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Robustly transitive maps with critical points and large dimensional central spaces

Juan C. Morelli

TL;DR

This work constructs $C^1$ robustly transitive endomorphisms on $\mathbb{T}^n$ with persistent critical points and a central $k$-dimensional space, for any $n\ge2$ and $1\le k\le n-1$, by perturbing an expanding base map $F$ on a $\mathbb{T}^m$ factor and employing a robustly minimal Iterated Function System to mix dynamics. The method combines a blending region (a blender) with a targeted local surgery that creates critical points without destroying transitivity, and it demonstrates robustness by local cone-field estimates and the Inclination Lemma. The results generalize previous one-dimensional-center constructions to arbitrary central-space dimension and establish the existence of robust singular dynamics in high dimensions within the isotopy class of $F\times Id$. The work thus advances understanding of transitivity and robustness under singularities in high-dimensional noninvertible systems and opens questions about extending the framework beyond product manifolds.

Abstract

Given any triplet of positive integers $n \geq 2$, $m$ and $k$ such that $n=m+k$, we exhibit a $C^1$ robustly transitive endomorphism of $\mathbb{T}^n$ with persistent critical points in the isotopy class of $F \times Id$, where $F$ is an expanding map of $\mathbb{T}^m$ and $Id$ is the identity of $\mathbb{T}^k$. Furthermore, if $k$ is small, the map is not only in the isotopy class but in fact a perturbation of $F \times Id$.

Robustly transitive maps with critical points and large dimensional central spaces

TL;DR

This work constructs robustly transitive endomorphisms on with persistent critical points and a central -dimensional space, for any and , by perturbing an expanding base map on a factor and employing a robustly minimal Iterated Function System to mix dynamics. The method combines a blending region (a blender) with a targeted local surgery that creates critical points without destroying transitivity, and it demonstrates robustness by local cone-field estimates and the Inclination Lemma. The results generalize previous one-dimensional-center constructions to arbitrary central-space dimension and establish the existence of robust singular dynamics in high dimensions within the isotopy class of . The work thus advances understanding of transitivity and robustness under singularities in high-dimensional noninvertible systems and opens questions about extending the framework beyond product manifolds.

Abstract

Given any triplet of positive integers , and such that , we exhibit a robustly transitive endomorphism of with persistent critical points in the isotopy class of , where is an expanding map of and is the identity of . Furthermore, if is small, the map is not only in the isotopy class but in fact a perturbation of .

Paper Structure

This paper contains 18 sections, 30 theorems, 27 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Sketch of the Construction.
  3. Preliminaries
  4. Normal Hyperbolicity.
  5. Iterated Function Systems.
  6. A low dimensional example.
  7. Construction of $f$.
  8. Dynamics of $f$.
  9. A singular endomorphism $F$ of $\hbox{$\mathbb{T}$}^3$.
  10. Construction of $F$
  11. Dynamics of $F$.
  12. A regular endomorphism $f$ of $\mathbb{T}^n$.
  13. Construction of $f$.
  14. Dynamics of $f$.
  15. A singular endomorphism $F$ of $\hbox{$\mathbb{T}$}^n$.
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $n$ and $k$ be natural numbers such that $n \geq 2$ and $1 \leq k \leq n-1$ and let $\mathbb{T}^n$ be the $n$-torus. Then, there exists a smooth map $F: \mathbb{T}^n \to \mathbb{T}^n$ satisfying:

Figures (3)

  • Figure 1: $\tilde{g}_a:S^1 \to S^1$ is almost a contraction on $S^1$.
  • Figure 2: Graph of $u_i$.
  • Figure 3: Graphs of $\psi$ and $\varphi'$ (following mo)

Theorems & Definitions (69)

  • Theorem 1.1
  • Proposition 2.1
  • Definition 2.1
  • Theorem 2.1
  • Definition 2.2
  • Theorem 2.2
  • Lemma 2.1
  • proof
  • Remark 2.1
  • Corollary 2.1
  • ...and 59 more