Dynamics of the horocycle flow for homogeneous
and non-homogeneous foliations by hyperbolic
surfaces∗
Fernando Alcalde Cuesta
GeoDynApp - ECSING group
Foliations 2014 - ICMT Madrid - September 1-5, 2014
∗
Joint work with Françoise Dal’Bo, Matilde Martı́nez, and Alberto Verjovsky.
Dynamics of the horocycle flow
Aim: Study of the dynamics of the horocycle flow on compact laminated spaces by hyperbolic surfaces, which was initiated by Matilde
Martı́nez and Alberto Verjovsky.
Dynamics of the horocycle flow
Aim: Study of the dynamics of the horocycle flow on compact laminated spaces by hyperbolic surfaces, which was initiated by Matilde
Martı́nez and Alberto Verjovsky.
Hedlund’s Theorem on the minimality of the horocycle flow
on the unitary tangent bundle X = Γ \PSL(2, R) of a compact
hyperbolic surface S = Γ \H.
Dynamics of the horocycle flow
Aim: Study of the dynamics of the horocycle flow on compact laminated spaces by hyperbolic surfaces, which was initiated by Matilde
Martı́nez and Alberto Verjovsky.
Hedlund’s Theorem on the minimality of the horocycle flow
on the unitary tangent bundle X = Γ \PSL(2, R) of a compact
hyperbolic surface S = Γ \H.
Hedlund’s Theorem for X = Γ \PSL(2, R) × G where G is a
connected Lie group and Γ is a cocompact discrete subgroup
of PSL(2, R) × G
Dynamics of the horocycle flow
Aim: Study of the dynamics of the horocycle flow on compact laminated spaces by hyperbolic surfaces, which was initiated by Matilde
Martı́nez and Alberto Verjovsky.
Hedlund’s Theorem on the minimality of the horocycle flow
on the unitary tangent bundle X = Γ \PSL(2, R) of
OFa compact
O
R
hyperbolic surface S = Γ \H.
P
RY
Hedlund’s Theorem for N
XT
=A
Γ \PSL(2, R) × G where G is a
E
M
E and Γ is a cocompact discrete subgroup of
connected Lie
ELgroup
PSL(2, R) × G .
Dynamics of the horocycle flow
Aim: Study of the dynamics of the horocycle flow on compact laminated spaces by hyperbolic surfaces, which was initiated by Matilde
Martı́nez and Alberto Verjovsky.
Hedlund’s Theorem on the minimality of the horocycle flow
on the unitary tangent bundle X = Γ \PSL(2, R) of
OFa compact
O
R
hyperbolic surface S = Γ \H.
P
RY
Hedlund’s Theorem for N
XT
=A
Γ \PSL(2, R) × G where G is a
E
M
E and Γ is a cocompact discrete subgroup of
connected Lie
ELgroup
PSL(2, R) × G .
The foliation point of view - the homogeneous case.
Dynamics of the horocycle flow
Aim: Study of the dynamics of the horocycle flow on compact laminated spaces by hyperbolic surfaces, which was initiated by Matilde
Martı́nez and Alberto Verjovsky.
Hedlund’s Theorem on the minimality of the horocycle flow
on the unitary tangent bundle X = Γ \PSL(2, R) of
OFa compact
O
R
hyperbolic surface S = Γ \H.
P
RY
Hedlund’s Theorem for N
XT
=A
Γ \PSL(2, R) × G where G is a
E
M
E and Γ is a cocompact discrete subgroup of
connected Lie
ELgroup
PSL(2, R) × G .
The foliation point of view - the homogeneous case.
Some progress in the non-homogeneous case
Dynamics of the horocycle flow
Hedlund’s Theorem
Description and statement
1 Hedlund’s Theorem
Description and statement
Proof
2 Hedlund’s Theorem for X = Γ \PSL(2, R) × G
PSL(2, R)-minimality and B-minimality
PSL(2, R)-minimality ⇒ B-minimality
B-minimality ⇒ U-minimality
3 The foliation point of view
Lie foliations
Martı́nez and Verjovsky’s question
4 Some progress in the non-homogeneous case
A classical example
Foliations with ’topologically non-trivial’ leaves
Dynamics of the horocycle flow
Hedlund’s Theorem
Description and statement
Hyperbolic surface S = Γ \H
Dynamics of the horocycle flow
Hedlund’s Theorem
Description and statement
Hyperbolic surface S = Γ \H
− H = { z ∈ C | Im(z) > 0 }
Dynamics of the horocycle flow
Hedlund’s Theorem
Description and statement
Hyperbolic surface S = Γ \H
− H = { z ∈ C | Im(z) > 0 }
with the Riemannian metric
ds 2 =
|dz|2
.
Im(z)2
Dynamics of the horocycle flow
Hedlund’s Theorem
Description and statement
Hyperbolic surface S = Γ \H
− H = { z ∈ C | Im(z) > 0 }
with the Riemannian metric
ds 2 =
|dz|2
.
Im(z)2
− Γ cocompact discrete (torsion-free) subgroup of
PSL(2, R) = { f (z) =
az + b
| a, b, c, d ∈ R, ad − bc = 1 }.
cz + d
Dynamics of the horocycle flow
Hedlund’s Theorem
Description and statement
Unitary tangent bundle T 1 S = Γ \T 1 H = Γ \PSL(2, R)
~v
f
f∗z (~v )
z
f (z)
PSL(2, R) y T 1 H is free and transitive
Dynamics of the horocycle flow
Hedlund’s Theorem
Description and statement
Geodesic flow:
λ 0
~v = γ̇(0)
γ(t)
z = γ(0)
γ̇(t)
gt (v ) = (γ(t), γ̇(t))
Γ \PSL(2, R) x D = { ( 0
a b
d
gt (Γ ( c
a b
d
)) = Γ ( c
1
λ
t/2
e
)( 0
) | λ>0 }
0
e −t/2
)
right action of the diagonal group D
Dynamics of the horocycle flow
Hedlund’s Theorem
Description and statement
Geodesic flow:
λ 0
~v = γ̇(0)
γ(t)
z = γ(0)
γ̇(t)
gt (v ) = (γ(t), γ̇(t))
Γ \PSL(2, R) x D = { ( 0
a b
d
gt (Γ ( c
a b
d
)) = Γ ( c
1
λ
t/2
e
)( 0
) | λ>0 }
0
e −t/2
)
right action of the diagonal group D
Horocycle flow:
z = h(0)
~v
~v 0
h(s)
a b
d
hs (Γ ( c
hs (v ) = (h(s), ~v 0 )
1 s
) | s∈R }
1
1 s
)
0 1
Γ \PSL(2, R) x U = { ( 0
a b
d
)) = Γ ( c
)(
right action of the unipotent group U
Dynamics of the horocycle flow
Hedlund’s Theorem
Description and statement
Affine action: Γ \PSL(2, R) x B = DU = affine group
gt ◦hs = h−se t ◦gt
Dynamics of the horocycle flow
Hedlund’s Theorem
Description and statement
Affine action: Γ \PSL(2, R) x B = DU = affine group
gt ◦hs = h−se t ◦gt
Theorem (Hedlund, 1936)
Let S = Γ \H be a compact hyperbolic surface. Then the U-action
X = T 1 S = Γ \PSL(2, R) x U
is minimal (with dense orbits).
Dynamics of the horocycle flow
Hedlund’s Theorem
Proof
1 Hedlund’s Theorem
Description and statement
Proof
2 Hedlund’s Theorem for X = Γ \PSL(2, R) × G
PSL(2, R)-minimality and B-minimality
PSL(2, R)-minimality ⇒ B-minimality
B-minimality ⇒ U-minimality
3 The foliation point of view
Lie foliations
Martı́nez and Verjovsky’s question
4 Some progress in the non-homogeneous case
A classical example
Foliations with ’topologically non-trivial’ leaves
Dynamics of the horocycle flow
Hedlund’s Theorem
Proof
First Step: B-action is minimal
Dynamics of the horocycle flow
Hedlund’s Theorem
Proof
First Step: B-action is minimal
X = Γ \PSL(2, R) x B
is minimal
Dynamics of the horocycle flow
Hedlund’s Theorem
Proof
First Step: B-action is minimal
X = Γ \PSL(2, R) x B
m
is minimal
duality
Γ y PSL(2, R)/B = ∂H
is minimal
Dynamics of the horocycle flow
Hedlund’s Theorem
Proof
First Step: B-action is minimal
X = Γ \PSL(2, R) x B
m
duality
Γ y PSL(2, R)/B = ∂H
z = h(0)
~v
is minimal
~v 0
is minimal
h(s)
ξ = v (+∞) = v 0 (+∞)
Dynamics of the horocycle flow
Hedlund’s Theorem
Proof
First Step: B-action is minimal
X = Γ \PSL(2, R) x B
m
is minimal
duality
Γ y PSL(2, R)/B = ∂H
- Γ non-elementary ⇒ Γ y Λ(Γ ) minimal.
is minimal
Dynamics of the horocycle flow
Hedlund’s Theorem
Proof
First Step: B-action is minimal
X = Γ \PSL(2, R) x B
m
duality
Γ y PSL(2, R)/B = ∂H
- Γ non-elementary ⇒ Γ y Λ(Γ ) minimal.
- Γ cocompact ⇒ Λ(Γ ) = ∂H.
is minimal
is minimal
Dynamics of the horocycle flow
Hedlund’s Theorem
Proof
Second step: B-action is minimal ⇒ U-action is minimal
Dynamics of the horocycle flow
Hedlund’s Theorem
Proof
Second step: B-action is minimal ⇒ U-action is minimal
X compact ⇒ ∃ M 6= ∅ minimal closed U-invariant (U-minimal
for short) subset of X .
