Abstract
We study the action of isometries on metric spaces. In particular, we consider the recurrent set of the bilateral shift operator on the Banach space L ∞ (Z), and prove that the set of periodic points is not dense in the recurrent set. Then we apply this result to investigating the dynamics of Teichmuller modular groups acting on infinite dimensional Teichmuller spaces as well as composition operators acting on Hardy spaces.
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