= 4e

−2kD(0,γ˜z(t))

≤ 4e

2kD(0,z)

e

−2kD(z,γ˜z(t))

= 41 + |z|

1 − |z|

e

−2kΩ(h(z),h(z)−t)

(4.5) ,

where we made consecutive use of (2.4), the triangle inequality and the

conformal invariance of the hyperbolic distance. On the other hand,

1 − |γ˜z(t)|

2 ≥

1 − |γ˜z(t)|

1 + |γ˜z(t)|

= e

−2kD(0,γ˜z(t))

≥

1 − |z|

1 + |z|

e

−2kΩ(h(z),h(z)−t)

(4.6) ,

where we followed a similar process as above. As a result, through (4.4),

(4.5) and (4.6) we obtain

(4.7) 1 − |z|

1 + |z|

λΩ(h(z) − t)

e

2kΩ(h(z),h(z)−t)

≤ |G(˜γz(t))| ≤ 4

1 + |z|

1 − |z|

λΩ(h(z) − t)

e

2kΩ(h(z),h(z)−t)

,

for all t ∈ [0, Tz). Therefore, lim supt→Tz

|G(˜γz(t))| is finite if and only if

lim sup

t→Tz

λΩ(h(z) − t)

e

2kΩ(h(z),h(z)−t)

< +∞

which leads to the desired result.

(b) Now suppose that (ϕt) is elliptic with spectral value µ ∈ C, Reµ > 0.

The proof requires similar steps. Again, for a fixed z ∈ D, the backward

orbit ˜γz is Lipschitz if and only if lim supt→Tz G(˜γz(t)) < +∞. However,

this time by (2.1), G(˜γz(t)) = −µh(˜γz(t))/h′

(˜γz(t)), for all t ∈ [0, Tz). By

the definition of the Koenigs function and (2.3), this results in

|G(˜γz(t))| = |µ||e

µth(z)|

λΩ(e

µth(z))

λD(˜γz(t)) = |µh(z)|e

Reµt(1−|γ˜z(t)|

2

)λΩ(e

µth(z)).

Continuing exactly as in the non-elliptic case, we are led to the desired

result. □

Lemma 4.3 gives a characterization of the Lipschitz condition for backward orbits. Other than just yielding a second characterization, the importance of Theorem 1.3 lies on the fact that is more easily checked through

intuition and geometric consideration. Via the last theorem, we are also able

to prove certain helpful corollaries. Firstly, we will show that regularity is

a sufficient condition for a backward orbit to be Lipschitz.

Proof of Corollary 1.4. Fix z ∈ D and let h be the Koenigs function of (ϕt)

with Ω = h(D). The regularity of the backward orbit dictates that Tz = +∞.

There are three distinct cases for the kind of ˜γz as evidenced by Proposition

2.3. Either (ϕt) is elliptic and ˜γz(t) converges non-tangentially to a point

on the unit circle (first kind), or (ϕt) is non-elliptic and ˜γz converges nontangentially to a point on the unit circle other than the Denjoy-Wolff point,

We now move on to Corollary 1.5 which gives a practical way of finding

backward orbits that are Lipschitz, even if they are not regular. Indeed,

if we construct a convex simply connected domain that is also convex in

the positive direction (respectively µ-spirallike), then through a Riemann

mapping we may construct the associated non-elliptic (respectively elliptic

with spectral value µ) semigroup in D and all its backward orbits will be

Lipschitz. Of course, in the elliptic case, the µ-spirallike domain can only

be convex if µ > 0 (i.e. its imaginary part is 0).

Proof of Corollary 1.5. Let h be the Koenigs function of (ϕt) and so Ω =

h(D). Fix z ∈ D and consider its backward orbit ˜γz. Once again, we will

exploit Theorem 1.3 to produce the desired result.

Part A: Non-elliptic semigroups:

Following a similar procedure as in the previous corollary, using Lemma

2.1, we get λΩ(h(z) − t) ≤

1

δΩ(h(z)−t)

, for all t ≥ 0. On the other side, using

the convexity of Ω and Lemma 2.2, we get

kΩ(h(z), h(z) − t) ≥

1

2

log t

δΩ(h(z) − t)

.

Combining we get

λΩ(h(z) − t)

e

2kΩ(h(z),h(z)−t)

≤

1

δΩ(h(z)−t)

t

δΩ(h(z)−t)

=

1

t

.

As a result,

lim sup

t→Tz

λΩ(h(z) − t)

e

2kΩ(h(z),h(z)−t)

≤

1

Tz

∈ [0, +∞),

which means that ˜γz is Lipschitz by Theorem 1.3.

Part B: Elliptic semigroups:

The proof consists of analogue arguments. Once again λΩ(e

µth(z)) ≤

1

δΩ(e

µth(z)) . Likewise, using Lemma 2.2 in the convex case,

kΩ(h(z), eµth(z)) ≥

1

2

log |e

µt − 1||h(z)|

δΩ(e

µth(z)) .

Combining, we obtain

e

ReµtλΩ(e

µth(z))

e

2kΩ(h(z),eµth(z)) ≤

e

Reµt

δΩ(e

µth(z))

|e

µt−1||h(z)|

δΩ(e

µth(z))

≤

e

Reµt

(e

Reµt − 1)|h(z)|

.

Evidently,

lim sup

t→Tz

e

ReµtλΩ(e

µth(z))

e

2kΩ(h(z),eµth(z)) ∈ [1/h(z), +∞),

regardless of if Tz is finite or infinite. In view of Theorem 1.3, we get the

desired result.


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