude by a theorem by [Melbourne & Zweim¨ uller,15] that

n− 1

α ⌊nt⌋−1

k=0

f ◦ T k

t L, M1

− →

n→+∞ (Zt)t.

7/11 Fran¸ coise P` ene FCLT for Billiards with cusp

SLIDE 17

Proof of the functional limit theorem

Let f ∈ Hη s.t.

M f dµ = 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} Set ˜ f := R(·)−1

k=0

f ◦ T k and Nn(x) = #{k = 1, ..., n − 1 : T k(x) ∈ M} ≈ nµ(M). n− 1

α

⌊nt⌋−1

k=0

f ◦T k(x) ≈ n− 1

α

N⌊nt⌋(x)−1

k=1

˜ f ◦F k(x) ≈ µ(M)

1 α

(nµ(M))

1 α

⌊nµ(M)t⌋−1

k=1

˜ f ◦F k(x) ◮ We prove that  n− 1

α

⌊nt⌋−1

k=0

˜ f ◦ F k  

t L, J1

− →

n→+∞

Zt/µ(M)

1 α

t

◮ If the sign of f is constant around each cusp, we conclude by a theorem by [Melbourne & Zweim¨ uller,15] that

n− 1

α ⌊nt⌋−1

k=0

f ◦ T k

t L, M1

− →

n→+∞ (Zt)t.

◮ We conclude the finite dimensional distributions convergence without any assumption on the sign of f .

7/11 Fran¸ coise P` ene FCLT for Billiards with cusp

SLIDE 18

Proof of the limit theorem for the induced system

n− 1

α ⌊nt⌋−1

k=0

˜ f ◦ F k

t L, J1

− →

n→+∞

Zt/µ(M)

1 α =

i : βi=β∗ σf ,iZ(i) t /µ(M)

1 α

t

8/11 Fran¸ coise P` ene FCLT for Billiards with cusp

SLIDE 19

Proof of the limit theorem for the induced system

n− 1

α ⌊nt⌋−1

k=0

˜ f ◦ F k

t L, J1

− →

n→+∞

Zt/µ(M)

1 α =

i : βi=β∗ σf ,iZ(i) t /µ(M)

1 α

t

1. ˜ f ≈ Z0 :=

i

Af ,i(R1Mi − ˜ µ(R1Mi)) , Mi:=T −1(d(q, cusp i) < ǫ)

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SLIDE 20

Proof of the limit theorem for the induced system

n− 1

α ⌊nt⌋−1

k=0

˜ f ◦ F k

t L, J1

− →

n→+∞

Zt/µ(M)

1 α =

i : βi=β∗ σf ,iZ(i) t /µ(M)

1 α

t

1. ˜ f ≈ Z0 :=

i

Af ,i(R1Mi − ˜ µ(R1Mi)) , Mi:=T −1(d(q, cusp i) < ǫ)

2. prove the convergence of

 n− 1

α

⌊nt⌋−1

k=0

Z0 ◦ F k  

t

. Due to a theorem of [Tyran-Kami´ nska,10], it is enough to prove:

8/11 Fran¸ coise P` ene FCLT for Billiards with cusp

SLIDE 21

Proof of the limit theorem for the induced system

n− 1

α ⌊nt⌋−1

k=0

˜ f ◦ F k

t L, J1

− →

n→+∞

Zt/µ(M)

1 α =

i : βi=β∗ σf ,iZ(i) t /µ(M)

1 α

t

1. ˜ f ≈ Z0 :=

i

Af ,i(R1Mi − ˜ µ(R1Mi)) , Mi:=T −1(d(q, cusp i) < ǫ)

2. prove the convergence of

 n− 1

α

⌊nt⌋−1

k=0

Z0 ◦ F k  

t

. Due to a theorem of [Tyran-Kami´ nska,10], it is enough to prove:

2.1

n−1

k=0

δ

j n , Z0◦Fj n 1 α

L

− →

n→+∞PPP

 (t, y) → α|y|−α−1 µ(M)

i : βi =β∗

|σf ,i|α1{yσf ,i >0}   2.2 limε→0 lim supn→+∞

maxk
k

j=0 ˜

Zj,n,ε

2 = 0, with

˜ Zj,n,ε :=

Zj n

1 α 1

|Zj |<εn

1 α − E˜

µ[...]

