General Mathematics Vol. 12, No. 2 (2004), 11–18
Quasisymmetric and quasimöbius maps on
locally convex spaces
Silviu Crăciunaş, Petru Crăciunaş
Abstract
We give the definitions of the quasisymmetric and quasimöbius
maps, maps defined on a locally convex spaces and some properties
of this applications.
2000 Mathematical Subject Classification: 30C65
Let E be a locally convex spaces. We denote by A, the family of
continuous semi-norms on E.
Definition 1.Let (x1 , x2 , x3 ) be a triple of distinct points in E. For α ∈ A,
the α–ration of (x1 , x2 , x3 ) is the number ρα (x1 , x2 , x3 ) defined by


 |x2 − x1 |α , if |x3 − x1 |α 6= 0



|x3 − x1 |α



0,
if |x2 − x1 |α = |x3 − x1 |α = 0
ρα (x1 , x2 , x3 ) =






∞
if |x2 − x1 |α 6= 0 , |x3 − x1 |α = 0


11
.
12
Silviu Crăciunaş and Petru Crăciunaş
Definition 2.Let (x1 , x2 , x3 , x4 ) be a quadruple of distinct points in E. For
α ∈ A, the α–cross ratio of (x1 , x2 , x3 , x4 ) is the number τα (x1 , x2 , x3 , x4 )
defined by

|x1 − x3 |α |x2 − x4 |α


·
if |x1 − x4 |α · |x2 − x3 |α 6= 0


|x
−
x
|
|x
−
x
|
1
4
α
2
3
α





 |x1 − x3 |α · |x2 − x4 |α = 0



0 if
 |x − x | · |x − x | = 0
.
τα (x1 , x2 , x3 , x4 ) =
1
4 α
2
3 α






 |x1 − x3 |α · |x2 − x4 |α 6= 0




∞
if


 |x − x | · |x − x | = 0
1
4 α
2
3 α
Proposition 1.Lets E, F two locally convex spaces and A, B the families
of continuous semi-norms on E and F respectivelly. For ϕ : B −→ A, and
a homeomorphism η : [0, ∞] −→ [0, ∞] (η(∞) = ∞), if f : E −→ F verifies
(1)
|x0 − x00 |ϕ(β) = 0 is equivalent with |f (x0 ) − f (x00 )|β = 0
for any x0 , x00 ∈ E, then for
η(ρϕ(β) (x1 , x2 , x3 )) ∈ {0, ∞}
we have
ρβ (f (x1 ), f (x2 ), f (x3 )) = η(ρϕ(β) (x1 , x2 , x3 )).
Proof. If
η(ρϕ(β) (x1 , x2 , x3 )) = 0
then
ρϕ(β) (x1 , x2 , x3 ) = 0
Quasisymmetric and quasimöbius maps on locally convex spaces
13
and using the definition 1,

 |x2 − x1 |ϕ(β) = 0
 |x − x |
3
1 ϕ(β) ≥ 0.
From (1) we have
|f (x2 ) − f (x1 )|β = 0,
|f (x3 ) − f (x1 )|β ≥ 0
and consequently,
ρβ (f (x1 ), f (x2 ), f (x3 )) = 0.
If
η(ρϕ(β) (x1 , x2 , x3 )) = ∞
then
ρϕ(β) (x1 , x2 , x3 ) = ∞
and from the definition 2,

 |x2 − x1 |ϕ(β) 6= 0
 |x − x |
3
1 ϕ(β) = 0.
From (1) we have

 |f (x2 ) − f (x1 )|β 6= 0
 |f (x ) − f (x )| = 0
3
1 β
and
ρβ (f (x1 ), f (x2 ), f (x3 )) = ∞.
Proposition 2. Lets E, F two locally convex spaces and A, B the families
of continuous semi-norms on E and F respectivelly. For ϕ : B −→ A, and
a homeomorphism η : [0, ∞] −→ [0, ∞] (η(∞) = ∞), if f : E −→ F verifies
|x0 − x00 |ϕ(β) = 0 is equivalent with |f (x0 ) − f (x00 )|β = 0
14
Silviu Crăciunaş and Petru Crăciunaş
for any x0 , x00 ∈ E, then for
η(τϕ(β) (x1 , x2 , x3 , x4 )) ∈ {0, ∞}
we have
τβ (f (x1 ), f (x2 ), f (x3 ), f (x4 )) = η(τϕ(β) (x1 , x2 , x3 , x4 )).
Proof. We have the implications
η(τϕ(β) (x1 , x2 , x3 , x4 )) = 0 which implies τϕ(β) (x1 , x2 , x3 , x4 ) = 0
and from the definition 2,

