1
.
On Topological Entropy of Billiard Tables with Small
Inner Scatterers
Yi-Chiuan Chen
Institute of Mathematics, Academia Sinica, Taiwan
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In 2002, Foltin proved that billiard flows on strictly convex tables with
sufficiently small inner circular scatterers generically possess positive
topological entropy. (Foltin 2002 Nonlinearity )
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Foltin’s essential idea: (in one scatterer case)
• Generically, there exist two period-2 orbits that induce in a subset of
the phase space ∂B × ∂M × [−π/2, π/2] a Markov partition with the
Markov graph below.
a
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11
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D
Y.-C. Chen / Chaos, Solitons and Fractals 28 (2006) 377–385
b
C
A
A
B
B
~ with four symbols and (b) the full shift with two symbols represented by M
Fig. 3. (a) A subshift of finite type r
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Objective:
Present a different approach to Foltin’s result: via the anti-integrable
limit.
Aubry & Abramovici 1990 Physica D
MacKay & Meiss 1992 Nonlinearity
Bolotin & MacKay 2000 Cel Mech Dyn Astron
Chen 2004 Dynamical Systems
Chen 2005 DCDS B
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Let e1, e2 ∈ {1, . . . , K}. A piecewise straight path Γ is called a n-link
basic anti-integrable (AI) orbit from one point scatterer Oe1 to another
Oe2 if it is a segment of billiard orbit starting from Oe1 having exactly n
number of bounces with the boundary ∂M before reaching Oe2 .
Ω3
O1
Ω1
7
O
1
Let e1, e2 ∈ {1, . . . , K}. A piecewise straight
path Γ is called a n-link
basic anti-integrable (AI) orbit from one point scatterer Oe1 to another
Oe2 if it is a segment of billiard orbit starting from Oe1 having exactly n
number of bounces with the boundary ∂M before reaching Oe2 .
Ω3
O1
Ω1
O1
Ω2
2
Ω1
7
O
1
Let e1, e2 ∈ {1, . . . , K}. A piecewise straight
path Γ is called a n-link
basic anti-integrable (AI) orbit from one point scatterer Oe1 to another
Oe2 if it is a segment of billiard orbit starting from Oe1 having exactly n
number of bounces with the boundary ∂M before reaching Oe2 .
Ω3
O1
Ω1
O1
O2
Ω9
Ω2
2
Ω1
O1
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Existence of non-degenerate basic AI-orbits
• The existence of a 0-link basic AI-orbit is obvious.
O1
O2
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O1
O2
Existence of non-degenerate basic AI-orbits
Define h1 : ∂M → R,
h1(φ) := h(Oe1 , φ) + h(φ, Oe2 ).
O1
O2
φ
5
11
O1
O2
φ
Existence
of non-degenerate
basic AI-orbits
Define h2 : (∂M )2 → R,
h2(φ1, φ2) := h(Oe1 , φ1) + h(φ1, φ2) + h(φ2, Oe2 ).
O1
O2
φ1
φ2
6
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Kozlov & Treshchev 1991 Billiards
Q2, := {(φ1, φ2) ∈ (∂M )2 : |φ1 − φ2| > }.
• Q2, is homeomorphic to the product of the circle T and the
one-dimensional disc.
• h2 is C 2 on Q2,.
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Kozlov & Treshchev 1991 Billiards
Q2, := {(φ1, φ2) ∈ (∂M )2 : |φ1 − φ2| > }.
• Q2, is homeomorphic to the product of the circle T and the
one-dimensional disc.
• h2 is C 2 on Q2,.
Proposition 2. The function h2 attains at least two critical values on
Q2, for sufficiently small . At least one of the two critical values
is a maximum, but not all of the critical values of h2 on Q2, are
isolated maxima. Hence there are at least two geometrically distinct
non-degenerate 2-link basic AI-orbits for generic setting of (M, O).
Proof: The vector grad h2 on the boundary of Q2, is directed
inwards for sufficiently small , then invoke the Lyusternik-Shnirel’man
theory.
