In order to analyse the way in which the size of the iterates of a transcendental entire function
f
f
can behave, we introduce the concept of the annular itinerary of a point
z
z
. This is the sequence of non-negative integers
s
0
s
1
…
s_0s_1\ldots
defined by
\[
f
n
(
z
)
∈
A
s
n
(
R
)
,
for
n
≥
0
,
f^n(z)\in A_{s_n}(R),\;\;\text {for }n\ge 0,
\]
where
A
0
(
R
)
=
{
z
:
|
z
|
>
R
}
A_0(R)=\{z:|z|>R\}
and
\[
A
n
(
R
)
=
{
z
:
M
n
−
1
(
R
)
≤
|
z
|
>
M
n
(
R
)
}
,
n
≥
1.
A_n(R)=\{z:M^{n-1}(R)\le |z|>M^n(R)\},\;\;n\ge 1.
\]
Here
M
(
r
)
M(r)
is the maximum modulus of
f
f
on
{
z
:
|
z
|
=
r
}
\{z:|z|=r\}
and
R
>
0
R>0
is so large that
M
(
r
)
>
r
M(r)>r
, for
r
≥
R
r\ge R
.
We consider the different types of annular itineraries that can occur for any transcendental entire function
f
f
and show that it is always possible to find points with various types of prescribed annular itineraries. The proofs use two new annuli covering results that are of wider interest.