Determine the entropy of higher-dimensional geodesic laminations

Determine the topological entropy of a geodesic lamination in higher dimensions in order to extend the vanishing-extremal-entropy result from convex cocompact representations into \(\mathsf{PO}(2,1)\) to general convex cocompact representations \(\rho:G\to\mathsf{PO}(d,1)\).

Background

The paper proves that when $\rho:G\to\mathsf{PO}(2,1)$ and $\sigma:G\to\mathsf{PO}(d,1)$ are convex cocompact representations with $L_{\rho\sigma}>1$, the extremal entropy satisfies $E_{\rho\sigma}(L_{\rho\sigma})=0$. The proof bounds this entropy by the topological entropy of a geodesic lamination arising as the stretch locus of an optimal Lipschitz map.

For surfaces, the relevant geodesic lamination has zero topological entropy by a result of Fathi. The authors explicitly identify the missing ingredient for extending the argument when the first representation also takes values in higher-dimensional $\mathsf{PO}(d,1)$: determining the topological entropy of geodesic laminations in higher dimensions.