Bifurcations of the Essential Dynamics of Lorenz maps on the real line and the bifurcation scenario

for the Linear Family ^{∗}

Rafael Labarca and Carlos Moreira

Dedicated to the memory of Professor Camilo Blázquez

1 Introduction

In a remarkable contribution a meteorologist, E.N. Lorenz [1], 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, [5], produced a work where he introduced symbolic dynamics in order to understand the topological equivalence classes for nearly similar attractors. At that time R.F. Williams,[7], 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,[6], 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. The class of maps introduced in this way is contained in the class of maps DM_{0} that we work with(see section 2.1).Associated to any f ∈ DM_{0} we have two kneading sequences (a_{f} , b_{f}) = I(f) (see section 2.3 for the definition of these sequences) that satisfies a_{f} = inf{σ^{k}(a_{f}) , k ∈ ℕ } , b_{f} = sup{σ^{k}(b_{f}) , k ∈ ℕ } and {a_{f} , b_{f} } ⊂ Σ_{a_{f} ,b_{f}} (here Σ_{a,b} denote the set ∩_{i=0}^{∞}σ^{−i}([a, b]) , see section 2.3 for details). These properties inspired ( [23])the following definitions. A sequence of two symbols a = (0 , …) ∈ Σ_{2} (resp. b = (1 , …) ∈ Σ_{2}) is called minimal ( resp. maximal) if a = inf{σ^{k}(a) , k ∈ ℕ} (resp. b = sup{σ^{k}(b) , k ∈ ℕ }. Here σ : Σ_{2} → Σ_{2} denotes the usual shift map and Σ_{2} denote the set of sequences θ : ℕ → {0, 1} . We will denote by Min_{2} (resp. Max_{2} ) the set of minimal(resp. maximal) sequences in Σ_{2} . The properties that satisfies the kneading sequences allow us to define the Lexicographical World as LW = {(a , b) ∈ Min_{2} ×Max_{2} , {a , b} ⊂ Σ_{a ,b} }. The itinerary I : DM_{0} → LW , f → (a_{f} , b_{f}) defines a continuous and surjective map(see section 2.4). We will say that the map f ∈ DM_{0} has essentially the same dynamics as the map g ∈ DM_{0} if I(f) = I(g) . It is clear that two topologically equivalent maps are essentially equivalent. Therefore, the lexicographical world provide a universal model for (essentially) equivalent dynamics in this context. This mean the following: given (a , b) ∈ LW then there is f ∈ DM_{0} such that Σ_{a ,b} = Σ_{a_{f} ,b_{f}} and a surjective map I_{f} : Γ_{f} → Σ_{a,b} such that I_{f} ∘ f = σ ∘ I_{f} and reciprocally ( see section 2.3 for the definition of the set Γ_{f} and the map I_{f} ). So, all the possible essential dynamics produced by elements in DM_{0} are modeled by elements in LW and it is natural to ask if there is a parameterized family α : U ⊂ ℝ^{k} → DM_{0} with the following property : given any (a, b) ∈ LW the set α^{−1}({(a, b)}) is non empty. We believe that the correct answer to this question is no.

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

In these directions are the main results of the present paper: we describe, in a precise way, which part of the lexicographical world is realized by the piecewise linear family

F : Q → DM_{0} , F (λ, μ) = F_{(λ,μ)} given by

F_{λ,μ}(x) = \{ \begin{matrix}(1 + λ)x + 1, & x < 0 \\ (1 + μ)x − 1, & x > 0 \end{matrix} \}

answering a question posed by Hubbard and Sparrow in [3]. Here Q = {(λ, μ) ∈ ℝ^{2}; −1 ≤ μ ≤ 1 and −1 ≤ λ ≤ 1 }. Our main results are that there is an uncountable set of sequences (a, b) ∈ LW such that (I ∘ F)^{−1}({(a, b)}) is empty( it is interesting to note that, in doing this, we prove that period doubling solenoid do not exists for this family of maps in an independent way from [12] and [15]) and the exact description of these sets when they are non empty.

This paper is organized as follows: in section 2 we state our results, in section 3 we introduce some symbolic dynamics and the lexicographical world and in section 4 we prove the results.

Some comments are in order : de Melo and Martens announced, in an unpublished paper [22], the existence of parametrized families of elements in DM_{0} 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. The exact way in which these families realized the dynamics is given in [?].

