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Hamiltonian pseudo-rotations of projective spaces

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Abstract

The main theme of the paper is the dynamics of Hamiltonian diffeomorphisms of \(\mathbb {C}{\mathbb {P}}^n\) with the minimal possible number of periodic points (equal to \(n+1\) by Arnold’s conjecture), called here Hamiltonian pseudo-rotations. We prove several results on the dynamics of pseudo-rotations going beyond periodic orbits, using Floer theoretical methods. One of these results is the existence of invariant sets in arbitrarily small punctured neighborhoods of the fixed points, partially extending a theorem of Le Calvez and Yoccoz and of Franks and Misiurewicz to higher dimensions. The other is a strong variant of the Lagrangian Poincaré recurrence conjecture for pseudo-rotations. We also prove the \(C^0\)-rigidity of pseudo-rotations with exponentially Liouville mean index vector. This is a higher-dimensional counterpart of a theorem of Bramham establishing such rigidity for pseudo-rotations of the disk.

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Acknowledgements

The authors are grateful to Barney Bramham, Weinmin Gong, Felix Schlenk, Sobhan Seyfaddini, Egor Shelukhin, Michael Usher, Joa Weber and the referees for useful remarks and discussions. Parts of this work were carried out while both of the authors were visiting the Lorentz Center (Leiden, Netherlands) for the Hamiltonian and Reeb Dynamics: New Methods and Applications workshop, during the first author’s visit to NCTS (Taipei, Taiwan) and the second author’s visit to UCSC (Santa Cruz, California). The authors would like to thank these institutes for their warm hospitality and support.

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Authors and Affiliations

  1. Department of Mathematics, University of Central Florida, Orlando, FL, 32816, USA

    Başak Z. Gürel

  2. Department of Mathematics, UC Santa Cruz, Santa Cruz, CA, 95064, USA

    Viktor L. Ginzburg

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  1. Viktor L. Ginzburg
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  2. Başak Z. Gürel
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Correspondence to Viktor L. Ginzburg.

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The work is partially supported by NSF CAREER award DMS-1454342 (BG) and NSF Grant DMS-1308501 (VG).

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Ginzburg, V.L., Gürel, B.Z. Hamiltonian pseudo-rotations of projective spaces. Invent. math. 214, 1081–1130 (2018). https://doi.org/10.1007/s00222-018-0818-9

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