Simple dynamical systems

Abstract

<div class="page" title="Page 1"><div class="layoutArea"><div class="column"><p><span>In this paper, we study the class of simple systems on </span><span>R </span><span>induced by homeomorphisms having finitely many non-ordinary points. We characterize the family of homeomorphisms on R having finitely many non-ordinary points upto (order) conjugacy. For </span><span>x,y </span><span>∈ </span><span>R</span><span>, we say </span><span>x </span><span>∼ </span><span>y </span><span>on a dynamical system (</span><span>R</span><span>,f</span><span>) if </span><span>x </span><span>and </span><span>y </span><span>have same dynamical properties, which is an equivalence relation. Said precisely, </span><span>x </span><span>∼ </span><span>y </span><span>if there exists an increasing homeomorphism </span><span>h </span><span>: </span><span>R </span><span>→ </span><span>R </span><span>such that </span><span>h </span><span>◦ </span><span>f </span><span>= </span><span>f </span><span>◦ </span><span>h </span><span>and </span><span>h</span><span>(</span><span>x</span><span>) = </span><span>y</span><span>. </span><span>An element </span><span>x </span><span>∈ </span><span>R </span><span>is </span><span>ordinary </span><span>in (</span><span>R</span><span>,f</span><span>) if its equivalence class [</span><span>x</span><span>] is a neighbourhood of it.</span></p><p><span><br /></span></p></div></div></div>

The class of simple dynamics systems

“…We first state the following results without proof. For a proof refer [2]. These results are easy to prove and it will be used to prove our main theorems.…”

“…In this Section, we will be discussing some basic results that will be used to prove our main theorems. Most of the results are available in [2].…”

“…See [2] and [8], for the following characterization theorem for the set N (f ) (and hence for S(f )).…”

“…A point which is not ordinary is called non-ordinary. The idea of special points and non-ordinary points are relatively new to the literature (see [1], [2], [8]). Recently, we studied the class of simple dynamical systems induced by homeomorphisms.…”

“…Recently, we studied the class of simple dynamical systems induced by homeomorphisms. The reader may refer [2] to get an idea of simple systems induced by homeomorphisms having finitely many non-ordinary points. Throughout this paper, we will be working with continuous self maps of the real line.…”