Universality in inverse limit spaces of the logistic family occurs with positive measure

Karen Brucks^{∗} Department of Mathematical Sciences University of Wisconsin at Milwaukee Milwaukee WI 53201, USA

Zoltán Buczolich^{†} Department of Analysis Eötvös Loránd University Rákóczi út 5 H-1088 Budapest, Hungary

April 9, 1999

Abstract

We prove that the trajectory of the turning point c = \frac{1}{2} under g_{a} = ax(1−x) at cutting times is dense in [g_{a}^{2}(c), g_{a}(a)] for a set of parameters a of positive Lebesgue measure. Hence the inverse limit space (I, g_{a}) has a universality property for a positive measure set of parameters a of the logistic family.

1 Introduction

Inverse limit spaces of one-dimensional maps commonly appear as attractors in dynamical systems [1, 2, 3, 4, 5, 26]. In 1967, R.F. Williams [26] showed that hyperbolic one-dimensional attractors are inverse limits of maps on branched one-manifolds. Such attractors have the property of being everywhere locally homeomorphic with the product of a Cantor set and an arc. By contrast in [2], there is exhibited a residual full Lebesgue measure set of slopes a such that if T_{a} is a tent map with such a slope, then the inverse limit space (I, T_{a}) is nowhere locally homeomorphic to the product of a Cantor set and an arc. Moreover, every open set in (I, T_{a}) contains a homeomorphic copy of every inverse limit space (I, T_{a_{i}}) appearing in this family of inverse limit spaces (the slopes a_{i} are allowed to vary). Hence, there is a form of self-similarity and local recapitulation of the entire family of inverse limits of tent maps within (I, T_{a}). We say such inverse limit spaces have the self-similarity property; we precisely define this property below. The family of tent maps is given by: T_{a} = min{ax, a(1 − x)} where a ∈ [1, 2] and x ∈ [0, 1].

∗Supported in part by a Fulbright grant while visiting the Eötvös Loránd University, Budapest Hungary.

†This author was supported by the Hungarian National Foundation of Scientific Research Grant No. T 016094 and FKFP B-07/1997.

1991 Mathematics Subject Classification: Primary 58F03 Secondary 54H20

When dealing with a one-parameter family of unimodal maps {f_{a} : I → I}_{a∈Λ} with common turning point c, each f_{a} has a kneading map Q_{a}(k) and a sequence of cutting times {S_{k}(a)}. These notions are reviewed in the next section. If f_{a}^{2}(c) < c < f_{a}(c), then I_{a} ≡ [f_{a}^{2}(c), f_{a}(c)] is the smallest f_{a}-forward invariant interval and is referred to as the core of the map f_{a}. We let \overline{U} denote the closure of a set U. To say that the trajectory of the turning point c under f_{a} at cutting times is dense in the core, means that

\overline{\{f_{a}^{S_{k}(a)}(c)\}}_{k≥0} = I_{a}.  (1)

For a sequence of parameters {a_{i}}_{i≥1}, the inverse limit space (I, f_{a_{i}}) is defined as (I, f_{a_{i}}) = {(x_{0}, x_{1}, x_{2}, ...) | f_{a_{i}}(x_{i}) = x_{i−1} for i ≥ 1}. For each parameter a let f̃_{a} denote the map f_{a} restricted to its core I_{a} and rescaled such that f̃_{a} maps I onto I. The inverse limit space (I, f_{a}) consists exactly of a homeomorphic copy of (I, f̃_{a}) and an infinite ray entwined with (I, f̃_{a}). Clearly any subcontinuum of (I, f̃_{a}) is contained in (I, f_{a}) since I_{a} ⊂ I; the converse is not necessarily true. However for any proper non-trivial (not an arc or a point) subcontinuum H ⊂ (I, f_{a}), H ⊂ (I, f̃_{a}) holds [10]. Hence, one does not lose or gain nontrivial proper suncontinua when passing from (I, f_{a}) to (I, f̃_{a}). We say an inverse limit space (I, f_{a}) has the self-similarity property provided that for every choice of parameters {a_{i}}_{i≥1} and every open set U ⊂ (I, f̃_{a}) there exists a subcontinuum H ⊂ U such that H is homeomorphic to the inverse limit space (I, f̃_{a_{i}}). We call (I, f) a unimodal inverse limit space provided f : I → I is unimodal.

The one-parameter family of unimodal maps given by g_{a}(x) = ax(1 − x), for a ∈ [0, 4], is commonly referred to as the logistic family. There are other equivalent models, we use this model as it makes our calculations easiest. The logistic family is a full family, i.e., all possible combinatorics obtained by unimodal maps occurs within the logistic family [22, Chapter 2]. Hence, we say that a unimodal inverse limit space (I, f) has the universality property provided that for every choice of parameters {a_{i}}_{i≥1}, a_{i} ∈ [0, 4] for all i, and every open set U ⊂ (I, f̃) there exists a subcontinuum H ⊂ U such that H is homeomorphic to the inverse limit space (I, g̃_{a_{i}}). Since not every one-parameter family of unimodal maps is full, universality is stronger than self-similarity. In this note, we prove that for a positive Lebesgue measure set of parameters a, the inverse limit space (I, g_{a}) has the universality property (and therfore also the self-similarity property). If the turning point c = \frac{1}{2} is periodic or attracted to a periodic orbit under some g_{a}, then all proper subcontinua of (I, g_{a}) are points or arcs and hence (I, g_{a}) does not have the universality property. Thus, the set of parameters a for which (I, g_{a}) has the universality property cannot have full Lebesgue measure.

In [2], the tent maps which are shown to have the self-similarity property are such that the trajectory of the turning point c at cutting times is dense in the core. Moreover, if a tent map T_{a} has this property, i.e., (1) holds with f_{a} = T_{a}, then in fact the inverse limit space (I, T_{a}) has the self-similarity property; this follows from the constructions in the proofs of [2, Theorem 4,

Theorem 5]. Set A = \{a ∈ [\sqrt{2}, 2] | \overline{\{T_{a}^{S_{k}(a)}(c)\}}_{k≥0} = [T_{a}^{2}(c), T_{a}(c)]\}. In [2], it is shown that A is large topologically, namely it is a residual set, and it is remarked that A has full Lebesgue measure in [\sqrt{2}, 2]. In [9] it was shown that D = \{a ∈ [\sqrt{2}, 2] | \overline{\{T_{a}^{n}(c)\}}_{n≥0} = [T_{a}^{2}(c), T_{a}(c)]\} has full Lebesgue measure in [\sqrt{2}, 2]. Clearly, A is a proper subset of D. Using the work of [9] it is not difficult to establish that A has full Lebesgue measure in [\sqrt{2}, 2], see e.g. [14, Proposition 2]. Using standard renormalization techniques, one obtains similar results for a ∈ [1, \sqrt{2}). Although here, the ‘core’ consists of a finite number of pairwise disjoint nondegenerate closed intervals, see e.g. [25]. In related work [11], it is shown that [\sqrt{2}, 2] \ D is σ-porous.

