Renormalization of two-dimensional piecewise linear maps: Abundance of 2-D strange attractors

For a two parameter family of two-dimensional piecewise linear maps and for every natural number $ n $ we prove not only the existence of intervals of parameters for which the respective maps are $ n $ times renormalizable but also we show the existence of intervals of parameters where the coexistence of at least $ 2^n $ strange attractors takes place. This family of maps contains the two-dimensional extension of the classical one-dimensional family of tent maps.


Introduction
Several papers in the last century (see [2], [7], [8]) have been devoted to analytically prove the existence of strange attractors. , if (x, y) ∈ T 0 (2 − x + y, 2 − x − y) , if (x, y) ∈ T 1 , defined on the triangle T = T 0 ∪ T 1 , where As a first approach to the study of the dynamics of T a(s),b(s) for s close to 2, certain family of piecewise linear maps was introduced in [10]. This family was defined on the triangle T by These maps can be seen as the composition of linear maps defined by the matrices with the fold of the whole plane along the line C = (x, y) ∈ R 2 : x = 1 given by , if x < 1, The triangle T is invariant for these maps Λ t whenever 0 ≤ t ≤ 1. If t ≤ 1/ √ 2, then the dynamics of Λ t is simple. However, if t > 1/ √ 2, then the eigenvalues of the matrix A t have modulus greater than one and this fact gives rise to richer dynamics. In particular, as was proved in [11] there appear strange attractors. In the case in which t > 1/ √ 2, the map Λ t is called, according to Section 4 in [10], an Expanding Baker Map (EBM for short).
To begin with, let us recall the definition of fold.
Definition 1.2. Let K ⊂ R 2 be a compact and convex domain with nonempty interior, P a point in K and L a line with L ∩ int(K) = ∅ and P / ∈ L. Then L divides K into two subsets denoted by K 0 and K 1 (K 0 the one containing P ). We define the fold F L,P as the map where Q denotes the symmetric point of Q with respect to L.
In the above conditions, the map F L,P is said to be a good fold if F L,P (K) = K 0 . Now, let us write L = L 1 and let L 2 be a line with L 2 ∩ int(K 0 ) = ∅ and P / ∈ L 2 . Then, L 2 divides K 0 into two subsets K 00 and K 01 (K 00 denotes the one containing P ). Let us assume that F L2,P (K 0 ) = K 00 (i.e, F L2,P is a good fold). Repeating these arguments, we may successively define a sequence of good folds F L1,P , . . . , F Ln,P where F L1,P : K → K 0 , We now recall the concept of EBM. and assume that A(K 0 n ...0 ) ⊂ K. We define the Expanding Baker Map associated to P, A, L 1 , . . . , L n as the map Γ : K → K given by For short, we shall denote Γ = EBM (K, L 1 , . . . , L n , P, A).
The choice of the family Λ t , 0 ≤ t ≤ 1, was motivated in [9]: the study of the dynamics exhibited by the family Λ t is mainly justified when one compares its attractors (numerically obtained in [9]) with the attractors (numerically studied in [14]) for the family T a(s),b(s) with s ∈ [0, 2] and (a(s), b(s)) ∈ G, see (1). Both families of maps display convex strange attractors, connected (but non simply-connected) strange attractors and non-connected strange attractors (these last ones formed by numerous connected pieces).
A first analytical proof on the existence of a convex strange attractor of Λ t was given in [11] for every t ∈ (t 0 , 1], where t 0 = 1 √ 2 (1 + √ 2) 1/4 . The appearance of attractors with several pieces suggested the definition of renormalizable EBM given in [12].
Definition 1.4. Let Γ be a map defined in certain domain K. We said that D ⊂ K is a restrictive In [12] (see the Main Theorem), it was proved that there exist three values of the parameter t, The proof of this result is consequence of a renormalization procedure which allows us to understand how connected attractors may break up giving rise to new attractors formed by an increasing number of pieces. Furthermore, the proper renormalization method is fruitful to explain the coexistence of attractors: the renormalization can be simultaneously used on two disjoint restrictive domains to get two different attractors. Numerical evidences allowed us to conjecture that these attractors are strange. In fact, [12] was finished with the following three conjectures that we are going to prove in the present paper. Conjecture 1. There exists a decreasing sequence {t n } n∈N , convergent to 1 √ 2 such that Λ t is a n times renormalizable EBM for every t ∈ ( 1 √ 2 , t n ).

