Fixed points for dissipative-repulsive systems and topological dynamics of mappings defined on N-dimensional cells

We prove a fixed point theorem for continuous mappings which satisfy a compression-expansion condition on the boundary of a N-dimensional cell of R^N. Our work is motivated by a previous theorem of Andres, Gaudenzi and Zanolin (1990) dealing with the existence of periodic solutions for dissipative-repulsive systems and is also related to some recent results by Zgliczyński and Gidea (2004) concerning covering relations for Markov partitions. Applications are given to the study of the iterates of a continuous map of R^N. In this case we obtain some theorems on the existence of periodic points and chaotic-like dynamics