Ergodic theory of dynamical systems i.e., the qualitative analysis of iterations of a single transformation is nowadays a well developed theory. In 1945 S. Ulam and J. von Neumann in their short note  suggested to study ergodic theorems for the more general situation when one applies in turn different transformations chosen at random. Their program was fulfilled by S. Kakutani  in 1951. ‘Both papers considered the case of transformations with a common invariant measure. Recently Ohno  noticed that this condition was excessive. Ergodic theorems are just the beginning of ergodic theory. Among further major developments are the notions of entropy and characteristic exponents.
KeywordsMarkov Chain Ergodic Theory Ergodic Theorem Characteristic Exponent Stochastic Flow
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