Extension, embedding and global stability in two dimensional monotone maps

Figure 2.1.  In part (a) of this figure, we illustrate our notation of projecting a point $(x, y)$ to the boundary $\partial \Omega.$ In part (b), we illustrate the notion of putting the invariant domain $\Omega$ inside an Origami domain

Figure 2.2.  This figure shows the possible options for the boundary of a convex $\Omega$ and a possible ray extension. Part (a) is based on the assumption that $f$ is non-decreasing and Part (b) is based on the assumption that $f$ is non-increasing. Note that the missing quarter in Part (a) is due to the fact that we cannot have $f(\uparrow)$ and $F(\downarrow, \uparrow)$ at the same time. Similarly for the missing quarter of Part (b)

Figure 2.3.  Grafting $\gamma_1$ along $\gamma_2$

Figure 2.4.  The various cases for the boundary pieces of an origami domain

Figure 2.5.  A semi-convex domain is shown in Part (a). Next is shown a piece $ C_1\cup C_2 $ of the boundary of $ \Omega $ oriented positively, with $ f(t) = F(r(t)) $ changing monotonicity from $ C_1 $ to $ C_2 $. Parts (b) and (c) show two different ray extensions, while (d) shows the adjustment to make to these extensions, before filling the sector. As explained in the text, (b) is not a valid extension if monotonicity is to be preserved in the extension

Figure 2.6.  Extending from semi-convex domain to the origami domain reduces to extending over sectors, with cases labeled (a) through (d)

Figure 2.7.  Filling a rectangle where $ F $ is known on three sides. Extending partially by $ \widetilde{F}_1 $ as shown in (b) then extending over the resulting sector

Figure 4.1.  Part (a) of this figure shows the invariant region $\Omega, $ while part (b) shows $T(\Omega)$ together with the extension through horizontal projections. The scale is missing to indicate the general form of the region when $0 < h < \min\{p, \frac{1}{2}\}.$