On the Set of Fixed Points and Periodic Points of Continuously Differentiable Functions

Abstract

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 “big” or “small” 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 [1–5]). 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 𝐹₁(𝑓)={𝑥∶𝑓(𝑥)=𝑥} denotes the set of fixed points of 𝑓. For 𝑓 and 𝑛∈ℕ we define 𝑓^𝑛(𝑥) by induction: 𝑓^𝑛(𝑥)=𝑓𝑓^𝑛−1(𝑥)with𝑓⁰(𝑥)=𝑥,𝑓¹(𝑥)=𝑓(𝑥). (1) The orbit of 𝑥 under 𝑓 is given by the sequence orb(𝑥,𝑓)={𝑓^𝑖(𝑥)}^∞_𝑖=0. For 𝑛∈ℕ, let 𝐹_𝑛(𝑓)={𝑥∶𝑓^𝑛(𝑥)=𝑥} and 𝑃^𝑛_𝑓 be the set of periodic points of order 𝑛; that is, 𝑃^𝑛_𝑓=𝐹_𝑛𝑓⧵∪_𝑘<𝑛𝐹_𝑘𝑓. (2) 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 𝑇 of the interval [𝑎,𝑏], ‖𝑇‖ is the length of the largest subinterval of 𝑇, 𝐵(𝑓,𝜖) is the open ball about 𝑓 with radius 𝜖, 𝐴^∘ denotes the set of interior points of 𝐴, and 𝜆(𝐼) is the length of the interval 𝐼.

2. Continuously Differentiable Functions

For 𝐼=[𝑎,𝑏], consider 𝐶(𝐼,𝐼) and 𝐶¹(𝐼,𝐼) to be the family of all continuous maps and continuously differentiable maps from 𝐼 into itself, respectively. Recall that the usual metrics 𝜌₀ and 𝜌₁ on 𝐶(𝐼,𝐼) and 𝐶¹(𝐼,𝐼), respectively, are given by 𝜌₀𝑓,𝑔=sup_𝑥∈𝐼𝑓(𝑥)(𝑥(),𝜌−𝑔)for𝑓,𝑔∈𝐶𝐼,𝐼₁𝑓,𝑔=𝜌₀𝑓,𝑔+𝜌₀𝑓^′,𝑔^′for𝑓,𝑔∈𝐶¹(𝐼,𝐼). (3)

It is well known that the metric spaces (𝐶(𝐼,𝐼);𝜌₀) and (𝐶¹(𝐼,𝐼);𝜌₁) are complete and hence Baire's category theorem holds in these spaces. We say that a typical function in 𝐶(𝐼,𝐼) or 𝐶¹(𝐼,𝐼) has a certain property if the set of those functions which does not have this property is of first category in 𝐶(𝐼,𝐼) or in 𝐶¹(𝐼,𝐼). (Some authors prefer using the term generic instead of typical.) It is known that typically continuous self-maps of an interval have 𝜎-perfect, measure zero sets of periodic points (see [2]). Here we show that typically members of (𝐶¹(𝐼,𝐼),𝜌₁) have a finite set of fixed points and a countable set of periodic points.

Lemma 1. Let 𝑛≥1 and 𝑓∈𝐶¹(𝐼,𝐼) so that 𝐹_𝑛(𝑓) is not finite. Then there exists 𝑥₀∈𝐼 so that 𝑓^𝑛(𝑥₀)=𝑥₀ and (𝑓^𝑛)^′(𝑥₀)=1.

Proof. Suppose 𝐹_𝑛(𝑓) is not finite, and then we can choose a strictly monotone sequence in 𝐹_𝑛(𝑓). Without loss of generality, assume {𝑥_𝑖}^∞_𝑖=1⊂𝐹_𝑛(𝑓) so that 𝑥_𝑖<𝑥_𝑖+1 for each 𝑖≥1. Let ℎ(𝑥)=𝑓^𝑛(𝑥)−𝑥, and then, for each 𝑖, ℎ(𝑥_𝑖)=ℎ(𝑥_𝑖+1)=0. Thus by Rolle’s Theorem we have 𝑦_𝑖, 𝑥_𝑖<𝑦_𝑖<𝑥_𝑖+1 so that ℎ^′(𝑦_𝑖)=0. Let 𝑥₀=lim_𝑖→∞𝑥_𝑖, then 𝑥₀∈𝐼, and lim_𝑖→∞𝑦_𝑖=𝑥₀, and from the continuity of 𝑓^𝑛 and (𝑓^𝑛)^′ it follows that 𝑓^𝑛(𝑥₀)=𝑥₀ and (𝑓^𝑛)^′(𝑥₀)=1.

