Bifurcation of the essential dynamics of Lorenz maps on the real line and the bifurcation scenario

for Lorenz like flows: The contracting case. ^{∗}

Rafael Labarca and Carlos Moreira

Dedicated to Professor Jacob Palis on his sixtieth’s birthday

1 Introduction

In a remarkable contribution, 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 Afrajmovich, Bykov and Shilnikov( [1], [2]) and Guckenheimer, Williams( [15], [36], [16]) 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, one can reduce the dynamics of the flow to that of an interval transformation with a discontinuity.These transformation, 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 [16] and [2],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. In this direction, de Melo and Martens showed([8]) 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 the present work we describe the complete bifurcation diagram for an explicit natural choice of such an essentially universal family, namely the quadratic family

F_{μ,ν}(x) = \{ \begin{matrix}−μ + x^{2} , x > 0 \\ ν − x^{2} , x < 0 \end{matrix} \} ,

∗Partially supported by Fondecyt grants # 1970720 and # 1961212, DICYT-USACH- Chile and PRONEX on Dynamical Systems. Brazil

Our geometrical construction has several important consequences.One of them is that axiom A maps are dense in this family (this is also proved in [8]; see [14] for similar result for the logistic family). It also follows that the set of parameters whose corresponding Lorenz map has entropy zero is arc connected( see section 4.6) . Also, the combinatorial bifurcation diagramme allow us to reobtain, in a constructive way, the result of the Melo and Martens on the universality of the quadratic family. Moreover, we completely describe the exceptional combinatorics not realized in the model(a countable set). Also, as lemma 4 in section 3.1 shows,secondary bifurcations do not appear for injective Lorenz maps( as defined in section 2.1)

After this paper had been completed we became aware of the work of St. Pierre( [33])about the bifurcation diagram asociated to periodic kneading sequences, which obtains, in an independent way, some of the results presented here. Also, Hubbard and Sparrow defined, in [18], a set of pair of sequences which model all the topological expanding dynamics exhibited by Lorenz maps.

On the other hand, starting with Arnold ([3]) there has been considerable interest in the bifurcation theory of the canonical family( see for instance [5] and the references there in). The most common approach, based on the notion of rotation number, is very useful when applied to homeomorphisms of the circle, but it seems delicate to apply it to non injective maps . In fact, a bifurcation diagram for the canonical family beyond the injectivity domain is still unknown. The best approximation seems to be the work by Boyland([5]).See also the work of Epstein, Keen and Tresser([10]).

Recently, in [21], we extended the Hubbard-Sparrow model for expanding maps to an universal model for the “essential dynamics” of Lorenz maps and we called it the Lexicographical World (in the sequel denoted by LW ). This model includes the dynamics of all the interpolated maps used by Boyland ([5], page 359 ) in his construction.

It is clear that the structure of the Lexicographical world has to reflect in the bifurcation theory associated to any parameterized family of allowed maps. This is an extremely interesting problem focused for several authors in this and other contexts, see for instance [24],[7],[6],[11], [3], [5] .

This paper is organized as follows : In section 2 we state our results. In section 3 we describe the lexicographical world, in Section 4 we prove our results about the quadratic family F_{μ,ν} , in section 5 we extend our results to more general contracting families and, finally, in section 6 we will relate them to three dimensional geometric vector fields.

2 Statement of the results

2.1 The Set DM_{0}.

In the sequel DM_{0} will denote the set of maps f : (ℝ \ {0}) → ℝ such that:

(1) The restriction maps f|_{(−∞,0)} : (−∞, 0) ↦ ℝ and f|_{(0,∞)} : (0, ∞) → ℝ are continuous and non-decreasing maps.

(2)

f(0^{+}) = lim_{x↓0} f(x) ∈] − ∞, 0]

and

f(0^{−}) = lim_{x↑0} f(x) ∈ [0, ∞[.

An element in DM_{0} will be called \underline{injective} if its restriction to the interval ]f(0^{+}) , f(0^{−})[ is an injective map.

We will say that f ∈ DM_{0} is \underline{increasing} if the restriction maps f|_{(−∞,0)} : (−∞, 0) ↦ ℝ and f|_{(0,∞)} : (0, ∞) → ℝ are increasing.

We call the elements in DM_{0} \underline{Lorenz maps}.

2.2 The Lexicographical Order

Let Σ_{2} denote the set of sequences θ : ℕ → {0, 1} endowed with the topology given by the metric

d(α , β) = ∑_{i=0}^{∞} \frac{\overline{d}(α_{i} , β_{i})}{2^{i}},

where

\overline{d}(α_{i} , β_{i}) = \{ \begin{matrix}0 , α_{i} = β_{i} \\ 1 , α_{i} ≠ β_{i} \end{matrix}

Let σ : Σ_{2} → Σ_{2} be the shift map σ(θ_{0} , θ_{1} , θ_{2} , …) = (θ_{1} , θ_{2} , …) . Let Σ_{0} and Σ_{1} denote the sets {θ ∈ Σ_{2} ; θ_{0} = 0} and {θ ∈ Σ_{2} ; θ_{0} = 1} respectively. It is clear that Σ_{2} = Σ_{0} ∪ Σ_{1} .

In Σ_{2} we consider the \underline{lexicographical order}: θ < α for any θ ∈ Σ_{0} and α ∈ Σ_{1} or θ < α if there is n ∈ ℕ such that θ_{i} = α_{i} for i = 0 , 1 , 2 , …, n − 1 and θ_{n} = 0 and α_{n} = 1 .

For a ≤ b in Σ_{2} let [a , b] denote the interval {θ ∈ Σ_{2}| a ≤ θ ≤ b }. Σ_{a ,b} will denote the set ⋂_{n=0}^{∞} σ^{−n}([a , b]) .

2.3 The Set Σ_{a_{f},b_{f}}.

For f ∈ DM_{0} let Γ_{f} = (ℝ \ ⋃_{j=0}^{∞} f^{−j}(0)) denote the set of “continuity” of the map f.

For x ∈ Γ_{f} we define I_{f}(x) ∈ Σ_{2} by

I_{f}(x)(i) = 0 if f^{i}(x) < 0 and I_{f}(x)(i) = 1 if f^{i}(x) > 0.

For x = 0 we define:

I_{f}(0^{+}) = lim_{x↓0 ,x∈Γ_{f}} I_{f}(x)

and

I_{f}(0^{−}) = lim_{x↑0 ,x∈Γ_{f}} I_{f}(x).

In the same way to any x ∈ ⋃_{j=0}^{∞} f^{−j}(0) such that f^{i}(x) ≠ 0, 0 ≤ i < n; f^{n}(x) = 0 we associate the sequences :

I_{f}(x^{+}) = (I_{f}(x)(0), …, I_{f}(x)(n − 1), I_{f}(0^{+}))

and

I_{f}(x^{−}) = (I_{f}(x)(0), …, I_{f}(x)(n − 1), I_{f}(0^{−})).

For x ∈ Γ_{f} we define I_{f}(x^{+}) = I_{f}(x^{−}) = I_{f}(x).

Let I_{f} = {I_{f}(x^{+}); x ∈ [f(0^{+}), f(0^{−})[} ∪ {I_{f}(x^{−}); x ∈]f(0^{+}), f(0^{−})]}.

Clearly σ(I_{f}) ⊂ I_{f}. Let us denote a_{f} = I_{f}((f(0^{+}))^{+}) and b_{f} = I_{f}((f(0^{−}))^{−})

Lemma 1 ([20]) I_{f} = ⋂_{n=0}^{∞} σ^{−n}([a_{f}, b_{f}]) = Σ_{a_{f} ,b_{f}}.

We observe that associated to any f ∈ DM_{0} we can define a continuous map

h : [f(0^{+}) , f(0^{−})] ∩ Γ_{f} → Σ_{a_{f} ,b_{f}} ⊂ Σ_{2},

such that h∘f = σ∘h. The map h is given by h(x) = I_{f}(x)and collapses intervals into points. This map cannot be extended, continuously, to the set ∪_{i=0}^{∞}f^{−i}(0). There are two kinds of intervals that the map h can collapses: The wandering intervals and the intervals that are contained in the stable manifold of periodic sinks. An interval I ⊂ [f(0^{+}) , f(0^{−})] is called a \underline{wandering interval} , for the map f, if for any x ∈ I we have that x is a wandering point. We will call a point x a \underline{nonwandering} point if for any neighborhood U_{x} of x and any positive integer N we can find n ≥ N such that f^{n}(U_{x}) ∩ U_{x} ≠ ∅. The set of nonwandering points of the map f is denoted by Ω_{f} . A point x ∉ Ω_{f} is called a \underline{wandering point}. Given any interval ,I, \underline{the orbit} of this interval is the sequence of iterations (f^{n}(I) , n ∈ ℕ). Concerning the existence of wandering intervals we have the following:

Lemma 2 ([21]) Let {φ_{λ} , λ ∈ ℝ} ⊂ DM_{0} be a one parameter family of C^{2} increasing maps such that for each λ there are sequences λ_{n} → λ and μ_{n} → λ with φ_{λ_{n}}(x) > φ_{λ}(x) and φ_{μ_{n}}(x) < φ_{λ}(x) , ∀x then there is a residual set of parameters λ for which φ_{λ} has no wandering intervals.

We observe that our two parameter family of contracting maps satisfy this property.

Definition 1 Given f, g ∈ DM_{0}. We will say that f has \underline{essentially} the same dynamics as g if I_{f} = I_{g}.

We note that in this situation, up to the existence of some intervals where the itineraries of the points are the same, the dynamics of the maps f and g are topologically equivalent(see [9]).

2.4 The Lexicographical World

Let Min_{2} = {a ∈ Σ_{0} ; σ^{k}(a) ≥ a , k ∈ ℕ } and Max_{2} = {b ∈ Σ_{1} ; σ^{k}(b) ≤ b , k ∈ ℕ }.

Definition 2 The set LW = {(a , b) ∈ Min_{2}×Max_{2} ; {a , b} ⊂ Σ_{a ,b}} will be called the \underline{lexicographical world}.

For a ∈ Min_{2} its LW -fiber is the set LW_{0}(a) = {b ∈ Max_{2} ; (a , b) ∈ LW }. For b ∈ Max_{2} its LW -fiber is the set LW_{1}(b) = {a ∈ Min_{2} ; (a , b) ∈ LW }.

Remark 1 It is clear that given (a , b) ∈ LW then Σ_{a,b} ≠ ∅.

Let us now consider (a, b) ∈ LW.