Set

B = \{a > 2 | \overline{\{g_{a}^{S_{k}(a)}(c)\}}_{k≥0} = [g_{a}^{2}(c), g_{a}(c)]\}.  (2)

If a ∈ B, then g_{a} is topologically conjugate to a tent map, since any unimodal map without wandering intervals, restrictive intervals, or periodic attractors is conjugate to a tent map, see e.g. [12, page 16]. If two unimodal maps, f : I → I and g : I → I, are topologically conjugate (meaning there is a homeomorphism h : I → I, such that h ∘ f = g ∘ h), then (I, f) is homeomorphic to (I, g) as topological spaces. Hence, if a ∈ B, then every open set U ⊂ (I, g_{a}) contains a homeomorphic copy of any inverse limit space (I, T̃_{a_{i}}), since (1) is preserved by topological conjugacy and therefore (I, g_{a}) would be homeomorphic to some (I, Tã) with ã ∈ A. However, there exist (I, g_{b}) which are not homeomorphic to any (I, T_{a}) and hence this observation alone does not immediately give that (I, g_{a}) has the weaker selfsimilarity property for a ∈ B. Using results of [4, 10] one can show that a ∈ B (or a ∈ A) is indeed sufficient for (I, g_{a}) (or (I, T_{a})) to have the universality property. Details are provided in the appendix of this note. Thus, in this note we prove the following.

Theorem 1 The set B, defined in (2), has positive Lebesgue measure, and thus universality in inverse limit spaces of the logistic family occurs with positive measure.

As a first attempt at proving Theorem 1, one might consider the map φ : a ↦ h_{top}(g_{a}), where h_{top}(g_{a}) denotes the topological entropy of g_{a}. The topological entropy of a tent map T_{a} is log(a) and topological entropy is preserved by topological conjugacy; thus if g_{a} is topologically conjugate to a tent map Tã, then h_{top}(g_{a}) = log(ã). Hence, if φ were nice, say absolutely continuous, then A having positive measure would give that B does also. However, it is unknown whether φ is absolutely continuous. It is known that φ^{−1} has very bad absolute continuity properties (even off the flat pieces) and it is conjectured that φ is not absolutely continuous. As is remarked in the introduction of [15, page 216], work of Bruin and Sands [15, 24] establish that for (Lebesgue) a.e. α ∈ [0, log(2)], if f is an S-unimodal map with h_{top}(f) = α, then f is Collet-Eckmann and the turning point is not typical for the absolutely continuous invariant probability measure (a.c.i.p.). On the other hand, Benedicks and Carleson [6, Theorem 3] show that there is a positive Lebesgue measure set of parameters a, which we denote by BC, for which the logistic map g_{a} is Collet-Eckmann and the turning point is typical for the a.c.i.p. Hence, {h_{top}(g_{a}) | a ∈ BC} has zero Lebesgue measure but BC has positive Lebesgue measure; thus the bad absolutely continuous properties of φ^{−1}. It is precisely the set BC that we use in this note to prove Theorem 1.

As already remarked, it is known that for a positive Lebesgue measure set of parameters a, the map g_{a} has an a.c.i.p., see e.g. [6, 21, 22]. For a history and discussion of this and related results see [22, Chapter V]. We are not so interested in the a.c.i.p., but rather in the set B defined above. Our work follows that of Benedicks and Carleson [6, 7] as presented in [22, Theorem 6.1]. Although [22, Theorem 6.1] deals with a class of families of maps, of which the logisitic is one, we are interested in Theorem 6.1 applied to the logistic family. It is shown that there exists a set of parameters of positive Lebesgue measure, denoted here by BC, such that for a ∈ BC the map g_{a} is Collet-Eckmann and hence has an a.c.i.p. [17]. We prove the following proposition, from which Theorem 1 is immediate.

Proposition 1 Set BC^{∗} = \{a ∈ BC | \overline{\{g_{a}^{S_{k}}(c)\}}_{k≥0} = [g_{a}^{2}(c), g_{a}(c)]\}. Then BC^{∗} has positive Lebesgue measure. Notice that BC^{∗} = BC ∩ B.

One of the main ingredients in our proof of Proposition 1 is Lemma 2 of Section 3 of this note. Lemma 2 can be extracted (with some additional work) from the proof of [22, Theorem 6.1] and the work in [6, 7], and hence (in this sense) is not new. However, this lemma is not explicitly stated in these works. As we feel Lemma 2 is of independent interest, we explicitly state it in this note. The purpose of this note is to show explicitly how one can extract this cutting time result, i.e., Theorem 1, from the work of Benedicks and Carleson [6, 7], since cutting times are not considered in [6, 7].

Acknowledgements: The authors thank Marcy Barge and Beverly Diamond for helpful discussions.

2 Preliminaries

A continuous map f : [0, 1] → [0, 1] is called unimodal if there exists a unique turning or critical point, c, such that f|_{[0,c)} is increasing, f|_{(c,1]} is decreasing, and f(0) = f(1) = 0. To avoid trivial cases, we assume that f(c) > c > f(f(c)). We denote the forward images of c by c_{i} = f^{i}(c). Clearly the interval [c_{2}, c_{1}] is forward invariant and f maps [c_{2}, c_{1}] onto itself; the interval [c_{2}, c_{1}] is called the core of the map f. The itinerary of a point x ∈ [0, 1] under the map f, denoted I(x, f), is a one-sided infinite or finite string (j ≥ 0) given by:

I(x, f)_{j} = \{ \begin{matrix}0 & if f^{j}(x) < c \\ ∗ & if f^{j}(x) = c \\ 1 & if f^{j}(x) > c. \end{matrix}

We take the convention that the itinerary terminates with the first ∗, past the j = 0 position, and hence in this case would be finite in length. The intinerary of c_{1} is called the kneading invariant of the map f. For a basic introduction to the combinatorics of itineraries or kneading theory, see e.g. [16, Part II]. The next two remarks are included for Lemma 1.

Remark 1: There is a natural order on itineraries that corresponds to retaining or reversing orientation, depending on which branch of the unimodal map is used. For example, if f is increasing on [0, c] and I(x, f) = 101 …, then orientation is preserved in a neighborhood of x under f^{3}. This ordering has the property that x < y implies I(x, f) ≤ I(y, f) [16, page 65]. Of course, we are assuming that f is a nice unimodal map, such as a tent or logistic map.

Remark 2: It is known that the map a ↦ I(c_{1}(a), g_{a}) is nondecreasing, this follows from work in [18] (again, there is a natural order on the collection of all kneading invariants for unimodal maps, see e.g. [16, Part II]). Thus if a < b, then for each I(x, g_{a}) there exists y ∈ [0, 1] such that I(y, g_{b}) = I(x, g_{a}) [16, Theorem II.3.8].

