Bifurcation of the Essential Dynamics of Lorenz maps on the real line and the bifurcation scenario for Lorenz like flows : Contributions to the study

of the Expanding Case ^{∗}

Rafael Labarca and Carlos Moreira

February 2, 2001

1 Introduction

In a remarkable contribution a meteorologist, E.N. Lorenz [16], showed numerical evidence of the existence of a strange attractor for a quadratic system of ordinary differential equations in three variables. Some time later J. Guckenheimer, [8], produced a work where he introduced symbolic dynamics in order to understand the topologically equivalence classes for nearly similar attractors. At that time R.F. Williams,[17], introduced a geometrical model in order to understand the dynamics of these Lorenz attractors. Using this geometrical model the dynamical behavior of the three dimensional vector field can be reduce to the dynamical behavior of a one-dimensional map with one discontinuity and Guckenheimer and Williams,[10], used this fact to shown uncountable many classes of non-equivalent geometric Lorenz attractors. The evidence of the non-equivalence where the kneading sequences associated to these one-dimensional maps. Later Afrajmovich, Bykov and Shilnikov,[1],studied a two parameter family of three dimensional vector fields that unfolds a codimension two bifurcation which appears in a vector field defined in a neighborhood of the origin 0 ∈ ℝ^{3}. This vector field has an hyperbolic singularity, at 0 ∈ ℝ^{3} ,which has a two dimensional stable manifold and a one dimensional unstable manifold. For the connected components Λ_{1} and Λ_{2} of the set W_{0}^{u} − {0}, the vector field X satisfies Λ_{1} ⊂ W_{0}^{s} and Λ_{2} ⊂ W_{0}^{s}. In that work they described part of the bifurcation theory that appears in a generic two parameter unfolding of the vector field X.

So, its seem to be natural to obtain a bifurcation theory for a generic unfolding of X using symbolic dynamics. In this direction de Melo and Martens announced, in an unpublished paper [6], the existence of parameterized families of contacting Lorenz-like flows that are topologically universal in the sense that given any geometric Lorenz flow then its dynamics is ” essentially ” the same as the dynamics of some element of the family. In [15] we describe, exactly, the way in which such a contracting family realize all the allowed dynamics for Lorenz-like maps. The same universality is not true when we work with families of expanding Lorenz-like flows as de Melo and Martens knew and as we will show in the present paper.

∗Partially supported by Fondecyt grants # 1970720, # 1990903, DICYT-USACH- Chile and PRONEX on Dynamical Systems. Brazil

Also we have to mention the work by Hubbard and Sparrow [11], were they defined a set of pair of sequences which model all the topological dynamics exhibed by expanding Lorenz maps. Recently, in [13], we extend 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 ). Now, it is clear that any parameterized family of Lorenz maps has a bifurcation theory induced by the lexicographical world ( an extremely interesting problem focused for several authors in this an another contexts, see for instance [5],[4],[9], [2], [3]) . So, it seems to be natural to develop a programme to obtain the bifurcation theories associated to some typical parameterized families of Lorenz maps. We developed such a programme. In [14] and [15] we describe the bifurcation theory associated to a two parameter family of linear maps and to a two parameter family of contracting maps, respectively. In certain form, both bifurcation theories are similar. In the present paper we describe the bifurcation theory associated to a typical two parameter family of expanding Lorenz maps and the associated bifurcation theory is dramatically different from the previous one’s.

Of course, there are other points of view which may be used to described the bifurcation theory associated to parameterized families of Lorenz maps. In fact, starting with the work of Arnold ([2])a lot of work have been produced with respect to the bifurcation theory of the canonical family( see for instance [3] and the references there in). The approaches,in this case, was done by using the rotation number. This approach seems to be useful when applied to homeomorphisms of the circle but looks very complicated when we deal with non injective maps and, in fact, a bifurcation diagram for the dynamics (of the canonical family) is unknown (at least for us). The better approximation to the bifurcation theory of the canonical map seems to be the work by Boyland ( [3]). We have to mention that the lexicographical world includes the dynamics of all the interpolated maps used by Boyland ([3] page 359 )in its construction and, then, it is also a model for these maps.

To be more specific,in the present paper we study the two parameter family of cuadratic expanding Lorenz maps :

G_{μ,ν}(x) = \{ \begin{matrix}−μ + \sqrt{x} , x > 0 \\ ν − \sqrt{−x} , x < 0 \end{matrix}

and we give the corresponding bifurcation theory in the parameter space.

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, in section 5 we generalize our results and, finally, in section 6 we will relate them to three dimensional geometric vector fields.

Some other comments are in order: there are many other authors that have investigated these families from the same and from other points of view (combinatorial, differentiable, topological or geometrical)( see the list of references).

2 Statement of our 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) > 1.

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 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 ([13]) 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 expanding 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 [7]).

2.4 The Lexicographical World

Let Min_{2} = {a ∈ Σ_{0} ; σ^{k}(a) ≥ a , k ∈ ℕ } and Max_{2} = {b ∈ Σ_{b} ; 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 ([14] There is f ∈ DM_{0} such that I_{f} = Σ_{a,b}.

We will call this result the \underline{realization lemma}. We observe that the maps, in this lemma, may be obtained as 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)}).

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-universal 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-universal family} if ∀a ∈ 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 -universal 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: There are LW -universal 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 expanding Lorenz maps is a-universal, is b-universal but it is not LW -universal. Moreover, the respective a , b , LW -bifurcation theories are given.