Finally, we have to mention that there are many authors which have investigated these families from the same and from another points of view (combinatorial, differentiable, geometrical or topological)( see the list of references). In particular, compare our results with those of Glendinning in [?].

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}} and the Essential Dynamics.

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 ([?]) 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 linear 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 [?]).

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 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 3.2 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-universal family} if ∀a ∈ Min_{2} there is a nonempty set , A(a) ⊂ U , such that a_{f} = a , ∀f ∈ 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 ∈ 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 linear 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 which we do not prove in this section are proved in [?].

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 section 4.1, primary sequences are associated with primary bifurcations.

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 consecutives 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) 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}.

The elements in (A_{∞}^{∞} \ A_{∞}^{0}) will be called \underline{secondary sequences}, as we will see in section 4.2 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} .

Lemma 4 For any a ∈ A_{∞}^{∞} and d ∈ B_{∞}^{∞} ∩ LW_{0}(a) we have that the map σ : Σ_{a ,d} → Σ_{a ,d} is topologically transitive.

As before let a ∈ A_{∞}^{∞} and d ∈ LW_{0}(a) be a sequence which is not periodic nor preperiodic. Set a_{last} = sup\{σ^{k}(a) ; σ^{k}(a) ∈ [a , 0\underline{1}] , k ∈ ℕ\} and d_{first} = inf\{σ^{k}(d) ; σ^{k}(d) ∈ [1\underline{0} , d] , k ∈ ℕ\}.

In the sequel int(F) will denote the interior of the set F .

Lemma 5 (1) If there is p ∈ ℕ such that a_{last} ∈ int(σ^{p}([a_{last} , 0d])) then σ_{a ,d} : Σ_{a ,d} → Σ_{a ,d} is topologically transitive.

(2) If there is p ∈ ℕ such that d_{first} ∈ int(σ^{p}([a_{last} 0d])) then σ_{a ,d} : Σ_{a ,d} → Σ_{a ,d} is topologically transitive.

(3) If there is p ∈ ℕ such that a_{last} ∈ int(σ^{p}([1a , d_{first}])) then σ_{a ,d} : Σ_{a ,d} → Σ_{a ,d} is topologically transitive.

(4) If there is p ∈ ℕ such that d_{first} ∈ int(σ^{p}([1a , d_{first}])) then σ_{a ,d} : Σ_{a ,d} → Σ_{a ,d} is topologically transitive.

Lemma 6 Assume that a ∈ Min_{2} , d ∈ Max_{2} satisfies (1) a < a(d) ≤ a_{last} or d_{first} ≤ b(a) < d ,

(2) There are p > 1 and q > 1 such that σ^{p}([a_{last} , 0d]) ⊂ [a_{last} , d_{first}] and σ^{q}([1a , d_{last}) ⊂ [a_{last} , d_{first}] . Then there are :

- an invariant set Λ ⊂ Σ_{a ,d},

- an open set U ⊂ Σ_{2} and an integer n_{1} ∈ ℕ such that :

σ^{n_{1}}(U ∩ Σ_{a ,d}) ⊂ Λ and U ∩ Λ = Φ

In particular, σ_{a ,d} : Σ_{a ,d} → Σ_{a ,d} is not a topological transitive map.

Remark 2 Given a ∈ A_{∞}^{∞} we have :

(i) σ_{a ,b_{+}\underline{a}} : Σ_{a ,b_{+}\underline{a}} → Σ_{a ,b_{+}\underline{a}} is not topologically transitive,

(ii) σ_{a_{−}b_{+}\underline{a} ,b_{+}\underline{a}} : Σ_{a_{−}b_{+}\underline{a} ,b_{+}\underline{a}} → Σ_{a_{−}b_{+}\underline{a} ,b_{+}\underline{a}} is not topologically transitive and

(iii) σ_{a_{−}\underline{b} ,b_{+}a_{−}\underline{b}} : Σ_{a_{−}\underline{b} ,b_{+}\underline{a}} → Σ_{a_{−}\underline{b} ,b_{+}a_{−}\underline{b}} is not topologically transitive.

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 Realization Lemma.

In this section we will prove a realization lemma with increasing maps.

