Exact dimensionality of projected measures for expanding rational semigroups

Abstract

We study the dimension theory of expanding rational semigroups. An expanding rational semigroup is naturally described by a skew product on the product of the symbolic space and the Riemann sphere. For an invariant Borel probability measure of this skew product, we consider its disintegration over a symbolic factor and study the push-forwards of the conditional measures under the second coordinate projection. We prove a Ledrappier--Young type formula for the projected conditional measures. In particular, if the original invariant measure is ergodic, then these projected conditional measures are exact dimensional for almost every fiber. Applying this result to the trivial factor yields exact dimensionality of the projected invariant measure. This result can be viewed as a backward analogue of the result of [D.-J. Feng and H. Hu. \textit{Comm. Pure Appl. Math.} \textbf{62} (2009), no. 11, 1435--1500].