3 Symbolic Dynamics and the Lexicographical World

Here we introduce some results and notations that are necessary for our results. The results are proved in [13] or in [15].

3.1 Dynamical properties for sequences in LW

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

\underline{Example}: For a_{1} = \underline{01} , 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}); a_{1} , a_{2} ∈ A_{n} are consecutive sequences }. Set A_{∞} = ⋃_{n=0}^{n= ∞} 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}.

In the proof of this lemma( see [15])we 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} .

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 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).

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} .

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 [11].

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 [?], [?] and the references there in ). For our family of expanding 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 Expanding Maps

4.1 The map G.

Let us define G_{(μ,ν)}: (ℝ \ {0}) → ℝ, (μ, ν) ∈ ℝ^{2}, the two parameter family of maps in DM_{0} :

G_{(μ,ν)}(x) = \{ \begin{matrix}−μ + x^{1/2}, x > 0 \\ ν − (−x)^{1/2}, x < 0. \end{matrix}

In this section we will provide the bifurcation theory associated to this family of elements in DM_{0}.

4.2 The Injective Maps

Let us define the set IM = {(μ, ν); G_{(μ,ν)}|_{[−μ,ν]} : [−μ, ν] → [−μ, ν] is an injective map}.It is nor hard to see that IM = \{(μ, ν); μ ≥ 0 , ν ≥ 0 and(\sqrt{ν} −1/2)^{2}+(\sqrt{μ}−1/2)^{2} ≥ 1/2 }.

4.3 Fixed points.

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

a) x_{±}(μ) = [\frac{1±\sqrt{1−4μ}}{2}]^{2} defined for μ ≤ 1/4.

b)y_{±}(ν) = − [\frac{1±\sqrt{1−4ν}}{2} ]^{2} defined for ν ≤ 1/4.

We have y_{+}(ν) ≤ y_{−}(ν) ≤ 0 ≤ x_{−}(μ) ≤ x_{+}(μ). Note: The maps x_{−}(μ) and y_{−}(ν) are defined for 0 ≤ μ ≤ \frac{1}{4}; 0 ≤ ν ≤ \frac{1}{4}.

4.4 Preimages of 0

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

ν − (−x)^{1/2} = y_{1}(ν) imply x = −(ν − y_{1}(ν))^{2} = −[ν + ν^{2}]^{2} = y_{2}(ν);

ν − (−x)^{1/2} = y_{n}(ν) imply x = −[ν − y_{n}(ν)]^{2} = y_{n+1}(ν), ν ≥ 0.

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

−μ + x^{1/2} = x_{1}(μ) imply x = (μ + x_{1}(μ))^{2} = (μ + μ^{2})^{2} = x_{2}(μ);

−μ + x^{1/2} = x_{n}(μ) imply x = (μ + x_{n}(μ))^{2} = x_{n+1}(μ), μ ≥ 0.

4.5 Primary Bifurcations

a) G(0^{+}) = fixed point of the left hand-side

−μ = − [\frac{1 ± \sqrt{1 − 4ν}}{2} ]^{2}

that is:

μ = [\frac{1 ± \sqrt{1 − 4ν}}{2} ]^{2} .

b) G(0^{−}) = fixed point of the right hand-side

ν = [\frac{1 ± \sqrt{1 − 4μ}}{2}]^{2}

Figure 1: (b)

Let L\underline{0} = \{(μ, ν) ∈ ℝ_{0}^{+} × ℝ_{0}^{+}; a(G_{(μ,ν)}) = \underline{0}\} and R\underline{1} = \{(μ, ν) ∈ ℝ_{0}^{+} × ℝ_{0}^{+}; b(G_{(μ,ν)}) = \underline{1}\}.

We observe that L\underline{0} = \{(μ, ν); 0 ≤ ν ≤ ν_{−}(μ), 0 ≤ μ ≤ 1/4 or 0 ≤ ν ≤ 1/4 for μ ≥ 1/4\} and R\underline{1} = \{(μ, ν); 0 ≤ μ ≤ μ_{−}(ν), 0 ≤ ν ≤ 1/4 or 0 ≤ μ ≤ 1/4 for ν ≥ 1/4\}

Here ν_{−}(μ) is given by μ = (^{\frac{1−\sqrt{1−4ν_{−}(μ)}}{2}})^{2} and μ_{−}(ν) is given by ν = (^{\frac{1−\sqrt{1−4μ_{−}(ν)}}{2}})^{2}.

c) G(0^{+}) = nth preimage of zero = y_{n}(ν).

These equations define the curves μ_{n}(ν) = −y_{n}(ν) that satisfies: (μ, ν) ∈ Graph(μ_{n}) imply a(G_{(μ,ν)}) = \underline{0_{n}1} . Moreover, the inverses ν̂_{n}(μ) = μ_{n}^{−1}(ν) converges (uniformly in the C^{1} topology on compact intervals) to the curve s(μ) given by:

s(μ) = \{ \begin{matrix}\frac{1−(2μ^{1/2}−1)^{2}}{4} , 0 ≤ μ ≤ 1/4 \\ \frac{1}{4}, μ ≥ 1/4 \end{matrix}

d) G(0^{−}) = nth preimage of zero = x_{n}(μ).

