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.

中文翻译：

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严格凸射影流形的叶片熵熵

假设$ N $是具有叶面$ \ mathscr {F} _N $的紧流形，其叶子是紧致的严格凸投影流形。假设$ M $是具有叶面$ \ mathscr {F} _M $的紧致流形，其叶子是尺寸大于或等于$ 3 $的紧致双曲流形。假设我们有一个保留叶子的同胚性$ f：（N，\ mathscr {F} _N）\ to（M，\ mathscr {F} _M）$这是$ C ^ 1 $-规则的（仅限于叶子）。在先前的情况下，存在定义明确的叶体积熵$ h（N，\ mathscr {F} _N）$和$ h（M，\ mathscr {F} _M）$的概念，并且它拥有$ h（M， \ mathscr {F} _M）\ leq h（N，\ mathscr {F} _N）$。另外，如果等式成立，那么叶子必须是同质的。