Fixed point index and decompositions of isolated invariant compacta

Let U⊂R2 be an open subset and let f :U→f(U)⊂R2 be a homeomorphism. Let M=M1∪⋯∪Mr⊂U be a disjoint union of discs that isolates the invariant compactum K. The aim of this paper is to study the dynamics of f in K and to use the fixed point index to detect, in a simple and geometric way, the existence of periodic orbits on which f follows a determined pattern. Our method allows us to compute the fixed point index of every iteration of f in a neighborhood of the periodic orbits following a given itinerary in classical and important semidynamical systems with chaotic dynamics.