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Partitions by congruent sets and optimal positions

Published online by Cambridge University Press:  16 June 2010

YU-MEI XUE
Affiliation:
School of Mathematics and System Sciences, LMIB, Beijing University of Aeronautics and Astronautics, Beijing 100191, PR China (email: yxue@buaa.edu.cn)
TETURO KAMAE
Affiliation:
5-9-6 Satakedai, Suita 565-0855, Japan (email: kamae@apost.plala.or.jp)

Abstract

Let X be a metrizable space with a continuous group or semi-group action G. Let D be a non-empty subset of X. Our problem is how to choose a fixed number of sets in {g−1DgG}, say σ−1D with στ, to maximize the cardinality of the partition ℙ({σ−1Dστ}) generated by them. Let An infinite subset Σ of G is called an optimal position of the triple (X,G,D) if holds for any k=1,2,… and τ⊂Σ with =k. In this paper, we discuss examples of the triple (X,G,D) admitting or not admitting an optimal position. Let X=G=ℝn  (n≥1) , where the action gG to xX is the translation xg. If D is the n-dimensional unit ball, then holds and the triple (X,G,D) admits an optimal position. In fact, if n≥2 and Σ is an infinite subset of G such that for some δ with 0<δ<1 , Σ⊂{x∈ℝn ; ‖x‖=δ}, and that any subset of Σ with cardinality n+1 is not on a hyperplane, then Σ is an optimal position of the triple (X,G,D) . We have determined the primitive factor of the uniform sets coming from these optimal positions. We also show that in the above setting with n=2 and the unit square D′ in place of the unit disk D, the maximal pattern complexity is unchanged and p*X,G,D′(k)=k2k+2 , but there is no optimal position.

Type
Research Article
Copyright
Copyright © Cambridge University Press 2010

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References

[1]Gjini, N., Kamae, T., Tan, B. and Xue, Y.-M.. Maximal pattern complexity for Toeplitz words. Ergod. Th. & Dynam. Sys. 26 (2006), 114.CrossRefGoogle Scholar
[2]Kamae, T.. Uniform set and complexity. Discrete Math. 309 (2009), 37383747.CrossRefGoogle Scholar
[3]Kamae, T. and Rao, H.. Maximal pattern complexity over letters. European J. Combin. 27 (2006), 125137.CrossRefGoogle Scholar
[4]Kamae, T., Rao, H., Tan, B. and Xue, Y.-M.. Language structure of pattern Sturmian words. Discrete Math. 306 (2006), 16511668.CrossRefGoogle Scholar
[5]Kamae, T., Rao, H., Tan, B. and Xue, Y.-M.. Super-stationary set, subword problem and the complexity. Discrete Math. 309 (2009), 44174427.CrossRefGoogle Scholar
[6]Kamae, T., Rao, H. and Xue, Y.-M.. Maximal pattern complexity for two-dimensional words. Theoret. Comput. Sci. 359 (2006), 1527.CrossRefGoogle Scholar
[7]Kamae, T. and Zamboni, L.. Sequence entropy and the maximal pattern complexity of infinite words. Ergod. Th. & Dynam. Sys. 22 (2002), 11911199.CrossRefGoogle Scholar
[8]Kamae, T. and Zamboni, L.. Maximal pattern complexity for discrete systems. Ergod. Th. & Dynam. Sys. 22 (2002), 12011214.CrossRefGoogle Scholar
[9]Xue, Y.-M.. Transformations with discrete spectrum and sequence entropy. Master’s Thesis, Osaka City University, 2000 (in Japanese).Google Scholar