ALLOWABLE MAPS

The allowable map types (INPUT FUNCTIONS) for the JULIA program are the functions and their iterates of any map given below.  Note that each is a mapping from the complex plane into the complex plane.

Complex Linear (complex coefficients)
z → az +b, with a ≠0

Complex Mobius (complex coefficients)
z → (az + b)/(cz +d), with ad-bc ≠0

Complex Quadratic (complex coefficients)
z → az² +bz + c, with a ≠0

Complex Cubic (without the z² or z terms) (complex coefficients)
z → az³ + b, with a ≠0

Real Affine Linear (real coefficients) -These map the REAL vector z = [x,y] to a REAL vector by first applying a 2x2 REAL matrix, and then adding the vector [e, f].  Thus it is the map [x, y] → [ax+by+e, cx+dy+f], i.e.,

Note: when a pre-image under this map is required, the determinant condition ad-bc ≠0 must be met. 

NOTE:  All complex coefficients will be entered as ordered pairs. Thus 3+5i is written (3,5).

The m-value determines the ITERATE to be used

When an INPUT FUNCTION is entered, the user must also input the m-value, a positive integer used to denote which iterate of the given map is to be used.  Whenever this map is called by the program, it will always take into account this iterate m-value.  A value m=1 should be used if you do not wish a higher iterate to be used. 

EXAMPLE
Entering the map (1,0)z² + (0,1), m=2 will indicate the second iterate of the map z² + i thus producing the map
z → (z² + i)² +i.  We will refer to this situation by saying that the map f(z)=z² + i is the sub-generator function for the INPUT FUNCTION (generator function) G(z)=(z² + i)² +i where m=2.  That is, G(z) is equal to the 2nd iterate of f(z).