  We consider a transformation of a normalized measure space such that the
image of any point is a finite set. We call such transformation
$m$-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. Some results
which have analogies in the classical ergodic theory we are proved using
standard methods. Other results have no analogies.
