We study the problem of finding algebraically stable models for non-invertible holomorphic fixed point germs
f
:
(
X
,
x
0
)
→
(
X
,
x
0
)
f\colon (X,x_0)\to (X,x_0)
, where
X
X
is a complex surface having
x
0
x_0
as a normal singularity. We prove that as long as
x
0
x_0
is not a cusp singularity of
X
X
, then it is possible to find arbitrarily high modifications
π
:
X
π
→
(
X
,
x
0
)
\pi \colon X_\pi \to (X,x_0)
such that the dynamics of
f
f
(or more precisely of
f
N
f^N
for
N
N
big enough) on
X
π
X_\pi
is algebraically stable. This result is proved by understanding the dynamics induced by
f
f
on a space of valuations associated to
X
X
; in fact, we are able to give a strong classification of all the possible dynamical behaviors of
f
f
on this valuation space. We also deduce a precise description of the behavior of the sequence of attraction rates for the iterates of
f
f
. Finally, we prove that in this setting the first dynamical degree is always a quadratic integer.
Local dynamics of non-invertible maps near normal surface singularities