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

Proof: Let a = a_{0} < a_{1} < … < (alast = a_{p}) < 0\underline{1} < 1\underline{0} < (blast = 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 a 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 8 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 a closed interval L_{s} such that

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

L_{p} ∩ L_{p+1} = {0} and 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 satisfy f(L_{i}) = L_{η(i)} and is increasing between L_{i} and L_{i+1} , i ≠ p , 0 ≤ i ≤ k − 1 □ .

Now we prove:

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.

Proof: Assume #(per(a)) = n and #(per(d) = m. Let c_{0} = a < c_{1} < … < c_{p} < 0\underline{1} < 1\underline{0} < c_{p+1} < … < c_{m+n−1} , denote the consecutive elements in \{σ^{j}(a) , σ^{j}(d) , j ∈ ℕ}. Associated to c_{j} consider closed intervals L_{s} ; 0 ≤ s ≤ m + n − 1 , such that

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

L_{P} ∩ L_{p+1} = {0} and L_{i} ∩ L_{i+1} = ∅, otherwise.

Let {i_{1} , i_{2} , …, i_{n}} and {j_{1} , j_{2} , …, j_{m}}, be the indices defined by c_{i_{s}} = σ^{t}(a) ; for some t and c_{j_{r}} = σ^{l}(d) for some l. Let η : {i_{1} , …, i_{n}} → {i_{1} , …, i_{n}} and ρ : {j_{1} , …, j_{m}} → {j_{1} , …, j_{m}} be the permutations defined by σ(a_{i_{s}}) = a_{η(i_{s})} and ρ(d_{j_{r}}) = d_{ρ(j_{r})} .

Now, define an increasing map f ∈ DM_{0} by f(L_{i_{r}}) = L_{η(i_{r})} and f(L_{j_{r}}) = L_{ρ(j_{r})} □.

By a limit process we obtain

Lemma 10 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}) .

As a consequence of these lemmas we get

Proposition 1 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 one obtained for expansive maps in [3].

3.3 The finite and the entropy zero cases.

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

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

Lemma 11 Given f ∈ DM_{0} a “essentially Morse-Smale” map we have that [f(0^{+}), f(0^{−})] ∩ Γ_{f} = ⋃_{i=0}^{per(a_{f})−1} 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.

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 ). Actually, see [?], this problem reduces to find the region where the previous definition applies. For our linear family of Lorenz maps we will solve this problem in section 4.1.

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

3.4 The topologically expansive cases

Definition 6 A map f ∈ DM_{0} will be called \underline{topologically expansive} if ⋃_{i=0}^{∞} f^{−i}(0) is a dense set in [f(0^{+}), f(0^{−})].

Let denote by T E_{0} = {f ∈ DM_{0}; f is a topologically expansive map} and T E = I(T E_{0}) ⊂ LW.

Note: It is clear that EMS_{0} ∩ T E_{0} = ∅ and, as a consequence, (LW \ T E) ≠ ∅.

In fact, we have the following result due to Hubbard and Sparrow ([3]).

Lemma 12 ([3]) (a, b) ∈ T E if and only if (a, b) ∈ LW and there are not finite sequences d = 1d_{1}, …, d_{k}; c = 0c_{1} … c_{p} such that a = cb or b = da.

4 Proof of Our Results

Let Q = {(λ, μ) ∈ ℝ^{2}; −1 ≤ μ ≤ 1 and −1 ≤ λ ≤ 1 }.

Let F : Q → DM_{0} be the two-parameter family, F_{λ,μ} = F_{(λ,μ)} given by

F_{λ,μ}(x) = \{ \begin{matrix}(1 + λ)x + 1, & x < 0 \\ (1 + μ)x − 1, & x > 0 \end{matrix}

4.1 Primary Bifurcations

Assume a ∈ A_{0}^{∞}. Lemma 13 There is a non empty region L_{a} ⊂ Q such that (λ, μ) ∈ Q imply a(λ, μ) = a(F_{λ,μ}) = a. Proof: It is not difficult to see that the regions are given by the next figure.

Figure 1:

Corollary 2 a) For any a ∈ (\overline{A_{0}^{∞}} \ A_{0}^{∞}) the set L_{a} ⊂ Q is a curve.

b) Let G = {(λ, μ) ; −1 ≤ μ ≤ 1 , −μ < λ ≤ 1 }. Then

Q \ G = {(λ , μ) ; F_{(λ,μ)}is an entropy zero map }.

