Trajectory of the turning point is dense for a co-σ-porous set of tent maps

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

Zoltán Buczolich^{†} Department of Analysis Eötvös Loránd University Múzeum krt. 6-8 H-1088 Budapest, Hungary

November 8, 1999

Abstract

It is known that for almost every (with respect to Lebesgue measure) a ∈ [\sqrt{2}, 2] the forward trajectory of the turning point of the tent map T_{a} with slope a is dense in the interval of transitivity of T_{a}. We prove that the complement of this set of parameters of full measure is σ-porous.

1 Introduction

For a ∈ (1, 2] set T_{a}(x) = ax for 0 ≤ x ≤ \frac{1}{2} and T_{a}(x) = a(1 − x) for \frac{1}{2} ≤ x ≤ 1. We refer to this family of maps as the family of tent maps. Other models are possible, but two tent maps with the same slope are conjugate via an affine transformation and hence the model does not matter. We choose this model as it makes our computations the easiest. The only measure we use is Lebesgue measure.

trajectory of the turning point is dense for a co-σ-porous set of tent maps 2

We restrict our attention to the parameters a from [\sqrt{2}, 2]. If \sqrt{2} < a^{m} ≤ 2 for some m ∈ {1, 2, 2^{2}, 2^{3}, …}, then the nonwandering set of T_{a} consists of m disjoint closed intervals and a finite number of periodic points [15, p.78]. Moreover, for such a the map T_{a}^{m} restricted to any one of those intervals is a tent map with slope a^{m}, so is affinely conjugate to T_{a^{m}}. Thus, getting corresponding results for smaller parameter values is easy. We work with T_{a} restricted to its core, [T_{a}^{2}(\frac{1}{2}), T_{a}(\frac{1}{2})]; the core is the smallest forward invariant interval containing the turning point \frac{1}{2}. In fact, T_{a} is transitive on the core. For more information on transitivity, nonwandering sets, and other related topics see [1]. The term trajectory will always refer to the forward trajectory.

∗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.

019476 and FKFP 0189/1997.

1991 Mathematics Subject Classification: Primary 58F03 Secondary 54H20, 26A21

In [3] it was proven that for almost every (with respect to Lebesgue measure) a ∈ [\sqrt{2}, 2], the T_{a} trajectory of the turning point \frac{1}{2} is dense in [T_{a}^{2}(\frac{1}{2}), T_{a}(\frac{1}{2})]. Letting D denote those parameters a ∈ [\sqrt{2}, 2] such that the closure of the trajectory of the turning point under T_{a} is [T_{a}^{2}(\frac{1}{2}), T_{a}(\frac{1}{2})], we prove:

Theorem 1 The set [\sqrt{2}, 2] \ D is σ-porous.

In Section 2 we give basic definitions related to porosity and σ-porosity. For a detailed survey of these concepts we refer to [17] and the appendix of [16]. Each σ-porous set in R is of the first category and of zero Lebesgue measure. These sets arise quite often as exceptional sets. For example, Preiss and Zajíček verified that the set of points of Fréchet-nondifferentiability of any continuous convex function on a Banach space with a separable dual is σ-porous [12]. However, Konjagin showed that the set E = {x ∈ R | ∑_{n=1}^{∞} |\frac{sin(n!πx)}{n}| ≤ 1\} is a closed non-σ-porous set of zero Lebesgue measure [17, Chapter 5]. This shows that the σ-ideal of σ-porous sets is a proper subset of the σ-ideal of measure zero first category sets. Therefore Theorem 1 strengthens the result in [3]. To obtain this stronger result, a more delicate (refined) study of the kneading properties of tent maps was necessary. Some of these techniques might be of independent interest.

We break the proof of Theorem 1 into two cases. In Section 4 we deal with the easier case; namely parameters a such that lim inf_{k→∞} Q_{a}(k) < ∞, where Q_{a}(k) denotes the kneading map of T_{a}. The remainder of the paper deals with parameters a such that lim_{k→∞} Q_{a}(k) = ∞. Kneading maps and Hofbauer towers are recalled in the next section.

When obtaining measure results for one parameter families of unimodal maps, one often deals with the piecewise monotone functions ξ_{n}(a) ≡ f_{a}^{n}(c), where {f_{a}} is the one parameter family of maps with common turning point c and n ∈ N. Given an n, the laps of ξ_{n}(a) are the maximal subintervals of monotonicity of ξ_{n}. In [3], where f_{a} = T_{a}, the main tool is the following: there exists ϵ > 0 such that for almost every a ∈ [\sqrt{2}, 2] and for every M ∈ N, there is an n ≥ M such that c ∈ ξ_{n}(J) and |ξ_{n}(J)| > ϵ for the lap J of ξ_{n} containing a; here the measure of ξ_{n}(J) is denoted |ξ_{n}(J)|. In less precise terms, one produces long stretches in the graphs of the ξ_{n}’s for almost all a and for arbitrarily large n. Similar such long stretches have been used in obtaining other measure results [4, 8, 14]. For our porosity results we need to produce long stretches with some additional measure properties/estimates on the associated laps in parameter space; this is done in Section 6.

For a given tent map, T_{a}, the levels of the Hofbauer tower will be denoted by D_{n}(a). As remarked above, we first treat the easier case, namely when lim inf_{k→∞} Q_{a}(k) < ∞ or equivalently (see Lemma 1) when lim inf_{n→∞} |D_{n}(a)| > 0. In this case, infinitely many of the levels in the Hofbauer tower contain a ‘long stretch’ (denoted by W in Section 4). This ‘long stretch’ is used to get the porosity result. Since lim_{k→∞} Q_{a}(k) = ∞ is equivalent to lim_{n→∞} |D_{n}(a)| = 0, it is more difficult to find the required ‘long stretches’ in this second case. To understand the dynamics behind the formal proof presented in Section 6 we provide an algorithm, named the substantial cut algorithm, which finds the required ‘long stretches’ suitable for the porosity estimates. This is done in Section 5. A second algorithm, called the greedy algorithm, is also discussed. This algorithm provides ‘long stretches’ in a ‘fast’ way, but these stretches are not suitable for the geometric estimates needed for porosity. However, the greedy algorithm can be used to obtain lower estimates of the length of levels of the Hofbauer tower at certain cutting times which we call substantial cuts (see Lemma 9).

2 Preliminaries

Let (X, ρ) be a compact metric space, E ⊂ X, x ∈ X, and δ > 0. Then E^{c} = X \ E and B(x, δ) = {y ∈ X | ρ(x, y) < δ}. We define γ(E, x, δ) to be the minimum of 1 and the number defined by

sup{2η | η > 0 and there exists y ∈ X such that B(y, η) ⊂ B(x, δ) ∩ E^{c}}.

If no such y exists, we set γ(E, x, δ) = 0. We can now define the porosity of E in X. For more detailed discussions on porosity see [13, 16, 17].

Definition 1 If x ∈ E, then we define the porosity of E in X at x to be

p(E, x) = lim sup_{δ→0^{+}} \frac{γ(E, x, δ)}{δ}.

If p(E, x) > 0, then E is said to be porous in X at x. We say that E is porous in X if p(E, x) > 0 for all x ∈ E. Any subset of X which can be written as a countable union of sets, each porous in X, is said to be σ-porous in X. If A ⊂ X is σ-porous, then we say X \ A is co-σ-porous.

Notice that if X contains no isolated points, then any countable subset of X is σ-porous. For E, F ⊂ X, we denote the Hausdorff distance between E and F by HD(E, F); so HD(E, F) = max{d_{E}(F), d_{F} (E)}, where d_{A}(B) = sup{ρ(A, x) | x ∈ B}. We denote the closure of a set U ⊂ X by \overline{U}.

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 invariant and f maps [c_{2}, c_{1}] onto itself; the interval [c_{2}, c_{1}] is called the core of the map f.

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.  (1)

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)},  (2)

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}).  (3)

The kneading map was introduced by Hofbauer, see e.g. [9, 10]. If Q(k) is defined for all k ∈ N, then

Q(k) < k  (4)

for all k ∈ N, [5, page 3]. The kneading map (or cutting times) determine the combinatorics of f completely. A survey of this tool can be found in [5]; our discussion follows [5].

Closely related to the kneading map is the Hofbauer-tower [9]. 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}. From (2) it follows that for k ≥ 1

D_{S_{k}} = [c_{S_{k}}, c_{S_{Q(k)}}].  (5)

We say a unimodal map f is locally eventually onto, leo, provided that for every ϵ > 0 there exists M ∈ N such that if U is an interval with |U| > ϵ and if n ≥ M, then f^{n}(U) = [c_{2}, c_{1}]. We say x ∈ I is periodic provided there exists n ∈ N such that f^{n}(x) = x. Similarly, we say x ∈ I is eventually periodic provided there exists n ∈ N and a periodic point y such that f^{n}(x) = y. When writing [a, b] we do not assume that a ≤ b. We denote the rationals by Q and the length of an interval U by |U|.

3 Tent Map Preliminaries

For each a ∈ [\sqrt{2}, 2], the map T_{a}|[c_{2}, c_{1}] is leo, see e.g. [2, Lemma 2]. Let P = \{a ∈ [\sqrt{2}, 2] | the turning point \frac{1}{2} is either periodic or eventually periodic\}. The set P is countable and contains no isolated points [2]; therefore P is σ-porous. Also, for a ∈ P, the kneading map Q_{a}(k) is defined for only finitely many k ∈ N, see e.g. [5, page 3]. Next, notice that for any a ∈ (1, 2] the z_{k}’s as defined in (1) are such that ẑ_{k} = 1 − z_{k} and lim_{k→∞} z_{k} = lim_{k→∞} ẑ_{k} = \frac{1}{2}. When more than one parameter value is being used, we may write D_{n}(a) for the levels in the Hofbauer tower for T_{a}, Q_{a}(k) for the kneading map of T_{a}, S_{k}(a) for the cutting times of T_{a}, or c_{n}(a) for T_{a}^{n}(c). Another notation for c_{n}(a) is ξ_{n}(a); the latter being used when one is interested in T_{a}^{n}(c) as a function of the parameter a.

Definition 2 Set

• D = \{a ∈ [\sqrt{2}, 2] | \overline{\{T_{a}^{n}(c)\}}_{n≥0} = [c_{2}(a), c_{1}(a)]\} and

• I = \{a ∈ [\sqrt{2}, 2] | lim_{k→∞} Q_{a}(k) = ∞\}.

As noted in the Introduction, it is easy to establish that \{a ∈ [\sqrt{2}, 2] | lim inf_{k→∞} Q_{a}(k) < ∞ and a ∉ D ∪ P} is σ-porous; this is done in Section 4. The more interesting/difficult case is to show that I is σ-porous; this is done in Sections 6 and 7.

It is known that for a ∈ D we have lim inf_{k→∞} Q_{a}(k) < 2, see e.g. [4, Lemma 3.5] (this is not an ‘if and only if’ statement). Hence, D ∩ I = ∅. Again, in [3] it is shown that D has full Lebesgue measure in [\sqrt{2}, 2] and hence we have that I has zero Lebesgue measure in [\sqrt{2}, 2]. On the other hand, I is dense in [\sqrt{2}, 2] and is uncountable. Since we could not find a proof of this fact in the literature, we include a proof for completeness (Lemma 5). We do not explicitly use Lemmas 1 and 5 in the paper, however we include them to give/recall facts about the set I.

