Dimensions of Julia sets of expanding rational semigroups

Details Dimensions of Julia sets of expanding rational semigroups Dimensions of Julia sets of expanding rational semigroups

2004-05-27

arxiv.org/abs/math/0405522v11

Kodai Mathematical Journal (2005) Vol. 28, No.2, pp390-422

Mathematics

Dynamical Systems

31 pages, Revised on July 1, 2005. Some typos are fixed. The title was changed. See also http://www.math.sci.osaka-u.ac.jp/~su

Scientific paper

We estimate the upper box and Hausdorff dimensions of the Julia set of an expanding semigroup generated by finitely many rational functions, using the thermodynamic formalism in ergodic theory. Furthermore, we show Bowen's formula, and the existence and uniqueness of a conformal measure, for a finitely generated expanding semigroup satisfying the open set condition. What we discuss is the `backward iterated function systems', and since the generator maps are not injective in general, we need some efforts to estimate the measure of disks.

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