Conjecture 2.
There is no value of t for which Λ t is infinitely many renormalizable.

Conjecture 3.
For each natural number n there exists an interval I n ⊂ ( 1 √ 2 , t n ) such that Λ t displays, at least, 2 n−1 different strange attractors.
The proofs of these conjectures are again strongly based on the notion of renormalization. From [12] we know that the renormalization of Λ t leads to an EBM with two folds. To be more precise, these two folds take place, respectively, along the lines Therefore, we shall be mainly interested in the study of the two-parameter family of EBMs defines expansive linear maps with fixed point in O. Let P denote the set of parameters given by The choice of P guarantees that T is invariant by Ψ a,b for every (a, b) ∈ P. See Section 2 for more details of the definition of Ψ a,b .
To begin with, we shall prove the existence of two regions P j , j = 1, 2 of P in which Ψ a,b has an invariant rectangle R a,b and, in addition, for each (a, b) ∈ P j , Ψ 2 a,b is conjugate on R a,b to a direct product of tent maps. As a consequence, it follows that the maps Ψ a,b exhibit strange attractors on Later, we shall find a third region P 3 contained in P such that for every (a, b) ∈ P 3 the map Ψ a,b can be renormalized on two different restrictive domains ∆ a,b and Π a,b at the same time. In this way, we can define two different renormalization operators H ∆ and H Π and prove the following result.
Theorem A. For every (a, b) ∈ P 3 the map Ψ a,b is simultaneously renormalizable in F on two different restrictive domains. Namely: i) The restriction of Ψ 4 a,b to ∆ a,b is conjugate by means of an affine change in coordinates to Ψ H∆(a,b) restricted to Ψ H∆(a,b) (T ).
ii) The restriction of Ψ 4 a,b to Π a,b is conjugate by means of an affine change in coordinates to Ψ HΠ(a,b) .
The operators H ∆ and H Π satisfy fruitful properties used along the rest of the paper. In particular, P 1 ⊂ H ∆ (P 3 ) and P 2 ⊂ H Π (P 3 ). Moreover, both maps H ∆ and H Π has a repelling fixed point The maps Λ t defined in (2) satisfy, according to [12], that ] with a = 16t 8 and b = 1/2t 3 . Notice that is a curve contained in P starting at (1, √ 2). By using the relative positions between the tangent vector to γ 0 at (1, √ 2) and the eigenvectors of DH ∆ at this point, one concludes the existence of n 0 ∈ N such that H n ∆ (γ 0 ) ∩ P 1 = ∅, for every n > n 0 . This is the main argument used in the proof of the following result.
Theorem B. It holds that: i) There exists a decreasing sequence {t n } n∈N converging to 1 √ 2 , such that Λ t is a n times renormalizable EBM for every t ∈ ( 1 √ 2 , t n ).
ii) For each natural number n there exists an interval I n such that Λ t displays, at least, 2 n different strange attractors whenever t ∈ I n .
It is clear that the first statement of this theorem gives an affirmative answer to Conjecture as an open problem to show that the interval of parameters I n in which the existence of at least 2 n strange attractors is proved can be constructed inside the set of parameters where, according to the first statement of Theorem B, the map Λ t is a n times renormalizable map. Although we think this stronger result is also true, we remark that the most important dynamical property, i.e. the coexistence of any arbitrarily large number of persistent (in an interval of parameters) different strange attractors is demonstrated along this paper.
The paper is organized as follows. In Section 2 the subsets of parameters P 1 and P 2 are constructed and the coexistence of strange attractors for such parameters is proved. In Section 3 the proof of Theorem A is given and useful properties of the renormalization operators are stated.
Finally, Section 4 is devoted to demonstrate Theorem B.

Regions of coexistence of strange attractors
In order to find regions of parameters for which the map Ψ a,b displays coexistence of strange attractors we shall look for values of the parameter (a, b) such that Ψ a,b may be conjugate to a direct product of one-dimensional tent maps.