Lemma 2. For each 𝑓∈𝐶¹(𝐼,𝐼) and 0<𝜖<1 there is a polynomial 𝑝∶𝐼→𝐼^𝑜 with 𝜌₁(𝑓,𝑝)<𝜖.

Proof. Let 𝑀=sup_𝑥∈𝐼{|𝑓^′(𝑥)|,|𝑎|,|𝑏|,1} where 𝐼=[𝑎,𝑏]. Take 𝛽=𝜖/8𝑀, 𝛼=𝛽(𝑎+𝑏)/2, and 𝑔(𝑥)=𝛼+(1−𝛽)𝑓(𝑥). It is easy to see that 0<𝛽<1, 𝑔(𝐼)⊂𝐼^𝑜, and 𝜌₁(𝑓,𝑔)<𝜖/4. Let 𝜖₁=12inf_{[]𝑥∈𝑎,𝑏}𝑔(𝑥)(𝑥),𝜖−𝑎,𝑏−𝑔8(),𝜖𝑏−𝑎8 (4) and 𝑠(𝑡) be a polynomial with 𝜌₀(𝑔^′,𝑠)<𝜖₁/(𝑏−𝑎+1). Then 𝑝(𝑥)=𝑔(𝑎)+∫^𝑥_𝑎𝑠(𝑡)𝑑𝑡 is a polynomial and 𝑔(𝑥)−𝑝(𝑥)=∫^𝑥_𝑎(𝑔^′(𝑡)−𝑠(𝑡))𝑑𝑡. Thus we have 𝜌₁𝑔,𝑝=𝜌₀𝑔^′,𝑠+𝜌₀≤𝜖𝑔,𝑝¹()+𝜖𝑏−𝑎+1¹(𝑏−𝑎)(𝑏−𝑎)+1=𝜖₁<𝜖4, (5) hence 𝜌₁(𝑓,𝑝)≤𝜌₁(𝑓,𝑔)+𝜌₁(𝑔,𝑝)<𝜖/2. From 𝑔(𝐼)⊂[𝑎+2𝜖₁,𝑏−2𝜖₁] and 𝜌₁(𝑔,𝑝)≤𝜖₁ it follows that 𝑝(𝐼)⊂𝐼^𝑜.

Lemma 3. Let 𝑛>2 be a positive integer and 𝑓∈𝐶¹(𝐼,𝐼). If 𝑥₀ is a periodic point of order 𝑛 with (𝑓^𝑛)^′(𝑥₀)=1, then 𝑜𝑟𝑏(𝑥₀,𝑓)∩{𝑎,𝑏}=∅.

Proof. We show that 𝑎∉orb(𝑥₀,𝑓). The case 𝑏∉orb(𝑥₀,𝑓) follows similarly. Let orb(𝑥₀,𝑓)={𝑎=𝑥₀,𝑥₁,…,𝑥_𝑛=𝑥₀} with 𝑥_𝑖=𝑓(𝑥_𝑖−1) for 1≤𝑖≤𝑛. Since 𝑛>2, there exist 𝑥_𝑖∈orb(𝑥₀,𝑓)∩(𝑎,𝑏) and 1≤𝑠≤𝑛−1 so that 𝑓^𝑠(𝑥_𝑖)=𝑎. Since 𝑓^𝑠∈𝐶¹(𝐼,𝐼) and (𝑓^𝑠)^′(𝑥₀)≠0, there exists 𝛿₁>0 so that 𝑎<𝑥_𝑖−𝛿₁<𝑥_𝑖<𝑥_𝑖+𝛿₁<𝑏 and 𝑓^𝑠 is strictly increasing or strictly decreasing on 𝐽=[𝑥_𝑖−𝛿₁,𝑥_𝑖+𝛿₁]. Let 𝑓^𝑠 be strictly increasing on 𝐽, and then for 𝑥_𝑖−𝛿₁≤𝑥≤𝑥_𝑖 we have 𝑓^𝑠(𝑥)≤𝑓^𝑠(𝑥_𝑖)=𝑎; hence, 𝑓^𝑠∣_{[𝑥𝑖−𝛿1,𝑥𝑖]}=𝑎, implying that (𝑓^𝑠)^′(𝑥_𝑖)=0, a contradiction to (𝑓^𝑛)^′(𝑥₀)=1. On the other hand when 𝑓^𝑠 is strictly decreasing on 𝐽, for 𝑥_𝑖≤𝑥≤𝑥_𝑖+𝛿₁ we have 𝑓^𝑠(𝑥)≤𝑓^𝑠(𝑥_𝑖)=𝑎; hence, 𝑓^𝑠∣_{[𝑥𝑖,𝑥𝑖+𝛿1]}=𝑎 implying that (𝑓^𝑠)^′(𝑥_𝑖)=0, a contradiction to (𝑓^𝑛)^′(𝑥₀)=1.

