Bounded and unbounded behavior for area-preserving rational pseudo-rotations

TL;DR

This paper classifies rational pseudo-rotations on the torus with full support invariant measures, revealing boundedness, annular behavior, or large continua of fixed points, and introduces new geometric and recurrence tools.

Contribution

It provides a new classification framework for rational pseudo-rotations and introduces geometric and recurrence tools applicable to broader dynamical systems.

Findings

Either all orbits are bounded or the dynamics is annular or contains a large continuum of fixed points.

In the analytic setting, the large continuum case is excluded.

New geometric results on chains of disks and a Poincaré recurrence theorem are developed.

Abstract

A rational pseudo-rotation $f$ of the torus is a homeomorphism homotopic to the identity with a rotation set consisting of a single vector $v$ of rational coordinates. We give a classification for rational pseudo-rotations with an invariant measure of full support, in terms of the deviations from the constant rotation $x\mapsto x+v$ in the universal covering. For the simpler case that $v=(0,0)$, it states that either every orbit by the lifted dynamics is bounded, or the displacement in some rational direction is uniformly bounded (implying that the dynamics is annular) or the set of fixed points of $f$ contains a large continuum which is the complement of a disjoint union of disks (i.e. a fully essential continuum). In the analytic setting, the latter case is ruled out. In order to prove this classification, we introduce tools that are of independent interest and can be applied in a more general setting: in particular, a geometric result about the quasi-convexity and existence of asymptotic directions for certain chains of disks, and a Poincar\'e recurrence theorem on the universal covering for irrotational measures.