Lemma 3 There is f ∈ DM_{0} such that I_{f} = Σ_{a,b}.

Proof: Let us consider the map g : (ℝ \ {0}) → ℝ given by

g(x) = \{ \begin{matrix}2x − 1, & x > 0 \\ 2x + 1, & x < 0 \end{matrix}

In this case I_{g} = Σ_{2}. Let x_{a} < 0 and x_{b} > 0 be the points such that I_{g}(x_{a}^{+}) = a, I_{g}(x_{b}^{−}) = b.

Let f : (ℝ \ {0}) → ℝ be the map defined by:

f(x) = \{ \begin{matrix}g(x), & x ≤ \overline{x_{b}} \\ x_{b}, & \overline{x_{b}} ≤ x < 0 \\ x_{a}, & 0 < x ≤ \overline{x_{a}} \\ g(x), & \overline{x_{a}} ≤ x \end{matrix}

here \overline{x_{b}} < 0 < \overline{x_{a}} are given by g(\overline{x_{b}}) = x_{b} and g(\overline{x_{a}}) = x_{a}.

We will call this result the \underline{realization lemma}. It is clear that the maps, in this construction, are non-decreasing but we want a realization lemma with increasing maps. In section 3 we prove a realization lemma with increasing maps.

Therefore, we have a surjective map I : DM_{0} → LW, I(f) = (a_{f}, b_{f}) and DM_{0} = ⋃_{(a,b)∈LW} I^{−1}({(a, b)}). Also, using C^{0} proximity on compact sets, this map is continuous.

In this context the next definition is natural.

Definition 3 Let α : U ⊂ ℝ^{k} → DM_{0} be a map.

1) We will say that α is an \underline{a-surjective family} if ∀a ∈ Min_{2} there is a nonempty set , A(a) ⊂ U , such that a_{f} = a , ∀f ∈ α(A(a)) .

2) We will say that α is an \underline{b-surjective family} if ∀b ∈ Max_{2} there is a nonempty set , B(b) ⊂ U , such that b_{f} = b , ∀f ∈ α(B(b)) .

3) We will say that α is an \underline{LW -surjective family} if ∀ (a, b) ∈ LW there exists a nonempty set A(a, b) ⊂ U such that I ∘ α(A(a, b)) = (a, b).

It is clear that associated to any map α, as above, we have an a(b , LW )- decomposition of its domain. We will call this a(resp. b , LW )-decomposition the a(resp. b , LW ) bifurcation theory defined by α.

Open Problem: Are there LW -surjective families?.

Certainly, this is a very hard problem and we believe that there is not a finite k ∈ ℕ with this property. In the present paper we prove the following

Theorem 1 The given two parameter family of quadratic contracting Lorenz maps is a-surjective, is b-surjective but it is not LW -surjective. Moreover, the respective a , b , LW -bifurcation diagrammes are given.

3 Symbolic Dynamics and the Lexicographical World

Here we introduce some results and notations that are necessary for our results. The results which we do not prove in this section are proved in [21]. In the sequel we will denote by \underline{a} = \underline{a_{0}, a_{1}, ...a_{n}} the periodic sequence (a_{0}, a_{1}, ..., a_{n}, a_{0}, a_{1}, ..., a_{n}, ....) ∈ Σ_{2}. Assume that we have a countable set of sequences, A. We will say that two elements a < b ∈ A are consecutive in A( or consecutive if the context is clear) if there not exist an element c ∈ A such that a < c < b

3.1 Dynamical properties for sequences in LW

Let \underline{a_{1}} ≤ \underline{a_{2}} be two periodic sequences in Σ_{0}. The periodic sequence m(a_{1} , a_{2}) = \underline{a_{1} a_{2}} will be called the \underline{average} of the sequences \underline{a_{1}} and \underline{a_{2}}.

\underline{Example}: For \underline{a_{1}} = \underline{01} , \underline{a_{2}} = \underline{011} we have m(a_{1} , a_{2}) = \underline{01011} .

Let A_{0} = \{\underline{0_{n}1} , \underline{01_{m}} ; n, m ∈ ℕ \ \{0\} \} and A_{n+1} = A_{n} ∪ \{ m(a_{1} , a_{2}); \underline{a_{1}} , \underline{a_{2}} ∈ A_{n} are consecutive sequences }. Set A_{∞} = ⋃_{n=0}^{∞} A_{n}. The elements in A_{∞} will be called \underline{primary sequences}. As we will see in the next section, primary sequences are associated with primary bifurcations.

The elements in \overline{A_{∞}} are characterized by the following property:

Lemma 4 a ∈ \overline{A_{∞}} if and only if

(∗) a ∈ Min_{2} and σ(a) ≥ σ(b),

for b = sup{σ^{k}(a) ; k ∈ ℕ}.

To prove this lemma, let us define the renormalization map R_{a,b} : Σ_{2} → Σ(a, b) , by R_{a,b}(a_{0}, a_{1}, ⋯) = (ã_{0}, ã_{1}, ⋯) where ã_{i} = a if a_{i} = 0 and ã_{i} = b if a_{i} = 1 . Here Σ(a, b) = {θ : ℕ → {a, b}}. We have Proposition 1 Assume a ∈ Σ_{0} satisfy (∗) then R_{0,01}(a) satisfy (∗) in Σ_{0,01} .

Proof: It is easy to see that 1.- α ≤ β imply R_{0,01}(α) ≤ R_{0,01}(β) and 2.- σ_{0,01} ∘ R_{0,01}(α) = R_{0,01}(σ(α)) , any α ∈ Σ_{2}. Now it is clear that if σ^{i}(a) ≥ a then R_{0,01}∘ σ^{i}(a) ≥ R_{0,01}(a) and σ_{0,01}^{i}(R_{0,01}(a)) ≥

R_{0,01}(a) any i ∈ ℕ.

If σ(a) ≥ σ(b) then R_{0,01}(σ(a)) ≥ R_{0,01}(σ(b)) and σ_{0,01}(R_{0,01}(a)) ≥ σ_{0,01}(R_{0,01}(b)). Clearly; R_{0,01}(b) = sup{σ_{0,01}^{k}(R_{0,01}(a)); k ∈ ℕ}. So we have the result announced

in the proposition. \underline{Proof of the Lemma} Assume a satisfy (*) and a = 00 · · ·.

Let b = sup{σ^{k}(a); k ∈ ℕ}. In this situation we cannot have b = (11 ⋯). In fact, otherwise σ(b) = (1 ⋯) > σ(a) = (0, ⋯), a contradiction. Since σ^{k}(b) ≤ b, any k ∈ ℕ, we have that b cannot have two consecutive 1′s

between its symbols . Therefore in this situation we have:

a = 0^{k_{1}}(01)^{k_{2}}0^{k_{3}}(01)^{k_{4}} ⋯ = R_{0,01}(0^{k_{1}}1^{k_{2}}0^{k_{3}}1^{k_{4}} ⋯) = R_{0,01}(ã)

and ã, clearly, satisfy (*). Similarly , if b = (11 ⋯) and a satisfy (*) then a = 01 ⋯ and b = R_{10,1}(b̃) some

b̃ such that ã = inf {σ^{k}(˜(b), k ∈ ℕ} satisfy (*). Since, the periodic sequences that satisfy (*) are obtained as

a = inf \{σ^{k} (R_{0,01}^{i_{1}} ∘ R_{10,1}^{i_{2}} ∘ ⋯ ∘ R_{10,1}^{i_{j}}(\underline{0})); k ∈ ℕ\} or
a = inf {σ^{k} (R_{0,01}^{i_{1}} ∘ R_{10,1}^{i_{2}} ∘ ⋯ ∘ R_{0,01}^{i_{j}}(1)); k ∈ ℕ} or
a = inf \{σ^{k} (R_{10,1}^{i_{1}} ∘ R_{0,01}^{i_{2}} ∘ ⋯ ∘ R_{10,1}^{i_{j}}(\underline{0})); k ∈ ℕ\} or
a = inf \{σ^{k} (R_{10,1}^{i_{1}} ∘ ⋯ ∘ R_{0,01}^{i_{j}}(\underline{1})); k ∈ ℕ\}

we get the result □. Let us denote A_{∞}^{0} = A_{∞} and define, for any a ∈ A_{∞}^{0} the set:

A_{1}^{1}(a) = \{c ∈ Σ_{0} ; c = \underline{a_{−}b_{+}a^{n}} or c = \underline{a_{−}b^{m}b_{+}} , for n , m ∈ ℕ\}.

Here for b = \underline{b_{0}b_{1} … b_{k}0} we have b_{+} = b_{0}b_{1} … b_{k}1 .

Observe that if c_{n} = \underline{a_{−}b_{+}a^{n}} then c_{n} → a_{−}b_{+}\underline{a} and if d_{m} = \underline{a_{−}b^{m}b_{+}} then d_{m} → a_{−}\underline{b}.

Set

A_{2}^{1}(a) = {m(a_{1} , a_{2}) ; a_{1} , a_{2} ∈ A_{1}^{1}(a)are consecutive sequences} ∪ A_{1}^{1}(a)

and in general

A_{n+1}^{1}(a) = {m(a_{1} , a_{2}) ; a_{1} , a_{2} ∈ A_{n}^{1}(a)are consecutive sequences} ∪ A_{n}^{1}(a) , n ≥ 2.

Let A_{∞}^{1}(a) = ⋃_{n=1}^{n=∞} A_{n}^{1}(a) and A_{∞}^{1} = A_{∞}^{0} ∪ ⋃_{a∈A_{∞}^{0}} A_{∞}^{1}(a)

At this stage we have to point out the following: Assume a ∈ A_{∞}^{0} . Let ã = sup{σ^{k}(a); k ∈ ℕ and σ^{k}(a) ∈ Σ_{0}} and b̃ = inf {σ^{k}(a); k ∈ ℕ and σ^{k}(a) ∈ Σ_{1}}. Consider Σ_{ã,b̃} = {θ, ℕ → {ã, b̃}} be the set of sequences of the two symbols ã and b̃. Replace 0 = ã and b̃ = 1 and define A_{0}(a) = \{\underline{0_{n}1}, \underline{01_{m}}; n, m ∈ ℕ \ \{0\}\} and A_{n+1}(a) = A_{n}(a) ∪ {m(a_{1}, a_{2}); a_{1}, a_{2} ∈ A_{n}(a) are consecutive sequences}. Set

A_{∞}(a) = ⋃_{n=0}^{∞} A_{n}(a) and A_{∞}^{⋆}(a) = {inf{σ^{k}(α); k ∈ ℕ}; α = R_{ã,b̃}(η) ; η ∈ A_{∞}(a)}.

