The Fibonacci Unimodal Map

The Fibonacci recurrence of the critical orbit appeared in the work of Branner and Hubbard on complex cubic polynomials [BH, §12] and in Yoccoz's work [Yl, Y2] on quadratic ones, as the "worst" pattern of recurrence. On the other hand, a real quadratic Fibonacci map was suggested by Hofbauer and Keller [HK] as a possible candidate for a map having a "wild" attractor (that is, a set A which is the w-limit set for Lebesgue almost every orbit but is strictly smaller than the w-limit set for a generic orbit). The w-limit set of the critical orbit in [HK] possesses all known topological properties of wild attractors (compare [BL2]). In fact, we will see below that the quadratic Fibonacci map does not have a wild attractor; however, the corresponding question for a map with a degenerate critical point remains open. Actually, the first indication of the Fibonacci map appeared in the numerical work of Tsuda [T], related to the Belousova-Zhabotinskii reaction, and also in numerical work of Shibayama [Sh] (more precisely, they studied the sequence of "Fibonacci bifurcations" creating the Fibonacci map). This paper will study topological, geometrical, and measure-theoretical properties of the real Fibonacci map. Our goal was to figure out if this type of recurrence really gives any pathological examples and to compare it with the infinitely renormalizable patterns of recurrence studied by Sullivan [S]. It turns out that the situation can be understood completely and is of quite regular nature. In particular, any Fibonacci map (with negative Schwarzian and nondegenerate critical point) has an absolutely continuous invariant measure (so, we deal with a "regular" type of chaotic dynamics). It turns out also that geometrical properties of the closure of the critical orbit are quite different from those of the Feigenbaum map: its Hausdorff dimension is equal to zero and its geometry is not rigid but depends on one parameter. Branner and Hubbard introduce the concept of a tableau in order to describe recurrence of critical orbits. Their "Fibonacci tableau" is a basic example, which corresponds to one particularly close and regular pattern of recurrence. If a complex quadratic map z 1-+ z2 + c realizes this Fibonacci tableau, then the