If
f
f
is an orientation preserving homeomorphism of the plane with an isolated fixed point at the origin 0 and
index(
f
,
0
) =
p
{\text {index(}}f,0{\text {) = }}p
, then
index(
f
n
,0)
{\text {index(}}{f^n}{\text {,0)}}
is always well defined provided that
p
≠
1
p \ne 1
. In this case, for each
n
≠
0
n \ne 0
,
index(
f
n
,0) = index(
f
,
o
) =
p
{\text {index(}}{f^n}{\text {,0) = index(}}f,o{\text {) = }}p
. If
index(
f
,
0
) = 1
{\text {index(}}f,0{\text {) = 1}}
, then there is an integer
p
p
(possibly
p
=
1
p = 1
) such that for those values of
n
n
for which
index(
f
n
,0)
{\text {index(}}{f^n}{\text {,0)}}
is defined (i.e 0 is an isolated fixed point of
f
n
{f^n}
),
index(
f
n
,0) = 1
{\text {index(}}{f^n}{\text {,0) = 1}}
or
index(
f
n
,0) =
p
{\text {index(}}{f^n}{\text {,0) = }}p
.