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†
• For every i, define yi−1
:= Γei t ∂Uei−1 , x†i := Γei t ∂Uei , where the
symbol t means “perpendicular” intersection with the auxiliary circles
∂Uei .
∂U1
†
y−1
x†0
y0†
7
∂U2
x†1
13
†
• For every i, define yi−1
:= Γei t ∂Uei−1 , x†i := Γei t ∂Uei , where the
symbol t means “perpendicular” intersection with the auxiliary circles
∂Uei .
∂U1
†
y−1
x†0
y0†
∂U2
x†1
• Let h∗ be the length of of the “coloured” piecewise straight path.
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h− (xi , yi , ρ) :=
h(Ψi , yi ),
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h+ (xi , yi , ρ) :=
i , Ψi ).
Smoothing broken h(x
billiard
orbits
Recall that ρ = (ρ1 , . . . , ρK ) and notice that ρei ∈ {ρ1 , . . . , ρK } ∀ i ∈ Z.
∂U0
y−1
∂U1
β
x0
Ψ0
(a)
y0
Ψ1
O1
†
y−1
x†0
y0†
x1
O2
x†1
(b)
• h∗ = h∗(yi−1, xi)
Figure 3: (a) Broken orbits determined by xi and yi . (b) One 0-link basic AIorbit connect O1 and O2 , two 1-link basic AI-orbits (one connecting O1 to itself,
the other connecting to O2 , one 2-link basic AI-orbit connecting O1 to itself.
Suppose we have a sequence of pair points xi and yi , i ∈ Z, such that xi
is connected backwards to yi−1 and forwards to yi by segments of orbits as
described in the preceding paragraph. Gluing together these segments of orbits,
h− (xi , yi , ρ) :=
h(Ψi , yi ),
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h+ (xi , yi , ρ) :=
i , Ψi ).
Smoothing broken h(x
billiard
orbits
Recall that ρ = (ρ1 , . . . , ρK ) and notice that ρei ∈ {ρ1 , . . . , ρK } ∀ i ∈ Z.
∂U0
y−1
∂U1
β
x0
Ψ0
y0
Ψ1
O1
†
y−1
x†0
y0†
x1
(a)
O2
x†1
(b)
• h∗ = h∗(yi−1, xi)
• h+Figure
= h+3:(x(a)
= |xdetermined
Broken
by xi
i, y
i, ρei )orbits
i − Ψi|
and yi . (b) One 0-link basic AIorbit connect O1 and O2 , two 1-link basic AI-orbits (one connecting O1 to itself,
the other connecting to O2 , one 2-link basic AI-orbit connecting O1 to itself.
Suppose we have a sequence of pair points xi and yi , i ∈ Z, such that xi
is connected backwards to yi−1 and forwards to yi by segments of orbits as
described in the preceding paragraph. Gluing together these segments of orbits,
h− (xi , yi , ρ) :=
h(Ψi , yi ),
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h+ (xi , yi , ρ) :=
i , Ψi ).
Smoothing broken h(x
billiard
orbits
Recall that ρ = (ρ1 , . . . , ρK ) and notice that ρei ∈ {ρ1 , . . . , ρK } ∀ i ∈ Z.
∂U0
y−1
∂U1
β
x0
Ψ0
(a)
y0
Ψ1
O1
†
y−1
x†0
y0†
x1
O2
x†1
(b)
• h∗ = h∗(yi−1, xi)
• h+Figure
= h+3:(x(a)
= |xdetermined
Broken
by xi and yi . (b) One 0-link basic AIi, y
i, ρei )orbits
i − Ψi|
+
AI-orbits (one connecting O1 to itself,
2 , two
• h−orbit
= hconnect
(xi, O
yi1, and
ρei )O=
|yi 1-link
− Ψibasic
|
the other connecting to O2 , one 2-link basic AI-orbit connecting O1 to itself.
Suppose we have a sequence of pair points xi and yi , i ∈ Z, such that xi
is connected backwards to yi−1 and forwards to yi by segments of orbits as
described in the preceding paragraph. Gluing together these segments of orbits,
h (xi , yi , ρ) :=
h(Ψi , yi ),
h+ (xi , yi , ρ) :=
h(xi , Ψi ).