These equations define the curves ν_{n}(μ) = x_{n}(μ) that satisfies: (μ, ν) ∈ Graph(ν_{n}) imply b(G_{(μ,ν)}) = \underline{1_{n}0}. Moreover, the inverses μ̂_{n}(ν) = ν_{n}^{−1}(μ) converges (uniformly in the C^{1} topology in compact intervals) to the curve s(ν) given by

s(ν) = \{ \begin{matrix}\frac{1−(2ν^{1/2}−1)^{2}}{4} , 0 ≤ ν ≤ 1/4 \\ \frac{1}{4}, ν ≥ 1/4 \end{matrix}

Figure 2: (a)

Figure 3:

Proposition 3 The graph of the curve μ_{n}(ν) is transversal to graph of the curve ν_{m}(μ) all m, n ∈ ℕ.

Proof We have μ_{1}(ν) = ν^{2} and ν_{1}(μ) = μ^{2} .So, the result is true for n = 1 and m = 1 .

For ν_{2}(μ) we have ν_{2}(μ) = (μ+ν_{1}(μ))^{2} , then ν′_{2}(μ) = 2(μ+ν_{1}(μ))(1+ν′_{1}(μ)) ≥ 2(1/4+1/2)(1+ν_{1}(μ)) = _{\overline{3}}2(1+ν_{1}(μ)) = _{\overline{3}}2(1+2μ) ≥ _{\overline{3}}2(1+1/2) ≥ (3/2)^{2} ≥ 2 . Hence, we conclude the result for n = 1 , m = 2 .

Assume the result is true for n = 1 and m = p with ν′_{p}(μ) > 1 . For ν_{p+1}(μ) we have ν_{p+1}(μ) = (μ + ν_{p}(μ))^{2} and ν′_{p+1}(μ) = 2(μ + ν_{p}(μ))(1 + ν′_{p}(μ)) ≥ 2(1/4 + 1/2)(1 + 1) > 4(3/4) = 3 . Therefore, we get the result for n = 1 and any m ∈ ℕ.

Assume that the result is true for n = k and any m ∈ ℕ with μ′_{k}(ν) > 1 and ν′_{m}(μ) > 1 . For μ_{k+1}(ν) = (ν + μ_{k}(ν))^{2} we have μ′_{k+1}(ν) = 2(ν + μ_{k}(ν))(1 + μ′_{k}(ν)) > 2(1/2 + 1/4)(1 + 1) > 3 .So, we conclude the result for n = k + 1 and any m ∈ ℕ □.

e) Let us now assume that G(0^{+}) < y_{1}(ν). In this situation there is a point,

Figure 4:

Figure 5:

\overline{x_{1}}(μ, ν) ≥ 0, such that −μ + \overline{x}_{1}^{1/2} = y_{1}(ν), that is \overline{x_{1}}(μ, ν) = (μ + y_{1}(ν))^{2}.

If we look for the condition G(0^{−}) = \overline{x_{1}}(μ, ν) we get the curve \overline{μ_{1}}(ν) = ν^{1/2} − y_{1}(ν). This curve is tangent to the curve C\underline{10} at ν = 0; transversally intersects the curves μ_{n}(ν), n ≥ 2 and the curve s(μ). We note that (μ, ν) ∈ Graph(\overline{μ_{1}}) imply b(G_{(μ,ν)}) = \underline{100}

In the same way, if we assume that G(0^{+}) < y_{n}(ν), n ≥ 2, we will find a point \overline{x_{n}}(μ, ν) ≤ \overline{x_{n−1}}(μ, ν) such that −μ + \overline{x_{n}}^{1/2} = y_{n}(ν), that is

\overline{x_{n}}(μ, ν) = (μ + y_{n}(ν))^{2}.

If we look for the condition G(0^{−}) = \overline{x_{n}}(μ, ν) we obtain a curve \overline{μ_{n}}(ν) = ν^{1/2} − y_{n}(ν), that is: \overline{μ_{n}}(ν) = ν^{1/2} + μ_{n}(ν). This curve is tangent to \overline{μ_{n−1}}(ν) at ν = 0; transversally intersects the curves μ_{m}(ν), m ≥ n + 1 and the curve s(μ).

We observe that (μ, ν) ∈ Graph(\overline{μ_{n}}) imply b(G_{(μ,ν)}) = \underline{10_{n+1}}

f) Let us now assume that G(0^{−}) > x_{1}(μ). In this situation we can find a point \overline{y_{1}}(μ, ν) such that ν − (−\overline{y_{1}})^{1/2} = x_{1}(μ), that is \overline{y_{1}}(μ, ν) = −[ν − x_{1}(μ)]^{2}.

If we look for the condition G(0^{+}) = \overline{y_{1}}(μ, ν) we obtain a curve \overline{ν_{1}}(μ) = μ^{1/2} + x_{1}(μ). This curve is tangent to the curve C\underline{01} at μ = 0; transversally intersects the

Figure 7:

curves ν_{n}(μ), n ≥ 2 and the curve s(ν).

Also we have (μ, ν) ∈ Graph(\overline{ν_{1})} imply a(G_{(μ,ν)}) = \underline{011}

Similarly: if we assume that G(0^{−}) > x_{n}(μ) we can find a point \overline{y_{n}}(μ, ν) such that ν − (−\overline{y_{n}})^{1/2} = x_{n}(μ) that is y_{n}(μ, ν) = −[ν − x_{n}(μ)]^{2}.

