Parameter Scaling for the Fibonacci Point

We prove geometric and scaling results for the real Fibonacci parameter value in the quadratic family z 7→ z + c. The principal nest of the Yoccoz parapuzzle pieces has rescaled asymptotic geometry equal to the filled-in Julia set of z 7→ z − 1. The modulus of two such successive parapuzzle pieces increases at a linear rate. Finally, we prove a “hairiness” theorem for the Mandelbrot set at the Fibonacci point when rescaling at this rate. Stony Brook IMS Preprint #1996/4 June 1996 In this paper, we focus on the small scale similarities between the dynamical space and parameter space for the Fibonacci point in the family of maps z 7→ z2 + c. There is a general philosophy in complex dynamics that the structure we see in the parameter space around the parameter value c should be the “same” as that around the critical value ‘c’ in dynamical space [DH85]. In the case where the critical point is pre-periodic, Tan Lei [Lei90] proved such asymptotic similarities by showing that the Mandelbrot set and Julia set exhibit the same limiting geometry. For parameters in which the critical point is recurrent (i.e., it eventually returns back to any neighborhood of itself), the Mandelbrot and Julia sets are much more complicated. Milnor, in [Mil89], made a number of conjectures (as well as pictures!) for the case of infinitely renormalizable points of bounded type. Dilating by factors determined by the renormalization, the resulting computer pictures demonstrate a kind of self-similarity, with each successive picture looking like a “hairier” copy of the previous. McMullen [McM94] has proven that, for these points, the Julia set densely fills the plane upon repeated rescaling, i.e., hairiness; and Lyubich has recently proven hairiness of the Mandelbrot set for Feigenbaum like points. We focus on a primary example of dynamics in which we have a recurrent critical point and the dynamics is non-renormalizable: the Fibonacci map. The dynamics of the real quadratic Fibonacci map, where the critical point returns closest to itself at the Fibonacci iterates, has been extensively studied (especially see [LM93]). Maps with Fibonacci type returns were first discovered in the cubic case by Branner and Hubbard [BH92] and have since been consistently explored because they are a fundamental combinatorial type of the class of non-renormalizable maps. The Fibonacci map was used by Lyubich and Milnor in developing the generalized renormalization procedure which has proven very fruitful. The Fibonacci map was also highlighted in the work of Yoccoz as it was in some sense the worst case in the proof of local connectivity of non-renormalizable Julia sets with recurrent critical point [Hub93], [Mil92]. The local connectedness proof of Yoccoz involves producing a sequence of partitions of the Julia set, now called Yoccoz puzzle pieces. These Yoccoz puzzle pieces are then shown to exhibit the divergence property and in particular nest down to the critical point, proving local connectivity there. Yoccoz then transfers this divergence property to the parapuzzle pieces around the parameter point to demonstrate that the Mandelbrot set is locally connected at this parameter value. Lyubich further explores the Yoccoz puzzle pieces of Fibonacci maps and demonstrates that the principal nest of Yoccoz puzzle pieces has rescaled asymptotic geometry equal to the filled-in Julia set of z 7→ z2 − 1 and that the moduli of successive annuli grow at a linear rate [Lyu93b].