Geodesic
Dynamics of finite-multivalued transformations
Lobachevskii journal of mathematics, Tome 17 (2005), pp. 47-60

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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.
Keywords: ergodic theory, dynamic system, self-similar set. 
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     author = {K. B. Igudesman},
     title = {Dynamics of finite-multivalued transformations},
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K. B. Igudesman. Dynamics of finite-multivalued transformations. Lobachevskii journal of mathematics, Tome 17 (2005), pp. 47-60. http://geodesic.mathdoc.fr/item/LJM_2005_17_a4/