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Lowering topological entropy over subsets

Published online by Cambridge University Press:  21 July 2009

Department of Mathematics, University of Science and Technology of China, Hefei, Anhui, 230026, People’s Republic of China (email:,
Department of Mathematics, University of Science and Technology of China, Hefei, Anhui, 230026, People’s Republic of China (email:,
School of Mathematical Sciences, Fudan University, Shanghai 200433, People’s Republic of China (email:


Let (X,T) be a topological dynamical system (TDS), and h(T,K) the topological entropy of a subset K of X. (X,T) is lowerable if for each 0≤hh(T,X) there is a non-empty compact subset with entropy h; it is hereditarily lowerable if each non-empty compact subset is lowerable; it is hereditarily uniformly lowerable if for each non-empty compact subset K and each 0≤hh(T,K) there is a non-empty compact subset KhK with h(T,Kh)=h and Kh has at most one limit point. It is shown that each TDS with finite entropy is lowerable, and that a TDS (X,T) is hereditarily uniformly lowerable if and only if it is asymptotically h-expansive.

Research Article
Copyright © Cambridge University Press 2009

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