Lemma 1 Fix a > \sqrt{2}. Let Q(k) be the kneading map for T_{a}. Then lim_{k→∞} Q(k) = ∞ if and only if lim_{n→∞} |D_{n}| = 0.

Proof: If lim_{n→∞} |D_{n}| = 0, then (3) and (5) imply that lim_{k→∞} Q(k) = ∞.

If lim_{k→∞} Q(k) = ∞, then (3) and (5) imply that lim_{k→∞} |D_{S_{k}}| = 0. But, |D_{n}| < |D_{S_{k+1}}| for S_{k} < n < S_{k+1} since T_{a}|D_{n} is monotone for such n. Thus, lim_{n→∞} |D_{n}| = 0. □

Lemma 1 holds for more general unimodal maps with some expansion properties.

Definition 3 For a ∈ [\sqrt{2}, 2] and n ∈ N let ω_{n}(a) be the maximal open interval in parameter space containing a such that ξ_{n} is monotone on ω_{n}(a); recall that ξ_{n}(a) ≡ T_{a}^{n}(c). Note that ω_{n}(a) is not defined for a ∈ P and large n. In Figure 1, with n = S_{k}, we have ω_{S_{k}}(a) = (u, v).

We recall two known lemmas, see e.g. [3, 14].

Lemma 2 Fix ϵ_{0} > 0 and a_{0} ∈ (\sqrt{2}, 2]. Then there exists K_{0} ∈ N such that for all k ≥ K_{0},

\frac{|ξ′_{k}(b)|}{|ξ′_{k}(a)|} ≤ 1 + ϵ_{0},

whenever a, b, belong to the same lap of ξ_{n}|[a_{0}, 2].

Lemma 3 There exist positive constants α and β such that for all k ≥ 2 and all a ∈ [\sqrt{2}, 2]

αa^{k} ≤ |ξ′_{k}(a)| ≤ βa^{k},

where ξ′_{k} is defined. Hence we also have that \frac{1}{α}a^{−k} ≥ |(ξ_{k}^{−1})′(ξ_{k}(a))| ≥ \frac{1}{β} a^{−k}, for any branch of ξ_{k}^{−1}.

For a discussion of Lemma 4 see [14, Chapter 3]. Lemma 4 is used only in the proof of Lemma 5.

Lemma 4 Fix a ∈ (\sqrt{2}, 2]. Then, n is a cutting time for T_{a} if and only if ξ_{n}(ω_{n}(a)) ∋ c and a > b where b is the unique point in ω_{n}(a) such that ξ_{n}(b) = c.

Often one works in Phase Space, i.e., one plots the ξ_{n}(a)’s as functions of the parameter a. Figure 1 is a piece of Phase Space. Let a be given and suppose that n = S_{k}(a) = S_{k} is a cutting time for T_{a}; let Q_{a}(k) = Q(k). Then, one of the pictures in Figure 1 hold. From Lemma 2, we see that for large n the graph of ξ_{n} is almost linear and hence for ease we draw linear functions in Figure 1; thus assume that k is large in Figure 1. We have that ω_{S_{k}}(a) = (u, v). The point b in Figure 1 is such that ξ_{S_{k}}(b) = c = \frac{1}{2}. Also, S_{k} is a cutting time for all a′ ∈ (b, v), D_{S_{k}}(a′) = [T_{a′}^{S_{k}}(c), T_{a′}^{S_{Q(k)}}(c)] for all a′ ∈ (u, v) \ {b}, and S_{k} is not a cutting time for all a′ ∈ (u, b). Again, for a discussion of these and related details see [14, Chapter 3].

[figure: Figure 1: Phase Space at cutting time S_{k}; c; ξ_{S_{k}}; ξ_{S_{Q(k)}}; c_{S_{k}}(a); c_{S_{Q(k)}}(a); a; b; •; •; u; v; ξ_{S_{Q(k)}}; b; a; •; •; ξ_{S_{k}}; c_{S_{Q(k)}}(a); c_{S_{k}}(a); v; u]

Lemma 5 The set I is dense in [\sqrt{2}, 2] and is uncountable.

Proof: Let U ⊂ [\sqrt{2}, 2] be an open interval. Choose a_{1} ∈ U \ P and a cutting time n_{1} = S_{k_{1}}(a_{1}) such that ω_{n_{1}}(a_{1}) ⊂ U. We can make such a choice due to P being countable and Lemma 3. Let ϵ_{1} > 0 and set J_{1} = {a ∈ ω_{n_{1}}(a_{1}) | n_{1} = S_{k_{1}}(a) and |c−c_{n_{1}}(a)| < ϵ_{1}}. Then for each a ∈ J_{1} we have that n_{1} is a cutting time for T_{a} and |c − c_{n_{1}}(a)| < ϵ_{1}. Notice that J_{1} is an open subinterval of U, recall Lemma 4.

Fix a_{2} ∈ J_{1} \ P. Then n_{1} = S_{k_{1}}(a_{2}). Set n_{2} = S_{k_{1}+1}(a_{2}). Choose 0 < ϵ_{2} < \frac{ϵ_{1}}{2} such that J_{2} ≡ {a ∈ ω_{n_{2}}(a_{2}) | n_{2} = S_{k_{1}+1}(a) and |c − c_{n_{2}}(a)| < ϵ_{2}} ⊂ J_{1} and such that the sets J_{1} and J_{2} share no boundary points. Then (again use Lemma 4) for each a ∈ J_{2} we have that n_{1} = S_{k_{1}}(a), n_{2} = S_{k_{1}+1}(a), |c − c_{n_{1}}(a)| < ϵ_{1}, and |c − c_{n_{2}}(a)| < ϵ_{2}. Also, J̄_{2} is a proper closed subinterval of J̄_{1}.

Fix a_{3} ∈ J_{2} \ P. Then n_{1} = S_{k_{1}}(a_{3}) and n_{2} = S_{k_{1}+1}(a_{3}). Set n_{3} = S_{k_{1}+2}(a_{3}). Choose 0 < ϵ_{3} < \frac{ϵ_{2}}{2} such that J_{3} ≡ \{a ∈ ω_{n_{3}}(a_{3}) | n_{3} = S_{k_{1}+2}(a) and |c − c_{n_{3}}(a)| < ϵ_{3}\} ⊂ J_{2} and the sets J_{2} and J_{3} share no boundary points. Then, for each a ∈ J_{3} we have that n_{1} = S_{k_{1}}(a), n_{2} = S_{k_{1}+1}(a), n_{3} = S_{k_{1}+2}(a), |c − c_{n_{1}}(a)| < ϵ_{1}, |c − c_{n_{2}}(a)| < ϵ_{2}, and |c − c_{n_{3}}(a)| < ϵ_{3}. Also, J̄_{3} is a proper closed subinterval of J̄_{2}.

Continue this process and set a_{∗} = ∩_{n≥1}J_{n}. Remember that if lim inf_{k→∞} Q_{a_{∗}}(k) < ∞, then (by (3)) there exists some δ > 0 such that for infinitely many k we have |c − c_{S_{k−1}}| > δ. Hence, we have that lim_{k→∞} Q_{a_{∗}}(k) = ∞ with a_{∗} ∈ U. By varying the choices of {a_{i}} and hence of the sequences of cutting times {n_{i}}, one can easily show that I is uncountable. □

Lemma 6 Let a, a′ ∈ [\sqrt{2}, 2] and L > 0. Then for all x ∈ [0, 1], |T_{a}^{L}(x) − T_{a′}^{L}(x)| ≤ |a − a′|\frac{a^{L}−1}{a−1}.

Proof: Clearly, |T_{a}(x) − T_{a′}(x)| ≤ |a − a′|. Thus,

|T_{a}^{L}(x) − T_{a′}^{L}(x)| ≤ |T_{a′}(T_{a′}^{L−1}(x)) − T_{a}(T_{a′}^{L−1}(x))| + |T_{a}(T_{a′}^{L−1}(x)) − T_{a}(T_{a}^{L−1}(x))|
≤ |a − a′| + a(|T_{a′}^{L−1}(x) − T_{a}^{L−1}(x)|)
≤ |a − a′|(1 + a + a^{2} + … + a^{L−1}) + a^{L}|T_{a′}^{0}(x) − T_{a}^{0}(x)|
= |a − a′|\frac{a^{L} − 1}{a − 1}.
□

Definition 4 For a ∈ [\sqrt{2}, 2] and each k ∈ N denote by ω′_{S_{k}}(a) that portion of ω_{S_{k}}(a) (the split being at a) for which ξ_{S_{k}}(ω′_{S_{k}}(a)) contains interior points of D_{S_{k}}(a). In Figure 1, ω′_{S_{k}}(a) = (u, a).

The next lemma is known, see e.g. [14, Proposition 28].

Lemma 7 Fix a ∈ (\sqrt{2}, 2] and let ω_{S_{k}} and ω′_{S_{k}} be as in Definitions 3 and 4. Then,

lim_{k→∞} \frac{HD(D_{S_{k}}(a), ξ_{S_{k}}(ω′_{S_{k}}(a)))}{|ξ_{S_{k}}(ω′_{S_{k}}(a))|} = 0.

4 Case 1: lim inf_{k→∞} Q(k) < ∞

Proposition 1 Set H = \{a ∈ [\sqrt{2}, 2] | lim inf_{k→∞} Q_{a}(k) < ∞ and a ∉ D ∪ P\}. Then, H is σ-porous.

Proof: For each l ∈ N set H_{l} = \{a ∈ [\sqrt{2}, 2]\D | lim inf_{k→∞} Q_{a}(k) = l\}. Next, we define for each l ∈ N and a ∈ H_{l} a set I_{a,l} as follows. Fix l and a ∈ H_{l}. Then for infinitely many k we have Q_{a}(k) = l. For each such k we have, by (3), that c_{S_{k−1}} ∈ (z_{l−1}, z_{l}]∪[ẑ_{l}, ẑ_{l−1}). If for infintely many k we have c_{S_{k−1}} ∈ (z_{l−1}, z_{l}], then choose a closed interval with rational endpoints, denoted I_{a,l}, such that I_{a,l} ⊂ (z_{l}, c) and T_{a}^{n}(c) ∉ I_{a,l} for all n ∈ N. If for infinitely many k we have c_{S_{k−1}} ∈ [ẑ_{l}, ẑ_{l−1}), then choose a closed interval with rational endpoints, denoted I_{a,l}, such that I_{a,l} ⊂ (c, ẑ_{l}) and such that T_{a}^{n}(c) ∉ I_{a,l} for all n ∈ N. (Since a ∉ D and T_{a} is leo, we can choose such I_{a,l}.)

For each l ∈ N set W_{l} = {I_{a,l} | a ∈ H_{l}}. Since the endpoints of all I_{a,l} are rational, for each l the set W_{l} is countable. For l ∈ N and W ∈ W_{l} set H_{l,W} = {a ∈ H_{l} | I_{a,l} = W }.

