Let $M$ be a closed, oriented, and connected Riemannian $n$-manifold, for $n\geq 2$, which is not a rational homology sphere. We show that, for a non-constant and non-injective uniformly quasiregular self-map $f:M\rightarrow M$, the topological entropy $h(f)$ is $\log \deg f$. This proves Shub’s entropy conjecture in this case.

中文翻译：

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均匀准正则动力学中的熵

让$M$是一个封闭的、有向的、连通的黎曼算子$n$-歧管，对于$n\geq 2$，这不是一个有理同调球。我们证明，对于非常量和非内射的一致拟正则自映射$f:M\右箭头 M$, 拓扑熵$h(f)$是$\log \度f$. 在这种情况下，这证明了 Shub 的熵猜想。