Entropy rigidity for foliations by strictly convex projective manifolds

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

Keywords

entropy rigidity, foliation, strictly convex projective structure, natural map

2010 Mathematics Subject Classification

Primary 53A20, 53C24. Secondary 57M50.

Full Text (PDF format)

The author was partially supported by the FNS grant no. 200020-192216.

Received 24 October 2020

Accepted 12 January 2021

Published 11 April 2021