Pearson-walk visualization of the characteristic function of the invariant measure for 1-d chaos

The Pearson-walk visualization of one-dimensional (1-d) chaos, which has been proposed in a qualitative fashion by the same authors recently (Physica 134A (1985) 123) is treated quantitatively. Continuity of the Pearson image is deduced for a map f(x) whose kth iterate fk(x) is continuous for any k. Then the Lyapunov exponent is used to describe the length of the Pearson image. The normalized Pearson-walk visualization is introduced to show that it can be related to the existence of the invariant measure. For a 1-d map that has a definite invariant measure it is shown that the characteristic function of the invariant measure is represented by a unique point in the normalized Pearson plane for a large iteration-number limit.