The Scenery Flow for Hyperbolic Julia Sets

We define the scenery flow space at a point z in the Julia setJ of a hyperbolic rational map T : C C with degree at least2, and more generally for T a conformal mixing repellor. We prove that, for hyperbolic rational maps, except for a fewexceptional cases listed below, the scenery flow is ergodic.We also prove ergodicity for almost all conformal mixing repellors;here the statement is that the scenery flow is ergodic for therepellors which are not linear nor contained in a finite unionof real-analytic curves, and furthermore that for the collectionof such maps based on a fixed open set U, the ergodic casesform a dense open subset of that collection. Scenery flow ergodicityimplies that one generates the same scenery flow by zoomingdown towards almost every z with respect to the Hausdorff measureH^d, where d is the dimension of J, and that the flow has a uniquemeasure of maximal entropy. For all conformal mixing repellors, the flow is loosely Bernoulliand has topological entropy at most d. Moreover the flow atalmost every point is the same up to a rotation, and so as acorollary, one has an analogue of the Lebesgue density theoremfor the fractal set, giving a different proof of a theorem ofFalconer. 2000 Mathematical Subject Classification: 37F15, 37F35, 37D20.