Positive entropy actions by higher-rank lattices

Aaron Brown, Homin Lee
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Abstract

We study smooth actions by lattices $\Gamma$ in higher-rank simple Lie groups $G$ assuming one element of the action acts with positive topological entropy and prove a number of new rigidity results. For lattices $\Gamma$ in $\mathrm{SL}(n,\mathbb{R})$ acting on $n$-manifolds, if the action has positive topological entropy we show the lattice must be commensurable with $\mathrm{SL}(n,\mathbb{Z})$. Moreover, such actions admit an absolutely continuous probability measure with positive metric entropy; adapting arguments by Katok and Rodriguez Hertz, we show such actions are measurably conjugate to affine actions on (infra)tori. In our main technical arguments, we study families of probability measures invariant under sub-actions of the induced $G$-action on an associated fiber bundle. To control entropy properties of such measures, in the appendix we establish certain upper semicontinuity of entropy under weak-$*$ convergence, adapting classical results of Yomdin and Newhouse.
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高阶网格的正熵作用
我们研究了高阶简单李群$G$中$\Gamma$网格的光滑作用,假设作用的一个元素具有正拓扑熵,并证明了一些新的刚性结果。对于作用于$n$-manifolds的$mathrm{SL}(n,\mathbb{R})$中的$\Gamma$网格,如果作用具有正拓扑熵,我们证明了该网格必须与$mathrm{SL}(n,\mathbb{Z})$可共轭。此外,这样的作用允许一个具有正度量熵的绝对连续概率度量;根据卡托克和罗德里格斯-赫兹的论证,我们证明了这样的作用与(下)环上的非线性作用是可测共轭的。在我们的主要技术论证中,我们研究了在相关纤维束上的诱导 $G$ 作用的子作用下不变的概率计量族。为了控制这些度量的熵属性,我们在附录中根据约姆丁和纽豪斯的经典结果,建立了弱$*$收敛下熵的某些上半连续性。
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