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

Marian Gidea; Yitzchak Shmalo

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arXiv · 1706.08960

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

Marian Gidea·Yitzchak Shmalo

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Abstract

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 the `windows' into cubical complexes, and we assign to the vertices of the cubes labels determined by the dynamics. In this way we encode the dynamics information into a combinatorial structure. We use a version of the Sperner Lemma saying that if the labeling satisfies certain conditions, then there exist fixed points/periodic orbits/orbits with prescribed itineraries. Our arguments are elementary.

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BibTeX RIS

Marian Gidea, Yitzchak Shmalo. 2017-06-27. Combinatorial approach to detection of fixed points, periodic orbits, and symbolic dynamics. https://arxiv.org/abs/1706.08960

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