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New classes of $C^1$ robustly transitive maps with persistent critical points

Cristina Lizana, Wagner Ranter

Abstract

We exhibit a new large class of $C^1$ open examples of robustly transitive maps displaying persistent critical points in the homotopy class of expanding endomorphisms acting on the two dimensional Torus and the Klein bottle.

New classes of $C^1$ robustly transitive maps with persistent critical points

Abstract

We exhibit a new large class of open examples of robustly transitive maps displaying persistent critical points in the homotopy class of expanding endomorphisms acting on the two dimensional Torus and the Klein bottle.

Paper Structure

This paper contains 16 sections, 13 theorems, 16 equations, 5 figures.

Table of Contents

  1. Preliminaries
  2. Iterated Function Systems - IFS
  3. Prototypes
  4. Construction of the examples
  5. Periodic case
  6. One-dimensional dynamics
  7. Two-dimensional dynamics
  8. Weak hyperbolicity
  9. Dynamics properties for IFS
  10. Creating critical points
  11. Proof of Theorem \ref{['thm A']} for the periodic case
  12. Irrational Case
  13. Sketch of the proof
  14. Step I: creating a blender
  15. Step II: crossing the blender
  16. ...and 1 more sections

Key Result

Theorem A

Let $L$ be a linear endomorphism whose eigenvalues are $\mu, \lambda \in \mathbb{R}$ so that $1<|\mu|\ll|\lambda|$. Then there are $C^1$-robustly transitive endomorphisms homotopic to $L$ displaying critical points that are persistent under small perturbations.

Figures (5)

  • Figure 1: Periodic case.
  • Figure 2: Irrational case.
  • Figure 3: The graphs of $f_1$ and $f_2$ with topological degree $\mu$.
  • Figure 4: The graphs of $\psi$ and $\varphi'$, respectively.
  • Figure 5: The blender for an iterated of $F$.

Theorems & Definitions (26)

  • Theorem A
  • Theorem B
  • Remark 1.2.1
  • Proposition 2.1.1
  • proof
  • Lemma 2.1.1: Unstable cone fields for $F$
  • proof
  • Corollary 2.1
  • Proposition 2.1.2
  • proof
  • ...and 16 more