This program is a demo of the Poincare recurrence effect that occurs 
when a transformation is applied repeatedly to a mathematical system.




When a transformation is applied repeatedly to a mathematical system,
and the system cannot leave a bounded region, it returns infinitely
often to states near its original state. This is known as the Poincare 
recurrence, after the French mathematician Henri Poincare 
(born at Nancy the 29th April 1854; died Paris 17th July 1912)  
More details in:
  J.P. Crutchield, J.D. Farmer, N.H. Packard and R.S. Shaw;
  'Chaos';Scientific American, 255 (December 1986), pp. 38-49.
The program implements a discrete version described in
  F.J. Dyson and H. Falk;
  'Period of a Discrete Cat Mapping'; The American Mathematical
  Monthly, 99 (#7, August-September 1992), pp. 603-614
The program creates a figure of a cat (fox ?) in a box of 20 by 20 
pixels. Another box at its right displays the resulting image after 
applying a certain number of transformations to the original.
Each transformation is defined by the matrix 
[[1 1] [1 2]] (you may try other matrices)
and puts pixel (x,y) from the image to position (x',y')
in the transformed image:
    [x']   [ 1  1 ] [x]
	[  ] = [      ] [ ]
    [y']   [ 1  2 ] [y]
Function keys F3 and F4 execute one or five transformations
more. F5 quits the program. F1 and F2 are not used.
The menu field corresponding to F2 displays the current number
of transformations applied to the original image.
Look at transformation 15 and 30.
I use a Mac version of this example in a Lab session of my course 
on Linear Algebra as an introduction to matrix powers and linear 
transformations.
Comments or improvements are wellcomed:
    
     Francesc Comellas, Dept. Matematica Aplicada i Telematica,
     Universitat Politecnica de Catalunya,
     Campus Nord, B3; 08071 BARCELONA, Catalonia, Spain.
     comellas@mat.upc.es