MEASURES OF CHAOS AND A SPECTRAL DECOMPOSITION OF DYNAMICAL-SYSTEMS ON THE INTERVAL

Let f : [0, 1] --> [0, 1] be continuous. For x, y is-an-element-of [0, 1], the upper and lower (distance) distribution functions, F(xy) and F(xy), are defined for any t greater-than-or-equal-to 0 as the lim su p and lim inf as n --> infinity of the average number of times that th e distance \f(i)(x) - f(i)(y)\ between the trajectories of x and y is less than t during the first n iterations. The spectrum of f is the sy stem SIGMA(f) of lower distribution functions which is characterized b y by the following properties: (1) The elements of SIGMA(f) are mutual ly incomparable; (2) for any F is-an-element-of SIGMA(f), there is a p erfect set P(F) not-equal empty set such that F(uv) = F and F(uv) = 1 for any distinct u, upsilon is-an-element-of P(F); (3) if S is a scra mbled set for f, then there are F, G in SIGMA(f) and a decomposition S = S(f) or S(G) (S(g) may be empty), such that F(uv) greater-than-or-e qual-to F if u, upsilon is-an-element-of S(F) and F(uv) greater-than-o r-equal-to G if u, upsilon is-an-element-of S(G). Our principal result s are: (1) If f has positive topological entropy, then SIGMA(f) is non empty and finite, and any F is-an-element-of SIGMA(f) is zero on an in terval [0, epsilon], where epsilon > 0 (and hence any P(F) is a scramb led set in the sense of Li and Yorke). (2) If f has zero topological e ntropy, then SIGMA(f) = {F} where F = 1. It follows that the spectrum of f provides a measure of the degree of chaos of f. In addition, a us eful numerical measure is the largest of the numbers integral-1/0 (1 - F(t)) dt, where F is-an-element-of SIGMA(f).