2017/12/31 by Viktor L. Ginzburg, Basak Z. Gurel · 1 citation
Mathematics · #math.DS #math.SG #msc:37J10 #msc:37J45 #msc:53D40
paper · pdf · doi:10.1007/s00222-018-0818-9
published as Inventiones mathematicae, 2018, https://doi.org/10.1007/s00222-018-0818-9 · 38 pages; final version (with minor revisions and updated references); published Online First in Inventiones mathematicae
arxiv created 2018/10/03 · arxiv updated 2018/10/04
The main theme of the paper is the dynamics of Hamiltonian diffeomorphisms of \mathbb C\mathbb Pn 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 C0-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.