Application of waist inequality to entropy and mean dimension

Abstract

Waist inequality is a fundamental inequality in geometry and topology. We apply it to the study of entropy and mean dimension of dynamical systems. We consider equivariant continuous maps π : ( X , T ) → ( Y , S ) \pi : (X, T) \to (Y, S) between dynamical systems and assume that the mean dimension of the domain ( X , T ) (X, T) is larger than the mean dimension of the target ( Y , S ) (Y, S) . We exhibit several situations for which the maps π \pi necessarily have positive conditional metric mean dimension. This study has interesting consequences to the theory of topological conditional entropy. In particular it sheds new light on a celebrated result of Lindenstrauss and Weiss [Israel J. Math. 115 (2000), pp. 1–24] about minimal dynamical systems non-embeddable in [ 0 , 1 ] Z [0,1]^{\mathbb {Z}} .