If we look for the condition G(0^{−}) = \overline{y_{n}}(μ, ν) we obtain a curve \overline{ν_{n}}(μ) = μ^{1/2} + x_{n}(μ) = μ^{1/2} + ν_{n}(μ). This curve is tangent to \overline{ν_{n−1}}(μ) at μ = 0; transversally intersects the curves ν_{n}(μ); m ≥ n + 1 and the curve s(ν). We note that (μ, ν) ∈ Graph(\overline{ν_{n}}) imply a(G_{(μ,ν)}) = \underline{01_{n+1}}.

Remark: a) Associated to any a ∈ A_{0} = \{\underline{0_{n}1}; \underline{01_{n}}; n ∈ ℕ\} we have constructed two curves C_{a}, C_{b}, b = b(a), such that (μ, ν) ∈ C_{a} imply a(G_{(μ,λ)}) = a and (μ, ν) ∈ C_{b} imply b(G_{(μ,λ)}) = b. These two curves transversally intersects at a point {P (a, b)} = C_{a} ∩ C_{b};

b) For a_{1}, a_{2} ∈ A_{0} such that C_{a_{1}} ∩ C_{b_{2}} ≠ ∅, where b_{2} = b(a_{2}) , it is not hard to prove that the intersection is transversal and contains a unique point.

g) Let us now ask for the sets L_{a_{−}\underline{b}} and R_{b_{+}\underline{a}} for a ∈ A_{0}, b = b(a).

Figure 9:

At this point we establishes one of the main differences between contracting and expanding families.

Let a ∈ A_{0}, R_{b_{+}\underline{a}} = \{(μ, ν); b(G_{(μ,ν)}) = b_{+}\underline{a}\} and L_{a_{−}\underline{b}} = \{(μ, ν); a(G_{(μ,λ)}) = a_{−}\underline{b}\}

Lemma 10 R_{b_{+}\underline{a}} is formed by

a) a noncompact set, R_{b_{+}\underline{a}}^{1}, whose boundary is formed by two curves, γ_{1}(b_{+}\underline{a}) and γ_{2}(b_{+}\underline{a}) such that:

– (i) γ_{2}(b_{+}\underline{a}) ⊂ C_{b}; P (a, b) ∈ γ_{2}(b_{+}\underline{a})

– (ii) P (a, b) ∈ \overline{γ_{1}(b_{+}\underline{a})} and γ_{1}(b_{+}\underline{a}) is tangent to C_{b} at P (a, b) and

b) a curve, C_{b_{+}\underline{a}}, which is tangent to γ_{1}(b_{+}\underline{a}) at a point Q(b_{+}\underline{a}) ≠ P (a, b) that satisfy C_{b_{+}\underline{a}} is tangent to C_{b} at (0, 0) ∈ \overline{C_{b_{+}\underline{a}}}.

Figure 10: Figure 11:

The figure (11) displays the geometry of this Lemma. Lemma 11 L_{a_{−}\underline{b}} is formed by

a) a noncompact set, L_{a_{−}\underline{b}}^{1}, whose boundary is formed by two curves, γ_{1}(a_{−}\underline{b)} and

γ_{2}(a_{−}\underline{b}) such that:

– (i) γ_{2}(a_{−}\underline{b}) ⊂ C_{a}; P (a, b) ∈ γ_{2}(a_{−}\underline{b})

– (ii) P (a, b) ∈ \overline{γ_{1}(a_{−}\underline{b})} and γ_{1}(a_{−}\underline{b}) is tangent to C_{a} at P (a, b) and b) a curve, C_{a_{−}\underline{b}}, which is tangent to γ_{1}(a_{−}\underline{b}) at a point Q(a_{−}\underline{b}) = Q(b_{+}\underline{a}), that

satisfy C_{a_{−}\underline{b}} is tangent to C_{a} at (0, 0) ∈ \overline{C_{a_{−}\underline{b}}}. The following figure (12) displays the geometry of this Lemma. Proof: The proof of these two lemmas can be carried out in the following steps.

(i) assume a = \underline{0_{n}1}. In this case consider the point P = P (\underline{0_{n}1}, \underline{10_{n}}) ∈ ℝ^{2}. (ii) let U(P) be a small neighborhood of the point P. (iii) the intersection C_{a}∩U(P) and C_{b}∩U(P) divides U(p) into two disjoint regions

U_{a}^{+}, U_{a}^{−}; U_{b}^{+} and U_{b}^{−}, respectively (see figure (13)).

Figure 13:

(iv) verify that there is a point SN(b) ∈ C_{a} ∩ U_{b}^{+} where we can find a saddle node for F^{#per(b)}, near x = ν. Moreover, for (μ, ν) ∈]SN(b), P (a, b)[⊂ C_{a} ∩ \overline{U_{b}^{+}} we have b(F_{(μ,ν)}) = b_{+}\underline{a}.

(v) verify that there is a point SN(a) ∈ C_{b} ∩ U_{a}^{+} where we can find a saddle-node for F^{#per(a)} near x = 0^{+}. Moreover, for (μ, ν) ∈]SN(a), P (a, b)[⊂ C_{b} ∩ U_{a}^{+} we have a(F_{(μ,ν)}) = a_{−}\underline{b}.

(vi) there is a point Q(a, b) ∈ U_{b}^{+} ∩ U_{a}^{+} such that – a) Q(a, b) ∈ \overline{SN(a)} ∩ \overline{SN(b)};

– b) the map F_{Q(a,b)}^{#per(a)} has a fixed point with derivative 1 which is expanding.

