A Full Description of ω -Limit Sets of Cournot Maps Having Non-Empty Interior and Some Economic Applications

: Given a continuous Cournot map F ( x , y ) = ( f 2 ( y ) , f 1 ( x )) deﬁned from I 2 = [ 0,1 ] × [ 0,1 ] into itself, we give a full description of its ω -limit sets with non-empty interior. Additionally, we present some partial results for the empty interior case. The distribution of the ω -limits with non-empty interior gives information about the dynamics and the possible outputs of each ﬁrm in a Cournot model. We present some economic models to illustrate, with examples, the type of ω -limits that appear.


Introduction
Topological structure and classification of ω-limits for general compact metric spaces remain an important problem and, in many aspects, an unknown subject. In order to better understand the main goal pursued in this paper in relation to the description of certain ω-limit sets for Cournot maps, and their presence in the numeric simulations of several economic models, let us first present the needed notions and their associated notation.
Given a compact metric space X, let C(X, X) denote the set of continuous maps from X into itself. The interior of a subset Y ⊂ X is denoted by Int(Y). We say that Y ⊂ X is a nowhere dense set whenever the interior of its closure is empty. Let ϕ ∈ C(X, X). For n ∈ N we put ϕ n = ϕ • ϕ n−1 , with ϕ 0 = Identity. The orbit of a point x ∈ X through ϕ is Orb ϕ (x) = {ϕ n (x)} ∞ n=0 . We say that Y ⊂ X is n-periodic if ϕ n (Y) = Y and ϕ k (Y) = Y for 0 < k < n; in this case, Orb ϕ (Y) = Y ∪ ϕ(Y) ∪ ... ∪ ϕ n−1 (Y) is the orbit of Y. If Y = {x} we say that it is a n-periodic point. The period of x is denoted by ord ϕ (x). When X = I := [0, 1] is the unit interval and Y is a subinterval of I, we have an n-periodic subinterval. If X = I 2 , and Y = J 1 × J 2 , with J i subintervals of I, i = 1, 2, we have an n-periodic rectangle.
The set of limit points of Orb ϕ (x) (i.e., its accumulation points or elements y in X, for which there exists an infinite subsequence n 1 < n 2 < . . . < n j < . . . of positive integers such that ϕ n j (x) converges to y), is called the ω-limit set of x by ϕ, and we denote it by ω ϕ (x). For any x ∈ X, ω ϕ (x) is non-empty, closed and strongly invariant (ϕ(ω ϕ (x)) = ω ϕ (x)) (see [1]). Moreover, for any n ∈ N ω ϕ (x) = n−1 j=0 ω ϕ n (ϕ j (x)). (1) We say that ϕ is transitive if, given two non-empty open sets U, V ⊂ X, there is n 0 ∈ N such that ϕ n 0 (U) ∩ V = ∅. If there is k 0 ∈ N such that ϕ n (U) ∩ V = ∅ for all n ≥ k 0 , then ϕ is called a (topologically) mixing map. Observe that topologically mixing implies transitivity (see [2] for a detailed explanation on properties of these types of maps). Moreover, ϕ is transitive if and only if there exists a point x ∈ X with ω ϕ (x) = X (see [1]).
The pair (X, ϕ) is called a discrete dynamical system. Given such a system, the main objective is to know the asymptotic behavior of the orbits Orb ϕ (x) for all x ∈ X and to analyze the topological structure of their ω-limit sets. This is a difficult problem, and we restrict our attention to a particular class of continuous two-dimensional maps.
Given f ∈ C(I, I), I = [0, 1], the topological structure of an ω-limit set L of f is well known: either L is a nowhere dense set or L = n i=1 J i , where J i are non-degenerate closed subintervals of I such that f (J i ) = J i+1(mod n) and J i ∩ J k = ∅ if i = k (for a proof, see [1]). Conversely, any set of the above forms can be realized as an ω-limit set for a suitable continuous interval map (see [3] or [4]).
Nevertheless, the description of ω-limit sets for continuous maps f : I n → I n , with n ≥ 2, is an open problem. In this direction, only a few results are known (see for instance, [5][6][7][8][9]). It seems to be more reasonable to focus our attention on special classes of two-dimensional continuous maps.
Following [10], we are interested in describing the topological structure of ω-limit sets of a particular class of C(I 2 , I 2 ) called antitriangular maps or Cournot maps. We say that F ∈ C(I 2 , I 2 ) is an antitriangular map (or Cournot map) if F(x, y) = ( f 2 (y), f 1 (x)), with f i ∈ C(I, I), i = 1, 2. It is immediate to see that for any n ≥ 0 it holds The set of antitriangular maps on I 2 will be denoted by C A (I 2 , I 2 ). This type of two-dimensional map appears closely related to an economical process called the Cournot duopoly (see [11]), in which two firms produce an identical good, and in each step they try to obtain the maximum profit according to the decision of the opposite firm in the last step. This economical process has been profusely studied in the literature (for instance, see [12][13][14][15][16][17],...).
The following result gives a description of ω-limit sets of antitriangular maps.
L is a finite union p i=1 R i , where R i = I i × J i ⊂ I 2 are non-degenerate periodic rectangles of F such that R i ∪ R j is not a rectangle and Int(R i ) ∩ Int(R j ) = ∅ for 1 ≤ i < j ≤ p.
But, as it was highlighted in [10], not every distribution of rectangles in the square can be realized as an ω-limit of a suitable antitriangular map. In this paper, our main goal is to give a full description of ω-limit sets with non-empty interior, and consider some questions about ω-limit sets with empty interior. This paper is organized as follows. After some preliminary results stated in Section 2, we proceed to develop in Section 3 a full description for the ω-limit sets of Cournot maps F(x, y) = ( f 2 (y), f 1 (x)) that have a non-empty interior. It is worth mentioning that this study is strongly related to the nature of the ω-limit sets associated with the onedimensional maps f 2 • f 1 and f 1 • f 2 , in connection with the property of either being or not being a mixing ω-limit set, see Definition 2. Our analysis will be applied and illustrated with several economic models, such as Puu's duopoly or Matsumoto-Nonaka's model, among others, in Section 4. Next, in Section 5, on the one hand we present some examples of ω-limit sets of Cournot maps having empty interior and we show the existence of connected limit sets with empty interior; on the other hand, in the non-empty interior case, we characterize the connected ω-limit sets ω F (x, y) of Cournot maps F by the inspection of their canonical projections π j (ω F (x, y)), j = 1, 2, and the intersection of the limit sets ω F 2 (x, y) and ω F 2 (F(x, y)). We also state some open problems with the hope of advancing the comprehension of the empty-interior case of Cournot maps.

