The scenery flow for hyperbolic Julia sets
Bedford, T.J. and Fisher, A.C. and Urbanski, M. (2002) The scenery flow for hyperbolic Julia sets. Proceedings of the London Mathematical Society, 3 (85). pp. 467492. (http://dx.doi.org/10.1112/S002461150201362X)
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We define the scenery flow space at a point z in the Julia set J of a hyperbolic rational map T : C > C with degree at least 2, and more generally for T a conformal mixing repellor. We prove that, for hyperbolic rational maps, except for a few exceptional 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 the repellors which are not linear nor contained in a finite union of realanalytic curves, and furthermore that for the collection of such maps based on a fixed open set U, the ergodic cases form a dense open subset of that collection. Scenery flow ergodicity implies that one generates the same scenery flow by zooming down towards almost every z with respect to the Hausdorff measure Hd, where d is the dimension of J, and that the flow has a unique measure of maximal entropy. For all conformal mixing repellors, the flow is loosely Bernoulli and has topological entropy at most d. Moreover the flow at almost every point is the same up to a rotation, and so as a corollary, one has an analogue of the Lebesgue density theorem for the fractal set, giving a different proof of a theorem of Falconer.


Item type: Article ID code: 4333 Dates: DateEvent2002PublishedSubjects: Social Sciences > Industries. Land use. Labor > Management. Industrial Management
Science > MathematicsDepartment: Strathclyde Business School > Management Science
Unknown DepartmentDepositing user: Strathprints Administrator Date deposited: 17 Oct 2007 Last modified: 08 Apr 2024 15:45 Related URLs: URI: https://strathprints.strath.ac.uk/id/eprint/4333