A formula for the boundary of chaos in the lexicographical scenario and applications to the bifurcation diagram of the standard two parameter family of quadratic increasing-increasing Lorenz maps

The Geometric Lorenz Attractor has been a source of inspiration for many mathematical studies. Most of these studies deal with the two or one dimensional representation of its first return map. A one dimensional scenario (the increasing-increasing one's) can be modeled by the standard two parameter family of contracting Lorenz maps. The dynamics of any member of the standard family can be modeled by a subshift in the Lexicographical model of two symbols. These subshifts can be considered as the maximal invariant set for the shift map in some interval, in the Lexicographical model. For all of these subshifts, the lower extreme of the interval is a minimal sequence and the upper extreme is a maximal sequence. The Lexicographical world (LW) is precisely the set of sequences (lower extreme, upper extreme) of all of these subshifts. In this scenario the topological entropy is a map from LW onto the interval \begin{document}$[0, \log{2}]$\end{document} . The boundary of chaos (that is the boundary of the set of \begin{document}$ (a, b) ∈ LW$\end{document} such that \begin{document}$h_{top}(a, b)>0$\end{document} ) is given by a map \begin{document}$ b = χ(a)$\end{document} , which is defined by a recurrence formula. In the present paper we obtain an explicit formula for the value \begin{document}$χ(a)$\end{document} for \begin{document}$a$\end{document} in a dense set contained in the set of minimal sequences. Moreover, we apply this computation to determine regions of positive topological entropy for the standard quadratic family of contracting increasing-increasing Lorenz maps.


(Communicated by Enrique Pujals)
Abstract. The Geometric Lorenz Attractor has been a source of inspiration for many mathematical studies. Most of these studies deal with the two or one dimensional representation of its first return map. A one dimensional scenario (the increasing-increasing one's) can be modeled by the standard two parameter family of contracting Lorenz maps. The dynamics of any member of the standard family can be modeled by a subshift in the Lexicographical model of two symbols. These subshifts can be considered as the maximal invariant set for the shift map in some interval, in the Lexicographical model. For all of these subshifts, the lower extreme of the interval is a minimal sequence and the upper extreme is a maximal sequence. The Lexicographical world (LW) is precisely the set of sequences (lower extreme, upper extreme) of all of these subshifts. In this scenario the topological entropy is a map from LW onto the interval [0, log 2]. The boundary of chaos (that is the boundary of the set of (a, b) ∈ LW such that htop(a, b) > 0) is given by a map b = χ(a), which is defined by a recurrence formula. In the present paper we obtain an explicit formula for the value χ(a) for a in a dense set contained in the set of minimal sequences. Moreover, we apply this computation to determine regions of positive topological entropy for the standard quadratic family of contracting increasing-increasing Lorenz maps.

