© Hindawi Publishing Corp. ON COMMON FIXED POINTS, PERIODIC POINTS, AND RECURRENT POINTS OF CONTINUOUS FUNCTIONS

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 ] ) : F m ( 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 ω f studied by Bruckner and Ceder and show that the set of recurrent points of such functions are closed intervals.


Introduction.
The subject of commuting continuous functions on an interval has been considered by a group of researchers in the 20th century. In the 1920s, J. F. Ritt in a sequence of papers investigated the algebraic properties of function composition as a binary operation on the set of rational complex functions. In particular, he showed that commuting polynomials always have a common fixed point. In 1954, Eldon Dyer asked whether two commuting continuous functions must have a common fixed point. The same question was asked by A. J. Shields in 1955 and L. Dubins in 1956. For sometimes, this conjecture was considered by a group of people and led them to some partial results. Boyce [4] and Huneke [6] answered this question in negative by the construction of a pair of commuting continuous functions which have no fixed point in common. In [2], we considered a generalization of the common fixedpoint problem and conjectured that in general two commuting continuous self-maps of the intervals do not have a common periodic point. To settle the conjecture, some partial results that gave more information about the structure of such pairs were obtained. However, in constructing such examples, we were convinced that the construction of such pair is extremely difficult, even though it seems that most pairs should not have any periodic points in common. Thus, in the Twenty-First Summer Symposium in Real Analysis, we posed Conjecture 1.1 and raised the possibility of using a Baire-category argument to settle the problem. It is also not difficult to show that commuting continuous self-maps of the intervals must have common recurrent points. However, this is not true in general for the case of metric spaces. In [1], we also constructed an example of a pair of commuting continuous self-maps of a compact metric space with no common periodic points, which also provided an answer to a question raised in [2]. Conjecture 1.1. Two commuting continuous functions on an interval will typically have disjoint sets of periodic points. Steele [7] considered the above conjecture and investigated the likelihood of such pair f and g when P (f ) is first category and pointed out the difficulties that one might face using a Baire-category argument.
In this paper, we first provide a natural setting for the study of this problem and give some results including a generalization of Steele's main theorem. We also consider the class of functions with continuous ω f (x) studied by Bruckner and Ceder [5] and show that the set of recurrent points of such functions is a closed interval. We begin with some preliminaries.
The class of continuous self-maps of a closed interval I is denoted by C(I, I). For f ∈ C(I, I) and any integer n ≥ 1, f n denotes the nth iterate of f . The orbit of x under f (i.e., the set {f k (x) : k ≥ 0}) is denoted by O(f , x). The set of cluster points of O(f , x) is denoted by ω(f , x). We simply use ω(x) instead of ω(f , x) when there is no room for confusion. We also use ω f (x) for the function n=1 F n (f ) the set of periodic points of f , and R(f ) the set of recurrent points of f , respectively. A subset Y of I is called invariant under g if g(Y ) ⊆ Y . A closed, invariant, nonempty subset of I is called minimal if it contains no proper subset that is also closed, invariant, and nonempty. Every closed, invariant, and nonempty subset of I contains a minimal set. If Y is a minimal set, then Y ⊆ R(f ), and if it is not the orbit of a periodic point, then it is a perfect set. A minimal set is also nowhere dense.
The open ball about x with radius is denoted by B (x), and the interior and closure of A are denoted by A o and A, respectively. The set of rational numbers is denoted by Q and the set of natural numbers is denoted by N. For two compact subsets A and B of I, by d(A, B) and d H (A, B), we mean the distance of the sets A and B with respect to the ordinary metric and Hausdorff metric, respectively. Even though the results given here are true on any closed interval I, we only prove them for I = [0, 1]. In approximating a continuous function with a polynomial, we consider its Bernstein polynomials restricted to I. These polynomials have the interesting property that for f ∈ C(I, I) all its Bernstein polynomials are also in C(I, I). In the sequel, we use Theorem 1.2 given below from [3]. The proof of Theorem 1.3 is also trivial, and thus, it is omitted.
Theorem 1.3. Let Ᏼ and ρ be as defined, then (Ᏼ,ρ) is a compact metric space. Proof. Suppose S is of first category and S = ∪ ∞ m=1 S m with each S m closed. Then, from the Baire-category theorem, it follows that S does not contain an interval. On the other hand, if S does not contain an interval, then for each m ≥ 1, S m is a nowhere dense set, implying that S is a first-category subset of I.

Corollary 2.2. Let f ∈ C(I, I). Then P (f ) is of first category if and only if P (f ) does not contain an interval.
: P (f ) and P (g) are both first category} is a residual subset of Ᏼ.

Proof. Let
. We show that A is a residual subset of Ᏼ by showing that A 1 and A 2 are firstcategory subsets of Ᏼ. We prove our claim for set A 1 ; the case for A 2 similarly follows. Suppose (f , g) ∈ A 1 , then P (f ) is not of first category in I. Thus it contains an interval, hence we have 1]. Suppose (f 1 ,g 1 ) ∈ B m,n and > 0, then we have Q n ⊆ F m (f 1 ). Let P be a polynomial with f 1 −P < , then (P , g 1 ) ∈ B ((f 1 ,g 1 )). It is easy to see that B m,n is a closed subset of Ᏼ and F m (P ) has finitely many elements for a polynomial P , thus F m (P ) does not contain Q n . This implies that B m,n = B m,n has no interior, hence B m,n is a nowhere dense subset of Ᏼ.

