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Partitions by congruent sets and optimal positions

Published online by Cambridge University Press:  16 June 2010

YU-MEI XUE and
TETURO KAMAE
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)
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Abstract

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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−1D; g∈G}, say σ−1D with σ∈τ, to maximize the cardinality of the partition ℙ({σ−1D; σ∈τ}) generated by them. Let \[ p_{X,G,D}^*(k)=\sup _{\tau \subset G,~\#\tau =k} \#\mathbb {P}(\{\sigma ^{-1}D;~\sigma \in \tau \})\quad (k=1,2,\ldots ). \] An infinite subset Σ of G is called an optimal position of the triple (X,G,D)if \[ \#\mathbb {P}(\{\sigma ^{-1}D;~\sigma \in \tau \})=p_{X,G,D}^*(k) \] holds for any k=1,2,… and τ⊂Σ with #τ=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 g∈G to x∈X is the translation x−g. If D is the n-dimensional unit ball, then \[ p_{X,G,D}^*(k)=2\sum _{i=0}^n{k-1\choose i} \] holds and the triple (X,G,D)admits an optimal position. In fact, if n≥2and 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+1is 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=2and the unit square D′ in place of the unit disk D, the maximal pattern complexity is unchanged and p*X,G,D′(k)=k2 −k+2 , but there is no optimal position.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 31 , Issue 2 , April 2011, pp. 613 - 629
Copyright
Copyright © Cambridge University Press 2010

References

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