Claim 1: Each H_{l,W} is \frac{|W |}{2}-porous. Proof: Fix l, W ∈ W_{l} and a ∈ H_{l,W}. Thus, I_{a,l} = W. Without loss of generality, assume that for infintely many k we have that c_{S_{k−1}} ∈ [ẑ_{l}, ẑ_{l−1}) and hence that I_{a,l} = W ⊂ (c, ẑ_{l}). Say, W = [w_{1}, w_{2}]. Set W′ = ξ_{S_{k−1}}^{−1}(W) ∩ ω_{S_{k−1}}(a). Say, W′ = [w′_{1}, w′_{2}]. Set Δ = |c_{S_{k−1}} − w_{1}| and Δ′ = |w′_{1} − a|. See Figure 2. From Lemma 2 we have that for large k, ξ_{S_{k−1}}|ω_{S_{k−1}}(a) is almost linear. Hence, \frac{Δ}{Δ′} ≈ \frac{|W |}{|W′|} and therefore \frac{|W′|}{Δ′} ≈ \frac{|W |}{Δ} ≥ |W |. Thus,

\frac{|W′|}{Δ′} > \frac{|W |}{2}.  (6)

It follows from the definitions of W = I_{a,l} and H_{l,W} that W′ ∩ H_{l,W} = ∅, since for each a_{0} ∈ W′ there exists n = S_{k−1} such that T_{a_{0}}^{n}(c) ∈ W. Hence, Claim 1 follows from (6). □

Lastly, as H = ∪_{l∈N} ∪_{W ∈W_{l}} H_{l,W}, we have that H is σ-porous. □

[figure: Figure 2: Construction of W = I_{a,l} and W′; c; ξ_{S_{k−1}}; ẑ_{l−1}(a); ẑ_{l}(a); w_{2}; w_{1}; a; •; c_{S_{k−1}}(a); w′_{1}; w′_{2}]

5 Substantial and cosubstantial cuts

In this section first we give the definition of the key-concepts of the next section the substantial and co-substantial cuts.

We also discuss two algorithms the greedy and the substantial algorithm. These algorithms both can help one to find long stretches and they are behind the dynamics of our proof, in fact, the arguments of the next section can be used to verify that the substantial cut algorithm indeed works. However we give this idea to help the understanding the technical details of the next section and to understand the dynamics behind the technicalities there.

In this section we assume that a ∈ I is fixed, that is, a ∈ [\sqrt{2}, 2] and lim_{k→∞} Q_{a}(k) = ∞. We will also assume that ϵ is a small positive constant.

First we give the definitions of substantial and cosubstantial cuts.

Definition 5 We call a cutting time S_{k} a substantial cutting time provided that |c − c_{S_{k}}| ≤ 3|c − c_{S_{k−1}}|.

Remark: As lim_{k→∞} Q(k) = ∞, it follows from (3) and (5) that there are infinitely many substantial cuts. If additionally, Q(k) is eventually nondecreasing, then it follows from [7, Lemma 2.4] that there exists K ∈ N such that for all k ≥ K, the cut S_{k} is a substantial cut.

Definition 6 We call a cutting time S_{k} a co-substantial cutting time provided that |c − c_{S_{Q(k)}}| ≤ 3|c − c_{S_{k−1}}|.

Next we give an informal discussion of a greedy algorithm. We make the greedy algorithm precise in Definition 7. Proposition 2 uses the greedy algorithm to produce long stretches. However, long stretches alone are not enough for our porosity result, Theorem 1, and hence we modify the greedy algorithm to obtain the substanial cut algorithm.

Assume S_{k_{0}} is a cutting time and J_{0} = [z_{k_{0}−1}, c] and l_{0} = S_{k_{0}}. Then T_{a}^{l_{0}}(J_{0}) = D_{S_{k_{0}}} = [c_{S_{k_{0}}} , c_{S_{Q(k_{0})}}]. Our target is to find an m > l_{0} and an interval I ⊂ J_{0} such that T_{a}^{m}|_{I} is monotone and |T_{a}^{m}(I)| > ϵ. Since T_{a} is leo it is clear that such an interval exists, the question is how to find it. This is when we can use the greedy algorithm. The interval J_{0} consists of two pieces J_{0}^{1} = [z_{k_{0}}, c] and J_{0}^{2} = [z_{k_{0}−1}, z_{k_{0}}] such that T_{a}^{l_{0}}(J_{0}^{1}) = [c_{S_{k_{0}}} , c] and T_{a}^{l_{0}}(J_{0}^{2}) = [c, c_{S_{Q(k_{0})}}]. Since T_{a}^{l_{0}+1} is not monotone on J_{0} we need to choose one of these pieces and we make a “greedy-choice”, that is, we take the bigger piece (i.e., the piece for which T_{a}^{l_{0}}(J_{0}^{i}) is bigger), and call this piece J_{1}. Thus we set J_{1} = [z_{k_{0}}, c] and t = k_{0} if |c − c_{S_{k_{0}}} | ≥ |c − c_{s_{Q(k_{0})}}| and J_{1} = [z_{k_{0}−1}, z_{k_{0}}] and t = Q(k_{0}) otherwise. Clearly, setting l_{1} = l_{0} + S_{Q(t+1)}, T_{a}^{l_{1}}|_{J_{1}} is monotone and T_{a}^{l_{1}}(J_{1}) = [c_{S_{t+1}}, c_{S_{Q(t+1)}}]. Then J_{1} can be split into two pieces J_{1}^{1} and J_{1}^{2} such that T_{a}^{l_{1}}(J_{1}^{1}) = [c_{S_{t+1}}, c] and T_{a}^{l_{1}}(J_{1}^{1}) = [c, c_{S_{Q(t+1)}}]. Again we are greedy and choose J_{2} = J_{1}^{1} if |c − c_{S_{t+1}}| ≥ |c − c_{S_{Q(t+1)}}| and J_{2} = J_{1}^{2} otherwise. We keep repeating this procedure to obtain a nested sequence of intervals J_{0} ⊃ J_{1} ⊃ ... ⊃ J_{n} ⊃ .... Then for some large n we set I = J_{n} and m = l_{n} to obtain |T_{a}^{m}(I)| > ϵ.

Considering T_{a}^{l_{n}}(J_{n}), there is a corresponding greedy algorithm which describes movements between between levels in the Hofbauer tower, corresponding to cutting times. For this algorithm, the interest is in which levels of the tower are visited and hence the algorithm is given as a function from N into the cutting times {S_{k}}. This algorithm is different than the usual action on the tower as described for example in [4].

Definition 7 Fix m ≥ 0. Define G_{m} : N → {S_{k}} by G_{m}(1) = S_{m} and if G_{m}(n) = S_{t}, then

G_{m}(n + 1) = \{ \begin{matrix}S_{t+1} & if |c − c_{S_{t+1}}| ≥ |c − c_{S_{Q(t+1)}}| \\ S_{Q(t+1)} & else. \end{matrix}

We call this algorithm the greedy algorithm. The name comes from the fact that at each cut we are greedy and take the larger piece of [c_{S_{t+1}}, c_{S_{Q(t+1)}}].

Lemma 8 is a technical lemma used in Proposition 2 and elsewhere in the paper.

Lemma 8 Let S_{k} be a substantial cut (cosubstantial cut) with Q(k) > 6. Then |c−c_{S_{k}}| < |c − c_{S_{Q(k)}}| (|c − c_{S_{Q(k)}}| < |c − c_{S_{k}}|).

Proof: We have that |c_{S_{k}} −c_{S_{Q(k)}}| > 8|c−c_{S_{k−1}}|, since Q(k) > 6. The lemma now follows from the definition of a substantial cut (cosubstantial cut). □

Proposition 2 Let S_{k} be a cutting time and and fix δ < min{|c − c_{1}|, |c − c_{2}|}. Set H_{1} = [c, c_{S_{Q(k)}}] and H_{2} = [c, c_{S_{k}}]. Then for i ∈ {1, 2}, there exists a closed interval I_{i} ⊂ H_{i} and l_{i} ≥ 1 such that

• T_{a}^{l_{i}}|I_{i} is monotone, and

• |T_{a}^{l_{i}}(I_{i})| > δ.

Moreover, when S_{k} is a substantial cut with Q(k) > 6, then 1 ≤ l_{1} ≤ S_{k−1}.

Proof: For i = 1, set G = G_{S_{Q(k)}} and for i = 2, set G = G_{S_{k}}.

Claim 1: Fix i ∈ {1, 2}. If G(n) = S_{m} with Q(m+1) > 1, then |c−c_{G(n)}| < |c−c_{G(n+1)}|. Proof: Since Q(m+1) > 1, we have that S_{Q(m+1)} > S_{1} = 2 and hence that a^{S_{Q(m+1)}} > 2. But, a^{S_{Q(m+1)}} > 2 implies that the |c_{S_{m+1}} − c_{S_{Q(m+1)}}| > 2|c − c_{S_{m}}|. It is now easy to check that the claim holds from the definition of G. This completes Claim 1.

Assume that i = 1; the case for i = 2 is similar. If as we apply the greedy algorithm G we arrive at a level of the tower S_{m} with Q(m + 1) ∈ {0, 1} then we are done since either |c − c_{1}| or |c − c_{2}| is contained in D_{S_{m+1}}.

Since lim_{k→∞} Q(k) = ∞, there exists t ≥ 1 such that |c − c_{S_{Q(k)+t}}| < |c − c_{S_{Q}(k)}|. Hence, if as we apply the greedy algorithm we have the condition ‘Q(m + 1) > 1’, then (due to Claim 1) we cannot get to a level in the tower above S_{Q(k)+t}. Thus, we arrive at any level of the tower at most once until we arrive at D_{2} or D_{1}, in which case we are done. If S_{k} is a substantial cut and Q(k) > 6, then from Claim 1 and Lemma 8 we cannot return to level S_{k} and hence cannot get above this level. Therefore l_{1} ≤ S_{k−1}. □

As previosuly remarked, for our porosity estimate we need more than just to find a long stretch. Assume that we have a small number η̃ > 0 given in advance and we also have a k_{0} such that S_{k_{0}} is a substantial cut and we want to find an interval I ⊂ [c, c_{S_{Q(k_{0})}}] and an l ≥ 1 such that T_{a}^{l}|_{I} is monotone, |T_{a}^{l}(I)| > ϵ and |I| > η̃Ẽ_{I} where Ẽ_{I} is the length of the shortest closed interval containing both I and c_{S_{k_{0}}} . For our porosity estimates we need |I| ≥ η̃Ẽ_{I} which we call the “metric assumption” for our porosity estimates; this assumption roughly means that we not only want an interval on which we have a long stretch, but we want the length of this interval being sufficiently large, compared to its distance from c_{S_{k_{0}}} . Of course the best being if this interval is a central branch (which we can have if a ∉ I), but for a ∈ I finding such intervals is more difficult. In an algorithm, which we will call the substantial cut algorithm, like in the greedy algorithm, we will define a nested sequence of intervals J_{0} ⊃ J_{1} ⊃ ... ⊃ J_{n} ⊃ ... such that for a sequence l_{0} = S_{k_{0}} < l_{1} < ... < l_{n} < ..., T_{a}^{l_{n}}|_{J_{n}} is monotone and T_{a}^{l_{n}}(J_{n}) corresponds to a cutting time level of the Hofbauer tower, J_{0} = [α_{0}, β_{0}] with T_{a}^{S_{k_{0}}} (α_{0}) = c_{S_{k_{0}}} and

T_{a}^{S_{k_{0}}} (β_{0}) = c_{S_{Q(k_{0})}}. For a large value of n we will be able to choose I = T_{a}^{S_{k_{0}}} (J_{n}).

To understand the dynamics behind the next technical section we need to see how the greedy algorithm is modified, that is, how we define J_{n+1} by selecting a proper piece of J_{n}.

