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A note on the hyperbolicity of the non-wandering sets of real quadratic maps

Diyath Pannipitiya

Abstract

The goal of this paper is to discuss about the hyperbolicity of the non-wandering set $\mathcal{NW}(f_c)$ of real quadratic function $f_c(x)=x^2+c$ when $c\in (-\infty, -2]$. Even though the results we present here are not new, it is not easier to find the proofs of them. We present two different ways to prove the hyperbolicity of $\mathcal{NW}(f_c)$ for the``considerably difficult case'' of when $c$ is closer to $-2$.

A note on the hyperbolicity of the non-wandering sets of real quadratic maps

Abstract

The goal of this paper is to discuss about the hyperbolicity of the non-wandering set of real quadratic function when . Even though the results we present here are not new, it is not easier to find the proofs of them. We present two different ways to prove the hyperbolicity of for the``considerably difficult case'' of when is closer to .
Paper Structure (3 sections, 2 theorems, 44 equations, 3 figures)

This paper contains 3 sections, 2 theorems, 44 equations, 3 figures.

Table of Contents

  1. Intnroduction
  2. $\mathcal{NW}(f_{-2})$ is not hyperbolic.
  3. $\mathcal{NW}(f_c)$ is hyperbolic for all $c<-2$.

Key Result

Lemma 3.3

If $J\Subset T$, then there exist $\Lambda \gneq 1$ for all $x,y\in J$ such that

Figures (3)

  • Figure 1: Graph of $f_c(x)=x^2+c$ when $c<-2$
  • Figure 2: Graph of $f_c(x)=x^2+c$ when $c<-2$
  • Figure 3: Graph of $f(x)=x^2-k$ maps $[\alpha,p]$ onto $[-p,p]$

Theorems & Definitions (9)

  • Definition 1.1
  • Definition 1.2
  • Definition 3.1
  • Definition 3.2
  • Lemma 3.3
  • proof
  • Definition 3.4
  • Definition 3.5
  • Lemma 3.6