Dynamics of riemannian 1-foliations on $3$-manifolds

TL;DR

This paper investigates the dynamical behavior of Riemannian 1-foliations on closed 3-manifolds, establishing nonhyperbolicity and analyzing recurrence, limit sets, and attractors.

Contribution

It provides a classification-based proof of nonhyperbolicity and detailed descriptions of dynamical features for Riemannian 1-foliations on 3-manifolds.

Findings

Proves nonhyperbolicity of Riemannian 1-foliations on 3-manifolds

Describes recurrence points and omega-limit sets in detail

Analyzes the structure of attractors in these foliations

Abstract

In this paper we study several dynamical properties of the riemannian $1$-dimensional foliation $\mathcal{L}$ on an oriented closed 3-manifold $M$. Carriere classified such pairs $(M,\mathcal{L})$. Using the classification we prove the nonhyperbolicity of $(M,\mathcal{L})$. Also we describe in detail recurrence points, $\omega$-limit sets and attractors.