Direct product of one-dimensional tent maps
Let us recall the family of one-dimensional tent maps {λ µ } µ∈[0,2] given by It is known that the interval I µ = [µ(2 − µ), µ] is an invariant set of λ µ for every µ ∈ (1, 2]. Moreover, I µ is a strange attractor for every µ ∈ ( √ 2, 2] and strange attractors with several pieces may be also obtained for values of the parameter µ ∈ (1, √ 2] by using renormalization techniques. Furthermore, these strange attractors are strongly topologically mixing according to the next definition. From now on, we shall denote by Γ µ the two dimensional map defined by Lemma 2.2. The following statements hold: a) For every µ ∈ ( √ 2, 2] the map Γ µ has a strongly topologically mixing strange attractor with two positive Lyapounov exponents.
b) For every natural number n and for every µ ∈ (2 1 2 n+1 , 2 1 2 n ] the map Γ µ is n times renormalizable and displays 2 n strange attractors with two positive Lyapounov exponents.
Proof. It is clear that is invariant by Γ µ . Moreover, since λ µ is strongly topologically mixing on I µ it is easy to deduce that Γ µ is strongly topologically mixing on S µ .
According to [4] the map Γ µ has a unique absolutely continuous and ergodic measure ν with support equal to S µ . This measure ν coincides with ν ×ν, where ν is the unique absolutely continuous and ergodic measure of λ µ . The rest of the proof of the first statement follows in the same way as Proposition 3 in [13].

The two-parameter family of EBMs
From now on, Ω(Q 1 , Q 2 , . . . , Q n ) denotes the polygonal set in R 2 with consecutive vertices Q 1 , Let us recall from [12] the two-parameter family of EBMs and P (see (5)) denotes the set of parameters given by The choice of P guarantees that T is invariant by Ψ a,b for every (a, b) ∈ P.
We shall also consider Then, we can split T in four domains, see Figure 2, As we have seen in [12], Lemma 4.4, for every Ψ a,b in F there exists a unique fixed point P a,b in int(T ) given by Figure 2: The smoothness domains for a map in F.
Let us now define the points As it has been proved in [12], Section 4, for every point (x, y) in the interior of T there exists a natural number n such that Ψ n a,b (x, y) belongs to certain invariant domain R 1 given by, see Figure   3, where

The region P 1
From now on, given two different points Q 1 and Q 2 in R 2 we shall denote by Q 1 Q 2 the straight segment joining Q 1 and Q 2 .
Given (a, b) ∈ P we may compute the image of R 1 by Ψ a,b . It is easy to check that F 1 belongs to T + 1 for every (a, b) ∈ P. Then, Let us define the family of maps Figure 4, Then, (a, b) ∈ P 1 if and only if F 2 belongs to T + 1 . In this case, it is easy to check that ). The aim of this section is to prove that for every (a, b) ∈ P 1 there exists certain domain where Ψ 2 a,b may be conjugate to a direct product of two one-dimensional tent maps. Let us compute It is easy to check that F 3 belongs to T − 1 and we define the point as the intersection between F 2 F 3 and L (b). In this situation, we consider the set, see Figure 5, being Proof. To prove that R a,b is strictly invariant we only need to check that F 3 belongs to the segment KF 2 , where F 2 is the symmetric point of F 2 with respect to L (b).
Since K is the middle point of the segment F 2 F 2 , it is easy to check that the abscise of F 2 is Finally, by comparing x 2 to the abscise of F 3 (see (14)) one may check that F 3 belongs to the segment KF 2 whenever Notice that this inequality holds for every (a, b) ∈ P 1 .
Now, let us note that from Section 4.1 in [12], R 1 captures the orbit of every point in int(T ) and therefore the same holds for R 2 (see (13)). On the other hand, R a,b ∩ T + 1 = R 2 ∩ T + 1 = Ω(F, F 1 , F 2 , K). In order to prove that R a,b captures the orbit of any point of T , it is enough to check that for every point (x, y) ∈ R 2 ∩ T − 1 there exists a natural number such that Ψ n a,b (x, y) ∈ Ω(F, F 1 , F 2 , K). This last claim immediately follows from the fact that, see (9), there is no orbit contained in T − 1 and also that R 2 is invariant.
Lemma 2.5. For every (a, b) ∈ P 1 there exists an affine change in coordinates such that Ψ 2 a,b restricted to R a,b transforms into the map Γ a 2 restricted to S a 2 (see (7) and (8)).
Proof. Let us first note that Ψ a,b displays a fixed point in the boundary of T − 1 given by On the other hand, the preimage of L (b) in T + 1 is given by Let us consider the change in coordinates It is a simple calculation to check that Therefore, ω a,b (R a,b ) = S a 2 (see (8)). Moreover, Let us consider a point (x, y) ∈ Ω(F, F 1 , K −1 , F −1 ). On one hand, and therefore On the other hand, (X, Y ) = ω a,b (x, y) satisfies X ≥ 1 and Y ≥ 1 and, consequently, Then, one may check that ω a,b • Ψ 2 a,b (x, y) = Γ a 2 (X, Y ). Now, let us consider a point (x, y) ∈ Ω(F 2 , K, F −1 , K −1 ) and let be the symmetric point of (x, y) with respect to L −1 . Then, Ψ 2 a,b (x, y) = Ψ 2 a,b ( x, y). Moreover, (X, Y ) = ω a,b (x, y) and ( X, Y ) = ω a,b ( x, y) are symmetric points in S a 2 with respect to Y = 1 and therefore Finally, let us consider a point (x, y) ∈ Ω(F, K, F 3 , K 1 ) and let be the symmetric point of (x, y) with respect to L (b). Then, y) and ( X, Y ) = ω a,b ( x, y) are symmetric points in S a 2 with respect to X = 1 and therefore For each natural number n let us define At this point, we can state the following result, whose proof is an easy consequence of Lemma 2.2, Lemma 2.4 and Lemma 2.5.
Proposition 1. For every n ∈ N and for every (a, b) ∈ P 1,n the map Ψ a,b exhibits 2 n−1 strongly topologically mixing strange attractors.

