Combinatorial approach to detection of fixed points, periodic orbits, and symbolic dynamics

Figure 1.  A sequence of correctly aligned windows. The first window $D_1$ in the sequence has marked its exit and entry set; $D_1$ is correctly aligned with $D_2$ under $f$, and $D_2$ is correctly aligned with $D_3$ under $f$

Figure 2.  Sperner labeling of a simplicial decomposition, and a completely labeled triangle in the simplicial decomposition

Figure 3.  Examples of simplicial decomposition of polytopes and of labelings; refer to Example 1

Figure 4.  A 2D window correctly aligned with itself under some map, and the corresponding labeling of $[0, 1]^2$ according to Condition O

Figure 5.  Coarse decomposition of $C$ and transformation into a polytope $\widetilde C$

Figure 6.  A 3D window correctly aligned with itself under some map, and a cubical decomposition satisfying Condition P; refer to Example 3

Figure 7.  A periodic sequence of correctly aligned windows $D_1, D_2, D_3$, where $D_1$ is correctly aligned with $D_2$, $D_2$ is correctly aligned with $D_3$, and $D_3$ is correctly aligned with $D_1$

Figure 8.  Top: The square $D_0$ and its image under $f^7(D_0)$, showing that $D_0$ is correctly aligned with itself under $f^7$ (top). Middle: $D_0$ is sub-divided into smaller squares of side $10^{-7}$, whose vertices are labeled. Zooming in, a completely labeled sub-square $D_1$ is shown. Bottom: The square $D_1$ is sub-divided into smaller squares of side $10^{-10}$, whose vertices are labeled. Zooming in, a completely labeled sub-square $D_2$ is shown

Figure 9.  Approximate period-$7$ orbit