1.
Introduction. In a remarkable contribution, around 1963, the meteorologist E.N. Lorenz [23] exhibited numerical evidence for the existence of a strange attractor in a quadratic system of ordinary differential equations in three variables. Some time later, around 1974-1979, Afrajmovich, Bykov and Shilnikov ( [1], [2]) and Guckenheimer, Williams ([10], [11], [31]) proposed the so called geometrical models for the behavior observed by Lorenz. An important feature of these models is the existence of a (partial) cross-section to the flow, as well as a smooth invariant foliation by curves. Using this foliation, one can reduce the dynamics of the flow to that of an interval transformation with a discontinuity. These transformations, generically, divide into two disjoint classes: the expanding ones (those whose derivative, from both sides, at the discontinuity is infinity) and the contracting ones (those whose derivative, from both sides, at the discontinuity is zero). As observed in [2] and [11], there exist uncountably many conjugacy classes of such transformations. In fact the moduli space is essentially 2-dimensional and can be parameterized by the admissible kneading sequences (forward itineraries of the discontinuity).
In view of these results, it is natural to look for a bifurcation theory of these transformations and flows using symbolic dynamics ( [5,15,18,19]). In this direction, de Melo and Martens ( [7]) and Labarca and Moreira ([18]) showed the existence of parameterized families of contracting Lorenz flows that are topologically universal in the sense that given any geometric Lorenz flow, its dynamics is "essentially" the same as the dynamics of some element of the family.
In fact, the two parameter family of quadratic lexicographical Lorenz Maps F µ,ν : (R \ {0}) → R given by is topologically universal. That is, for almost any continuous map g : (R \ {0}) → R which is an increasing-increasing and such that g(0 − ) ≥ 0 and g(0 + ) ≤ 0 there is a parameter value (µ, ν) such that the dynamics of F µ,ν (x) is essentially the dynamics of g(x). Hence, up to topologically semi-conjugacy, the two parameter family of quadratic Lorenz Maps F µ,ν (x) represents almost all the interesting dynamics of increasing-increasing one dimensional maps with one discontinuity. Therefore, it is a very interesting problem to know the bifurcation theory associated to the family F µ,ν (x) (form now and on we will call the family F µ,ν (x) the standard family of quadratic increasing-increasing Lorenz Maps, or by short the standard family).
We recall that an isentrope of the topological entropy is a region of the parameter space (µ, ν) such that h top (F µ,ν ) = h 0 = constant (here h top mean topological entropy). A way to understand the bifurcation theory associated to the standard family, is to understand the structure of the different isentropes of the family. As John Milnor ([26]) pointed out, it is not an easy task (see for instance [6] and [28]).
In the present work we advance a step further in these questions and in the programme established by Labarca and Moreira (see [18]).
To be more specific (for the necessary definitions see section 2): we obtain an explicit formula for the value χ(c) for any c ∈ [a − b + a, a] for any a ∈ A ∞ ∞ and we apply it for the computation of regions of positive entropy for the standard family.
2. Preliminaries. It is well known that one of the purposes of the topological theory of Dynamical Systems is to find universal models describing the topological dynamics of a large class of systems (see for instance [3], [8], [30]).
One of these universal models is the shift on n-symbols σ : Σ n → Σ n where Σ n is the set of sequences {θ : N 0 → {0, 1, 2, . . . , n − 1}} and σ is the shift map defined by (σ(θ))(i) = θ(i + 1). Here Σ n is endowed with a certain topology. This model has been introduced to study one dimensional dynamics, by Metropolis, Stein, Stein at [24] and [25] (actually, for n = 2 with a different, said "naive" presentation) and eventually stated formally by Milnor and Thurston at [27], where the notion of a signed order in the shift space (Σ n , σ) was also defined.
In fact, several signed orders can be defined in Σ n in a different way (as Milnor and Thurston did). Let us doing this here. Let 0 = x 0 < x 1 < x 2 < . . . < x 2n−1 = 1 be 2n points in the unit interval [0, 1]. Let I j = [x 2j , x 2j+1 ] for j = 0, 1, 2, . . . , n − 1; and T : n−1 j=0 I j → [0, 1] be a map such that its restriction to I j is linear and onto [0, 1], for any j = 0, 1, . . . , n − 1. The restriction of the map T to any interval I j can be either orientation preserving or orientation reversing. Hence, we may define 2 n piecewise linear maps T : n−1 j=0 I j → [0, 1] as before. Let Lin(n) denote the set formed by these 2 n maps. Associated to any T ∈ Lin(n) we have its maximal invariant In this set, we consider the topology induced by the euclidean topology of the interval [0, 1].
It is not hard to see that the set Λ T is bijective to Σ n . In fact, the itinerary map where, from now and on, we denote θ = (θ 0 θ 1 θ 2 . . .). Hence, by using the bijective map I T : Λ T → Σ n we can induce in Σ n : Let us denote by Σ n (T ) the ordered, compact topological space (Σ n , τ T , ≤ T ). In this way, we have introduced 2 n of these ordered compact metric spaces. These models has been extensively used to obtain a great amount of information about maps defined in an interval (see for instance [3,6,8,13,15,16,17,18,27,29]); vector fields on three dimensional manifolds (see for instance [5,11,12,14,19,20,32]) among other kinds of dynamical systems.
In the special case of one dimensional dynamics, the shift of two symbols may be used to study increasing (decreasing) maps with one discontinuity like the Lorenz maps, unimodal maps like the quadratic family or increasing-decreasing (decreasingincreasing) maps with one discontinuity. Namely, for n = 2 the ordered metric compact space (Σ 2 , τ T , ≤ T ) corresponding to the increasing-increasing map T is known as the lexicographical space which generates the lexicographical world (see for instance [15,16,17,18,21,22]) which is denoted LW.
In the present work we deal with the lexicographical world. That is, here we consider the set Σ 2 with the topology induced by the map T : I 0 ∪ I 1 → [0, 1] such that T | I0 and T | I1 are increasing maps.
Let a denote the finite string a = a 0 a 1 . . . a n and a be the infinite sequence a = a 0 a 1 . . . a n = a 0 a 1 . . . a n , a 0 a 1 . . . a n , ... · · · . For example, if a = 0011 then a = 0011 = 00110011001100110011 . . .. Let denotes the sets of maximal and minimal sequences in the lexicographical order.
-If a = 0a 1 a 2 . . . a n−1 1 then a − will denote the string a − = 0a 1 a 2 . . . a n−1 0 and -If a 1 , a 2 are two sequences then we define the sequence m(a 1 , a 2 ) by m(a 1 , a 2 ) = a 1 a 2 . For instance for a 1 = 001 and a 2 = 01 we have m(a 1 , a 2 ) = 00101.

