SRB measures for polygonal billiards with
contracting reflection laws
Gianluigi Del Magno
(joint with P. Duarte, J. Lopes Dias, J.P. Gaivão, D. Pinheiro)
UFBA, Salvador
Corinaldo, 22-06-15
Billiard
N
1. No friction
✓
✓+
2. Specular reflection law
θ− = θ+
2
Billiard map
(s2 , θ2 )
(s1 , θ1 )
• M = {(s, θ) : s ∈ [0, 1) and θ ∈ (−π/2, π/2)}
• Billiard map:
Φ(s1 , θ1 ) = (s2 , θ2 )
• invariant probability:
dµ = c cos θdsdθ
3
Singularities
The map Φ is piecewise smooth.
⇡
2
M
N
✓
⇡
2 0
s
1
• N singular set,
• Φ : M \ N → Φ(M \ N ) diffeomorphism.
4
Dynamics vs geometry
Chaotic
Chaotic
Non-chaotic
5
Polygonal billiards are not hyperbolic
For every polygon P , the billiard map ΦP has zero Lyapunov exponents:
χ− (x) = χ+ (x) = 0
a.e. x ∈ M.
Top Lyapunov exponent:
χ+ (x) = lim sup
n→+∞
1
log kDx ΦnP k.
n
Since µ is smooth,
χ− (x) = −χ+ (x)
a.e.
6
Contracting reflection law:
f : [−π/2, π/2] → [−π/2, π/2],
• f is a C 2 embedding
✓
• f (0) = 0
• 0 < f 0 (θ) ≤ λf < 1
for |θ| < π/2
Example: f (θ) = λf · θ,
N
✓+
θ+ = f (θ− )
0 < λf < 1.
7
Generalized billiard map:
ΦP,f = Rf ◦ ΦP ,
Rf (s, θ) = (s, f (θ))
• ΦP,f piecewise smooth map,
• ΦP,f (M ) ⊂ [0, 1) × −
• Maximal invariant set:
⇡
2
λf π λf π
2 , 2
ΛP,f :=
T
n
n≥0 ΦP,f (M )
M
f⇡
2
✓
⇤P,f
f⇡
⇡
2 0
2
s
1
8
Dominated splitting
Definition
A map Ψ has dominated splitting on an invariant set Σ if there exist
a continuous invariant splitting T Σ = E ⊕ F and constants 0 < µ < 1
and K > 0 such that
kDx Ψn |E k
≤ Kµn ,
x ∈ Σ, n ≥ 1.
kDx Ψn |F k
Examples:
• kDx Ψn |E k = 2−n ,
kDx Ψn |F k ≥ 2n (uniformly hyperbolic)
• kDx Ψn |E k ≤ 2−n ,
kDx Ψn |F k = 1
• kDx Ψn |E k = 2n ,
kDx Ψn |F k ≥ 3n
9
Theorem (Markarian, Pujals, Sambarino)
For every polygon P and every contracting reflection law f , the billiard
map ΦP,f has dominated splitting on ΛP,f :
Tx ΛP,f = E(x) ⊕ F (x),
kDx ΦnP,f |E k
≤ Kµn ,
kDx ΦnP,f |F k
x ∈ ΛP,f , n ≥ 1,
and F (x) = span[(1, 0)]
(horizontal direction θ = const.).
parallel trajectories remain
parallel
10
Proposition (DDDGP)
There exists a smooth norm | · | equivalent to k · k such that
|Dx Φn |E | ≤ λnf ,
|Dx Φn |F | =
n
Y
µi (x),
i=1
where
µi (x) =
cos θi+
cos θi−
=
cos f (θi− )
cos θi−
no expansion µi = 1
or weak expansion µi → 1
≥1
and
θi± = θ± (Φi (x)).
uniform expansion
µi · µi+1 ≥ µ > 1
11
Uniform hyperbolicity
Theorem (DDDGP)
For every polygon P without ‘parallel sides’ and every contracting
reflection law f , we have ΛP,f 6= ∅ and the splitting T ΛP,f = E ⊕ F is
uniformly hyperbolic: ∃ C > 0 and µ > 1 such that for n ≥ 1,
kDx ΦnP,f |E k ≤ Cµ−n ,
parallel sides
kDx ΦnP,f |F k ≥ Cµn .
no parallel sides
no parallel sides
12
Mechanism producing chaos
d
dispersion mechanism
d+
d−
=
d+
cos f (θ− )
cos θ−
≥1
13
Examples
Contracting reflection law f (θ) = λθ, 0 < λ < 1.
λ=0.3
λ=0.7
λ=0.65
λ=0.8
14
SRB measures
Question: Does Φ = ΦP,f have SRB measures?
e ⊂Λ .
• Local unstable manifold W u (x) exists ∀x ∈ Λ
P,f
W u (x) is a C 1 -curve such that 1) Φ−n (W u (x)) is a curve ∀n ≥ 1,
2) `(Φ−n (W u (x)) → 0 as n → +∞ exponentially fast.
• Unstable partition: ξ = {ξ(x)}, ξ(x) =

W u (x)
e
if x ∈ Λ,
{x}
otherwise.
