Fixed points with rotation as obstructions to topological transitivity. I (Q1262572)

The author presents a new type of fixed point theorem which deals with the following situation: S is a connected orientable surface which has a non-compact boundary component, \(\partial\), h: \(S\to S\) is an orientation-preserving homeomorphism with \(h(\partial)=\partial\). Let F be an isolated fixed point of h in the interior of S. Call \({\mathcal F}\) the set of those fixed points of h different from F which are Nielsen equivalent to F in S but not Nielsen equivalent to \(\partial\) in \(S\setminus F\). The author assumes that F is a ``fixed point with rotation'' which means that the following two conditions hold: a) There is an injective continuous map f: [0,\(\infty)\to S\) with \(f(0)=F\) and \(h(W)=W\), where \(W=f([0,\infty))\). b) There is a compact arc, \(\beta\), in \(S\setminus F\) from \(\partial\) to \(P\in W\) such that, if w is the part of W between P and h(P), the arc \(\beta \cup w\cup [h(\beta)]^{-1}\) represents 0 in \(\pi_ 1(S,\partial)\) but not in \(\pi_ 1(S\setminus F,\partial)\). If h induces the identity on \(\pi_ 1(S\setminus F,\partial)\), if cl W is nowhere dense in S, if \(S\setminus cl W\) is connected, and if \(\partial \cap cl W=\emptyset\), then the conclusion is that \({\mathcal F}\neq \emptyset\).