abstract:

We show that if a compact, connected, and oriented $n$-manifold $M$ without boundary admits a non-constant non-injective uniformly quasiregular self-map, then the dimension of the real singular cohomology ring $H^*(M;\Bbb{R})$ of $M$ is bounded from above by $2^n$. This is a positive answer to a dynamical counterpart of the Bonk-Heinonen conjecture on the cohomology bound for quasiregularly elliptic manifolds. The proof is based on an intermediary result that, if $M$ is not a rational homology sphere, then each such uniformly quasiregular self-map on $M$ has a Julia set of positive Lebesgue measure.

中文翻译：

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均匀拟正则椭圆流形的锐上同调界

摘要：

我们证明，如果一个无边界的紧致、连通和有向的 $n$-流形 $M$ 承认一个非常量非单射一致拟正则自映射，那么实奇异上同调环的维数 $H^*(M ;\Bbb{R})$ 的 $M$ 以 $2^n$ 为界。这是对拟正则椭圆流形上同调界的 Bonk-Heinonen 猜想的动力学对应物的肯定回答。证明基于中间结果，如果 $M$ 不是有理同调球体，则 $M$ 上的每个此类一致拟正则自映射都有一个 Julia 集的正 Lebesgue 测度。