Dynamics of the horocycle flow
Hedlund’s Theorem
Proof
Second step: B-action is minimal ⇒ U-action is minimal
X compact ⇒ ∃ M 6= ∅ minimal closed U-invariant (U-minimal
for short) subset of X .
Aim
M is B-invariant
Dynamics of the horocycle flow
Hedlund’s Theorem
Proof
Second step: B-action is minimal ⇒ U-action is minimal
X compact ⇒ ∃ M 6= ∅ minimal closed U-invariant (U-minimal
for short) subset of X .
Aim
M is B-invariant + X x B minimal ⇒ M = X .
Dynamics of the horocycle flow
Hedlund’s Theorem
Proof
Second step: B-action is minimal ⇒ U-action is minimal
X compact ⇒ ∃ M 6= ∅ minimal closed U-invariant (U-minimal
for short) subset of X .
Aim
M is B-invariant + X x B minimal ⇒ M = X .
Consider x = Γ f ∈ M represented by f ∈ PSL(2, R) so that
xU = Γ fU = M
Dynamics of the horocycle flow
Hedlund’s Theorem
Proof
Second step: B-action is minimal ⇒ U-action is minimal
X compact ⇒ ∃ M 6= ∅ minimal closed U-invariant (U-minimal
for short) subset of X .
Aim
M is B-invariant + X x B minimal ⇒ M = X .
Consider x = Γ f ∈ M represented by f ∈ PSL(2, R) so that
xU = Γ fU = M
m
∃ γn ∈ Γ
∃ un = (
1 tn
0 1
) with tn → +∞ such that
Dynamics of the horocycle flow
Hedlund’s Theorem
Proof
Second step: B-action is minimal ⇒ U-action is minimal
X compact ⇒ ∃ M 6= ∅ minimal closed U-invariant (U-minimal
for short) subset of X .
Aim
M is B-invariant + X x B minimal ⇒ M = X .
Consider x = Γ f ∈ M represented by f ∈ PSL(2, R) so that
xU = Γ fU = M
m
∃ γn ∈ Γ
∃ un = (
1 tn
0 1
) with tn → +∞ such that
γn fun → f
Dynamics of the horocycle flow
Hedlund’s Theorem
Proof
Second step: B-action is minimal ⇒ U-action is minimal
X compact ⇒ ∃ M 6= ∅ minimal closed U-invariant (U-minimal
for short) subset of X .
Aim
M is B-invariant + X x B minimal ⇒ M = X .
Consider x = Γ f ∈ M represented by f ∈ PSL(2, R) so that
xU = Γ fU = M
m
∃ γn ∈ Γ
∃ un = (
1 tn
0 1
) with tn → +∞ such that
fn =: f −1 γn fun → I
Dynamics of the horocycle flow
Hedlund’s Theorem
Proof
Thus, we have:
x ∈ M and xfn = Γγn fun = Γ fun = xun ∈ xU ⊂ M
Dynamics of the horocycle flow
Hedlund’s Theorem
Proof
Thus, we have:
x ∈ M and xfn = Γγn fun = Γ fun = xun ∈ xU ⊂ M
m
fn ∈ HM =: {f 0 ∈ PSL(2, R) | Mf 0 ∩ M =
6 ∅}
Dynamics of the horocycle flow
Hedlund’s Theorem
Proof
Thus, we have:
x ∈ M and xfn = Γγn fun = Γ fun = xun ∈ xU ⊂ M
m
fn ∈ HM =: {f 0 ∈ PSL(2, R) | Mf 0 ∩ M =
6 ∅}
Lemma
HM is a closed U-bi-invariant subset of PSL(2, R).
Dynamics of the horocycle flow
Hedlund’s Theorem
Proof
Lemma
There exists k ∈ N such that fn ∈
/ B for n ≥ k.
Dynamics of the horocycle flow
Hedlund’s Theorem
Proof
Lemma
There exists k ∈ N such that fn ∈
/ B for n ≥ k.
Proof.
Dynamics of the horocycle flow
Hedlund’s Theorem
Proof
Lemma
There exists k ∈ N such that fn ∈
/ B for n ≥ k.
Proof. Assume on the contrary that for all k ∈ N, there exists
nk ≥ k such that
fnk ∈ B
Dynamics of the horocycle flow
Hedlund’s Theorem
Proof
Lemma
There exists k ∈ N such that fn ∈
/ B for n ≥ k.
Proof. Assume on the contrary that for all k ∈ N, there exists
nk ≥ k such that
fnk ∈ B
f −1 γnk f = fnk un−1
∈B
k
⇔
Dynamics of the horocycle flow
Hedlund’s Theorem
Proof
Lemma
There exists k ∈ N such that fn ∈
/ B for n ≥ k.
Proof. Assume on the contrary that for all k ∈ N, there exists
nk ≥ k such that
fnk ∈ B
⇔
f −1 γnk f = fnk un−1
∈B
k
⇔
γnk ∈ Γ ∩ fBf −1
Dynamics of the horocycle flow
Hedlund’s Theorem
Proof
Lemma
There exists k ∈ N such that fn ∈
/ B for n ≥ k.
Proof. Assume on the contrary that for all k ∈ N, there exists
nk ≥ k such that
fnk ∈ B
⇔
f −1 γnk f = fnk un−1
∈B
k
⇔
γnk ∈ Γ ∩ fBf −1
⇒
[γnk , γnk 0 ] ∈ Γ ∩ fUf −1
∀ k, k 0 ∈ N.
Dynamics of the horocycle flow
Hedlund’s Theorem
Proof
Lemma
There exists k ∈ N such that fn ∈
/ B for n ≥ k.
Proof. Assume on the contrary that for all k ∈ N, there exists
nk ≥ k such that
fnk ∈ B
⇔
f −1 γnk f = fnk un−1
∈B
k
⇔
γnk ∈ Γ ∩ fBf −1
⇒
[γnk , γnk 0 ] ∈ Γ ∩ fUf −1
⇓
∀ k, k 0 ∈ N.
Γ has no parabolic elements
[γnk , γnk 0 ] = I
∀ k, k 0 ∈ N.
Dynamics of the horocycle flow
Hedlund’s Theorem
Proof
⇓
f −1 γnk f = u
λnk
0
0
λ−1
nk
u −1 with u ∈ U and λnk → ∞
Dynamics of the horocycle flow
Hedlund’s Theorem
Proof
⇓
f −1 γnk f = u
λnk
0
0
λ−1
nk
u −1 with u ∈ U and λnk → ∞
⇓
fnk = f −1 γnk funk = u
λnk
0
0
λ−1
nk
u −1 unk
Dynamics of the horocycle flow
Hedlund’s Theorem
Proof
⇓
f −1 γnk f = u
λnk
0
0
λ−1
nk
u −1 with u ∈ U and λnk → ∞
⇓
fnk = f −1 γnk funk = u
λnk
0
0
λ−1
nk
u −1 unk → I
Dynamics of the horocycle flow
Hedlund’s Theorem
Proof
⇓
f −1 γnk f = u
λnk
0
0
λ−1
nk
u −1 with u ∈ U and λnk → ∞
⇓
fnk = f −1 γnk funk = u
λnk
0
0
λ−1
nk
u −1 unk → I
⇓
fnk (e1 ) = (λnk , 0) → e1 = (1, 0),
which is not possible.
Dynamics of the horocycle flow
Hedlund’s Theorem
Proof
Put
fn =
an bn
cn dn
Dynamics of the horocycle flow
Hedlund’s Theorem
Proof
Put
fn =
an bn
cn dn
∈
/B
⇔
cn 6= 0
Dynamics of the horocycle flow
Hedlund’s Theorem
Proof
Put
fn =
an bn
cn dn
∈
/B
⇔
cn 6= 0
For every α ∈ R∗+ , take
un0
in U
=
1
0
α−an
cn
1
un00
=
n
1 − α1 (bn + dn α−a
cn )
0
1
Dynamics of the horocycle flow
Hedlund’s Theorem
Proof
Put
fn =
an bn
cn dn
∈
/B
⇔
cn 6= 0
For every α ∈ R∗+ , take
un0
=
1
0
α−an
cn
1
un00
=
n
1 − α1 (bn + dn α−a
cn )
0
1
in U so that
un0 fn un00
=
α
0
cn α−1
∈ HM
Dynamics of the horocycle flow
Hedlund’s Theorem
Proof
Put
fn =
an bn
cn dn
∈
/B
⇔
cn 6= 0
For every α ∈ R∗+ , take
un0
=
1
0
α−an
cn
1
un00
=
n
1 − α1 (bn + dn α−a
cn )
0
1
in U so that
un0 fn un00
=
α
0
cn α−1
⇓
∈ HM
cn → 0
α 0
∈ HM
0 α−1
Dynamics of the horocycle flow
Hedlund’s Theorem
Proof
This means that
M
α 0
0 α−1
∩M=
6 ∅
∀α ∈ R∗+
Dynamics of the horocycle flow
Hedlund’s Theorem
Proof
This means that
M
M
α 0
0 α−1
∩M=
6 ∅
⇓ M is minimal
α 0
=M
0 α−1
∀α ∈ R∗+
∀α ∈ R∗+
Dynamics of the horocycle flow
Hedlund’s Theorem
Proof
This means that
M
M
α 0
0 α−1
∩M=
6 ∅
⇓ M is minimal
α 0
=M
0 α−1
So M is B-invariant and hence M = X .
