Positive entropy actions by higher-rank lattices

We study smooth actions by lattices $\Gamma$ in higher-rank simple Lie groups $G$ assuming one element of the action acts with positive topological entropy and prove a number of new rigidity results. For lattices $\Gamma$ in $\mathrm{SL}(n,\mathbb{R})$ acting on $n$-manifolds, if the action has positive topological entropy we show the lattice must be commensurable with $\mathrm{SL}(n,\mathbb{Z})$. Moreover, such actions admit an absolutely continuous probability measure with positive metric entropy; adapting arguments by Katok and Rodriguez Hertz, we show such actions are measurably conjugate to affine actions on (infra)tori. In our main technical arguments, we study families of probability measures invariant under sub-actions of the induced $G$-action on an associated fiber bundle. To control entropy properties of such measures, in the appendix we establish certain upper semicontinuity of entropy under weak-$*$ convergence, adapting classical results of Yomdin and Newhouse.