Let f^{n} be some iterate of f and let J be any maximal subinterval on which f^{n}|_{J} is monotone. Then f^{n} : J → [0, 1] is called a branch of f^{n}. A branch f^{n} : J → [0, 1] is called a central branch if c ∈ ∂J. Hence there are always two central branches, the images of which are the same. An iterate n is called a cutting time if the image of the central branch of f^{n} contains c. The cutting times are denoted by S_{0}, S_{1}, S_{2}, … (S_{0} = 1 and S_{1} = 2). If f^{S_{k}} : J → [0, 1] is the left central branch of f^{S_{k}}, then there is a unique point z_{k} ∈ J such that

f^{S_{k}}(z_{k}) = c.  (3)

By construction, z_{k} has the property that ∪_{0<j≤S_{k}}f^{−j}(c) ∩ (z_{k}, c) = ∅ and is therefore called a closest precritical point. The point ẑ_{k}, defined analogously for the right central branch of f^{S_{k}}, is also a closest precritical point. It can be proven that the difference of two consecutive cutting times is again a cutting time. Hence we can write

S_{k} − S_{k−1} = S_{Q(k)},

where Q : N → N is an integer function, called the kneading map. An equivalent statement is

c_{S_{k}} ∈ (z_{Q(k+1)−1}, z_{Q(k+1)}] ∪ [ẑ_{Q(k+1)}, ẑ_{Q(k+1)−1}).  (4)

The kneading map was introduced by Hofbauer, see e.g. [19, 20]. The kneading map (or cutting times) determine the combinatorics of f completely. A survey of this tool can be found in [13]; our discussion follows [13].

Closely related to the kneading map is the Hofbauer-tower [19, 20]. Given a unimodal map f, the associated Hofbauer-tower is the disjoint union of intervals {D_{n}}_{n≥1}, where D_{1} = [0, c_{1}] and for n ≥ 2,

D_{n+1} = \{ \begin{matrix}f(D_{n}) & if c ∉ D_{n} \\ [c_{n+1}, c_{1}] & if c ∈ D_{n}. \end{matrix}

Notice that the image of either central branch f^{n} : J → [0, 1] is such that f^{n}(J) = D_{n}. Thus, n is a cutting time if and only if c ∈ D_{n}. Also, setting ν = min{x < f(c) | f^{n−1}|[x, f(c)] is monotone}, we have that

f^{n−1}([ν, f(c)]) = D_{n}.  (5)

When more than one parameter value is being used, we may write D_{n}(a) for the levels in the Hofbauer tower for g_{a}, S_{k}(a) for the cutting times of g_{a}, or c_{n}(a) for g_{a}^{n}(c). Another notation for c_{n}(a) is ξ_{n}(a); the latter being used when one is interested in g_{a}^{n}(c) as a function of the parameter a. The rationals are denoted by Q. The Lebesgue measure of a set U is denoted |U|.

For a discussion of Lemma 1 see [24, Chapter 3]; we sketch a proof.

Lemma 1 Fix n ∈ N and let U be a maximal open subinterval of monotonicity for the map ξ_{n}(a) ≡ g_{a}^{n}(c) such that ξ_{n}(U) contains the turning point c = \frac{1}{2} in its interior. Let a_{∗} ∈ U be such that ξ_{n}(a_{∗}) = c, set W = (a_{∗}, 4] ∩ U, and set V = [0, a_{∗}] ∩ U. Then n is a cutting time for each a ∈ W, and n is not a cutting time for each a ∈ V.

Proof: Fix b ∈ W and set c_{m} = c_{m}(b) = g_{b}^{m}(c) for m ∈ N. Without loss of generality, assume that ξ_{n}(W) ⊂ (c, 1]. As above, let ν = min{x < c_{1} | g_{b}^{n−1}|[x, c_{1}] is monotone}. It suffices to show that c ∈ g_{b}^{n−1}((ν, c_{1})) = D_{n}(b), recall (5).

We have that I(c_{1}(a), g_{a})_{j=0,...,n} = I(c_{1}(b), g_{b})_{j=0,...,n} for all a ∈ W and I(x, g_{b})_{j=0,...,n−1} = I(c_{1}(b), g_{b})_{j=0,...,n−1} for all x ∈ (ν, c_{1}(b)]. Since ξ_{n}(W) ⊂ (c, 1), we have c_{n}(b) > c. These two observations give that (g_{b}^{n−1})′ > 0 on the maximal subinterval of monotonicity of g_{b}^{n−1} containing c_{1}(b). Also, as a_{∗} < b, there exists x̃ ∈ [0, 1] with I(x̃, g_{b}) = I(c_{1}(a_{∗}), g_{a_{∗}}), see Remark 2. Combining these facts and using the monotonicity of intineraries (see Remark 1) leads to x̃ ∈ (ν, c_{1}(b)). Thus, g_{b}^{n}(x̃) = c and hence c ∈ D_{n}(b). Therefore n is a cutting time for g_{b}. Similarly one can argue that n is not a cutting time for any a ∈ V. □

3 Tools

Again, Lemma 2 can be extracted from the proof of [22, Theorem 6.1] and the work in [6, 7]. Providing a proof for Lemma 2 would require a reworking of most of the proof of [22, Theorem 6.1] and hence would be lengthy. In the interest of producing a short note, we do not include a proof but rather sketch how to extract the lemma from these works.

Lemma 2 There exists a closed interval I_{0} ⊂ [0, 1] \ {c} with |I_{0}| > 0 and such that for a set of parameters BC_{0} ⊂ BC we have for each a_{0} ∈ BC_{0} an infinite sequence n_{1} < n_{2} < n_{3} < … and subintervals ω_{n_{i}}^{a_{0}} such that

1. ω_{n_{i}}^{a_{0}} ⊂ ω_{n_{i}−1}^{a_{0}}, 2. lim_{i→∞} |ω_{n_{i}}^{a_{0}}| = 0,

3. a_{0} ∈ ω_{n_{i}}^{a_{0}},

4. ξ_{n_{i}}(ω_{n_{i}}^{a_{0}}) ⊇ I_{0},

5. ξ_{n_{i}}|ω_{n_{i}}^{a_{0}} is monotone,

6. there exists C > 0 such that \frac{|ξ′_{n_{i}} (a)|}{|ξ′_{n_{i}} (b)|} < C for all a, b ∈ ω_{n_{i}}^{a_{0}}, and

7. |BC_{0} ∩ (4 − ϵ, 4]| > 0 for all ϵ > 0.

In fact, C is independent of a_{0} and n_{i}; note that C > 1.

Here we give a short outline to explain why Lemma 2 is true and how it relates to the proof of the Benedicks and Carleson result as proved in [22, pages 402-433]. Recall that BC denotes the set of parameters constructed in the proof of [22, Theorem 6.1], explicitly BC = E of [22, Theorem 6.1]. To each a ∈ BC there are intervals ω_{n}^{a} associated. These intervals are nested, their lengths tend to zero, and ω_{0}^{a} is a small left neighborhood of 4 = a_{∗}. A neighborhood U_{Δ} ≡ (c − e^{−Δ}, c + e^{−Δ}) of the turning point is considered, where Δ is some large integer. Assume now that a_{0} ∈ BC is fixed and for ease of notation we will write ω_{n} for ω_{n}^{a_{0}}. In [22, pages 414-416] free returns of ω_{n}’s to U_{Δ} are defined. If for some n ∈ N we have a free return, then ξ_{n}(ω_{n−1}) ∩ U_{Δ} ≠ ∅ and ω_{n} is defined as a suitable subinterval of ω_{n−1} (this subinterval might equal ω_{n−1}). If for n we do not have a free return, then ω_{n} = ω_{n−1}.

