<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>

In this article, subsets of T 2 that can arise as sets of all periodic points of a continuous 2-dimensional toral automorphism are characterized. Here the torus T 2 is viewed as [0, 1) × [0, 1) as a group under coordinatewise addition modulo 1.

No abstract

We call a point "dynamically special" if it has a dynamical property, which no other point has. We prove that, for continuous self maps of the real line, all dynamically special points are in the closure of the union of the full orbits of periodic points, critical points and limits at infinity. We completely describe the set of dynamically special points of real polynomial functions. The following characterization for the set of special points is also obtained: A subset of R is the set of dynamically special points for some continuous self map of R if and only if it is closed. Mathematics Subject Classification (2000). Primary 54H20; Secondary 37E05.