∀α ∈ R∗+
∀α ∈ R∗+
Dynamics of the horocycle flow
Hedlund’s Theorem
Proof
This means that
M
M
α 0
0 α−1
∩M=
6 ∅
⇓ M is minimal
α 0
=M
0 α−1
∀α ∈ R∗+
∀α ∈ R∗+
So M is B-invariant and hence M = X .
Theorem (Hedlund, 1936)
Let S = Γ \H be a compact hyperbolic surface. Then the
U-action X = T 1 S = Γ \PSL(2, R) x U is minimal.
Dynamics of the horocycle flow
Hedlund’s Theorem for X = Γ \PSL(2, R) × G
1 Hedlund’s Theorem
Description and statement
Proof
2 Hedlund’s Theorem for X = Γ \PSL(2, R) × G
PSL(2, R)-minimality and B-minimality
PSL(2, R)-minimality ⇒ B-minimality
B-minimality ⇒ U-minimality
3 The foliation point of view
Lie foliations
Martı́nez and Verjovsky’s question
4 Some progress in the non-homogeneous case
A classical example
Foliations with ’topologically non-trivial’ leaves
Dynamics of the horocycle flow
Hedlund’s Theorem for X = Γ \PSL(2, R) × G
Theorem (A.-Dal’Bo)
Let G be a connected Lie group. Let X = Γ \PSL(2, R) × G be the
quotient of the Lie group PSL(2, R) × G by a cocompact discrete
subgroup Γ .
Dynamics of the horocycle flow
Hedlund’s Theorem for X = Γ \PSL(2, R) × G
Theorem (A.-Dal’Bo)
Let G be a connected Lie group. Let X = Γ \PSL(2, R) × G be the
quotient of the Lie group PSL(2, R) × G by a cocompact discrete
subgroup Γ . Then
X x U is minimal
Dynamics of the horocycle flow
Hedlund’s Theorem for X = Γ \PSL(2, R) × G
Theorem (A.-Dal’Bo)
Let G be a connected Lie group. Let X = Γ \PSL(2, R) × G be the
quotient of the Lie group PSL(2, R) × G by a cocompact discrete
subgroup Γ . Then
X x U is minimal
m
X x PSL(2, R) is minimal
Dynamics of the horocycle flow
Hedlund’s Theorem for X = Γ \PSL(2, R) × G
Theorem (A.-Dal’Bo)
Let G be a connected Lie group. Let X = Γ \PSL(2, R) × G be the
quotient of the Lie group PSL(2, R) × G by a cocompact discrete
subgroup Γ . Then
X x U is minimal
m
X x PSL(2, R) is minimal
m
Γ y G is minimal
Dynamics of the horocycle flow
Hedlund’s Theorem for X = Γ \PSL(2, R) × G
Theorem (A.-Dal’Bo)
Let G be a connected Lie group. Let X = Γ \PSL(2, R) × G be the
quotient of the Lie group PSL(2, R) × G by a cocompact discrete
subgroup Γ . Then
X x U is minimal
m
X x PSL(2, R) is minimal
m
Γ y G is minimal
m
p2 (Γ ) = G .
Dynamics of the horocycle flow
Hedlund’s Theorem for X = Γ \PSL(2, R) × G
Theorem (A.-Dal’Bo)
Let G be a connected Lie group. Let X = Γ \PSL(2, R) × G be the
quotient of the Lie group PSL(2, R) × G by a cocompact discrete
subgroup Γ . Then
X x U is minimal
m
X x PSL(2, R) is minimal
m
Γ y G is minimal
m
p2 (Γ ) = G .
p1
PSL(2, R) ←− PSL(2, R) × G
p2
−→ G
Dynamics of the horocycle flow
Hedlund’s Theorem for X = Γ \PSL(2, R) × G
Examples
A. Borel: ∃ Γ ⊂ PSL(2, R) × PSL(2, R) cocompact discrete
Dynamics of the horocycle flow
Hedlund’s Theorem for X = Γ \PSL(2, R) × G
Examples
A. Borel: ∃ Γ ⊂ PSL(2, R)×PSL(2, R) = H cocompact discrete
⇒ X = Γ \H compact 6-manifold
Dynamics of the horocycle flow
Hedlund’s Theorem for X = Γ \PSL(2, R) × G
Examples
A. Borel: ∃ Γ ⊂ PSL(2, R)×PSL(2, R) = H cocompact discrete
⇒ X = Γ \H compact 6-manifold
Γ is irreducible (i.e. p1 (Γ ) = p2 (Γ ) = PSL(2, R))
Dynamics of the horocycle flow
Hedlund’s Theorem for X = Γ \PSL(2, R) × G
Examples
A. Borel: ∃ Γ ⊂ PSL(2, R)×PSL(2, R) = H cocompact discrete
⇒ X = Γ \H compact 6-manifold
Γ is irreducible (i.e. p1 (Γ ) = p2 (Γ ) = PSL(2, R))
⇓
X is U-minimal
Dynamics of the horocycle flow
Hedlund’s Theorem for X = Γ \PSL(2, R) × G
Examples
A. Borel: ∃ Γ ⊂ PSL(2, R)×PSL(2, R) = H cocompact discrete
⇒ X = Γ \H compact 6-manifold
Γ is irreducible (i.e. p1 (Γ ) = p2 (Γ ) = PSL(2, R))
⇓
X is U-minimal
Breuillard et al.: ∃ Γ ⊂ PSL(2, R) × SO(3) = H cocompact
discrete such that
p2 (Γ ) = SO(3)
Dynamics of the horocycle flow
Hedlund’s Theorem for X = Γ \PSL(2, R) × G
Examples
A. Borel: ∃ Γ ⊂ PSL(2, R)×PSL(2, R) = H cocompact discrete
⇒ X = Γ \H compact 6-manifold
Γ is irreducible (i.e. p1 (Γ ) = p2 (Γ ) = PSL(2, R))
⇓
X is U-minimal
Breuillard et al.: ∃ Γ ⊂ PSL(2, R) × SO(3) = H cocompact
discrete such that
p2 (Γ ) = SO(3) ⇒ X = Γ \H is U-minimal
Dynamics of the horocycle flow
Hedlund’s Theorem for X = Γ \PSL(2, R) × G
PSL(2, R)-minimality and B-minimality
1 Hedlund’s Theorem
Description and statement
Proof
2 Hedlund’s Theorem for X = Γ \PSL(2, R) × G
PSL(2, R)-minimality and B-minimality
PSL(2, R)-minimality ⇒ B-minimality
B-minimality ⇒ U-minimality
3 The foliation point of view
Lie foliations
Martı́nez and Verjovsky’s question
4 Some progress in the non-homogeneous case
A classical example
Foliations with ’topologically non-trivial’ leaves
Dynamics of the horocycle flow
Hedlund’s Theorem for X = Γ \PSL(2, R) × G
PSL(2, R)-minimality and B-minimality
p1
PSL(2, R) ←− PSL(2, R) × G
∪
∪
p1 (Γ )
←−
Γ
p2
−→
G
∪
−→ p2 (Γ )
PSL(2, R)-minimality:
X = Γ \PSL(2, R) × G x PSL(2, R) is minimal
m
duality
Γ y G is minimal
Dynamics of the horocycle flow
Hedlund’s Theorem for X = Γ \PSL(2, R) × G
PSL(2, R)-minimality and B-minimality
p1
PSL(2, R) ←− PSL(2, R) × G
∪
∪
p1 (Γ )
←−
Γ
p2
−→
G
∪
−→ p2 (Γ )
PSL(2, R)-minimality:
X = Γ \PSL(2, R) × G x PSL(2, R) is minimal
m
duality
Γ y G is minimal
m
p2 (Γ ) = G
Dynamics of the horocycle flow
Hedlund’s Theorem for X = Γ \PSL(2, R) × G
PSL(2, R)-minimality and B-minimality
B-minimality:
X = Γ \PSL(2, R) × G x B is minimal
m
duality
Γ y ∂H × G is minimal
Dynamics of the horocycle flow
Hedlund’s Theorem for X = Γ \PSL(2, R) × G
PSL(2, R)-minimality and B-minimality
B-minimality:
X = Γ \PSL(2, R) × G x B is minimal
m
duality
Γ y ∂H × G is minimal
Proposition
Let G be a connected Lie group and let Γ be a discrete subgroup
of PSL(2, R) × G . The action X x B is minimal if and only if
p2 (Γ ) = G and p1 (Γ ) y ∂H is minimal.
Dynamics of the horocycle flow
Hedlund’s Theorem for X = Γ \PSL(2, R) × G
PSL(2, R)-minimality and B-minimality
B-minimality:
X = Γ \PSL(2, R) × G x B is minimal
m
duality
Γ y ∂H × G is minimal
Proposition
Let G be a connected Lie group and let Γ be a discrete subgroup
of PSL(2, R) × G . The action X x B is minimal if and only if
p2 (Γ ) = G and p1 (Γ ) y ∂H is minimal.
Aim
If Γ is cocompact, then X x B is minimal if and only if p2 (Γ ) = G .