 |x1 − x3 |ϕ(β) · |x2 − x4 |ϕ(β) = 0
 |x − x |
1
4 ϕ(β) · |x2 − x3 |ϕ(β) ≥ 0.
Using (1) we have

 |f (x1 ) − f (x3 )|β · |f (x2 ) − f (x4 )|β = 0
 |f (x ) − f (x )| · |f (x ) − f (x )| ≥ 0
1
4 β
2
3 β
and
τβ (f (x1 ), f (x2 ), f (x3 ), f (x4 )) = 0.
Also if
η(τϕ(β) (x1 , x2 , x3 , x4 )) = ∞
then
τϕ(β) (x1 , x2 , x3 , x4 ) = ∞
and

 |x1 − x3 |ϕ(β) · |x2 − x4 |ϕ(β) 6= 0
 |x − x |
1
4 ϕ(β) · |x2 − x3 |ϕ(β) = 0
which implies
Quasisymmetric and quasimöbius maps on locally convex spaces
15

 |f (x1 ) − f (x3 )|β · |f (x2 ) − f (x4 )|β 6= 0
 |f (x ) − f (x )| · |f (x ) − f (x )| = 0.
1
4 β
2
3 β
Finally,
τβ (f (x1 ), f (x2 ), f (x3 ), f (x4 )) = ∞.
Definition 3. Lets E, F two locally convex spaces, A, B the families of
continuous semi-norms on E and F respectively and D ⊂ E a open set. We
say that f : D −→ F is a quasisymmetric map if there exists ϕ : B −→ A
and a homeomorphism η : [0, ∞] −→ [0, ∞] (η(∞) = ∞) such that
(i) |x1 − x2 |ϕ(β) = 0 is equivalent with |f (x1 ) − f (x2 )|β = 0
(ii) ρβ (f (x1 ), f (x2 ), f (x3 )) ≤ η(ρϕ(β) (x1 , x2 , x3 ))
for any x1 , x2 , x3 ∈ D.
Definition 4.Lets E, F two locally convex spaces, A, B the families of
continuous semi-norms on E and F , respectively and D ⊂ E a open set.
We say that f : D −→ F is a quasiimöbius map if there exists ϕ : B −→ A
and a homeomorphism η : [0, ∞] −→ [0, ∞] (η(∞) = ∞) such that
(i) |x1 − x2 |ϕ(β) = 0 is equivalent with |f (x1 ) − f (x2 )|β = 0
(ii) τβ (f (x1 ), f (x2 ), f (x3 ), f (x4 )) ≤ η(τϕ(β) (x1 , x2 , x3 , x4 ))
for any x1 , x2 , x3 , x4 ∈ D.
Proposition 3.If T ∈ Isom (E, F ), then there exists ϕ : B −→ A so that
T is quasisymmetric and quasimöbius for η : [0, ∞] −→ [0, ∞], η(t) = t.
16
Silviu Crăciunaş and Petru Crăciunaş
Proof. Let β ∈ B. The application x −→ |T x|β is a continuous semi-norm
on E. There exists α ∈ A so that |x|α = |T x|β . We define ϕ : B −→ A,
ϕ(β) = α.
If x1 , x2 ∈ E
|T (x1 ) − Tx2 |β = |T (x1 − x2 )|β = |x1 − x2 |ϕ(β)
and (i) is satisfied.
Also, we have,
ρβ (T (x1 ), T (x2 ), T (x3 )) = ρϕ(β) (x1 , x2 , x3 )
τβ (T (x1 ), T (x2 ), T (x3 ), T (x4 )) = τϕ(β) (x1 , x2 , x3 , x4 )
for any x1 , x2 , x3 , x4 ∈ E and the conditions (ii) from the two definitions
are true for η(t) = t.
Proposition 4.If
|x1 − x3 |α · |x1 − x4 |α + |x1 − x4 |α · |x2 − x3 |α 6= 0 ,
then
τα (x1 , x2 , x3 , x4 ) =
Proof. If
1
.
τα (x1 , x2 , x4 , x3 )


