Lyapunov exponents and complexity for interval maps

Abstract

The amount of irregularity in trajectories of dynamical systems can be quantified in various ways. From a geometrical point of view, Lyapunov exponents measure the dependence of the future behaviour on small changes in the system's initial conditions. A statistician might use entropy as a measure of his uncertainty in predicting the future of the system from its past. Algorithmic complexity measures the amount of information needed to reproduce a finite section of the trajectory with a fixed precision on a universal computer. In this note we examine the irregularity of trajectories produced by interval transformations. For very stable (contracting) and for very unstable (expanding) maps we see that the above three measures of irregularity essentially coincide. For some classes of maps in between, however, we find out that algorithmic complexity can describe aspects of irregularity to which entropy and Lyapunov exponents are unsensitive. The dynamics of these maps are so subtle that up to now rigorous results are available only for unimodal interval maps with negative Schwarzian derivative. In the sequel the following notations are used constantly: A transformation T : [0, 1] [0,1] is called piecewise monotone, if there exists a partition Z of [0,1] into finitely many intervals such that Tz = Tlz : Z --* [0,1] is monotone and continuous. A piecewise monotone T is called piecewisc C r, if each Tz is of class C r and has bounded r-th derivative. The endpoints of the intervals Z E Z are called the critical points of T. For a piecewise monotone T let Zn = Z V T 1 Z Y . . . V T ( I )Z be the partition of [0, 1] into monotonicity and continuity intervals of T ~. Z,[x] denotes that interval in Z= which contains x, Z~[x] := n=>0Z,[x]. Z~[x] is an interval containing x, often it is just the set {x}. The Lebesgue measure on [0, 1] is denoted by m, and by a measure we always mean a Borel probability measure. In order to avoid frequent normalizing by ~ 1 in Section 3, we take henceforth all logarithms to base 2.