Distortion of S-unimodal maps


For $I = [-1, 1]$ let $f: I \to I$ be a $C^3$ transformation such that $f(x) = h(x^2)$ where $h$ is monotone and has a negative (non-positive) Schwarzian derivative. It is known (see [11]) that the behavior of such maps falls into one of three categories: (a) There is an attracting fixed point or periodic orbit that attracts almost every point, (b) there is an attracting cantor set that attracts almost every point and upon which f is homeomorphic, or (c) the map $f$ is sensitive to initial conditions. It is also known that for families of unimodal maps typified by the quadratic family $Q_\lambda(x) = \lambda – (\lambda + 1)x^2$ there is a positive measure set of parameter values for which case (c) holds and $f$ possesses a finite absolutely continuous ergodic invariant measure (see [2], [12], [15], [16], [32]). It is not true that sensitivity to initial conditions is sufficient for the existence of such a measure, even for the quadratic family $Q_\lambda$ (see [17]).

In this paper we first develop techniques for establishing distortion estimates for maps with a negative Schwarzian derivative and for quadratic maps composed with maps of negative Schwarzian derivative. Using these techniques we establish two results: (1) In case (b) all of the homeomorphic cantor sets have Lebesgue measure zero, and (2) in case (c) almost every point has the same $\omega$-limit set which is either a finite union of intervals or an absorbing non-homeomorphic cantor set.


John Guckenheimer

Stewart Johnson