Dynamics of the horocycle flow
Hedlund’s Theorem for X = Γ \PSL(2, R) × G
PSL(2, R)-minimality ⇒ B-minimality
1 Hedlund’s Theorem
Description and statement
Proof
2 Hedlund’s Theorem for X = Γ \PSL(2, R) × G
PSL(2, R)-minimality and B-minimality
PSL(2, R)-minimality ⇒ B-minimality
B-minimality ⇒ U-minimality
3 The foliation point of view
Lie foliations
Martı́nez and Verjovsky’s question
4 Some progress in the non-homogeneous case
A classical example
Foliations with ’topologically non-trivial’ leaves
Dynamics of the horocycle flow
Hedlund’s Theorem for X = Γ \PSL(2, R) × G
PSL(2, R)-minimality ⇒ B-minimality
Definition
An element (f , g ) ∈ PSL(2, R) × G is said to be semi-parabolic if
f 6= I belongs to U up to conjugation.
Dynamics of the horocycle flow
Hedlund’s Theorem for X = Γ \PSL(2, R) × G
PSL(2, R)-minimality ⇒ B-minimality
Definition
An element (f , g ) ∈ PSL(2, R) × G is said to be semi-parabolic if
f 6= I belongs to U up to conjugation.
Proposition
If X = Γ \PSL(2, R) × G is compact, then Γ does not contain semiparabolic elements.
Dynamics of the horocycle flow
Hedlund’s Theorem for X = Γ \PSL(2, R) × G
PSL(2, R)-minimality ⇒ B-minimality
Definition
An element (f , g ) ∈ PSL(2, R) × G is said to be semi-parabolic if
f 6= I belongs to U up to conjugation.
Proposition
If X = Γ \PSL(2, R) × G is compact, then Γ does not contain semiparabolic elements.
Proof. The proof is based on the following lemma:
Dynamics of the horocycle flow
Hedlund’s Theorem for X = Γ \PSL(2, R) × G
PSL(2, R)-minimality ⇒ B-minimality
Definition
An element (f , g ) ∈ PSL(2, R) × G is said to be semi-parabolic if
f 6= I belongs to U up to conjugation.
Proposition
If X = Γ \PSL(2, R) × G is compact, then Γ does not contain semiparabolic elements.
Proof. The proof is based on the following lemma:
Lemma
If Γ contains a semi-parabolic element, then the geodesic flow on
X = Γ \PSL(2, R) × G has divergent positive semi-orbits.
Dynamics of the horocycle flow
Hedlund’s Theorem for X = Γ \PSL(2, R) × G
PSL(2, R)-minimality ⇒ B-minimality
Definition
An element (f , g ) ∈ PSL(2, R) × G is said to be semi-parabolic if
f 6= I belongs to U up to conjugation.
Proposition
If X = Γ \PSL(2, R) × G is compact, then Γ does not contain semiparabolic elements.
Proof. The proof is based on the following lemma:
Lemma
If Γ contains a semi-parabolic element, then the geodesic flow on
X = Γ \PSL(2, R) × G has divergent positive semi-orbits.
Proof sketch.
∃ (fuf −1 , g ) ∈ Γ semi-parabolic ⇒ xD + diverges for x = Γ (f , g 0 )
Dynamics of the horocycle flow
Hedlund’s Theorem for X = Γ \PSL(2, R) × G
PSL(2, R)-minimality ⇒ B-minimality
Proposition
If X = Γ \PSL(2, R) × G is compact, then
X x B is minimal
⇔
p2 (Γ ) = G
Dynamics of the horocycle flow
Hedlund’s Theorem for X = Γ \PSL(2, R) × G
PSL(2, R)-minimality ⇒ B-minimality
Proposition
If X = Γ \PSL(2, R) × G is compact, then
X x B is minimal
⇔
p2 (Γ ) = G
⇔
Γ y G is minimal
⇔
X x PSL(2, R) is minimal.
Dynamics of the horocycle flow
Hedlund’s Theorem for X = Γ \PSL(2, R) × G
PSL(2, R)-minimality ⇒ B-minimality
Proposition
If X = Γ \PSL(2, R) × G is compact, then
X x B is minimal
⇔
p2 (Γ ) = G
⇔
Γ y G is minimal
⇔
X x PSL(2, R) is minimal.
Proof
Dynamics of the horocycle flow
Hedlund’s Theorem for X = Γ \PSL(2, R) × G
B-minimality ⇒ U-minimality
1 Hedlund’s Theorem
Description and statement
Proof
2 Hedlund’s Theorem for X = Γ \PSL(2, R) × G
PSL(2, R)-minimality and B-minimality
PSL(2, R)-minimality ⇒ B-minimality
B-minimality ⇒ U-minimality
3 The foliation point of view
Lie foliations
Martı́nez and Verjovsky’s question
4 Some progress in the non-homogeneous case
A classical example
Foliations with ’topologically non-trivial’ leaves
Dynamics of the horocycle flow
Hedlund’s Theorem for X = Γ \PSL(2, R) × G
B-minimality ⇒ U-minimality
Proof. Similar to the case where G is trivial:
∃ (fn , gn ) in HM = { h ∈ PSL(2, R) × G | Mh ∩ M =
6 ∅}
converging to (I , e):
Dynamics of the horocycle flow
Hedlund’s Theorem for X = Γ \PSL(2, R) × G
B-minimality ⇒ U-minimality
Proof. Similar to the case where G is trivial:
∃ (fn , gn ) in HM = { h ∈ PSL(2, R) × G | Mh ∩ M =
6 ∅}
converging to (I , e):
xU = Γ (f , g )U = M ⇒ ∃ γn = (γ1n , γ2n ) ∈ Γ ∃ un ∈ U :
(fn , gn ) = (f −1 γ1n fun , g −1 γ2n g ) → (I , e)
Dynamics of the horocycle flow
Hedlund’s Theorem for X = Γ \PSL(2, R) × G
B-minimality ⇒ U-minimality
Proof. Similar to the case where G is trivial:
∃ (fn , gn ) in HM = { h ∈ PSL(2, R) × G | Mh ∩ M =
6 ∅}
converging to (I , e):
xU = Γ (f , g )U = M ⇒ ∃ γn = (γ1n , γ2n ) ∈ Γ ∃ un ∈ U :
(fn , gn ) = (f −1 γ1n fun , g −1 γ2n g ) → (I , e)
Bi-invariance of HM :
Lemma
HM is a closed U-bi-invariant subset of PSL(2, R) × G .
Dynamics of the horocycle flow
Hedlund’s Theorem for X = Γ \PSL(2, R) × G
B-minimality ⇒ U-minimality
Geometric consequence of the Γ -cocompactness:
Lemma
There exists k ∈ N such that fn ∈
/ B for n ≥ k.
Dynamics of the horocycle flow
Hedlund’s Theorem for X = Γ \PSL(2, R) × G
B-minimality ⇒ U-minimality
Geometric consequence of the Γ -cocompactness:
Lemma
There exists k ∈ N such that fn ∈
/ B for n ≥ k.
Proof. Since X = Γ \PSL(2, R) × G is compact, there are no
semi-parabolic elements in Γ .
Dynamics of the horocycle flow
Hedlund’s Theorem for X = Γ \PSL(2, R) × G
B-minimality ⇒ U-minimality
Geometric consequence of the Γ -cocompactness:
Lemma
There exists k ∈ N such that fn ∈
/ B for n ≥ k.
Proof. Since X = Γ \PSL(2, R) × G is compact, there are no
semi-parabolic elements in Γ .
Using HM is U-bi-invariant and passing to the limit, we have:
α 0
α−1
(( 0
), e) ∈ HM
⇔
α 0
α−1
M( 0
)∩M=
6 ∅
∀α ∈ R∗+
Dynamics of the horocycle flow
Hedlund’s Theorem for X = Γ \PSL(2, R) × G
B-minimality ⇒ U-minimality
Geometric consequence of the Γ -cocompactness:
Lemma
There exists k ∈ N such that fn ∈
/ B for n ≥ k.
Proof. Since X = Γ \PSL(2, R) × G is compact, there are no
semi-parabolic elements in Γ .
Using HM is U-bi-invariant and passing to the limit, we have:
α 0
α−1
(( 0
), e) ∈ HM
α 0
α−1
⇔
M( 0
)∩M=
6 ∅
∀α ∈ R∗+
By minimality, this implies
α 0
α−1
M( 0
)=M
∀α ∈ R∗+ .
Dynamics of the horocycle flow
The foliation point of view
1 Hedlund’s Theorem
Description and statement
Proof
2 Hedlund’s Theorem for X = Γ \PSL(2, R) × G
PSL(2, R)-minimality and B-minimality
PSL(2, R)-minimality ⇒ B-minimality
B-minimality ⇒ U-minimality
3 The foliation point of view
Lie foliations
Martı́nez and Verjovsky’s question
4 Some progress in the non-homogeneous case
A classical example
Foliations with ’topologically non-trivial’ leaves
Dynamics of the horocycle flow
The foliation point of view
Remark
The two examples
PSL(2, R)
/ PSL(2, R) × G
π
X = Γ \PSL(2, R) × PSL(2, R)
are G -Lie foliations.