The region P 2
Let us recall the set R 1 = Ω(C 1 , D 1 , E 1 , F 1 ) introduced in (11). Now, we define the family of maps F 2 = {Ψ a,b ∈ F : (a, b) ∈ P 2 } where, see Figure 6, Let us compute These points respectively belong to T + 1 and T + 0 . Let be the intersection between D 2 D 3 and C. We define the invariant set, see Figure 7, being The aim of this section is to prove that for every (a, b) ∈ P 2 there exists certain domain where Ψ 2 a,b may be conjugate to a direct product of two one-dimensional tent maps. Proof. We only need to prove that D 3 belongs to the segment N D 2 , where D 2 is the symmetric point of D 2 with respect to C.
It is easy to check that the abscise of D 2 is By comparing x 2 to the abscise of D 3 (see (19)) one may check that D 3 belongs to the segment N D 2 for every (a, b) ∈ P 2 .
Remark 2.7. As in the previous case, see Lemma 2.4, we could prove that the set R a,b (see Figure   7) captures any orbit starting from the interior of T . However, the proof is not so easy as the one given before. In any case, we want to stress that this kind of proofs is not necessary in order to ensure the existence of strange attractors for Ψ a,b contained in R a,b .
Lemma 2.8. For every (a, b) ∈ P 2 there exists an affine change in coordinates such that Ψ 2 a,b restricted to R a,b transforms into the map Γ a 2 restricted to S a 2 (see (7) and (8)).
Proof. Let us first note that Ψ a,b displays a fixed point in the boundary of T + 0 given by On the other hand, the preimage of C in T + 1 is given by Let us consider the change in coordinates It is a simple calculation to check that Therefore, τ a,b (R a,b ) = S a 2 (see (8)). Moreover, Let us consider a point (x, y) ∈ Ω(D, D 1 , N −1 , D −1 ). On one hand, and therefore On the other hand, (X, Y ) = τ a,b (x, y) satisfies X ≥ 1 and Y ≥ 1 and, consequently, Then, one may check that τ a,b • Ψ 2 a,b (x, y) = Γ a 2 (X, Y ). be the symmetric point of (x, y) with respect to C. Then, Ψ 2 a,b (x, y) = Ψ 2 a,b ( x, y). Moreover, (X, Y ) = τ a,b (x, y) and ( X, Y ) = τ a,b ( x, y) are symmetric points in S a 2 with respect to X = 1 and therefore For each natural number n let us define As a consequence of Lemma 2.2, Lemma 2.6 and Lemma 2.8 we have the following result.
Proposition 2. For every n ∈ N and for every (a, b) ∈ P 2,n the map Ψ a,b exhibits 2 n−1 strongly topologically mixing strange attractors.