4.2 Secondary Bifurcations

In the sequel we are only interesting in the bifurcation scenario in D = \overline{G}.

Lemma 14 There is (a, b) ∈ (T E \ I ∘ F (D)). As a consequence: the linear family do not contains a universal model for topologically expanding maps.

Proof: Let us consider a_{0} = \underline{01}, b_{0} = \underline{10}, a_{1} = \underline{(a_{0})_{−}(b_{0})_{+}} = \underline{0011}, b_{1} = \underline{(b_{0})_{+}(a_{0})_{−}} = \underline{1100} and a_{n+1} = \underline{(a_{n})_{−}(b_{n})_{+}}, b_{n+1} = \underline{(b_{n})_{+}(a_{n})_{−}}; n ≥ 1. The sequences a = lim_{n→∞} a_{n}, b = lim_{n→∞} b_{n} satisfies the conditions of the Lemma in III.5.1. It is easy to see that associated to a_{n} and b_{n} there are curves T E(a_{n}), T E(b_{n}) that satisfies

(1) (λ, μ) ∈ T E(a_{n}) implies a(F_{λ,μ}) = a_{n} and (λ, μ) ∈ T E(b_{n}) implies b(F_{λ,μ}) = b_{n}

In fact; for a_{0} = \underline{01} we have T E(\underline{01}) = \{(0, μ); 0 ≤ μ ≤ 1\} and for b_{0} = \underline{10} we have T E(\underline{10}) = \{(λ, 0); 0 ≤ λ ≤ 1\}.

Let A_{μ}^{0} = [0^{+}, μ] and its second image F^{2}(A_{μ}^{0}) = [−λ, (1+λ)(μ(1+μ)−1)+1]. When (1 + λ)(μ(1 + μ) − 1) + 1 = 0 we get the sequence b_{1} = \underline{1100}. That is

T E(\underline{1100}) = \{(\frac{μ(μ + 1)}{1 − μ(1 + μ)}; μ); 0 ≤ μ ≤ −\frac{1 + \sqrt{3}}{2}\}

In a symmetric way we get the curve T E(\underline{0011}) = \{(λ, μ(λ)), 0 ≤ λ ≤ −\frac{1+\sqrt{3}}{2}\}.

Clearly, T E(\underline{1100}) ⊂ \{(λ, μ); μ < λ\} and T E(\underline{0011}) ⊂ \{(λ, μ); μ > λ\}. Moreover λ′(0) = μ′(0) = 1.

Now, for (1 + λ)(μ(1 + μ) − 1) + 1 ≤ μ we have F^{2}(A_{μ}^{0}) ⊂ [−λ, μ] and we can define F_{1} : [0^{+}, μ] → [−λ, μ] by

F_{1}(x) = (1 + λ)(1 + μ)x − λ.

Similarly, for B_{λ}^{0} = [−λ, 0^{−}] we have F^{2}(B_{λ}^{0}) = [(1 + μ)[1 − λ(1 + λ)] − 1, μ]. For −λ ≤ (1 + μ)[1 − λ(1 + λ)] − 1 we have F^{2}(B_{λ}^{0}) ⊂ [−λ, μ] and we can define F_{1} : [−λ, 0^{+}] → [−λ, μ] by

F_{1}(x) = (1 + μ)(1 + λ)x + μ.

Using the map F_{1} we can start again and we will obtain T E(\underline{11010010}) which is a curve {(λ(μ); μ); 0 ≤ μ ≤ μ_{1}} where λ(μ) satisfy the equation

(1 + λ)^{3}(1 + μ)^{3}μ − (1 + λ)^{2}(1 + μ)^{2}λ − (1 + λ)(1 + μ)λ + μ = 0

and

8(1 + μ_{1})^{3}μ_{1} − 4(1 + μ_{1})^{2} − 2(1 + μ_{1}) + μ_{1} = 0.

Inductively we get the result.

(2) (0, 0) ∈ \overline{T E(a_{n})} and (0, 0) ∈ \overline{T E(b_{n})}

(3) T E(a_{n}) ⊂ {(λ, μ) ∈ [0, 1] × [0, 1]; λ < μ}

and T E(b_{n}) ⊂ {(λ, μ) ∈ [0, 1] × [0, 1]; μ < λ}

(4) T E(a_{n}) → T E(a) and T E(b_{n}) → T E(b) with

T E(a) ⊂ {(λ, μ) ∈ [0, 1] × [0, 1]; λ < μ}
T E(b) ⊂ {(λ, μ) ∈ [0, 1] × [0, 1]; μ < λ}

(5) (λ, μ) ∈ T E(a) imply a(F_{(λ,μ)}) = a and(λ, μ) ∈ T E(b) imply b(F_{(λ,μ)}) = b.

