TY - JOUR

T1 - Ergodic theory for markov fibred systems and parabolic rational maps

AU - Aaronson, Jon

AU - Denker, Manfred

AU - Urbański, Mariusz

PY - 1993/6

Y1 - 1993/6

N2 - A parabolic rational map of the Riemann sphere admits a nonatomic-conformai measure on its Julia set where h = the Hausdorff dimension of the Julia set and satisfies 1/2 < h < 2. With respect to this measure the rational map is conservative, exact and there is an equivalent σ-finite invariant measure. Finiteness of the measure is characterised. Central limit theorems are proved in the case of a finite invariant measure and return sequences are identified in the case of an infinite one. A theory of Markov fibred systems is developed, and parabolic rational maps are considered within this framework.

AB - A parabolic rational map of the Riemann sphere admits a nonatomic-conformai measure on its Julia set where h = the Hausdorff dimension of the Julia set and satisfies 1/2 < h < 2. With respect to this measure the rational map is conservative, exact and there is an equivalent σ-finite invariant measure. Finiteness of the measure is characterised. Central limit theorems are proved in the case of a finite invariant measure and return sequences are identified in the case of an infinite one. A theory of Markov fibred systems is developed, and parabolic rational maps are considered within this framework.

UR - http://www.scopus.com/inward/record.url?scp=84968503810&partnerID=8YFLogxK

U2 - 10.1090/S0002-9947-1993-1107025-2

DO - 10.1090/S0002-9947-1993-1107025-2

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AN - SCOPUS:84968503810

SN - 0002-9947

VL - 337

SP - 495

EP - 548

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

IS - 2

ER -