There are some free returns for which there is a substantial escape [22, page 415]; at these substantial escapes |ξ_{n}(ω_{n−1})| ≥ \sqrt{|U_{Δ}|} >> |U_{Δ}|. For almost every parameter value in BC there are infinitely many substantial escapes. This can be verified by scrutinizing the proofs of Lemmas 6.7-9 in [22], or by following Section 2.2 of [7, pages 81-86]. (The notation of [7] differs slightly from that of [22]; for example, escape positions are discussed instead of substantial escapes, etc..) We denote by BC′ those parameter values in BC for which there are infinitely many substantial escapes. Assume now that a_{0} ∈ BC′.

Given ϵ_{1} > 0 and using that |ω_{n}| → 0, choose and fix a large n′ such that there is a substantial escape at n′ and |ω_{n′}| < ϵ_{1}. For ease of notation, set ω = ω_{n′}. It is not difficult to see that there is a constant ζ_{1} (not depending on n′ or on a_{0}) such that |ξ_{n}(ω)| < ζ_{1}|ξ_{n−1}(ω)| for all n ≤ n′. Let k_{0} ≤ n′ be maximal such that |ξ_{k_{0}}(ω)| ≤ |U_{Δ}|. Then |ξ_{k_{0}}(ω)| > \frac{|U_{Δ}|}{ζ_{1}} = μ.

Using the argument of the Main Distortion Lemma [22, pages 421-424] and knowing that for parameters in BC the parameter and space derivatives are comparable [22, Prop. 6.2, page

406], one can infer that \frac{|ξ′_{k_{0}} (a)|}{|ξ′_{k_{0}} (b)|} < C for a, b ∈ ω. This distortion estimate clearly implies that ξ_{k_{0}} is monotone on ω.

By Corollary (Jakobson) [22, page 403], we have that |BC ∩ (4 − ϵ, 4])| > 0 for all ϵ > 0. Now choose finitely many subintervals I^{l} such that each is of length \frac{μ}{3} and their union covers [0, 1]. Then clearly there exists l_{0} such that I^{l_{0}} ⊂ ξ_{k_{0}}(ω).

Since there are only finitely many intervals I^{l}, one can choose one of these intervals as I_{0} and choose a measurable subset BC_{0} of BC′ ⊂ BC such that conditions 1-7 of Lemma 2 hold for any a_{0} ∈ BC_{0}, for the intervals ω_{n_{i}} defined by using the above substantial escape intervals and a sequence of ϵ_{1,i}’s tending to zero.

Lemma 3 Let I_{0} be as in Lemma 2. Fix d_{l} ∈ (0, c) and d_{r} ∈ (c, 1) such that |c − d_{l}| = |c − d_{r}| ≤ \frac{1}{100^{100}} . Choose k_{1} ∈ N and I_{1} ⊂ I_{0} such that g_{4}^{k_{1}}|I_{1} is monotone and g_{4}^{k_{1}}(I_{1}) = [0, 1]. Set k_{2} = min{k | g_{4}^{k}((c, d_{l})) ∋ c}. Then there exists ϵ > 0 such that for every a ∈ [4 − ϵ, 4] we have the following:

1. there exists I_{a} ⊂ I_{1} such that g_{a}^{k_{1}}(I_{a}) ⊃ [d_{l}, d_{r}] and g_{a}^{k_{1}}|I_{a} is monotone,

2. g_{a}^{k_{2}+i}(c) < c for 0 ≤ i ≤ 2,

3. g_{a}^{k_{2}}((d_{l}, c)) ∋ c, and

4. g_{a}^{k_{2}}|[d_{l}, c] is monotone.

Each of I_{1} and I_{a} are chosen to be closed intervals. As logistic maps are symmetric about the turning point c, from (3) and (4) we also have that g_{a}^{k_{2}}((c, d_{r})) ∋ c, and g_{a}^{k_{2}}|[c, d_{r}] is monotone.

Proof: The lemma follows from the smoothness in both variables of the function g(a, x) = ax(1 − x), g_{4}(c) = 1, and g_{4}^{n}(c) = 0 for all n > 1. □

Lemma 4 Let ϵ be as in Lemma 3. Fix a_{0} ∈ (4 − ϵ, 4) and p, q ∈ Q with (p − \frac{p+q}{4} , q + \frac{p+q}{4} ) ⊂ (c_{2}(a_{0}), c_{1}(a_{0})) \ \{c\}. Let I_{a_{0}} be chosen by item (1) of Lemma 3. Then, we can define nondegenerate closed intervals I_{l} ≡ I_{l}(a_{0}, p, q), I_{r} ≡ I_{r}(a_{0}, p, q) ⊂ I_{a_{0}} and δ_{a_{0}} > 0 with the following properties:

1. either (p, q) ⊂ (c, c_{1}(a)) for every a ∈ (a_{0} − δ_{a_{0}}, a_{0} + δ_{a_{0}}), or (p, q) ⊂ (c_{2}(a), c) for every a ∈ (a_{0} − δ_{a_{0}}, a_{0} + δ_{a_{0}}),

2. g_{a_{0}}^{k_{1}}(I_{l}) ⊂ (d_{l}, c) and g_{a_{0}}^{k_{1}}(I_{r}) ⊂ (c, d_{r}),

3. if a ∈ (a_{0}−δ_{a_{0}}, a_{0}+δ_{a_{0}}), then g_{a}^{k_{1}}(I_{l}) ⊂ (d_{l}, c), g_{a}^{k_{1}}(I_{r}) ⊂ (c, d_{r}), and g_{a}^{k_{1}}|J_{a_{0}} is monotone, where J_{a_{0}} is the shortest closed interval containing both I_{l} and I_{r},

4. if a ∈ (a_{0} − δ_{a_{0}}, a_{0} + δ_{a_{0}}), then g_{a}^{k_{1}+k}(I_{l}), g_{a}^{k_{1}+k}(I_{r}) ⊂ (0, c) for k = 1, …, k_{2}; if additionally (p, q) ⊂ (c_{2}(a_{0}), c), then g_{a}^{k_{1}+k_{2}+1}(I_{l}), g_{a}^{k_{1}+k_{2}+1}(I_{r}) ⊂ (c, 1), and

5. if a ∈ (a_{0} − δ_{a_{0}}, a_{0} + δ_{a_{0}}), then g_{a}^{k_{1}+k_{2}+i}(I_{l}), g_{a}^{k_{1}+k_{2}+i}(I_{r}) ⊂ (p, q), where i = 1 if (p, q) ⊂ (c, c_{1}(a)) and i = 2 if (p, q) ⊂ (c_{2}(a), c).

