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5 Examples

IMG predefines a large collection of sphere machines, as well as the generic constructions of polynomials.

5.1 Examples of groups

DeclareGlobalFunction("PoirierExamples");

5.1-1 PoirierExamples

The examples from Poirier's paper [Poi]. See details under PolynomialSphereMachine (3.3-1); in particular, PoirierExamples(1) is the Douady rabbit map.

DeclareGlobalFunction("DBRationalIMGGroup");

5.1-2 DBRationalIMGGroup

Returns: An IMG group from Dau's database.

This function returns the iterated monodromy group from a database of groups associated to quadratic rational maps. This database has been compiled by Dau Truong Tan [Tan02].

When called with no arguments, this command returns the database contents in raw form.

The argments can be a sequence; the first integer is the size of the postcritical set, the second argument is an index for the postcritical graph, and sometimes a third argument distinguishes between maps with same post-critical graph.

If the argument is a rational map, the command returns the IMG group of that map, assuming its canonical quadratic rational form form exists in the database.

gap> DBRationalIMGGroup(z^2-1);
IMG((z-1)^2)
gap> DBRationalIMGGroup(z^2+1); # not post-critically finite
fail
gap> DBRationalIMGGroup(4,1,1);
IMG((z/h+1)^2|2h^3+2h^2+2h+1=0,h~-0.64)

DeclareGlobalFunction("PostCriticalMachine");

5.1-3 PostCriticalMachine

Returns: The Mealy machine of f's post-critical orbit.

This function constructs a Mealy machine P on the alphabet [1], which describes the post-critical set of f. It is in fact an oriented graph with constant out-degree 1. It is most conveniently passed to Draw (fr: Draw).

The attribute Correspondence(P) is the list of values associated with the stateset of P.

gap> z := Indeterminate(Rationals,"z");;
gap> m := PostCriticalMachine(z^2);
<Mealy machine on alphabet [ 1 ] with 2 states>
gap> Display(m);
   |  1
---+-----+
 a | a,1
 b | b,1
---+-----+
gap> Correspondence(m);
[ 0, infinity ]
gap> m := PostCriticalMachine(z^2-1);; Display(m); Correspondence(m);
   |  1
---+-----+
 a | c,1
 b | b,1
 c | a,1
---+-----+
[ -1, infinity, 0 ]

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