\underline{Lemma 2} Ā_{∞}^{⋆}(a) = Ā_{∞}^{1}(a).

As before, let σ_{ã,b̃} : Σ_{ã,b̃} → Σ_{ã,b̃} be the shift map. We have

\underline{Lemma 3} α ∈ Ā_{∞}^{⋆}(a) if and only if

(⋆) α ∈ Min_{2}(ã, b̃) and σ_{ã,b̃}(α) ≥ σ_{ã,b̃}(β) for β = sup{σ_{ã,b̃}^{k}(α); k ∈ ℕ}

Inductively, for any a ∈ A_{∞}^{n}(a) ,let

A_{1}^{n+1}(a) = \{c ∈ Σ_{0} ; c = \underline{a_{−}b_{+}a^{j}}or c = \underline{a_{−}b^{k}b_{+}} for j, k ∈ ℕ \}

Now , we define

A_{2}^{n+1}(a) = {m(a_{1} , a_{2}) ; a_{1} , a_{2} ∈ A_{1}^{n+1}(a) are consecutive sequences} ⋃ A_{1}^{n+1}(a)

and

A_{m+1}^{n+1}(a) = {m(a_{1} , a_{2}) ; a_{1} , a_{2} ∈ A_{m}^{n+1}(a) are consecutive sequences} ⋃ A_{m}^{n+1}(a) , m ≥ 2.

As before, define A_{∞}^{n+1}(a) = ⋃_{m=1}^{∞} A_{m}^{n+1}(a) ; A_{∞}^{n+1} = ⋃_{a∈A_{∞}^{n}} A_{∞}^{n+1}(a)∪A_{∞}^{n} and finally, A_{∞}^{∞} = ⋃_{j=0}^{∞} A_{∞}^{j}.

\underline{Note} A similar construction; as we did in Lemma 2 and Lemma 3, for a ∈ A_{∞}^{∘}, we can do for any a ∈ A_{∞}^{∞}.

The elements in (A_{∞}^{∞} \ A_{∞}^{0}) will be called \underline{secondary sequences}. As we will see in the next section secondary sequences are associated with secondary bifurcations.

Let denote by B_{∞}^{∞} the set {sup{σ^{k}(a) , k ∈ ℕ} , a ∈ A_{∞}^{∞} }. We will denote by b(a) the sequence sup{σ^{k}(a) , k ∈ ℕ } for a ∈ Σ_{0} and by a(b) , the sequence inf{σ^{k}(b) , k ∈ ℕ } for b ∈ Σ_{1} . Clearly, b(a) ∈ Max_{2} and a(b) ∈ Min_{2} .

It is clear that Σ_{a ,\underline{1}} ≠ ϕ for any a ∈ Σ_{0} . Hence we can define maps φ , ψ , χ : Σ_{0} → Σ_{1} by :

φ(a) = inf{b ∈ Σ_{1}| Σ_{a ,b} ≠ ϕ },
ψ(a) = inf{b ∈ Σ_{1} ; Σ_{a ,b} contains ∞ -elements}

and

χ(a) = inf{b ∈ Σ_{1} ; Σ_{a ,b} is uncountable}.

Clearly, a_{1} ≤ a_{2} imply φ(a_{1}) ≤ φ(a_{2}) , ψ(a_{1}) ≤ ψ(a_{2}) and χ(a_{1}) ≤ χ(a_{2}) and for all c ∈ Σ_{1} such that c < φ(a) we have Σ_{a ,c} = Φ.

\underline{Examples} : For any 0\underline{01} ≤ a ≤ \underline{01} we have φ(a) = \underline{10} , ψ(\underline{01}) = χ(\underline{01}) = 1\underline{10}. Also φ(\underline{0}) = ψ(\underline{0}) = χ(\underline{0}) = 1\underline{0} ; φ(0\underline{1}) = ψ(0\underline{1}) = \underline{1} ; φ(\underline{0_{n}1}) = \underline{10_{n}} , ψ(\underline{0_{n}1}) = χ(\underline{0_{n}1}) = 1\underline{10_{n}} and φ(\underline{01_{m}}) = \underline{1_{m}0} , ψ(\underline{01_{m}}) = χ(\underline{01_{m}}) = \underline{11_{m}0}.

3.2 The Morse-Smale and the Entropy Zero cases

Definition 4 a.- We will call a map f ∈ DM_{0} \underline{Morse-Smale} if a_{f} ∈ A_{∞}^{0} and b_{f} = φ(a_{f}).

b.- We will call a map f ∈ DM_{0} \underline{essentially Morse-Smale} if a_{f} ∈ A_{∞}^{∞} and b_{f} = φ(a_{f}).

We will denote by MS_{0} ⊂ DM_{0} the set of Morse-Smale and essentially Morse-Smale maps . We call these maps Morse-Smale because its dynamics essentially reduces to a periodic orbit.

Lemma 5 ([21]) Given f ∈ DM_{0} be a “Morse-Smale” map we have that [f(0^{+}), f(0^{−})] = ⋃_{i=0}^{per(a_{f})−1} \overline{I_{i}}, where x ∈ I_{0} implies I_{f}(x) = a_{f} and I_{f}|_{I_{j}} is constant and equal to a_{j} = σ^{j}(a_{f}) for 0 < j ≤ per(a_{f}) − 1.

Lemma 6 Let a ∈ A_{∞}^{0}. There is an injective f ∈ DM_{0} such that a_{f} = a and b_{f} = φ(a).Moreover, this map satisfy f(f(0^{+})) = f(f(1^{−})).

Proof: Let a = a_{0} < a_{1} < … < (alast = a_{p}) < 0\underline{1} < 1\underline{0} < (bfirst = b_{p+1}) < b_{p+2} < … < b_{k} = b(a), denote the consecutive elements in {σ^{j}(a); j ∈ ℕ}. Let ℓ < 0 < r and I = [ℓ, r]. Consider closed intervals L_{0}, L_{1}, …, L_{k} such that ⋃_{i=0}^{k} L_{i} = [ℓ, r]; left(L_{0}) = ℓ, right(L_{0}) = left(L_{1}), …, right(L_{p}) = 0 = left(L_{p+1}), …, right(L_{k−1}) = left(L_{k}); right(L_{k}) = r. Here left[x, y] = x and right[x, y] = y.

Now, define an injective map f that sends L_{p+1} → L_{0}, L_{p+2} → L_{1}, …, L_{k} → L_{k−p−1} and L_{0} → L_{k−p}, L_{1} → L_{k−p+1}, …, L_{p} → L_{k}. This map satisfy the lemma. □

The next result follows immediately from this lemma:

Corollary 1 Let a ∈ \overline{A_{∞}^{0}} \ A_{∞}^{0} . There is an injective map f ∈ DM_{0} such that a_{f} = a and b_{f} = φ(a) □.

We note that these maps can be considered as bijective maps on the circle.

Lemma 7 Let a ∈ A_{∞}^{∞}. There is f ∈ DM_{0} an increasing map, such that a_{f} = a and φ(a) = b_{f} .

Proof: Let

a = a_{0} < a_{1} < … < a_{p} < 0\underline{1} < 1\underline{0} < a_{p+1} < … < a_{k}.

denote the consecutive elements in {σ^{j}(a) , j ∈ ℕ}. Associated with a_{s} , 0 ≤ s ≤ k ,consider closed intervals L_{s} , s = 0, … k , such that

L_{0} ≤ L_{1} ≤ … ≤ L_{p} ≤ L_{p+1} ≤ … ≤ L_{k},

L_{i} ∩ L_{i+1} = one point or L_{i} ∩ L_{i+1} = ∅, otherwise.

Let η : {0 , 1 , …, k} → {0 , 1 , …, k} be the permutation defined by σ(a_{j}) = a_{η(j)} . Clearly, η(p) = k and η(p + 1) = 0 . Now, define an increasing map f ∈ DM_{0} which satisfies f(L_{i}) = L_{η(i)} and is increasing between L_{i} and L_{i+1} when L_{i} ∩ L_{i+1} = ∅ □

In a similar way we obtain

Lemma 8 Associated to any a ∈ Min_{2} there is an increasing map f ∈ DM_{0} such that a_{f} = a and b_{f} = φ(a_{f}) .

In general we get

Lemma 9 For a ∈ A_{∞}^{∞} and d ∈ B_{∞}^{∞} ∩ LW_{0}(a) . There is an increasing map f ∈ DM_{0} such that a_{f} = a , b_{f} = d.

As a consequence of these lemmas we have

Proposition 2 Given (a, b) ∈ LW there is an increasing map f ∈ DM_{0} such that I(f) = (a_{f} , b_{f}) = (a, b) .

This result is a generalization of a similar result obtained for expansive maps in [18].

Definition 5 We will call a map f ∈ DM_{0} an \underline{entropy zero} map if b_{f} ≤ χ(a_{f}) .

One of the most interesting problems related with the bifurcation theory associated to a parameterized family of dynamical systems {f_{λ}; λ ∈ U ⊂ ℝ^{k} }, is to describe the set {λ ∈ U ; f_{λ} is an entropy zero map } (see for instance [26], [4] and the references there in ). For our quadratic family of Lorenz maps we will prove some results, in this direction, in section 4.6.

We observe that any Morse-Smale or essentially Morse-Smale map is an entropy zero map.

4 Proof of the Results.

4.1 The Contracting Family.

Let us consider the two parameter family F_{(λ,μ)} : (ℝ \ {0}) → ℝ, μ ≥ 0 , ν ≥ 0 , of maps in DM_{0}

F_{μ,ν}(x) = \{ \begin{matrix}−μ + x^{2} , x > 0 \\ ν − x^{2} , x < 0 \end{matrix}

In this section we will provide the bifurcation theory associated to this family.

4.2 Primary Bifurcations.

We call the bifurcations in this section \underline{primary bifurcations} because they appears as bifurcation curves for injective maps in DM_{0} and can be studied by using rotation numbers( see for instance [5], [3], [13] and [17]).

4.2.1 Fixed points:

The fixed points of the map F_{(μ,ν)} are given by

(a) y_{±}(ν) = \frac{−1±\sqrt{1+4ν}}{2} defined for ν ≥ −1/4 and

(b) x_{±}(μ) = \frac{1±\sqrt{1+4μ}}{3} defined for μ ≥ −1/4.

We have y_{−}(ν) ≤ y_{+}(ν) ≤ 0 ≤ x_{−}(μ) ≤ x_{+}(μ).

