Multivalued dynamic systems with weights

We consider $m$-valued transformations of the probability space $(X,\mathcal B,\mu)$ endowed with a set of weights $\Bigl\{\alpha_j\colon X\to(0,1],\ \sum_{j=1}^m\alpha_j\equiv1\Bigr\}$. For this case we introduce analogs of the basic notions of the ergodic theory, namely, the measure invariance, ergodicity, Koopman and Frobenius–Perron operators. We study the properties of these operators, prove ergodic theorems, and give some examples. We also propose a technique for reducing some problems of the fractal geometry to those of the functional analysis.