We propose a general framework for proving that a compact, infinite-dimensional map has an invariant set on which the dynamics is semiconjugated to a subshift of finite type. The method is then applied to certain Poincaré map of the Kuramoto-Sivashinsky PDE on the line with odd and periodic boundary conditions and with parameter . We give a computer-assisted proof of the existence of symbolic dynamics and countable infinity of periodic orbits with arbitrary large periods.