On the loss of upper semi-continuity of metric entropy for $C^r$ diffeomorphisms

In this article, we give an upper bound estimate for the quantitative loss of the upper semi-continuity of the metric entropy for $C^r(r>1)$ diffeomorphisms. Building on earlier entropy estimates and reparametrization methods, we optimize the upper bound estimate with respect to both dimension and asymptotic Lipschitz constant. Motivated by Buzzi’s examples, we show that the estimate is sharp.