Published online by Cambridge University Press: 28 December 2020
Let
$(X,T)$
be a topological dynamical system consisting of a compact metric space X and a continuous surjective map
$T : X \to X$
. By using local entropy theory, we prove that
$(X,T)$
has uniformly positive entropy if and only if so does the induced system
$({\mathcal {M}}(X),\widetilde {T})$
on the space of Borel probability measures endowed with the weak* topology. This result can be seen as a version for the notion of uniformly positive entropy of the corresponding result for topological entropy due to Glasner and Weiss.