# Attractors of Cartan foliations

## TL;DR

This paper investigates the existence of attractors in Cartan foliations, linking geometric structures to dynamical properties and providing conditions for attractor existence through holonomy groups.

## Contribution

It introduces a framework connecting Cartan geometries with attractor existence, reducing the problem to holonomy group actions and providing new criteria for attractor existence.

## Key findings

- Existence of attractors is linked to the structure Lie group action.
- Conditions on linear holonomy groups ensure minimal attractors.
- Examples illustrate the theoretical results.

## Abstract

The paper is focused on the existence problem of attractors for foliations. Since the existence of an attractor is a transversal property of the foliation, it is natural to consider foliations admitting transversal geometric structures. As transversal structures are chosen Cartan geometries due to their universality. The existence problem of an attractor on a complete Cartan foliation is reduced to a similar problem for the action of its structure Lie group on a certain smooth manifold. In the case of a complete Cartan foliation with a structure subordinated to a transformation group, the problem is reduced to the level of the global holonomy group of this foliation. Each countable automorphism group preserving a Cartan geometry on a manifold and admitting an attractor is realized as the global holonomy group of some Cartan foliation with an attractor. Conditions on the linear holonomy group of a leaf of a reductive Cartan foliation sufficient for the existence of an attractor (and a global attractor) which is a minimal set are found. Various examples are considered.