The famous Oseledets theorem states that if gn is a station-
ary sequence of m × m matrices, then with probability 1 there is a (random) basis in R m such that for any vector x the asymptotic behaviour of ||gn . . . g1 x|| is the same as that for one of the vectors from this basis. The fact that the sequence is stationary is crucial for the existence of such a basis. I shall consider the product of non-identically distributed independent matrices and will explain under what conditions one can prove the existence of distinct Lyapunov exponents as well as the Oseledetss dichotomy (or rather multihotomy) of the space.

Endoscopy structures of automorphic forms was one of the
basic structures discovered through the Arthur-Selberg trace formula method to establish the Langlands functoriality for classical groups.

In this talk, we will discuss my recent work on characterization of the endoscopy structure in terms of the order of pole at s=1 of certain L-functions, and in terms of a family of periods of automorphic forms, which was discovered jointly with David Ginzburg. At the end, I may discuss how to contruct the
endoscopy transfer by integral operators, which is
a joint work with Ginzburg and Soudry.