Lemma 4. For 𝑘≥1, the set 𝔼_𝑘=𝑓∈𝐶¹(𝐼,𝐼)∶𝐹_𝑘𝑓∩𝑥∶𝑓^𝑘′(𝑥)=1≠∅ (6) is closed in 𝐶¹(𝐼,𝐼).

Proof. Let 𝑔_𝑛∈𝔼_𝑘 and 𝜌₁(𝑔_𝑛,𝑔)→0. Then there exists {𝑡_𝑛}^∞_𝑛=1⊂𝐼 such that 𝑔^𝑘_𝑛(𝑡_𝑛)=𝑡_𝑛 and (𝑔^𝑘_𝑛)^′(𝑡_𝑛)=1. Without loss of generality, we may assume that lim_𝑛→∞𝑡_𝑛=𝑡₀, then 𝑡₀∈𝐼. Let 𝜖>0 be arbitrary. Due to the uniform continuity of 𝑔^𝑘 on 𝐼 there exists a positive integer 𝑛₁ such that, for 𝑛≥𝑛₁, |𝑔^𝑘(𝑡_𝑛)−𝑔^𝑘(𝑡₀)|<𝜖. Since 𝜌₀(𝑔_𝑛,𝑔)→0, there exists a positive integer 𝑛₂ such that, for 𝑛≥𝑛₂,  𝜌₀(𝑔^𝑘_𝑛,𝑔^𝑘)<𝜖. Choose the positive integer 𝑛₃ so that |𝑡_𝑛−𝑡₀|<𝜖 for 𝑛≥𝑛₃, and let 𝑚₁=max(𝑛₁,𝑛₂,𝑛₃). Then for 𝑚≥𝑚₁ we have 𝑔^𝑘𝑡₀−𝑡₀≤𝑔^𝑘𝑡₀−𝑔^𝑘𝑡_𝑚+𝑔^𝑘𝑡_𝑚−𝑔^𝑘_𝑚𝑡_𝑚+𝑔^𝑘_𝑚𝑡_𝑚−𝑡₀≤𝜖+𝜌₀𝑔^𝑘,𝑔^𝑘_𝑚+𝑡_𝑚−𝑡₀≤3𝜖, (7) Implying that 𝑔^𝑘(𝑡₀)=𝑡₀. We also have 𝑔^𝑘_𝑛^′𝑡_𝑛−𝑔^𝑘′𝑡_𝑛≤lim_𝑛→∞𝜌₁𝑔^𝑘_𝑛,𝑔^𝑘=0. (8) Thus we have lim_𝑛→∞|1−(𝑔^𝑘)^′(𝑡_𝑛)|=|1−(𝑔^𝑘)^′(𝑡₀)|=0 and 𝑔∈𝔼_𝑘.

Theorem 5. There exists a residual subset 𝑀₁ of 𝐶¹(𝐼,𝐼) such that, for every 𝑓∈𝑀₁, 𝐹₁(𝑓) is finite.

