Hamiltonian Pseudo-rotations of Projective Spaces

TL;DR

This paper investigates Hamiltonian pseudo-rotations of complex projective spaces, revealing invariant sets, recurrence properties, and $C^0$-rigidity, thus extending known results to higher dimensions using Floer theory.

Contribution

It introduces new dynamical properties of Hamiltonian pseudo-rotations in higher dimensions, including invariant sets and rigidity results, expanding the understanding beyond classical two-dimensional cases.

Findings

Existence of invariant sets near fixed points

A strong variant of the Lagrangian Poincaré recurrence conjecture

$C^0$-rigidity for pseudo-rotations with exponentially Liouville mean index

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 Poincar\'e 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.