Assume J_{n} = [α_{n}, β_{n}] and we chose our notation so that α_{n} is closer to α_{0} than β_{n}. To satisfy our “metric assumption” we will have to control our greed and sometimes we will have to keep the shorter piece of J_{n} in order to stay sufficiently close to α_{0}. To be more precise assume that T_{a}^{l_{n}}(J_{n}) = [c_{S_{t}}, c_{S_{Q(t)}}] for an integer t. If T_{a}^{l_{n}}(α_{n}) = c_{S_{Q(t)}} we say that we are in a co-active situation. Otherwise, when T_{a}^{l_{n}}(α_{n}) = c_{S_{t}}, we are not in a co-active situation. Choose γ_{n} ∈ [α_{n}, β_{n}] such that T_{a}^{l_{n}}(γ_{n}) = c. In the non co-active case we check whether the cut at [c_{S_{t}}, c_{S_{Q(t)}}] is substantial, or not; if it is then we set J_{n+1} = [γ_{n}, β_{n}] (that is, we are greedy and keep the longer piece; recall Lemma 8); if it is not a substantial cut then we set J_{n+1} = [α_{n}, γ_{n}] (that is, in order to satisfy the metric assumption we choose the piece closer to α_{0} even if it is smaller than the other piece; since we do not have a substantial cut this smaller piece is still relatively long). In the co-active case we check whether the cut at [c_{S_{t}}, c_{S_{Q(t)}}] is co-substantial, or not; if it is then we set J_{n+1} = [γ_{n}, β_{n}] (that is, we are greedy and keep the longer piece); if it is not a co-substantial cut then we set J_{n+1} = [α_{n}, γ_{n}] (that is, in order to satisfy the metric assumption we choose the piece closer to α_{0} even if it is smaller than the other piece; since we do not have a co-substantial cut this smaller piece is still relatively long).

Next we give a formal definition of the substantial cut algorithm.

Let k_{0} ∈ N be fixed and assume S_{k_{0}} is a substantial cut. Set Γ(0) = 0, J_{0} = [z_{k_{0}−1}, c] and l_{0} = S_{k_{0}}. (The auxilary function Γ tells us whether we are at a co-active (Γ = 1) or a non-co-active (Γ = 0) cut.) Then, T_{a}^{l_{0}}(J_{0}) = D_{S_{k_{0}}} = [c_{S_{k_{0}}} , c_{S_{Q(k_{0})}}], and T_{a}^{l_{0}}|J_{0} is monotone. Let J_{1} ⊂ J_{0} be such that T_{a}^{l_{0}}(J_{1}) = [c, c_{S_{Q(k_{0})}}]. Set k_{1} = Q(k_{0}) + 1 and l_{1} = S_{k_{0}} + S_{Q(k_{1})}. Then, T_{a}^{l_{1}}(J_{1}) = D_{S_{k_{1}}} = [c_{S_{k_{1}}} , c_{S_{Q(k_{1})}}], and T_{a}^{l_{1}}|J_{1} is monotone. Set Γ(1) = 1.

Assume that n ≥ 1 and that we have constructed finite sequences {Γ(i)}_{i=0}^{n}, {k_{i}}_{i=0}^{n}, {l_{i}}_{i=0}^{n}, and closed nested intervals J_{0} ⊃ J_{1} ⊃ … ⊃ J_{n} such that

• T_{a}^{l_{i}}|J_{i} is monotone for 0 ≤ i ≤ n,

• T_{a}^{l_{i}}(J_{i}) = D_{S_{k_{i}}} for 0 ≤ i ≤ n,

• l_{i} = l_{i−1} + S_{Q(k_{i})} for 1 ≤ i ≤ n,

• k_{i} ∈ {k_{i−1} + 1, Q(k_{i−1}) + 1} for 1 ≤ i ≤ n,

• k_{i} = k_{i−1} + 1 ⇒ T_{a}^{l_{i−1}}(J_{i}) = [c, c_{S_{k_{i−1}}} ], for 1 ≤ i ≤ n,

• k_{i} = Q(k_{i−1}) + 1 ⇒ T_{a}^{l_{i−1}}(J_{i}) = [c, c_{S_{Q(k_{i−1})}}], for 1 ≤ i ≤ n, and

• Γ(i) tells us whether the cut at S_{k_{i}} is co-active or not.

We want to define k_{n+1}, l_{n+1}, and J_{n+1}. There are two options, which we call Option A and Option B.

Option A: Set k_{n+1} = k_{n} + 1 and l_{n+1} = l_{n} + S_{Q(k_{n+1})}. Let J_{n+1} ⊂ J_{n} be such that T_{a}^{l_{n+1}}|J_{n+1} is monotone and T_{a}^{l_{n+1}}(J_{n+1}) = D_{S_{k_{n+1}}} .

Option B: Set k_{n+1} = Q(k_{n}) + 1 and l_{n+1} = l_{n} + S_{Q(k_{n+1})}. Let J_{n+1} ⊂ J_{n} be such that T_{a}^{l_{n+1}}|J_{n+1} is monotone and T_{a}^{l_{n+1}}(J_{n+1}) = D_{S_{k_{n+1}}} .

If Γ(n) = 0 and S_{k_{n}} is a substantial (not substantial) cut then set Γ(n + 1) = 1 and use option B (set Γ(n + 1) = 0 and use option A) to define k_{n+1} l_{n+1}, and J_{n+1}.

If Γ(n) = 1 and S_{k_{n}} is a co-substantial (not co-substantial) cut then set Γ(n + 1) = 1 and use option A (set Γ(n + 1) = 0 and use option B) to define k_{n+1} l_{n+1}, and J_{n+1}.

The above definition of Γ(n + 1) explains our name for the function Γ. If Γ(n) = 1, then we need to check whether the cut at S_{k_{n}} is co-substantial or not (co-active case). If Γ(n) = 0, then we need to check whether the cut at S_{k_{n}} is substantial or not (not co-active case).

If for some j we have |D_{S_{k_{j}}} | > ϵ, then the algorithm terminates at this step and J_{j} can be chosen as I′, and l_{j} as m.

It is worth while to compare the substantial cut algorithm and the greedy algorithm after this formal definition.

Assume k_{n}, l_{n}, and J_{n} are defined. Recall that D_{S_{kn}} = [c_{S_{kn}} , c_{S_{Q(kn)}}]. During the greedy algorithm use option A if |c − c_{S_{kn}} | ≥ |c − c_{S_{Q(kn)}}| and use option B otherwise. This means that we are greedy, we always want to follow the ‘larger piece’ at each cut.

In the substantial cut algorithm, to satisfy the metric assumption (that is we need the long stretch relatively close to c) we allow the usage of option A for nonsubstantial cuts. At nonsubstantial cuts it may still happen that |c − c_{S_{kn}} | < |c − c_{S_{Q(kn)}}|, but at these steps, to obtain the metric estimate, we limit our ‘greed’ and choose the piece which stays close to c. Studying the proof of Lemma 12 of Section 6 one can verify that being nongreedy at these steps yields the suitable estimate for the metric assumption.

Finally we show that substantial cuts are interesting for other reasons as well. It is obvious that at a cutting time |D_{S_{k}}| can be arbitrarily small. On the other hand for substantial cuts we have:

Lemma 9 Let S_{k} be a substantial cut and fix δ = min{|c − c_{j}| | 1 ≤ j ≤ 6}. Then

|D_{S_{k}}| ≥ a^{−S_{k−1}}δ.  (7)

Proof: If Q(k) ≤ 6, then (7) follows from (5), the definition of δ, and a^{−S_{k−1}} < 1. Assume that Q(k) > 6. Then, from Proposition 2, there exists L ⊂ D_{S_{k}} and l ≤ S_{k−1} such that T_{a}^{l}|L is monotone and |T_{a}^{l}(L)| > δ. Now, L ⊂ D_{S_{k}}, a^{l}|L| ≥ δ, and l ≤ S_{k−1} imply the result. □

6 Tools for the case lim_{k→∞} Q(k) = ∞

Throughout this section a ∈ I is fixed; recall a ∈ (\sqrt{2}, 2). In this section we give the technical details of the estimates corresponding to this case. Behind the “dynamics” of our argument there is the substantial cut algorithm. Our argument is based on induction, the key result of this section is Proposition 3 which roughly will state that if we can find a sufficiently long stretch with good metric estimates for all suitable “lower level” substantial and co-substantial cuts in the Hofbauer tower, then we can find the requiered long stretch at the “next level” as well.

Choose K_{1} ∈ N such that

• k > K_{1} and l′ ≥ min{Q(k + 1), Q(Q(k) + 1)} imply that S_{Q(l′)} ≥ 20, and

• k > K_{1} implies that Q(k) > 100.

As a > \sqrt{2}, we have that a^{S_{Q(l′)}} > 2^{10} > 1000 for l′ satisfying the above inequality. Since a ∉ P, there exists ϵ_{1} > 0 such that |c_{S_{k}} − c_{S_{Q(k)}}| > ϵ_{1} for k ≤ K_{1}. Also, throughout this section we assume that 0 < ϵ < min\{|c − c_{1}|, |c − c_{2}|, \frac{ϵ_{1}}{2} \} is fixed and η ∈ (0, \frac{1}{2}).

In the next definition we introduce auxiliary points y_{k}, and ȳ_{k} for substantial cuts. These points will help us in our induction for the estimates needed for the “metric assumption”, we will use them to show that if we have good metric properties at “lower levels” of the tower then we have good properties at the “next level” as well.

Definition 8 If S_{k} is a substantial cut for T_{a}, define y_{k} to be the unique point on the same side of c as c_{S_{k}} such that |y_{k} − c| = 4a^{−S_{Q(k)}}|c − c_{S_{Q(k)}}|. The point ȳ_{k} is defined similarly to y_{k} but with |ȳ_{k} − c| = 2|y_{k} − c| = 8a^{−S_{Q(k)}}|c − c_{S_{Q(k)}}|. It is easy to check that y_{k} and ȳ_{k} ∉ [c, c_{S_{k}}].

The next definition will give our metric assumption (based on the auxiliary point y_{k}) which we can use in our induction. The ϵ − η-good substantial cuts will provide for us long stretches with good “metric properties”.

Definition 9 A substantial cut S_{k} is said to be ϵ − η-good provided that there exist I ⊂ [c, c_{S_{Q(k)}}] and l ≥ 1 such that T_{a}^{l}|I monotone, |T_{a}^{l}(I)| > ϵ, and |I| ≥ ηE_{I}, where E_{I} is the length of the shortest closed interval containing both y_{k} and I. (See Figure 3.)

Figure 3: Construction of E_{I} c_{S_{k}} c  c_{S_{Q(k)}} I • y_{k}  E_{I}

Definition 10 A substantial cut S_{k} is strongly ϵ − η-good, provided there exist I ⊂ [c, c_{S_{Q(k)}}] and l ≥ 1 such that T_{a}^{l}|I is monotone, |T_{a}^{l}(I)| > ϵ, and |I| ≥ ηĒ_{I}, where Ē_{I} is the length of the shortest closed interval containing both ȳ_{k} and I.

The next three definitions are the cosubstantial versions of the previous two.

Definition 11 If S_{k} is a cosubstantial cut for T_{a}, define y′_{k} to be the unique point on the same side of c as c_{S_{Q(k)}} such that |y′_{k} − c| = 4a^{−S_{Q(k)}}|c − c_{S_{k}}|. The point ȳ′_{k} is defined similarly to y′_{k} but with |ȳ′_{k} − c| = 2|y′_{k} − c| = 8a^{−S_{Q(k)}}|c − c_{S_{k}}|. Again, it is easy to check that y′_{k} and ȳ′_{k} ∉ [c, c_{S_{Q(k)}}].

