Dynamics of infinite-multivalued transformations

Details Dynamics of infinite-multivalued transformations Dynamics of infinite-multivalued transformations

2004-12-08

arxiv.org/abs/math/0412158v1

Mathematics

Dynamical Systems

16 pages, latex

Scientific paper

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.

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