• Conditional measures wrt ξ:
Z
ν(A) =
νxξ (A)dν(x)
∀ Borel A.
15
Definition
SRB measure ν is a Φ-invariant probability such that
νxξ Lebξu (x)
ν-a.e. x.
Basin of the measure ν:
X
1 n−1
ϕ(T k (x)) =
x ∈ M : lim
n→+∞ n
k=0
(
B(ν) =
+
)
Z
ϕdν
∀ϕ ∈ C(M ) .
16
Theorem (DDDG)
For every polygon P without ‘parallel sides’ and every contracting
reflection law f ,
1. ∃ ergodic SRB measures ν1 , . . . , νn ,
2. ν SRB =⇒ ν = α1 ν1 + · · · + αn νn with αi ≥ 0 and
P
i αi
= 1,
3. periodic orbits are dense in ΛP,f ,
4. B(ν1 ) ∪ · · · ∪ B(νn ) ⊂ M has full Lebesgue measure dsdθ.
17
5. spectral ergodic decomposition:
a) νi = νi,1 + · · · + νi,mi
b) Φ∗ νi,j = νi,j+1
c) (Φmi , ν̂i,j ) is Bernoulli
6. if νi is mixing, then νi has exponential decay of correlations:
∀ϕ, ψ Hölder, ∃ 0 < Λ(ϕ, ψ) < 1 and K(ϕ, ψ) > 0 such that
Z
Z
Z
ϕ(Φn (x))ψ(x))dνi − ϕ(x)dνi ψ(x)dνi ≤ Λn (ϕ, ψ)K(ϕ, ψ).
Used results of Pesin (92) and Sataev (92) for Parts 1-3 and 5. Ad hoc
proof for Part 4. Used Chernov-Zhang (99) for Part 5 .
18
General result on SRB measures (Pesin + Sataev)
Hypotheses:
H1: T : M \ N → M map with singular set N ,
H2: ∃ families of cones {C s } and {C u } on M \ N and constants C > 0
and λ > 1 such that
1. DT (C u ) ⊂ C u and DT −1 (C s ) ⊂ C s ,
2. kDx T −n (v)k ≤ Cλ−n kvk, v ∈ C s (x) and
kDx T n (v)k ≥ Cλn kvk, v ∈ C u (x),
H3: ∃K > 0, a ∈ (0, 1), b > 0 and 0 > 0 such that
LebW (W ∩ T −n (N )) ≤ Kb (an + LebW (W ))
for every u-manifold W , every n ≥ 1 and every ∈ (0, 0 ).
19
Theorem (Pesin (92), Sataev (92))
If T satisfies H1-H3, then ∃ ergodic SRB measures ν1 , . . . , νn concenT
trated on Λ := n≥0 T n (M ) with the following properties:
1. if ν is SRB, then ν = α1 ν1 + · · · + αn νn with αi ≥ 0 and
P
i αi
= 1,
2. B(νi ) has positive Lebesgue measure,
3. spectral ergodic decomposition of each νi ,
4. periodic points of T are dense in Λ.
20
Construction of SRB measures
• W = W u (x) local unstable manifold,
• ν0 probability measure on M ,
ν(A) = LebW (A ∩ W ),
A Borel set.
• sequence of probabilities:
µn =
X
1 n−1
T k ν.
n k=0 ∗
• Every weak-* limit point of {µn } is an SRB measure.
21
n-step expansion condition (for billiards)
W is a u-curve and W1 , . . . , Wm largest connected components of W
where Φn is smooth.
λn (Wi ) is the least expansion of Φn along Wi .
H(δ) is the set of horizontal segments of length ≤ δ.
n-step expansion (Chernov-Zhang): ∃n > 0 such that
lim inf sup
δ→0
X
W ∈H(δ) i
1
< 1.
λn (Wi )
22
Nk := N ∪ Φ−1 (N ) ∪ · · · ∪ Φ−k+1 (N ) singular set of Φk .
bk maximal number of smooth curves of the singular set of Φk meeting
at a point.
σk = Cλk least expansion rate inside C u of Φk .
Nk
Condition A: ∃n > 0 such that bn < σn
Condition A
=⇒
bk
n-step expansion
23
Theorem (DDDG)
Let P be a convex polygon P without π/2 angles. Then there exists
λP > 0 such that if f is a contracting reflection law f with λf ≤ λP ,
then bn grows linearly in n and Condition A is satisfied.
Theorem (DDDG)
The n-step condition is satisfied for every polygon (including nonconvex) and every contracting refection law.
24
Strongly contracting reflection laws
If λf 1, then the map ΦP,f is ‘close’ to the 1-dim map ΨP :
e
d
c
a
′
s
s
b
a
b
c d
e
a
ΨP piecewise linear expanding map.
25
Regular polygons
Ψ̂P reduced map.
Equilateral triangle
Ψ̂P (x) = −2x mod 1
Regular 2n + 1-gon
Ψ̂P = −βn (x − 1/2) mod 1
π
βn = −1/ cos 2n+1
26
Thank you!
27