Notice that each of g_{a_{0}}^{k_{1}}|I_{∗}, g_{a_{0}}^{k_{1}+k_{2}}|I_{∗}, and g_{a_{0}}^{k_{1}+k_{2}+i}|I_{∗} are monotone, where ∗ = r or l. Also, as I_{a_{0}} ⊂ I_{1} ⊂ I_{0}, we have that I_{r}, I_{l} ⊂ I_{0}.

Proof: Again, consider the function of two variables, g(a, x) = ax(1 − x), which is continuous in both variables. The continuity of g(a, x) gives (1). Item (2) follows from (1) of Lemma 3. Item (1) of Lemma 3 gives (3). Items (1)-(4) of Lemma 3 give (4) holding for a = a_{0}; then use the continuity of g(a, x) to obtain (4) holding for a ∈ (a_{0} − δ_{0}, a_{0} + δ_{0}). It remains to prove (5).

From (2)-(4) of Lemma 3 we have that g_{a_{0}}^{k_{2}}|[d_{l}, c] is monotone, g_{a_{0}}^{k_{2}}(c) < c, and [g_{a_{0}}^{k_{2}}(c), c] ⊂ g_{a_{0}}^{k_{2}}([d_{l}, c]). Let z^{∗} be the unique point in (d_{l}, c) such that g_{a_{0}}^{k_{2}}(z^{∗}) = c. If (p, q) ⊂ (c, c_{1}(a_{0})), then g_{a_{0}}^{k_{2}+1}([z^{∗}, c]) ⊃ (p, q) since (by (2) of Lemma 3) g_{a_{0}}^{k_{2}+1}(c) < c. If (p, q) ⊂ (c_{2}(a_{0}), c), let z^{∗∗} be the unique point in (z^{∗}, c) such that g_{a_{0}}^{k_{2}+1}(z^{∗∗}) = c; such a z^{∗∗} exists by (2) of Lemma 3. Then, g_{a_{0}}^{k_{2}+2}([z^{∗}, z^{∗∗}]) = (c_{2}(a_{0}), c_{1}(a_{0})) ⊃ (p, q). Thus, (5) holds for a = a_{0}. Again, use the continuity of g(a, x) to get that (5) holds for all a ∈ (a − δ_{0}, a + δ_{0}). □

Lemma 5 Let ϵ be as in Lemma 3. Fix a_{0} ∈ (4 − ϵ, 4) ∩ BC_{0}, let {n_{i}} be as in Lemma 2 for the fixed a_{0}, and let δ_{0} be as in Lemma 4. Let M ∈ N be such that i ≥ M implies that ω_{n_{i}}^{a_{0}} ⊂ (4−ϵ, 4)∩(a_{0} −δ_{a_{0}}, a_{0} +δ_{a_{0}}). Fix i ≥ M. Set ω_{l} = ξ_{n_{i}}^{−1}(I_{l})∩ω_{n_{i}}^{a_{0}} and ω_{r} = ξ_{n_{i}}^{−1}(I_{r})∩ω_{n_{i}}^{a_{0}}; I_{r} and I_{l} are as in Lemma 4. Then, 1. n_{i} + k_{1} is a cutting time either for all a ∈ ω_{l} or for all a ∈ ω_{r},

2. if n_{i} + k_{1} is a cutting time for all a ∈ ω_{l}, then n_{i} + k_{1} + k_{2} + i is a cutting time for all a ∈ ω_{l}, where i = 1 if (p, q) ⊂ (c, c_{1}(a_{0})) and i = 2 if (p, q) ⊂ (c_{2}(a_{0}), c),

3. if n_{i} + k_{1} is a cutting time for all a ∈ ω_{r}, then n_{i} + k_{1} + k_{2} + i is a cutting time for all a ∈ ω_{r}, where i = 1 if (p, q) ⊂ (c, c_{1}(a_{0})) and i = 2 if (p, q) ⊂ (c_{2}(a_{0}), c).

Proof: We have that ξ_{n_{i}}|ω_{n_{i}}^{a_{0}} is monotone, I_{0} ⊂ ξ_{n_{i}}(ω_{n_{i}}^{a_{0}}) (follows from Lemma 2), I_{0} ⊃ I_{l}, I_{r} (follows from Lemmas 3 and 4), and thus ω_{l}, ω_{r} ⊂ ω_{n_{i}}^{a_{0}}.

Item (3) of Lemma 4 gives that ξ_{n_{i}+k_{1}}(ω_{l}) ⊂ (d_{l}, c) and ξ_{n_{i}+k_{1}}(ω_{r}) ⊂ (c, d_{r}). Let Γ_{a_{0}} be the shortest closed interval containing both ω_{l} and ω_{r}. Thus, c ∈ ξ_{n_{i}+k_{1}}(Γ_{a_{0}}) \ (ξ_{n_{i}+k_{1}}(ω_{l}) ∪ ξ_{n_{i}+k_{1}}(ω_{r})). Moreover, ξ_{n_{i}+k_{1}}|Γ_{a_{0}} is monotone; for if not, then there exists a ∈ (a_{0} − δ_{a_{0}}, a_{0} + δ_{a_{0}}) ∩ Γ_{a_{0}} and 0 < j < k_{1} such that ξ_{n_{i}+j}(a) = c. But then, x ≡ ξ_{n_{i}}(a) ∈ J_{a_{0}} (J_{a_{0}} from Lemma 4-3) and g_{a}^{j}(x) = g_{a}^{n_{i}+j}(c) = ξ_{n_{i}+j}(a) = c, contradicting item (1) of Lemma 3. Summing up, we have that ξ_{n_{i}+k_{1}}|Γ_{a_{0}} is monotone, Γ_{a_{0}} ⊃ ω_{l}, ω_{r}, and c ∈ ξ_{n_{i}+k_{1}}(Γ_{a_{0}})\(ξ_{n_{i}+k_{1}}(ω_{l})∪ξ_{n_{i}+k_{1}}(ω_{r})). Thus from Lemma 1 we have that n_{i} + k_{1} is a cutting time for all a ∈ ω_{l}, or for all a ∈ ω_{r}, i.e.,

(1) holds.

We next prove (2) as the proof of (3) is similar. Assume that n_{i} + k_{1} is a cutting time for all a ∈ ω_{l} and that (p, q) ⊂ (c, c_{1}(a)) for every a ∈ (a_{0} −δ_{a_{0}}, a_{0} +δ_{a_{0}}); the case for (p, q) ⊂ (c_{2}(a), c) being similar.

From the above, we have that ξ_{n_{i}+k_{1}}|Γ_{a_{0}} is monotone and that there is a unique a_{∗} ∈ Γ_{a_{0}} with ξ_{n_{i}+k_{1}}(a_{∗}) = c. Lemma 1 implies that a_{∗} is to the left of ω_{l}. Let E denote the shortest closed interval containing both a_{∗} and ω_{l}; then E ⊂ Γ_{a_{0}}. Notice that ξ_{n_{i}+k_{1}}(E) ⊂ [d_{l}, c], since ξ_{n_{i}+k_{1}}(ω_{l}) ⊂ (d_{l}, c), ξ_{n_{i}+k_{1}}(a_{∗}) = c, and ξ_{n_{i}+k_{1}}|E is monotone. See Figure 1-a.

