Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-01T21:13:44.308Z Has data issue: false hasContentIssue false
  • English
  • Français

Decompositions of factor codes and embeddings between shift spaces with unequal entropies

Published online by Cambridge University Press:  20 November 2012

SOONJO HONG ,
UIJIN JUNG  and
IN-JE LEE
SOONJO HONG
Affiliation:
Centro de Modelamiento Matemático, Universidad de Chile, Av. Blanco Encalada 2120, Piso 7, Santiago de Chile, Chile (email: hsoonjo@dim.uchile.cl)
UIJIN JUNG
Affiliation:
Centro de Modelamiento Matemático, Universidad de Chile, Av. Blanco Encalada 2120, Piso 7, Santiago de Chile, Chile (email: hsoonjo@dim.uchile.cl) School of Mathematics, Korea Institute for Advanced Study, 85 Hoegiro, Dongdaemun-gu, Seoul 130-722, South Korea (email: uijin@kias.re.kr)
IN-JE LEE*
Affiliation:
306-1 Bolli-dong, Dalseo-gu, Daegu 704-913, South Korea (email: ijlee@kaist.ac.kr)
*
Current address: The Jesuit Novitiate of St. Stanislaus in Korea, 1196-43 Gyeongsu-daero, Jangan-gu, Suwon-si, Gyeonggi-do, South Korea (e-mail: paulusisj@gmail.com)
Get access Rights & Permissions [Opens in a new window]

Abstract

Given a factor code between sofic shifts X and Y, there is a family of decompositions of the original code into factor codes such that the entropies of the intermediate subshifts arising from the decompositions are dense in the interval from the entropy of Y to that of X. Furthermore, if X is of finite type, we can choose those intermediate subshifts as shifts of finite type. In the second part of the paper, given an embedding from a shift space to an irreducible sofic shift, we characterize the set of the entropies of the intermediate subshifts arising from the decompositions of the given embedding into embeddings.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 33 , Issue 1 , February 2013 , pp. 144 - 157
Copyright
Copyright © Cambridge University Press 2013

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]Adler, R. and Marcus, B.. Topological entropy and equivalence of dynamical systems. Mem. Amer. Math. Soc. 20(219) (1979), iv+84 pp.Google Scholar
[2]Boyle, M.. Lower entropy factors of sofic systems. Ergod. Th. & Dynam. Sys. 3 (1983), 541–557.CrossRefGoogle Scholar
[3]Boyle, M.. Eventual extensions of finite codes. Proc. Amer. Math. Soc. 104 (1988), 965–972.CrossRefGoogle Scholar
[4]Boyle, M.. Factoring factor maps. J. Lond. Math. Soc. (2) 57 (1998), 491–502.CrossRefGoogle Scholar
[5]Boyle, M., Kitchens, B. and Marcus, B.. A note on minimal covers for sofic systems. Proc. Amer. Math. Soc. 95 (1985), 403–411.CrossRefGoogle Scholar
[6]Boyle, M. and Krieger, W.. Almost Markov and shift equivalent sofic systems. Dynamical Systems (College Park, MD, 1986–87) (Lecture Notes in Mathematics, 1342). Springer, Berlin, 1988, pp. 33–93.Google Scholar
[7]Boyle, M. and Tuncel, S.. Infinite-to-one codes and Markov measures. Trans. Amer. Math. Soc. 285 (1984), 657–684.CrossRefGoogle Scholar
[8]Denker, M., Grillenberger, C. and Sigmund, K.. Ergodic Theory on Compact Spaces (Lecture Notes in Mathematics, 527). Springer, New York, 1976.CrossRefGoogle Scholar
[9]Jung, U.. Open maps between shift spaces. Ergod. Th. & Dynam. Sys. 29 (2009), 1257–1272.CrossRefGoogle Scholar
[10]Jung, U.. On the existence of open and bi-continuing codes. Trans. Amer. Math. Soc. 363 (2011), 1399–1417.CrossRefGoogle Scholar
[11]Jung, U. and Lee, I.. Bi-resolving graph homomorphisms and extensions of bi-closing codes, Acta. Appl. Math. to appear, a special issue dedicated to K. H. Kim, arXiv:0904.3042.Google Scholar
[12]Kitchens, B., Marcus, B. and Trow, P.. Eventual factor maps and compositions of closing maps. Ergod. Th. & Dynam. Sys. 11 (1991), 85–113.CrossRefGoogle Scholar
[13]Krieger, W.. On the periodic points of topological Markov chains. Math. Z. 169 (1979), 99–104.CrossRefGoogle Scholar
[14]Krieger, W.. On the subsystems of topological Markov chains. Ergod. Th. & Dynam. Sys. 2 (1982), 195–202.CrossRefGoogle Scholar
[15]Lind, D. and Marcus, B.. An Introduction to Symbolic Dynamics and Coding. Cambridge University Press, Cambridge, 1995.CrossRefGoogle Scholar
[16]Lindenstrauss, E.. Lowering topological entropy. J. Anal. Math. 67 (1995), 231–267.CrossRefGoogle Scholar
[17]Marcus, B.. Sofic systems and encoding data. IEEE Trans. Inform. Theory 31 (1985), 366–377.CrossRefGoogle Scholar
[18]Nasu, M.. Constant-to-one and onto global maps of homomorphisms between strongly connected graphs. Ergod. Th. & Dynam. Sys. 3 (1983), 387–413.CrossRefGoogle Scholar
[19]Trow, P.. Decompositions of finite-to-one factor maps. Israel J. Math. 91 (1995), 129–155.CrossRefGoogle Scholar
[20]Trow, P.. Decompositions of factor maps involving bi-closing maps. Monatsh. Math. 125 (1998), 165–172.CrossRefGoogle Scholar
[21]Williams, R. F.. Classification of subshifts of finite type. Ann. Math. (2) 98 (1973), 120–153; errata, Ann. Math. (2) 99 (1974), 380–381.CrossRefGoogle Scholar