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Differentiability with respect to parameters of average values in probabilistic contracting dynamical systems†

Published online by Cambridge University Press:  19 September 2008

W. Douglas Withers
W. Douglas Withers
Affiliation:
Department of Mathematics, U.S. Naval Academy, Annapolis, Maryland 21402, USA
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Abstract

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We consider a dynamical system consisting of a compact subset of RN or CN with several contracting maps chosen with prescribed probabilities, which may depend on position. We show that if the maps and the probabilities are Cl+α functions of the spatial variable and an external parameter, then the average value of a Cl+α function is a differentiate function of the parameter. One implication of this theorem is that for certain families of complex functions dependent on a parameter the reciprocal of the dimension of an invariant measure on the Julia set is a harmonic function of the parameter.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 10 , Issue 3 , September 1990 , pp. 599 - 610
Copyright
Copyright © Cambridge University Press 1990

References

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