4.2.2 Preimages of 0

(a) ν − x^{2} = 0 imply x = −\sqrt{ν} = y_{1}(ν), ν ≥ 0

ν − x^{2} = −\sqrt{ν} imply x = −\sqrt{ν + \sqrt{ν}} = y_{2}(ν)

ν − x^{2} = y_{n}(ν) imply x = −\sqrt{ν − y_{n}(ν)} = y_{n+1}(ν), n ≥ 2;

(b) −μ + x^{2} = 0 imply x = \sqrt{μ} = x_{1}(μ), μ ≥ 0

−μ + x^{2} = \sqrt{μ} imply x = \sqrt{μ + \sqrt{μ}} = x_{2}(μ)

−μ + x^{2} = y_{n}(μ) imply x = \sqrt{μ + x_{n}(μ)} = x_{n+1} , n ≥ 2.

4.2.3 Curves of bifurcation (first generation)

(a) F (0^{−}) = fixed point of the right hand side

ν = \frac{1 ± \sqrt{1 + 4μ}}{2}

(b) F (0^{+}) = fixed point of the left hand side

−μ = \frac{−1 ± \sqrt{1 + 4ν}}{2}

Figure 1:

We note(see figure 2 and 3) that:

(μ, ν) ∈ ℝ^{2} such that ν ≥ \frac{1 + \sqrt{1 + 4μ}}{2}, μ ≥ 0 imply b(F_{(μ,ν)}) = \underline{1}

Figure 2:

Figure 3:

(μ, ν) ∈ ℝ^{2} such that μ ≥ \frac{1 + \sqrt{1 + 4ν}}{2} , ν ≥ 0 imply a(F_{(μ,ν)}) = \underline{0}

Given a ∈ Min_{2} and b ∈ Max_{2} let denote by R_{b} = {(μ, ν) ∈ ℝ_{0}^{+} × ℝ_{0}^{+}; b(F_{(μ,ν)}) = b}

L_{a} = {(μ, ν) ∈ ℝ_{0}^{+} × ℝ_{0}^{+}; a(F_{(μ,ν)}) = a}

(c) F (0^{+}) = n-th preimage of zero = y_{n}(ν). These equation define the curves μ_{n}(ν) = −y_{n}(ν) that satisfies:

(μ, ν) ∈ \{(μ_{n}(ν); ν); ν ≥ 0\} imply a(F_{(μ,ν)}) = \underline{0_{n}1}.

These curves converges to the curve (μ(ν) , ν) (see figure 4) where μ(ν) is given by

μ(ν) = \frac{1 + \sqrt{1 + 4ν}}{2} , ν ≥ 0 .

Figure 4:

In fact, we note that μ_{1}(ν) = \sqrt{ν} , μ_{2}(ν) = −y_{2}(ν) = \sqrt{ν + \sqrt{ν}} , …, μ_{n+1}(ν) = −y_{n+1}(ν) = \sqrt{ν + μ_{n}(ν)} .

Hence, taking the limit when n → ∞, we get μ(ν) = \sqrt{ν + μ(ν)} , so μ^{2} = ν + μ and, consequently, μ(ν) = \frac{1 + \sqrt{1 + 4ν}}{2} as announced.

We also observe that the curves (μ_{n}(ν), , ν) transversally intersect the curve

(μ , \frac{1 + \sqrt{1 + 4μ}}{2} ) , any n ∈ ℕ.

(d) F (0^{−}) = n-th preimage of zero = x_{n}(μ).

These equation define the curves ν_{n}(μ) = x_{n}(μ) that satisfies:

(μ, ν) ∈ \{(μ, ν_{n}(μ)); μ ≥ 0\} imply b(F_{(μ,ν)}) = \underline{1_{n}0}.

These curves converges to the curve (μ , ν(μ)) (see figure 5 ) where ν(μ) given by

ν(μ) = \frac{1 + \sqrt{1 + 4μ}}{2} , μ ≥ 0 .

Figure 5:

Let us now prove the following Proposition 3 The curve C\underline{10} = \{(μ , ν_{1}(μ)) , μ ∈]0 , ∞[ \} transversally intersect the curve C\underline{0_{n}1} = \{μ_{n}(ν) , ν) , ν ≥ 0 \} , ∀n ∈ ℕ.

Proof: For C\underline{10} we have C\underline{10} = \{(μ, \sqrt{μ)}; μ > 0\}. Hence C\underline{10} is described by μ(ν) = ν^{2} , ν > 0.

Now C\underline{01} is described as μ_{1}(ν) = \sqrt{ν}. We have C\underline{01} ∩ C\underline{10} = \{(1, 1)\}. and μ1^{′}(ν) =

\frac{1}{2\sqrt{ν}}; (ν^{2})′ = 2ν. So μ_{1}(ν) is transversal to μ(ν) = ν^{2} at ν = 1. Note μ1^{′}(ν) ≤ \frac{1}{2}

and μ′(ν) ≥ 2, all ν ≥ 1 = ν_{1}.

Inductively, assume that C\underline{10} transversally intersect C\underline{0n1} at a point (ν_{n}^{2}, ν_{n}) with

μn^{′}(ν) ≤ 1/2 and μ′(ν) ≥ 2, all ν ≥ ν_{n} ≥ ν_{n−1} .

For the curve C\underline{0_{n+1}1} we have μ_{n+1}(ν) = \sqrt{ν + μ_{n}(ν)} and μn+1^{′}(ν) = \frac{1 + μ′_{n}(ν)}{2μ_{n+1}(ν)}

Let (ν_{n+1}^{2}, ν_{n+1}) ∈ C\underline{0_{n+1}1} ∩ C\underline{10} . We have μ′(ν) = 2ν ≥ 2ν_{n+1} > 2ν_{n} ≥ 2ν_{n−1} ≥

⋯ ≥ 2 and μn+1^{′}(ν) = \frac{1 + μn^{′}(ν)}{2μ_{n+1}(ν)} ≤ \frac{3/2}{4} < 1/2, any ν ≥ ν_{n+1} > ν_{n}. So, we get the

result. □

Let us now prove

Proposition 4 The curve C\underline{1_{m}0} = \{(μ , ν_{m}(μ)) ; μ ∈]0 , ∞[ \} transversally intersect the curve C\underline{0_{n}1} , ∀n ∈ ℕ.

Proof: For m = 2 we have ν_{2}(μ) = \sqrt{μ + \sqrt{μ}} then ν2^{′}(μ) = \frac{1 + \frac{1}{2\sqrt{μ}}}{2ν_{2}(μ)} =

\frac{1 + ν1^{′}(μ)}{2ν_{2}(μ)} .

Since \frac{1 + \frac{1}{2\sqrt{μ}}}{2ν_{2}(μ)} < 1 we get ν′_{2}(μ) < 1 . Let denote by μ̃_{2}(ν) its inverse map.

We have μ̃_{2}′(ν) > 1 . Since μ′_{n}(ν) < 1 we obtain the result in this case.

Inductively, assume ν′_{m}(μ) < 1 and that {(μ , ν_{m}(μ)) } transversally intersect the curve C\underline{0_{n}1}.

Since νm+1^{′}(μ) = \frac{1 + νm^{′}(μ)}{2ν_{m+1}(μ)} < \frac{1 + 1}{2ν_{m+1}(μ)} < 1, we get the result. □

(e) Let us now assume that F (0^{+}) < y_{1}(ν).

In this situation there is \overline{x_{1}}(μ, ν) such that −μ + \overline{x_{1}}^{2} = y_{1}(ν); that is \overline{x_{1}}(μ, ν) = \sqrt{μ − \sqrt{ν}}.

If we ask for the condition F (0^{−}) = \overline{x_{1}}(μ, ν) we get the curve \overline{μ_{1}}(ν) = ν^{2}+\sqrt{ν}(see figure 6). This curve is tangent to μ_{1}(ν) at ν = 0 and transversally intersect μ_{n}(ν), n ≥ 2 and μ(ν).

In the same way: if we assume that F (0^{+}) < y_{n}(ν), n ≥ 2 we will find \overline{x_{n}}(μ, ν) such that −μ + \overline{x_{n}}^{2} = y_{n}(ν) that is \overline{x_{n}}(μ, ν) = \sqrt{μ + y_{n}(ν)}.

If we look for the condition F (0^{−}) = \overline{x_{n}}(μ, ν) we get a curve \overline{μ_{n}}(ν) = ν^{2} − y_{n}(ν). This curve is tangent to μ_{n}(ν) at ν = 0 and transversally intersect μ_{j}(ν), j ≥ n + 1.

Figure 6:

We note that (μ, ν) ∈ \{(\overline{μ_{n}}(ν), ν); ν ≥ 0\} imply b(F_{(μ,ν)}) = \underline{10_{n+1}}.

(f) In a similar way if we assume F (0^{−}) > x_{1}(μ) we will find \overline{y_{1}}(μ, ν) such that ν − \overline{y_{1}}^{2} = x_{1}(μ) that is: \overline{y_{1}}(μ, ν) = −\sqrt{ν − x_{1}(μ)}.

The condition F (0^{+}) = \overline{y_{1}}(μ, ν) imply the existence of a curve, \overline{ν_{1}}(μ), such that \overline{ν_{1}}(μ) = μ^{2} + x_{1}(μ) = μ^{2} + \sqrt{μ}, μ ≥ 0. This curve(see figure 7) is tangent to ν_{1}(μ) at ν = 0 and transversally intersect ν_{n}(μ); n ≥ 2 and ν(μ).

In a similar way, if we assume that F (0^{−}) > x_{n}(μ), n ≥ 2 we will find \overline{y_{n}}(μ, ν) such that ν − \overline{y_{n}}^{2} = x_{n}(μ), that is \overline{y_{n}}(μ, ν) = \sqrt{ν − x_{n}(μ)}.

If we ask for the condition F (0^{+}) = \overline{y_{n}}(μ, ν) we get the curve \overline{ν_{n}}(μ) = μ^{2} +x_{n}(μ), μ ≥ 0. This curve is tangent to ν_{n}(μ) at ν = 0 and transversally intersect ν_{j}(μ); j ≥ n + 1.

We note that (μ, ν) ∈ \{(μ, \overline{ν_{n}}(μ)); μ > 0\} imply a(F_{(μ,ν)}) = \underline{01_{n+1}}.

Note 1 In this section we have proved :

1) That associated to any sequence a ∈ A_{0} = \{\underline{0_{n}1}; \underline{01_{n}}/n ∈ ℕ\}, we have two curves, C(a) and C(b(a)) , such that (μ, ν) ∈ C(a) imply a(F_{(μ,ν)}) = a and (μ, ν) ∈ C(b(a)) imply b(F_{(μ,ν)}) = b(a).