(vii) there is a curve, C_{b_{+}\underline{a}}, tangent to C_{b} at P (a, b); P (a, b) ∈ (\overline{C_{b_{+}\underline{a}}} \ C_{b_{+}\underline{a}}); such that (μ, ν) ∈ C_{b_{+}\underline{a}} imply b(F_{(μ,ν)}) = b_{+}\underline{a}. This curve is unbounded and coincides with the saddle-node curve, SN(b), up to Q(a, b).

(viii) there is a curve, C_{a_{−}\underline{b}}, tangent to C_{a} at P (a, b); P (a, b) ∈ (\overline{C_{a_{−}\underline{b}}} \ C_{a_{−}\underline{b}}); such that (μ, ν) ∈ C_{a_{−}\underline{b}} imply a(F_{(μ,ν)}) = a_{−}\underline{b}. This curve is unbounded and coincides with the saddle-node curve, SN(a), up to Q(a, b).

(ix) the existence of the other component of the curves, C_{b_{+}\underline{a}} and C_{a_{−}\underline{b}}, can be now easily computed.

Corollary 2 (The geometry of sets L_{a} and R_{b}, a ∈ A_{0}, b = b(a)).

a) L_{a} = \{(μ, ν); a(F_{(μ,ν)}) = a\} is unbounded and (\overline{L_{a}} \ L_{a}) ⊂ C_{a} ∪ γ_{1}(b_{+}\underline{a}) (see figure(14)).

(b) R_{b} = \{(μ, ν); b(F_{(μ,ν)}) = b\} is unbounded and (\overline{R_{b}} \ R_{b}) ⊂ C_{b} ∪ γ_{1}(a_{−}\underline{b}) (see figure(15)).

Figure 14:

4.6 The average

Let a_{1}, a_{2} ∈ A_{0} be two consecutive sequences, a_{1} < a_{2}. Let b_{1} = b(a_{1}) < b_{2} = b(a_{2}). Denote a = m(a_{1}, a_{2}) and b = m(b_{1}, b_{2}) = b(a).

Lemma 12 There are curves C_{a} and C_{b} such that:

(i) (0, 0) ∈ \overline{C_{a}}, (0, 0) ∈ \overline{C_{b}}.

(ii) C_{a} is tangent to C_{a_{i}} at (0, 0), i = 1, 2 and C_{b} is tangent to C_{b_{i}} at (0, 0) , i = 1, 2.

(iii) C_{a} ∩ C_{b} = {P (a, b)} and C_{a} transversally intersects C_{b} at P (a, b).

The figure(16) displays the geometry of the curves C_{a} and C_{b}. Note: (i) As in the previous section we obtain a similar result for L_{a_{−}\underline{b}}, R_{b_{+}\underline{a}}, L_{a} and R_{b}.

(ii) In this way we have obtained the geometry of the sets L_{a_{−}\underline{b}}, R_{b_{+}\underline{a}}, L_{a} and R_{b} for a ∈ A_{1} = A_{0} ∪ {m(a_{1}, a_{2}); a_{1}, a_{2} ∈ A_{0} are consecutive sequences}.

Inductively, we can obtain the geometry of the sets L_{a_{−}\underline{b}}, R_{b_{+}\underline{a}}, L_{a} and R_{b} for a ∈ A_{n+1} = A_{n} ∪ {m(a_{1}, a_{2}) ; a_{1}, a_{2} ∈ A_{n} are consecutive sequences}.

4.7 The Closure of the Primary Bifurcations

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. Set b_{n} = b(a_{n}), 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 and ψ_{b}: (0, ∞) → ℝ satisfies

φ_{a}(μ) = ψ_{b}(μ) for μ ≥ μ_{a};

The maps φ_{a}|_{(0,μ_{a})} and ψ_{b}|_{(0,μ_{a})} are tangent at μ_{a}. The curves C_{a_{n}} converges to C_{a} and the curves C_{b_{n}} converges to C_{b}.

The intersections {P (a_{n}, b_{n})} = C_{a_{n}}∩C_{b_{n}} converges to {P (a, b)} = {(μ_{a}, φ_{a}(μ_{a}))}. Moreover, (μ, ν) ∈ C_{a} imply a(F_{(μ,ν)}) = a and (μ, ν) ∈ C_{b} imply b(F_{(μ,ν)}) = b.

In fact, the curve C_{a} is the limit of the sequence of curves C_{a_{n}} and the curve C_{b} is the limit of the sequence of curves C_{b_{n}}.

4.8 Secondary Bifurcations

Let a ∈ A_{∞}. Associated with a define A(a) = \{c ∈ Σ_{0}; c = \underline{a_{−}b_{+}a^{m}} or c = \underline{a_{−}b^{m}b_{+}}; m ∈ ℕ}. Denote b = b(a).

Lemma 13 For any c ∈ A(a) we can find a curve, C_{a}, such that

(i) \{(0, 0), P (a, b)\} ⊂ (\overline{C_{c}} \ C_{c});

(ii) C_{c} is located in the bounded region limited by C_{a} and C_{a_{−}\underline{b}} and

(iii) (μ, ν) ∈ C_{c} imply a(F_{(μ,λ)}) = c.

Figure 18:

The following figures describe these properties of the curves C_{c}.

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

Lemma 14 For any d ∈ B(b) we can find a curve, C_{d}, such that

(i) \{(0, 0), P (a, b)\} ⊂ (\overline{C_{d}} \ C_{d});

(ii) C_{d} is located in the bounded region limited by C_{b} and C_{b_{+}\underline{a}} and

(iii) (μ, ν) ∈ C_{d} imply b(F_{(μ,λ)}) = d.

