Minimal Diffeomorphisms cannot satisfy Generalized Dominated Splitting

TL;DR

This paper introduces a weaker form of dominated splitting called generalized dominated splitting and proves that diffeomorphisms with this property cannot be minimal, extending previous results in dynamical systems.

Contribution

The paper defines generalized dominated splitting and generalizes Zhang's result to show such diffeomorphisms are not minimal.

Findings

Generalized dominated splitting is a weaker condition than classical dominated splitting.

Diffeomorphisms with nontrivial global generalized dominated splitting cannot be minimal.

The result extends the understanding of the structure of dynamical systems with dominated-like splittings.

Abstract

We introduce a new notion called generalized dominated splitting which is weaker than classical dominated splitting. We use this notion to generalize a result of Zhang\cite{Zh}: every diffeomorphism with nontrivial global generalized dominated splitting can not be minimal.