International Journal of Mathematics and Mathematical Sciences
Volume 2003 (2003), Issue 39, Pages 2465-2473
doi:10.1155/S0161171203205366
On common fixed points, periodic points, and recurrent points of continuous functions
Berks-Lehigh Valley College, Pennsylvania State University, Reading 19610-6009, PA, USA
Received 15 May 2002; Revised 27 November 2002
Copyright © 2003 Aliasghar Alikhani-Koopaei. 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
It is known that two commuting continuous functions on an interval need not have a common fixed point. However, it is not known if such two functions have a common periodic point. we had conjectured that two commuting continuous functions on an interval will typically have disjoint sets of periodic points. In this paper, we first prove that S is a nowhere dense subset of [0,1] if and only if {f∈C([0,1]):Fm(f)∩S¯≠∅} is a nowhere dense subset of C([0,1]). We also give some results about the common fixed, periodic, and recurrent points of
functions. We consider the class of functions f with continuous ωf studied by Bruckner and Ceder and show that the set of recurrent points of such functions are closed intervals.