Proof. If 𝐹₁(𝑓) has infinitely many elements, then from Lemma 1 it follows that there exists a point 𝑥₀∈𝐹₁(𝑓) so that 𝑓^′(𝑥₀)=1. Let 𝔼₁=𝑓∈𝐶¹(𝐼,𝐼)∶∃𝑥₀∈𝐼∩𝐹₁𝑓,𝑓^′𝑥₀=1. (9) From Lemma 4 the set 𝔼₁ is closed in 𝐶¹(𝐼,𝐼). To show that 𝔼₁ has no interior point, let 𝑓∈𝔼₁ and 𝜖>0, and then, by Lemma 2, there exists a polynomial 𝑝 such that 𝜌₁(𝑓,𝑝)<𝜖/2 and 𝑝(𝐼)⊂[𝑎+𝜖₁,𝑏−𝜖₁] for some 0<𝜖₁<𝜖/2. Let 𝑀𝑝=𝑥∶𝑝^′(𝑥)=1=𝑡₁,𝑡₂,…,𝑡_𝑚. (10) Choose 0<𝜖₂<𝜖₁ so that 𝑝(𝑥)−𝑥≠𝜖₂ for all 𝑥∈𝑀(𝑝) and take ℎ=𝑝−𝜖₂. Then ℎ∈𝐶¹(𝐼,𝐼),  𝜌₁(ℎ,𝑓)<𝜖,  𝑀(ℎ)={𝑥∶ℎ^′(𝑥)=1}=𝑀(𝑝), and ℎ(𝑥)≠𝑥 on 𝑀(ℎ), implying that 𝔼₁ is of first category and 𝑀₁=𝐶¹(𝐼,𝐼)⧵𝔼₁ is residual.

Theorem 6. The set of functions 𝑓∈𝐶¹(𝐼,𝐼) with 𝑓(𝑎)=𝑏, 𝑓(𝑏)=𝑎, and (𝑓²)^′(𝑎)=1 is a nowhere dense subset of 𝐶¹(𝐼,𝐼).

Proof. Let 𝐴={𝑓∈𝐶¹(𝐼,𝐼)∶𝑓(𝑎)=𝑏,𝑓(𝑏)=𝑎,and(𝑓²)^′(𝑎)=1}. It is easy to see that 𝐴 is a closed subset of 𝐶¹(𝐼,𝐼). To show that 𝐴 is nowhere dense, let 𝑓∈𝐶¹(𝐼,𝐼) and 𝜖>0. Choose 0<𝛾<min{𝜖/(4(𝑏−𝑎)+4),1} and 𝑎<𝑐<𝑏 such that |𝑓(𝑡)−𝑓(𝑎)|<(𝑏−𝑎)[1−𝜖/4] for 𝑡∈[𝑎,𝑐]. Let ℎ(𝑥)=𝑓(𝑥)+∫^𝑥_𝑎𝑘(𝑡)𝑑𝑡, where 𝑘(𝑡)=−𝛾,𝑡=𝑎,0,𝑡=𝑎+𝑐−𝑎4,𝛾3(),𝑡=𝑎+𝑐−𝑎2,0,𝑡=𝑐,𝑡=𝑏,linear,otherwise. (11)
It is clear that ℎ(𝑎)=𝑓(𝑎)=𝑏,  ℎ(𝑏)=𝑓(𝑏)=𝑎,  ℎ^′(𝑎)=𝑓^′(𝑎)−𝛾, and ℎ^′(𝑏)=𝑓^′(𝑏). It is easy to see that ℎ∈𝐶¹(𝐼,𝐼), (ℎ²)^′(𝑎)=(ℎ²)^′(𝑏)≠1, and 𝜌₁(𝑓,ℎ)<𝜖/2.

Theorem 7. Let 𝜖>0, 𝑓∈𝐶¹(𝐼,𝐼), and 𝐴_𝑛={𝑥∶(𝑓^𝑛)^′(𝑥)=1}∩𝐹_𝑛(𝑓) be a finite set, and 𝑜𝑟𝑏(𝑥,𝑓)⊂(𝑎,𝑏) for 𝑥∈𝐴_𝑛. Then there exists a function 𝑔∈𝐶¹(𝐼,𝐼) with 𝜌₁(𝑓,𝑔)<𝜖/4 such that {𝑥∶(𝑔^𝑛)^′(𝑥)=1}∩𝐹_𝑛(𝑔)=∅.