We have that ξ_{n_{i}+k_{1}+k_{2}}|E is monotone; for else, there exists a ∈ E ⊂ Γ_{a_{0}} and 0 < j < k_{2} such that ξ_{n_{i}+k_{1}+j}(a) = c. Set x = ξ_{n_{i}+k_{1}}(a) ∈ (d_{l}, c). Then, g_{a}^{j}(x) = ξ_{n_{i}+k_{1}+j}(a) = c, contradicting (4) of Lemma 3.

Next, ξ_{n_{i}+k_{1}}(a_{∗}) = c and (2) of Lemma 3 imply that g_{a_{∗}}^{k_{2}}(c) = ξ_{n_{i}+k_{1}+k_{2}}(a_{∗}) < c. Also,

(4) of Lemma 4 implies that ξ_{n_{i}+k_{1}+k_{2}}(ω_{l}) ⊂ (0, c). These two observations and ξ_{n_{i}+k_{1}+k_{2}}|E monotone, imply that ξ_{n_{i}+k_{1}+k_{2}}(E) ⊂ (0, c) and hence that ξ_{n_{i}+k_{1}+k_{2}+1}|E is monotone. Lastly, note that ξ_{n_{i}+k_{1}+k_{2}+1}(ω_{l}) ⊂ (p, q) ⊂ (c, c_{1}(a_{0})) by (5) of Lemma 4 and that ξ_{n_{i}+k_{1}+k_{2}+1}(a_{∗}) < c from (2) of Lemma 3. Thus, recall Lemma 1, we have (2) holding. See Figure 1-b. □

c  c E a_{∗}  ω_{l}  ω_{l} a_{∗} qp d_{l}  ξ_{n_{i}+k_{1}}  ξ_{n_{i}+k_{1}+k_{2}+1} Figure 1-a: Piece of ξ_{n_{i}+k_{1}}  Figure 1-b: Piece of ξ_{n_{i}+k_{1}+k_{2}+1}

4 Gathering

Let p < q be rational numbers in [0, 1] with c = \frac{1}{2} ∉ (p, q). Let ϵ > 0 be as in Lemma 3. Set

Ω_{p,q} = {a ∈ BC_{0} | there exists S_{k}(a) such that c_{S_{k}(a)} ∈ (p, q)}, and
Ω_{p,q}^{0} = {a ∈ BC_{0} ∩ (4 − ϵ, 4] | (p, q) ⊂ (ξ_{2}(a), ξ_{1}(a))}.  (6)

Note that both Ω_{p,q} and Ω_{p,q}^{0} are Lebesgue measurable sets.

Lemma 6 Let a_{0} ∈ (4−ϵ, 4)∩BC_{0} be a Lebesgue density point of Ω_{p,q}^{0}. Then the upper density of Ω_{p,q} at a_{0} is positive.

Proof: Choose i ≥ M from Lemma 5 and ω_{n_{i}}^{a_{0}} from Lemma 2. Let I_{l}, I_{r} be as in Lemma 4 (for the fixed a_{0}), ω_{l} and ω_{r} be as in Lemma 5, and C be as in Lemma 2. Then, from 5-6 of Lemma 2, we have for a suitable constant K (in fact, K = \frac{|I_{0}|}{C^{2}} works) that

\frac{|ω_{l}|}{|ω_{n_{i}}^{a_{0}}|} ≥ K \frac{|I_{l}|}{|I_{0}|} ≡ η > 0,

and

\frac{|ω_{r}|}{|ω_{n_{i}}^{a_{0}}|} ≥ K \frac{|I_{r}|}{|I_{0}|} ≡ η > 0.

We have, by Lemma 5, either

ω_{l} ∩ BC_{0} ⊂ Ω_{p,q}

or

ω_{r} ∩ BC_{0} ⊂ Ω_{p,q}.

Hence, since a_{0} is a density point of BC_{0} and a_{0} ∈ ω_{n_{i}}^{a_{0}}, the upper density of Ω_{p,q} at a_{0} is positive. □

Proof of Proposition 1: Observe that almost every point of Ω_{p,q}^{0} is a Lebesgue density point of Ω_{p,q}^{0} and hence, by Lemma 6, almost all of these points are upper density points of Ω_{p,q}. Since Ω_{p,q} is Lebesgue measurable, almost every of its upper density points belong to Ω_{p,q}. Thus, almost every point of Ω_{p,q}^{0} is in Ω_{p,q}. Therefore, setting Ω_{p,q}^{1} = {a ∈ Ω_{p,q}^{0} | a ∉ Ω_{p,q}}, we have that |Ω_{p,q}^{1}| = 0. Set

Ω^{1} = ⋃_{{p<q, c∉(p,q) \atop p,q∈Q}} Ω_{p,q}^{1}.

Then |Ω^{1}| = 0.

Assume that a ∈ (BC_{0} ∩ (4 − ϵ, 4]) \ Ω^{1}. If (p, q) ⊂ (ξ_{2}(a), ξ_{1}(a)) and c ∉ (p, q), then a ∈ Ω_{p,q}^{0}. As a ∉ Ω^{1}, we have that a ∉ Ω_{p,q}^{1}. Hence, a ∈ Ω_{p,q} and thus there exists S_{k}(a) such that c_{S_{k}(a)} ∈ (p, q). This implies a ∈ BC^{∗}. □

5 Appendix

The purpose of this appendix is to prove Proposition 2 below. One can prove this proposition using Lemma 8 below and either [10, Proposition 4] or [4, Lemma 1.2]. We use [10, Proposition 4]. Using arguments similar to those here, one can prove that a ∈ A guarantees that (I, T_{a}) has the universality property.

Proposition 2 Let a_{0} ∈ B. Then (I, g_{a_{0}}) has the universality property, i.e., for every choice of parameters {a_{i}}_{i≥1}, with a_{i} ∈ [2, 4] for all i, and open set U ⊂ (I, g̃_{a_{0}}) there exists a subcontinuum H ⊂ U such that H is homeomorphic to (I, g̃_{a_{i}}).

First some preliminaries needed to state [10, Proposition 4], which appears here as Proposition 3. We need to generalize and slightly modify some of our earlier definitions and state some new ones.

Given continuous maps f_{i} : I_{i} → I_{i−1}, the associated inverse limit space with bonding maps f_{i} is

(I_{i}, f_{i}) = {(x_{1}, x_{2}, x_{3}, …) | x_{i−1} = f_{i}(x_{i}) for i ≥ 2}.

