%0 Journal Article
%T On the Set of Fixed Points and Periodic Points of Continuously Differentiable Functions
%A Aliasghar Alikhani-Koopaei
%J International Journal of Mathematics and Mathematical Sciences
%D 2013
%I Hindawi Publishing Corporation
%R 10.1155/2013/929475
%X In recent years, researchers have studied the size of different sets related to the dynamics of self-maps of an interval. In this note we investigate the sets of fixed points and periodic points of continuously differentiable functions and show that typically such functions have a finite set of fixed points and a countable set of periodic points. 1. Introduction and Notation The set of periodic points of self-maps of intervals has been studied for different reasons. The functions with smaller sets of periodic points are more likely not to share a periodic point. Of course, one has to decide what ˇ°bigˇ± or ˇ°smallˇ± means and how to describe this notion. In this direction one would be interested in studying the size of the sets of periodic points of self-maps of an interval, in particular, and other sets arising in dynamical systems in general (see [1¨C5]). For example, typically continuous functions have a first category set of periodic points (see [1, 5]). This result was generalized in [2] for the set of chain recurrent points. At times, even the smallness of these sets in some sense could be useful. For example, in [6] we showed that two commuting continuous self-maps of an interval share a periodic point if one has a countable set of periodic points. Schwartz (see [7]) was able to show that if one of the two commuting continuous functions is also continuously differentiable, then it would necessarily follow that the functions share a periodic point. Schwartz's result along with the results given in [6] may suggest that continuously differentiable functions have a countable set of periodic points. This is not true in general. However, in this note we show that typically such functions have a finite set of fixed points and a countable set of periodic points. Here denotes the set of fixed points of . For and we define by induction: The orbit of under is given by the sequence . For , let and be the set of periodic points of order ; that is, Two functions on a given interval are said to be of the same monotone type if both are either strictly increasing or strictly decreasing on that interval. Here, for a partition of the interval , is the length of the largest subinterval of , is the open ball about with radius , denotes the set of interior points of , and is the length of the interval . 2. Continuously Differentiable Functions For , consider and to be the family of all continuous maps and continuously differentiable maps from into itself, respectively. Recall that the usual metrics and on and , respectively, are given by It is well known that the metric
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