Moreover, C(a) and C(b(a)) transversally intersect at the point {P (a, b)} = C_{a} ∩ C(b(a)).

2) For (μ, ν) ∈ C(a) we have that 0 is a fixed point of F_{(μ,ν)}^{#per(a)} that attracts a small right neighborhood of it.

3) For (μ, ν) ∈ C(b(a)) we observe that x = ν is a fixed point of F_{(μ,ν)}^{#per(b(a))} that attracts a small left neighborhood of it.

4.2.4 Transversality of the bifurcation curves.

Let us denote

φ_{1}(μ) = \sqrt{μ}, φ_{2}(μ) = \sqrt{μ + φ_{1}(μ)}, ⋯, φ_{n+1}(μ) = \sqrt{μ + φ_{n}(μ)}

and

ψ_{1}(ν) = \sqrt{ν}ψ_{2}(ν) = \sqrt{ν + ψ_{2}(ν)}; ⋯; ψ_{n+1}, (ν) = \sqrt{ν_{n} + ψ_{n}(ν)}.

Define G_{n}(μ, ν) = −μ + φ_{n}(ν) and H_{n}(μ, ν) = ν − φ_{n}(μ).

As we saw, in 4.2.3, we have that G_{n}^{−1}(0) = C\underline{0_{n}1} and H_{n}^{−1}(0) = C\underline{1_{n}0}.

Let

L_{n}(μ, ν) = −μ + [G_{n}(μ, ν)]^{2} =
= −μ + (−μ + ψ_{n}(ν))^{2} = −μ + μ^{2} − 2μψ_{n}(ν) + ψ_{n}^{2}(ν).

and

M_{n}(μ, ν) = ν − [H_{n}(μ, ν)]^{2} = ν − [ν − φ_{n}(μ)]^{2} =

= ν − ν^{2} + 2νφ_{n}(μ) − φ_{n}^{2}(μ).

We have that L_{n}^{−1}(0) is transversal to H_{m}^{−1}(0), any n, m ∈ ℕ. In fact, we have

\frac{∂L_{n}}{∂μ} = −1 + 2μ − 2ψ_{n}(ν).
\frac{∂L_{n}}{∂ν} = 2ψ_{n}(ν) · ψ′_{n}(ν) − 2μψ′_{n}(v).
\frac{∂H_{n}}{∂μ} = −φ′_{n}(μ).
\frac{∂H_{n}}{∂ν} = 1

Then

\frac{−\frac{∂L_{n}}{∂μ}}{\frac{∂L_{n}}{∂ν}} = \frac{1 + 2(ψ_{n}(ν) − μ)}{2ψ′_{n}(ν)[ψ_{n}(ν) − μ]}

and

\frac{−\frac{∂H_{n}}{∂μ}}{\frac{∂H_{n}}{∂ν}} = φ′_{n}(μ) < 1.

Since L_{n}(μ, ν) = 0 implies \sqrt{μ} = ψ_{n}(ν) − μ, then

\frac{\frac{∂L_{n}}{∂μ}}{\frac{∂L_{n}}{∂ν}} = \frac{1}{ψ′_{n}(ν)} · [1 + \frac{1}{2\sqrt{μ}}] > 1.

Hence L_{n}^{−1}(0) is transversal to H_{n}^{−1}(0). Let us now see that M_{m}^{−1}(0) is transversal to L_{n}^{−1}(0). In fact

\frac{−\frac{∂M_{m}}{∂μ}}{\frac{∂M_{m}}{∂ν}} = \frac{2φm^{′}(μ)[φ_{m}(μ) − ν]}{1 + 2[φ_{m}(μ) − ν]} = \frac{2φm^{′}(μ) · \sqrt{ν}}{1 + 2\sqrt{ν}} .

And, consequently M_{m}^{−1}(0) is transversal to L_{n}^{−1}(0) for any n, m ∈ ℕ(see figure 8)

\underline{Remark} We note that L_{n}^{−1}(0) ”begin” at the intersection G_{n−1}^{−1}(0) ∩ H_{1}^{−1}(0)and it is tangent to H_{1} = 0 . In the same way, if N_{n}(μ, ν) = F (L_{n}), then N_{n} is transversal to H_{n}^{−1}(0)and M_{n}^{−1}(0) any m ≥ 0, and N_{n}^{−1}(0) ”begin” in the intersection G_{n−1}^{−1}(0) ∩ H_{2}(0), and so on.

4.2.5 Primary saddle-nodes.

In this section we will prove

Proposition 5 Associated with any a ∈ A_{0} and b = b(a) there is a curve SN(a, b) ⊂ ℝ^{2} such that:

(i) there are values ν_{a,b} and μ_{a,b} such that F_{(μ_{a,b},ν_{a,b})}^{#per(a)} has a fixed point with derivative equal to 1 and (μ_{a,b}, ν_{a,b}) ∈ SN(a, b);

(ii) SN(a, b) \ {(μ_{a,b}, ν_{a,b})} = SN(a) ∪ SN(b) are two curves which satisfies:

(a) SN(a) transversally intersects C(b) at a point \{(\overline{μ}_{a,b}, \overline{ν}_{a,b})\} = SN(a) ∩ C(b). SN(b) transversally intersects C(a) at a point {(μ̃_{a,b}, ν̃_{a,b})} = SN(b) ∩ C(a);

(b) For any (μ, ν) ∈ SN(a) there is a saddle-node periodic point which correspond to an unfolding of the fixed point of F_{(μ,λ)}^{#per(a)}, (μ, ν) ∈ C(a).

(c) For any (μ, ν) ∈ SN(b) there is a saddle-node periodic point which correspond to an unfolding of the fixed point of F_{(μ,ν)}^{#per(b)}, (μ, ν) ∈ C(b).

Proof: Initially, let us consider a = \underline{01}. In this situation the curve C_{a} = C\underline{01} = {(μ , μ^{2}) , μ ≥ 0} represent the fact that F (0^{+}) = y_{1}(ν) ,hence F (F (0^{+})) = F (y_{1}(ν)) = 0 that is, ”0^{+}” is a periodic point whose period is #(a) . Also, for b = \underline{10} the curve C_{b} = C\underline{01} = \{(μ , \sqrt{μ}) , μ ≥ 0\} represent the fact that F (0^{−}) = x_{1}(μ) and, then, F (F (0^{−})) = F (x_{1}(μ)) = 0 , that is ”0^{−}” is a periodic point (of the map F ) whose period is #(b) .

Let us consider the point (1 , 1) ∈ C\underline{01} ∩ C\underline{10} . The graph of the map F_{(1,1)} is as in left hand side of figure 9 and the graph of the restriction of the map F_{(1,1)}^{2} is as in the right hand side of figure 9.

Now, let H = \{(μ , ν) ; μ ≥ 0 , ν ≥ 0\}. We have that H\C\underline{01} = \{(μ , ν) , , μ^{2} < ν\}∪\{(μ , ν) , μ^{2} > ν\} = A_{1}∪A_{2} and H \C\underline{10} = \{(μ , ν) , \sqrt{μ} < ν\}∪\{(μ , ν) , \sqrt{μ} > ν} = B_{1} ∪ B_{2} . So, we have the following components of the set H \ (C_{a} ∪ C_{b}) = A_{1} ∩ B_{1} ∪ A_{1} ∩ B_{2} ∪ A_{2} ∩ B_{1} ∪ A_{2} ∩ B_{2} where A_{1} ∩ B_{1} = {(μ , ν) , ν > μ^{2} and ν > \sqrt{μ}\} , A_{1} ∩ B_{2} = \{(μ , ν) , \sqrt{μ} > ν > μ^{2}\} , A_{2} ∩ B_{1} = \{(μ , ν) , μ^{2} > ν > \sqrt{μ}\} and A_{2} ∩ B_{2} = \{(μ , ν) , μ^{2} > ν and \sqrt{μ} > ν\}. The next figures displays the first return map, to the interval [0 , ν] for (μ , ν) ∈ A_{i} ∩ B_{j} , i = j = 1, 2 and located near the point (1, 1) .

Figure 10: A_{1} ∩ B_{1} A_{1} ∩ B_{2}.

Figure 11: A_{2} ∩ B_{1} A_{2} ∩ B_{2}.

Now, it is clear that we can find, in the interior of the set A_{1} ∩ B_{2} , a point (μ_{a,b} , ν_{a,b}) such that the first return map to the interval [0 , ν] looks like in figure 11.

Let us now consider the family F_{(μ , μ^{2}+ρ)} , ρ ∈ ℝ. Clearly, there is a unique ρ_{a,b} such that, for μ = μ_{a,b} we have ν_{a,b} = μ^{2} + ρ. Now it is not hard to prove, for the one-parameter family F_{(μ , μ^{2}+ρ)} , ρ fixed, the following

Lemma 10 (i) Assume ρ < ρ_{a,b} and near ρ_{a,b}. In this case the first return map has the following generic unfolds :

Figure 13:

(ii) Assume ρ > ρ_{a,b} and near ρ_{a,b}. In this case the first return map has the following generic unfold :

Figure 14:

(iii) The curves SN(b) = μ\underline{10} ; sn(ρ) and SN(a) = μ\underline{01} ; sn(ρ) are given in figure 15,

(iv) The curve SN(b) intersects the curve {(μ , μ^{2} + ρ) , μ ≥ 0} , ρ < ρ_{a,b} fixed , in a unique point.

Figure 16:

(v) The curve SN(a) intersects the curve {(μ , μ^{2} + ρ) , μ ≥ 0} , ρ > ρ_{a,b} fixed , in a unique point(see figure 16). □

This complete the proof in the case a = \underline{01} . The other cases follows in a similar way.

Figure 17 displays these essential facts concerning the curve SN(a, b).□

Remark 2 Given a ∈ A_{0}, b = b(a) we define L_{a} = {(μ, ν); a(F_{(μ,ν)}) = a} and R_{b} = {(μ, ν); b(F_{(μ,ν)}) = b}.

It is not hard to prove that the sets L_{a} and R_{b} looks like as in figure 18

4.2.6 Let us consider

A_{1} = {m(a_{1}, a_{2}); a_{1}, a_{2} ∈ A_{0} are consecutive sequences } ∪ A_{0}.