Renormalization scheme. Proof of Theorem A
In [12] the authors studied a one-parameter family of EBMs {Λ t } t (see also (2)). In particular, they proved that there exist three intervals of parameters I 3 ⊂ I 2 ⊂ I 1 such that Λ t is a n times renormalizable EBM for every t ∈ I n , n = 1, 2, 3 (see Main Theorem in [12]). Furthermore, for every t ∈ I n , n = 1, 2, 3, the map Λ t is a n times renormalizable EBM in 2 n−1 different restrictive domains.
In any case, the renormalization scheme requires a change in coordinates in certain triangles ∆ a,b and Π a,b which are invariant for some power of Λ t .
The aim of this section is to extend this renormalization process to the two-parameter family F. To this end, we shall find a set of parameters P 3 ⊂ P such that, if (a, b) ∈ P 3 , then Ψ a,b is renormalizable in F. Namely, for each (a, b) ∈ P 3 , we shall prove that the restriction of Ψ 4 a,b to each one of two different restrictive domains is conjugate by means of an affine change in coordinates to a EBM which belongs to F.

The two restrictive domains
Let us consider Ψ a,b ∈ F and recall that Ψ a,b displays a unique fixed point P a,b = (x a,b , y a,b ) ∈ int(T ) (see (10)). Let us denote by P the symmetric point of P a,b with respect to C. We shall denote by and Π a,b , see Figure 8, given by ∆ a,b = Ω(P a,b , P , Q) , Π a,b = Ω(P a,b , P , H 1 ) .
Let us introduce the change in coordinates In new coordinates, P a,b transforms into O and the distance between P a,b and C is one. Now, the critical segments are given by being Note that γ a,b > 0 and it is easy to check that one may easily calculate its iterates by Ψ a,b . In fact, the action of Ψ a,b on this region consists in a rotation (centered at P a,b ) by an angle π 2 and an expansion by a factor a.
For any point A ∈ R 2 we shall write A 1 = Ψ a,b (A) and, in general, From now on, it will be very useful Figure 9. If we denote by ∆ i = Ψ Since a > 1, the critical line C always cuts the interior of ∆ 1 in points We shall impose that M 1 belongs to T + 1 , i.e., a ≤ γ a,b or, equivalently, Now, We shall impose that H 2 = (a, −a) and M 2 = (a 2 , 0) belong to the regions T − 1 and T + 1 respectively.
These conditions are respectively given by 2a ≥ γ a,b and a 2 ≤ γ a,b or, equivalently, In this case, the critical line L (b) cuts the interior of ∆ 2 at points H 2 and K 2 given by In this way, (Ω(P, H 2 , K 2 , M 2 )) = Ω(P, H 3 , K 3 , M 3 ) .
On the other hand, if b > 2(1+a+a 2 ) a(2+a+a 2 ) then H 4 / ∈ P Q and it is evident that ∆ a,b is not invariant by Ψ 4 a,b . Finally, if b < 2+a 2 +a 3 1+a+a 3 then a 2 > γ a,b and thus M 2 ∈ T − 1 and does not belong to ∆ 4 . Now, we shall prove that Ψ 4 a,b is conjugate to the map Ψ a ,b given in (31).
Let us define the points H and K in ∆ a,b such that Ψ a,b (H) = H 1 and Ψ a,b (K) = K 1 (see Figure   9). In coordinates (X, Y ), these points are given by We may also define the points H and K in ∆ a,b such that Ψ 2 a,b ( H) = H 2 and Ψ 2 a,b ( K) = K 2 .
These points are given by, see Figure 9, in such a way that ∆ 4 = Ψ Remark 3.2. Let us note that, given (a, b) ∈ P ∆ , then (a , b ) ∈ P.
We shall now impose that the point H 4 belongs to the segment P H. This is equivalent to .
Thus, the following result is consequence of the construction of P ∆ and Proposition 3.
For any point A ∈ R 2 we still write A 1 = Ψ a,b (A) and A i = Ψ a,b (A i−1 ).
From now on, it will be very useful Figure 12 As we have done in the previous subsection, let us suppose that M 1 ∈ T + 1 and H 2 ∈ T − 1 .
Equivalently, conditions (26) and (27) hold. We denote by J 1 and H 2 the intersection between the critical line and the lines L(M 1 , 0) and L(P, −1) respectively. In coordinates (X, Y ) these points are given by Now, Let us now impose that Π 2 does not intersect the critical set. This is equivalent to assume that M 2 ∈ T + 1 , i.e., condition (28) holds.
Under this assumption, Let us now impose that Π 3 does not intersect the critical set. This is equivalent to assume that H 4 ∈ P Q, i.e., condition (29) holds. In this case, Finally, in order to determine the set of values where Π 4 = Ψ 4 a,b (Π) ⊂ Π, we need to impose that H 5 ∈ P H 1 , i.e., a 3 γ a,b ≤ 2 or, equivalently, Since a > 1, it is clear that condition (36) implies condition (29). In this way, we define the set of parameters, see Figure 13, Proposition 5. For every (a, b) ∈ P Π the domain Π a,b is invariant by Ψ 4 a,b . Furthermore, for every (a, b) ∈ P Π the map Ψ a,b is renormalizable in Π a,b . Namely, Ψ 4 a,b is conjugate by means of an affine change in coordinates φ a,b to the map being a = a 4 , b = a −1 γ a,b and B a the matrix introduced in (4).
Proof. As a consequence of the construction of P Π , if (a, b) ∈ P Π then Π a,b is invariant by Ψ 4 a,b .
Now, we shall prove that Ψ 4 a,b is conjugate to the map Ψ a ,b given in (38). Let us define the points H 1 and J in Π a,b such that Ψ a,b ( H 1 ) = H 2 and Ψ a,b (J) = J 1 (see Figure   12). In coordinates (X, Y ), these points are given by in such a way that Π 4 = Ψ 4 a,b (Π 0 ) where Π 0 = Ω(P, H 1 , J, M ).
Moreover, since Ψ k a,b (Π 0 ) does not leave the region Remark 3.3. Let us note that, given (a, b) ∈ P Π , then (a , b ) ∈ P.
Remark 3.4. Let us remark that condition (28) is not necessary to guarantee the invariance of Π a,b .
In fact, it holds that Ψ  However, if condition (28) is not satisfied, the map Ψ a,b is not renormalizable in F : the renormalized map is a EBM with three folds.

