Recurrent and periodic points for isometries of L^\infty spaces Ege FujikawaKatsuhiko Matsuzaki 37F3047A1647B38Teichmuller spaceTeichmuller modular groupbilateral shift operatorcomposition operatorHardy space 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 $L^{\infty}(\mathbb{Z})$, 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 Teichm\"uller modular groups acting on infinite dimensional Teichm\"uller spaces as well as composition operators acting on Hardy spaces. Indiana University Mathematics Journal 2006 text pdf 10.1512/iumj.2006.55.2660 10.1512/iumj.2006.55.2660 en Indiana Univ. Math. J. 55 (2006) 975 - 998 state-of-the-art mathematics http://iumj.org/access/