2004/12/08 by Konstantin Igudesman, Igudesman, Konstantin
Mathematics · #28A80 #28D05 #37A05 #Dynamical Systems (math.DS) #FOS: Mathematics #math.DS #msc:28A80 #msc:28D05 #msc:37A05
paper · pdf · doi:10.48550/arxiv.math/0412158
16 pages, latex
arxiv created 2004/12/08 · arxiv updated 2009/12/01
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.