The scenery flow for hyperbolic Julia sets

T.J. Bedford, A.C. Fisher, M. Urbanski

Research output: Contribution to journalArticle

9 Citations (Scopus)

Abstract

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 real-analytic 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.
LanguageEnglish
Pages467-492
Number of pages25
JournalProceedings of the London Mathematical Society
Volume3
Issue number85
DOIs
Publication statusPublished - 2002

Fingerprint

Hyperbolic Set
Julia set
Rational Maps
Ergodicity
Density Theorem
Fractal Set
Hausdorff Measure
Topological Entropy
Henri Léon Lebésgue
Open set
Bernoulli
Corollary
Union
Entropy
Analogue
Imply
Curve
Subset

Keywords

  • hausdorff measure
  • julia set
  • mathematics
  • reliability management

Cite this

Bedford, T.J. ; Fisher, A.C. ; Urbanski, M. / The scenery flow for hyperbolic Julia sets. In: Proceedings of the London Mathematical Society. 2002 ; Vol. 3, No. 85. pp. 467-492.
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The scenery flow for hyperbolic Julia sets. / Bedford, T.J.; Fisher, A.C.; Urbanski, M.

In: Proceedings of the London Mathematical Society, Vol. 3, No. 85, 2002, p. 467-492.

Research output: Contribution to journalArticle

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