Auxiliary Results
In this section, some auxiliary results are summed up since they will be used later.
For the next results, recall that by a bitransitive map f we understand a continuous map for which f and f 2 are transitive.  The following result allows us to construct ω-limit sets from mixing sets (we will say that an invariant closed set Y ⊂ X is mixing for a continuous map ϕ : X → X whenever ϕ| Y is mixing); it can be easily derived from Theorem 6 in [6] where the property is established in the frame of mixing compact ω-limit sets in the Euclidean space R n . Theorem 2 ([6] Theorem 6). Let f , g ∈ C(I, I) and let x ∈ I such that ω f (x) is mixing. Then for any y ∈ I there are x 1 , y 1 ∈ I such that ω f ×g (x 1 , The following lemma states the relationship between two ω-limit sets from a continuous map defined in a compact metric space.
The case in which Int(Y) ∩ Int(Z) = ∅ can be even more specific. The following result deals with interval maps having two different ω-limits sets with non-empty interiors sharing periodic points. According to the structure of the ω-limit sets L with non-empty interior, corresponding to interval maps f ∈ C(I, I), described in Section 1, we know that L = p s=1 I s and f (I s ) = I s+1 mod(p) , s = 1, . . . , p. Then, we say that L is mixing for f whenever the transitive restricted map f p | I s is topologically mixing for s = 1, . . . , p. Otherwise, L is said to be no mixing.
Lemma 4. Let f ∈ C(I, I). Let A = p i=1 I i and B = q j=1 J j be two different ω-limit sets of f with non-empty interiors. Suppose p ≥ q. Suppose A ∩ B = ∅ and Int(A) ∩ B = ∅. Then, either p = q or p = 2q, in both cases A is mixing, and the number of connected components of A ∪ B is q.
Proof. If A ∩ B = ∅ and Int(A) ∩ B = ∅, then it is easily seen that A and B share endpoints, which are evidently periodic points of f . Let (A ∪ B) 1 < ... < (A ∪ B) c be the c connected components of A ∪ B (given J, K ⊂ I, J < K means that x < y for all x ∈ J, y ∈ K; realize that this number c is finite). Define Obviously, a i 1 = a i 2 , b j 1 = b j 2 for all i 1 , i 2 ∈ {1, ..., p} and j 1 , j 2 ∈ {1, ..., q}, since f maps connected components on connected components and A, B are periodic sets of f . Moreover, since p ≥ q, either a 1 = b 1 or a 1 = b 1 + 1.
If b 1 ≥ 2, it is not restrictive to assume that we have in (A ∪ B) 1 the disposition .., q}, = 1, 2, . . . Notice that c < p because b 1 ≥ 2 and q ≤ p. Since f maps connected components on connected components, and c is the number of such connected components, we have The first case is not allowable because A is a p-periodic orbit and c < p; in the second case, we distinguish two situations: , and due to the fact that f c (J j s ) = J j s for all s, we deduce that f c (I i ) = I i , which again contradicts that c < p.
Therefore, we have proved that b 1 = 1. This gives c = q. Notice that the possibility a 1 = b 1 + 1 can occur.
Due to the structure of the connected components (A ∪ B) 1 , . . . , (A ∪ B) q mentioned above, observe that ). This means that each I i has an endpoint as a fixed point of f p . Then Lemma 1 and Lemma 2 imply that A is mixing.

Remark 2.
Realize that if p = q in the above Lemma 4, we also obtain that B is mixing, but in the case p = 2q it is not guaranteed that B is mixing, because in this situation we only know that its endpoints are fixed by f 2q but we have no information about the action of f q on these endpoints.
To describe the ω-limit sets with non-empty interior, we will use the following results, whose proofs can be consulted in [18]. Among them, we include the characterization of finite limit sets, that is, periodic orbits of Cournot maps; in fact, the movement of periodic rectangles realized as ω-limit sets of Cournot maps can be 'interpreted' as the evolution of periodic points, and in this sense the distribution of periodic points can show us in some cases the possible way to distribute periodic rectangles. Lemma 6 ([18] Proposition 3.8). Let F ∈ C A (I 2 , I 2 ). Let M ⊂ I 2 be a periodic orbit of F with Then there are d, k 1 , k 2 ∈ N and there are two finite sets H, V ⊂ I holding the following properties:

2.
For all (x, y) ∈ M, either dk be a periodic orbit for F ∈ C A (I 2 , I 2 ). Then

3.
If n ∈ {2m : m ∈ N, m ≥ 2}, and M does not satisfy the equivalent conditions of Lemma 5, the distribution of points of M is described by Lemma 6. See Figure 3.

Description of the Non-Empty Interior Case
In this section, the starting point for the analysis lies in fixing the type of ω-limit sets obtained in the one-dimensional case ( f 1 • f 2 and f 2 • f 1 ) and, then, to study the type of ω-limit sets generated in the two-dimensional case for the antitriangular map F(x, y) = ( f 2 (y), f 1 (x)).
Let F ∈ C A (I 2 , I 2 ) be an antitriangular map such that F(x, y) = ( f 2 (y), f 1 (x)) where f 1 , f 2 ∈ C(I, I). Let L = ω F (x, y) be an ω-limit with non empty interior. Then, according to where k is the number of different rectangles in the decomposition. As general observations, we can assume that where I i and J j are closed non-degenerate periodic subintervals of f 2 • f 1 and f 1 • f 2 respectively, with periods p and q, respectively. In our next development, we put I s = f 2 (J s ), J t = f 1 (I t ), s = 1, . . . , q, t = 1, . . . , p. Moreover, by Proposition 1, Let π 1 and π 2 be the projection maps on the first and second coordinate, respectively. Let J be an interval appearing in A ∪ B; for j = 1, 2, we define p j (J) = {R ⊆ L : π j (R) = J} the number of rectangles R in L, such that π j (R) = J.
In order to simplify the classification of ω-limit sets with non-empty interior, we consider the following definition. Definition 1. We say that two ω-limit sets there exists a permutation σ of k = k 1 = k 2 elements such that π j (L 1 ) = π j (L 2 ) and p j (π j ( R σ(i) )) = p j (π j (R i )) for each j = 1, 2 and i = 1, . . . , k.
From now we say that two ω-limit sets are different if they are not equivalent (for an example of equivalent limit sets, see Figure 4). Finally, we recall the definition of the mixing ω-limit set.