Statement of the results.
3.1. The map χ. We regard the definition of the maps ϕ, ψ, χ from [17].
Let A 1 = {m(a 1 , a 2 ), a 1 < a 2 are consecutive sequences in A 0 } ∪ A 0 and A n+1 = {m(a 1 , a 2 ); a 1 < a 2 are consecutive sequences in A n } ∪ A n . So, we have: A n . For a ∈ A ∞ let us define: Now, associated to any a ∈ A 1 ∞ we construct: Similarly, for any n ≥ 2, we may define: Remark 2. There are minimal sequences a ∈ Min 2 such that a / ∈ A ∞ ∞ . For instance: a = 00111, a = 000111, a = 001011011.
Our main results here are Theorems A and B: Theorem 3.6 (Theorem A).

For any
Remark 3. This result was included as part of the PH.D. Thesis of S. Aranzubia ( [4]) at the Departamento de Matemática y Ciencia de la Computación of the Universidad de Santiago de Chile.
Concerning the size of the set A ∞ ∞ in Σ 0 we have: , a] and J(a) = [a − (b(a)) + a, a] we have that: Remark 4. These two results imply that we have an explicit formula for χ(c), for c in a dense set of minimal sequences.
We want to acknowledge to an unknown referee who asked about the size of the set A ∞ ∞ . In the way of answering this question we were forced to formalize the proof of the result in Theorem B.

3.3.
Application of the Theorem A to the standard quadratic family F (µ,ν) . As an application, of our main result, we will prove several results concerning the values of the topological entropy for the standard quadratic family. Initially let us define some special itineraries.
Proposition 2. For any ν > 0 the map associated to the intersection of the curves , t ≥ 0} has entropy zero.
Proposition 3. For any µ > 0 the map associated to the intersection of the curves , t ≥ 0} has entropy zero.
which has a non-empty interior.
Let us now assume that the result is true for any a ∈ A n and let us prove the result for a ∈ A n+1 .
For any a ∈ A n+1 , we have that a ∈ A n or a = (a 1 ) n+1 a 2 or a = (a 1 ) n a 2 (a 1 ) n−1 a 2 or a = (a 1 ) n−1 a 2 (a 1 ) n−1 a 2 (a 1 ) n−2 a 2 or a = (a 1 ) n−1 a 2 (a 1 ) n−2 a 2 (a 1 ) n−2 a 2 , and so on; where a 1 and a 2 are two consecutives periodic sequences in A 0 .
For a ∈ A n the result was already proved. Assume that we have: a = (a 1 ) n+1 a 2 .

Remark 12.
We note that if d ∈ Max 2 satisfies d > b + a and c is a sequence such Also, for any c such that c < a − b + a, we may find a natural number n such that c < α n = a − b + a n < a − b + a and χ(c) ≤ χ(a − b + a n ) = χ(α n ) = (b + a n+1 ) + a − b + a n < b + a = χ(a − b + a). In this case: Λ[a − b + a n , χ(a − b + a n )] ⊂ int (Λ[c, b + a]) and, as a > 0.
In particular, for c and d such that In a similar way, for any c and d such that In fact in [4] was proved that Where #(a)= period of the sequence a.

5.
Proof of the Theorem B. The topology, in the Lexicographical space, is equivalent to the topology induced by the metric d : With this topology the Lexicographical space is compact and complete. Moreover, for a sequence of intervals I n = [α n , β n ] ⊂ Σ 2 such that 1. Proof. Initially, let us assume that α = 0 or α = 01. In this case for a n = 0 n 1 and α n = 01 n , we have that d(a n , 0) → 0, n → ∞ and d(α n , 01) → 0, n → ∞. Therefore 0 ∈ I ∞ and 01 ∈ I ∞ .
Therefore we obtain α ∈ I ∞ as we announced.
In the case 01 n < α < 01 n+1 , we proceed in a similar way. This complete the proof of the proposition 5.1.
We proceed in a similar way for 0011 < α < 001101. Then, we proved the result for the case a = 01. Let us now prove the result in the general case. Let us take any a ∈ A ∞ and let us prove that J ∞ ∩ I(a) is a dense set in I(a) ∩ Min 2 .
We have Hence, let us assume α ∈ I(a)∩Min 2 . Without loss we may assume that a − b < α < a − b + a. Otherwise α ∈ J(a) and we complete the proof.
In this case the interval So, let assume that α / ∈ J(ρ) for any ρ ∈ A ∞ ∞ . In this case let us construct a sequence of interval (I j (a)) ∞ j=1 such that |I j+1 (a)| ≤

