Hamiltonian Pseudo-rotations of Projective Spaces

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 Franks 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.

38 pages; final version (with minor revisions and updated references); published Online First in Inventiones mathematicae

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• A Hölder-type inequality for the $C^0$ distance and Anosov-Katok pseudo-rotations
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• On periodic points of Hamiltonian diffeomorphisms of $\mathbb{C}{\text{P}}^d$ via generating functions
• On the generic behavior of the spectral norm
• Hamiltonian no-torsion
• On the Barcode Entropy of Reeb Flows
• On the Hofer-Zehnder conjecture for non-contractible periodic orbits in Hamiltonian dynamics
• Invariant Sets and Hyperbolic Closed Reeb Orbits
• On the dynamics characterization of complex projective spaces
• On the Hofer-Zehnder conjecture for semipositive symplectic manifolds
• A counterexample to Lagrangian Poincaré recurrence in dimension four
• On the Filtered Symplectic Homology of Prequantization Bundles