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.
We give an explicit description of all finite Borel measures on Noetherian topological spaces X, and characterize them as objects dual to a space of functions on X. We use these results to study the asymptotic behavior of continuous dynamical systems on Noetherian spaces.
In this article we prove an analogue of the equidistribution of preimages theorem from complex dynamics for maps of good reduction over nonarchimedean fields. While in general our result is only a partial analogue of the complex equidistribution theorem, for most maps of good reduction it is a complete analogue. In the particular case when the nonarchimedean field in question is equipped with the trivial absolute value, we are able to supply a strengthening of the theorem, namely that the preimages of any tame valuation equidistribute to a canonical measure.
We show by explicit example that local intersection multiplicities in holomorphic dynamical systems can grow arbitrarily fast, answering a question of V. I. Arnold. On the other hand, we provide results showing that such behavior is exceptional, and that typically local intersection multiplicities grow subexponentially.Date: August 21, 2017.
We study the sequence of attraction rates of iterates of a dominant superattracting holomorphic fixed point germ f : (C 2 , 0) → (C 2 , 0). By using valuative techniques similar to those developed by Favre-Jonsson, we show that this sequence eventually satisfies an integral linear recursion relation, which, up to replacing f by an iterate, can be taken to have order at most two. In addition, when the germ f is finite, we show the existence of a bimeromorphic model of (C 2 , 0) where f satisfies a weak local algebraic stability condition.
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