Definition 12 A cosubstantial cut S_{k} is said to be ϵ − η-good provided that there exist I ⊂ [c, c_{S_{k}}] and l ≥ 1 such that T_{a}^{l}|I monotone, |T_{a}^{l}(I)| > ϵ, and |I| ≥ ηE_{I}, where E_{I} is the length of the shortest closed interval containing both y′_{k} and I.

Definition 13 A cosubstantial cut S_{k} is strongly ϵ − η-good, provided there exists I ⊂ [c, c_{S_{k}}] and l ≥ 1 such that T_{a}^{l}|I is monotone, |T_{a}^{l}(I)| > ϵ, and |I| ≥ ηĒ_{I}, where Ē_{I} is the length of the shortest closed interval containing both ȳ′_{k} and I.

Lemma 10 Let S_{k} be a substantial cut. Set p_{0} = Q(k) + 1 and q_{0} = Q(p_{0}). If S_{p_{0}} is not a cosubstantial cut, then there exists l ≥ 1 such that S_{q_{0}+l} is a substantial cut, L ≡ S_{q_{0}+l} = S_{q_{0}} + S_{Q(q_{0}+1)} + … + S_{Q(q_{0}+l)} ≤ S_{Q(k)}, and S_{q_{0}+l′} is not a substantial cut for 1 ≤ l′ < l.

What is the dynamics behind this lemma? On the next figure we show some levels of the Hofbauer tower. The top level is corresponding to the cut at S_{k}. To satisfy our “metric assumptions” in the substantial cut algorithm we want to stay “close to” c_{S_{k}}, marked by an arrow. Of course, due to the substantial cut, we need to throw away the small piece containing c_{S_{k}}, and we follow the iterated T_{a} images of c, instead (the other point marked by an arrow on the top level). At level S_{p_{0}} we mark by an arrow the T_{a}^{S_{q_{0})}} image of c, which actually equals c_{S_{Q(p_{0})}}. This is the “coendpoint” of the Hofbauer-tower level at S_{p_{0}}. We drop now down to the bottom level on the figure, S_{Q(p_{0})}, and we also picture the image of the level S_{p_{0}} as a subset of the bottom level. By our assumption we do not have a cosubstantial cut at level S_{p_{0}}, that is, the “copiece” of length Δ is sufficiently long. Now, starting from the bottom level, we move up in the tower until at level S_{q_{0}+l} we have again a substantial cut. At the non substantial cuts we just simply follow the piece which contains the c_{S_{q_{0}} +j} non-“coendpoint”, j = 1, ..., l −1; these endpoints are marked by an arrow again. Finally we have a substantial cut at level S_{q_{0}+l}, and the whole procedure starts again...

[figure: c; c_{S_{Q(p_{0})}} = c_{S_{q_{0}}} = T_{a}^{S_{Q(p_{0})}}(c); c_{S_{Q(Q(p_{0}))}}; c_{S_{Q(q_{0}+1)}}; c_{S_{q_{0}+1}} = T_{a}^{S_{q_{0}} +S_{Q(q_{0}+1)}}(c); c_{S_{q_{0}+l}} = T_{a}^{S_{q_{0}} +S_{Q(q_{0}+1)}+...+S_{Q(q_{0}+l)}}(c); c_{S_{Q(q_{0}+l)}}; c_{S_{Q(k)}}; c; c_{S_{k}}; ⋮; ⋮; Δ; Figure 4: Some Hofbauer tower levels; c; c_{S_{Q(p_{0})}} = c_{S_{q_{0}}} = T_{a}^{S_{Q(p_{0})}}(c); c_{S_{p_{0}}}; Δ; ⋮; c_{S_{p_{0}}}]

Proof: Set Δ = |c − c_{S_{Q(Q(k)+1)}}| = |c − c_{S_{q_{0}}} |. From (4), q_{0} = Q(Q(k) + 1) ≤ Q(k). Since S_{p_{0}} is not a cosubstantial cut, it follows that

Δ > 3|c − c_{S_{Q(k)}}|.  (8)

Hence, q_{0} < Q(k), else q_{0} = Q(k) and Δ = |c − c_{S_{Q(k)}}| contradicting (8). If there does not exist such an l, then |c−c_{S_{q_{0}+1}}| > 3|c−c_{S_{q_{0}}} | = 3Δ, |c−c_{S_{q_{0}+2}}| > 3|c−c_{S_{q_{0}+1}}| > 3^{2}Δ, …, |c − c_{S_{Q(k)}}| > 3^{Q(k)−q_{0}}Δ, contradicting (8). □

Lemma 11 Let S_{k} be a cosubstantial cut with Q(k) ≥ 6. Set p_{0} = k +1 and q_{0} = Q(p_{0}). If S_{p_{0}} is not a cosubstantial cut, then there exists l ≥ 1 such that S_{q_{0}+l} is a substantial cut, L = S_{q_{0}+l} = S_{q_{0}} + S_{Q(q_{0}+1)} + … + S_{Q(q_{0}+l)} ≤ S_{Q(k)}, and S_{q_{0}+l′} is not a substantial cut for 1 ≤ l′ < l.

Proof: Set Δ = |c − c_{S_{k}}|. Since S_{k} is a cosubstantial cut and Q(k) ≥ 6, we have Δ ≥ |c − c_{S_{k−1}}| and thus, from (3), q_{0} = Q(k + 1) ≤ Q(k). Next, S_{p_{0}} not a cosubstantial cut implies that |c − c_{S_{q_{0}}} | > 3Δ, and S_{k} a cosubstantial cut implies that |c − c_{S_{Q(k)}}| ≤ Δ; thus q_{0} = Q(k+1) < Q(k). Again, if there does not exist such an l, then Δ ≥ |c−c_{S_{Q(k)}}| ≥ 3^{Q(k)−q_{0}}|c − c_{S_{q_{0}}} | > 3^{Q(k)−q_{0}} 3Δ; a contradiction. □

The next lemma shows that if in our substantial cut algorithm a substantial cut is not followed by a cosubstantial cut and the later substantial cut (which exists by Lemmas 10 and 11) is ϵ − η-good then the original substantial cut is strongly ϵ − η-good.

Lemma 12 Assume that k > K_{1}. If S_{k} is a substantial cut, S_{p_{0}} is not a cosubstantial cut, where p_{0} = Q(k) + 1, and S_{q_{0}+l} is ϵ − η-good, where q_{0} = Q(p_{0}) and l is as in Lemmas 10 then the cut at S_{k} is strongly ϵ − η-good.

Of course, the above lemma has a ‘cosubstantial’ version as well:

Lemma 13 Assume that k > K_{1}. If S_{k} is a cosubstantial cut, S_{p_{0}} is not a cosubstantial cut, where p_{0} = k + 1, and S_{q_{0}+l} is ϵ − η-good, where q_{0} = Q(p_{0}) and l is as in Lemma 11, then the cut at S_{k} is strongly ϵ − η-good.

Next we give the proof of Lemma 12 and in paranthetical remarks we show what should be modified for the proof of Lemma 13.

Proof: We use the notation of Lemma 10 (Lemma 11 for proving Lemma 13). Thus, S_{q_{0}} < S_{Q(k)} and l ≥ 1 is minimal such that S_{q_{0}+l} is a substantial cut.

Set C = c_{S_{Q(k)}}. (To prove Lemma 13 set C = c_{S_{k}}.) Let h = |c − C| and for 0 ≤ i ≤ l, set h_{i} = |c − c_{S_{q_{0}+i}}|. Then,

3h < h_{0} and 3h_{l′−1} < h_{l′}  (9)

for 1 ≤ l′ ≤ l − 1. From the definition of the z_{k}’s (see (1)) and the definition of q_{0} we have that either [c, C] ∋ z_{q_{0}} or that [c, C] ∋ ẑ_{q_{0}}. Recall that L = S_{q_{0}+l}. Without loss of generality, assume that [c, C] ∋ z_{q_{0}} and therefore that [c, C] ∋ z_{q_{0}+l−1}. Recall that T_{a}^{L} maps [c, z_{q_{0}+l−1}] onto [c_{S_{q_{0}+l}}, c_{S_{Q(q_{0}+l)}}] in a one to one manner with T_{a}^{L}(z_{q_{0}+l−1}) = c_{S_{Q(q_{0}+l)}}. Denote by ϕ the linear mapping that is the extension of T_{a}^{L}|[c, z_{q_{0}+l−1}] onto R. It is clear that the absolute value of the slope of ϕ is a^{L}. Clearly, z_{q_{0}+l} = ϕ^{−1}(c) and |c − z_{q_{0}+l}| = a^{−L}|c − c_{S_{q_{0}+l}}|. Set w = ϕ^{−1}(y_{q_{0}+l}); recall Definition 8 for y_{q_{0}+l}. Then, w is on the same side of c as ȳ_{k}. (In the proof of Lemma 13 w is on the same side of c as ȳ′_{k}.)

Using |c − c_{S_{q_{0}+l}}| + |c − c_{S_{Q(q_{0}+l)}}| = a^{S_{Q(q_{0}+l)}}|c − c_{S_{q_{0}+l−1}}| and |c − c_{S_{q_{0}+l}}| ≤ 3|c − c_{S_{q_{0}+l−1}}|, we have that h_{l−1}a^{S_{Q(q_{0}+l)}} = |c − c_{S_{Q(q_{0}+l)}}| + |c − c_{S_{q_{0}+l}}| ≤ |c − c_{S_{Q(q_{0}+l)}}|(1 +

\frac{3a^{−S_{Q(q_{0}+l)}}}{1−3a^{−S_{Q(q_{0}+l)}}} ) = |c − c_{S_{Q(q_{0}+l)}}|\frac{1}{1−3a^{−S_{Q(q_{0}+l)}}} . Hence,

h_{l−1}a^{S_{Q(q_{0}+l)}}(1 − 3a^{−S_{Q(q_{0}+l)}}) ≤ |c − c_{S_{Q(q_{0}+l)}}| ≤ h_{l−1}a^{S_{Q(q_{0}+l)}}.  (10)

From the definition of y_{q_{0}+l} and (10) we have that |c−y_{q_{0}+l}| = 4a^{−S_{Q(q_{0}+l)}}|c−c_{S_{Q(q_{0}+l)}}| ≥ 4h_{l−1}(1 − 3a^{−S_{Q(q_{0}+l)}}). Thus, (use that |c − y_{q_{0}+l}| = a^{S_{q_{0}+l}}|w − z_{q_{0}+l}| and L = S_{q_{0}+l})

|w − z_{q_{0}+l}| ≥ 4a^{−L}h_{l−1}(1 − 3a^{−S_{Q(q_{0}+l)}}).  (11)

On the other hand, |c − z_{q_{0}+l}| = a^{−L}|c − c_{S_{q_{0}+l}}| ≤ a^{−L} \frac{3a^{−S_{Q(q_{0}+l)}}}{1−3a^{−S_{Q(q_{0}+l)}}} |c − c_{S_{Q(q_{0}+l)}}|

≤ a^{−L} \frac{3a^{−S_{Q(q_{0}+l)}}}{1−3a^{−S_{Q(q_{0}+l)}}} a^{S_{Q(q_{0}+l)}}h_{l−1} = a^{−L} \frac{3}{1−3a^{−S_{Q(q_{0}+l)}}} h_{l−1}, i.e.,

|c − z_{q_{0}+l}| ≤ a^{−L} \frac{3}{1 − 3a^{−S_{Q(q_{0}+l)}}} h_{l−1}.  (12)

Thus, using (9), (11) and (12), we have that |c − w| ≥ a^{−L}h_{l−1}(4(1 − 3a^{−S_{Q(q_{0}+l)}}) − \frac{3}{1−3a^{−S_{Q(q_{0}+l)}}} ) ≥ a^{−L}h_{l−1}((4)(0.99) − \frac{3}{0.99}) > 0.9a^{−L}h_{l−1} ≥ (0.9)3^{l}a^{S_{Q(k)}−L}a^{−S_{Q(k)}}h ≡ A. Here we have used that q_{0} + l ≥ min{Q(k), Q(k + 1), Q(Q(k) + 1)} and hence that S_{Q(q_{0}+l)} ≥ 20, i.e., a^{−S_{Q(q_{0}+l)}} < 0.001. More succinctly,

|c − w| ≥ (0.9)3^{l}a^{(S_{Q(k)}−L)}a^{−S_{Q(k)}}h ≡ A.  (13)

Since the assumptions of Lemma 10 hold (in the proof of Lemma 13 we use Lemma 11) hold, we have that L = S_{q_{0}+l} ≤ S_{Q(k)}. If L < S_{Q(k)}, then S_{Q(k)} − L ≥ S_{Q(q_{0}+l+1)} > 20, since q_{0} + l + 1 ≥ min{Q(k + 1), Q(Q(k) + 1)} and k > K_{1}. Hence if S_{Q(k)} ≠ L, then S_{Q(k)} − L ≥ 20. We use this fact in the next claim.

