Maximal Invariant Sets for Restrictions of Tent and Unimodal Maps

Carlos Gustavo Tamm de Araujo Moreira

Instituto de Matemática Pura e Aplicada

Estrada Dona Castorina, 110 22460-320, Rio de Janeiro-RJ E-mail: gugu@impa.br

Abstract

The aim of this paper is to study the restriction of a tent map f: I → I to a subinterval J ⊂ I, and to characterize when the maximal invariant of f|_{J} is a countable set. We shall extend our results for a certain class of unimodal maps, via a semi-conjugacy. In the case of tent maps there is a critical case where the maximal invariant set is a Cantor set with Hausdorff dimension zero. The caracterization of this critical case leads to interesting arithmetical phenomena, particularly in the case of the complete tent map.

1 Introduction

The classical family of tent maps is the family (T_{λ})_{1<λ≤2} defined by

T_{λ}(x) = \begin{cases}λx, if 0 ≤ x ≤ \frac{1}{2} \\ λ(1 − x), if \frac{1}{2} ≤ x ≤ 1 \end{cases} .

It is the simplest continuous family of unimodal maps, and gives topological models for many differentiable unimodal maps (for instance, the complete tent map T_{2} is conjugated to f(x) = x^{2} − 2 in [-2,2] via the homeomorphism h: [0, 1] → [−2, 2], h(x) = 2 cos(πx)).

The aim of this work is to study restrictions of a tent map T_{λ} to closed subintervals J ⊂ [0, 1], and characterize when the maximal invariant set K = K_{λ,J} of T_{λ}|_{J} is an uncountable set, when it has positive Hausdorff dimension (see definition below) and when it has positive Lebesgue measure (case in which it must be a nondegenerate interval). Our main result is that, given λ ∈ (1, 2], there is an explicitly given interval [a(λ), b(λ)] ⊂ [0, 1] such that K is uncountable if and only if J contains [a(λ), b(λ)] and it has positive Hausdorff dimension if and only if the interior of J contains [a(λ), b(λ)]. In the case λ = 2 we relate a(λ) and b(λ) with the classical Morse-Thue constant and use it to show their transcendence.

We also discuss generalizations of our results to C^{2} unimodal maps, via the study of (semi) conjugacies between them and tent maps.

Let us finish this introduction with some definitions:

Given a metric space X and a positive real number α, we define the Hausdorff measure of dimension α associated to a covering U = (U_{λ})_{λ∈L} of X by bounded sets U_{λ} by m_{α}(U) = ∑_{λ∈L}(diam U_{λ})^{α}, where diam U_{λ} denotes the diameter of U_{λ}, and, if we define the norm of a covering U by ||U|| = sup_{U∈U}(diam U), then the Hausdorff measure of dimension α of the set X is m_{α}(X) = lim inf_{U covering of X ||U||→0} m_{α}(U).

It is not difficult to see that there is a unique d ∈ [0, +∞] such that if α > d then m_{α}(X) = 0 and if α < d then m_{α}(X) = +∞. This number d is called the Hausdorff dimension of X. It is easy to see that if X ⊂ ℝ then its Hausdorff dimension d =: HD(X) belongs to [0, 1].

Given a Cantor set K ⊂ ℝ, a gap of K is a connected component of the complement of K. Given a bounded gap U, let U_{L} and U_{R} the nearest gaps respectively at the left and at the right of U that are bigger than U, and let I_{L} and I_{R} be the intervals between U_{L} and U and between U and U_{R}, respectively. We define τ(U) = min{|I_{L}|/|U|, |I_{R}|/|U|}, and we define the thickness of K by τ(K) = inf{τ(U), U bounded gap of K}. Its importance follows from Newhouse’s gap Lemma (see [P-T]), according to which if K_{1} and K_{2} are Cantor sets on the real line such that τ(K_{1}) · τ(K_{2}) ≥ 1 then either K_{1} is contained in a gap of K_{2} , K_{2} is contained in a gap of K_{1} or K_{1} ∩ K_{2} ≠ ∅. It is possible to prove that we always have HD(K) ≥ log 2/ log (2 + \frac{1}{τ(K)}). In particular, if τ(K) > 0 then HD(K) > 0. The equality holds for the affine Cantor sets K_{α} with α ∈ (0, \frac{1}{2}), which are defined as the maximal invariant

set of φ_{α}: [0, α] ∪ [1 − α, 1] → [0, 1], given by φ_{α}(x) = \begin{cases}1 − x/α, if x ∈ [0, α] \\ (x − 1 + α)/α, if x ∈ [1 − α, 1] \end{cases} ,

for which τ(K_{α}) = \frac{α}{1−2α} and HD(K_{α}) = − log 2/ log α.

