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A diffeomorphism with global dominated splitting can not be minimal

Pengfei Zhang

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.

A diffeomorphism with global dominated splitting can not be minimal

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.

Paper Structure

This paper contains 3 sections, 2 theorems, 6 equations.

Table of Contents

  1. Introduction
  2. partial hyperbolicity and quasi-hyperbolic strings
  3. dominated splitting and minimality

Key Result

Proposition 2.1

Let $f$ be a diffeomorphism on $M$ and $\mathcal{W}$ be an $f$-invariant foliation tangent to a distribution $E\subset TM$ such that $Df$ is uniformly expanding (or uniformly contracting) on $E$. Then there exists a nonrecurrent point of $f$. Moreover the set $\{z\in M: z\notin\omega(z)\}$ of points

Theorems & Definitions (3)

  • Proposition 2.1
  • Proposition 2.2
  • Remark 1