/G =
PSL(2,R)
SO(3)
Dynamics of the horocycle flow
The foliation point of view
In fact, X is the unitary tangent bundle of the G -Lie foliation
H
/ PSL(2, R)/PSO(2, R) × G
/G =
π
M = Γ \PSL(2, R)/PSO(2, R) × PSL(2, R)
according to Fedida’s description
PSL(2,R)
SO(3)
Dynamics of the horocycle flow
The foliation point of view
In fact, X is the unitary tangent bundle of the G -Lie foliation
H
/ PSL(2, R)/PSO(2, R) × G
/G =
PSL(2,R)
SO(3)
π
M = Γ \PSL(2, R)/PSO(2, R) × PSL(2, R)
according to Fedida’s description with holonomy representation
given by
h = p2 |Γ : Γ → p2 (Γ ) ⊂ G
Dynamics of the horocycle flow
The foliation point of view
Lie foliations
1 Hedlund’s Theorem
Description and statement
Proof
2 Hedlund’s Theorem for X = Γ \PSL(2, R) × G
PSL(2, R)-minimality and B-minimality
PSL(2, R)-minimality ⇒ B-minimality
B-minimality ⇒ U-minimality
3 The foliation point of view
Lie foliations
Martı́nez and Verjovsky’s question
4 Some progress in the non-homogeneous case
A classical example
Foliations with ’topologically non-trivial’ leaves
Dynamics of the horocycle flow
The foliation point of view
Lie foliations
Definition
Let H be a connected Lie group with a epimorphism ρ : H → G and
let Γ be a cocompact discrete subgroup of H.
Dynamics of the horocycle flow
The foliation point of view
Lie foliations
Definition
Let H be a connected Lie group with a epimorphism ρ : H → G and
let Γ be a cocompact discrete subgroup of H. As before, we have a
G -Lie foliation
K = Ker ρ
/ H
ρ
π
M = Γ \H
/ G
Dynamics of the horocycle flow
The foliation point of view
Lie foliations
Definition
Let H be a connected Lie group with a epimorphism ρ : H → G and
let Γ be a cocompact discrete subgroup of H. As before, we have a
G -Lie foliation
K0 ⊂ K = Ker ρ
/ H/K0
ρ
π
M = Γ \H/K0
/ G
Dynamics of the horocycle flow
The foliation point of view
Lie foliations
Definition
Let H be a connected Lie group with a epimorphism ρ : H → G and
let Γ be a cocompact discrete subgroup of H. As before, we have a
G -Lie foliation
K0 ⊂ K = Ker ρ
/ H/K0
ρ
/ G
π
M = Γ \H/K0
A G -Lie foliation constructed by this method is called homogeneous.
Dynamics of the horocycle flow
The foliation point of view
Lie foliations
Definition
Let H be a connected Lie group with a epimorphism ρ : H → G and
let Γ be a cocompact discrete subgroup of H. As before, we have a
G -Lie foliation
K0 ⊂ K = Ker ρ
/ H/K0
ρ
/ G
π
M = Γ \H/K0
A G -Lie foliation constructed by this method is called homogeneous.
Examples
There are non-homogeneous examples constructed by G. Hector, S.
Matsumoto and G. Meigniez whose leaves are hyperbolic surfaces.
Dynamics of the horocycle flow
The foliation point of view
Lie foliations
Proposition
Let F be G -Lie foliation by hyperbolic surfaces of a compact connected manifold M, and let X = T 1 F be its unitary tangent bundle.
Dynamics of the horocycle flow
The foliation point of view
Lie foliations
Proposition
Let F be G -Lie foliation by hyperbolic surfaces of a compact connected manifold M, and let X = T 1 F be its unitary tangent bundle.
Then
∼ PSL(2, R) × G
e =
F is homogeneous
⇔ X
Dynamics of the horocycle flow
The foliation point of view
Lie foliations
Proposition
Let F be G -Lie foliation by hyperbolic surfaces of a compact connected manifold M, and let X = T 1 F be its unitary tangent bundle.
Then
∼ PSL(2, R) × G
e =
F is homogeneous
⇔ X
Proof. Any semi-simple ideal in a Lie algebra is a direct summand.
Dynamics of the horocycle flow
The foliation point of view
Lie foliations
Proposition
Let F be G -Lie foliation by hyperbolic surfaces of a compact connected manifold M, and let X = T 1 F be its unitary tangent bundle.
Then
∼ PSL(2, R) × G
e =
F is homogeneous
⇔ X
Proof. Any semi-simple ideal in a Lie algebra is a direct summand.
Theorem
Let X = T 1 F be the unitary tangent bundle of a homogeneous
G -Lie foliation F by hyperbolic surfaces of a compact manifold. If
F is minimal, then X x U is minimal.
Dynamics of the horocycle flow
The foliation point of view
Lie foliations
Proposition
Let F be G -Lie foliation by hyperbolic surfaces of a compact connected manifold M, and let X = T 1 F be its unitary tangent bundle.
Then
∼ PSL(2, R) × G
e =
F is homogeneous
⇔ X
Proof. Any semi-simple ideal in a Lie algebra is a direct summand.
Theorem
Let X = T 1 F be the unitary tangent bundle of a homogeneous
Riemannian foliation F by hyperbolic surfaces of a compact manifold.
If F is minimal, then X x U is minimal (by using Molino’s theory).
Dynamics of the horocycle flow
The foliation point of view
Martı́nez and Verjovsky’s question
1 Hedlund’s Theorem
Description and statement
Proof
2 Hedlund’s Theorem for X = Γ \PSL(2, R) × G
PSL(2, R)-minimality and B-minimality
PSL(2, R)-minimality ⇒ B-minimality
B-minimality ⇒ U-minimality
3 The foliation point of view
Lie foliations
Martı́nez and Verjovsky’s question
4 Some progress in the non-homogeneous case
A classical example
Foliations with ’topologically non-trivial’ leaves
Dynamics of the horocycle flow
The foliation point of view
Martı́nez and Verjovsky’s question
Let M be a compact connected manifold, and let F be a minimal
foliation of M by hyperbolic surfaces.
Dynamics of the horocycle flow
The foliation point of view
Martı́nez and Verjovsky’s question
Let M be a compact connected manifold, and let F be a minimal
foliation of M by hyperbolic surfaces.
Martı́nez and Verjovsky’s question
Let X = T 1 F be the unitary tangent bundle of F. Is it true that
X x U is minimal?
Dynamics of the horocycle flow
The foliation point of view
Martı́nez and Verjovsky’s question
Let M be a compact connected manifold, and let F be a minimal
foliation of M by hyperbolic surfaces.
Martı́nez and Verjovsky’s question
Let X = T 1 F be the unitary tangent bundle of F. Is it true that
X x U is minimal?
Example
Let Γ be a cocompact Fuchsian group.
Dynamics of the horocycle flow
The foliation point of view
Martı́nez and Verjovsky’s question
Let M be a compact connected manifold, and let F be a minimal
foliation of M by hyperbolic surfaces.
Martı́nez and Verjovsky’s question
Let X = T 1 F be the unitary tangent bundle of F. Is it true that
X x U is minimal?
Example
Let Γ be a cocompact Fuchsian group. Consider the diagonal action
Γ y PSL(2, R) × ∂H
given by
γ(f , ξ) = (γf , γ(ξ))
and the quotient X = Γ \PSL(2, R) × ∂H.
Dynamics of the horocycle flow
The foliation point of view
Martı́nez and Verjovsky’s question
Then X is the unitary tangent bundle of the transversely homographic
foliation F induced by
H
/ H × ∂H p2
/ ∂H
π
M = Γ \H × ∂H
Dynamics of the horocycle flow
The foliation point of view
Martı́nez and Verjovsky’s question
Then X is the unitary tangent bundle of the transversely homographic
foliation F induced by
H
/ H × ∂H p2
/ ∂H
π
M = Γ \H × ∂H
X x PSL(2, R) is minimal
Dynamics of the horocycle flow
The foliation point of view
Martı́nez and Verjovsky’s question
Then X is the unitary tangent bundle of the transversely homographic
foliation F induced by
H
/ H × ∂H p2
/ ∂H
π
M = Γ \H × ∂H
X x PSL(2, R) is minimal ⇔ Γ y ∂H is minimal
Dynamics of the horocycle flow
The foliation point of view
Martı́nez and Verjovsky’s question
Then X is the unitary tangent bundle of the transversely homographic
foliation F induced by
H
/ H × ∂H p2
/ ∂H
π
M = Γ \H × ∂H
X x PSL(2, R) is minimal ⇔ Γ y ∂H is minimal
X x B is not minimal
Dynamics of the horocycle flow
The foliation point of view
Martı́nez and Verjovsky’s question
Then X is the unitary tangent bundle of the transversely homographic
foliation F induced by
H
/ H × ∂H p2
/ ∂H
π
M = Γ \H × ∂H
X x PSL(2, R) is minimal ⇔ Γ y ∂H is minimal
X x B is not minimal as dual to the diagonal action
Γ y ∂H × ∂H
given by
γ(ξ1 , ξ2 ) = (γ(ξ1 ), γ(ξ2 ))
Dynamics of the horocycle flow
The foliation point of view
Martı́nez and Verjovsky’s question
Martı́nez and Verjovsky’s question
Let X = T 1 F be the unitary tangent bundle of F. Is it true that
X x U is minimal if and only if X x B is minimal?
Dynamics of the horocycle flow
The foliation point of view
Martı́nez and Verjovsky’s question
Martı́nez and Verjovsky’s question
Let X = T 1 F be the unitary tangent bundle of F. Is it true that
X x U is minimal if and only if X x B is minimal?
Theorem (Martı́nez-Verjovsky)
Let (M, F) be a compact lamination by hyperbolic surfaces, and let
X = T 1 F its unitary tangent bundle. Assume that F is minimal.
Dynamics of the horocycle flow
The foliation point of view
Martı́nez and Verjovsky’s question
Martı́nez and Verjovsky’s question
Let X = T 1 F be the unitary tangent bundle of F. Is it true that
X x U is minimal if and only if X x B is minimal?
Theorem (Martı́nez-Verjovsky)
Let (M, F) be a compact lamination by hyperbolic surfaces, and let
X = T 1 F its unitary tangent bundle. Assume that F is minimal.
