Iterated Monodromy for a Two-Dimensional Map
James Belk and Sarah Koch
Abstract. We compute the iterated monodromy group for a postcritically
finite endomorphism F of
P2.
The postcritical set is the union of six lines, and
the wreath recursion for the group closely reflects the dynamics of F on these
lines.
Introduction
In [BN], L. Bartholdi and V. Nekrashevych solved the twisted rabbit problem
with iterated monodromy groups. Their work has brought new tools to bear in
the fields of dynamics and algebra. In [N2], V. Nekrashevych uses a more general
notion of iterated monodromy group to obtain combinatorial models for Julia sets
of certain maps of several complex variables. Other than this, little has been
done with iterated monodromy groups in dimensions greater than one. Here we
compute the iterated monodromy group for a postcritically finite endomorphism
F :
P2

P2.
The ideas used in this computation could generalize to calculate the
iterated monodromy groups for other maps
Pn

Pn.
Let F :
C2

C2
be the following rational function:
F (x, y) = 1
y2
x2
, 1
1
x2
.
Then F extends to a holomorphic endomorphism of the complex projective plane
P2,
i.e. an everywhere-defined holomorphic map
P2

P2.
In homogeneous coordinates,
this endomorphism is given by F (x : y : z) =
(x2

y2
:
x2

z2
:
x2).
Topologically, the map F is a branched cover of degree four, with fibers of the
form {(x, y), (−x, y), (x, −y), (−x, −y)}. The critical locus of F is the union of the
complex lines x = 0 and y = 0 in
C2,
as well as the line at infinity L∞ :=
P2
\
C2,
and F restricts to a covering map on the complement of these lines.
The postcritical locus of F is the forward orbit of the critical locus. A map is
called postcritically finite if the postcritical locus is an algebraic set, i.e. the union
of finitely many algebraic varieties. (Postcritically finite endomorphisms were first
studied by Fornæss and Sibony in [FS].) Our map F is postcritically finite, and
1991 Mathematics Subject Classification. Primary 20D99; Secondary 32A99.
Key words and phrases. Iterated monodromy groups, postcritically finite endomorphisms.
Belk is partially supported by a MSPRF from the NSF.
Koch is partially supported by a MSPRF from the NSF.
1
Contemporary Mathematics
Volume 510, 2010
1
http://dx.doi.org/10.1090/conm/510/10013
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