No 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>

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.

In this paper, we provide a characterization for the subshifts of finite type (SFT) in terms of Cellular automata (CA). In addition, we prove that 1. The following are equivalent for a non-singleton subshift of finite type XF. a) XF is transitive and Per(XF), the set of periodic points of XF, is cofinite b) XF is weak mixing c) XF is mixing. 2. For non-singleton sofic shifts, only the statements (a) and (b) are equivalent.

In this article, we prove that the group of all increasing homeomorphisms on R has exactly five normal subgroups, and the group of all homeomorphisms on R has exactly four normal subgroups. There are several results known about the group of homeomorphisms on R and about the group of increasing homeomorphisms on R ([2], [6], [7] and [8]), but beyond this there is virtually nothing in the literature concerning the topological structure in the aspects of topological dynamics. In this paper, we analyze this structure in some detail.