Definition 2.
We say that the ω-limit set A is mixing if ( f 2 • f 1 ) p | I i is mixing for all i ∈ {1, ..., p}. In other cases, we say that A is no mixing.
In the next development, the strategy followed describing the ω-limit sets with nonempty interiors consists in distinguishing the cases in which A, B are or are not mixing.
We can distinguish two general subcases: . Therefore, f 1 (A) = B and f 2 (B) = A, and there exists a bijective correspondence between the intervals of A and B and, hence, p = q. If p = 1 the distribution is trivial, we only have a unique rectangle R = I 1 × J 1 . For p ≥ 2 we reason as follows. Assuming an ordering such that R 1 = I 1 × J 1 , then In particular, if f 1 (I 1 ) = J 1 then p 2 (J 1 ) ≥ 2 since p ≥ 2, although it is also possible to find examples for which p 2 (I 1 ) = 1. Observe that under the previous assumptions, according to (4) the condition f 1 (I 1 ) = J t determines the images of all intervals I i and J j for i, j = 1, . . . , p. In fact, f 1 (I i ) = J i+t−1 (mod p) and f 2 (J i ) = I i+2−t (mod p) for any i ∈ {1, . . . p}. Thus, two different distributions (except equivalences) are possible: (D1) If t = 1, there are k = p = q rectangles and there exists an arrangement of the indexes such that R i = I i × J i for i = 1, . . . , k. It is clear that p 1 (I i ) = p 2 (J i ) = 1 for any i = 1, . . . , k. See Figure 5. (D2) If t = 1, there are k = 2p = 2q rectangles and f 1 (I i ) = J i (mod p) and f 2 (J i ) = I i+1 (mod p) , where p ≥ 2. Also, it is held that p 1 (I i ) = k p = 2 and p 2 (J i ) = k p = 2 for i = 1, . . . p, see Figure 6. Observe that if the number of rectangles k is odd, then only Distribution D1 is possible.
Notice that B = f 1 (A) and A = f 2 (B). Moreover, B ∩ Int(B) = ∅ and A ∩ Int(A) = ∅. In this case, the number of rectangles, k, is always even, in particular The ω-limit set L is given by As we have signaled above, the rectangles of the ω-limit set are described by Figure 7.
The number of connected components depends on the number of connected components of A ∪ A and B ∪ B which is less than or equal to p and q, respectively, see Lemma 4. Consequently, rectangles can be disjoint, have a common vertex or two common vertices, see . Notice that the case for two common vertices can only be realized by a unique distribution, given precisely by Figure 10; moreover, observe that R i ∩ R j = ∅ if and only if I i ∩ I j = ∅ and J i ∩ J j = ∅.  In the left side: p = q = 3, = 3 and k = 2 · = 6; p 1 (π 1 (R)) = 2 and p 2 (π 2 (R)) = 2 for each rectangle R in L. In the right side: p = q = 2, = 2, k = 2 = 4; again p 1 (π 1 (R)) = 2 and p 2 (π 2 (R)) = 2.

A Mixing and B No Mixing
Since B is no mixing and by Definition 2, Lemma 1 and Lemma 2, there are closed intervals J 1 i and J 2 i whose intersection is a fixed point of ( f 1 • f 2 ) q and such that and j = 1, 2. Now, it is not possible that f 1 (A) ∩ Int(B) = ∅ due to Lemma 3 and the different mixing character of the intervals. Assume that L = L 1 ∪ L 2 and that the rectangles of L 1 and L 2 are of type I i × J r j and type (I ) r j × J i (with i ∈ {1, . . . , p}, j ∈ {1, . . . , q}, r = 1, 2), respectively, where we have implicitly used Proposition 1 as well as the notation Next, by assuming that . . , q and s, t ∈ {1, 2}, s = t, due to the fact that B is not mixing; then both rectangles If q is even, = mq, m even, now each rectangle of the ω-limit L is the half part of a rectangle I i × J k or I j × J i , and the number of rectangles of L n is equal to , n = 1, 2, so |L| = 2 (notice that, as a consequence of the parity of , Thus, we assume that q is even. The number of rectangles of L must be a multiple of 4. Since L 1 has an even number of rectangles, the following cases sum up the possibilities. (D5) f 1 (A) ∩ B = ∅. In this case, the ω-limit is a set L which is the union of L 1 and L 2 both with the same number of rectangles. We also have that L i has an even number of rectangles for i = 1, 2. Observe that: p 1 (π 1 (R)) = p and p 2 (π 2 (R)) = q for each entire rectangle R having the form and each projection of a rectangle of L 1 must have a common point with the projection of another rectangle; similarly, p 2 (π 2 (R)) = p and p 1 (π 1 (R)) = q for each entire rectangle R ∈ L 2 (having the form (I ) j × (J ) i = (I ) 1 j ∪ (I ) 2 j × ((J ) i )) and each projection of a rectangle must have a common point with the projection of another rectangle. See Figure 12. Here, p = 2, q = 3, = 6; each 'entire' rectangle R in L 1 verifies p 1 (π 1 (R)) = 3 = p and p 2 (π 2 (R)) = 2 = q , whereas each 'entire' rectangle Notice that p = 2q, = p, q = 2; moreover, the number of rectangles in L = L 1 ∪ L 2 is just 4q.

