One-Dimensional Dynamics

TL;DR: In this paper, the authors present some specific conjectures about the measure-theoretic properties of a certain class of one-dimensional quadratic functions with negative Schwarzian derivatives.

Abstract: At a similar conference sponsored by the New York Academy of Sciences two years ago, we surveyed a range of questions concerning the iteration of quadratic functions.’ These past two years have seen substantial progress on several of these questions, and much of that work is described in this volume. Here we summarize those aspects of these developments which are primarily topological in nature. This summary enables us to present some specific conjectures about the measure-theoretic properties of a certain class of one-dimensional maps. We restrict our attention to the set of C’ mapsf: I I. I = [ O , l ] , which have negative Schwarzian derivatives. Singer introduced the Schwarzian derivative Sf = (j”’/f’) ( 3 / 2 ) (f “/f ’)* into the study of the iteration of one-dimensional maps.2 The class of maps with negative Schwarzian derivatives has good “expansive” properties for iteration. Precisely,fhas a negative Schwarzian derivative if and only if, for every 4-tuple of points xI < x2 < x3 < x4 contained in an interval on which f is monotone, the cross ratio ( f ( x l ) , f ( x 2 ) , f ( x , ) , f ( x , ) ) is larger than the cross ratio (xI . x2, x3, xq). Note that this geometric property is preserved by composition, so the functions with negative Schwarzian derivatives form a semigroup. We shall say that a fixed point x of a map f is stable if there is an interval J consisting of points whose trajectories tend to x. This admits the possibility that x might be the limit of trajectories on only one side of x. The cross ratio property of functions with negative Schwarzian derivatives prevents the existence of a stable fixed point x sandwiched between two other fixed points y < x < z, which points have the property that f is increasing on the interval [y, z ] . Expressed somewhat differently, this yields Singer’s main result2: iff has a negative Schwarzian derivative, then every stable periodic orbit y offhas a critical point offor an endpoint of I whose trajectory approaches y. For “quadratic” like maps with a single critical point and a negative Schwarzian derivative, this means that there can be, a t most, one stable periodic orbit. This prompts us to study the class of maps C defined to be those f: I I such that I . f is C’ and has a negative Schwarzian derivative 2. f has a single critical point in I that is assumed to be nondegenerate 3 . f ( 0 ) = f( 1 ) = 0. If f’(0) < 1 , then f has no other fixed point. Note that the quadratic mapsf(x) = ax( l x) are in C for 0 < a 5 4 . The maps f E C have a stronger property than the uniqueness of stable periodic orbits described above. I f f E C and U is a neighborhood of its critical point such that I U does not contain a stable periodic orbit, then the set E,, = (x I f ” ( x ) E I U for all n 2 01 has Lebesgue measure ~ e r o . ~ . ~ Together with Singer’s theorem, this yields the property that i f f € P, then almost all trajectories have the same forward limit set