8/11 Fran¸ coise P` ene FCLT for Billiards with cusp

SLIDE 22

Proof of the limit theorem for the induced system

n− 1

α ⌊nt⌋−1

k=0

˜ f ◦ F k

t L, J1

− →

n→+∞

Zt/µ(M)

1 α =

i : βi=β∗ σf ,iZ(i) t /µ(M)

1 α

t

1. ˜ f ≈ Z0 :=

i

Af ,i(R1Mi − ˜ µ(R1Mi)) , Mi:=T −1(d(q, cusp i) < ǫ)

2. prove the convergence of

 n− 1

α

⌊nt⌋−1

k=0

Z0 ◦ F k  

t

. Due to a theorem of [Tyran-Kami´ nska,10], it is enough to prove:

2.1

n−1

k=0

δ

j n , Z0◦Fj n 1 α

L

− →

n→+∞PPP

 (t, y) → α|y|−α−1 µ(M)

i : βi =β∗

|σf ,i|α1{yσf ,i >0}   proof : ˜ µ(R > y) ∼ Cy −α + use a theorem of [P. & Saussol,19+]. 2.2 limε→0 lim supn→+∞

maxk
k

j=0 ˜

Zj,n,ε

2 = 0, with

˜ Zj,n,ε :=

Zj n

1 α 1

|Zj |<εn

1 α − E˜

µ[...]

proof : check a criterion of [Billingsley,99].

8/11 Fran¸ coise P` ene FCLT for Billiards with cusp

SLIDE 23

Probability of a long excursion in a cusp area

Cusp with horizontal tangent line : z±(s) ≈ ±c±sβ/β, α =

β β−1

◮ Key estimate for long excursions : Let x ∈ M with R(x) = N and with successive reflections: T n(x) =

(sn, z±(sn)), ei(π±vn)

, n = 1, ...N − 1 Then sβ

n sin vn = c N−α + O(s2β−1 n

) .

9/11 Fran¸ coise P` ene FCLT for Billiards with cusp

SLIDE 24

Probability of a long excursion in a cusp area

Cusp with horizontal tangent line : z±(s) ≈ ±c±sβ/β, α =

β β−1

◮ Key estimate for long excursions : Let x ∈ M with R(x) = N and with successive reflections: T n(x) =

(sn, z±(sn)), ei(π±vn)

, n = 1, ...N − 1 Then sβ

n sin vn = c N−α + O(s2β−1 n

) . ◮ Probability of a long excursion : µ

m≥y

m

n=0 T n(R = m)

∼µ
(s, v) : s ≤ c y 1−α

sin

1 β v

=c′y 1−α.

9/11 Fran¸ coise P` ene FCLT for Billiards with cusp

SLIDE 25

Probability of a long excursion in a cusp area

Cusp with horizontal tangent line : z±(s) ≈ ±c±sβ/β, α =

β β−1

◮ Key estimate for long excursions : Let x ∈ M with R(x) = N and with successive reflections: T n(x) =

(sn, z±(sn)), ei(π±vn)

, n = 1, ...N − 1 Then sβ

n sin vn = c N−α + O(s2β−1 n

) . ◮ Probability of a long excursion : µ

m≥y

m

n=0 T n(R = m)

∼µ
(s, v) : s ≤ c y 1−α

sin

1 β v

=c′y 1−α.

So  

m≥y

µ(R ≥ m)   + yµ(R ≥ y) =

m≥y

(m + 1)µ(R = m) ∼ c′y 1−α.

9/11 Fran¸ coise P` ene FCLT for Billiards with cusp

SLIDE 26

Probability of a long excursion in a cusp area

Cusp with horizontal tangent line : z±(s) ≈ ±c±sβ/β, α =

β β−1

◮ Key estimate for long excursions : Let x ∈ M with R(x) = N and with successive reflections: T n(x) =

(sn, z±(sn)), ei(π±vn)

, n = 1, ...N − 1 Then sβ

n sin vn = c N−α + O(s2β−1 n

) . ◮ Probability of a long excursion : µ

m≥y

m

n=0 T n(R = m)

∼µ
(s, v) : s ≤ c y 1−α

sin

1 β v

=c′y 1−α.

So  

m≥y

µ(R ≥ m)   + yµ(R ≥ y) =

m≥y

(m + 1)µ(R = m) ∼ c′y 1−α. We do not know if (m + 1)µ(R = m) is decreasing, so we cannot conclude by usual Tauberian theorems.

9/11 Fran¸ coise P` ene FCLT for Billiards with cusp

SLIDE 27

Probability of a long excursion in a cusp area

Cusp with horizontal tangent line : z±(s) ≈ ±c±sβ/β, α =

β β−1

◮ Key estimate for long excursions : Let x ∈ M with R(x) = N and with successive reflections: T n(x) =

(sn, z±(sn)), ei(π±vn)

, n = 1, ...N − 1 Then sβ

n sin vn = c N−α + O(s2β−1 n

) . ◮ Probability of a long excursion : µ

m≥y

m

n=0 T n(R = m)

∼µ
(s, v) : s ≤ c y 1−α

sin

1 β v

=c′y 1−α.

