Embedding asymptotically expansive systems

Embedding asymptotically expansive systems A topological dynamical system is said asymptotically expansive when entropy and periodic points grow subexponentially at arbitrarily small scales. We prove a Krieger like embedding theorem for asymptotically expansive systems with the small boundary property. We show that such a system (X, T) embeds in the K-full shift if $$ h_{top}(T)<\log K$$ h t o p ( T ) < log K and $$\sharp Per_n(X,T)\le K^n$$ ♯ P e r n ( X , T ) ≤ K n for any integer n. The embedding is in general not continuous (unless the system is expansive and X is zero-dimensional) but the induced map on the set of invariant measures is a topological embedding. It is shown that this property implies asymptotical expansiveness. We prove also that the inverse of the embedding map may be continuously extended to a faithful principal symbolic extension. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Monatshefte f�r Mathematik Springer Journals

Embedding asymptotically expansive systems

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Publisher
Springer Vienna
Copyright
Copyright © 2017 by Springer-Verlag GmbH Austria
Subject
Mathematics; Mathematics, general
ISSN
0026-9255
eISSN
1436-5081
D.O.I.
10.1007/s00605-017-1079-1
Publisher site
See Article on Publisher Site

Abstract

A topological dynamical system is said asymptotically expansive when entropy and periodic points grow subexponentially at arbitrarily small scales. We prove a Krieger like embedding theorem for asymptotically expansive systems with the small boundary property. We show that such a system (X, T) embeds in the K-full shift if $$ h_{top}(T)<\log K$$ h t o p ( T ) < log K and $$\sharp Per_n(X,T)\le K^n$$ ♯ P e r n ( X , T ) ≤ K n for any integer n. The embedding is in general not continuous (unless the system is expansive and X is zero-dimensional) but the induced map on the set of invariant measures is a topological embedding. It is shown that this property implies asymptotical expansiveness. We prove also that the inverse of the embedding map may be continuously extended to a faithful principal symbolic extension.

Journal

Monatshefte f�r MathematikSpringer Journals

Published: Jul 3, 2017

References

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