It is not hard to prove(see 4.2.4) that associated with a = m(a_{1}, a_{2}), b = b(a) we can find two curves C(a), C(b), tangent to C(a_{2}) and C(b(a_{2})) respectively, such that:

(μ, ν) ∈ C(a) imply a(F_{(μ,λ)}) = a and (μ, ν) ∈ C(b) imply b(F_{(μ,λ)}) = b.

Figure 18:

Also, there are saddle-nodes associated to C(a) and C(b) as in 4.2.5. The geometry of the sets R_{b} and L_{a} is similar to the geometry of the sets described in the remark in 4.2.5.

Similarly, we consider the set

A_{n+1} = A_{n} ∪ {m(a_{1}, a_{2}); a_{1} < a_{2} are consecutive sequences inA_{n}}.

As before associated with a ∈ A_{n+1}, a = m(a_{1}, a_{2}),we are able to find the curves C(a), C(b(a)) and SN(a, b)(see figure 19).

Figure 19:

4.2.6 Let A_{∞} = ⋃_{n=0}^{∞} A_{n}. Assume a ∈ \overline{A_{∞}} is a sequence which is not periodic nor eventually periodic. Let b = sup{σ^{k}(a); k ∈ ℕ} and a_{n} ∈ A_{∞}, a_{n} → a. let b_{n} = b(a_{n}). It is clear that we have b_{n} → b.

In this situation we can find values of the parameter (μ_{a}, ν_{a}) and curves C(a), C(b) such that: C(a) = {(μ, ν); ν = φ_{a}(μ)}; C(b) = {(μ, ν); ν = ψ_{b}(μ) where φ_{a} : (0, ∞) → ℝ, ψ_{b} : (0, ∞) → ℝ satisfies

- φ_{a}(μ) = ψ_{b}(μ), 0 < μ ≤ μ_{a}.

- The maps φ_{a}|_{(μ_{a},∞)} and φ_{b}|_{(μ_{a},∞)} are tangent at μ_{a}.

Moreover

(i) the curves C(a_{n}) converges to C(a) and the curves C(b_{n}) converges to C(b);

(ii) the intersections C(a_{n}) ∩ C(b_{n}) converges to {(μ_{a}, ν_{a})};

(iii) (μ, ν) ∈ C(a) imply a(F_{(μ,λ)}) = a and(μ, ν) ∈ C(b) imply b(F_{(μ,λ)}) = b.

Figure 20:

Maybe, the unique point which is not easy to agree in this result is related with the coincidence of the curves C(a) and C(b) along {(μ, φ_{a}(μ)); 0 < μ ≤ μ_{a}}, but this is an easy consequence of the next result.

Let r > 0 be a small number. Let B_{r}^{+} = \{(μ, ν) ∈ ℝ^{2}, \sqrt{μ^{2} + ν^{2}} ≤ r, μ ≥ 0, ν ≥ 0}. Let m(A) denotes the Lebesgue measure of the set A ⊂ ℝ^{2}.

Lemma 11 m({(μ, ν) ∈ B_{r}^{+}; F_{(μ,ν)} ∉ MS_{0}}) = 0.

Proof: In fact, the proof can be done in a similar way as was done in [19] or in [30].

Remark 3 1) In fact, Arnold ([3] and Boyland ( [5]), obtained a similar result for the structure of the irrational Arnold’s Tongues of the canonical family

f_{λ ,b}(x) = x + λ + \frac{b}{2π} sin(2πx) .

2) A similar result(as in lemma 13) was proved by Swiatek in [34] with respect to parameterized families of homeomorphisms of the circle.

4.3 Secondary Bifurcations

We call the bifurcations in this section \underline{secondary} because this kind of changes in the dynamic cannot appear for injective maps in DM_{0} . Also, it is not possible to understand this scenario only by using rotation numbers as it is possible to see by comparing our results with those of [5].

4.3.1 Let B_{∞} = {b(a); a ∈ A_{∞}}. Let us now consider a ∈ A_{∞}, b ∈ B_{∞} such that C(a) ∩ C(b) ≠ ∅.It is not hard to prove that ( see section 4.2.3) that:

a) C_{a} ∩ C_{b} = {P (a, b)} is a unique point.

b) define A(a, b) = \{c ∈ Σ_{0} ; c = \underline{a_{−}b_{+}a^{n}}; n ∈ ℕ or c = \underline{a_{−}b^{m}b_{+}}; m ∈ ℕ\} and B(a, b) = \{d ∈ Σ_{1}; d = \underline{b_{+}a^{n}a_{−}}, n ∈ ℕ or d = \underline{b_{+}a_{−}b^{m}}; m ∈ ℕ\}.

((i) Associated with c = \underline{a_{−}b_{+}} there is a curve C_{c}, tangent to C_{b}; P (a, b) ∈ \overline{C_{c}} and such that (μ, ν) ∈ C_{c} imply a(F_{(μ,ν)}) = c;

((ii) Associated with d = \underline{b_{+}a} there is a curve C_{d}, tangent to C_{a}; P (a, b) ∈ \overline{C_{d}} and such that (μ, ν) ∈ C_{d} imply b(F_{(μ,ν)}) = d;

Figure 21: (b)(i) and (b)(ii)

(iii) Associated with c = \underline{a_{−}b_{+}a^{n}}, n ≥ 1, there is a curve C_{c}, tangent to C_{a};

P (a, b) ∈ \overline{C_{c}} and such that (μ, ν) ∈ C_{c} imply a(F_{(μ,ν)}) = c;

(iv) Associated with d = \underline{b_{+}a^{n}a_{−}}, n ≥ 1, there is a curve C_{d}, tangent to C_{a};

P (a, b) ∈ \overline{C_{c}} and such that (μ, ν) ∈ C_{d} imply b(F_{(μ,λ)}) = d;

(v) Associated with c = \underline{a_{−}b^{n}b_{+}}, n ≥ 1, there is a curve C_{c}, tangent to C_{b};

P (a, b) ∈ \overline{C_{c}} and such that (μ, λ) ∈ C_{c} imply a(F_{(μ,λ)}) = c;

(vi) Associated with d = \underline{b_{+}a_{−}b^{n}}, n ≥ 1, there is a curve,C_{d}, tangent to C_{b};

P (a, b) ∈ \overline{C_{d}} and such that (μ, λ) ∈ C_{d} imply b(F_{(μ,λ)}) = d;

In fact, let denote by ξ_{b} : [μ_{b}, ∞[→ ℝ and ξ_{a} : [μ_{a}, ∞[→ ℝ the maps that satisfies:

Figure 22: (b)(iii) and (b)(iv)

Figure 23: (b)(v) and (b)(vi)

(i) {(μ; ξ_{b}(μ)}; μ ≥ μ_{b}} is the curve which represents the fact that x = v is a periodic point with itinerary b;

(ii) {(μ; ξ_{a}(μ)}; μ > μ_{a}} is the curve which represents the fact that x = −μ is a periodic point with itinerary a.

Let denote C_{a} ∩ C_{b} = \{(\overline{μ_{b}}, \overline{ν_{b}})\}.

Now, associated with c_{n} = \underline{b_{+}a^{n}a_{−}}, n ≥ 0 we can obtain maps ξ_{n} : [\overline{μ_{b}}, ∞[→ ℝ such that

(i) ξ′_{n}(\overline{μ_{b}}) = ξ′_{a}(\overline{μ_{b}}) − ξ′_{b}(μ_{b}); ξ_{n}(\overline{μ_{b}}) = 0

(ii) \{(μ; ξ_{b}(μ)+ξ_{n}(μ)); μ ≥ \overline{μ_{b}}\} is the curve which represents the fact that x = v is a periodic point with itinerary c_{n}.

In a similar way, associated with d_{m} = \underline{b_{+}a_{−}b^{m}}, m ≥ 1, there are maps ϕ_{m} : [\overline{μ_{b}}, ∞[→ ℝ such that

(i) ϕ_{m}(\overline{μ_{b}}) = 0; ϕ_{m}(\overline{μ_{b}}) = 0.

(ii) \{(μ, ξ_{b}(μ) + ϕ_{m}(μ)); μ ≥ \overline{μ_{b}}\} is the curve which represents the fact that x = v is a periodic point with itinerary d_{m}.

(c) Associated with c = a_{−}\underline{b} there is a curve C_{c}, tangent to SN(b) at the point P_{c} ∈ C_{c}, transversal to C_{b} at the point Q_{c}, such that (μ, λ) ∈ C_{c} imply a(F_{(μ,λ)}) = c.

Moreover let L_{c} be the open rectangle bounded by C_{a}, SN(b), C_{c} and C_{b}. We have L_{c} ⊂ R_{b} and (μ, ν) ∈ L_{c} imply a(F_{(μ,λ)}) = c. Also, (μ, λ) ∈ \overline{L_{c}} ∩ [C_{b} ∪ SN(b) \ C_{a}] imply a(F_{(μ,λ)}) = c;

(d) Associated with d = b_{+}a_{−}\underline{b} there is a curve C_{d}, tangent to C_{a_{−}\underline{b}} at the point Q_{a_{−}\underline{b}} ∈ \overline{C_{d}} ∩ C_{a_{−}\underline{b}} and such that (μ, ν) ∈ C_{d} imply b(F_{(μ,λ)}) = d;

Figure 24: (c) and (e)

Figure 25: (d) and (f)

(e) Associated with d = b_{+}\underline{a} there is a curve, C_{d}, tangent to SN(a) at the point P_{d} ∈ C_{d}, transversal to C_{a} at the point Q_{d} such that (μ, ν) ∈ C_{d} imply b(F_{(μ,λ)}) = d. Moreover, let R_{d} be the open rectangle bounded by C_{a}, SN(a), C_{b} and C_{d}. We have R_{d} ⊂ L_{a} and (μ, ν) ∈ R_{d} imply b(F_{(μ,λ)}) = d. Also,

(μ, λ) ∈ \overline{R_{d}} ∩ [C_{a} ∪ SN(a) \ C_{b}] imply b(F_{(μ,λ)}) = d;

(f) Associated with c = a_{−}b_{+}\underline{a} there is a curve C_{c}, tangent to C_{b_{+}\underline{a}}at the point Q_{b_{+}\underline{a}} ∈ \overline{C_{c}} ∩ C_{b_{+}\underline{a}}, and such that (μ, ν) ∈ C_{c} imply a(F_{(μ,λ)}) = c.

Figures (21),(22),(23),(24) and (25) displays the properties in 4.3.1.

