Let (X, B, lambda, T) be a dynamical system and Log( n)x be the n-times iterated logarithm. In the first half of this thesis we will prove that given p > 0, and n ∈ N there exists an increasing sequence of non negative integers nk and a function f ∈ LLogpn L(X) such that ANf( x) =1Nk=0 N-1Tnkx fails to converge a.e. but ANg( x) converges a.e. for all g ∈ LLogqn L(X) with q > p. In the second half of this thesis we extend a theorem of Bellow and Calderon, which states that the sequence of convolution powers munf( x) = k∈Z mun(k) f(Tkx) converges a.e, when mu is a strictly aperiodic probability measure on Z such that the expectation E(mu) = 0 and the second moment of mu, m2(mu) infinity.
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