Claim 1: We have ȳ_{k} ∈ [w, c]. Proof: First assume that either l ≥ 2, or that S_{Q(k)} ≠ L. Then, use (13), |c − w| ≥ A > 8a^{−S_{Q(k)}}h = |ȳ_{k} − c| and therefore that ȳ_{k} ∈ [w, c].

Next assume that l = 1 and S_{Q(k)} = L. Then, c_{S_{q_{0}+l}} = c_{S_{Q(k)}} and |c − c_{S_{q_{0}+l}}| ≤ h (here we have equality when S_{k} is a substantial cut and strict inequality when S_{k} is a cosubstantial cut). Hence, |c − z_{q_{0}+l}| ≤ a^{−L}h. From (11) we have that |w − z_{q_{0}+l}| ≥ 4a^{−L}(0.99)h_{0} ≥ 12a^{−L}(0.99)h > 11a^{−L}h. Lastly, |c − z_{q_{0}+l}| ≤ a^{−L}h and |w − z_{q_{0}+l}| > 11a^{−L}h imply that |c−w| ≥ 10a^{−L}h = 10a^{−S_{Q(k)}}h. Thus, |c−ȳ_{k}| = 8a^{−S_{Q(k)}}h < |c−w|. Claim 1 is complete.

(The proof of Claim 1 with ȳ_{k} replaced by ȳ′_{k} gives the next claim, which is used in the proof of Lemma 13.

Claim 2: We have ȳ′_{k} ∈ [w, c].)

By assumption S_{q_{0}+l} is ϵ − η-good. Hence, choose I ⊂ [c, c_{S_{Q(q_{0}+l)}}] and l′ ≥ 1 with T_{a}^{l′}|I monotone, |T_{a}^{l′}| > ϵ, and |I| ≥ ηE_{I}, where E_{I} is the length of the shortest closed interval containing both y_{q_{0}+l} and I. Set J = ϕ^{−1}(I).

Then J ⊂ [z_{q_{0}+l}, c_{S_{Q(k)}}] ⊂ [c, c_{S_{Q(k)}}]. From linearity of ϕ, |I| ≥ ηE_{I}, and the definition of w we obtain that |J| ≥ ηĒ_{J}, where Ē_{J} is the length of the shortest closed interval containing both w and J. Let E_{J} be the length of the shortest closed interval containing both ȳ_{k} and J. From E_{J} < Ē_{J} and |J| ≥ ηĒ_{J}, we have that |J| ≥ ηE_{J}. Lastly, since |T_{a}^{L+l′}(J)| = |T_{a}^{l′}(I)| > ϵ and T_{a}^{L+l′}|J is monotone, it follows that S_{k} is strongly ϵ − η-good.

(In the proof of Lemma 13 an argument is similar to the above paragraph is used with J ⊂ [c, c_{S_{k}}].) □

The next two lemmas are the versions of Lemmas 12 and 13 which work in the case when a substantial or a cosubstantial cut is followed by an ϵ − η-good cosubstantial cut. In this case we obtain again that the original cut is ϵ − η-good.

Lemma 14 Assume that k > K_{1}. If S_{k} is a substantial cut, S_{p_{0}} is a cosubstantial cut that is ϵ − η-good, where p_{0} = Q(k) + 1, and S_{q_{0}} < S_{Q(k)} with q_{0} = Q(p_{0}), then the cut at S_{k} is strongly ϵ − η-good.

Lemma 15 Assume that k > K_{1}. If S_{k} is a cosubstantial cut, S_{p_{0}} is a cosubstantial cut that is ϵ − η-good, where p_{0} = k + 1, and S_{q_{0}} < S_{Q(k)} with q_{0} = Q(p_{0}), then the cut at S_{k} is strongly ϵ − η-good.

Again we give a proof of Lemma 14 and point out in some paranthetical remarks the differences of the proof of Lemma 15.

Proof: From the definition of q_{0} we have that q_{0} + 1 ≥ min{Q(k + 1), Q(Q(k) + 1)} and hence, since k > K_{1}, S_{Q(q_{0}+1)} > 20. Thus, S_{Q(k)} − S_{q_{0}} ≥ S_{q_{0}+1} − S_{q_{0}} = S_{Q(q_{0}+1)} > 20. Set C = c_{S_{Q(k)}}. (In the proof of Lemma 15 set C = c_{S_{k}}.) Put h = |c − C|. An argument similar to that for (10) gives

ha^{S_{q_{0}}} (1 − 3a^{−S_{q_{0}}} ) ≤ |c − c_{S_{p_{0}}} | ≤ ha^{S_{q_{0}}} .  (14)

Denote by ϕ the linear mapping that is the extension of T_{a}^{S_{q_{0}}} |[c, C] onto R. It is clear that the absolute value of the slope of ϕ is a^{S_{q_{0}}} .

Again, as in the proof of Lemmas 12 and 13, without loss of generality assume that z_{q_{0}} = ϕ^{−1}(c) ∈ [c, C]. Set w = ϕ^{−1}(y′_{p_{0}}). Again, |c − z_{q_{0}}| = a^{−S_{q_{0}}} |c − c_{S_{q_{0}}} | and w is on the same side of c as ȳ_{k}. (In the proof of Lemma 15 w is on the same side of c as ȳ′_{k}.) The definition of y′_{p_{0}} and (14) give

|c − y′_{p_{0}}| ≥ 4h(1 − 3a^{−S_{q_{0}}} ) ≥ 4(0.99)h.  (15)

Again in (15), k > K_{1} and q_{0} ≥ min{Q(k + 1), Q(Q(k) + 1)} implies that S_{Q(q_{0})} ≥ 20, i.e., a^{−S_{Q(q_{0})}} < 0.001. From |w − z_{q_{0}}| = a^{−S_{q_{0}}} |c − y′_{p_{0}}| and (15) we have

|w − z_{q_{0}}| ≥ 4(0.99)a^{−S_{q_{0}}} h.  (16)

Since S_{p_{0}} is a cosubstantial cut, we have that |c − c_{S_{q_{0}}} | ≤ 3h. Hence,

|c − z_{q_{0}}| ≤ 3ha^{−S_{q_{0}}} .  (17)

Thus, from (16) and (17) we have

|c − w| ≥ (0.9)a^{−S_{q_{0}}} h = (0.9)a^{(S_{Q(k)}−S_{q_{0}} )}a^{−S_{Q(k)}}h > 8a^{−S_{Q(k)}}h,  (18)

remember that S_{Q(k)} − S_{q_{0}} > 20.

Then |c − ȳ_{k}| = 8a^{−S_{Q(k)}}h and (18) imply that ȳ_{k} ∈ [c, w]. (In the proof of Lemma 15 |c − ȳ′_{k}| = 8a^{−S_{Q(k)}}h and (18) imply that ȳ′_{k} ∈ [c, w].) Lastly an argument similar to that in the proof of Lemmas 12 and 13 now implies that S_{k} is strongly ϵ − η-good. □

In the next proposition we put together the above four lemmas to obtain the induction property saying that if all “earlier” substantial and cosubstantial cuts are ϵ−η-good then the “next” one is also.

Proposition 3 Assume that k > K_{1} and let S_{k} be either a substantial or cosubstantial cut. If all substantial or cosubstantial cuts S_{k′} with S_{Q(k′)} < S_{Q(k)} are ϵ − η-good, then S_{k} is ϵ − η-good.

Proof: Assume that all substantial or cosubstantial cuts S_{k′} with S_{Q(k′)} < S_{Q(k)} are ϵ−ηgood; when we refer to ‘the hypothesis’ we are referring to precisely this assumption. Let p_{0} and q_{0} be as in Lemma 10 or as in Lemma 11, depending on whether S_{k} is a substantial or a cosubstantial cut.

Case 1: Assume that S_{p_{0}} is not a cosubstantial cut. Then, from Lemma 10 or Lemma 11, S_{q_{0}+l} is a substantial cut and L = S_{q_{0}+l} = S_{q_{0}} + S_{Q(q_{0}+1)} + … + S_{Q(q_{0}+l)} ≤ S_{Q(k)}. Hence, S_{Q(q_{0}+l)} < S_{Q(k)} and therefore (by hypothesis) S_{q_{0}+l} is ϵ − η-good. Thus, by Lemmas 12 and 13, S_{k} is strongly ϵ − η-good. Case 1 is complete.

If S_{k} is a substantial cut, then (using (4)) q_{0} = Q(Q(k) + 1) ≤ Q(k). Since k > K_{1}, Q(k) > 6 and hence |c_{S_{k}} − c_{S_{Q(k)}}| > 8|c − c_{S_{k−1}}|. If S_{k} is a cosubstantial cut, then |c − c_{S_{Q(k)}}| ≤ |c − c_{S_{k−1}}|. Joining these two facts, gives that |c − c_{S_{k}}| ≥ |c − c_{S_{k−1}}|. Thus, by (3), q_{0} = Q(k + 1) ≤ Q(k). Therefore, whether S_{k} is a substantial or cosubstantial cut, we have

S_{q_{0}} ≤ S_{Q(k)}.

Case 2: Assume that S_{p_{0}} is a cosubstantial cut and that S_{q_{0}} < S_{Q(k)}. Then, by hypothesis, S_{p_{0}} is ϵ − η-good, and therefore, by Lemmas 14 and 15, S_{k} is strongly ϵ − ηgood. Case 2 is complete.

Case 3: Assume that S_{p_{0}} is a cosubstantial cut and that S_{q_{0}} = S_{Q(k)}, i.e., q_{0} = Q(k). For ease of notation, set S_{q_{0}} = M.

If S_{k} is a substantial cut, set C = c_{S_{Q(k)}} and h = h′ = |c − c_{S_{Q(k)}}|. If S_{k} is a cosubstantial cut, set C = c_{S_{k}}, h = |c − c_{S_{k}}| , and h′ = |c − c_{S_{Q(k)}}|. In both cases h′ ≤ h and, since q_{0} = Q(k), c_{S_{p_{0}}} = T_{a}^{M}(C).