Remark 4 (i) Let γ(a, a_{−}b_{+}\underline{a}) be the curve which start in P (a, b), follows C_{a} up to Q_{b_{+}\underline{a}} and then follows C_{a_{−}b_{+}\underline{a}}. We have that the sequence of curves

\{C\underline{a_{−}b_{+}a^{m}}; m ∈ ℕ\} converges to γ(a, a_{−}b_{+}\underline{a});

(ii) Let γ(a, b_{+}\underline{a}) be the curve which start in P (a, b), follows C_{a} up to Q_{b_{+}\underline{a}} and then follows C_{b_{+}\underline{a}}. We have that the sequence of curves \{C\underline{b_{+}a^{m}a_{−}}, m ∈ ℕ\} converges to γ(a, b_{+}\underline{a});

(iii) Let γ(b, a_{−}\underline{b}) be the curve which start in P (a, b), follows C_{b} up to Q_{a_{−}\underline{b}} and then follows C_{a_{−}b}. We have that the sequence of curves \{C\underline{a_{−}b^{m}b_{+}}; m ∈ ℕ\} converges to γ(b, a_{−}\underline{b});

(iv) Let γ(b, b_{+}a_{−}\underline{b}) be the curve which start in P (a, b), follows C_{b} up to Q_{a_{+}\underline{b}} and then follows C_{b_{+}a_{−}\underline{b}}. We have that the sequence of curves \{C\underline{b_{+}a_{−}b^{m}}, m ∈ ℕ\} converges to γ(b, b_{+}a_{−}\underline{b});

(v) The same construction that we did here can be done for any a ∈ A_{∞}^{∞}, b ∈ B_{∞}^{∞} such that C_{a} ∩ C_{b} ≠ ∅.

4.3.2 We will say that f ∈ DM_{0} satisfy the \underline{axiom A} if:

(i) All the attracting periodic point of f are hyperbolic and

(ii) The complement of the stable manifolds of the attracting periodic orbits, in [f(0^{+}), f(0^{−})], is an hyperbolic set.

Lemma 12 Let a ∈ A_{∞}^{∞} and b ∈ β_{∞}^{∞} such that C_{a}∩C_{b} ≠ ∅. There is a neighborhood U(a, b) ⊂ ℝ^{2} of the point P (a, b) such that

m({(μ, λ) ∈ U(a, b); F_{(μ,ν)} do not satisfy axiomA}) = 0.

That is; the measure of the bifurcation in U(a, b) in zero.

Proof This result follows as similar results in [19] or [30].

4.4 Doubling Periods

Let us consider a ∈ A_{∞}^{∞} and b ∈ β_{∞}^{∞} such that C_{a} ∩ C_{b} ≠ ∅.

We will use a_{0} = a, b_{0} = b. Let us construct the following sequences:

a_{n+1} = \underline{(a_{n})_{−}(b_{n})_{+}} ; and b_{n+1} = \underline{(b_{n})_{+}(a_{n})_{−}}.

We have

a_{0} > a_{1} > a_{2} > … > lim_{n→∞} a_{n} = c

b_{0} < b_{1} < b_{2} < … < lim_{n→∞} b_{n} = d

We will call the sequence (a_{n}, b_{n}) the doubling period sequence with starting point (a_{0}, b_{0}). The pair (c, d) will be called the \underline{target} of the doubling period with starting point (a_{0}, b_{0}).

Lemma 13 Associated with c (resp. d) there is a curve C_{c} ( resp. C_{d}) which is the limit of the curves C_{a_{n}} (resp. C_{b_{n}}). Moreover C_{c} ∩ C_{d} contains a unique point P (c, d).

For (μ, λ) ∈ C_{c} we have a(F_{(μ,λ)}) = c and for (μ, λ) ∈ C_{d} we have b(F_{(μ,λ)}) = d.

4.5 Tails

Lemma 14 a) Assume a ∈ A_{∞}^{∞} and d ∈ Max_{2} is a non-periodic sequence such that C_{a} ∩ C_{d} ≠ ∅.

There is a curve C_{a_{−}d}, which “born” in {P (a, d)} = C_{a} ∩ C_{d}, tangent to C_{d} such that P (a, d) ∈ \overline{C_{a_{−}d}} and (μ, λ) ∈ C_{a_{−}d} imply a(F_{(μ,λ)}) = a_{−}d.

b) Assume b ∈ β_{∞}^{∞} and c ∈ Max_{2} is a non-periodic sequence such that C_{c}∩C_{b} ≠ ∅.

There is a curve C_{b_{+}c}, which “born” in {P (c, d)} = C_{c} ∩ C_{b}, tangent to C_{c} at this point such that P (c, d) ∈ \overline{C_{b_{+}c}} and (μ, λ) ∈ C_{b_{+}c} imply b(F_{(μ,d)}) = b_{+}c.

We let the proof of this lemma to the reader.

Figure 26:

Figure 27:

The figures (26)and (27) displays these facts.

4.6 Entropy zero

In this section we will make (following [21]) some remarks concerning the set

EZ = {(μ, λ); h(F_{(μ,λ)}) = 0}.

Here, h(f) means the topological entropy of the map f ∈ DM_{0} .

a) Assume a ∈ A_{∞}^{∞}, b = b(a). We have

int(L_{a} ∩ R_{b}) ⊂ int(EZ)

In fact, in this situation the set Σ_{a,b} is formed by the iterations σ^{i}(a) , i = 1 , ⋯, #per(a) − 1 and their preimages under the map σ.

b) Assume a ∈ A_{∞}^{∞}, b = b(a) and {P (a, b)} = C_{a} ∩ C_{b} then P (a, b) ∈ int(EZ).

In fact, there is a neighborhood of the point P (a, b), which is covered by the union L_{a} ∪ L_{a_{−}\underline{b}} ∪ \{L_{c}; c ∈ A(a, b) ∪ B(a, b)\} ∪ \{C_{c}; c ∈ \overline{A(a, b) ∪ B(a, b)}\}. ( here A(a, b) and B(a, b) where defined in 4.3.1. It is clear that for any c ∈ \overline{A(a, b) ∪ B(a, b)}\} we have that the set Σ_{c,d} is formed by the iterations σ^{i}(c) i = 1 , ⋯, #per(c) − 1 and their preimages under the map σ, here d = sup{σ^{i}(c); i ∈ ℕ}.

c) Assume a ∈ A_{∞}^{∞}, b = b(a) then

int(L_{a_{−}\underline{b}}) ∪ R_{b_{+}\underline{a}}) ⊂ int(EZ).

In fact, if (μ, ν) ∈ int(L_{a_{−}\underline{b}}) we have that a(μ, ν) = a and b(μ, ν) = sup\{σ^{i}(a) i ∈ ℕ}.

d) Assume a ∈ A_{∞}^{∞}, b = b(a) then

\overline{L_{a}} ∩ C_{b_{+}\underline{a}} ⊂ EZ \ int(EZ)

and

\overline{R_{b}} ∩ C_{a_{−}\underline{b}} ⊂ EZ \ int(EZ)

In fact, given any (μ, ν) ∈ \overline{L_{a}} ∩ C_{b_{+}\underline{a}} we have that there is a small neighborhood V (μ, ν) of the point (μ, ν) which is divided by the curve C_{b_{+}\underline{a}} in two disjoint pieces, one of them, included in int(EZ) and the other included in the interior of the positive entropy set.

All the next results, in this section, follows in a similar way.

e) Let a ∈ A_{∞}^{∞}, b = b(a) and (P (a, b), Q_{a_{−}\underline{b}}) ⊂ C_{b} be the open interval whose boundary points are P (a, b) and Q_{a_{−}\underline{b}} then

(P (a, b), Q_{a_{−}b}) ⊂ int(EZ).

f) Let a ∈ A_{∞}^{∞}, b = b(a) and (P (a, b), Q_{b_{+}\underline{a}}) ⊂ C_{a} be the open interval whose boundary points are P (a, b) and Q_{b_{+}\underline{a}} then (P (a, b), Q_{b_{+}\underline{a}}) ⊂ int(EZ). g) Also, we have

{(μ, 0); 0 < μ < 1} ⊂ int(EZ)

and

{(0, ν); 0 < ν < 1} ⊂ int(EZ).

Moreover:

(1, 0) ∈ [EZ \ int(EZ)]

and

(0, 1) ∈ [EZ \ int(EZ)].

h) Let a ∈ A_{∞}^{∞}, b = b(a). For any (μ, ν) such that

a(F_{(μ,λ)}) ≤ a_{−}\underline{b} and b(F_{(μ,λ)}) > b

or

a(F_{(μ,λ)}) < a_{−}\underline{b} and b(F_{(μ,λ)}) ≥ b

we have that (μ, λ) ∈ int{(μ, λ); h(F_{(μ,λ)}) > 0} i) Let a ∈ A_{∞}^{∞}, b = b(a). For any (μ, ν) such that

a(F_{(μ,λ)}) ≤ a and b(F_{(μ,λ)}) > b_{+}\underline{a}

or

a(F_{(μ,λ)}) < a and b(F_{(μ,λ)}) ≥ b_{+}\underline{a}

We have that (μ, λ) ∈ int{(μ, λ); h(F_{(μ,λ)}) ≠ 0}.

j) Let a ∈ A_{∞}^{∞}, b = b(a). Denote by B(a, b) ∈ ℝ^{2} the point where the curves C_{a} and C_{b} born. Let SN(a, b) = C_{a} ∩ SN(b); \overline{SN}(a, b) = C_{b} ∩ SN(a). We have that the intervals [B(a, b), SN(a, b)[⊂ C_{a} and [B(a, b), \overline{SN}(a, b)[ are contained in int({(μ, λ); h(F_{(μ,λ)}) = 0}).

k) Let

C\underline{0} = \{(\frac{1 + \sqrt{1 + 4ν}}{2} , ν); ν ≥ 0\}

and

C\underline{1} = \{(μ, \frac{1 + \sqrt{1 + 4μ}}{2}); μ ≥ 0\} .

We have,

(C\underline{0} \ \{(1, 0)\}) ∪ (C\underline{1} \ \{(0, 1)\}) ⊂ int(\{(μ, z); h(F_{(μ,λ)}) ≠ 0\}).

l) Let

L\underline{0} = \{(μ, ν); μ ≥ \frac{1 + \sqrt{1 + 4ν}}{2} , ν ≥ 0\},

R\underline{1} = \{(μ, ν); ν ≥ \frac{1 + \sqrt{1 + 4ν}}{2} , μ ≥ 0\},

L_{0\underline{1}} = \{(μ , ν) , a(μ , ν) = 0\underline{1} \},

and

R_{1\underline{0}} = \{(μ , ν) , b(μ , ν) = 1\underline{0} \}.

