The invariant measure is uniform in the y direction, and in the x direction is exactly given by the "two scale factor Cantor set" with the probabilities p1=a, p2=1-a, and the lengths l1=b and l2=c. If we use the values 0.6 and 0.4 for the two probabilities and 0.25 and 0.4 for the lengths, the expected dimensions can be calculated from equation (10.24) or read off the figures in the pdf file. Evaluate various dimensions using the box counting method, and compare with these expectations. (Remember, of course, that to get the dimension for the bakers' map you must add one to the dimension of the line set, to take into account the continuous distribution of points in the ydirection.
Having exact values for the dimensions allows us to get some further insights into the limitations of the box counting method. For example compare the success of the box counting algorithm in computing D(q=0) = DC and the information density D(q=1).Return to text.