A and B No Mixing
Realize that if A and B are no mixing, then f 1 (A) and f 2 (B) are also no mixing. We can assume that, except for equivalences, there are closed intervals I 1 i and I 2 i , whose intersection is a fixed point of ( f 2 • f 1 ) p and such that We also assume that R 1 = I 1 1 × J 1 1 . Let us assume also that I 1 i < I 2 i and J 1 i < J 2 i , where I < J means that x < y for each x ∈ I and y ∈ J. Now, according to Lemma 4, only the cases f 1 (A) ∩ B = ∅ and f 1 (A) ∩ Int(B) = ∅ are admissible (notice that the last one is equivalent to f 1 (A) = B). Recall that = lcm(p, q), p and q being the number of intervals in A and B, respectively. Moreover, thanks to the equivalence relation given in Definition 1, we can assume without loss of generality that 1. Case Firstly, we claim that the number of rectangles in the orbit of ; on the other hand, we also deduce that p|m and q|m, and consequently |m. From the above reasoning, either m = or m = 2 . If it were m = , with even, (a, b) and p| 2 , q| 2 , in contradiction with the definition of the lowest common multiple. If it were m = , with odd, = 2s , contrary to our hypothesis. Therefore, (a, b) is F-periodic and its period is 2 . As a consequence, Orb F (I 1 × J 1 ) has 2 different rectangles (maybe with common vertexes). Since F 2 (I 1 × J 1 ) = I 1 × J 1 , and is the first time s in which we have that Orb F (I 1 × J 1 ) has either or 2 elements. But the case is forbidden because then F (a, b) = (a, b) due to the fact that (a, b), F(a, b), . . . , are the 'middle points' of I 1 × J 1 and its respective images. This ends the claim.
The last claim does not imply necessarily that the number k of rectangles in the ω-limit L = ω F (x, y) = Orb F (I 1 1 × J 1 1 ) must be 2 . It will depend on the parity of k 1 := p and k 2 := q . Realize that k 1 and k 2 cannot be even simultaneously, at least some of them are odd. If k 1 is odd and k 2 is even (the case k 1 even and k 2 odd is analogous), then , and really we have 2 different rectangles in L because the parts I 1 1 × J 1 1 and I 2 1 × J 1 1 gives rise to the union of I 1 × J 1 1 . The distribution that appears is equal to the case mixing-no mixing previously considered, distribution D5. If k 1 and k 2 are both odd integers, then = pk 1 = qk 2 and F 2 (I 1 , and in this case we find 4 different rectangles (notice that F 2j (I 1 1 × J 1 1 ) = I 1 1 × J 1 1 for j = 1, . . . , − 1); moreover, the rectangles in the ω-limit L are distributed in two groups of rectangles, namely L 1 and L 2 , with |L 1 | = |L 2 | = 2 . The distribution of L 2 mimics the distribution of L 1 , shifting the position of the axes. See Figure 14. Thus, let us assume k 1 and k 2 are odd numbers. Then, structures that are made up of two rectangles with a common point appear, which allows us to consider different orientations, namely, (+) and (−) see Figure 15. For this case, there must coexist these types of structures with both orientations, but, how many of them? In order to describe and count the structures with negative orientation, we proceed as follows. Recall that we assume Moreover, p ≥ q, = lcm(p, q) and k 1 = p , k 2 = q are odd integers. If the ω-limit set L is written as L = L 1 ∪ L 2 , take into account that we are going to describe the structures with negative orientation in L 1 , for subrectangles contained in bigger rectangles of type I i × J j , and that we start to iterate with the subrectangle I 1 1 × J 1 1 . For i = 1, . . . , p define σ(j) ∈ {1, 2} as the value for which 1+i (mod(p)) .
Notice that ( 1+n (mod(p)) . Similarly, we extend τ as Here, , and the same time This implies that σ( ) = τ( ) = 2 because k 1 and k 2 are odd integers, and we can observe that from this point the values of the orbit of I 1 1 × J 1 1 belong to the same structure. Notice also that if i ∈ {1, . . . , } then 1+i (mod(q)) . As we see, this again guarantees that our extensions, σ, τ : N → {1, 2} N are well defined.

2.
Case Firstly, let us show that the point (a, b) ∈ I 1 × J 1 , given by a = I 1 1 , is F-periodic and its (minimal) period m can be or 2 . Indeed, it is clear , respectively, as a consequence of the no mixing property satisfied by A and B, and the fact that = p = q. On the other hand, if n < were the order of (a, b), ) would imply that |n, a contradiction. Therefore, m is either or 2 . Consequently, we must distinguish two cases according to whether the order m of the aforementioned F-periodic (a, b) is or 2 : (i.O) If m = , we affirm that the number k of rectangles in L = ω F (x, y) is k = , and in fact ω F (x, y) is composed of rectangles of type I i × J j . To prove it, recall that we are assuming that (x, y) ∈ R 1 = I 1 1 × J 1 1 ⊂ I 1 × J 1 . Notice that F (a, b) = (a, b) ∈ I 1 × J 1 , so F (I 1 1 × J 1 1 ) = I i 1 × J j 1 for some i, j ∈ {1, 2}. If i = j = 1, then we would have F 2 (I 1 1 × J 1 1 ) = F (I 1 1 × J 1 1 ) = I 1 1 × J 1 1 , but at the same time, for the decomposition of A and B obtained for their no mixing property, If i = j = 2, now F (I 1 1 × J 1 1 ) = I 2 1 × J 2 1 and since also F 2 (I 1 1 × J 1 1 ) = I 2 1 × J 2 1 , we would obtain F s (I 2 1 × J 2 1 ) = I 2 1 × J 2 1 for all s ≥ 1, a new contradiction (at least, F 2 (I 2 1 × J 2 1 ) = I 1 1 × J 1 1 ). Therefore, i = j, which implies that either F (I 1 , and thus all the rectangle I 1 × J 1 is included in Orb F (I 1 1 × J 1 1 ). From here, and taking into account that an entire rectangle only contains at most a unique point of OrbF(a, b), we deduce that |L| = , and ω F (x, y) is composed of k = rectangles of type I i × J j . Even more, its distribution follows the pattern of Distribution D1, and with equivalence we can reorder the intervals in such a way that f 1 (I i ) = J i+1(modp) and f 2 (J i ) = I i+1(modp) for i = 1, . . . , p. See Figure 16. Finally, notice that only the case odd is permitted here, since if is even, we would have (a, b) = F (a, b) and then ( f 2 • f 1 ) 2 (a) = a, ( f 1 • f 2 ) 2 (b) = b, impossible (realize that a and b are -periodic points of f 2 • f 1 and f 1 • f 2 , respectively). (ii.E) If m = 2 , we assert that |L| = 4 and, to be more precise, L = ω F (x, y) is composed of 4 rectangles having type I u i , J v j for some i, j ∈ {1, . . . , 2 }, u, v ∈ {1, 2}, such that each of them are included in I i × J j , with a common vertex given by a point of the orbit of (a, b). To prove it, first notice that Orb F (a, b) =: , so from Lemma 5 and Theorem 3-(2) we deduce that for any i ∈ {1, . . . , 2 } there exist r, s ∈ {1, . . . , 2 }, r = s, such that x i = x r , y i = y s . This implies that the F-orbit of (a, b) is included in 2 entire rectangles of type I r × J s . On the other hand, F 2 (I 1 . . , } and, according to the distribution provided for Theorem 3 to the orbit of (a, b), each entire rectangle I i × J j only contains at most a unique point of Orb F (a, b). All the above details allow us to conclude that k = |L| = 4 and each rectangle I i × J j containing a point of Orb F (a, b) is divided into two different subintervals with disjoint interiors and a point in Orb F (a, b) as common vertex, see Figure 17, and therefore we obtain a structure of the type described in Figure 15. Observe that if the position of the intervals is shifted, for example if I 1 i > I 2 i , for any i, then the orientation of every structure in the column i changes. This case is the same as that obtained in Distribution D2, but changing the entire rectangle by two subrectangles of the type given in Figure 15.