Simultaneous renormalizations. The region P 3
To begin with, we note that, see (34) and (37), P Π = P ∆ . From now on, let us denote this set of parameters by P 3 . According to Remark 3.2 and Remark 3.3 we may define the maps As a consequence of Proposition 4 and Proposition 5 we have the following result.
Theorem 3.5. For every (a, b) ∈ P 3 the map Ψ a,b is simultaneously renormalizable in F on two different restrictive domains. Namely: i) The restriction of Ψ 4 a,b to ∆ a,b is conjugate by means of an affine change in coordinates to Ψ H∆(a,b) restricted to Ψ H∆(a,b) (T ).
ii) The restriction of Ψ 4 a,b to Π a,b is conjugate by means of an affine change in coordinates to Ψ HΠ(a,b) .
Let us consider the domain of parameters that D is chosen in such a way that the maps, see (39), (40) and (25), are well defined in D.   Proof. The first statement is immediate from the definition of H and the fact that  To prove the last statement we begin by observing that, from the definition of H, it is clear that n∈N H −n (W ) is contained in the fiber V 1 . The restriction of H −1 to this fiber is a contractive map given by H −1 (1, b) = 1, 2(b+1) 2+b whose fixed point is P * . Therefore, n∈N H −n (W ) = {P * } .
In order to obtain the family of curves F announced in the third statement we use the Linearization Theorem of Sternberg [15]. According to this result, there exist a neighborhood U of P * , a neighborhood V of O and a C ∞ -diffeomorphism h : U → V such that for every (a, b) ∈ H −1 (U ), where L is the linear map L(x, y) = (λ 1 x, λ 2 x).
The family F of curves η s given by y = sx ν with ν = log λ2 log λ1 and s ∈ R ∪ {∞} (η ∞ is, by definition, the straight line x = 0) satisfies    Then one may check that P 3 ⊂ H(P 3 ), see Figure 14(b), and hence a chain like the one given in (42) may be defined. Therefore,the family F of EBM can be also renormalized any finite number of times on the restrictive domain Π. using Proposition 2, it is easy to obtain a new interval of parameters I n for which Λ t has at least 2 n strange attractors whenever t ∈ I n .