Entropy rigidity for foliations by strictly convex projective manifolds

Abstract

Let $N$ be a compact manifold with a foliation ${\mathcal{F}}_N$ whose leaves are compact strictly convex projective manifolds. Let $M$ be a compact manifold with a foliation ${\mathcal{F}}_M$ whose leaves are compact hyperbolic manifolds of dimension bigger than or equal to $3$. Suppose to have a foliation-preserving homeomorphism $f:(N,{\mathcal{F}}_N)\to (M,{\mathcal{F}}_M)$ which is $C^1$-regular when restricted to leaves. In the previous situation there exists a well-defined notion of foliated volume entropies $h(N,{\mathcal{F}}_N)$ and $h(M,{\mathcal{F}}_M)$ and it holds $h(M,{\mathcal{F}}_M)\le h(N,{\mathcal{F}}_N)$. Additionally, if equality holds, then the leaves must be homothetic.

11 pages, small changes and some typos corrected, To appear on Pure and Applied Mathematics Quarterly