Abstract
LetT be a homeomorphism of a metrizable compactX, the sequencec k/k tends to 0 andc k tends to infinity. We’ll study the limit behaviour of the distributions of the sums (1/c k) ∑ =0 -1 F oT i whereF is from a space of continuous functions—the central limit problem and the speed of convergence in the ergodic theorem. The main attention is given to the case whereX is the unit circle andT is an irrational rotation; in this case we consider the spaces of absolutely continuous, Lipschitz, andk-times differentiable functionsF.
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