The figures (19) and (20) describe these properties of the curves C_{d}.

Remark 2 1) We observe that: given c ∈ A(a) and d = sup σ^{k}(c); k ∈ ℕ then C_{c} ∩ C_{d} = ∅.

2) For any c ∈ A(a) define A(c) = \{γ ∈ Σ_{0}; γ = \underline{c_{−}d_{+}c^{m}} or γ = \underline{c_{−}d^{m}d_{+}} , m ∈ ℕ}, where d = a(c) . The same result is true for any γ ∈ A(c) (that is, C_{γ} is a curve located in a region bounded by C_{c} and C_{c_{−}\underline{d}} ; \{(0, 0) , P (a, b) \} ⊂ (overlineC_{γ} \ C_{γ})).

3) For elements c ∈ (\overline{A(a)} \ A(a)) we obtain the corresponding curve C_{c}, as the limit of the curves C_{c_{n}} c_{n} ∈ A(a) and c_{n} → c.

4) Given a ∈ A_{∞} , c ∈ A(a) and γ ∈ A(c) we can continue as in 2) above. Figure 19: Figure 20:

4.9 Tongues

Consider a ∈ A_{∞} and c ∈ A(a) or c = a. Lemma 15 Associated with any of these sequences there is a tongue L_{c_{−}b_{+}\underline{a}} = \{(μ, ν); a(F_{(μ,ν)}) = c_{−}b_{+}\underline{a}\} contained in R_{b_{+}\underline{a}}. Also, there are : a point Q(c) ∈ C_{c}, a curve C_{c_{−}b_{+}\underline{a}} and a point Q̃(c_{−}b_{+}\underline{a}) ∈ C_{c_{−}b_{+}\underline{a}} such that

(i) \{P (a, b), (0, 0)\} ⊂ \overline{C_{c_{−}b_{+}\underline{a}}}; (ii) [Q(c), P (a, b)] ⊂ C_{c}, [Q(c), Q̃(c_{−}b_{+}\underline{a})] ⊂ R_{b_{+}\underline{a}}, [Q̃(c_{−}b_{+}\underline{a}), P (a, b)] ⊂ C_{c_{−}b_{+}\underline{a}}

and

∂L_{c_{−}b_{+}\underline{a}} = [Q(c), P (a, b)] ∪ [Q(c), Q̃(c_{−}b_{+}\underline{a})] ∪ [Q̃(c_{−}b_{+}\underline{a}), P (a, b)].

The figure (21) displays these facts

Figure 21:

Now, consider d ∈ B(b) or d = b, b = b(a).

Lemma 16 Associated with any of these sequences there is a tongue R_{d_{+}a_{−}\underline{b}} = \{(μ, ν); b(F_{(μ,ν)}) = d_{+}a_{−}\underline{b}\} contained in L_{a_{−}\underline{b}}. Also, there are: a point Q(d) ∈ C_{d}, a curve C_{d_{+}a_{−}\underline{b}} and a point Q̃(d_{+}a_{−}\underline{b}) ∈ C_{d_{+}a_{−}\underline{b}} such that

(i) \{P (a, b), (0, 0)\} ⊂ C_{d_{+}a_{−}\underline{b}}

(ii) [Q(d), P (a, b) ⊂ C_{d}, [Q(d), Q̃(d_{+}a_{−}\underline{b})] ⊂ C_{a_{−}\underline{b}},

[Q̃(c_{−}b_{+}\underline{a}), P (a, b)] ⊂ C_{c_{−}b_{+}\underline{a}} and

∂R_{d_{+}a_{−}\underline{b}} = [Q(d), P (a, b)] ∪ [Q(d), Q̃(d_{+}a_{−}\underline{b})] ∪ [Q̃(d_{+}a_{−}\underline{b}), P (a, b)].

4.10 Completing the secondary bifurcations

Let a ∈ A_{∞}, A_{1}(a) = A(a) and define A_{2}(a) = A_{1}(a) ∪ {m(a_{1}, a_{2}); a_{1}, a_{2} ∈ A_{1}(a) are consecutive sequences}.

Inductively, A_{n+1}(a) = A_{n}(a) ∪ {m(a_{1}, a_{2}); a_{1}, a_{2} ∈ A_{1}(a) are consecutive sequences} and A_{∞}(a) = ⋃_{n=1}^{∞} A_{n}(a).

For any c ∈ A_{∞}(a) we have a curve, C_{c}, as in 4.8, also a tongue, L_{c_{−}b_{+}\underline{a}}, as in 4.9.

Take a ∈ A_{∞}, b = b(a), B_{1}(b) = B(b) and B_{2}(b) = B_{1}(b) ∪ {m(d_{1}, d_{2}); d_{1}, d_{2} ∈ B_{1}(b) are consecutive sequences}. Inductively, B_{n+1}(b) = B_{n}(b)∪{m(d_{1}, d_{2}); d_{1}, d_{2} ∈ B_{n}(b) are consecutive sequences} and B_{∞}(b) = ⋃_{n=1}^{∞} B_{n}(b).

For any d ∈ B_{∞}(b) we have a curve, C_{d}, as in 4.8 and a tongue, L_{d_{+}a_{−}\underline{b}}, as in 4.9.

