Fixed point index of iterations of local homeomorphisms of the plane: a Conley index approach

Planar isolated and stable fixed points have index =1

“…The techniques of the proof of Theorem 3 suggest the possibility of proving, for arbitrary homeomorphisms, that planar isolated and stable fixed points have index=1 by using the results of [26] instead of Brouwer's lemma on translation arcs.…”

Local Fixed Point Indices of Iterations of Planar Maps

“…Unfortunately, it is usually difficult to establish the exact form of the indices for a given map. Nevertheless, during the last years the description of indices was given for many important classes of maps such as: planar homeomorphisms [14,16,20,22]; R 3 -homeomorphisms [15,23]; smooth maps [4,9,18,24]; and holomorphic maps [3,6,25,26]. In this paper we give the restrictions for indices of some class of planar maps.…”

“…What is more, Conley index approach turned out to be very fruitful in this problem, by this method Ruiz del Portal and Salazar reproved the formula (1.1), found the form of indices also in reversing orientation case [22] and described the behavior of f near p [20,21] (the reader may consult [17] for general theory of Conley index).…”

Indices of the iterates of R3-homeomorphisms at Lyapunov stable fixed points

“…When a fixed point is an isolated invariant set of an orientation preserving planar homeomorphism, the problem of the computation of the indices of its iterates was solved by Le Calvez and Yoccoz [13,14] and, by the authors, in the orientation reversing case [18]. Later Le Calvez solved the general problem in the orientation preserving case using the Carathéodory's theory of prime ends [15] and the authors, in [19], the general case for orientation reversing planar homeomorphisms.…”