It has the metric d(x, y) = ∑_{i} \frac{|x_{i}−y_{i}|}{2^{i}}. The i-th projection is denoted as π_{i} : (I_{i}, f_{i}) → I_{i}. In general, (I_{i}, f_{i}) is a continuum, i.e. a compact connected metric space. It is one-dimensional and chainable, see [23]. For i > j set f_{i,j} = f_{j+1} ∘ ⋯ ∘ f_{i} : I_{i} → I_{j}; f_{i,i} denotes the identity map.

c_{i} f_{i} I_{i−1}  I_{i} Figure 2

For any (I_{i}, f_{i}) we assume each f_{i} : I_{i} → I_{i−1} is unimodal and onto. For convenience, let c represent the turning point of each f_{i}; allowing c to vary with i. Here we relax the definition of unimodal maps by not assuming that f_{i}(∂I_{i}) ⊂ ∂I_{i−1}.

Let (I_{i}, f_{i}) be given and for i ≥ 1 and z ∈ I_{i} set,

ν̂_{i}(z) = \{ \begin{matrix}1 & if z ∈ (c, f_{i+1}(c)], \\ ∗ & if z = c, \\ 0 & if z ∉ [c, f_{i+1}(c)]. \end{matrix}

For i ≥ 1 and z ∈ I_{i} let ν̂|_{i}(z) = ν̂_{1}(f_{i,1}(z)), …, ν̂_{i−1}(f_{i}(z)), ν̂_{i}(z) for z ∈ I_{i}. Set C_{1} = ∂I_{1} ∪ {c} and for i ≥ 2 set

C_{i} = ∂I_{i} ∪ ⋃_{1≤j≤i} f_{i,j}^{−1}(c) ⊂ I_{i}.

Each C_{i} defines a partition of I_{i}, which we denote by P_{i}. When more than one inverse limit space is being discussed, we use C_{i}^{f} and C_{i}^{g} etc.

Definition 1 We say (I_{i}, f_{i}) is combinatorially equivalent to (J_{i}, g_{i}), written (I_{i}, f_{i}) ≈ (J_{i}, g_{i}), provided that for each i there exists a monotone bijection h_{i} : C_{i}^{f} → C_{i}^{g} such that ν̂|_{i}(α) = ν̂|_{i}(h_{i}(α)) for each α ∈ C_{i}^{f} and such that h_{i}(c^{f}) = c^{g}.

[figure: I_{i}; I_{i−1}; I_{i−2}; I_{i−3}; f_{i−2}; f_{i−1}; f_{i}; f_{i,i−3}; f_{i}; f_{i−1}; f_{i−2}; c_{i}; c_{i−1}; Figure 3]

For ease of notation, in the statement of Proposition 3 it is assumed that I_{i} = [f_{i+2,i}(c), f_{i+1,i}(c)] and J_{i} = [g_{i+2,i}(c), g_{i+1,i}(c)] for all i. Assuming no f_{i} nor g_{i} is the full unimodal map, we have that f_{i+3,i}(c) and g_{i+3,i}(c) are in the interior of I_{i} respectively J_{i}. Using results of [8] there is no loss of generality by assuming that no f_{i} nor g_{i} is a full unimodal map (we say f : I → I is a full unimodal map provided that f maps the closure of each component of I \ {c} onto I).

Proposition 3 [10, Proposition 4] Let (I_{i}, f_{i}) and (J_{i}, g_{i}) be given such that

(I_{i}, f_{i}) ≈ (J_{i}, g_{i}).  (7)

Suppose that a_{i−3}^{f}, b_{i−3}^{f} ∈ C_{i−3}^{f} are such that f_{i,i−3}(c) ∈ [a_{i−3}^{f}, b_{i−3}^{f}] = cl P_{i−3}^{f} for some P_{i−3}^{f} ∈ P_{i−3}^{f}. Write A_{i−3}^{f} = f_{i−3,1}(a_{i−3}^{f}) and B_{i−3}^{f} = f_{i−3,1}(b_{i−3}^{f}). Let a_{i−3}^{g}, b_{i−3}^{g}, A_{i−3}^{g} and B_{i−3}^{g} be the corresponding points for (J_{i}, g_{i}). If for all i sufficiently large

(1 − \frac{1}{i^{2}} ) \frac{|B_{i−3}^{f} − f_{i,1}(c)|}{|B_{i−3}^{f} − A_{i−3}^{f}|} ≤ \frac{|B_{i−3}^{g} − g_{i,1}(c)|}{|B_{i−3}^{g} − A_{i−3}^{g}|} ≤ (1 + \frac{1}{i^{2}} ) \frac{|B_{i−3}^{f} − f_{i,1}(c)|}{|B_{i−3}^{f} − A_{i−3}^{f}|},  (8)

then (I_{i}, f_{i}) is homeomorphic to (J_{i}, g_{i}).

Lemma 7 Let a ∈ B, Then lim inf Q_{a}(k) < 2.

Lemma 7 is immediate from [12, Lemma 3.5].

Remark 1 Let a ∈ B. Then lim_{n→∞} z_{n}(a) = \frac{1}{2} and lim_{n→∞} ẑ_{n}(a) = \frac{1}{2}. This follows from the fact that any logistic map has no wandering interval. For details see [22].

Lemma 8 Fix a_{0} ∈ B. There exists Γ ∈ (c_{2}(a_{0}), c) such that for every z ∈ [c_{2}(a_{0}), Γ] and ϵ > 0 there exist positive integers m_{1} < m_{2} < m_{3} < ⋯ and real numbers d_{i}, e_{i}, i ≥ 1, such that for all i:

1. c_{2}(a_{0}) < d_{i} < e_{i} ,

2. g_{a_{0}}^{S_{m_{i}+1}}(c_{2}(a_{0})) ∈ (z − ϵ, z + ϵ),

3. g_{a_{0}}^{S_{m_{i}+1}}(d_{i}) = c_{2} and g_{a_{0}}^{S_{m_{i}+1}}(e_{i}) = c,

4. g_{a_{0}}^{S_{m_{i}+1}}|[c_{2}(a_{0}), d_{i}] is strictly decreasing,

5. g_{a_{0}}^{S_{m_{i}+1}}|[d_{i}, e_{i}] is strictly increasing, and

6. lim_{i→∞} d_{i} = lim_{i→∞} e_{i} = c_{2}(a_{0}).

