Goto Chapter: Top 1 2 3 4 5 6 Bib Ind
 [Top of Book]  [Contents]   [Previous Chapter]   [Next Chapter] 

2 IMG package
 2.1 A brief mathematical introduction
 2.2 An example session

2 IMG package

2.1 A brief mathematical introduction

This chapter assumes that you have no familiarity with iterated monodromy groups. If you do, and wish to see their usage within GAP through a sample session, please skip to Section 2.2. For a more thourough introduction on self-similar groups and iterated monodromy groups, see [BGN03] or [Nek05].

Basic knowledge of the package FR is assumed, however. Please refer to its online documentation, or the same references as above.

The fundamental objects are Thurston maps: branched self-coverings of the sphere S^2. These are continuous maps f:S^2-> S^2 that, locally, are modelled on the complex map z↦ z^n. The critical points are those points z at which the map is modelled on z↦ z^n for some n>1. The post-critical set is the union P of strictly forward orbits of critical points. It is assumed finite.

Two Thurston maps f:(S^2,P)->(S^2,P) and g:(S^2,Q)->(S^2,Q) are combinatorially equivalent if they are isotopic through isotopies (S^2,P)->(S^2,Q) that are constant on P.

Denote by G the fundamental group π_1(S^2∖ P,*) at a basepoint *. Then combinatorial equivalence classes of Thurston maps are classified by wreath recursions for G, namely homomorphisms ϕ:G-> G≀ Sym_d. They are encoded, in IMG, by a new kind of FR machine, called a sphere machine. Generators of the machine correspond to loops in the fundamental group of the sphere (punctured at post-critical points), that circle once counter-clockwise around a post-critical point.

By a fundamental theorem of Thurston, every Thurston map (apart from a few low-complexity examples) is either combinatorially equivalent to a rational map, or is obstructed: there exists a system of curves on S^2∖ P with some appropriate non-contraction property.

The operations in this package let one manipulate Thurston maps, and in particular

2.2 An example session

This is a brief introduction describing some of the simpler features of the IMG package. It assumes you have some familiarity with the theory of groups defined by automata, and of holomorphic dynamical systems; if not, a brief mathematical introduction may be found in Section 2.1. We show here and comment a typical use of the package.

The package is installed by unpacking the archive in the pkg/ directory of your GAP installation. It can also be placed in a local directory, which must be added to the load-path by invoking gap with the -l option.

gap> LoadPackage("img");
true

Many maps and sphere machines are predefined by the IMG, see Chapter 5.

We may start by defining a machine by its polynomial:

gap> basilica := PolynomialSphereMachine(2,[1/3]);
<sphere machine with alphabet [ 1 .. 2 ] and adder FRElement(...,f3) on Group( [ f1, f2, f3 ] )/[ f\
3*f2*f1 ]>
gap> Display(basilica);
 G  |      1         2
----+---------+---------+
 f1 | f1^-1,2   f3^-1,1
 f2 |    f1,1    <id>,2
 f3 |    f3,2    <id>,1
----+---------+---------+
Adding element: FRElement(...,f3)
Relators: [ f3*f2*f1 ]
gap> P1MapBySphereMachine(basilica);
<z^2-1._z>

We have just created the basilica machine, with group G=⟨ f1,f2,f3|f3f2f1⟩, and computed the corresponding rational map.

We now compute the mating of the basilica with itself: that is the Thurston map which acts as z^2-1 on the upper hemisphere, by angle doubling on the equator, and also as z^2-1 on the lower hemisphere. It turns out that this map is obstructed, i.e. does not have a realization as a rational map. The obstruction consists of a curve separating the points -1 and 0 on both hemispheres:

gap> basilica2 := Mating(basilica,basilica);
<sphere machine with alphabet [ 1 .. 2 ] on Group( [ f1, f2, g1, g2 ] ) / [ f2*f1*g2*g1 ]>
gap> P1MapBySphereMachine(last);
rec(
  machine := <sphere machine with alphabet [ 1, 2 ] on Group( [ f1, f2, g1, g2 ] ) / [ f2*f1*g2*g1 \
]>, matrix := [ [ 1 ] ], multicurve := [ f1*g1^G ] )
 [Top of Book]  [Contents]   [Previous Chapter]   [Next Chapter] 
Goto Chapter: Top 1 2 3 4 5 6 Bib Ind

generated by GAPDoc2HTML