Without loss of generality, assume that z_{q_{0}} ∈ [c, C]. Denote by ψ_{0} the linear mapping that is the extension of T_{a}^{M}|[c, C] onto R. Thus, ψ_{0}(c) = c_{S_{q_{0}}} and ψ_{0}([c, C]) = [c_{S_{p_{0}}} , c_{S_{q_{0}}} ]. Clearly, v_{0} ≡ z_{q_{0}} = ψ_{0}^{−1}(c) and

|c − z_{q_{0}}| = a^{−M}|c − c_{S_{q_{0}}} | = a^{−M}h′.  (19)

Set p_{1} = p_{0} + 1 and q_{1} = Q(p_{1}). For S_{k} a substantial cut, S_{p_{0}} a cosubstantial cut, and q_{0} = Q(k) we obtain that |c − c_{S_{p_{0}}} | > |c − c_{S_{Q(k)}}| and hence (recall (3)) q_{1} = Q(p_{0} + 1) ≤ Q(Q(k) + 1) = q_{0}. Also, for S_{k} a cosubstantial cut, S_{p_{0}} a cosubstantial cut, and q_{0} = Q(k), we have that |c − c_{S_{p_{0}}} | > |c − c_{S_{k}}|and therefore (recall (3)) q_{1} = Q(p_{0} + 1) ≤ Q(k + 1) = Q(p_{0}) = q_{0}. Either way, we have

q_{1} ≤ q_{0}.

Assume that for r ≥ 1 we have defined p_{r} = p_{r−1} + 1 and q_{r} = Q(p_{r}), and that M ≡ S_{q_{0}} = S_{q_{1}} = … = S_{q_{r}}. Let v_{r−1} ∈ [c, C] be the unique point such that ψ_{r−1} ≡ T_{a}^{rM} maps [v_{r−1}, C] linearly onto [c, c_{S_{p_{r−1}}} ]. Note that the absolute value of the slope of ψ_{r−1} is a^{rM}. Set h_{r−1} = |c − c_{S_{p_{r−1}}} |.

It follows from M ≡ S_{q_{0}} = S_{q_{1}} = … = S_{q_{r}}, that S_{p_{r}} is a cosubstantial cut (in fact S_{p_{i}} is a cosubstantial cut for 0 ≤ i ≤ r).

Let ψ_{r} denote the linear extension of T_{a}^{M} ∘ ψ_{r−1}|[v_{r−1}, C]. Set v_{r} = ψ_{r}^{−1}(c). Then,

|v_{r} − v_{r−1}| = |ψ_{r}^{−1}(c) − ψ_{r}^{−1}(c_{S_{qr}} = c_{S_{Q(k)}})| = a^{−(r+1)M}h′.  (20)

Clearly, ψ_{r} maps [v_{r}, C] linearly onto [c, c_{S_{pr}}] and ψ_{r} = T_{a}^{(r+1)M} on [v_{r−1}, C]. From (19), (20), |c − v_{r−1}| = |c − z_{q_{0}}|(1 + a^{−M} + a^{−2M} + … + a^{−(r−1)M}), and |c − v_{r}| = |c − v_{r−1}| + |v_{r} − v_{r−1}| = |c − v_{r−1}| + a^{−(r+1)M}h′, we obtain

|c − v_{r}| ≤ ∑_{j=1}^{∞} a^{−jM}h′ = \frac{a^{−M}}{1 − a^{−M}} h′ < (1.1)a^{−M}h.  (21)

As previously done, for the numerical estimate of 1.1 in (21) we use that k > K_{1} and hence that M = S_{q_{0}} ≥ 20; also recall that h′ ≤ h.

Note that h_{r} ≡ |ψ_{r}([v_{r}, C])| = |c − c_{S_{pr}} | = a^{M}h_{r−1} − h′. If h_{r} ≥ ϵ, then the cut at S_{k} is ϵ − η-good. To see this, set J = [v_{r}, C]. Then T_{a}^{(r+1)M}|J is monotone and |T_{a}^{(r+1)M}(J)| = h_{r} ≥ ϵ. We need that \frac{|J|}{E_{J}} > η, where E_{J} is the length of the shortest closed interval containing both y_{k} and J when S_{k} is a substantial cut or containing both y′_{k} and J when S_{k} is a cosubstantial cut. We have E_{J} = h+|c−y_{k}| if S_{k} is a substantial cut and E_{J} = h+|c−y′_{k}| if S_{k} is a cosubstantial cut. Either way we have that E_{J} = h+4a^{−M}h. Next, from (21) we obtain that |c − v_{r}| ≤ (0.002)h. Hence, |J| = h − |c − v_{r}| ≥ 0.99h. Putting these equations together we have that \frac{|J|}{E_{J}} ≥ \frac{0.99h}{h+4^{−M} h} > 0.8. Thus, \frac{|J|}{E_{J}} > η, since η < \frac{1}{2}.

Therefore, it remains to deal with the case where for some r ≥ 1 we have q_{i} = q_{0} for 1 ≤ i ≤ r, q_{r+1} < q_{0}, and h_{r} < ϵ. In this case we still have that S_{p_{r}} is a cosubstantial cut. Moreover, ϵ_{1} > 2ϵ > 2h_{r} > |c_{S_{pr}} − c_{S_{qr}} | gives (from the definition of ϵ_{1}) that p_{r} > K_{1}. Thus, applying either Lemma 13 or Lemma 15 we obtain that the cosubstantial cut at S_{p_{r}} is strongly ϵ − η-good.

More precisely, if S_{p_{r+1}} is not a cosubstantial cut, then we apply Lemma 13 with k = p_{r} in the statement of the lemma. To do this we need that S_{q_{r}+l′′} is ϵ − η-good, where to avoid confusion we have replaced l in Lemma 13 with l′′. However, from Lemma 11, we have that q_{r+1} + l′′ ≤ q_{r} = Q(k) and hence, use (4), Q(q_{r+1} + l′′) < Q(k). Thus, by hypothesis, S_{q_{r+1}+l′′} is ϵ − η-good.

Next, if S_{p_{r+1}} is a cosubstantial cut, then we apply Lemma 15 with k = p_{r} in the statement of the lemma. To do this we need q_{r+1} < Q(k) = q_{0} (which we have) and S_{p_{r+1}} is ϵ − η-good. However, q_{r+1} < Q(k) and the hypothesis give that S_{p_{r+1}} is ϵ − ηgood. Thus, we have that S_{p_{r}} is strongly ϵ − η-good.

Let ȳ′_{p_{r}} be as in Definition 11, i.e., ȳ′_{p_{r}} is the point on the same side of c as c_{S_{qr}} such that |c − ȳ′_{p_{r}}| = 8a^{−S_{qr}} |c − c_{S_{pr}} | = 8a^{−M}h_{r}. Set w = ψ_{r}^{−1}(ȳ′_{p_{r}}). Then,

|v_{r} − w| = a^{−(r+1)M}|c − ȳ′_{p_{r}}| = 8a^{−(r+2)M}h_{r}.  (22)

Using (21) we have that |C −v_{r}| = h−|c−v_{r}| ≥ h−(1.1)ha^{−M} = h(1−(1.1)a^{−M}). This and h_{r} = a^{(r+1)M}|C − v_{r}| give h_{r} ≥ a^{(r+1)M}h(1 − (1.1)a^{−M}) > (0.99)a^{(r+1)M}h. Hence, (also use (22))

|v_{r} − w| ≥ 8a^{−(r+2)M}(0.99)a^{(r+1)M}h > 7a^{−M}h.  (23)

It follows from (21) and (23) that w and v_{r} lie on opposite sides of c. Hence, (again, use (21) and (23))

|c − w| = |v_{r} − w| − |c − v_{r}| > 5a^{−M}h.  (24)

Since the cosubstantial cut at S_{p_{r}} is strongly ϵ − η-good, we may choose I ⊂ [c, c_{S_{pr}}] and l ≥ 1 such that T_{a}^{l}|I is monotone, |T_{a}^{l}(I)| > ϵ, and |I| > ηE_{I}, where E_{I} is the length of the shortest closed interval containing both ȳ′_{p_{r}} and I.

Set J = ψ_{r}^{−1}(I). Then J ⊂ [v_{r}, C] and denoting the length of the shortest closed interval containing both w and J by E_{J} we have, by the linearity of ψ_{r}^{−1}, that |J| > ηE_{J}. However, |T_{a}^{(r+1)M+l}(J)| > ϵ and T_{a}^{(r+1)M+l}|J is monotone, since ψ_{r}|J = T_{a}^{(r+1)M}|J. From (24) we have that |c−w| > 4a^{−M}h = 4a^{−M}|c−C|. But 4a^{−M}|c−C| equals |c−y_{k}| when S_{k} is a substantial cut and equals |c−y′_{k}| when S_{k} is a cosubstantial cut. Therefore the cut at S_{k} is ϵ − η-good. □

Definition 14 Fix δ < min{|c−c_{1}|, |c−c_{2}|}. For each k ∈ N such that S_{k} is a substantial cut, set A_{k,δ} = {I ⊂ [c, c_{S_{Q(k)}}] | there exists l ≥ 1 with T_{a}^{l}|I monotone and |T_{a}^{l}(I)| > δ}.

Definition 15 Fix δ < min{|c−c_{1}|, |c−c_{2}|}. For each k ∈ N such that S_{k} is a cosubstantial cut, set B_{k,δ} = {I ⊂ [c, c_{S_{k}}] | there exists l ≥ 1 with T_{a}^{l}|I monotone and |T_{a}^{l}(I)| > δ}.

Remark: It follows from Proposition 2 that each of A_{k,δ} and B_{k,δ} is nonempty.

Lemma 16 Fix δ < min{|c − c_{1}|, |c − c_{2}|} and suppose that S_{k} is a substantial cut (cosubstantial cut). Then there exists 0 < γ < 1 such that for any I ∈ A_{k,δ} (B_{k,δ}) we have that |I| < γE_{I}, where E_{I} is the length of the shortest closed interval containing both y_{k} (y′_{k}) and I.

Proof: We do the case when S_{k} is a substantial cut, the case of a cosubstantial cut being similar. Let I and E_{I} be as above. Set g(x) = \frac{x}{x+|c−y_{k}|} for x ∈ [0, 1]. On [0, 1] the function g(x) has a maximum when x = 1; hence for x ∈ [0, 1], we have g(x) ≤ \frac{1}{1+|c−y_{k}|}.

This, |c − y_{k}| + |I| ≤ E_{I}, and |I| ≤ 1, give that \frac{|I|}{E_{I}} ≤ \frac{|I|}{|I|+|c−y_{k}|} ≤ \frac{1}{1+|c−y_{k}|}. Thus,

|I| ≤ E_{I} \frac{1}{1 + |c − y_{k}|}.

The lemma now follows. □

Definition 16 For each substantial cut S_{k}, set η_{k} = inf{γ < 1 | if I ∈ A_{k,ϵ}, then |I| ≤ γE_{I}}. Clearly, η_{k} > 0 since A_{k,ϵ} ≠ ∅.

Definition 17 For each cosubstantial cut S_{k}, set β_{k} = inf{β < 1 | if I ∈ B_{k,ϵ}, then |I| ≤ βE_{I}}. Again, β_{k} > 0.

