## Konstantin B. Igudesman

# Dynamics of finite-multivalued transformations

## (Lobachevskii Journal of Mathematics, Vol.17, pp.47 - 60)

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.

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