Consider the orbits of the Gauss map

Since the map has as its derivative a ``sawtooth function'' with slope 1, the discontinuous derivative of the Gauss map is

provided is not an integer. (The points are points of discontinuity of the Gauss map, so its derivative obviously does not exist at those points.)

The following explorations are meant to convince you that the claims made about the Gauss map in this paper are in fact true. The links below will call up the Maple Form Interface. Please read the details in this document before calling up the Maple Form Interface if you are not familiar with Maple.

1. This code simply generates the first N iterates of the orbit orb(x0) of the Gauss map. Initially, and N=5. Try changing x0 to rational or irrational values. Verify that is a fixed point of the Gauss map. (Various values are included in the code to simplify typing. To use these values of x0, delete the # mark at the beginning of the line with the desired value of x0 and insert a # mark at the beginning of the previous value assigned to x0. The # mark has the effect of ``commenting out'' a line of Maple code so that anything to the right of # will be ignored when the code is executed.)
2. This code calculates the finite time analog of the Lyapunov exponent of an orbit orb(x0) of the Gauss map. Initially, and N=200. Change x0 to various rational, irrational and reduced quadratic irrational values. Do the values of behave as expected? (Various values are included in the code to simplify typing. To use these values of x0, delete the # mark at the beginning of the line with the desired value of x0 and insert a # mark at the beginning of the previous value assigned to x0. The # mark has the effect of ``commenting out'' a line of Maple code so that anything to the right of # will be ignored when the code is executed.)