### Abstract

Original language | English |
---|---|

Pages (from-to) | 467-492 |

Number of pages | 25 |

Journal | Proceedings of the London Mathematical Society |

Volume | 3 |

Issue number | 85 |

DOIs | |

Publication status | Published - 2002 |

### Fingerprint

### Keywords

- hausdorff measure
- julia set
- mathematics
- reliability management

### Cite this

*Proceedings of the London Mathematical Society*,

*3*(85), 467-492. https://doi.org/10.1112/S002461150201362X

}

*Proceedings of the London Mathematical Society*, vol. 3, no. 85, pp. 467-492. https://doi.org/10.1112/S002461150201362X

**The scenery flow for hyperbolic Julia sets.** / Bedford, T.J.; Fisher, A.C.; Urbanski, M.

Research output: Contribution to journal › Article

TY - JOUR

T1 - The scenery flow for hyperbolic Julia sets

AU - Bedford, T.J.

AU - Fisher, A.C.

AU - Urbanski, M.

PY - 2002

Y1 - 2002

N2 - 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.

AB - 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.

KW - hausdorff measure

KW - julia set

KW - mathematics

KW - reliability management

UR - http://plms.oxfordjournals.org/archive/

UR - http://dx.doi.org/10.1112/S002461150201362X

U2 - 10.1112/S002461150201362X

DO - 10.1112/S002461150201362X

M3 - Article

VL - 3

SP - 467

EP - 492

JO - Proceedings of the London Mathematical Society

JF - Proceedings of the London Mathematical Society

SN - 0024-6115

IS - 85

ER -