So, we conclude (a, b) ∈ [T E \ I ∘ F (D)] as we announced in the lemma.

3.5.3 Some remarks concerning the family F_{λ,μ}.

a) Let A_{0}, A_{1}, …, A_{∞} be defined as in I.5.(c). Let a ∈ A_{∞}. There is a parameter value (−μ_{a}, μ_{a}) ∈ D such that a(−μ_{a}, μ_{a}) = a; b(−μ_{a}, μ_{a}) = b(a).

b) Associated to any a ∈ A_{∞} there are two curves T E(a), T E(b(a)) such that

(λ, μ) ∈ T E(a) imply a(F_{(λ,μ)}) = a

and

(λ, μ) ∈ T E(b(a)) imply b(F_{(λ,μ)}) = b(a)

Moreover

T E(a) ∩ T E(b(a)) = {(−μ_{a}, μ_{a})}

For example, take a = \underline{01_{n}}, b = \underline{1_{n}0}. In this case we have

1 →^{f} μ →^{f} (1 + μ)μ − 1 →^{f} (1 + μ)^{2}μ − (1 + μ) − 1 →^{f} …

→^{f} (1 + μ)^{k}μ − (1 + μ)^{k−1} − … − (1 + μ) − 1.

Therefore

f^{k+1}(1) = ξ_{k+1}(μ) = (1 + μ)^{k}μ − (1 + μ)^{k−1} − … − 1

Observe:

1) ξ_{k+1}(0) = 0 − \underbrace{1 − 1 … − 1}_{k} = −k

2) ξ_{k+1}(1) = 2^{k} − (2^{k−1} + … + 1) = 2^{k} − \frac{2^{k}−1}{2−1} = 1.

3) ξ′_{k+1}(μ) = k(1 + μ)^{k−1}μ + (1 + μ)^{k} − (k − 1)(1 + μ)^{k−2} − 1 or, since

ξ_{k+1}(μ) = (1 + μ)^{k}μ − \frac{(1+μ)^{k}−1}{1+μ−1} = (1 + μ)^{k}μ − \frac{(1+μ)^{k}−1}{μ}, then

ξ′_{k+1}(μ) = k(1 + μ)^{k−1}μ + (1 + μ)^{k} − \frac{k(1 + μ)^{k−1}μ − [(1 + μ)^{k} − 1]}{μ^{2}}, μ ≠ 0.

4) Let μ_{k} ⋔ ξ_{k}(μ_{k}) = 0. Clearly μ_{k} → 1.

The equation h_{n} = μ_{n−1} correspond to b = \underline{1_{n}o}. These curve satisfies h_{n} → −1 with n → ∞ and so on.

(c) The curve T E(a) transversally intersects the curves μ = μ_{0} for μ_{a} ≤ μ_{0} ≤ 1.

The curve T E(b(a)) transversally intersects the curves λ = λ_{0} for − μ_{a} ≤ λ_{0} ≤ 1.

Moreover,for a_{n} = 0_{n}1 …, we have that T E(a_{n}) converges to {(1, μ); −1 ≤ μ ≤ 1} in the C^{1}-topology. For b_{n} = 1_{n}0 …, we have that T E(b_{n}) converges to {(λ, 1); −1 ≤ λ ≤ 1} in the C^{1}-topology.

(d) The curves T E(10_{n} …) converges to {(1, −1)} and the curves T E(01_{n} …) converges to {(−1, 1)} with n → ∞.

(e) Given a ∈ A_{∞}, b = b(a), we can find curves T E(b_{+}\underline{a}), T E(b_{+}a_{−}\underline{b}), T E(a_{−}b_{+}\underline{a}) and T E(a_{−}\underline{b}) such that:

(1) (−μ_{a}, μ_{a}) ∈ \overline{T E(c)} for c = b_{+}\underline{a}, b_{+}a_{−}\underline{b}, a_{−}b_{+}\underline{a}, a_{−}\underline{b};

(2) T E(a_{−}b_{+}\underline{a}) and T E(b_{+}\underline{a}) are tangents to T E(a) at (−μ_{a}, μ_{a});

(3) T E(a − \underline{b}) and T E(b_{+}a_{−}\underline{b}) are tangents to T E(b) at (−μ_{a}, μ_{a})

Figure (1) represents these essential facts concerning the geometry of these curves.

