Multivalued dynamic systems with weights

We consider m-valued transformations of a probability space (X, \(\mathcal{B}\), µ) endowed with a set of weights \(\left\{ {\alpha _j :X \to (0,1],\sum\limits_{j = 1}^m {\alpha _j } \equiv 1} \right\}\). 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.