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.

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

Received 24 October 2020

Accepted 12 January 2021

Published 11 April 2021