See Figure 4.

g_{a_{0}}^{S_{m_{i}} +1} c Γ z c_{2}  d_{i}  e_{i}  c  c_{1} Figure 4: Piece of g_{a_{0}}^{S_{m_{i}} +1}

Proof: Let W ∋ \frac{1}{2} be an open interval such that W ⊂ (z_{1}(a), ẑ_{1}(a)) for all a ∈ [2, 4]; since S_{1}(a) = 2 for all a, we have z_{1}(a) ∈ g_{a}^{−2}(c), recall display (3). We let c_{n} = c_{n}(a_{0}), Q(k) = Q_{a_{0}}(k), S_{k} = S_{k}(a_{0}), and z_{n} = z_{n}(a_{0}) for each n, k. Fix p ∈ N such that Q(p + 1) ∈ {0, 1} and (z_{p−1}, ẑ_{p−1}) ⊂ W; we can choose such a p due to Remark 1 and Lemma 7. Set Δ = \overline{W} ∩ [c_{S_{Q(p)}}, c]. Notice that g_{a_{0}}^{2}|Δ is strictly monotone. Let Γ be such that

g_{a_{0}}^{2}(Δ) = [c_{2}, Γ].  (9)

By the choice of p, if for some m we have c_{S_{m}} ∈ (z_{p−1}, z_{p}), then (use (4)) Q(m + 1) = p and therefore Q(Q(m + 1) + 1) ∈ {0, 1}. Thus, c_{S_{m}} ∈ (z_{p−1}, z_{p}) implies that

c_{S_{m+1}} ∈ (c, c_{S_{Q(p)}}) and c_{S_{Q(m+1)}} ∈ (0, z_{2}) ∪ (ẑ_{2}, c_{1}).  (10)

It follows from a_{0} ∈ B and (10) that if w ∈ Δ and δ > 0, then there exist positive integers m_{1} < m_{2} < m_{3} < ⋯ such that for each i we have:

c_{S_{m_{i}}} ∈ (z_{p−1}, z_{p}),
c_{S_{m_{i}+1}} = g_{a_{0}}^{S_{p}}(c_{S_{m_{i}}}) ∈ (w − δ, w + δ), and  (11)
c_{S_{Q(m_{i}+1)}} ∈ (0, z_{2}) ∪ (ẑ_{2}, c_{1}).

Let z ∈ [c_{2}, Γ] and ϵ > 0. Choose w ∈ Δ and δ > 0 such that g_{a_{0}}^{2}((w−δ, w+δ)) ⊂ (z−ϵ, z+ϵ). Let m_{1} < m_{2} < m_{3} < ⋯ be as in (11) for this choice of w and δ. Then for each i there exists α_{i} < z_{m_{i}+1} such that:

g_{a_{0}}^{S_{m_{i}+1}+2}(α_{i}) = c and g_{a_{0}}^{S_{m_{i}+1}+2} : [α_{i}, z_{m_{i}+1}] → [c_{2}, c],

(12)

g_{a_{0}}^{S_{m_{i}+1}+2}(c) ∈ (z − ϵ, z + ϵ) and g_{a_{0}}^{S_{m_{i}+1}+2} : [z_{m_{i}+1}, c] → [c_{2}, c_{S_{m_{i}+1}+2}].

Moreover, g_{a_{0}}^{S_{m_{i}+1}+2}|[α_{i}, z_{m_{i}+1}] is strictly decreasing and g_{a_{0}}^{S_{m_{i}+1}+2}|[z_{m_{i}+1}, c] is strictly increasing for each i. Of course, g_{a_{0}}^{S_{m_{i}+1}+2}(z_{m_{i}+1}) = c_{2} for each i.

For each i set d_{i} = g_{a_{0}}^{2}(z_{m_{i}+1}) and e_{i} = g_{a_{0}}^{2}(α_{i}). Properties 1-4 now easily follow from (12) and (z_{p−1}, ẑ_{p−1}) ⊂ W. Property 6 follows from Remark 1 and lim_{i→∞} m_{i} = ∞. □

Proof of Proposition 2: Fix a_{0} ∈ B and choose a sequence of parameters {a_{i}}_{i≥1} with a_{i} ∈ [2, 4] for all i. If a_{i} = 4 for some i’s, then replacing such a_{i}’s by sufficiently close ã_{i}’s results in an inverse limit space that is homeomorphic to (I, g̃_{a_{i}}) [8]. Thus we may assume that a_{i} ≠ 4 for all i.

Repeated use of Lemma 8 allows us to choose positive integers n_{1} < n_{2} < n_{3} < ⋯ and real numbers b_{1} > b_{2} > b_{3} > ⋯ such that the following hold.

1. For each i, g_{a_{0}}^{S_{n_{i}}} maps [c_{2}(a_{0}), b_{i+1}] onto [c_{2}(a_{0}), b_{i}]. Set I_{i} = [c_{2}(a_{0}), b_{i}] and h_{i} = g_{a_{0}}^{S_{n_{i}}} |I_{i} for all i. Set m_{i} = (c_{2}(a_{0}) + b_{i})/2, and denote by τ_{i} the reflection of I_{i} about m_{i}, that is, τ_{i}(x) = 2m_{i} − x. Set f_{i} = τ_{i} ∘ h_{i} ∘ τ_{i+1}. It is easy to see that the inverse limit spaces (I_{i+1}, f_{i}) and (I_{i+1}, h_{i}) are homeomorphic.

2. For each i there exists a unique turning point c_{i} ∈ I_{i} such that g_{a_{0}}^{S_{n_{i}}} (c_{i+1}) = c_{2}(a_{0}), g_{a_{0}}^{S_{n_{i}}} |[c_{2}(a_{0}), c_{i+1}] is strictly decreasing, and g_{a_{0}}^{S_{n_{i}}} |[c_{i+1}, b_{i+1}] is strictly increasing. Recall Figure 4. Hence f_{i} : I_{i+1} → I_{i} is unimodal and onto.

3. Setting g_{i} = g_{a_{i}} for i ≥ 1, we have the hypotheses of Proposition 3 holding, i.e, (I, g_{i}) is combinatorially equivalent to (I_{i+1}, f_{i}) and (8) of Proposition 3 holds.

Thus (I, g_{i}) is homeomorphic to (I_{i+1}, f_{i}). Let X be the subcontinuum of (I, g̃_{a_{0}}) such that

x ∈ X ⟺ π_{N_{i}}(x) ∈ [c_{2}(a_{0}), b_{i+1}],

for i = 0, 1, 2, ⋯ with N_{0} = 0 and N_{i} = ∑_{k=1}^{i} n_{k}. Then X is homeomorphic to (I_{i+1}, f_{i}) and hence to (I, g_{i}).

This paragraph follows [2, Corollary 6]. Let ĝ_{a_{0}} be the induced homeomorphism on (I, g̃_{a_{0}}) given by ĝ_{a_{0}}((x_{1}, x_{2}, x_{3}, ⋯)) = (g_{a_{0}}(x_{1}), x_{1}, x_{2}, x_{3}, ⋯). Let U ⊂ (I, g̃_{a_{0}}) be open. Without loss of generality assume that U is of the form [U_{1} × U_{2} × ⋯ × U_{n} × I × ⋯] ∩ (I, g̃_{a_{0}}). Due to 3 above, to show that there exists H ⊂ U with H homeomorphic to (I, g̃_{a_{i}}) it suffices to show there exists a subcontinuum D of (I, g̃_{a_{0}}) such that D is homeomorphic to (I, g̃_{a_{0}}) and π_{1}(D) ⊂ U_{n}, since then π_{n}(ĝ_{a_{0}}^{n−1}(D)) ⊂ U_{n} and thus ĝ_{a_{0}}^{n−1}(D) ⊂ U. Repeating the construction given in 1-3 above and choosing n_{1} appropriately, we can guarantee that X ≡ D is such that π_{1}(D) ⊂ U_{n}. □

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