Hamiltonian pseudo-rotations of projective spaces

Abstract

The main theme of the paper is the dynamics of Hamiltonian diffeomorphisms of CP n with the minimal possible number of periodic points (equal to n + 1 by Arnold's conjecture), called here Hamiltonian pseudorotations. 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.

Approximate Identities and Lagrangian Poincaré Recurrence

“…One interesting class of γ-a.i. 's, relevant for what follows, is identified in [16]. These are (Hamiltonian) pseudo-rotations of CP n , i.e., Hamiltonian diffeomorphisms of CP n with minimal possible number of periodic points, equal to n + 1 by the Arnold conjecture, [28,39].…”

“…Among pseudo-rotations are the Anosov-Katok pseudo-rotations from Example 2.2 and true rotations (i.e., isometries) of CP n with finitely many fixed points. The following theorem is proved in a slightly different form in [16]: Theorem 3.2 (γ-convergence, Thm. 5.1, [16]).…”

“…The following theorem is proved in a slightly different form in [16]: Theorem 3.2 (γ-convergence, Thm. 5.1, [16]). Let ϕ be a pseudo-rotation of CP n .…”

“…The notion of an approximate identity and several variants of the definition are spelled out in Section 2. Approximate identities can have interesting and non-trivial dynamics, and the main examples of such maps are the so-called pseudo-rotations; see [3,5,10,16,17]. Yet, in some way, these maps resemble actions of compact abelian groups on manifolds.…”

On the dynamics characterization of complex projective spaces

“…There are several definitions of pseudo-rotations, but roughly speaking pseudorotations are Hamiltonian diffeomorphisms with finite and minimal possible number of periodic points. They have been studied in [GG18a] and the references therein.…”