# Finiteness of partially hyperbolic attractors with one-dimensional center

## TL;DR

This paper demonstrates that in the space of $C^1$ partially hyperbolic diffeomorphisms with one-dimensional center, having finitely many attractors is a dense and open property, supported by robust geometric features.

## Contribution

It establishes the generic finiteness of attractors for a broad class of partially hyperbolic systems using new geometric techniques.

## Key findings

- Finiteness of attractors is dense and open in the specified class.
- Robust geometric properties of laminations are key to the proof.
- Generic diffeomorphisms either have finitely many attractors or exhibit Newhouse phenomena.

## Abstract

We prove that the set of diffeomorphisms having at most finitely many attractors contains a dense and open subset of the space of $C^1$ partially hyperbolic diffeomorphisms with one-dimensional center. This is obtained thanks to a robust geometric property of partially hyperbolic laminations that we show to hold after perturbations of the dynamics. This technique also allows to prove that $C^1$-generic diffeomorphisms far from homoclinic tangencies in dimension $3$ either have at most finitely many attractors, or satisfy Newhouse phenomenon.