Proof. Let 𝐴_𝑛={𝑥∶(𝑓^𝑛)^′(𝑥)=1}∩𝐹_𝑛(𝑓)={𝑧_𝑗}^𝑙_𝑗=1. Let 𝐵={𝑡_𝑗}^𝑘_𝑗=1 be the elements of 𝐴_𝑛 with distinct orbits, that is, orb(𝑡_𝑖,𝑓)∩orb(𝑡_𝑗,𝑓)=∅ for 𝑖≠𝑗 and ∪^𝑘_𝑗=1orb(𝑡_𝑗,𝑓)⊇𝐴_𝑛. It is clear that each 𝑡_𝑗∈𝐵 is a periodic point of 𝑓 with some period 𝑚 where 𝑚 is a factor of 𝑛. This suggests that if one can construct a function 𝑔 that is sufficiently close to 𝑓, and either 𝑡_𝑗∉𝑃^𝑚_𝑓 or (𝑓^𝑚)^′(𝑡_𝑗)≠1, then 𝑥∶𝑔^𝑛′(𝑥)=1∩𝐹_𝑛𝑔⊆𝐴_𝑛⧵orb𝑡_𝑗,𝑓. (12) By repeating this process for each 𝑡_𝑗∈𝐵 in a finite number of steps we can construct the desired function 𝑔. Thus for convenience, we assume that, for 1≤𝑗≤𝑘,  𝑧_𝑗∈𝑃^𝑛_𝑓. Consider the partition 𝑇 of [𝑎,𝑏] obtained by {𝑓^𝑖(𝑡_𝑗),1≤𝑖≤𝑛,1≤𝑗≤𝑘}, 𝜖₁=(1/4)min{𝜖,‖𝑇‖}, and choose a positive 𝛿 less than 𝜖₁ such that, for each 𝑖, 1≤𝑖≤𝑛 and each 𝑗,  1≤𝑗≤𝑘, and if |𝑡−𝑡_𝑗|<𝛿, then |𝑓^𝑖(𝑡)−𝑓^𝑖(𝑡_𝑗)|<𝜖₁. Given that, for 𝑥∈𝐴_𝑛, orb(𝑥,𝑓)⊂(𝑎,𝑏), (𝑓^𝑛)^′ is continuous, and (𝑓^𝑛)^′(𝑥)=1 on 𝐵, we may choose the nondegenerate closed intervals 𝐻_𝑗, 1≤𝑗≤𝑘 of length less than 𝛿 and the positive numbers 𝛾_𝑗 such that(i)𝑡_𝑗∈(𝐻_𝑗)^∘ is the midpoint of 𝐻_𝑗 and 𝑓^𝑖(𝐻_𝑗)∩orb(𝑡_𝑗,𝑓)={𝑓^𝑖(𝑡_𝑗)} for 0≤𝑖≤𝑛,(ii)𝑓 is strictly monotone on each 𝐻_𝑗,𝑖=𝑓^𝑖(𝐻_𝑗) for 0≤𝑖≤𝑛−1, 1≤𝑗≤𝑘, where 𝐻_𝑗,0=𝐻_𝑗,(iii)the intervals 𝐻_𝑗,𝑖=𝑓^𝑖(𝐻_𝑗) are mutually disjoint for 1≤𝑖≤𝑛, 1≤𝑗≤𝑘,(iv)𝑎<min(𝐻_𝑗,𝑖)−𝛾_𝑗𝜆(𝐻_𝑗,𝑖)<max(𝐻_𝑗,𝑖)+𝛾_𝑗𝜆(𝐻_𝑗,𝑖)<𝑏, for 0≤𝑖≤𝑛−1,(v)max_{1≤𝑗≤𝑘}{𝛾_𝑗,(𝑏−𝑎)𝛾_𝑗}<(1/8)[min_{0≤𝑖≤𝑛,1≤𝑗≤𝑘}{𝜖/𝑘,𝜆(𝑓^𝑖(𝐻_𝑗)),𝛽_𝑗,𝑖}], where 𝛽_𝑗,𝑖=inf{|𝑓^′(𝑥)|∶𝑥∈𝑓^𝑖(𝐻_𝑗)}.