Then X x B is minimal if and only if F is not defined by a continuous
locally free B-action.
Dynamics of the horocycle flow
The foliation point of view
Martı́nez and Verjovsky’s question
Martı́nez and Verjovsky’s question
Let X = T 1 F be the unitary tangent bundle of F. Is it true that
X x U is minimal if and only if X x B is minimal?
Theorem (Martı́nez-Verjovsky)
Let (M, F) be a compact lamination by hyperbolic surfaces, and let
X = T 1 F its unitary tangent bundle. Assume that F is minimal.
Then X x B is minimal if and only if F is not defined by a continuous
locally free B-action.
Proof sketch
Dynamics of the horocycle flow
Some progress in the non-homogeneous case
1 Hedlund’s Theorem
Description and statement
Proof
2 Hedlund’s Theorem for X = Γ \PSL(2, R) × G
PSL(2, R)-minimality and B-minimality
PSL(2, R)-minimality ⇒ B-minimality
B-minimality ⇒ U-minimality
3 The foliation point of view
Lie foliations
Martı́nez and Verjovsky’s question
4 Some progress in the non-homogeneous case
A classical example
Foliations with ’topologically non-trivial’ leaves
Dynamics of the horocycle flow
Some progress in the non-homogeneous case
A classical example
1 Hedlund’s Theorem
Description and statement
Proof
2 Hedlund’s Theorem for X = Γ \PSL(2, R) × G
PSL(2, R)-minimality and B-minimality
PSL(2, R)-minimality ⇒ B-minimality
B-minimality ⇒ U-minimality
3 The foliation point of view
Lie foliations
Martı́nez and Verjovsky’s question
4 Some progress in the non-homogeneous case
A classical example
Foliations with ’topologically non-trivial’ leaves
Dynamics of the horocycle flow
Some progress in the non-homogeneous case
A classical example
Example
The Hirsch foliation F is a codimension-one foliation by hyperbolic
surfaces on a compact 3-manifold
Dynamics of the horocycle flow
Some progress in the non-homogeneous case
A classical example
Example
The Hirsch foliation F is a codimension-one foliation by hyperbolic
surfaces on a compact 3-manifold
Dynamics of the horocycle flow
Some progress in the non-homogeneous case
A classical example
Example
The Hirsch foliation F is a codimension-one foliation by hyperbolic
surfaces on a compact 3-manifold
Dynamics of the horocycle flow
Some progress in the non-homogeneous case
A classical example
leaf without holonomy
leaf with holonomy
Dynamics of the horocycle flow
Some progress in the non-homogeneous case
A classical example
Transverse dynamics of F is represented by the doubling map
f (x) = 2x (mod 1)
Dynamics of the horocycle flow
Some progress in the non-homogeneous case
A classical example
Transverse dynamics of F is represented by the doubling map
f (x) = 2x (mod 1)
⇓
F is minimal
Dynamics of the horocycle flow
Some progress in the non-homogeneous case
A classical example
Transverse dynamics of F is represented by the doubling map
f (x) = 2x (mod 1)
⇓
F is minimal
The leaves are neither homeomorphic to the plane nor to the
cylinder
Dynamics of the horocycle flow
Some progress in the non-homogeneous case
A classical example
Transverse dynamics of F is represented by the doubling map
f (x) = 2x (mod 1)
⇓
F is minimal
The leaves are neither homeomorphic to the plane nor to the
cylinder
⇓ Martı́nez-Verjovsky theorem
X = T 1 F x B is minimal
Dynamics of the horocycle flow
Some progress in the non-homogeneous case
A classical example
Question
Is the unitary tangent bundle of the Hirsch foliation U-minimal?
Dynamics of the horocycle flow
Some progress in the non-homogeneous case
A classical example
Question
Is the unitary tangent bundle of the Hirsch foliation U-minimal?
Answer
Yes.
Dynamics of the horocycle flow
Some progress in the non-homogeneous case
A classical example
Question
Is the unitary tangent bundle of the Hirsch foliation U-minimal?
Answer
Yes.
The proof depends on the following fact:
Dynamics of the horocycle flow
Some progress in the non-homogeneous case
A classical example
Question
Is the unitary tangent bundle of the Hirsch foliation U-minimal?
Answer
Yes.
The proof depends on the following fact:
Remark [S. Matsumoto]
The horocyclic flow on each leaf of F has no minimal sets.
Dynamics of the horocycle flow
Some progress in the non-homogeneous case
Foliations with ’topologically non-trivial’ leaves
1 Hedlund’s Theorem
Description and statement
Proof
2 Hedlund’s Theorem for X = Γ \PSL(2, R) × G
PSL(2, R)-minimality and B-minimality
PSL(2, R)-minimality ⇒ B-minimality
B-minimality ⇒ U-minimality
3 The foliation point of view
Lie foliations
Martı́nez and Verjovsky’s question
4 Some progress in the non-homogeneous case
A classical example
Foliations with ’topologically non-trivial’ leaves
Dynamics of the horocycle flow
Some progress in the non-homogeneous case
Foliations with ’topologically non-trivial’ leaves
Theorem (A.-Dal’Bo-Martı́nez-Verjovsky)
Let (M, F) be a minimal compact lamination by hyperbolic surfaces.
If F admits a leaf without holonomy which is neither homeomorphic
to the plane nor to the cylinder, then the horocycle flow on the
unitary tangent bundle X = T 1 F is minimal.
Dynamics of the horocycle flow
Some progress in the non-homogeneous case
Foliations with ’topologically non-trivial’ leaves
Theorem (A.-Dal’Bo-Martı́nez-Verjovsky)
Let (M, F) be a minimal compact lamination by hyperbolic surfaces.
If F admits a leaf whose holonomy covering is neither homeomorphic
to the plane nor to the cylinder, then the horocycle flow on the
unitary tangent bundle X = T 1 F is minimal.
Dynamics of the horocycle flow
Some progress in the non-homogeneous case
Foliations with ’topologically non-trivial’ leaves
Theorem (A.-Dal’Bo-Martı́nez-Verjovsky)
Let (M, F) be a minimal compact lamination by hyperbolic surfaces.
If F admits a leaf whose holonomy covering is neither homeomorphic
to the plane nor to the cylinder, then the horocycle flow on the
unitary tangent bundle X = T 1 F is minimal.
Corollary
For the non-homogeneous Lie foliations constructed by G. Hector, S.
Matsumoto and G. Meigniez, the horocycle flow is minimal.
Dynamics of the horocycle flow
Some progress in the non-homogeneous case
Foliations with ’topologically non-trivial’ leaves
Proof skecht. Before we start, we need the following definition:
Dynamics of the horocycle flow
Some progress in the non-homogeneous case
Foliations with ’topologically non-trivial’ leaves
Proof skecht. Before we start, we need the following definition:
Definition
Consider a complete hyperbolic surface L = Γ \H which can be
partitioned into countably many pair of pants (pants decomposition)
whose boundary components have uniformly bounded lengths (good
pants decomposition). Its fundamental group Γ (which is purely
hyperbolic of the first kind) is called tight by S. Matsumoto.
Dynamics of the horocycle flow
Some progress in the non-homogeneous case
Foliations with ’topologically non-trivial’ leaves
First Step: Obtaining good pants decompositions of the leaves.
There is a leaf without holonomy L = Γ \H which is neither
homeomorphic to the plane nor to the cylinder
Dynamics of the horocycle flow
Some progress in the non-homogeneous case
Foliations with ’topologically non-trivial’ leaves
First Step: Obtaining good pants decompositions of the leaves.
There is a leaf without holonomy L = Γ \H which is neither
homeomorphic to the plane nor to the cylinder (⇔ its fundamental group is neither cyclic nor trivial)
Dynamics of the horocycle flow
Some progress in the non-homogeneous case
Foliations with ’topologically non-trivial’ leaves
First Step: Obtaining good pants decompositions of the leaves.
There is a leaf without holonomy L = Γ \H which is neither
homeomorphic to the plane nor to the cylinder (⇔ its fundamental group is neither cyclic nor trivial)
m
L contains a pair of pants
Dynamics of the horocycle flow
Some progress in the non-homogeneous case
Foliations with ’topologically non-trivial’ leaves
First Step: Obtaining good pants decompositions of the leaves.
There is a leaf without holonomy L = Γ \H which is neither
homeomorphic to the plane nor to the cylinder (⇔ its fundamental group is neither cyclic nor trivial)
m
L contains a pair of pants
⇓
L without holonomy and F minimal
L contains countable many pairs of pants Pn approaching every
end
Dynamics of the horocycle flow
Some progress in the non-homogeneous case
Foliations with ’topologically non-trivial’ leaves
First Step: Obtaining good pants decompositions of the leaves.
There is a leaf without holonomy L = Γ \H which is neither
homeomorphic to the plane nor to the cylinder (⇔ its fundamental group is neither cyclic nor trivial)
m
L contains a pair of pants
⇓
L without holonomy and F minimal
L contains countable many pairs of pants Pn approaching every
end
S
⇓ L is quasi-isometric to n Pn
L has a good pants decomposition
Dynamics of the horocycle flow
Some progress in the non-homogeneous case
Foliations with ’topologically non-trivial’ leaves
First Step: Obtaining good pants decompositions of the leaves.
There is a leaf without holonomy L = Γ \H which is neither
homeomorphic to the plane nor to the cylinder (⇔ its fundamental group is neither cyclic nor trivial)
m
L contains a pair of pants
⇓
L without holonomy and F minimal
L contains countable many pairs of pants Pn approaching every
end
S
⇓ L is quasi-isometric to n Pn
L has a good pants decomposition (⇔ Γ is tight)
Dynamics of the horocycle flow
Some progress in the non-homogeneous case
Foliations with ’topologically non-trivial’ leaves
First Step: Obtaining good pants decompositions of the leaves.
