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Embedding ℤk-actions in cubical shifts and ℤk-symbolic extensions

Published online by Cambridge University Press:  26 March 2010

YONATAN GUTMAN
YONATAN GUTMAN*
Affiliation:
Institute of Mathematics, The Hebrew University, Jerusalem, Israel (email: [email protected])
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Abstract

Mean dimension is an invariant which makes it possible to distinguish between topological dynamical systems with infinite entropy. Extending in part the work of Lindenstrauss we show that if (X,ℤk) has a free zero-dimensional factor then it can be embedded in the ℤk-shift on ([0,1]d)ℤk, where d=[C(k) mdim(X,ℤk)]+1 for some universal constant C(k), and a topological version of the Rokhlin lemma holds. Furthermore, under the same assumptions, if mdim(X,ℤk)=0, then (X,ℤk) has the small boundary property. One of the applications of this theory is related to Downarowicz’s entropy structure, a master invariant for entropy theory, which captures the emergence of entropy on different scales. Indeed, we generalize this invariant and prove the Boyle–Downarowicz symbolic extension entropy theorem in the setting of ℤk-actions. This theorem describes what entropies are achievable in symbolic extensions.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 31 , Issue 2 , April 2011 , pp. 383 - 403
Copyright
Copyright © Cambridge University Press 2010

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References

[1]Boyle, M. and Downarowicz, T.. The entropy theory of symbolic extensions. Invent. Math. 156(1) (2004), 119–161.Google Scholar
[2]Boyle, M., Fiebig, D. and Fiebig, U.. Residual entropy, conditional entropy and subshift covers. Forum Math. 14(5) (2002), 713–757.CrossRefGoogle Scholar
[3]Downarowicz, T.. Entropy of a symbolic extension of a dynamical system. Ergod. Th. & Dynam. Sys. 21(4) (2001), 1051–1070.CrossRefGoogle Scholar
[4]Downarowicz, T.. Entropy structure. J. Anal. Math. 96 (2005), 57–116.CrossRefGoogle Scholar
[5]Engelking, R.. Dimension Theory (North-Holland Mathematical Library, 19). North-Holland, Amsterdam, 1978, Translated from the Polish and revised by the author.Google Scholar
[6]Flores, A.. Über n-dimensionale Komplexe, die im ℝ2n+1 absolut selbstverschlungen sind. Erg. Math. Kolloqu. 6 (1935), 4–7.Google Scholar
[7]Gromov, M.. Topological invariants of dynamical systems and spaces of holomorphic maps. I. Math. Phys. Anal. Geom. 2(4) (1999), 323–415.CrossRefGoogle Scholar
[8]Grünbaum, B.. Convex Polytopes (With the cooperation of Victor Klee, M. A. Perles and G. C. Shephard) (Pure and Applied Mathematics, 16). Interscience, New York, 1967.Google Scholar
[9]Lightwood, S. J.. Morphisms from non-periodic -subshifts. I. Constructing embeddings from homomorphisms. Ergod. Th. & Dynam. Sys. 23(2) (2003), 587–609.CrossRefGoogle Scholar
[10]Lindenstrauss, E.. Mean dimension, small entropy factors and an embedding theorem. Publ. Math. Inst. Hautes Études Sci. 89(1) (2000), 227–262, 1999.Google Scholar
[11]Lindenstrauss, E. and Weiss, B.. Mean topological dimension. Israel J. Math. 115 (2000), 1–24.CrossRefGoogle Scholar
[12]Shub, M. and Weiss, B.. Can one always lower topological entropy? Ergod. Th. & Dynam. Sys. 11(3) (1991), 535–546.Google Scholar