RENDICONTI DEL SEMINARIO MATEMATICO

122 D. Papini - F. ZanolinFigure 6: Suppose that a continuous mapping ψ transforms the generalized rectangleR (the “fat” cheese-like object) to the worm-like set ψ(R). The two components of the[ ·] − -set of R as well as their images under ψ are represented by segments (arcs) witha darker color at the contour of the corresponding figures. The stretching property isvisualized by the fact that there is a “crossing” of the “worm” through the “cheese”.To put emphasis on the role of the compact set K in the stretching definition, sometimeswe also write(D,K,ψ) : Â⊳̂B.Using a result about plane continua previously applied in a different context also byConley [11] and Butler [6] (for a proof, see [63] as well as [59, 60]), the followingfixed point theorem was obtained.THEOREM 1. Let ̂R = (R,R − ) be an oriented rectangle in X. If (D,K,ψ): ̂R⊳̂R, then there is w ∈ D (actually w ∈ K) such that ψ(w) = w.An illustration of Theorem 1 is given in Figure 6.Of course, not all the crossings fit to our purposes. For instance, Figure 7 showsan example of nonexistence of fixed points.