In this paper, we prove Ruelle’s inequality for the entropy and Lyapunov exponents of random diffeomorphisms. Y. Kefer has studied this problem in ergodic case, but his theorem and proof seem to be incorrect. A new discussion is given in this paper with some new ideas and methods to be introduced, especially for f∈Diff~2(M), we introduce a new definition of C~2-norm |f|c~2 and the concept of relation number r(f) which play an important role both in this paper and in general studies.
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