A diffeomorphism with global dominated splitting can not be minimal

TL;DR

This paper proves that a diffeomorphism with a nontrivial dominated splitting on a closed manifold cannot be minimal, using Mane's argument and Liao's selecting lemma.

Contribution

It establishes a new restriction on minimal dynamics by showing dominated splitting prevents minimality on closed manifolds.

Findings

Diffeomorphisms with dominated splitting are not minimal.

The proof employs Mane's argument and Liao's selecting lemma.

Provides a theoretical result linking dominated splitting and minimality.

Abstract

Let M be a closed manifold and f be a diffeomorphism on M. We show that if f has a nontrivial dominated splitting TM=E\oplus F, then f can not be minimal. The proof mainly use Mane's argument and Liao's selecting lemma.