The Converse Result for the Non-Empty Interior Case
Until now, we have described the structure of ω-limit sets of Cournot maps with nonempty interior. Of course, given one of these structures L, up to equivalences, it is natural to ask whether it is possible or not to find a Cournot map F and a point (x, y) ∈ I 2 such that ω F (x, y) = L. The answer is the affirmative. The general strategy for this construction relies on the fact that we can take advantage of our knowledge of interval maps f : I → I exhibiting union of intervals m i=1 I i as ω-limit sets for suitable points x ∈ I. In order to illustrate it, we do the corresponding converse result for Distributions D1 (with k odd), with the hope that the other cases can be managed for the reader without any difficulty.
(D1) Assume that L = k i=1 (I i × J i ), k odd, and I i , J i verify (D1), that is: To simplify the notation, let α i = i and β j = j, that is, (2) . We then define f 1 ∈ C(I, I) in such a way that and linear in the rest of I i , i = 1, . . . , k. Also, we define a map f 2 ∈ C(I, I) holding f 2 (J i ) = I i+1 in a linear way (change of variables) for i = 1, . . . , k. Then, it is straightforward to see that Orb F ( Since k is odd, from the definitions of f j we deduce that f 2 • f 1 is mixing on A, f 1 • f 2 is mixing on B, and f 1 (A) = B, f 2 (B) = A. According to Theorem 2, there exists (x, y) ∈ I 2 such that ω F 2k (x, y) = I 1 .

Applications to Economic Models
In the seminal paper of Day [19] it was highlighted that complex behavior that can be described as chaotic can emerge from simple economic structures. This fact has had as a consequence for the increasing interest in economic dynamics, which is supported by the large amount of related literature published since then.
In this frame, oligopoly structure plays an important role. The assumption that only a few firms can compete makes sense in some markets in which the goods are perfectly substitutable [20]. In the decision process, each firm adjusts its output (product decision) based on the expectation over the other firm's output decisions. Under this premise, firms take their decisions simultaneously. In an oligopoly the supply side is (not perfectly) competitive-as we often say "Cournot competition". A duopoly is an oligopoly in which the number of firms is equal to two. Following Friedman [21], "a duopoly is a market in which two firms sell a product to a large number of consumers. Each consumer is too small to affect the market price for the product, that is, on the buyers' side, the market is competitive." Different theoretical models of duopoly have been considered [21]. In 1838, Cournot [22] proposed a non-collusive duopoly model. In this duopoly model, firms decide their output levels simultaneously, that is, firms compete in quantities and it is assumed that price is given by the market. Moreover, each firm seeks to maximize the profit taking into account the decision of the competitor. Cournot duopoly was considered in [17], where the complexity of the dynamics was established. Later, Dana and Montrucchio [13] gave more details about the dynamical properties of the Cournot model. More recently, Pražák and Kovárník have considered in [15] a nonlinear inverse demand function and have shown bifurcation diagrams of the duopoly model. Antitriangular maps are closely related to Cournot duopoly models. In this section we deal with the type of ω-limits of several Cournot duopoly models in which dynamics is led by antitriangular maps.
Let us notice the relevance of this topic by highlighting its possible interest in implications for strategic decisions. In this direction, cyclical phenomena have an important role in economics in the analysis of the dynamics and fluctuations of the economic activity in a model. A cycle of length n is the orbit of a periodic point. In the matter at hand, that is, ω-limits of non-empty interior of Cournot maps, the full description of the structure is useful in the sense that it provides information about chaotic orbits (realize that ω-limits with non-empty interior appear when the Cournot map F is chaotic). A cyclical movement appears if, in the definition of cycle, we replace points by rectangles, since in each step the point of the chaotic orbit belongs to the corresponding rectangle and this turns on the fact that in each step we have boundaries (restrictions) for the values of the outputs because the boundaries of the rectangles limit the output values of the production firms when initial conditions have been fixed. In this way, the restriction of the regions of ω-limits in Cournot duopolies plays an important role in the analysis of dynamics. Therefore, the distribution of the ω-limits gives us information about the possible layouts in the rectangles giving the feasible region in which both firms take their output values, the rectangles being covered following a "cyclical process".

