Pseudo-rotations with sufficiently Liouvillean rotation number are \(C^0\) -ri

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• 作者：Barney Bramham
• 刊名：Inventiones Mathematicae
• 出版年：2015
• 出版时间：February 2015
• 年：2015
• 卷：199
• 期：2
• 页码：561-580
• 全文大小：236 KB
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• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1432-1297

文摘

It is an open question in smooth ergodic theory whether there exists a Hamiltonian disk map with zero topological entropy and (strong) mixing dynamics. Weak mixing has been known since Anosov and Katok first constructed examples in 1970. Currently all known examples with weak mixing are irrational pseudo-rotations with Liouvillean rotation number on the boundary. Our main result however implies that for a dense subset of Liouville numbers (strong) mixing cannot occur. Our approach involves approximating the flow of a suspension of the given disk map by pseudoholomorphic curves. Ellipticity of the Cauchy–Riemann equation allows quantitative \(L^2\) -estimates to be converted into \(C^0\) -estimates between the pseudoholomorphic curves and the trajectories of the flow on growing time scales. Arithmetic properties of the rotation number enter through these estimates.