There is a leaf without holonomy L = Γ \H which is neither
homeomorphic to the plane nor to the cylinder (⇔ its fundamental group is neither cyclic nor trivial)
m
L contains a pair of pants
⇓
L without holonomy and F minimal
L contains countable many pairs of pants Pn approaching every
end
S
⇓ L is quasi-isometric to n Pn
Any leaf has a good pants decomposition
Dynamics of the horocycle flow
Some progress in the non-homogeneous case
Foliations with ’topologically non-trivial’ leaves
Second step: Finding sufficient conditions for the U-minimality.
Dynamics of the horocycle flow
Some progress in the non-homogeneous case
Foliations with ’topologically non-trivial’ leaves
Second step: Finding sufficient conditions for the U-minimality.
Proposition
Let (M, F) be a minimal compact lamination such that
Dynamics of the horocycle flow
Some progress in the non-homogeneous case
Foliations with ’topologically non-trivial’ leaves
Second step: Finding sufficient conditions for the U-minimality.
Proposition
Let (M, F) be a minimal compact lamination such that
i) the affine action X = T 1 F x B is minimal,
Dynamics of the horocycle flow
Some progress in the non-homogeneous case
Foliations with ’topologically non-trivial’ leaves
Second step: Finding sufficient conditions for the U-minimality.
Proposition
Let (M, F) be a minimal compact lamination such that
i) the affine action X = T 1 F x B is minimal,
ii) the horocycle flow is transitive, i.e. ∃ x ∈ X such that xU = X .
Dynamics of the horocycle flow
Some progress in the non-homogeneous case
Foliations with ’topologically non-trivial’ leaves
Second step: Finding sufficient conditions for the U-minimality.
Proposition
Let (M, F) be a minimal compact lamination such that
i) the affine action X = T 1 F x B is minimal,
ii) the horocycle flow is transitive, i.e. ∃ x ∈ X such that xU = X .
Let M 6= ∅ be a U-minimal set in T 1 F.
Dynamics of the horocycle flow
Some progress in the non-homogeneous case
Foliations with ’topologically non-trivial’ leaves
Second step: Finding sufficient conditions for the U-minimality.
Proposition
Let (M, F) be a minimal compact lamination such that
i) the affine action X = T 1 F x B is minimal,
ii) the horocycle flow is transitive, i.e. ∃ x ∈ X such that xU = X .
Let M 6= ∅ be a U-minimal set in T 1 F.
If there is a real
number t0 > 0 such that gt0 (M) = M, then M = X .
Proof
Dynamics of the horocycle flow
Some progress in the non-homogeneous case
Foliations with ’topologically non-trivial’ leaves
Condition (i): All the leaves are ’topologically non-trivial’
⇓
1
Martı́nez-Verjovsky theorem
X = T F x B is minimal
Dynamics of the horocycle flow
Some progress in the non-homogeneous case
Foliations with ’topologically non-trivial’ leaves
Condition (i): All the leaves are ’topologically non-trivial’
⇓
Martı́nez-Verjovsky theorem
1
X = T F x B is minimal
Condition (ii): Since F is minimal, it is enough to prove the
horocycle flow is transitive in restriction to some leaf L.
Dynamics of the horocycle flow
Some progress in the non-homogeneous case
Foliations with ’topologically non-trivial’ leaves
Condition (i): All the leaves are ’topologically non-trivial’
⇓
Martı́nez-Verjovsky theorem
1
X = T F x B is minimal
Condition (ii): Since F is minimal, it is enough to prove the
horocycle flow is transitive in restriction to every leaf L.
Aim
If L has a good pants decomposition, then for every x ∈ T 1 L,
either xU = T 1 L or there is a real number t0 > 0 such that
gt0 (x) ∈ xU (so any proper U-minimal set M 6= ∅ in T 1 L
verifies gt0 (M) = M).
Dynamics of the horocycle flow
Some progress in the non-homogeneous case
Foliations with ’topologically non-trivial’ leaves
Third Step: Using Matsumoto’s idea to prove U-minimality.
Dynamics of the horocycle flow
Some progress in the non-homogeneous case
Foliations with ’topologically non-trivial’ leaves
Third Step: Using Matsumoto’s idea to prove U-minimality.
Fundamental lemma
Let L = Γ \H be a non-compact hyperbolic surface. Assume
there are sequences of
closed geodesics Cn with uniformly bounded lengths
real numbers tn → +∞ such that
gtn (x) ∈ Cn
for some x ∈ T 1 L. Then there is a real number t0 > 0 such
that
gt0 (x) ∈ xU.
Dynamics of the horocycle flow
Some progress in the non-homogeneous case
Foliations with ’topologically non-trivial’ leaves
Proposition
Assume L has a good pants decomposition. Then, for any
tangent vector x ∈ T 1 L, either xU = T 1 L or there is a real
number t0 > 0 such that gt0 (x) ∈ xU.
Dynamics of the horocycle flow
Some progress in the non-homogeneous case
Foliations with ’topologically non-trivial’ leaves
Proposition
Assume L has a good pants decomposition. Then, for any
tangent vector x ∈ T 1 L, either xU = T 1 L or there is a real
number t0 > 0 such that gt0 (x) ∈ xU. In particular, if M 6= ∅
is a proper U-minimal set in in T 1 L, then gt0 (M) = M for
some t0 > 0.
Dynamics of the horocycle flow
Some progress in the non-homogeneous case
Foliations with ’topologically non-trivial’ leaves
Proposition
Assume L has a good pants decomposition. Then, for any
tangent vector x ∈ T 1 L, either xU = T 1 L or there is a real
number t0 > 0 such that gt0 (x) ∈ xU. In particular, if M 6= ∅
is a proper U-minimal set in in T 1 L, then gt0 (M) = M for
some t0 > 0.
Using the duality between the U-action and the linear Γ -action
and the classification of the limit points of Γ , we can deduce:
Theorem (S. Matsumoto)
For any hyperbolic surface with a good pants decomposition,
the horocycle flow has no minimal sets.
Proof
Dynamics of the horocycle flow
Some progress in the non-homogeneous case
Foliations with ’topologically non-trivial’ leaves
Theorem (A.-Dal’Bo-Martı́nez-Verjovsky)
Let (M, F) be a minimal compact lamination by hyperbolic surfaces.
If F admits a leaf without holonomy which is neither homeomorphic
to the plane nor to the cylinder, then the horocycle flow on the
unitary tangent bundle X = T 1 F is minimal.
Dynamics of the horocycle flow
Some progress in the non-homogeneous case
Foliations with ’topologically non-trivial’ leaves
Theorem (A.-Dal’Bo-Martı́nez-Verjovsky)
Let (M, F) be a minimal compact lamination by hyperbolic surfaces.
If F admits a leaf whose holonomy covering is neither homeomorphic
to the plane nor to the cylinder, then the horocycle flow on the
unitary tangent bundle X = T 1 F is minimal.
Dynamics of the horocycle flow
Some progress in the non-homogeneous case
Foliations with ’topologically non-trivial’ leaves
GRACIAS
Dynamics of the horocycle flow
Some progress in the non-homogeneous case
Foliations with ’topologically non-trivial’ leaves
Proof (‘if’ part). It is enough to show that
∆ = p1 (Γ ) y ∂H
is minimal. We distinguish three cases:
Dynamics of the horocycle flow
Some progress in the non-homogeneous case
Foliations with ’topologically non-trivial’ leaves
Proof (‘if’ part). It is enough to show that
∆ = p1 (Γ ) y ∂H
is minimal. We distinguish three cases:
If ∆ = p1 (Γ ) is discrete, then
X compact ⇒ ∆\H compact
and therefore
Dynamics of the horocycle flow
Some progress in the non-homogeneous case
Foliations with ’topologically non-trivial’ leaves
Proof (‘if’ part). It is enough to show that
∆ = p1 (Γ ) y ∂H
is minimal. We distinguish three cases:
If ∆ = p1 (Γ ) is discrete, then
X compact ⇒ ∆\H compact
and therefore
− ∆ is non-elementary ⇒ ∆ y Λ(∆) is minimal.
Dynamics of the horocycle flow
Some progress in the non-homogeneous case
Foliations with ’topologically non-trivial’ leaves
Proof (‘if’ part). It is enough to show that
∆ = p1 (Γ ) y ∂H
is minimal. We distinguish three cases:
If ∆ = p1 (Γ ) is discrete, then
X compact ⇒ ∆\H compact
and therefore
− ∆ is non-elementary ⇒ ∆ y Λ(∆) is minimal.
− ∆ is cocompact ⇒ Λ(∆) = ∂H.
Dynamics of the horocycle flow
Some progress in the non-homogeneous case
Foliations with ’topologically non-trivial’ leaves
Proof (‘if’ part). It is enough to show that
∆ = p1 (Γ ) y ∂H
is minimal. We distinguish three cases:
If ∆ = p1 (Γ ) is discrete, then
X compact ⇒ ∆\H compact
and therefore
− ∆ is non-elementary ⇒ ∆ y Λ(∆) is minimal.
− ∆ is cocompact ⇒ Λ(∆) = ∂H.
If ∆ = p1 (Γ ) is non-discrete dense, then ∆ y Λ(∆) = ∂H is
minimal again.