Puu's Duopoly
The following duopoly model was proposed by Puu [23]. In this model, the goods produced for both firms are equivalent and the demand function is isoelastic. It is assumed that the price is given by where x and y are the outputs of firms X and Y respectively. The costs are assumed to be linear with the form c 1 (x) = c 1 x and c 2 (y) = c 2 y, where c 1 and c 2 , the marginal costs, will be the model parameters. Obviously, now profits are given by Considering that the objective of the firms is to maximize their own profits, the equations ∂Π 1 ∂x = 0 and ∂Π 2 ∂y = 0 give us Finally, the reaction map is which is an antitriangular map in which has been introduced the max function due to the fact that productions cannot be negative. It is assumed without loss of generality that c 1 ≥ c 2 , and we can compute ω-limit sets, see some examples in Figure 18 (we have used the software Mathematica for generating Figures 18-23). More details about the dynamics of the model can be found in [24].

M. Kopel introduced in [14]
a general Cournot duopoly model in which two firms, X and Y, plan their productions, x t , y t , at time t, following the system of difference equations: where λ i ∈ [0, 1] and µ i > 0 for i = 1, 2. The positive parameters µ i measure the intensity of the effect that one firm's action has on the other firm, see also [25] for more details. Then, the two-dimensional map R λ 1 ,λ 2 ,µ 1 ,µ 2 (x, y) : describes the simultaneous choices of both firms. Observe that taking λ 1 = λ 2 = 1, a Cournot duopoly model is obtained, given as In Figure 19, we have shown some ω-limits for the map (7). We have selected some of them. To get it, we have used random initial conditions, with 6 decimal digits by default.

Matsumoto-Nonaka's Model
In [26] a two-market model consisting of two firms that produce differentiated goods is introduced, see also [27]. The first firm produces good x in the first market and the second one produces good y in the second market. In this model, different externalities affect the dynamics.
As in the previous subsection, in [26] a more general model is introduced. Nevertheless, we are interested in the particular case in which inverse demand functions are given by where p 1 and p 2 are the market prices of goods x and y, respectively, and α ∈ [1, 2] ⊂ R and β ∈ [0, 2] ⊂ R. Each firm decides future production depending on the other firm's choice and production externalities. Cost functions are given by and the profit function of each firm is given by As assumed above, each firm tends to maximize its profit. In order to do this the firms can choose their production levels, which would affect to the other firm. The first firm maximizes the profit with respect to x and the same occurs for the second firm with respect to y. The reaction functions of the firms are given by Now, the function is the reaction function for the outcome (x, y). The iterations are given by x t+1 = f α (y t ) = (αy t − α + 1) 2 , where α ∈ [1, 2] ⊂ R and β ∈ [0, 2] ⊂ R. Thus, the reaction map F α,β is a Cournot map 1]. In this situation, we obtain some ω-limits, the results are shown in Figure 20.

Cournot Duopoly When the Competitors Operate under Capacity Constraints
In [28] A. Norin and T. Puu introduced a Cournot duopoly where the competitors have capacity constraints. The production cost of the firms are given by where u and v denote the capacity limits and x and y denote the supplies of the two competitors. Observe that when x = y = 0 then the total costs are zero and tend to infinity when production approaches capacity limits. The price p is given by p = 1 x+y and the profits are To maximize the profits (10) we get the following equations From the previous system of Equation (11), solving the first for x and the second for y (see [28] for details), the following reaction functions are obtained since clearly productions cannot be negative. Hence, the reaction map we construct is a Cournot map Different ω-limits for the model are shown in Figure 21, where the initial conditions are x 0 = 0.2 and y 0 = 0.3. We have computed orbits of length equal to 200,000 and we have drawn the last 100,000 points.

A Recurrent Class of Two-Dimensional Endomorphisms
The class of maps given by F(x, y) = (y, h(x)), where h(x) is a continuous function, piecewise continuously differentiable, has been considered among others in [29], and in this framework this kind of map provides us another different example to which we can apply this analysis.

Some Advances in the Empty-Interior Case
For a Cournot map, we already know the topological structure of an ω-limit set with empty interior, see Theorem 1. Essentially, the situation is similar to the interval case, the ω-limit set is a nowhere dense set. In the interval case this means that the set is totally disconnected. This changes for Cournot maps, we can find ω-limit sets with empty interior and locally connected, even connected.
In view of the last example, we stress that it is possible to find ω-limit sets of antitriangular maps consisting of the union of the diagonal and the graph of an interval map (for instance, see Figure 26).

Proposition 2.
Let f ∈ C(I, I) be a transitive map. Let x 0 ∈ I such that ω f (x 0 ) = I. Then, can be realized as the ω-limit set of the Cournot map F(x, y) = (y, f (x)).

Remark 3.
If in the above result, we choose a map f ∈ C(I, I) and a point x 0 such that ω f (x 0 ) is a Cantor set C; notice that we are able to find totally disconnected ω−limit sets of the form For instance, if we consider the well-known parabola f (x) = λ ∞ x(1 − x) of type 2 ∞ (its set of periods is given by all the powers of 2), where λ ∞ = 3.569... is the limit value of the period doubling bifurcation or route to chaos of the unimodal family of parabolas f λ (x) = λx(1 − x), λ ∈ [0, 4] (for instance, see [30] for more information), we can choose a point x 0 in such a way that ω f (x 0 ) = C 0 is homeomorphic to the classical ternary Cantor set and, in this case for F(x, y) = (y, f (x)), we find that Figure 27.
In the same direction, we find the following result on a peculiar type of ω-limit set for Cournot maps (see Figure 28).
Proof. Since f is transitive, there is x 0 ∈ I such that ω f (x 0 ) = I. Consider the antitriangular map F(x, y) = ( f (y), f (x)). Since f is totally transitive, it is easy to check that The graphs Γ f and Γ * f corresponding to f = T 2 , the second iterate of the tent map.
In our third example we stress the fact that, contrary to the first examples, there exist ω-limit sets of antitriangular maps which are not connected. To see it, define the map (see Figure 29) where f 2 was defined in the second example. In this case, there exists z 0 ∈ [ 1 3 , 2 3 ] such that ω f 3 (z 0 ) = 1 3 , 2 3 . By defining F 3 (x, y) = (y, f 3 (x)), a direct inspection of the orbit of (0, z 0 ) gives us the limit set Figure 29. The map f 3 and, in red, the ω-limit set ω F 3 ((0, z 0 )).
Our following example exhibits an ω-limit set with empty interior and homeomorphic to C 2 := (x, y) : x 2 + y 2 = 1 ∪ (x, y) : (x − 2) 2 + y 2 = 1 , namely C 2 is the union of two circles joined by a point. We define F 4 (x, y) = (y, f 4 (x)), where Since f 4 is a transitive map, take a point w 0 ∈ I such that ω f 4 (w 0 ) = I. Then by direct computation we find that where ∆ is the diagonal of the unit square and Γ f 4 = {(x, f 4 (x)) : x ∈ I} is the graph of f 4 (see Figure 30). Figure 30. The map f 4 and the ω-limit set ω F 4 ((w 0 , w 0 )) homeomorphic to C 2 .
In fact, we can generalize the above example to find ω-limit sets homeomorphic to the union of n-circles, n ≥ 2, attach one of them to the other by a point, or C n = (x, y) ∈ R 2 : (x − 2j) 2 + y 2 = 1, for some j = 0, 1, . . . , n − 1 .
This is a substantial difference of the ω-limit sets of antitriangular maps with respect to the one dimensional case-the limit sets can be connected and, at the same time, exhibit empty interior.

