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For a dynamical system (X, f ) and a function ϕ : X → R N the rotation set is defined. The case when (X, f ) is a transitive subshift of finite type and ϕ depends on the cylinders of length 2 is studied. Then the rotation set is a convex polyhedron. The rotation vectors of periodic points are dense in the rotation set. Every interior point of the rotation set is a rotation vector of an ergodic measure.

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Abstract. We prove an almost sure invariance principle and a central limit theorem for the process (F°/") B > 0 , where / is a map of an interval with a non-positive Schwarzian derivative whose trajectories of critical points stay far from the critical points, and F is a measurable function with bounded p-variation ( p > 1).The almost sure invariance principle implies the Log-log laws, integral tests and a distributional type of invariance principle for the process (F°f) n3:0 .