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Equicontinuity of the Hutchinson operator $F$ and sensitivity of $F_-$

Aliasghar Sarizadeh

TL;DR

The paper investigates the relationship between equicontinuity of the Hutchinson operator $F$ and sensitivity of the Hutchinson operator $F_-$ for an iterated function system on the circle. It constructs a circle $\mathrm{IFS}$ with an exceptional minimal set and uses a circle map with an attracting fixed point to produce backward-minimal dynamics, yielding equicontinuity of $F$ while $F_-$ remains sensitive. It also presents a concrete non-symmetric example $\mathcal{F}=\{f_1,f_2,f_3,f_4\}$, where $f_1=R_\alpha$ is an irrational rotation and the other maps are piecewise-affine, showing that both $F$ and $F_-$ are equicontinuous on $S^1$. Overall, the results demonstrate that equicontinuity of $F$ does not constrain the sensitivity of $F_-$ and highlight nuanced forward/backward dynamics in circle IFS.

Abstract

For an iterated function system $ \mathcal{F} = \{ f_1, \dots, f_k \} $ of homeomorphisms on a compact metric space $(X, d)$, write $ \mathcal{F}_-= \{ f_1^{-1}, \dots, f_k^{-1} \} $. The objective of this paper is to illustrate an iterated function system $\mathcal{F}$ of homeomorphisms on the circle that the Hutchinson operator of $\mathcal{F}$ is equicontinuous, but the Hutchinson operator of $\mathcal{F}_-$ is sensitive.

Equicontinuity of the Hutchinson operator $F$ and sensitivity of $F_-$

TL;DR

The paper investigates the relationship between equicontinuity of the Hutchinson operator and sensitivity of the Hutchinson operator for an iterated function system on the circle. It constructs a circle with an exceptional minimal set and uses a circle map with an attracting fixed point to produce backward-minimal dynamics, yielding equicontinuity of while remains sensitive. It also presents a concrete non-symmetric example , where is an irrational rotation and the other maps are piecewise-affine, showing that both and are equicontinuous on . Overall, the results demonstrate that equicontinuity of does not constrain the sensitivity of and highlight nuanced forward/backward dynamics in circle IFS.

Abstract

For an iterated function system of homeomorphisms on a compact metric space , write . The objective of this paper is to illustrate an iterated function system of homeomorphisms on the circle that the Hutchinson operator of is equicontinuous, but the Hutchinson operator of is sensitive.
Paper Structure (2 sections, 2 theorems, 3 equations)

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

Table of Contents

  1. Introduction and preliminaries
  2. Proof of the main results

Key Result

Theorem A

There exists an IFS $\mathcal{F}$ of homeomorphisms on the circle that the Hutchinson operator of $\mathcal{F}$ is equicontinuous, for which the Hutchinson operator of $\mathcal{F}_-$ is sensitive.

Theorems & Definitions (4)

  • Theorem A
  • Theorem B
  • proof : Proof of Theorem \ref{['T1']}
  • proof : Proof of Theorem \ref{['T2']}