Random Processes in Hyperbolic Spaces Hyperbolic Brownian ...

19 Hyperbolic Geometry
Since, in view of (2.15), we have that
it follows that
1 − |φ(x)| 2 = 1 − |φ0(σ(x))| 2 = − 4[σ(x)]n+1 4xn+1
=
,
|σ(x) − en+1|
2 |σ(x) − en+1|
2
|φ(y) − φ(x)|
lim
=
y→x |y − x|
1 − |φ(x)|2
2xn+1
. (2.16)
In general, given a positive continuous function λ(x) on D ∈ Rn , a metric ρ(x, y) = inf
γ λ(γ(t)) |γ′ (t)|dt
and a bijection f of D onto D1 ∈ Rn such that
|f(y) − f(x)|
lim
= µ(x),
y→x |y − x|
it is possible to define a positive a continuous function λ1(x) = λ(x)/µ(x) on D1 and a metric
ρ1(x, y) = inf
γ λ1(γ(t)) |γ ′ (t)|dt on D1. It follows that f is an isometry of (D, ρ) onto (D1, ρ1)
(see e.g. Beardon (2) page 7).
Since φ0 ◦ σ is a bijection of H n+1 onto B n+1 that satisfies (2.16), taking into account (2.14), we
define λ(x) = 1
1−|φ(x)|2
2
, µ(x) = and λ1(x) =
xn+1 2xn+1
1−|φ(x)| 2 . It follows that φ = φ0◦σ is an isometry
between (H n+1 , η) and (B n+1 , η1) where η1 is the metric obtained from the differential
ds = 2|dx|
. (2.17)
1 − |x| 2
The upper-half plane H 2 = {z = x + iy ∈ C : Im[z] > 0} with the metric η derived from the
differential ds = |dz|
Im[z] and the disk B2 = {z = x+iy ∈ C : |z| < 1} with the metric η1 derived from
the differential ds = 2|du|
1−|u| 2 are two models for the hyperbolic plane. We give now a description of
the hyperbolic metric in the upper-half plane H 2 and in the disk B 2 .
Now let
g(z) =
az + b
cz + d
(2.18)
be a Möbius transformation of the form (2.6), such that a, b, c, d are real and ad − bc > 0. Since
for every z = x + iy ∈ H 2 we have
g(z) =
ax + b + iay
cx + d + icy
= (ax + b)(cx + d) + acy2
(cx + d) 2 + c 2 y 2
it follows that g(z) maps H 2 onto itself. Observing that
y(ad − bc)
+ i
(cx + d) 2 + c2y2 |g ′
(z)| =
a(cz + d) − c(az + b)
(cz + d) 2
=
ad − bc
(cz
+ d) 2
=
ad − bc
(cx + d) 2 + c2y2