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.

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Contemporary Mathematics

Volume 510, 2010

1

http://dx.doi.org/10.1090/conm/510/10013