Proposition 4.
For each n ≥ 1, there exists an antitriangular map Φ n such that ω Φ n (X n ) is homeomorphic to C n for some X n ∈ I 2 .
Proof. We distinguish two cases according to the parity of n.
If n = 2k, define (see Figure 31 for n = 6) n+1 for j = 0, 1, . . . , n 2 , Since ϕ n is transitive, take a point x 0 ∈ I for which ω ϕ n (x 0 ) = I. Define Φ n (x, y) = (y, ϕ n (x)). Then, as an immediate consequence of iterating (x 0 , x 0 ) successively and to consider the transitivity of the point x 0 , we obtain where ∆ is the diagonal of the unit square and Γ ϕ n = {(x, ϕ n (x)) : x ∈ I} is the graph of ϕ n . Figure 31. The map ϕ 6 and the ω-limit set ω Φ 6 ((x 0 , x 0 )) homeomorphic to C 6 . Now, let us present some other connected ω-limit sets whose interiors are empty, and where its aspect is similar to a lattice structure. Example 1. Consider a totally transitive map f : I → I, for instance the tent map. Let x 0 ∈ I such that ω f (x 0 ) = I and define the Cournot map F(x, y) = (y, f (x)). On the other hand, take a periodic point y 0 of period N ≥ 1. Then, if {y 0 , y 1 , . . . , y N−1 } is the periodic orbit, it is easily seen, with the help of Remark 1, that ω F (x 0 , y 0 ) = (I × {y 0 , y 1 , . . . , y N−1 }) ∪ ({y 0 , y 1 , . . . , y N−1 } × I).
In Figure 34 you can see the lattice generated by a periodic point of period 2. According to the aspect of the drawn ω-limit sets of a Cournot map with non-empty interior, we are left the following result concerning the connectivity of such a ω-limit set. Before we state a topological result (see ( [31], p. 27)) which implies, as an easy consequence, that the union of two closed sets in the interval has an empty interior if each of them has in turn an empty interior. Lemma 7. If K is a closed set in a topological space (X, T ), and A ⊂ X is any subset of X, then Int(K ∪ Int(A)) = Int(K ∪ A).
We present here a characterization for connected ω-limit sets of Cournot maps that have non-empty interior. Realize that we can justify it in view of our analysis developed in Section 3, nevertheless we give the corresponding theoretical proof. Proposition 5. Let F(x, y) = ( f 2 (y), f 1 (x)), (x, y) ∈ I 2 be a Cournot map. Let ω F (x, y) be an ω-limit set with non-empty interior. Then, ω F (x, y) is connected if and only if (1) π j (ω F (x, y)) is connected, where π j is meant the canonical projection to the j-th coordinate, j = 1, 2.
Proof. Assume that ω F (x, y) is connected. Since π j is continuous, j = 1, 2, we have (1). On the other hand, if ω F 2 (x, y) ∩ ω F 2 (F(x, y)) = ∅, then, taking into account that ω F (x, y) = ω F 2 (x, y) ∪ ω F 2 (F(x, y)), we would have a decomposition of ω F (x, y) into two closed disjoint sets, which contradicts that ω F (x, y) is connected. Now, we assume that (1) and (2) hold, and we are going to prove that in these circumstances ω F (x, y) is connected. By Proposition 1 and (1 ) is a connected set in the real line, therefore an interval, which is possibly degenerate. If the interval is degenerate, then both ω f 2 • f 1 (x) and ω f 2 • f 1 ( f 2 (y)) are singletons (in fact, the same point), and it is easily seen that ω F (x, y) is reduced to a fixed point, therefore a connected set. So, next we assume that π 1 (ω F (x, y) is a closed interval with nonempty interior. According to Lemma 7, either ω f 2 • f 1 (x) or ω f 2 • f 1 ( f 2 (y)) has non-empty interior. Without loss of generality, assume that ω Necessarily, either m = 1 or m = 2: if m ≥ 3, then if we denote I i = [a i , b i ], i = 1, . . . , m, the omega-limit set ω f 2 • f 1 ( f 2 (y)) must contain the intervals [b i , a i+1 ] for i = 1, . . . , m − 1, but this is impossible, as was shown in Lemma 4.
Applying the maps f 1 , f 2 , we find Then, by Theorem 2 there exists (x 1 , y 1 ) ∈ I 2 such that: is mixing, j = 1, 2): in this case, since ω F (x, y) has non-empty interior and we can use Lemma 3, it can be noticed that the we obtain Distribution D6, which contradicts our hypothesis (2) on the intersection of ω F 2 (x, y) and ω F 2 (F(x, y)); in this case, reasoning as above, we arrive at Distribution D4, see Figure 10, a connected set with non-empty interior.
(ii) Let m = 1. Notice that if Int(ω F (x, y)) = ∅, also ω f 1 • f 2 (y) has non-empty interior, as a consequence of Proposition 1 and the fact that in a Cartesian product the connected components of a set are the product of connected components of each space, see [31].
Here, additionally, we can distinguish two different cases depending on the nature of ω f 1 • f 2 (y) = B.
(ii.1) If B = J is an interval with f 2 (B) = f 2 (J) = I 1 , we obtain: either Distribution D1, with k = 1, if I 1 is mixing, therefore a connected ω-limit set holding (1) and (2); or the Distribution that we have found in case A, B no mixing, two rectangles tied by the vertex obtained with the 'middle' points (fixed points) of I 1 and J 1 .
Lemma 4 both A = I 1 and J 1 are mixing. Reasoning as in Case (i.2), we can find (x 1 , y 1 ) such that ω f 2 • f 1 (x) = I 1 and ω f 1 • f 2 (y) = J, and it is immediate to see that ω F will be the union of two rectangles tied by a common vertex, a connected set.