Let 𝑥₀=𝑡_𝑖 for some 𝑖, 𝛾, and 𝐽=[𝑐,𝑑] be the associated 𝛾_𝑖 and 𝐻_𝑖, respectively. Define 𝑔_{𝐽,𝑥0,𝛾}(𝑥)=−𝛾3𝑥,if𝑥=𝑐+⁰−𝑐2,3𝑥0,if𝑥=𝑐+⁰−𝑐4,𝛾,if𝑥=𝑥₀,0,if𝑥=𝑥₀+𝑑−𝑥₀4,−𝛾3,if𝑥=𝑥₀+𝑑−𝑥₀2,[]⧵(),𝑟0,if𝑥∈𝑎,𝑏𝑐,𝑑linear,otherwise,_{𝐽,𝑥0,𝛾}(𝑥)=𝑥𝛾,if𝑥=𝑐+⁰−𝑐4,0,if𝑥=𝑥₀,−𝛾𝑥₀−𝑐𝑑−𝑥₀if𝑥=𝑥₀+3𝑑−𝑥₀4,[]⧵(),ℎ0if𝑥∈𝑎,𝑏𝑐,𝑑linear,otherwise,₁(𝑥)=^𝑥_𝑎𝑔_{𝐽,𝑥0,𝛾}(𝑡)ℎ𝑑𝑡,₂(𝑥)=^𝑥_𝑎𝑟_{𝐽,𝑥0,𝛾}(𝑡)𝑑𝑡. (13) We have ℎ_𝑚(𝑐)=ℎ_𝑚(𝑑)=ℎ^′_𝑚(𝑐)=ℎ^′_𝑚(𝑑)=0 for 𝑚=1,2, ℎ₁(𝑥₀)=0, ℎ^′₁(𝑥₀)=𝛾, ℎ₂(𝑥₀)=(𝑥₀−𝑐)𝛾/2, and ℎ^′₂(𝑥₀)=0. By considering four different cases, we construct a function ℎ∈𝐶¹(𝐼,𝐼)∩𝐵(𝑓,𝜖) so that, for Cases A and B, 𝐹_𝑛(ℎ)∩{𝑥∶(ℎ^𝑛)^′(𝑥)=1}∩𝐽=∅ and 𝐹_𝑛(ℎ)∩{𝑥∶(ℎ^𝑛)^′(𝑥)=1}∩𝐽={𝑥₀} for Cases C and D. From conditions (iv) and (v) we have ℎ(𝐼)⊂𝐼, and from condition (v) it follows that ℎ is of the same monotone type as 𝑓 on each 𝑓^𝑠(𝐽),  0≤𝑠<𝑛.
Case A. Let 𝑓^𝑛(𝑥)<𝑥 for 𝐽⧵{𝑥₀} and 𝑓^𝑛(𝑥₀)=𝑥₀.(i)If 𝑓 is strictly increasing on 𝐽, then by taking ℎ(𝑥)=𝑓(𝑥)−ℎ₂(𝑥) we have ℎ(𝑥)<𝑓(𝑥) on 𝐽^𝑜 and  ℎ(𝑥)=𝑓(𝑥) on 𝐼⧵𝐽^𝑜 and the function ℎ^𝑛−1 is also strictly increasing on 𝑓(𝐽), and as a result ℎ^𝑛(𝑥)<𝑓^𝑛(𝑥)≤𝑥 on 𝐽^𝑜. Thus 𝐹_𝑛(ℎ)∩𝐽=∅.(ii)If 𝑓 is strictly decreasing on 𝐽, then by taking ℎ(𝑥)=𝑓(𝑥)+ℎ₂(𝑥) we have ℎ(𝑥)>𝑓(𝑥) on 𝐽^𝑜 and ℎ(𝑥)=𝑓(𝑥) on 𝐼⧵𝐽^𝑜 and the function ℎ^𝑛−1 is also strictly decreasing on 𝑓(𝐽) and as a result, ℎ^𝑛(𝑥)<𝑓^𝑛(𝑥)≤𝑥 on 𝐽^𝑜. Thus 𝐹_𝑛(ℎ)∩𝐽=∅.
Case B. Let 𝑓^𝑛(𝑥)>𝑥 for 𝐽⧵{𝑥₀} and 𝑓^𝑛(𝑥₀)=𝑥₀.
If 𝑓 is strictly increasing on 𝐽, take ℎ(𝑥)=𝑓(𝑥)+ℎ₂(𝑥), and if 𝑓 is strictly decreasing on 𝐽, take ℎ(𝑥)=𝑓(𝑥)−ℎ₂(𝑥). Then similar to Case A, we may show that ℎ^𝑛(𝑥)>𝑓^𝑛(𝑥)≥𝑥 for 𝑥∈𝐽^𝑜; hence, 𝐹_𝑛(ℎ)∩𝐽=∅.
