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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: gyonatan@math.huji.ac.il)
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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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