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Natural invariant measures, divergence points and dimension in one-dimensional holomorphic dynamics

Published online by Cambridge University Press:  01 August 2009

WILLIAM INGLE ,
JACIE KAUFMANN  and
CHRISTIAN WOLF
WILLIAM INGLE
Affiliation:
Department of Mathematics, Wichita State University, Wichita, KS 67260, USA (email: ingle@math.wichita.edu, kaufmann@math.wichita.edu, cwolf@math.wichita.edu)
JACIE KAUFMANN
Affiliation:
Department of Mathematics, Wichita State University, Wichita, KS 67260, USA (email: ingle@math.wichita.edu, kaufmann@math.wichita.edu, cwolf@math.wichita.edu)
CHRISTIAN WOLF
Affiliation:
Department of Mathematics, Wichita State University, Wichita, KS 67260, USA (email: ingle@math.wichita.edu, kaufmann@math.wichita.edu, cwolf@math.wichita.edu)
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Abstract

In this paper we discuss the dimension-theoretical properties of rational maps on the Riemann sphere. In particular, we study the existence and uniqueness of generalized physical measures for several classes of maps including hyperbolic, parabolic, non-recurrent and topological Collet–Eckmann maps. These measures have the property that their typical points have maximal Hausdorff dimension. On the other hand, we prove that the set of divergence points (the set of points which are non-typical for any invariant measure) also has maximal Hausdorff dimension. Finally, we prove that if (fa)a is a holomorphic family of stable rational maps, then the dimension d(fa) is a continuous and plurisubharmonic function of the parameter a. In particular, d(f) varies continuously and plurisubharmonically on an open and dense subset of Ratd, the space of all rational maps with degree d≥2.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 29 , Issue 4 , August 2009 , pp. 1235 - 1255
Copyright
Copyright © Cambridge University Press 2009

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