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Robust transitivity without sectional-hyperbolicity

A. Arbieto, W. Britto, C. A. Morales, E. Rego

Abstract

For any integer $n \geq 5$, we construct an $n$-dimensional $C^1$ vector field exhibiting a robustly transitive singular attractor which is not sectional-hyperbolic. Nevertheless, the attractor is singular-hyperbolic. This provides the first such examples improving some features of the constructions in [17, 32].

Robust transitivity without sectional-hyperbolicity

Abstract

For any integer , we construct an -dimensional vector field exhibiting a robustly transitive singular attractor which is not sectional-hyperbolic. Nevertheless, the attractor is singular-hyperbolic. This provides the first such examples improving some features of the constructions in [17, 32].
Paper Structure (9 sections, 1 theorem, 27 equations, 1 table)

This paper contains 9 sections, 1 theorem, 27 equations, 1 table.

Table of Contents

  1. Introduction
  2. Proof of the theorem
  3. Choice of toral endomorphisms
  4. The solenoid
  5. Suspension
  6. Insertion of a singularity
  7. Singular- but not sectional-hyperbolicity
  8. Robust transitivity
  9. Declaration of competing interest

Key Result

Theorem 1

For every integer $n\geq5$ there is an $n$-dimensional $C^1$ vector field exhibiting a robustly transitive singular attractor which is not sectional-hyperbolic. Moreover, this attractor is singular-hyperbolic.

Theorems & Definitions (4)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem