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Textile systems and one-sided resolving automorphisms and endomorphisms of the shift

Published online by Cambridge University Press:  01 February 2008

MASAKAZU NASU
MASAKAZU NASU*
Affiliation:
19-8, 9-chōme, Takaya-Takamigaoka, Higashi-Hiroshima 739-2115, Japan (email: [email protected])
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Abstract

Two results on textile systems are obtained. Using these we prove that for any automorphism φ of any topologically-transitive subshift of finite type, if φ is expansive and φ or φ−1 has memory zero or anticipation zero, then φ is topologically conjugate to a subshift of finite type. Moreover, this is generalized to a result on chain recurrent onto endomorphisms of topologically-transitive subshifts of finite type. Using textile systems and textile subsystems, we develop a structure theory concerning expansiveness with the pseudo orbit tracing property on directionally essentially weakly one-sided resolving automorphisms and endomorphisms of subshifts.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 28 , Issue 1 , February 2008 , pp. 167 - 209
Copyright
Copyright © Cambridge University Press 2007

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References

[1]Akatsuka, K.. The 6-scope cellular automata whose global maps are bijective. Master’s Thesis, Mie University, Tsu, 1989(in Japanese).Google Scholar
[2]Aoki, N. and Hiraide, K.. Topological Theory of Dynamical Systems. North-Holland, Amsterdam, 1994.Google Scholar
[3]Ashley, J.. LR conjugacies of shifts of finite type are uniquely so. Symbolic Dynamics and its Applications (New Haven, CT, 1991) (Contemporary Mathematics, 135). American Mathematical Society, Providence, RI, 1992, pp. 57–84.CrossRefGoogle Scholar
[4]Blanchard, F. and Maass, A.. Dynamical properties of expansive one-sided cellular automata. Israel J. Math. 99 (1997), 149–174.CrossRefGoogle Scholar
[5]Blanchard, F. and Tisseur, P.. Some properties of cellular automata with equicontinuity points. Ann. Inst. H. Poincaré Probab. Statist. 36 (2000), 569–582.CrossRefGoogle Scholar
[6]Boyle, M.. Some sofic shifts cannot commute with nonwandering shifts of finite type. Illinois J. Math. 48 (2004), 1267–1277.CrossRefGoogle Scholar
[7]Boyle, M., Fiebig, D. and Fiebig, U.. A dimension group for local homeomorphisms and endomorphisms of onesided shifts of finite type. J. Reine Angew. Math. 487 (1997), 27–59.Google Scholar
[8]Boyle, M. and Kitchens, B.. Periodic points for onto cellular automata. Indag. Math. (N.S.) 10 (1999), 483–493.CrossRefGoogle Scholar
[9]Boyle, M. and Krieger, W.. Periodic points and automorphisms of the shift. Trans. Amer. Math. Soc. 302 (1987), 125–149.CrossRefGoogle Scholar
[10]Boyle, M. and Lee, B.. Jointly periodic points in cellular automata: computer explorations and conjectures. Experimental Math., to appear.Google Scholar
[11]Boyle, M. and Lind, D.. Expansive subdynamics. Trans. Amer. Math. Soc. 349 (1997), 55–102.CrossRefGoogle Scholar
[12]Boyle, M. and Maass, A.. Expansive invertible onesided cellular automata. J. Math. Soc. Japan 52 (2000), 725–740; J. Math. Soc. Japan 56 (2004), 309–310 (Erratum).CrossRefGoogle Scholar
[13]Fiebig, D.. Personal communication, 1996.Google Scholar
[14]Hedlund, G. A.. Endomorphisms and automorphisms of the shift dynamical system. Math. Systems Theory 3 (1969), 320–375.CrossRefGoogle Scholar
[15]Kitchens, B.. Symbolic Dynamics, One-sided, Two-sided and Countable State Markov Chains. Springer, Berlin, 1998.Google Scholar
[16]K˚urka, P.. Languages, equicontinuity and attractors in cellular automata. Ergod. Th. & Dynam. Sys. 17 (1997), 417–433.CrossRefGoogle Scholar
[17]Lind, D. and Marcus, B.. An Introduction to Symbolic Dynamics and Coding. Cambridge University Press, Cambridge, 1995.CrossRefGoogle Scholar
[18]Nasu, M.. Local maps inducing surjective global maps of one-dimensional tessellation automata. Math. Systems Theory 11 (1978), 327–351.CrossRefGoogle Scholar
[19]Nasu, M.. Uniformly finite-to-one and onto extensions of homomorphisms between strongly connected graphs. Discrete Math. 39 (1982), 171–179.CrossRefGoogle Scholar
[20]Nasu, M.. Textile systems for endomorphisms and automorphisms of the shift. Mem. Amer. Math. Soc. 546 (1995).Google Scholar
[21]Nasu, M.. Maps in symbolic dynamics. Lecture Notes of The Tenth KAIST Mathematics Workshop 1995. Ed. G. H. Choe. Korea Advanced Institute of Science and Technology, Mathematics Research Center, Taejon, 1996.Google Scholar
[22]Nasu, M.. Endomorphisms of expansive systems on compact metric spaces and the pseudo-orbit tracing property. Trans. Amer. Math. Soc. 352 (2000), 4731–4757.CrossRefGoogle Scholar
[23]Nasu, M.. The dynamics of expansive invertible onesided cellular automata. Trans. Amer. Math. Soc. 354 (2002), 4067–4084.CrossRefGoogle Scholar
[24]Nasu, M.. Expansive automorphisms and expansive endomorphisms of the shift (a survey). Dynamics of Complex Systems (Kyoto, 2004) RIMS Kokyuroku (1404) (2004), 68–82.Google Scholar
[25]Nasu, M.. Nondegenerate q-biresolving textile systems and expansive automorphisms of onesided full shifts. Trans. Amer. Math. Soc. 358 (2006), 871–891.CrossRefGoogle Scholar
[26]Reddy, W. L.. Lifting expansive homeomorphisms to symbolic flows. Math. Systems Theory 2 (1968), 91–92.CrossRefGoogle Scholar
[27]Walters, P.. On the Pseudo-orbit Tracing Property and its Relation to Stability (Lecture Notes in Mathematics, 668). Springer, Berlin, 1978, pp. 231–244.Google Scholar