Pseudo-rotations vs. Rotations

TL;DR

This paper investigates the fixed-point data of Hamiltonian pseudo-rotations in projective spaces, proving conjectures about their similarity to true rotations and exploring the existence of ghost pseudo-rotations, using combinatorial methods.

Contribution

It proves the conjecture that pseudo-rotations share fixed-point data with true rotations in certain cases and introduces an index divisibility theorem linking spectral properties to fixed-point data.

Findings

Confirmed the fixed-point data match between pseudo-rotations and rotations in specific cases.

Identified the concept of ghost pseudo-rotations with similar fixed-point data but different dynamics.

Developed a combinatorial index divisibility theorem related to spectral properties.

Abstract

Continuing the study of Hamiltonian pseudo-rotations of projective spaces, we focus on the conjecture that the fixed-point data set (the actions and the linearized flows at one-periodic orbits) of a pseudo-rotation exactly matches that data for a suitable unique true rotation even though the two maps can have very different dynamics. We prove this conjecture in several instances, e.g., for strongly non-degenerate pseudo-rotations of ${{\mathbb C}{\mathbb P}}^2$ with some notable exceptions, which we call ghost pseudo-rotations. The existence of ghost pseudo-rotations is a completely open question. The conjecture is closely related to the properties of the action and index spectra of pseudo-rotations, and ghost pseudo-rotations, if they exist, satisfy all known restrictions on the fixed-point data for pseudo-rotations but these data are distinctly different from the data for any rotation. The main new ingredient of the proofs is purely combinatorial and of independent interest. This is the index divisibility theorem connecting the divisibility properties of the Conley--Zehnder index sequence for the iterates of a map with the properties of its spectrum.