(4) For a ∈ A_{∞} let

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

.

(i) Associated with any c = \underline{a_{−}b_{+}a^{n}} we can find a direction v_{n} ∈ ℝ^{2}, v_{n} → v_{a}, v_{a} = the direction of the curve T E(a) at (−μ_{a}, μ_{a}); – Associated with any c = \underline{a_{−}b^{n}b_{+}} we can find a direction v_{n} ∈ ℝ^{2}, v_{n} → v_{b},

v_{b} = the direction of the curve T E(b) at (−μ_{a}, μ_{a})

(ii) Associated with any c = \underline{a_{−}b_{+}a^{n}} (c = \underline{a_{−}b^{m}b_{+}}) we can find two curves T E(c) and T E(\underline{b_{+}a^{n}a}) (resp. T E(c), T E(\underline{b_{+}a_{−}b^{m}})) tangents to the direction v_{n} (resp. v_{m}) and located is different sides of the direction v_{n} (resp. v_{m}) that satisfies

(λ, μ) ∈ T E(c) imply a(F_{(λ,μ)}) = c
(λ, μ) ∈ T E(\underline{b_{+}a^{n}a}) imply b(F_{(λ, μ)}) = \underline{b_{+}a^{n}a}
[resp. (λ, μ) ∈ T E(c) imply a(F_{(λ,μ)}) = c and
(λ, μ) ∈ T E(\underline{b_{+}a_{−}b^{m}}) imply b(F_{(λ,μ)}) = \underline{b_{+}a_{−}b^{m}}]

Figure (2) represents these essential facts.

(iii) Given c ∈ A_{1}^{1}(a), , b = b(c), we can find curves T E(b_{+}\underline{c}), T E(b_{+}c_{−}\underline{b}), T E(c_{−}b_{+}\underline{c}) and T E(c_{−}\underline{b}) such that:

(a) (λ, μ) ∈ T E(b_{+}\underline{c}) imply b(F_{(λ,μ)}) = b_{+}\underline{c};

∗ (λ, μ) ∈ T E(b_{+}c_{−}\underline{b}) imply b(F_{(λ,μ)}) = b_{+}c_{−}\underline{b};

∗ (λ, μ) ∈ T E(c_{−}b_{+}\underline{c}) imply a(F_{(λ,μ)}) = c_{−}b_{+}\underline{c};

∗ (λ, μ) ∈ T E(c_{−}\underline{b}) imply a(F_{(λ,μ)}) = c_{−}\underline{b};

(b) These four curves are tangents to the corresponding direction associated to c;

(c) (−μ_{a}, μ_{a}) ∈ \overline{T E(d)} for d = b_{+}\underline{c}, b_{+}c_{−}\underline{b}, c_{−}b_{+}\underline{c} and c_{−}\underline{b}.

Figure 2:

Figure 3:

Figure 4:

Figure (3) represents these essential facts.

Remark: ∀ a ∈ A_{∞} we have that inf A_{1}^{1}(a) = a_{−}b_{+}\underline{a} and sup A_{1}^{1}(a) = a_{−}\underline{b}.

(5) Given a ∈ A_{∞} let A_{2}^{1}(a) = A_{1}^{1}(a) ∪ {m(a_{1}, a_{2}) ; a_{1}, a_{2} ∈ A_{1}^{1}(a) are consecutive sequences}.

(i) Associated with c = m(a_{1}, a_{2}) we can find a direction, v_{c} ∈ ℝ^{2}, located between v_{a_{1}} and v_{a_{2}};

(ii) Associated with c = m(a_{1}, a_{2}) we can find two curves, T E(c) and T E(b(c)), tangents to the direction v_{c} and located in different sides of the direction v_{c} that satisfies:

(λ, μ) ∈ T E(c) imply a(F_{(λ,μ)}) = c;
(λ, μ) ∈ T E(b(c)) imply b(F_{(λ,μ)}) = b(c);

(iii) Given c ∈ A_{2}^{1}(a) we can continuous as in (4.iii).

