Hyperbolic Tessellation and Colorings of Trees

Kim, Dong Han; Lim, Seonhee

doi:10.1155/2013/706496

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“…Factor complexity was generalized from sequences to vertex colorings of a regular tree in [8] and [7]. For a graph X , let us denote its vertex set by V X and its set of oriented edges by E X .…”

Section: Introductionmentioning

confidence: 99%

Continued fraction algorithm for Sturmian colorings of trees

Kim¹,

Lim²

2017

Ergod. Th. Dynam. Sys.

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Factor complexity $b_{n}(\unicode[STIX]{x1D719})$ for a vertex coloring $\unicode[STIX]{x1D719}$ of a regular tree is the number of classes of $n$-balls up to color-preserving automorphisms. Sturmian colorings are colorings of minimal unbounded factor complexity $b_{n}(\unicode[STIX]{x1D719})=n+2$. In this article, we prove an induction algorithm for Sturmian colorings using colored balls in a way analogous to the continued fraction algorithm for Sturmian words. Furthermore, we characterize Sturmian colorings in terms of the data appearing in the induction algorithm.

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Section: Introductionmentioning

confidence: 99%

Continued fraction algorithm for Sturmian colorings of trees

Kim¹,

Lim²

2017

Ergod. Th. Dynam. Sys.

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Subword complexity and Sturmian colorings of regular trees

Kim¹,

Lim²

2013

Ergod. Th. Dynam. Sys.

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Abstract. In this article, we study subword complexity of colorings of regular trees. We characterize colorings of bounded subword complexity and study Sturmian colorings, which are colorings of minimal unbounded subword complexity.We classify Sturmian colorings using their type sets. We show that any Sturmian coloring is a lifting of a coloring on a quotient graph of the tree which is a geodesic or a ray with loops possibly attached, thus a lifting of an "infinte word". We further give a complete characterization of the quotient graph for eventually periodic ones.

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Hyperbolic Tessellation and Colorings of Trees

Cited by 2 publications

References 10 publications

Continued fraction algorithm for Sturmian colorings of trees

Continued fraction algorithm for Sturmian colorings of trees

Subword complexity and Sturmian colorings of regular trees

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