After t steps, the dot will be at position Mod[m t, n] where n is the total number of positions, and m is the number of positions moved at each step. The repetition period is given by n/GCD[m, n]. The picture on page 613 shows the values of m and n for which this is equal to n.
An alternative interpretation of the system discussed here involves arranging the possible positions in a circle, so that at each step the dot goes a fraction m/n of the way around the circle. The repetition period is maximal when m/n is a fraction in lowest terms. The picture below shows the repetition periods as a function of the numerical size of the quantity m/n.