Let
T
T
be an operator on
C
(
X
)
C(X)
,
X
X
compact, with
‖
T
‖
⩽
1
\left \| T \right \| \leqslant 1
, and suppose
T
T
has a nowhere vanishing invariant function
ψ
−
1
{\psi ^{ - 1}}
. The operator
R
R
defined by
R
f
=
T
(
f
ψ
−
1
)
ψ
Rf = T(f{\psi ^{ - 1}})\psi
is (a) "locally" a Markov operator, and (b) (locally) strongly ergodic iff
T
T
is. This is used to prove Sine’s local strong ergodicity theorem without assuming that
T
T
is positive.