Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A shadowable chain recurrent set with an attached hyperbolic singularity

Sogo Murakami

TL;DR

The paper addresses whether chain recurrent sets that contain an attached hyperbolic singularity can exhibit the standard shadowing property. It builds a $C^\infty$ flow on $S^4$ whose nonwandering/chain recurrent set contains such an attached singularity, yet retains standard shadowing on the chain recurrent set by modeling a factor of a modified suspension flow of a Smale horseshoe. A central tool is a factor transfer theorem (Theorem \ref{'thm.factor'}) that carries standard shadowing from a model flow to its quotient when a closed orbit is collapsed to a singularity. By wiring the model through a carefully constructed $X$ on $\mathbb{R}^4$ and a projection to $S^4$, the authors obtain a counterexample to a conjecture of Arbieto, López, Rego and Sánchez, showing that shadowing can persist in the presence of attached singularities. The work advances understanding of shadowing in nonuniformly hyperbolic dynamics and provides a method to generate flows with shadowing on ${\rm CR}$ sets containing singularities.

Abstract

We prove that every factor map between topological flows preserves the standard shadowing property if it is injective except for a closed orbit that shrinks to a singularity. As an application, we construct a $C^\infty$-flow on a four-dimensional sphere whose nonwandering set contains an attached hyperbolic singularity yet possesses the standard shadowing property. This gives a counterexample to a conjecture given by Arbieto, López, Rego and Sánchez (Math. Annalen 390:417-437).

A shadowable chain recurrent set with an attached hyperbolic singularity

TL;DR

The paper addresses whether chain recurrent sets that contain an attached hyperbolic singularity can exhibit the standard shadowing property. It builds a flow on whose nonwandering/chain recurrent set contains such an attached singularity, yet retains standard shadowing on the chain recurrent set by modeling a factor of a modified suspension flow of a Smale horseshoe. A central tool is a factor transfer theorem (Theorem \ref{'thm.factor'}) that carries standard shadowing from a model flow to its quotient when a closed orbit is collapsed to a singularity. By wiring the model through a carefully constructed on and a projection to , the authors obtain a counterexample to a conjecture of Arbieto, López, Rego and Sánchez, showing that shadowing can persist in the presence of attached singularities. The work advances understanding of shadowing in nonuniformly hyperbolic dynamics and provides a method to generate flows with shadowing on sets containing singularities.

Abstract

We prove that every factor map between topological flows preserves the standard shadowing property if it is injective except for a closed orbit that shrinks to a singularity. As an application, we construct a -flow on a four-dimensional sphere whose nonwandering set contains an attached hyperbolic singularity yet possesses the standard shadowing property. This gives a counterexample to a conjecture given by Arbieto, López, Rego and Sánchez (Math. Annalen 390:417-437).
Paper Structure (5 sections, 13 theorems, 68 equations, 4 figures)

This paper contains 5 sections, 13 theorems, 68 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Construction of $X$
  4. A chain recurrent set with an attached singularity
  5. Proof of Theorem \ref{['thm.factor']}

Key Result

Theorem 1.1

There is a $C^\infty$ flow $\phi$ on $S^4$ such that a hyperbolic singularity is attached to ${\rm CR}(\phi)$ and $\phi$ has the standard shadowing property on ${\rm CR}(\phi)$. Moreover, the chain recurrent set ${\rm CR}(\phi)$ coincides with the nonwandering set $\Omega(\phi)$ of $\phi$.

Figures (4)

  • Figure 1: $3$-fold Smale horseshoe with $f(H_i) = V_i$ for $i = 0,1,2$.
  • Figure 2: A family of diffeomorphisms $\{f_t\}_{0 \leq t \leq 1}$. The slope of the curve at $(t, (z, w))$ is $V(z, w, t)$.
  • Figure 3: Bounded $\widetilde{S}$.
  • Figure 4: Unbounded $\widetilde{S}$.

Theorems & Definitions (15)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Proposition 2.1
  • Proposition 2.2
  • Lemma 3.1
  • Lemma 3.2
  • Claim 1
  • Lemma 3.3
  • Proposition 4.1
  • ...and 5 more