Abstract.  

The topological mean dimension is a topological invariant introduced by Gromov, which is zero for topological systems with finite dimension or finite topological entropy.  For a topological system (X,T), we consider the induced map $T_*$ on the set $\mathcal M(X)$ of Borel probability measures. It is well known that $T_*$ has infinite topological entropy, if  $T$ has positive topological entropy. We show that the topological mean dimension of $T_*$ is also infinite. This answers a question of B. Kloeckner. Moreover we give precise rates of divergence  of $h_W(T_*,\epsilon)$ when $\epsilon$ goes to zero, where $h_W(T_*,\epsilon)$ denotes the Bowen metric entropy with respect to the Wasserstein distance $W$.  Joint work with Ruxi Shi.