4.11 The Doubling Period Sequences

Let a ∈ A_{∞}; c ∈ A_{∞}(a). Define

a_{0} = c, b_{0} = b(c); a_{1} = \underline{(a_{0})_{−}(b_{0})_{+}},
b_{1} = \underline{(b_{0})_{+}(a_{0})_{−}} and a_{n+1} = \underline{(a_{n})_{−}(b_{n})_{+}},
b_{n+1} = \underline{(b_{n})_{+}(a_{n})_{−}}, n ≥ 1.

For any a_{n}, we have a curve, C_{a_{n}}, as in 4.8 and a tongue, L_{(a_{n})_{−}(a(a_{n}))_{+}} \underline{b_{n}}, as in 4.9.

For any b_{n}, we have a curve, C_{b_{n}}, as in 4.8 and a tongue, R_{(b_{n})_{+}(a(b_{n}))_{−}} \underline{b_{n}}, as in 4.9.

For a = lim_{n→∞} a_{n} and b = lim_{n→∞} b_{n} we have a curve as in 4.8.

4.12 \underline{0} and \underline{1} tongues.

Let a ∈ Min_{2} be any periodic sequence. Associated with the sequence ã = a_{−}\underline{1} there is a tongue Lã ⊂ R\underline{1} a curve Cã ⊂ R\underline{1} and an interval Iã ⊂ \{(1/4 , ν); ν ∈ [1/4 , 3/4] } such that:

• ∂\overline{Lã} = C_{a} ∩ Lã ∪ Cã ∪ Iã.

• (μ , ν) ∈ Lã imply a(μ, ν) = ã.

• (μ , ν) ∈ Iã ∪ Cã imply a(μ, ν) = ã.

• (0, 0) ∈ \overline{Cã} and \overline{C_{a}} is tangent to \overline{Cã} at (0, 0) .

The next figure(23) represent these facts. Let b ∈ Max_{2} be any periodic sequence.

Associated with the sequence b̃ = b_{+}\underline{0} there is a tongue Rb̃ ⊂ L\underline{0} a curve Cb̃ ⊂ L\underline{0} and an interval Ib̃ ⊂ \{(μ , 1/4); μ ∈ [1/4 , 3/4] \} such that:

• ∂\overline{Rb̃} = C_{b} ∩ Rb̃ ∪ Cb̃ ∪ Ib̃.

• (μ , ν) ∈ Rb̃ imply b(μ, ν) = b̃.

• (μ , ν) ∈ Ib̃ ∪ Cb̃ imply b(μ, ν) = b̃.

• (0, 0) ∈ \overline{Cb̃} and \overline{C_{b}} is tangent to \overline{Cb̃} at (0, 0) . The next figure(24) represent these facts.

Figure 23:

4.13 Completing the bifurcation diagram

Assume a ∈ Min_{2}, b ∈ Max_{2} , b ≠ b(a) and C_{a} ∩ C_{b} ≠ ∅.

(i) Let us assume that a and b are periodic sequences. In this situation we have:

– a) there are two curves, C_{a_{−}\underline{b}} and C\underline{a_{−}b_{+}}, tangents to C_{a} at P (a, b)((0, 0)) such that P (a, b) ∈ \overline{C_{a_{−}\underline{b}}} (or C\underline{a_{−}b_{+}}) and (0, 0) ∈ \overline{C_{a_{−}\underline{b}}} (or C\underline{a_{−}b_{+}}).

– b) there are two curves, C_{b_{+}\underline{a}} and C\underline{b_{+}a_{−}}, tangents to C_{b} at P (a, b)((0, 0)) such that P (a, b) ∈ \overline{C_{b_{+}\underline{a}}} (or P (a, b) ∈ \overline{C\underline{b_{+}a_{−}}}) and (0, 0) ∈ \overline{C_{b_{+}\underline{a}}} (or \overline{C\underline{b_{+}a}}).

– c) there is a curve, C_{a_{−}b_{+}\underline{a}}, tangent to C_{a} at P (a, b) and (0, 0) such that P (a, b) ∈ \overline{C_{a_{−}b_{+}\underline{a}}} and (0, 0) ∈ \overline{C_{a_{−}b_{+}\underline{a}}}.

– d) there is a curve, C_{b_{+}a_{−}\underline{b}}, tangent to C_{b} at P (a, b) and (0, 0) such that P (a, b) ∈ \overline{C_{b_{+}a_{−}\underline{b}}} and (0, 0) ∈ \overline{C_{b_{+}a_{−}\underline{b}}}.

Figure 25: (a) and (c)

(ii) Let us now assume that b is a periodic sequence and a it is not a periodic sequence. In this situation there is a curve, C_{b_{+}a}, tangent to C_{b} at P (a, b) and (0, 0). The figure (27) represent these facts.

Figure 26: (b) and (d)

(iii) Let us assume that a is a periodic sequence and b is not a periodic sequence. In this situation there is a curve, C_{a_{−}\underline{b}}, tangent to C_{a} at P (a, b) and (0, 0). The figure (28) represent these facts.

Figure 27: (ii)

4.14 Measure of the bifurcation set

Set π = {(μ, ν); μ ≥ 0, ν ≥ 0} : AA = {(μ, ν) ∈ π; G_{(μ,ν)} satisfy the Axiom A} and B = π \ AA.

For a ∈ A_{∞} let Δ_{a} = L_{a} ∪ L_{a_{−}\underline{b}} ∪ R_{b_{+}\underline{a}}, where b = b(a) and Δ_{∞} = ⋃_{a∈A_{∞}} Δ_{a} ∪ L\underline{0} ∪ R\underline{1}.

