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Dynamics of finite-multivalued transformations. (English) Zbl 1089.37001

Summary: We consider a transformation of a normalized measure space such that the image of any point is a finite set. We call such a transformation an \(m\)-transformation. In this case, the orbit of any point looks like a tree. In the study of \(m\)-transformations, we are interested in the properties of the trees. An \(m\)-transformation generates a stochastic kernel and a new measure. Using these objects, we introduce analogies of some main concept of ergodic theory: ergodicity, Koopman and Frobenius-Perron operators etc. We prove ergodic theorems and consider examples. We also indicate possible applications to fractal geometry and give a generalization of our construction.

MSC:

37A05 Dynamical aspects of measure-preserving transformations
28D05 Measure-preserving transformations
28A80 Fractals
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