Igudesman, Konstantin B. Dynamics of finite-multivalued transformations. (English) Zbl 1089.37001 Lobachevskii J. Math. 17, 47-60 (2005). 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. Cited in 2 Documents MSC: 37A05 Dynamical aspects of measure-preserving transformations 28D05 Measure-preserving transformations 28A80 Fractals Keywords:selfsimilar set; transformation of a normalized measure space; \(m\)-transformation; ergodic theory; fractal geometry PDFBibTeX XMLCite \textit{K. B. Igudesman}, Lobachevskii J. Math. 17, 47--60 (2005; Zbl 1089.37001) Full Text: arXiv EuDML EMIS