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notice that billiard
ρei ∈ {ρ1 , . . orbits
. , ρK } ∀ i ∈ Z.
Recall that ρ Smoothing
= (ρ1 , . . . , ρK ) andbroken
∂U0
y−1
∂U1
β
x0
Ψ0
y0
Ψ1
O1
†
y−1
x†0
y0†
x1
(a)
O2
x†1
(b)
Define a map F (·, ρ) = {Fi(·, ρ)}i∈Z on
Q
i∈Z(∂Uei−1
× ∂Uei ) by
Figure 3: (a) Broken orbits
+ determined by xi and yi .−(b) One 0-link basic AIForbit
ρ) :=O1 D
(xi−1
, yi−1
, ρAI-orbits
h (x
)
i(z,connect
zi (h
ei−1 ) + (one
i−1, yi−1
and
O2 , two
1-link
basic
connecting
O,1 ρtoei−1
itself,
AI-orbit
O1 to
the other connecting+h
to ∗O(y
2 , one,2-link
x ) +basic
h+(x
, y , ρconnecting
) + h−(x
, yitself.
, ρ )),
i−1
i
i
i
ei
i
i
ei
Suppose we have a sequence of pair points xi and yi , i ∈ Z, such that xi
where
z = {zi}backwards
, xand
i∈Z := {(y
i)}i∈
Z.
is connected
toi−1
yi−1
forwards
to yi by segments of orbits as
described in the preceding paragraph. Gluing together these segments of orbits,
we get broken billiard orbits with broken points xi and yi . Then consider the
h− (xi , yi , ρ) :=
h(Ψi , yi ),
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h+ (xi , yi , ρ) :=
i , Ψi ).
Smoothing broken h(x
billiard
orbits
Recall that ρ = (ρ1 , . . . , ρK ) and notice that ρei ∈ {ρ1 , . . . , ρK } ∀ i ∈ Z.
∂U0
y−1
∂U1
β
x0
Ψ0
y0
(a)
Ψ1
O1
†
y−1
x†0
O2
y0†
x1
x†1
(b)
Proposition 3. For non-zero ρ, a zero {(yi−1, xi)}i of F (·, ρ) corresponds to a unique orbit connecting points in the order . . ., x0, Ψ0,
Figure 3: (a) Broken orbits determined by xi and yi . (b) One 0-link basic AIy0, xorbit
.. O , two 1-link basic AI-orbits (one connecting O to itself,
1, Ψ
1, y1,O. . and
connect
1
2
1
the other connecting to O2 , one 2-link basic AI-orbit connecting O1 to itself.
Suppose we have a sequence of pair points xi and yi , i ∈ Z, such that xi
is connected backwards to yi−1 and forwards to yi by segments of orbits as
described in the preceding paragraph. Gluing together these segments of orbits,
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Theorem 5.
• Suppose every n-link basic AI-orbit is non-degenerate for n ≤ 2.
†
• Given a 2-AI-orbit determined by z † = {(yi−1
, x†i )}i∈Z, on each ∂Uei
there exist subsets ∆yi containing yi†, ∆xi containing x†i , and exists
Q
1
ρ0 > 0 such that F is C on the subset i∈Z(∆yi−1 × ∆xi ) × [0, ρ0)K .
• Moreover, F (·, ρ) has a unique simple zero on the subset, in particular, F (z †, 0) = 0.
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Remarks:
• 3-link, 4-link, 5-link, . . ., n-link AI-orbits;
larger topological entropy for smaller inner scatterers
• replace circular scatterers by convex ones
x̂a
x̂
U
x̂c
Aρ
O
ψ
β
x
y
Figure 7: Local orbits inside U . Aρ = dark-shaded domain, Bρ = light-shaded
domain.
Moreover, if x and y satisfy the above inequality when ρ = ρ0 then they remain
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Theorem 6. For any positive integer N , there is ρ0 and there is an
open and dense subset of (M, O) whose billiard flow with strictly convex inner scatterers of diameters ρ1, . . . , ρK has topological entropy at
least N if max1≤e≤K {ρe} < ρ0.