In the same way, as we did with A_{1}^{1}andA_{2}^{1} we can continuous with the set A_{n+1}^{1}(a) = A_{n}^{1}(a)∪{m(a_{1}, a_{2}) ; a_{1}, a_{2} ∈ A_{n}^{1} are consecutive sequences} , n ≥ 2. Let A_{∞}^{1}(a) = ⋃_{n=1}^{∞} A_{n}^{1}(a) and A_{∞}^{1} = ⋃_{a∈A_{∞}^{0}} A_{∞}^{1}(a) ∪ A_{∞}^{0}. Inductively, we make the same construction for elements in A_{∞}^{n+1}, where: A_{∞}^{n+1} = ⋃_{a∈A_{∞}^{n}} A_{∞}^{n}(a) ∪ A_{∞}^{n}. Let A_{∞}^{∞} = ⋃_{n=0}^{∞} A_{∞}^{n}. The same construction, as we did in (4.iii), holds for any c ∈ A_{∞}^{∞}.

(6) Let us now consider a ∈ A_{∞}^{∞} and b ∈ B_{∞}^{∞} such that T E(a) ∩ T E(b) ≠ ∅ we have:

(i) The intersection between these curves is transversal and contains a unique point P (a, b);

(ii) Define A(a, b) = \{c ∈ Σ_{0}; c = \underline{a_{−}b_{+}a^{n}}; n ∈ ℕor c = \underline{a_{−}b^{m}b_{+}}; m ∈ ℕ}. Associated with any c ∈ A(a, b) we can make the same construction as we did in (4) and (5) before.

(7) Let a ∈ A_{∞}^{∞} and d ∈ Max_{2} be a sequence which is eventually periodic such that T E(d) ∩ T E(a) ≠ ∅. We have:

(i) The intersection between these two curves is transversal and contains a unique point P (a, d);

(ii) We can find a curve T E(a_{−}d), tangent to T E(d) and such that P (a, d) ∈ \overline{T E(a_{−}d)} which satisfy:

(λ, μ) ∈ T E(a_{−}d) imply a(F_{(λ,μ)}) = a_{−}d.

Figure (4) represents these facts.

(8) Let b ∈ B_{∞}^{∞} and c ∈ Min_{2} be an eventually periodic sequence such that T E(c) ∩ T E(b) ≠ ∅. We have that:

(i) The intersection between these two curves is transversal and contains a unique point P (c, b);

(ii) We can find a curve, T E(b_{+}c), tangent to T E(c) such that P (c, b) ∈ \overline{T E(b_{+}c)} which satisfy

(λ, μ) ∈ T E(b_{+}c) imply b(F_{(λ,μ)}) = b_{+}c.

Figure (5) represents these facts Figure 5:

Figure 6:

(9) Let a ∈ A_{∞}^{∞} and d ∈ Max_{2} be a sequence which is not periodic nor eventually periodic such that T E(a) ∩ T E(d) ≠ ∅. We have that:

(i) The intersection between these two curves is transversal and contains a unique point P (a, d);

(ii) We can find a curve, T E(a_{−}d), tangent to T E(d) and such that P (a, d) ∈ \overline{T E(a_{−}d)} which satisfy:

(λ, μ) ∈ T E(a_{−}d) imply a(F_{(λ,μ)}) = a_{−}d.

(10) Let b ∈ B_{∞}^{∞} and c ∈ Min_{2} be a sequence which is not periodic nor eventually periodic such that T E(b) ∩ T E(c) ≠ ∅. We have that:

(i) The intersection between these two curves is transversal and contains a unique point P (c, b); (ii) We can find a curve, T E(b_{+}c), tangent to T E(c) and such that P (c, b) ∈

\overline{T E(b_{+}c)} which satisfy:

(λ, μ) ∈ T E(b_{+}c) imply b(F_{(λ,μ)}) = b_{+}c.

(11) F_{(λ,μ)} is conjugated to F_{(λ̃,μ̃)} if and only if

a(F_{(λ,μ)}) = a(F_{(λ̃,μ̃)})

and

b(F_{(λ,μ)}) = b(F_{(λ̃,μ̃)}).

In fact, any (λ, μ) ∈ D represents a unique equivalence class of elements in F (D).

Remark 3 Theorem 1 is a consequence of the results in section 4.

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