2 The Case of Tent Maps

Let 1 < λ ≤ 2. The tent map T_{λ}: [0, 1] → [0, 1] is defined by

T_{λ}(x) = \begin{cases}λx, & if x ≤ \frac{1}{2} \\ λ(1 − x), & if x ≥ \frac{1}{2} \end{cases}

Since T_{λ}([0, 1]) = [0, \frac{λ}{2}], if M_{J} := ⋂_{n=0}^{∞} T^{−n}(J) is the maximal invariant of T_{λ}|_{J}, we have M_{J} = M_{J∩[0, \frac{λ}{2} ]} ∪ (J ∩ T_{λ}^{−1}(M_{J∩[0, \frac{λ}{2} ]})). So, we may restrict our attention to the case J ⊂ [0, \frac{λ}{2}]. Moreover, if x ≥ λ(1 − \frac{λ}{2}) = T_{λ}(\frac{λ}{2}) and x ∈ [0, \frac{λ}{2}] then T_{λ}(x) ≥ λ(1 − \frac{λ}{2}), so the interval L = [λ(1 − \frac{λ}{2}), \frac{λ}{2}] is T_{λ}-invariant, and M_{J∩[0, \frac{λ}{2} ]} = J ∩ ⋃_{r=0}^{∞} λ^{−r}M_{J∩L}, and so we may restrict our attention to the case J ⊂ [λ(1 − \frac{λ}{2}), \frac{λ}{2}].

Figure 1 above shows the dynamics of T_{λ} and the interval L.

Let J = [λ(1 − \frac{λ}{2}) + ε, \frac{λ}{2}]. If x ∈ M_{J} then x ≤ \frac{λ}{2} − \frac{ε}{λ}, since if x > \frac{λ}{2} − \frac{ε}{λ} then T_{λ}(x) < λ(1 − \frac{λ}{2}) + ε ⇒ M_{J} = M_{[λ(1− \frac{λ}{2} )+ε, \frac{λ}{2} − \frac{ε}{λ}]}. On the other hand, if J = [λ(1 − \frac{λ}{2}), \frac{λ}{2} − ε] then, since [λ(1 − \frac{λ}{2}) + ελ, \frac{λ}{2} − ε] = J ∩ T_{λ}(J), we have M_{J} ⊆ ⋃_{r=0}^{∞} λ^{−r}M_{[λ(1− \frac{λ}{2} )+ελ, \frac{λ}{2} −ε]}, and so we may restrict our attention to the intervals J_{ε} = [λ(1 − \frac{λ}{2}) + ελ, \frac{λ}{2} − ε] =: [a_{ε}, b_{ε}]. Indeed, the discussion above implies that if J = [λ(1− \frac{λ}{2})+δ, \frac{λ}{2} −η], then J ⊃ J_{ε}, where ε = max\{\frac{δ}{λ}, η\} and M_{J} ⊆ ⋃_{r=0}^{∞} λ^{−r}M_{J_{ε}} (see Figure 2). T_{λ}|_{J_{ε}} is conjugated to the function ψ_{λ,c}: [0, \frac{1}{λ}] ∪ [1 − \frac{c}{λ}, 1] defined by

ψ_{λ,c}(x) = \begin{cases}1 − λx, & if x ∈ [0, \frac{1}{λ}] \\ λ(x − 1 + \frac{c}{λ}), & if x ∈ [1 − \frac{c}{λ}, 1], \end{cases}

where

c = \frac{(\frac{b_{ε}}{λ} − a_{ε}) · λ}{b_{ε} − a_{ε}} = \frac{b_{ε} − λa_{ε}}{b_{ε} − a_{ε}} = \frac{\frac{λ}{2} − ε − λ^{2}(1 + ε − \frac{λ}{2})}{\frac{λ}{2} − ε − λ(1 + ε − \frac{λ}{2})}

= \frac{λ(λ − 1)^{2} − 2(λ^{2} + 1)ε}{λ(λ − 1) − 2(λ + 1)ε}

= λ − 1 − \frac{4ε}{λ(λ − 1) − 2(λ + 1)ε} < λ − 1 for ε > 0.

by the affine conjugacy h(x) = \frac{b_{ε}−x}{b_{ε}−a_{ε}} (see Figure 3).

We shall study the maximal invariant of ψ_{λ,c} in [0, 1]. Let

ψ̃_{λ,c} = \begin{cases}c − λ^{2}x, & if x ∈ [0, \frac{c}{λ^{2}} ] \\ 1 − λ + λ^{2}x, & if x ∈ [\frac{λ−1}{λ^{2}}, \frac{1}{λ}] \\ λ(x − 1 + \frac{c}{λ}), & if x ∈ [1 − \frac{c}{λ}, 1] \end{cases} = \begin{cases}ψ_{λ,c}^{2}, & if x ∈ [0, \frac{1}{λ}] ∩ ψ_{λ,c}^{−2}([0, 1]) \\ ψ_{λ,c}, & if x ∈ [1 − \frac{c}{λ}, 1] \end{cases} .