We have

int(L\underline{0} \ R_{1\underline{0}}) ∪ int(R\underline{1} \ L_{0\underline{1}}) ⊂ int(\{(μ, ν); h(F_{(μ,λ)}) ≠ 0\}).

¿From all of the above results it is clear that EZ is an arc-connected set. We do not know if it is locally connected.

4.7 Non-zero measure for the Bifurcation

The result in this subsection is a consequence of a result proved by Rovella in [32]. We established it here only for the sake of completness.

Let a ∈ Min_{2} and b ∈ Max_{2} be eventually periodic sequences such that C_{a}∩C_{b} ≠ ∅. Let C_{a} ∩ C_{b} = {P (a, b)} and U(a, b) be any neighborhood of P (a, b). We have

Lemma 15 m({(μ, ν) ∈ U(a, b); F_{(μ,ν)} is not AxA}) > 0.

As a consequence of this result we get

Corollary 2 Let (μ, ν) ∈ (EZ \int(EZ)) and U(μ, λ) be any neighborhood of (μ, λ). We have

m({(α, β) ∈ U(μ, λ); F_{(α,β)} is not Ax.A}) > 0.

Proof: (μ, λ) can be approximated by a sequences (μ_{n}, ν_{n}) as in the Lemma 17.

4.8 Tales in L\underline{0} and R\underline{1}

a) Associated to b = 1\underline{0} there is a curve C_{1\underline{0}} ⊂ L\underline{0} such that (1, 0) ∈ \overline{C_{1\underline{0}}} ;this curve is tangent to C\underline{0} at (1, 0) and (μ, ν) ∈ C_{1\underline{0}} imply b(F_{(μ,λ)}) = 1\underline{0}.

b) Associated to a = 0\underline{1} there is a curve C_{0\underline{1}} ⊂ R\underline{1} that satisfy (0, 1) ∈ \overline{C_{0\underline{1}}} ; it is tangent to C\underline{1} and (0, 1); (μ, ν) ∈ C_{0\underline{1}} imply a(F_{(μ,λ)}) = 0\underline{1};

c) Let φ_{1\underline{0}} :]1, ∞[→ ℝ denote the map that satisfy Graph(φ_{1\underline{0}}) = C_{1\underline{0}}.

We have

\{(μ, ν); b(F_{(μ,λ)})\} = 1\underline{0}\} = \{(μ, ν); 0 < ν ≤ φ_{1\underline{0}}(μ); μ > 1\}.

d) Let φ_{0\underline{1}} :]1, ∞[→ ℝ denote the map that satisfy Graph(φ_{0\underline{1}}) = C_{0\underline{1}}.

We have

\{(μ, ν); a(F_{(μ,λ)}) = 0\underline{1}\} = \{(μ, ν); 0 < μ ≤ φ_{0\underline{1}}(ν), ν > 1\}.

e) Let b ∈ B_{∞}^{∞} and \{P (\underline{0}, b)\} = C\underline{0} ∩ C_{b}.

There is a curve C_{b_{+}\underline{0}} ⊂ L\underline{0} such that P (\underline{0}, b) ∈ \overline{C_{b_{+}\underline{0}}}; C_{b_{+}\underline{0}} is tangent to C\underline{0} at P (\underline{0}, b) and (μ, λ) ∈ C_{b_{+}\underline{0}} imply b(F_{(μ,λ)}) = b_{+}\underline{0}.

f) Let P (\underline{0}, b) = (μ_{\underline{0},b}, ν_{u0,b}) and φ_{b_{+}\underline{0}} :]μ_{\underline{0},b}, ∞[→ ℝ be the map that satisfy Graph(φ_{b_{+}\underline{0}}) = C_{b_{+}\underline{0}}.

Let φ_{b} : [μ\underline{0}, ∞[→ ℝ be the map that satisfy Graph(φ_{b}) = L\underline{0} ∩ C_{b}.

Clearly φ_{b}(μ) < φ_{b_{+}\underline{0}}(μ), μ > μ_{\underline{0},b}.

We have \{(μ, ν); b(F_{(μ,λ)}) = b_{+}\underline{0}\} = \{(μ, ν); φ_{b}(μ) < ν ≤ φ_{b_{+}\underline{0}}(μ)\}.

g) Let a ∈ A_{∞}^{∞} and \{P (a, \underline{1})\} = C_{a} ∩ C\underline{1}. There is a curve C_{a_{−}\underline{1}} ⊂ R\underline{1} such that P (a, \underline{1}) ∈ \overline{C_{a_{−}\underline{1}}}; C_{a_{−}\underline{1}} is tangent to C\underline{1} at P (a, \underline{1}) and (μ, λ) ∈ C_{a_{−}\underline{1}} imply a(F_{(μ,λ)}) = a_{−}\underline{1}.

h) Let P (a, \underline{1}) = (μ_{a,\underline{1}}, ν_{a,\underline{1}}) and φ_{a_{−}\underline{1}} :]ν_{a,\underline{1}}, ∞[→ ℝ, φ_{a} : [ν_{a,\underline{1}}, ∞[→ ℝ be the maps that satisfies:

Graph(φ_{a_{−}\underline{1}}) = C_{a_{−}\underline{1}} and Graph(φ_{a}) = R\underline{1} ∩ C_{a}.

Clearly φ_{a}(ν) < φ_{a_{−}\underline{1}}(ν),for any ν > ν_{a_{−},\underline{1}}.

We have \{(μ, ν); a(F_{(μ,ν)}) = a_{−}\underline{1}\} = \{(μ, ν); φ_{a}(ν) < μ ≤ φ_{a_{−}\underline{1}}(ν), ν > ν_{a,\underline{1}}\}.

4.9 Bifurcations in R\underline{1} ∪ L\underline{0}

In the same way as was proved in [19] or as in [30] we can prove.

Lemma 16 m(\{(μ, ν) ∈ R\underline{1} ∪ L\underline{0}; F_{(μ,ν)} do not satisfy Ax.A\}) = 0.

4.10 Density of Axiom A Systems.

A consequence of our’s and Rovella’s result is:

Lemma 17 {(μ, ν); F_{(μ,ν)} satisfy Ax.A} is a dense set in ℝ_{0}^{+} × ℝ_{0}^{+}.

4.11 The Global Picture

Taking together all the results in this section the global picture of the bifurcation diagramme is as in figure 28.

5 The general case

Let us consider the five parameter family of maps F_{(α,c_{1},c_{2},μ,ν)}: (R \ {0}) → ℝ, μ ≥ 0 , ν ≥ 0 α > 1, c_{1} > 0 , c_{2} > 0 in DM_{0} given by:

F_{((α,c_{1},c_{2},μ,ν)}(x) = \{ \begin{matrix}−μ + c_{1}x^{α}, x > 0 \\ ν − c_{2}(−x)^{α}, x < 0 \end{matrix}

We have to observe that all the previous results, for the quadratic family, are (still) true when we fix generic values of the parameter α , c_{1} and c_{2} . That is, the bifurcation diagramme associated to any generic two parameter family F_{(μ,ν)} = F_{(α,c_{1},c_{2},μ,ν)}is,up to an homeomorphism, the same as the given for the quadratic family.

6 Application to Geometric Vector fields

Assume X_{0}: U ⊂ ℝ^{3} → ℝ^{3} is a three dimensional vector field that satisfies:

a) 0 ∈ U is an hyperbolic singularity of the vector field X, whose eigenvalues satisfies −λ_{1} < −λ_{2} < 0 < λ_{3};

b) The components, γ_{1} and γ_{2}, of the set (W_{0}^{u} \ {0}) satisfies γ_{1} ⊂ W_{0}^{s}, γ_{2} ⊂ W_{0}^{s};

c) There is a transversal section, Σ ⊂ U, such that: Σ is transversal to W_{0}^{s} and Σ \ W_{0}^{s} = Σ_{1} ∪ Σ_{2} are two disjoint sets in Σ that satisfies:

π_{1}(Σ_{1}) ⊂ Σ_{1} and π_{2}(Σ_{2}) ⊂ Σ_{2}

where π_{i}: Σ_{i} → Σ , i = 1, 2 is the first return map associated to the respective cross section.

Let U_{0} ⊂ 𝔛^{r}(U, ℝ^{3}) be a neighborhood of the vector field X_{0} in 𝔛^{r}(U, ℝ^{3}) = {X: U ⊂ ℝ^{3} → ℝ^{3}; X is C^{r}} with the usual C^{r}-topology.

In the above conditions there are codimension one submanifold, N_{1}, N_{2} ⊂ U_{0}; such that:

a) Y ∈ N_{i} if and only if γ_{i}(Y) ⊂ W_{σ(Y)}^{s} here σ(Y) is the hyperbolic singularity, near 0, that correspond to the vector field Y. Also, (W_{σ(Y)} \ {σ(Y)}) = γ_{1}(Y) ∪ γ_{2}(Y), where γ_{i}(Y) is the natural extension of the component γ_{i};

b) (U_{0} \ N_{i}) = U_{1}^{i} ∪ U_{2}^{i} where Y ∈ U_{1}^{1} if and only if π_{1}(Σ_{1}) ⊂ Σ_{1} and Y ∈ U_{2}^{2} if and only if π_{2}(Σ_{2}) ⊂ Σ_{2} and

c) N_{1} is transversal to N_{2} .

Finally we will assume that Σ can be foliated by one-dimensional submanifolds {F_{x}; x ∈ Σ} such that

(i) x ∈ W_{Y}^{s} ∩ Σ imply F_{x} = W_{Y}^{s} ∩ Σ and

(ii) π_{i}(F_{x}) ⊂ F_{π_{i}(x)}, x ∈ Σ_{i}.

In this situation the interesting part of the bifurcation theory, for elements in U_{0}, is located in U_{1}^{2} ∩ U_{2}^{1}. In this set we can apply the results in Sections 3,4 and 5 to obtain, bis a bis, similar results for generic two parameter families of vector fields in U_{0}.

ACKNOWLEDGMENTS

Part of this paper is an outgrowth of research during a visit of the authors to ICTP (Trieste - Italy), a visit of Rafael Labarca to IMPA-Brazil and a visit of Carlos Moreira to the Universidad de Santiago de Chile. We thanks to these institutions for their support and hospitality while preparing this paper.

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R. Labarca Departamento de Matematica y CC Universidad de Santiago de Chile Casilla 307 Correo 2 Santiago - Chile and C. Moreira IMPA Estrada Dona Castorina 110 CEP 22460-320 Jardim Botanico Rio de Janeiro Brazil