Lemma 2.4. Let S ⊂ [0, 1] be a nowhere dense set and let m be a positive integer. Then the set
Proof. First we show that A m is a closed subset of C([0, 1]). For this, let {g n } ⊆ A m be a sequence and {g n } converges uniformly to g; we show that g ∈ A m . Since F m (g n )∩S = ∅, take y n ∈ F m (g n )∩S. Without loss of generality, we may assume lim n→∞ y n = y 0 , thus y 0 ∈ S. Let > 0 be arbitrary. Choose k large enough so that |g m (y k )−g m (y 0 )| < , g m k −g m < , and |y k −y 0 | < . Then we have Since was arbitrary, we have g m (y 0 ) = y 0 , implying that To this end, let P be a Bernstein polynomial of f so that f − P < /4m and let E 1 = F 1 (P ) ∩ S = {z 1 ,z 2 ,...,z k }. Due to the uniform continuity of P , for > 0, there exists δ k > 0 so that for t 1 . For each z i ∈ E 1 , we have one of the following cases.
By combining Lemmas 2.4 and 2.5, we have the following theorem.   = [0, 1], and let a, b ∈ I, a = b. If > 0  is a real number, then there exists a function h ∈ C(I, I) such that h(a) = f (a), Proof. We can assume < 1. Then, for Thus h ∈ C(I, I).
then ᐆ is a nonempty, closed, and nowhere dense subset of Ᏼ.
Proof. It is easy to see that (f , f ) ∈ ᐆ when f is the identity function, so ᐆ is nonempty. Suppose (f n ,g n ) ∈ ᐆ and (f n ,g n ) → (f , g). Then we have f n → f , g n → g, and f n • g n = g n • f n for each n. We show that (f , g) ∈ ᐆ. Let x ∈ I and > 0. Since f and g are uniformly continuous, we may choose δ, so that 0 < δ < and for every x 1 , Since f n → f and g n → g, there exists N ∈ N such that n > N implies f n − f < δ, g n − g < δ. Thus, if n > N, |f (g(x)) − g(f (x))| ≤ |f (g(x)) − f (g n (x))| + |f (g n (x)) − f n (g n (x))| + |g n (f n (x)) − g(f n (x))| + |g(f n (x)) − g(f (x))| ≤ 2δ + 2 ≤ 4 . Since > 0 and x were arbitrary, we have f • g = g • f , so (f , g) ∈ ᐆ and ᐆ is closed. Now we show that ᐆ is nowhere dense. Let > 0, (f , g) ∈ ᐆ. Suppose f or g is not the identity function, hence there exists x 0 so that for each x ∈ I, and let h 1 ∈ C(I, I) be a function so that h 1 (x 0 ) = f (x 0 ), but h 1 (g(x 0 )) ≠ f (g(x 0 )) and f −h 1 < . From Lemma 2.10, with a = x 0 and b = g(x 0 ), it follows that such h 1 ∈ C(I, I) exists. Then we have (h 1 ,h 2 ) ∈ B ((f , g) h 2 (x 0 )). Thus ᐆ is the union of a nowhere dense set and a singleton {(x, x)}, thus it is a nowhere dense subset of Ᏼ.
Theorem 2.12. Let f and g be two commuting continuous self-maps of the unit interval. Then we have one of the following holds: particular, Theorem 2.12 is also true if in its statement we replace A m,n with F(f ) or F(g).
The family of all ω-limit sets of a continuous function with the Hausdorff metric forms a metric space. Bruckner and Ceder [5] introduced a kind of chaos in terms of the map ω f : x → ω(f , x). In the same paper, they proved Lemma 2.15 and used it to prove Theorem 2.16 which characterizes the continuity of ω f .
Theorem 2.16. The following conditions are equivalent: (1) ω f is continuous; Using the above theorem, we show that the recurrent set of a continuous function defined on a closed interval with continuous ω f is a closed interval. Lemma 2.17. For any natural number n, ω f n is continuous when ω f is continuous.
Proof. From the lower semicontinuity of ω f , it follows that {f k } ∞ k=1 is equicontinuous, implying that {f nk } ∞ k=1 is equicontinuous for each n ∈ N. Hence ω f n is also lower semicontinuous. Thus the lemma follows from Theorem 2.16.

Theorem 2.18. Suppose f is a continuous self-map of the unit interval with
n=1 be a sequence in R(f ) with lim n→∞ x n = x 0 . We show that x 0 ∈ R(f ). Let be an arbitrary positive number. From the continuity of ω f and the fact that lim n→∞ x n = x 0 , there exists N 1 ∈ N so that |x n − x 0 | < and d H (ω(f , x n ), ω(f , x 0 )) < for all n ≥ N 1 . Since {x n } ⊆ R(f ), we have x n ∈ ω(f , x n ) for each n. Thus, for each n ≥ N 1 , there exists z n ∈ ω(f , x 0 ) so that |x n − z n | < , implying that |z n − x 0 | < 2 . Hence d(x 0 ,ω(f ,x 0 )) ≤ 2 , for every > 0, implying that x 0 ∈ ω(f , x 0 ), hence x 0 ∈ R(f ). Thus we have shown that R(f ) is a closed and connected subset of the real line, hence it is a closed interval. Remark 2.19. Due to the fact that R(f ) = ∩ ∞ m=1 ∪ ∞ n=m {x ∈ I : |f n (x) − x| ≤ 1/m}, we see that the recurrent set of continuous self-maps of a closed interval is an F σ δ set. A question of considerable interest is the classification of such maps with F σ recurrent set. This class of functions contains nomadic functions. A continuous self-map of an interval is called nomadic if it has a dense orbit at some point. It is also known that if R(f ) is closed, then f has topological entropy zero. Thus Theorem 2.18 provides another subclass of the class of continuous self-maps of the interval with an F σ recurrent set as well as zero topological entropy.

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management. For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy. Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made. Until now, computational methods and models are central to the analysis of economic and financial decisions.
However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models. In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches. For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset. Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results. The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making. In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: • Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning • Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management • Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation