Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-24T15:26:52.986Z Has data issue: false hasContentIssue false
  • English
  • Français

Uniform sets and super-stationary sets over general alphabets

Published online by Cambridge University Press:  14 March 2011

TETURO KAMAE
TETURO KAMAE*
Affiliation:
Satakedai 5-9-6, 565-0855, Japan (email: [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

Uniform sets and super-stationary sets over the binary alphabet have been extensively studied. In this paper, they are generalized to general alphabets. We generalize the fact that any uniform set contains a super-stationary set so that any uniform complexity is realized by a super-stationary set. This gives a formula to calculate the uniform complexity functions. We also give characterizations of the class of super-stationary sets in general settings in two somewhat different ways than in the binary case. Super-stationary sets are considered as phenomena which are independent of the time scale, but sensitive only to the direction of time, or dependent just on the order of events in time series. Hence, characterizations of super-stationary sets give insights into what is time, what looks like a history without a description of time duration, or what remains meaningful after we lose the quantitative sense of time.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 31 , Issue 5 , October 2011 , pp. 1445 - 1461
Copyright
Copyright © Cambridge University Press 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]McCutcheon, R.. Elementary Methods in Ergodic Ramsey Theory (Lecture Notes in Mathematics, 1722). Springer, New York, 1999, Ch. 2.CrossRefGoogle Scholar
[2]Kamae, T. and Zamboni, L.. Sequence entropy and the maximal pattern complexity of infinite words. Ergod. Th. & Dynam. Sys. 22 (2002), 1191–1199.CrossRefGoogle Scholar
[3]Kamae, T. and Zamboni, L.. Maximal pattern complexity for discrete systems. Ergod. Th. & Dynam. Sys. 22 (2002), 1201–1214.CrossRefGoogle Scholar
[4]Kamae, T., Rao, H., Tan, B. and Xue, Y.-M.. Language structure of pattern Sturmian words. Discrete Math. 306 (2006), 1651–1668.CrossRefGoogle Scholar
[5]Gjini, N., Kamae, T., Tan, B. and Xue, Y.-M.. Maximal pattern complexity for Toeplitz words. Ergod. Th. & Dynam. Sys. 26 (2006), 1–14.CrossRefGoogle Scholar
[6]Kamae, T., Rao, H. and Xue, Y.-M.. Maximal pattern complexity for 2-dimensional words. Theoret. Comput. Sci. 359 (2006), 15–27.CrossRefGoogle Scholar
[7]Kamae, T. and Rao, H.. Maximal pattern complexity over ℓ letters. European J. Combin. 27 (2006), 125–137.CrossRefGoogle Scholar
[8]Kamae, T.. Uniform set and complexity. Discrete Math. 309 (2009), 3738–3747.CrossRefGoogle Scholar
[9]Kamae, T., Rao, H., Tan, B. and Xue, Y.-M.. Super-stationary set, subword problem and the complexity. Discrete Math. 309 (2009), 4417–4427.CrossRefGoogle Scholar
[10]Xue, Y.-M. and Kamae, T.. Partitions by congruent sets and optimal positions. Ergod. Th. & Dynam. Sys. (doi 10.1017/S0143385709001175) to appear.Google Scholar
[11]Kamae, T. and Salimov, P. V.. On maximal pattern complexity of some of automatic words. Ergod. Th. & Dynam. Sys. (doi 10.1017/S0143385710000453) to appear.Google Scholar