Dynamics of the horocycle flow
Some progress in the non-homogeneous case
Foliations with ’topologically non-trivial’ leaves
If ∆ is neither discrete nor dense, then there is f ∈ PSL(2, R)
such that
∆ ⊂ fPSO(2, R)f −1
or
∆ ⊂ fBf −1
Dynamics of the horocycle flow
Some progress in the non-homogeneous case
Foliations with ’topologically non-trivial’ leaves
If ∆ is neither discrete nor dense, then there is f ∈ PSL(2, R)
such that
∆ ⊂ fPSO(2, R)f −1
⇓
X is not compact
or
∆ ⊂ fBf −1
Dynamics of the horocycle flow
Some progress in the non-homogeneous case
Foliations with ’topologically non-trivial’ leaves
If ∆ is neither discrete nor dense, then there is f ∈ PSL(2, R)
such that
∆ ⊂ fPSO(2, R)f −1
or
∆ ⊂ fBf −1
∆ has no parabolic elements
⇓
∆ is abelian,
Dynamics of the horocycle flow
Some progress in the non-homogeneous case
Foliations with ’topologically non-trivial’ leaves
If ∆ is neither discrete nor dense, then there is f ∈ PSL(2, R)
such that
∆ ⊂ fPSO(2, R)f −1
or
∆ ⊂ fBf −1
Γ has no semi-parabolic elements
⇓
∆ is abelian,
Dynamics of the horocycle flow
Some progress in the non-homogeneous case
Foliations with ’topologically non-trivial’ leaves
If ∆ is neither discrete nor dense, then there is f ∈ PSL(2, R)
such that
∆ ⊂ fPSO(2, R)f −1
or
∆ ⊂ fBf −1
Γ has no semi-parabolic elements
⇓
∆ is abelian,
which also contradicts the compactness of X .
Back
Dynamics of the horocycle flow
Some progress in the non-homogeneous case
Foliations with ’topologically non-trivial’ leaves
Proof sketch. The proof is based on the two following lemmas:
Dynamics of the horocycle flow
Some progress in the non-homogeneous case
Foliations with ’topologically non-trivial’ leaves
Proof sketch. The proof is based on the two following lemmas:
Lemma
The leaves of F are the orbits of a continuous locally free B-action
if and only if there is a B-minimal set M in X such that
| Ty1 F ∩ M |= 1
∀x ∈ M ,
where y = π(x) is the image of x by the projection π : X → M.
Dynamics of the horocycle flow
Some progress in the non-homogeneous case
Foliations with ’topologically non-trivial’ leaves
Proof sketch. The proof is based on the two following lemmas:
Lemma
The leaves of F are the orbits of a continuous locally free B-action
if and only if there is a B-minimal set M in X such that
| Ty1 F ∩ M |= 1
∀x ∈ M ,
where y = π(x) is the image of x by the projection π : X → M.
Lemma
Let M be a B-minimal set of X . If
| Ty1 F ∩ M |≥ 2
then M = X , namely X x B is minimal.
∀x ∈ M ,
Dynamics of the horocycle flow
Some progress in the non-homogeneous case
Foliations with ’topologically non-trivial’ leaves
Proof sketch. The proof is based on the two following lemmas:
Lemma
The leaves of F are the orbits of a continuous locally free B-action
if and only if there is a B-minimal set M in X such that
| Ty1 F ∩ M |= 1
∀x ∈ M ,
where y = π(x) is the image of x by the projection π : X → M.
Lemma
Let M be a B-minimal set of X . If
| Ty1 F ∩ M |≥ 2
then M = X , namely X x B is minimal.
Back
∀x ∈ M ,
Dynamics of the horocycle flow
Some progress in the non-homogeneous case
Foliations with ’topologically non-trivial’ leaves
Proof.
If M 6= X , then C = { t ∈ R | gt (M) = M } is a discrete
subgroup of R, so trivial or cyclic.
Dynamics of the horocycle flow
Some progress in the non-homogeneous case
Foliations with ’topologically non-trivial’ leaves
Proof.
If M 6= X , then C = { t ∈ R | gt (M) = M } is a discrete
subgroup of R, so trivial or cyclic.
By hypothesis, C is cyclic and hence the B-invariant set
[
[
MD =
gt (M) =
gt (M)
t∈R
is closed.
t∈[0,t0 ]
Dynamics of the horocycle flow
Some progress in the non-homogeneous case
Foliations with ’topologically non-trivial’ leaves
Proof.
If M 6= X , then C = { t ∈ R | gt (M) = M } is a discrete
subgroup of R, so trivial or cyclic.
By hypothesis, C is cyclic and hence the B-invariant set
[
[
MD =
gt (M) =
gt (M)
t∈R
t∈[0,t0 ]
is closed. Then
MD = X
by B-minimality.
Dynamics of the horocycle flow
Some progress in the non-homogeneous case
Foliations with ’topologically non-trivial’ leaves
Proof.
If M 6= X , then C = { t ∈ R | gt (M) = M } is a discrete
subgroup of R, so trivial or cyclic.
By hypothesis, C is cyclic and hence the B-invariant set
[
[
MD =
gt (M) =
gt (M)
t∈R
t∈[0,t0 ]
is closed. Then
MD = X
by B-minimality.
There is t ∈ [0, t0 ] such that gt (x) ∈ M.
Dynamics of the horocycle flow
Some progress in the non-homogeneous case
Foliations with ’topologically non-trivial’ leaves
Proof.
If M 6= X , then C = { t ∈ R | gt (M) = M } is a discrete
subgroup of R, so trivial or cyclic.
By hypothesis, C is cyclic and hence the B-invariant set
[
[
MD =
gt (M) =
gt (M)
t∈R
t∈[0,t0 ]
is closed. Then
MD = X
by B-minimality.
There is t ∈ [0, t0 ] such that gt (x) ∈ M. Since gt (xU) =
gt (x)U and M is U-invariant, gt (xU) ⊂ M and therefore
X = M.
Back
Dynamics of the horocycle flow
Some progress in the non-homogeneous case
Foliations with ’topologically non-trivial’ leaves
The horocycle flow is dual to the linear action
Γ y PSL(2, R)/U = R2 − {0}/{±I } = E
Dynamics of the horocycle flow
Some progress in the non-homogeneous case
Foliations with ’topologically non-trivial’ leaves
The horocycle flow is dual to the linear action
Γ y PSL(2, R)/U = R2 − {0}/{±I } = E
Definition
Let Γ be a non-elementary Fuchsian group. A limit point ξ ∈ Λ(Γ )
is said to be horocyclic if any horodisc centered at ξ contains points
of the orbit Γ z for any z ∈ H.
Dynamics of the horocycle flow
Some progress in the non-homogeneous case
Foliations with ’topologically non-trivial’ leaves
The horocycle flow is dual to the linear action
Γ y PSL(2, R)/U = R2 − {0}/{±I } = E
Definition
Let Γ be a non-elementary Fuchsian group of the first kind. A point
ξ ∈ ∂H is said to be horocyclic if any horodisc centered at ξ contains
points of the orbit Γ z for any z ∈ H.
Dynamics of the horocycle flow
Some progress in the non-homogeneous case
Foliations with ’topologically non-trivial’ leaves
The horocycle flow is dual to the linear action
Γ y PSL(2, R)/U = R2 − {0}/{±I } = E
Definition
Let Γ be a non-elementary Fuchsian group of the first kind. A point
ξ ∈ ∂H is said to be horocyclic if any horodisc centered at ξ contains
points of the orbit Γ z for any z ∈ H.
Theorem
For each v ∈ T 1 H, the following conditions are equivalent:
i) the point ξ = v (+∞) is horocyclic,
ii) the horocycle orbit of the projected point x ∈ T 1 L is dense,
iii) 0 belongs to Γυ where υ is the element of E corresponding to
the horocycle passing through v .
Dynamics of the horocycle flow
Some progress in the non-homogeneous case
Foliations with ’topologically non-trivial’ leaves
Proof.
From the previous theorem, any proper U-minimal set M =
6 ∅
1
in T L contains a horocycle centered at non-horocyclic limit
point ξ of Γ .
Dynamics of the horocycle flow
Some progress in the non-homogeneous case
Foliations with ’topologically non-trivial’ leaves
Proof.
From the previous theorem, any proper U-minimal set M =
6 ∅
1
in T L contains a horocycle centered at non-horocyclic limit
point ξ of Γ .
According to the proposition, for any point x in this horocycle,
there is t0 > 0 such that gt0 (x) ∈ xU = M.
Dynamics of the horocycle flow
Some progress in the non-homogeneous case
Foliations with ’topologically non-trivial’ leaves
Proof.
From the previous theorem, any proper U-minimal set M =
6 ∅
1
in T L contains a horocycle centered at non-horocyclic limit
point ξ of Γ .
According to the proposition, for any point x in this horocycle,
there is t0 > 0 such that gt0 (x) ∈ xU = M.
By duality, we deduce that
e t0 /2 (Γυ) = Γυ
Dynamics of the horocycle flow
Some progress in the non-homogeneous case
Foliations with ’topologically non-trivial’ leaves
Proof.
From the previous theorem, any proper U-minimal set M =
6 ∅
1
in T L contains a horocycle centered at non-horocyclic limit
point ξ of Γ .
According to the proposition, for any point x in this horocycle,
there is t0 > 0 such that gt0 (x) ∈ xU = M.
By duality, we deduce that
e t0 /2 (Γυ) = Γυ
and therefore 0 ∈ Γυ, which contradits the assumption that ξ
is non-horocyclic.
Back