Case C. Let 𝑓^𝑛(𝑥)<𝑥 for 𝑐<𝑥<𝑥₀,  𝑓^𝑛(𝑥)>𝑥 for 𝑥₀<𝑥<𝑑, and 𝑓^𝑛(𝑥₀)=𝑥₀.(i)If 𝑓 is strictly increasing on 𝐽, then by taking ℎ(𝑥)=𝑓(𝑥)+ℎ₁(𝑥) we have ℎ(𝑥)<𝑓(𝑥) for 𝑐<𝑥<𝑥₀ and ℎ(𝑥)>𝑓(𝑥) for 𝑥₀<𝑥<𝑑, and the function ℎ^𝑛−1 is also strictly increasing on 𝑓(𝐽). Thus ℎ^𝑛(𝑥)<𝑓^𝑛(𝑥)<𝑥 for 𝑐<𝑥<𝑥₀ and ℎ^𝑛(𝑥)>𝑓^𝑛(𝑥)>𝑥 for 𝑥₀<𝑥<𝑑; hence, 𝐹_𝑛(ℎ)∩𝐽={𝑥₀}.(ii)If 𝑓 is strictly decreasing on 𝐽, then by taking ℎ(𝑥)=𝑓(𝑥)−ℎ₁(𝑥) we have ℎ(𝑥)>𝑓(𝑥) for 𝑐<𝑥<𝑥₀ and  ℎ(𝑥)<𝑓(𝑥) for 𝑥₀<𝑥<𝑑, and the function ℎ^𝑛−1 is also decreasing on 𝑓(𝐽). Thus ℎ^𝑛(𝑥)<𝑓^𝑛(𝑥)<𝑥 for 𝑐<𝑥<𝑥₀ and ℎ^𝑛(𝑥)>𝑓^𝑛(𝑥)>𝑥 for 𝑥₀<𝑥<𝑑; hence, 𝐹_𝑛(ℎ)∩𝐽={𝑥₀}.
Case D. Let 𝑓^𝑛(𝑥)>𝑥 for 𝑐<𝑥<𝑥₀,  𝑓^𝑛(𝑥)<𝑥 for 𝑥₀<𝑥<𝑑, and 𝑓^𝑛(𝑥₀)=𝑥₀.
If 𝑓 is strictly increasing on 𝐽, take ℎ(𝑥)=𝑓(𝑥)−ℎ₁(𝑥), and if 𝑓 is strictly decreasing on 𝐽, take ℎ(𝑥)=𝑓(𝑥)+ℎ₁(𝑥). Then similar to Case C, we may show that ℎ^𝑛(𝑥)>𝑓^𝑛(𝑥)>𝑥 for 𝑐<𝑥<𝑥₀ and ℎ^𝑛(𝑥)<𝑓^𝑛(𝑥)<𝑥 for 𝑥₀<𝑥<𝑑. Thus 𝐹_𝑛(ℎ)∩𝐽={𝑥₀}.
Note that ℎ^𝑛 is strictly increasing on 𝐽∪𝑓^𝑛(𝐽), so we have 𝐹_𝑛(ℎ)∩𝐽⧵𝑓^𝑛(𝐽)∪𝑓^𝑛(𝐽)⧵𝐽=∅. (14) Thus for 𝑥∈𝐼 either orb(𝑥,ℎ)=orb(𝑥,𝑓) or orb(𝑥,ℎ)∩𝐽≠∅, of which in such case 𝐹_𝑛(ℎ)∩𝐽={𝑥₀}, ℎ^′(𝑥₀)=𝑓^′(𝑥₀)±𝛾, and ℎ^′(ℎ^𝑖(𝑥₀))=𝑓^′(𝑓^𝑖(𝑥₀)) for 1≤𝑖≤𝑛−1. Thus ℎ^𝑛′𝑥₀=ℎ^′ℎ^𝑛−1𝑥₀ℎ^′ℎ^𝑛−2𝑥₀⋯ℎ^′ℎ𝑥₀ℎ^′𝑥₀=𝑓^′𝑓^𝑛−1𝑥₀𝑓^′𝑓^𝑛−2𝑥₀×𝑓^′𝑓^𝑛−3𝑥₀⋯𝑓^′𝑓𝑥₀ℎ^′𝑥₀=𝑓^𝑛′𝑥₀ℎ^′𝑥₀𝑓^′𝑥₀=𝑓^′𝑥₀±𝛾𝑓^′𝑥₀=𝛾1±𝑓^′𝑥₀≠1. (15) We have 𝐹_𝑛(ℎ)∩𝑥∶ℎ^𝑛′(𝑥)=𝐹=1_𝑛𝑓∩𝑥∶𝑓^𝑛′(𝑥)=1⧵𝑥₀. (16) Also 𝜌₀𝑓^′,ℎ^′≤sup_𝑥∈𝐽𝑔(𝑥,|𝑟(𝑥≤𝜖))|,𝜌8𝑘₀𝑓,ℎ≤sup_𝑥∈𝐽ℎ₁(𝑥),ℎ₂(𝑥≤𝜖).8𝑘 (17) Hence 𝜌₁(𝑓,ℎ)≤𝜌₀(𝑓,ℎ)+𝜌₀(𝑓^′,ℎ^′)≤𝜖/4𝑘.
Doing this recursively for each 𝑡_𝑗, 1≤𝑗≤𝑘, we get a function 𝑔∈𝐵(𝑓,𝜖/4) such that 𝐹_𝑛(𝑔)∩{𝑥∶(𝑔^𝑛)^′(𝑥)=1}=∅.