(See Figure 4).

Clearly the maximal invariants of ψ̃_{λ,c} and ψ_{λ,c} in [0, 1] are equal. Let λ − 1 > p_{λ} = \frac{1}{λ+1} be the fixed point of ψ_{λ,c}. We have the following possibilities:

i) c ≥ p_{λ}: In this case the maximal invariant of ψ_{λ,c} contains (indeed is equal to) a Cantor set. Indeed, if Ĩ = [\frac{c−p_{λ}}{λ^{2}} , p_{λ}] then the maximal invariant of ψ̃_{λ,c}|Ĩ is a regular Cantor set, with positive Hausdorff dimension and positive thickness (see Figure 5).

ii) c < \frac{λ−1}{λ^{2}}: In this case the maximal invariant of ψ_{λ,c} is countable. Indeed, in this case [0, c] is ψ̃_{λ,c}-invariant, and the maximal invariant M_{[0,1]} is equal to ⋃_{r=0}^{∞} ψ̃_{λ,c}^{−r}(M_{[0,c]}), but M_{[0,c]} = \{\frac{c}{λ^{2}+1}\}, since ψ̃_{λ,c}|_{[0,c]} = c − λ^{2}x, if x ∈ [0, \frac{c}{λ^{2}} ], and is not defined otherwise (see Figure 6). iii) \frac{λ−1}{λ^{2}} ≤ c ≤ \frac{1}{λ+1} = p_{λ}: In this case [0, c] is still ψ̃_{λ,c}-invariant, and M_{[0,1]} = ⋃_{r=0}^{∞} ψ̃_{λ,c}^{−r}(M_{[0,c]}).

Therefore, M_{[0,1]} is a Cantor set (resp. countable) iff M_{[0,c]} is a Cantor set (resp. countable). On the other hand, ψ̃_{λ,c}|_{[0,c]} is conjugated by h(x) = \frac{x}{c} to ψ_{λ^{2},λ^{2}− \frac{λ−1}{c}}. We have now a renormalization operator F (λ, c) = (λ^{2}, λ^{2} − \frac{λ−1}{c}) (see Figure 7).

Lemma 2.1. Given λ, there is a unique value of c, denoted by c(λ), such that we can iterate infinitely many times the renormalization operator F beginning with (λ, c). This value is given by the expression c(λ) = (λ − 1)\frac{α(λ)}{1+α(λ)}, where α(λ) = ∑_{r=1}^{∞} \frac{1}{∏_{j=1}^{r}(λ^{2^{j}} −1)}.

Remark. The characterization of the maximal invariant set of ψ_{λ,c} will be as follows:

• λ − 1 > c > c(λ) ⇒ M_{[0,1]} is a Cantor set with positive Hausdorff dimension. (Since the case i) will occur after a certain number of renormalizations)

• c < c(λ) ⇒ M_{[0,1]} is countable. (Since the case ii) will occur after a certain number of renormalizations)

We shall also prove that if c = c(λ) then M_{[0,1]} is a Cantor set with Hausdorff dimension zero (and so with thickness 0).

Proof. Suppose that we can iterate F infinitely many times. Let

F^{n}(λ, c) = (λ_{n}, c_{n}) ⇒ λ_{n+1} = λ_{n}^{2}, c_{n+1} = λ_{n}^{2} − \frac{λ_{n} − 1}{c_{n}} ⇒
λ_{n} = λ_{0}^{2^{n}}, c_{n} = \frac{λ_{n} − 1}{λ_{n}^{2} − c_{n+1}} ⇒ c_{0} = \frac{λ_{0} − 1}{λ_{0}^{2} − _{\frac{λ_{0}^{2}−1}{λ_{0}^{4}−_{\frac{λ^{4}_{0}−1}{^{...}λ{2^{n} \atop 0} − \frac{λ{2^{n} \atop 0} −1}{λ{2^{n+1} \atop 0} −c_{n+1}}}}}}}

Since c_{n} ∈ [0, 1], ∀n, λ_{0}^{2^{n}} → ∞ and, if λ_{n} ≥ 2, q_{n}(c) := \frac{λ_{n}−1}{λ_{n}^{2}−c} is \frac{1}{2} contracting in [0, 1], c_{0} must be equal to the generalized continued fraction

\frac{λ_{0} − 1}{λ_{0}^{2} − _{\frac{λ_{0}^{2}−1}{λ_{0}^{4}− \frac{λ^{4}_{0}−1}{...}}}} .

We have

x_{n} := \frac{b_{0}}{a_{1} + _{\frac{b_{1}}{a_{2}+_{\frac{b_{2}}{^{...}+ \frac{b_{n−1}}{an}}}}}} = \frac{p_{n}}{q_{n}} ,

where p_{n} and q_{n} are defined by

\begin{cases}p_{−1} = 1, p_{0} = 0, p_{n+2} = a_{n+2}p_{n+1} + b_{n+1}p_{n} \\ q_{−1} = 0, q_{0} = 1, q_{n+2} = a_{n+2}q_{n+1} + b_{n+1}q_{n} \end{cases}

This can be proved by induction:

x_{n+1} = \frac{(a_{n} + \frac{b_{n}}{a_{n+1}} ) p_{n−1} + b_{n−1}p_{n−2}}{(a_{n} + \frac{b_{n}}{a_{n+1}} ) q_{n−1} + b_{n−1}q_{n−2}}

= \frac{a_{n+1}(a_{n}p_{n−1} + b_{n−1}p_{n−2}) + b_{n}p_{n−1}}{a_{n+1}(a_{n}q_{n−1} + b_{n−1}q_{n−2}) + b_{n}q_{n−1}} = \frac{a_{n+1}p_{n} + b_{n}p_{n−1}}{a_{n+1}q_{n} + b_{n}q_{n−1}} = \frac{p_{n+1}}{q_{n+1}} .

In our case, p_{n} and q_{n} satisfy the recurrence Z_{n+2} = λ_{0}^{2^{n+2}}Z_{n+1} − (λ_{0}^{2^{n+1}} − 1)Z_{n}, ∀n ≥ 0. Then we have Z_{n+2} − (λ_{0}^{2^{n+2}} − 1)Z_{n+1} = Z_{n+1} − (λ_{0}^{2^{n+1}} − 1)Z_{n} ⇒ Z_{n} − (λ_{0}^{2^{n}} − 1)Z_{n−1} is constant ⇒ Z_{n+2} = (λ_{0}^{2^{n+2}} − 1)Z_{n+1} + c, ∀n ≥ 0 ⇒ Z_{n} = ∏_{j=1}^{n}(λ_{0}^{2^{j}} − 1) · w_{n}, where w_{n} satisfies w_{n+2} = w_{n+1}+_{\frac{c}{∏_{j=1}^{n+2}(λ_{0}^{2^{j}} −1)}}, ∀n ≥ 0 ⇒ w_{n} = w_{1}+c ∑_{r=2}^{n} ∏_{j=1}^{r}(_{\frac{1}{λ_{0}^{2^{j}} −1}}), where c = Z_{1}−(λ_{0}^{2}−1)Z_{0}, w_{1} = \frac{Z_{1}}{λ_{0}^{2}−1}.

Applying this results for p_{n} and q_{n} we obtain p_{n} = ∏_{j=1}^{n}(λ_{0}^{2^{j}} − 1) · w_{n} and q_{n} = ∏_{j=1}^{n}(λ_{0}^{2^{j}} −1)w̃_{n}, where w_{n} = w_{1}+c ∑_{r=2}^{n} ∏_{j=1}^{r}(_{\frac{1}{λ_{0}^{2^{j}} −1}}) and w̃_{n} = w̃_{1}+c̃∑_{r=2}^{n} ∏_{j=1}^{r}(_{\frac{1}{λ_{0}^{2^{j}} −1}}). We have c = p_{1} − (λ_{0}^{2} − 1)p_{0} = (λ_{0} − 1) − (λ_{0}^{2} − 1) · 0 = λ_{0} − 1, and w_{1} = \frac{p_{1}}{λ_{0}^{2}−1} = \frac{1}{λ_{0}+1} ⇒ w_{n} = (λ_{0} − 1) ∑_{r=1}^{n} ∏_{j=1}^{r}(_{\frac{1}{λ_{0}^{2^{j}} −1}}). On the other hand, c̃ = q_{1} − (λ_{0}^{2} − 1)q_{0} = λ_{0}^{2} − (λ_{0}^{2} − 1) = 1, and w_{1} = \frac{q_{1}}{λ_{0}^{2}−1} = \frac{λ_{0}^{2}}{λ_{0}^{2}−1} ⇒ w̃_{n} = 1 + ∑_{r=1}^{n} ∏_{j=1}^{r}(_{\frac{1}{λ_{0}^{2^{j}} −1}}). Since \frac{p_{n}}{q_{n}} = \frac{w_{n}}{w̃_{n}}, if we define α(λ) = ∑_{r=1}^{∞} ∏_{j=1}^{r}(\frac{1}{λ^{2^{j}} −1}), we have c(λ) = lim_{n→∞} \frac{p_{n}}{q_{n}} = \frac{(λ−1)α(λ)}{α(λ)+1}.

Let us consider the case c = c(λ) in more detail: Let c_{−1} = 1, c_{0} = c(λ), λ_{n} = λ_{0}^{2^{n}}, c_{n+1} = λ_{n}^{2} − \frac{λ_{n}−1}{c_{n}}, ∀n ≥ 0 and d_{n} = ∏_{r=0}^{n} c_{r}. Then, for each n ≥ 0, the interval [0, d_{n−1}] is invariant by ψ_{λ,c}^{2^{n}}, which is conjugated by the diffeomorphism h_{n}(x) = \frac{x}{d_{n}} to ψ_{λ_{n},c_{n}}. We can decompose ψ_{λ,c}^{2^{n}} in his two components φ_{n}^{(1)} := ψ_{λ,c}^{2^{n}}|_{[0, \frac{d_{n−1}}{λn}]} and φ_{n}^{(2)} := ψ_{λ,c}^{2^{n}}|_{[d_{n−1}(1− \frac{cn}{λn}),d_{n−1}]}, which are expansive affine maps.

Let K be the maximal invariant set of ψ_{λ,c} in [0, 1]. Clearly K is a closed set. To prove that K is a Cantor set, it is enough to prove that K has no isolated points. Let δ = max_{x∈K}(min_{y∈K} |y − x|). We want to prove that δ = 0.

Let K_{n} = K ∩ [0, d_{n−1}]. We have K_{0} = K, and K_{n} = (φ_{n}^{(2)})^{−1}(K_{n+1}) ∪ ⋃_{j=0}^{∞}(φ_{n}^{(1)})^{−j}(K_{n+1}), that is, K_{n} is the union of countable many congruent or reduced copies of K_{n+1}, which shows that δ_{n} ≤ δ_{n+1}, where δ_{n} := max_{x∈K_{n}}(min_{y∈K_{n}} |y − x|). Since δ_{n} ≤ d_{n−1}, we have lim_{n→∞} δ_{n} = 0 ⇒ δ = δ_{0} = 0, because δ_{0} ≤ δ_{n}, ∀n ∈ N.

Let us prove now that the Hausdorff dimension of K is 0. Let n ∈ N and define

c̃_{λ}(n) = \frac{λ − 1}{λ^{2} − \frac{λ^{2}−1}{..._{\overline{λ^{2^{n}} −1}}}} .

Then, if c = c̃_{λ}(n), ψ_{λ,c} is n times renormalizable, and after n renormalizations we obtain the set K_{n} which is the maximal invariant set of ψ_{λ^{2^{n}},1}, such that K̃ is the union of countable many diffeomorphic images of K_{n}, and so has the same Hausdorff dimension of K_{n}, where K̃ is the maximal invariant set of ψ_{λ,c} in [0, 1]. On the other hand, K_{n} is an affine Cantor set and his Hausdorff dimension h satisfies (λ^{2^{n}})^{−h} + (λ^{2^{n}})^{−h} = 1 ⇒ h = \frac{log 2}{2^{n} log λ} (see [P-T]). Since c(λ) < c̃_{λ}(n), ∀n ∈ N, we have HD(K) ≤ −\frac{log 2}{2^{n} log λ}, ∀n ∈ N ⇒ HD(K) = 0. Our last claims follow from the fact that if c_{1} < c_{2} then there is an affine image of K_{λ,c_{1}} contained in K_{λ,c_{2}}, where K_{λ,c} is the maximal invariant of ψ_{λ,c} in [0, 1]. Indeed, the restriction of ψ_{λ,c_{2}} to [ε, 1 − λε] is conjugated to ψ_{λ, \frac{c_{2}−λ^{2}ε−ε}{1−(1+λ)ε}} (see Figure 8). Taking ε = \frac{c_{2}−c_{1}}{1+λ^{2}−c_{1}(1+λ)} we are done. Remark. If c > c(λ) then c > c̃_{λ}(n) for some n, and so HD(K) ≥ HD(K_{n}) = \frac{log 2}{2^{n} log λ} > 0

We have proved the following

Theorem 2.2. Let λ ∈ (1, 2] and [a, b] ⊂ [0, 1]. Let ε = max\{0, \frac{λ}{2} − b, \frac{a}{λ} − (1 − \frac{λ}{2})\}, and let c = \frac{\frac{λ}{2} −ε−λ^{2}(1+ε− \frac{λ}{2} )}{\frac{λ}{2} −ε−λ(1+ε− \frac{λ}{2} )} = λ − 1 − \frac{4ε}{λ(λ−1)−2(λ+1)ε}.

If c(λ) = \frac{(λ−1)α(λ)}{1+α(λ)}, where α(λ) = ∑_{r=1}^{∞} \frac{1}{∏_{j=1}^{r}(λ^{2^{j}} −1)}, and if K is the maximal invariant of T_{λ} in [a, b], then:

i) λ − 1 > c > c(λ) ⇒ K is a Cantor set with positive Hausdorff dimension;

ii) c = c(λ) ⇒ K is a Cantor set with Hausdorff dimension 0; iii) c < c(λ) ⇒ K is countable.

Remark 0: If c = λ − 1, which is equivalent to ε = 0, then K is an interval.

Remark 1: Conditions i), ii) and iii) are respectively equivalent to

i’) [a(λ), b(λ)] ⊂ (a, b);

ii’) [a(λ), b(λ)] ⊂ [a, b] and a = a(λ) or b = b(λ);

iii’) [a(λ), b(λ)] ⊄ [a, b]

where, given ε > 0 for which c = c(λ), a(λ) = a_{ε} = \frac{λ(1−c(λ))}{1+λ^{2}−c(λ)(1+λ)} and b(λ) = b_{ε} = \frac{λ(λ−c(λ))}{1+λ^{2}−c(λ)(1+λ)}.

Remark 2: c = λ − 1 − \frac{4ε}{λ(λ−1)−2(λ+1)ε} is a strictly decreasing function of ε, whose largest value is λ − 1 > c(λ), so for each λ ∈ (1, 2], there is an unique ε > 0 such that c = c(λ).

3 The Case λ = 2:

For λ = 2 we have an interesting arithmetic phenomenum involving the binary representation of b(λ). Let x ∈ [0, 1] such that T_{2}^{n}(x) ≠ 0, ∀n ∈ N (this is equivalent to x ≠ \frac{K}{2^{n}}, ∀K, n ∈ N). Then we can associate to x the sequence (σ_{i})_{i≥0}, where

σ_{i} = \begin{cases}0, & if T_{2}^{i}(x) ∈ (0, 1/2) \\ 1, & if T_{2}^{i}(x) ∈ (1/2, 1) \end{cases} .

If the representation of x in basis two is x = ∑_{j=0}^{∞} \frac{μ_{j}}{2^{j+1}} then σ_{0} = μ_{0} and, for n ≥ 1,

σ_{n} = \begin{cases}μ_{n}, & if μ_{n−1} = 0 \\ 1 − μ_{n}, & if μ_{n−1} = 1. \end{cases}

and so we can reobtain the μ_{i}’s from the σ_{i}’s. The restriction of T_{2} to [a(2), b(2)] is conjugated to ψ_{2,c(2)}, and b(2) correspond to 0 by this conjugacy. Let us study the orbit of 0 by ψ_{2,c(2)}.

Let

τ_{n} := \begin{cases}0, & if ψ_{2,c(2)}^{n}(0) ∈ [0, \frac{1}{2}] \\ 1, & if ψ_{2,c(2)}^{n}(0) ∈ (\frac{1}{2}, 1] \end{cases}

τ̃_{n} := \begin{cases}0, & if (ψ_{2,c(2)}^{2})^{n}(0) ∈ [0, \frac{1}{4}) \\ 1, & if (ψ_{2,c(2)}^{2})^{n}(0) ∈ [\frac{1}{4}, \frac{1}{2}] \end{cases}

then:

i) τ̃_{n} is well-defined, because [0, c(2)] is invariant by ψ_{2,c(2)}^{2} ⇒ (ψ_{2,c(2)}^{2})^{n}(0) ∈ [0, c(2)] ⊂ [0, \frac{1}{2}], ∀n ≥ 0;

ii) τ̃_{n} = 0 ⇔ (τ_{2n} = 0 and τ_{2n+1} = 1), and τ̃_{n} = 1 ⇔ (τ_{2n} = 0 and τ_{2n+1} = 0);

iii) τ̃_{n} = τ_{n}, ∀n ≥ 0, because, by the construction, ψ_{2,c(2)}^{2}|_{[0,c(2)]} is conjugated to ψ_{4,c(4)}, and the combinatorial behavior of the orbit of 0 by ψ_{λ,c(λ)} does not depend on λ.

The properties i), ii) and iii) allow us to obtain the sequence τ_{n} recursively by the rules:

τ_{0} = 0; τ_{n} = 0 ⇒ τ_{2n} = 0 and τ_{2n+1} = 1; τ_{n} = 1 ⇒ τ_{2n} = 0 and τ_{2n+1} = 0.

We have (τ_{n}) = 0100010101000100 ….

Since the conjugacy reverses orientation, the sequence (σ_{i}) associated to b(2) is given by σ_{i} = 1 − τ_{i}, and the sequence μ_{i} is given by μ_{0} = σ_{0} = 1,

μ_{n+1} = \begin{cases}σ_{n+1}, & if μ_{n} = 0 \\ 1 − σ_{n+1}, & if μ_{n} = 1. \end{cases}

So, (σ_{n}) = 1011101010111011 …, and (μ_{n}) = 1101001100101101 …, that is, b(2) = 0, 1101001100101101 … in basis 2, and a(2) = 2 − 2b(2) = 0, 010110011010010 ⋯ = ∑_{i=0}^{∞} \frac{λ_{i}}{2^{i+1}} , where λ_{i} = 1 − μ_{i+1}, ∀i.

This shows, in particular, that a(2) and b(2) have both 50% of zeros and 50% of ones in their binary representations, but do not have three consecutive zeros nor three consecutive ones.

It is easy to prove by induction that b(2) = 2m, where

m = 0, 0110100110010110 ⋯ = ∑_{i=1}^{∞} \frac{m_{i}}{2^{i}} ,

and the digits m_{i} of the binary representation of m form the Morse-Thue sequence defined by m_{1} = 0,

m_{k} = 0 ⇒ m_{2k−1} = 0, m_{2k} = 1

m_{k} = 1 ⇒ m_{2k−1} = 1, m_{2k} = 0.

This number m is called the Morse-Thue constant, and is equal to the external angle in the Mandelbrot set corresponding to the Feigenbaum map of the quadratic family (see [S] pp. 279, 299; [D]). We have b(2) = 2m = f(\frac{1}{2}), where f(x) = ∑_{i=0}^{∞} m_{i+1}x^{i} satisfies f(x) = (1 − x)f(x^{2}) + \frac{x}{1−x^{2}} , and so, by the results of [L-Po], we know that b(2) (and so a(2)) is a transcendental number. Since we have b(2) = \frac{4−2c(2)}{5−3c(2)}, where

c(2) = \frac{α(2)}{1 + α(2)}, α(2) = ∑_{r=1}^{∞} \frac{1}{∏_{j=1}^{r}(2^{2^{j}} − 1)},

we obtain

m = \frac{1}{2}b(2) = \frac{2 − c(2)}{5 − 3c(2)} = \frac{2 + α(2)}{5 + 2α(2)},

which gives an explicit expression for m in terms of a series.

4 The Case of Unimodal Maps

In this section we consider C^{2} unimodal maps with non-flat critical point. These maps have no wandering intervals, by Theorem A of Chapter IV of [M-S]. Theorem B of the same chapter implies that these maps have only a finite number of periodic attractors. In what follows a pre-periodic point is a point with finite positive orbit, i.e., a point that lands on a periodic orbit.

We may suppose by the same initial considerations that we have done for tent maps that our unimodal maps f: [0, 1] → [0, 1] satisfy f(c) = 1, where c is the critical point, and f(1) = 0 (the tent map T_{λ} restricted to L = [λ(1 − \frac{λ}{2}), \frac{λ}{2}] satisfies this combinatorial condition after an affine conjugacy).

We have now two cases:

a) The topological entropy of f is 0: In this case f is combinatorially equivalent to g_{a}(x) = ax(1 − x) for some a ≤ c, where cx(1 − x) is the Feigenbaum map of the quadratic family (which is a full family of unimodal maps). We have two subcases.

a.i) a < c: In this case g_{a}(x) has a finite number of periodic points which form the non-wandering set of g_{a}. In this case, if J ⊅ [0, 1], the maximal invariant set of f|_{J} is infinite if and only if J contains a recurrent interval (that is, there is an interval I_{0} ⊂ I and an m ∈ N such that f^{m}(I_{0}) ⊂ I_{0}, f^{k}(I_{0}) ⊂ I, for k = 1, 2, …, m and I_{0} is contained in the basin of attraction of some periodic point of period m or f^{m}|_{I_{0}} = id).

a.ii) a = c: In this case the closure of the orbit of the critical point is a Cantor set K, and the ω-limit of any point x can be a periodical point if x is pre-periodic or else must be equal to K. Here if J ≠ [0, 1] and J does not contain any recurrent interval then the maximal invariant of f|_{J} must be finite (see Ex. 5.3 of Chapter II of [M-S]).

b) The topological entropy of f is strictly positive: In this case we can use Proposition 4.3 of Chapter III of [M-S] to obtain a semi-conjugacy between f and T_{exp(h_{t}(f))}, where h_{t}(f) is the topological entropy of f. This semi-conjugacy can collapse an interval only if it is contained into X, where X is the set of points which are eventually mapped into a restrictive interval (see Section II.5 of [M-S]) or into the basin of a periodic attractor. Let λ = h_{t}(f). We have the following commutative diagram:

[0, 1]  ⟶^{f}  [0, 1]
h ↓  ↓ h
[λ(1 − \frac{λ}{2}), \frac{λ}{2}] ⟶^{T_{λ}} [λ(1 − \frac{λ}{2}), \frac{λ}{2}]

h is a monotonous semi-conjugacy, and if x is not a pre-periodic point of T_{λ} then h^{−1}(x) is a single point. Let I be an interval. We shall prove that the maximal invariant of f|_{I} is uncountable if and only if I contains the orbit of some recurrent interval or the maximal invariant of T_{λ}|_{h(I)} is uncountable. For this, let a(λ) and b(λ) as before, J = [a(λ), b(λ)] and J̃ = h^{−1}(J). Since a(λ) and b(λ) are not pre-periodic, we have Ĩ ⊄ J̃ → h(Ĩ) ⊄ h(J̃) = J. Thus, if T_{λ}|_{h(I)} is uncountable then J ⊂ h(I) ⇒ K is contained in the maximal invariant of T_{λ}|_{h(I)}, where K is the maximal invariant of T_{λ}|_{J} ⇒ h^{−1}(K) is contained in the maximal invariant of f|_{I} (because x ∈ h^{−1}(K) ⇒ T_{λ}^{n}(h(x)) ∈ J, ∀n ∈ N ⇒ h(f^{n}(x)) ∈ J, ∀n ∈ N ⇒ f^{n}(x) ∈ h^{−1}(J), ∀n ∈ N ⇒ f^{n}(x) ∈ I, ∀n ∈ N). Therefore in this case the maximal invariant of f|_{I} is uncountable. If I contains the orbit of some recurrent interval then clearly the maximal invariant of f|_{I} has non-empty interior. If h(I) ⊅ J then the turning point of T_{λ} does not belong to the maximal invariant of T_{λ}|_{h(I)}, and so the pre-images of a restrictive interval are disjoint of the maximal invariant of f|_{I}. Moreover, the maximal invariant of T_{λ}|_{h(I)} is countable, and consists only of preperiodic points. Thus the maximal invariant of f|_{I} is uncountable if and only if there is a periodic point p of period m and a closed interval I_{0}, with p ∈ I_{0} such that f^{m}(I_{0}) ⊂ I_{0} and f^{k}(I_{0}) ⊂ I, for 0 ≤ k < m, I_{0} being collapsed by the semi-conjugacy.

b.i) For unimodal maps f without recurrent intervals (for maps of the quadratic family or, more generally, with negative Schwarzian derivative, it means that f is nonrenormalizable, and that the critical point does not belong to the immediate basin of attraction of a periodic point), the map h is a conjugacy, so the maximal invariant set of f|_{I} is uncountable if and only if I ⊃ h^{−1}([a(λ), b(λ)]). Here λ = h_{t}(f) and a(λ) and b(λ) were defined in the end of Section 2.

Under these hypotheses, if f is a Collet-Eckmann map (i.e., if there are C > 0, r > 1 such that |Df^{n}(f(C))| ≥ Cr^{n} for any n ∈ ℕ, where C is the critical point of f) then h is a Höldercontinuous conjugacy, and the maximal invariant set of f|_{I} has positive Hausdorff dimension if and only if the same occurs for T_{λ}|_{h(I)}. In particular, the maximal invariant set of f|_{h^{−1}([a(λ),b(λ)]} is a Cantor set with zero Hausdorff dimension. We do not know if it is true in more general settings, and in particular if we drop the hypothesis of f being Collet-Eckmann.

Acknowledgement: I would like to thank Welington de Melo for helpful conversations about this subject, and to the Referee for his helpful corrections and suggestions.

References:

[D] A. Douady; Julia sets and the Mandelbrot set. In H. -O. Peitgen and P.H. Richter: The beauty of Fractals (Springer, Berlin), pp. 161-173.

[L-Po] J.H. Loxton and A.J. Van der Poorten, Arithmetic properties of the solutions of a class of functional equations, J. -Reine-Angew. -Math. 330 (1982), pp. 159–172.

[M-S] W. de Melo and S. Van Strien, One-Dimensional Dynamics, Ergebrisse der Mathematik und ihrer Grenzgebiete 3. Folge-Band 25, Springer-Verlag, 1991.

[P-T] J. Palis and F. Takens, Hyperbolicity and sensitive chaotic dynamics at homoclinic bifurcations: fractal dimensions and infinitely many attractors, Cambridge Univ. Press (1992).

[S] M. Schroeder, Fractal, Chaos, Power Laws - W.H. Freeman and Company, New York, 1991.