Let us consider
So there is n ∈ N such that a − b n b + < α < a − b n−1 b + or a − b + a n−1 < α < a − b + a n Without loss let us assume that a − bb + < α < a − b + or a − b + < α < a − b + a is the case.
In the case a − bb + < α < a − b + we must have: In this case let I 2 (a) = [a − bb + , a − bb + aa − b + ] we have I 2 (a) ⊂ I 1 (a) and In this case let we have I 2 (a) ⊂ I 1 (a) and ∞ . This complete the second step of our inductive construction. Let us proceed, for the sake of completitude, with the third step for our inductive construction.
Let us consider the case I 1 (a) = [a − b, a − b + a] and I 2 (a) = [a − bb + , a − bb + aa − b + ]. So, we assume that a − bb + < α < a − bb + aa − b + is the case.
In this situation we may have: Let us assume that a − bb + < α < (a − b + ) − b + a − is the case. In this situation let us consider the sequence a − bb + a − b + ∈ A ∞ ∞ . We have Assume that a − bb + < α < a − bb + a − b + is the case. In this situation we must have a − bb In this situation let us consider In this situation let us consider Without loss, let us assume that n = 1 and that we have In this situation we must have a − bb + a − b + a < α < (a − bb + a)(b + aa − b + )a − bb + a Hence we must have either In this situation let us consider Let us assume that a − bb + a − b + a < α < a − bb + a − b + aa − bb + a is the case. In this situation we must have Let us assume now that This complete the inductive step in the case a − bb Let us finally, consider the case In this situation let us consider the set We have: Without loss, let us assume that n = 1 and that we have From now on we continue as in the previous case.
So this complete the inductive step, and the proof of the result.
6. Proof of the results on topological entropy for the standard quadratic family of Lorenz maps. In this section we will prove the results established in section 3.3.
The figure 2 represents the graph of the curves in proposition 11.
Proof. Let us remind that the quadratic family is defined by: Figure 9 shows the graph of the map F µ,ν (x).
Is not hard to verify that the curves ν = x n (µ, ν) and µ = 1+ √ 1+4ν 2 transversally intersects at the point (µ n , ν n ) such that I(µ n ) = 0 and I(v n ) = 10 n . The figure 17 shows the graph of the map F (µn,νn) . For F (µn,νn) we have that h top (F µn,νn ) = log(x n ), where x n is the greatest real root of the polynomial: (−1) n+1 (x n+1 −x n−1 −1), we note that: x so x = 1, which is a contradiction with the assumption about the value of the point x. Therefore x = 1.
Lemma 6.3. For any ν > 0 the map associated to the intersection of the curves , t ≥ 0} has entropy zero.
Proof. Let us consider the curve µ = t + 1 + √ 1 + 4ν 2 , t ≥ 0, which is a translation of the curve µ = 1 + √ 1 + 4ν 2 . Let us consider the associated map: Figure 19 shows the graph of the map F µ(t),0 (x) Figure 19. Graph of the map F µ(t),0 (x) We observe that for any t such that 0 ≤ t < ν 2 the corresponding map F µ(t),ν has a graph like in the figure 20.
In this case if we solve the equation we obtain: x = √ t. In this situation the map: , ν] has positive topological entropy. Moreover, for the parameter value ν = √ t, the corresponding map F (µ(t), √ t) has zero topological entropy when restricted to the interval ([y(ν), ν] \ {0}). Figure 21 shows the graph of the map F (µ(t),  We observe that any x such that 0 < x < √ t satisfy: lim Let us now establish the following: Corollary 3. The map F µ,ν , for parameter values (µ, ν) in the curve t + 1 + √ 1 + 4ν 2 , ν with ν > √ t has positive topological entropy.
Proof. The proof is similar to that of lemma 6.3.

Corollary 4.
For parameter values (µ, ν) in the curve µ, t + 1 + √ 1 + 4µ 2 with µ > √ t we have that the corresponding map F (µ,ν) has positive entropy Proof. Similar to that of the corollary 3 Remark 16. The figure 1 shows the picture that we have, up to now, with respect to the topological entropy In what follows let us denote by B(0, 1) or simply B the region of the parameter values (µ, ν) such that (µ, ν) belongs to the region bounded by the curves: The figure 2 represents the region B(0, 1). We observe that this region contains the interesting part of the set H 0 = {(µ, ν) : h top (µ, ν) = 0}. Moreover we can connect the bubble with the part of H 0 which is outside of the bubble through the points (0, 1) and (1, 0). Proof. Similar to the proof of the lemma 6.5 The figure 4 shows region B 2 (0010, 01, 1101, 1) Remark 17. We observe that for parameter values (µ, ν) such that b(µ, ν) = 1101 and 0010 ≤ a(µ, ν) = a < 001101, we have h top (F µ,ν | Λ(µ,ν) ) > 0.
See figure 6 shows the region B 1 ∪ B 2 ∪ B 3 Remark 19.