Theorem 8. Typically continuously differentiable self-maps of intervals have a countable set of periodic points.

Proof. Take 𝐻_𝑛=𝑓∈𝐶¹(𝐼,𝐼)∶𝐹_𝑛𝑓,𝐵isnotfinite₁=𝑓∈𝐶¹(𝐼,𝐼)∶𝑎∈𝑃²_𝑓∩𝑥∶𝑓^2′(𝑥)(𝑎),𝐹=1with𝑓=𝑏₁=𝑓∈𝐶¹(𝐼,𝐼)∶∃𝑥₀𝑥∈𝐼suchthat₀∈𝐹₁𝑓and𝑓^′𝑥₀,𝔼=1_𝑛=𝑓∈𝐶¹(𝐼,𝐼)∶𝐹_𝑛𝑓∩𝑥∶𝑓^𝑛′(𝑥)=1≠∅. (18)
The sets 𝐹₁ and 𝐵₁ are first category sets (see Theorems 5 and 6). For 𝑛≥1 from Lemmas 4 and 1 we have that 𝔼_𝑛 is closed and 𝐻_𝑛⊆𝔼_𝑛. Without loss of generality we may assume that 𝑓∈𝐶¹(𝐼,𝐼) is neither a constant function nor a polynomial of first degree, and then by Lemma 2 we may choose a polynomial 𝑔∈𝐶¹(𝐼,𝐼) of degree 𝑛≥2 such that 𝑔(𝐼)⊂𝐼^𝑜 and 𝜌₁(𝑓,𝑔)<𝜖/4. Using Theorem 7 we can construct ℎ∈𝐶¹(𝐼,𝐼) so that 𝜌₁(𝑔,ℎ)<𝜖/4 and the set 𝐴_𝑛=𝐹_𝑛(ℎ)∩{𝑥∶(ℎ^𝑛)^′(𝑥)=1}=∅, thus 𝜌₁(𝑓,ℎ)<𝜖 and ℎ∉𝔼_𝑛, hence 𝔼^𝑜_𝑛=∅. This implies that for each 𝑘≥2 the set 𝔼_𝑘 is nowhere dense. Thus 𝐹=(∪^∞_𝑘=2𝔼_𝑘)∪𝐹₁∪𝐵₁ is a first category set, so 𝑀=𝐶¹(𝐼,𝐼)⧵𝐹 is a residual set, and, for 𝑓∈𝑀, 𝑃(𝑓)=∪^∞_𝑛=1𝐹_𝑛(𝑓) is countable.

Acknowledgments

The author would like to thank professor David Preiss for his valuable comments and suggestions as well as the anonymous referees for their comments.