Remark 4.
We emphasize that the proof strengthens our study in Section 3, because inside it we have found the only possible connected ω-limit sets with non-empty interior that appeared in that section. Moreover, notice that the conditions are always necessary, even in the general frame of a compact metric space and a continuous map ϕ : X → X that has the form ϕ(x, y) = (ϕ 2 (y), ϕ 1 (x)), with ϕ j : X → X, j = 1, 2. It remains for us to study whether, in fact, they are or are not also sufficient conditions in order to characterize the connectivity of ω-limit sets of Cournot maps with empty interior.
Additionally, we can find another non-connected ω-limit set ω F (x, y) verifying, nevertheless, the property (1), see Figure 36 for the non-empty interior case. With regards to the empty interior case, consider the transitive map f : [0, 1] → [0, 1] given by f ( It is easily seen that f is transitive but not totally transitive. Let {y 1 , y 2 } = { 1 6 , 5 6 } be a periodic orbit having order two, and define the Cournot map F(x, y) = (y, f (x)). If x 0 has a dense orbit by f , so ω f (x 0 ) = [0, 1], we find ω F (x 0 , y 1 ) = [0, 1 2 1 2 ] . Then ω F (x 0 , y 1 ) is not connected, however π j (ω F (x 0 , y 1 )) = [0, 1] (see Figure 36). Finally, realize that, according to our description of ω-limit sets of Cournot maps with nonempty interior, we are in a position to assert that both conditions are also sufficient in order to establish the connectivity of ω F (x, y) in the non-empty interior case, which is an open question in the case of empty interior.
We finish the section by mentioning the notion of minimal set and some associated results, for the case of Cournot maps, extracted from [18]. A minimal set M ⊂ I 2 of a Cournot map F ∈ C A (I 2 , I 2 ) is a non-empty, closed and invariant set by F, which does not contain a proper subset with the same three properties. In [18] it is proved that a minimal set M of F cannot be of the form C 1 × C 2 , where C 1 and C 2 are Cantor sets. In the same paper, it is shown that, nevertheless, given two solenoidal sets Q 1 and Q 2 having periodic decompositions that are relatively prime, we can find a Cournot map F(x, y) = (g(y), f (x)) and x, y ∈ I such that ω F (x, y) = (Q 1 × f (Q 2 )) ∪ (Q 2 × f (Q 1 )) is a minimal set of F (for the notion of solenoidal set-a class of Cantor set-the reader is referred to [18] and references therein). An interesting question would be to find, if possible, a Cournot map that has the product C × C as an admissible ω-limit set, where C is a Cantor set.

Conclusions
Despite the difficulty in analyzing ω-limit sets for two-dimensional maps, in the case of Cournot maps we have been able to completely describe the structure of their omega limits with non-empty interior. As a complement, we have shown that these structures arise naturally in some economic models, which can be formulated by Cournot maps, for instance, the Puu's duopoly or the Matsumoto-Nonaka's model. Cyclical phenomena in economics attract great interest. Following Burns and Mitchell (1946), [32], "business cycles are a type of fluctuation found in the aggregate economic activity of nations that organize their work mainly in business enterprises-a cycle consists of expansions occurring at about the same time in many economic activities, followed by similarly general recessions, contractions, and revivals which merge into the expansion phase of the next cycle". In mathematical terms, the orbit of a periodic point is called a cycle of length n. In the case at hand, we can observe that replacing points in the orbit by rectangles, a "cyclical movement" applies, in which the rectangle limits the values in each step, thus, a more general cyclical process can be considered and the classification of the orbits with non-empty interior that have been studied throughout this paper contributes to shedding light on economic dynamics.
Obviously, it is an open question to give a complete and detailed description of ω-limit sets of Cournot maps with empty interior. In this paper, we have provided some interesting examples showing that the limit set can be homeomorphic to the circle, even to the union of k circles attached between them by common tangent points; moreover, a difference with the one-dimensional case appears, now the limit set can be connected even though it itself has an empty interior. Another task to be carried out is to find in the non-empty interior case a general formulation for all possible realignments and counting of negative orientations which have appeared in the case A and B no mixing, to try to obtain a closed expression for this number of negative orientations.
Realize that Cournot maps can be generalized in the following way: in the n-dimensional cube I n = [0, 1] n , we define F(x 1 , x 2 , . . . , x n ) = f σ(1) (x σ(1) ), f σ(2) (x σ(2) ), . . . , f σ(n) (x σ(n) ) , where σ is a (cyclic) permutation of {1, 2, . . . , n}, and each f σ(j) is a continuous function defined from I into itself. These maps can be named permuted direct product maps (see for instance, [33] or [34]). Then, an interesting question would be the study of ω-limit sets with non-empty interior for permuted direct product maps. For instance, for n = 3, we would have a union of cubes moving in I 3 according to specific rules to be determined.
Another interesting question is to prove whether the conditions (1) and (2) in Proposition 5 are sufficient or not in order to establish that the corresponding omegalimit is connected. Our conjecture is the affirmative and, even more, we can address the question to the general frame of a product X 1 × X 2 of compact metric spaces and, if we generalize the definition of a Cournot map as ϕ : X 1 × X 2 → X 1 × X 2 , ϕ(x, y) = (ϕ 2 (y), ϕ 1 (x)), with ϕ i : X i → X j , i = j, to ask for a characterization of connected ω-limit sets in terms of their projections and the intersection of the omega-limits ω ϕ 2 (x, y) and ω ϕ 2 (ϕ(x, y)).

Conflicts of Interest:
The authors declare no conflict of interest.