Fixed point index and decompositions of isolated invariant compacta (Q1878522)

Let \(U\subset \mathbb{R}^2\) be an open set and let \(f:U\rightarrow f(U)\subset\mathbb{R}^2\) be a homeomorphism. Let \(M=M_1\cup\dots\cup M_r\subset U\) be a disjoint union of discs that isolates the invariant compactum \(K\), i.e., \(K=\{x\in M\mid \exists\, \sigma:\mathbb{Z}\rightarrow M\) such that \(\sigma(0)=x\) and \(f(\sigma(i))=\sigma(i+1)\) for all \(i\in\mathbb{Z}\}\). The aim of this paper is to provide techniques to study the dynamics of \(f\) in \(K\) and, using fixed point index theory, to detect the periodic orbits of \(f\) in \(K\) following a given itinerary \(\tau\). Moreover, in some cases containing classical and important discrete dynamical systems the author computes the fixed point index of every iteration of \(f\) in small neighborhoods of the detected periodic points.