Lemma 17 a) m(Δ_{∞} \ AA) = 0 b) B contains open sets in π.

This result follows as in lemma ...in [12].

4.15 Entropy zero

For a ∈ A_{∞} let Γ_{a} = L_{a} ∪R_{b} ∪L_{a_{−}b_{+}\underline{a}} ∪R_{b_{+}a_{−}\underline{b}}, where b = b(a) and Γ_{∞} = ⋃_{a∈A_{∞}} Γ_{a} ∪ L_{0\underline{1}} ∪ R_{1\underline{0}}.

Figure 28: (iii)

Let π^{+} = {(μ, ν) ∈ π; μ ≠ 0 and ν ≠ 0}.

Lemma 18 EZ ∩ π^{+} = Γ_{∞}, here EZ denotes the set of parameter values (μ , ν) where the respective map G_{(μ,ν)} has entropy zero.

This result follows from theorem 1 in [13]

4.16 The Global Picture

Taking together all the results in this section the global picture of the bifurcation diagramme is the following:

5 The general case

Let us consider the five parameter family of maps G_{(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:

G_{((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 observe that all the previous results, for the expanding family, are (still) true when we fix the parameters α , c_{1} and c_{2} . That is, we can obtain the a and b - decomposition, associated to any two parameter family F_{(μ,ν)} = F_{(α,c_{1},c_{2},μ,ν)} in the same way as for the previous family. Nevertheless, the situation for topological equivalence is dramatically different as we will show in the next subsection.

5.1 Topologically equivalence for families

Assume F = {F_{(μ,ν)}; (μ, ν) ∈ π} and G = {G_{(μ,ν)}; (μ, ν) ∈ π} are two-parameter families of maps in DM_{0}.

Definition 6 We will say that F and G are \underline{equivalent} if there is a bijection ϕ: π → π such that ∀(μ, ν) ∈ π the map F_{(μ,ν)} is topologically equivalent to G_{ϕ(μ,ν)}.

For the five parameter family of maps G_{(α,c_{1},c_{2},μ,ν)} we have Lemma 19 There are parameter values (α, c_{1}, c_{2}) (α̃, c̃_{1}, c̃_{2}) such that the two parameter families F_{(μ,ν)} = G_{(α,c_{1},c_{2},μ,ν)} and H_{(μ,ν)} = G_{(α̃,c̃_{1},c̃_{2},μ,ν)} are not equivalent. Proof.

(i) We have G′(x) = 1 for x > 0 if and only if x(α, c_{1}) = (αc_{1})\frac{1}{1 − α}.

We have G′(x) = 1 for x < 0 if and only if y(α, c_{2}) = −(αc_{2})\frac{1}{1 − α}. (ii) The equation for a fixed point of the saddle node type, for x < 0, are

ν−c_{2}(−x)^{α} = x and x = −(αc_{2})\frac{1}{1 − α} , that is; ν = ν_{sn} = c_{2}(αc_{2})\frac{α}{1 − α} (1−α),

is the parameter value where the map G_{(μ,ν)} presents a saddle-node fixed point

at x(α, c_{2}) = −(αc_{2})\frac{1}{1 − α}.

(iii) Now we ask for the curve G(0^{+}) = −μ = fixed point in x < 0. We have the equations −μ = y_{2}(v) and ν − c_{2}(−y_{2}(v))^{α} = y_{2}(v), that is, ν − c_{2}μ^{α} = −μ, therefore; ν(μ) = c_{2}μ^{α} − μ.

This curve is tangent to the curve ν = ν_{sn} at the value μ = μ_{max} = (αc_{2})\frac{1}{1 − α} . (iv) Figure 29 displays the region of the parameters (μ, ν) such that a(μ, ν) = \underline{0}.

This region is given by {(μ, ν); 0 ≤ ν ≤ ν(μ); 0 ≤ μ ≤ μ_{max}} or 0 ≤ ν ≤ ν_{sn} for μ ≥ μ_{max}.

Figure 29:

(v) The set \{(μ, ν), b(μ, ν) = \underline{10}\} contains the curve ν\underline{10} = (\frac{μ}{c_{1}} )^{1/α}. See Figure 30 for this picture.

Figure 30:

(vi) We have ν\underline{10}(μ) = ν_{sn} if and only of

μ = μ̄ = c_{1}c_{2}^{α}(1 − α)^{α} · (αc_{2})\frac{α^{2}}{1 − α} .

(vii) In this situation

\frac{μ̄}{μ_{max}} = \frac{c_{1}c_{2}^{α}(1 − α)^{α}(αc_{2})\frac{α^{2}}{1 − α}}{(αc_{2})\frac{1}{1 − α}} = \frac{c_{1}}{c_{2}} \frac{1 − α}{α^{1+α}} = S(c_{1}, c_{2}, α).

It is clear that there are values (c_{1}, c_{2}, α) and (C̃_{1}, C̃_{2}, α̃) such that

S(c_{1}, c_{2}, α) < 1 < S(c̃_{1}, c̃_{2}, α̃).

In the first case the curve ν_{10}(μ) is like in figure 31 In the second case the curve ν_{10}(μ) is like in figure 32 Clearly, each case define a two parameter families as announced in the lemma .

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};

Figure 32:

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 RL to IMPA-Brazil and a visit of CM 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