Area-preserving irrotational diffeomorphisms of the torus with sublinear diffusion

TL;DR

This paper constructs a smooth, area-preserving, ergodic diffeomorphism of the torus with zero rotation set, where almost all lifted orbits are unbounded and accumulate in all directions, revealing complex dynamical behavior.

Contribution

It introduces a new example of a smooth, area-preserving, ergodic torus diffeomorphism with zero rotation set and unbounded orbits, expanding understanding of toral dynamics.

Findings

Constructed a Bernoulli, ergodic diffeomorphism with zero rotation set.

Almost every lifted orbit is unbounded and accumulates in all directions.

Demonstrated complex orbit behavior in area-preserving toral dynamics.

Abstract

We construct a $C^\infty$ area-preserving diffeomorphism of the two-dimensional torus which is Bernoulli (in particular, ergodic) with respect to Lebesgue measure, homotopic to the identity, and has a lift to the universal covering whose rotation set is $\{(0,0)\}$, which in addition has the property that almost every orbit by the lifted dynamics is unbounded and accumulates in every direction of the circle at infinity.