So  

m≥y

µ(R ≥ m)   + yµ(R ≥ y) =

m≥y

(m + 1)µ(R = m) ∼ c′y 1−α. We do not know if (m + 1)µ(R = m) is decreasing, so we cannot conclude by usual Tauberian theorems. But, by an analogous argument, we conclude µ (R ≥ y) ∼ β−1c′y −α .

9/11 Fran¸ coise P` ene FCLT for Billiards with cusp

SLIDE 28

˜ f (x) ≈ cR(x)

Cusp with horizontal tangent line : z±(s) ≈ ±c±sβ/β, α =

β β−1

◮ Key estimate for long excursions : Let x ∈ M with R(x) = N and with successive reflections: T n(x) =

(sn, z±(sn)), ei(π±vn)

, n = 1, ...N − 1 Then sβ

n sin vn = c N−α + O(s2β−1 n

) .

10/11 Fran¸ coise P` ene FCLT for Billiards with cusp

SLIDE 29

˜ f (x) ≈ cR(x)

Cusp with horizontal tangent line : z±(s) ≈ ±c±sβ/β, α =

β β−1

◮ Key estimate for long excursions : Let x ∈ M with R(x) = N and with successive reflections: T n(x) =

(sn, z±(sn)), ei(π±vn)

, n = 1, ...N − 1 Then sβ

n sin vn = c N−α + O(s2β−1 n

) . ◮ Approximation of ˜ f (x) = R(x)−1

k=0

f (T k(x)) by R(x) : (vk+2 − vk) ∼ (c+ + c−)sβ−1

k

∼˜ c/(R(x)| sin vk|

1 α ). So, ∀n < R(x): 10/11 Fran¸ coise P` ene FCLT for Billiards with cusp

SLIDE 30

˜ f (x) ≈ cR(x)

Cusp with horizontal tangent line : z±(s) ≈ ±c±sβ/β, α =

β β−1

◮ Key estimate for long excursions : Let x ∈ M with R(x) = N and with successive reflections: T n(x) =

(sn, z±(sn)), ei(π±vn)

, n = 1, ...N − 1 Then sβ

n sin vn = c N−α + O(s2β−1 n

) . ◮ Approximation of ˜ f (x) = R(x)−1

k=0

f (T k(x)) by R(x) : (vk+2 − vk) ∼ (c+ + c−)sβ−1

k

∼˜ c/(R(x)| sin vk|

1 α ). So, ∀n < R(x):

n−1

k=1

f (T k(x)) ∼

n/2

k=0

(f (P, ei(π+v2k+1)) + f (P, ei(π−v2k+1)) ∼

n/2

k=0

(f (P, ei(π+v2k+1)) + f (P, ei(π−v2k+1))R(x)| sin v2k+1|

1 α (v2k+3 − v2k+1)

˜ c ∼ R(x) ˜ c

[0,vn]

(f (P, ei(π−v)) + f (P, e−i(π−v)))| sin v|

1 α dv 10/11 Fran¸ coise P` ene FCLT for Billiards with cusp

SLIDE 31

Main theorem and other references

Let f ∈ Hγ,

M f dµ = 0.

◮ [Jung, P. & Zhang,19+] If sign(f ) is constant around each optimal cusp, then

n− 1

α ⌊nt⌋−1

k=0

f ◦ T k

t L, M1

− →

n→+∞ (Zt)t.

We have seen that : If x ∈ M and if R(x) is large : ∀n ≤ R(x),

n−1

k=1

f (T k(x)) ∼ R(x) ˜ c−1If (vn), with If (s) :=

[0,s]

(f (P, ei(π−v)) + f (P, e−i(π−v)))| sin v|

1 α dv .

◮ [Melbourne & Varandas,19+] (1 cusp)

◮

L, M1

− →

n→+∞ iff If is monotonous. (via (n− 1

α nt−1

k=0 ˜

f ◦ F k)t

L, M1

− →

n→+∞)

◮

L, M2

− →

n→+∞ if ∀v ∈ [0, π], If (v) ∈ [0, If (π)].

◮ [Jung,Melbourne,P` ene,Varandas & Zhang,19+] if ∃v ∈ [0, π] s.t. If (v) ∈ [0, If (π)], then

L, M2

− →

n→+∞.

11/11 Fran¸ coise P` ene FCLT for Billiards with cusp