 |x1 − x3 |α · |x2 − x4 |α 6= 0

 |x − x | · |x − x | 6= 0
1
4 α
2
3 α
then
|x1 − x3 |α
|x1 − x4 |α
¶−1
µ
|x1 − x4 |α |x2 − x3 |α
=
·
=
|x1 − x3 |α |x2 − x4 |α
τα (x1 , x2 , x3 , x4 ) =
·
|x2 − x4 |α
=
|x2 − x3 |α
1
.
τα (x1 , x2 , x4 , x3 )
Quasisymmetric and quasimöbius maps on locally convex spaces
If
17


 |x1 − x3 |α · |x2 − x4 |α = 0

 |x − x | · |x − x | 6= 0
1
4 α
2
3 α
then
τα (x1 , x2 , x3 , x4 ) = 0
and
τα (x1 , x2 , x4 , x3 ) = ∞ ,
and if


 |x1 − x3 |α · |x2 − x4 |α 6= 0

 |x − x | · |x − x | = 0
1
4 α
2
3 α
then
τα (x1 , x2 , x3 , x4 ) = ∞
and
τα (x1 , x2 , x4 , x3 ) = 0.
In the last two cases, we have also
τα (x1 , x2 , x3 , x4 ) =
1
.
τα (x1 , x2 , x4 , x3 )
Proposition 5.If f : D −→ F is a quasimöbius map then we have
−1
η −1 (τϕ(β)
(x1 , x2 , x3 , x4 ) ≤ τβ (f (x1 ), f (x2 ), f (x3 ), f (x4 )) ≤ η(τϕ(β) (x1 , x2 , x3 , x4 )),
for any x1 , x2 , x3 , x4 ∈ D with
|x1 − x3 |ϕ(β) · |x2 − x4 |ϕ(β) + |x1 − x4 |ϕ(β) · |x2 − x3 |ϕ(β) 6= 0 .
18
Silviu Crăciunaş and Petru Crăciunaş
Proof. Firstly, we have
|x1 − x3 |ϕ(β) · |x2 − x4 |ϕ(β) + |x1 − x4 |ϕ(β) · |x2 − x3 |ϕ(β) 6= 0
is equivalent with
|f (x1 ) − f (x3 )|β · |f (x2 ) − f (x4 )|β + |f (x1 ) − f (x4 )|β · |f (x2 ) − f (x3 )|β 6= 0
and using the previous proposition, we can write:
−1
η −1 (τϕ(β)
(x1 , x2 , x3 , x4 ) =
µ
η
≤
1
1
¶=
≤
1
η(τϕ(β) (x1 , x2 , x4 , x3 ))
τϕ(β) (x1 , x2 , x3 , x4 )
1
= τβ (f (x1 ), f (x2 ), f (x3 ), f (x4 )).
τβ (f (x1 ), f (x2 ), f (x3 ), f (x4 ))
≤ η(τϕ(β) (x1 , x2 , x3 , x4 ).
References
[1] Crăciunaş, S., Quasicoformité dans les espaces localements convexes,
Bull. Math. de la Soc. Math. de Roumanie, Tome 34(82), 1 (1990),
8-16
[2] Väisälä, J., Quasimöbius maps, J. Anal. Math. 44 (1984/85), 218-134.
Department of Mathematics
“Lucian Blaga” University of Sibiu
Str. Dr. I. Raţiu, No. 5 - 7
550012 - Sibiu, Romania
E-mail: silviu.craciunas@ulbsibiu.ro