Differentiability with respect to parameters of average values in probabilistic contracting dynamical systems
- 19 September 1990
- journal article
- research article
- Published by Cambridge University Press (CUP) in Ergodic Theory and Dynamical Systems
- Vol. 10 (3) , 599-610
- https://doi.org/10.1017/s0143385700005769
Abstract
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.Keywords
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