Weak containment rigidity for distal actions

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• 作者：Adrian Ioana^a ; ^{aioana@ucsd.edu" class="auth_mail" title="E-mail the corresponding author} ; Robin Tucker-Drob^b ; ^{rtuckerd@math.tamu.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Weak containment ; Weak equivalence ; Distal action ; Compact action ; Strong ergodicity ; Spectral gap ; Factor ; Rigidity
• 刊名：Advances in Mathematics
• 出版年：2016
• 出版时间：22 October 2016
• 年：2016
• 卷：302
• 期：Complete
• 页码：309-322
• 全文大小：384 K

文摘

We prove that if a measure distal action α of a countable group Γ is weakly contained in a strongly ergodic probability measure preserving action β of Γ, then α is a factor of β. In particular, this applies when α is a compact action.

As a consequence, we show that the weak equivalence class of any strongly ergodic action completely remembers the weak isomorphism class of the maximal distal factor arising in the Furstenberg–Zimmer Structure Theorem.