Discrete Dynamics in Nature and Society
Volume 2012 (2012), Article ID 970934, 7 pages
http://dx.doi.org/10.1155/2012/970934
Research Article

Uniform Convergence and Transitive Subsets

Lei Liu,1,2 Shuli Zhao,2 and Hongliang Liang2

1School of Mathematics, Sichuan University, Sichuan, Chengdu 610064, China
2Department of Mathematics, Shangqiu Normal University, Henan, Shangqiu 476000, China

Received 26 September 2011; Accepted 9 December 2011

Academic Editor: Recai Kilic

Copyright © 2012 Lei Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let ( 𝑋 , 𝑑 ) be a metric space and a sequence of continuous maps 𝑓 𝑛 ∶ 𝑋 → 𝑋 that converges uniformly to a map 𝑓 . We investigate the transitive subsets of 𝑓 𝑛 whether they can be inherited by 𝑓 or not. We give sufficient conditions such that the limit map 𝑓 has a transitive subset. In particular, we show the transitive subsets of 𝑓 𝑛 that can be inherited by 𝑓 if 𝑓 𝑛 converges uniformly strongly to 𝑓 .