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

  • * Corresponding author: Marian Gidea

    * Corresponding author: Marian Gidea 
Research of M.G. and Y.S. was partially supported by NSF grant DMS-0635607, NSF grant DMS-1700154, and by the Alfred P. Sloan Foundation grant G-2016-7320.
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  • We present a combinatorial approach to rigorously show the existence of fixed points, periodic orbits, and symbolic dynamics in discrete-time dynamical systems, as well as to find numerical approximations of such objects. Our approach relies on the method of 'correctly aligned windows'. We subdivide 'windows' into cubical complexes, and we assign to the vertices of the cubes labels determined by the dynamics. In this way, we encode the information on the dynamics into combinatorial structure. We use a version of Sperner's Lemma to infer that, if the labeling satisfies certain conditions, then there exist fixed points/periodic orbits/orbits with prescribed itineraries. The method developed here does not require the computation of algebraic topology-type invariants, as only combinatorial information is needed; our arguments are elementary.

    Mathematics Subject Classification: Primary: 37C25, 37B10, 37C50; Secondary: 55M20, 05A99.


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  • 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

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