# Rotational deviations and invariant pseudo-foliations for periodic point free torus homeomorphisms

## TL;DR

This paper investigates the relationship between bounded rotational deviations and invariant pseudo-foliations in certain torus homeomorphisms, introducing new tools like the $ ilde
ho$-centralized skew-product to analyze their dynamics.

## Contribution

It establishes a characterization linking bounded rotational deviations to the invariance of pseudo-foliations in periodic point free torus homeomorphisms, using novel mathematical constructs.

## Key findings

- Bounded rotational deviations imply the existence of invariant pseudo-foliations.
- Introduction of the $ ilde
ho$-centralized skew-product as a new analytical tool.
- Equivalence between bounded deviations and pseudo-foliation invariance under mild conditions.

## Abstract

This article deals with directional rotational deviations for non-wandering periodic point free homeomorphisms of the 2-torus which are homotopic to the identity. We prove that under mild assumptions, such a homeomorphism exhibits uniformly bounded rotational deviations in some direction if and only it leaves invariant a pseudo-foliation, a notion which is a slight generalization of classical one-dimensional foliations. To get these results, we introduce a novel object called $\tilde\rho$-centralized skew-product and their associated stable sets at infinity.