Theorem 2 Set η^{∗} = inf{η_{k} | S_{k} is a substantial cut}. Then, η^{∗} > 0.

Proof: Set

α = min{η_{j} | j ≤ K_{1} and S_{j} is a substantial cut },
β = min{β_{j} |j ≤ K_{1} and S_{j} is a cosubstantial cut },
τ = \frac{1}{2} min\{α, β, \frac{1}{2}\}, and
B = {k | S_{k} is a substantial or cosubstantial cut that is ϵ − τ-good}.

If either or both α, β are not defined (i.e., taking the minimum of an empty set), then delete them from the definition of τ. We show that B = ∅ and hence η^{∗} ≥ τ > 0, proving the theorem.

Suppose to the contrary that B is not empty. Then, choose k′ ∈ B such that Q(k′) ≤ Q(l) for all l ∈ B. Since k′ ∈ B, S_{k′} is either a substantial or cosubstantial cut that is not ϵ − τ-good. First suppose that k′ > K_{1}. From the definition of B, all substantial or cosubstantial cuts S_{m} with Q(m) < Q(k′) are ϵ − τ-good and therefore, by Proposition 3, S_{k′} is ϵ − τ-good; a contradiction. Thus, k′ ≤ K_{1}. But then, the definitions of α, β, and τ give that S_{k′} is also ϵ − τ-good. Again, a contradiction. Hence, B = ∅. □

Corollary 1 There exists η̃ > 0 such that every substantial cut S_{k} is ϵ − η̃-good.

Proposition 4 Let δ > 0. Then there exists a closed interval J′ ⊂ (a − δ, a), k, l ∈ N, and η_{1} ∈ (0, 1) such that

1. J′ ⊂ ω_{S_{k}}(a),

2. ξ_{S_{k}+l′}|J′ is monotone for 0 < l′ ≤ l,

3. |ξ_{S_{k}+l}(J′)| > \frac{ϵ}{2}, and

4. \frac{|J′|}{E_{J′}} ≥ η_{1}, where E_{J′} is the length of the shortest closed interval containing J′ and a.

Moreover, η_{1} is independent of δ (depending only on a and ϵ).

Proof: Let η̃ be as in Corollary 1 and without loss of generality assume that η̃ < \frac{1}{10}. We write c_{S_{k}} for c_{S_{k}}(a) and c_{S_{Q(k)}} for c_{S_{Q(k)}}(a). For any k, one endpoint of ξ_{S_{k}}(ω′_{S_{k}}(a)) is c_{S_{k}} and we denote the other endpoint as c′_{S_{k}}; here we take ω′_{S_{k}}(a) to be closed. In Figure 1, c′_{S_{k}} = ξ_{S_{k}}(u). Choose k ∈ N such that S_{k} is a substantial cut, ω′_{S_{k}}(a) ⊂ (a − δ, a],

k > K_{0} from Lemma 2, \frac{(1+ϵ_{0})a^{−S_{k}}}{αη̃(a−1)} < min\{0.1, \frac{ϵ}{8}\} (here α is from Lemma 3), and

\frac{|c′_{S_{k}} − c_{S_{Q(k)}}|}{|c_{S_{k}} − c′_{S_{k}}|} < \frac{η̃}{1000}.  (25)

For (25) use Lemma 7. From the definition of η̃, S_{k} is ϵ − η̃-good and hence choose I ⊂ [c, c_{S_{Q(k)}}] and l ≥ 1 such that T_{a}^{l}|I is monotone, |T_{a}^{l}(I)| > ϵ, and

|I| ≥ η̃E_{I},  (26)

where E_{I} is the length of the shortest interval containing both y_{k} and I. Let I′ ⊂ I be concentric with I and such that |I′| = 0.8|I|. From (25) and (26) it follows that I′ ⊂ ξ_{S_{k}}(ω′_{S_{k}}(a)) = [c_{S_{k}}, c′_{S_{k}}]. In the sequel we will write ξ_{S_{k}}^{−1} for the well defined (since ξ_{S_{k}} is strictly monotone on ω′_{S_{k}}(a)) inverse of ξ_{S_{k}} restricted to ω′_{S_{k}}(a). Set J′ = ξ_{S_{k}}^{−1}(I′). Since ω′_{S_{k}}(a) ⊂ (a − δ, a], we obtain 1, i.e. J′ ⊂ ω_{S_{k}}(a).

Let D_{I′} be the smallest closed interval containing c_{S_{k}} and I′; denote the length of D_{I′} by E_{I′}. It is not difficult to see that D_{I′} ⊂ ξ_{S_{k}}(ω′_{S_{k}}(a)). Notice that

E_{I′} < E_{I} ≤ \frac{|I|}{η̃}.  (27)

For x ∈ ξ_{S_{k}}(ω′_{S_{k}}(a)) = [c_{S_{k}}, c′_{S_{k}}] set e(x) = ξ_{S_{k}}^{−1}(x). From Lemma 2 we have

\frac{1}{1 + ϵ_{0}} |ξ′_{S_{k}}(a)| ≤ |ξ′_{S_{k}}(e(x))| ≤ (1 + ϵ_{0})|ξ′_{S_{k}}(a)|.

for any x ∈ ξ_{S_{k}}(ω′_{S_{k}}(a)). Similarly we have,

\frac{1}{1 + ϵ_{0}} |(e′(c_{S_{k}})| ≤ |e′(x)| ≤ (1 + ϵ_{0})|e′(c_{S_{k}})|.  (28)

for any x ∈ ξ_{S_{k}}(ω′_{S_{k}}(a)). From Lemma 3 and (28) we have

\frac{1 + ϵ_{0}}{α} a^{−S_{k}} ≥ |e′(x)| ≥ \frac{1}{β(1 + ϵ_{0})}a^{−S_{k}}  (29)

for any x ∈ ξ_{S_{k}}(ω′_{S_{k}}(a)) ⊃ D_{I′}.

Assume that a′ ∈ ξ_{S_{k}}^{−1}(D_{I′}) and let x′ = ξ_{S_{k}}(a′). Then, from (29),

|a − a′| = |e(c_{S_{k}}) − e(x′)|
≤ ∫_{D_{I′}} |(e′(x)| dx
≤ \frac{1 + ϵ_{0}}{α} a^{−S_{k}}E_{I′}.  (30)

From (30) and Lemma 6 we have

|T_{e(x)}^{l′}(x) − T_{a}^{l′}(x)| ≤ |a − e(x)|\frac{a^{l′} − 1}{a − 1}
< |a − e(x)|\frac{a^{l′}}{a − 1}
< \frac{1 + ϵ_{0}}{α} a^{−S_{k}}E_{I′} \frac{a^{l′}}{a − 1},  (31)

for any x ∈ D_{I′} and 0 < l′ ≤ l.

Claim 1: For 0 < l′ ≤ l we have ξ_{S_{k}+l′}|J′ is monotone, i.e., 2 holds. Proof: Suppose to the contrary that there exists some x ∈ I′ = ξ_{S_{k}}(J′) such that ξ_{S_{k}+l′}(e(x)) = T_{e(x)}^{l′}(x) = c. Then from (27) and (31)

|T_{a}^{l′}(x) − c| = |T_{a}^{l′}(x) − T_{e(x)}^{l′}(x)|
< \frac{1 + ϵ_{0}}{α} a^{−S_{k}}E_{I′} \frac{a^{l′}}{a − 1}
< \frac{1 + ϵ_{0}}{α} a^{−S_{k}} \frac{|I|}{η̃} \frac{a^{l′}}{a − 1}
< 0.1|I|a^{l′}.  (32)

Since x ∈ I′, it follows that [x − 0.1|I|, x + 0.1|I|] ⊂ I. Next, since T_{a}^{l′}|I is monotone, we obtain that

[T_{a}^{l′}(x) − 0.1a^{l′}|I|, T_{a}^{l′}(x) + 0.1a^{l′}|I|] ⊂ T_{a}^{l′}(I)  (33)

for 0 < l′ ≤ l. Now, (32) and (33) imply that c ∈ T_{a}^{l′}(I). However, by assumption, T_{a}^{l′} is monotone on I for 0 < l′ ≤ l and hence c ∉ T_{a}^{l′}(I). This contradiction completes Claim 1. □

Claim 2: We have |ξ_{S_{k}+l}(J′)| > \frac{ϵ}{2}, i.e., 3 holds. Proof: It follows from Claim 1 that ξ_{S_{k}+l′}|J′ is monotone for all 0 < l′ ≤ l and hence, in particular that, ξ_{S_{k}+l}|J′ is monotone. Choose a_{1} and a_{2} such that J′ = [a_{1}, a_{2}]. Then, [ξ_{S_{k}}(a_{1}), ξ_{S_{k}}(a_{2})] = I′ and

|T_{a}^{l}(ξ_{S_{k}}(a_{1})) − T_{a}^{l}(ξ_{S_{k}}(a_{2}))| = a^{l}|I′|
= a^{l} \frac{|I′|}{|I|}|I| = \frac{|I′|}{|I|}|T_{a}^{l}(I)|
≥ \frac{8}{10}ϵ.  (34)

For i = 1, 2 (recall (27) and (31)),

|T_{a}^{l}(ξ_{S_{k}}(a_{i})) − ξ_{S_{k}+l}(a_{i})| = |T_{a}^{l}(ξ_{S_{k}}(a_{i})) − T_{a_{i}}^{l}(ξ_{S_{k}}(a_{i}))| < \frac{1 + ϵ_{0}}{α} a^{−S_{k}}E_{I′} \frac{a^{l}}{a − 1}
≤ \frac{1 + ϵ_{0}}{α} a^{−S_{k}} \frac{|I|}{η̃} \frac{a^{l}}{a − 1} = \frac{1 + ϵ_{0}}{α} a^{−S_{k}} \frac{1}{η̃} \frac{1}{a − 1}|T_{a}^{l}(I)|
≤ \frac{1 + ϵ_{0}}{α} a^{−S_{k}} \frac{1}{η̃} \frac{1}{a − 1} < \frac{ϵ}{8}.  (35)

Lastly, (34) and (35) imply that |ξ_{S_{k}+l}(J′)| = |ξ_{S_{k}+l}(a_{1}) − ξ_{S_{k}+l}(a_{2})| > \frac{ϵ}{2}. Claim 2 is now complete. □

Again, |I′| = 0.8|I| ≥ 0.8η̃E_{I} > 0.8η̃E_{I′} and hence (recall (29))

|J′| = ∫_{I′} |e′(x)| dx ≥ \frac{1}{β(1 + ϵ_{0})}a^{−S_{k}}|I′|
> \frac{1}{β(1 + ϵ_{0})}a^{−S_{k}}0.8η̃E_{I′}.  (36)

Letting D_{J′} denote the shortest interval containing a and J′ and setting E_{J′} = |D_{J′}| we have (recall (29))

E_{J′} ≤ ∫_{D_{I′}} |e′(x)| dx ≤ \frac{1 + ϵ_{0}}{α} a^{−S_{k}}E_{I′}.  (37)

Joining (36) and (37) we obtain

\frac{|J′|}{E_{J′}} > \frac{0.8η̃α}{β(1 + ϵ_{0})^{2}} ≡ η_{1}.  (38)

Thus, 4 holds. □

7 Proof of Theorem 1

We have already shown that P and H = \{a ∈ [\sqrt{2}, 2] | lim inf_{k→∞} Q_{a}(k) < ∞ and a ∉ D ∪ P} are σ-porous. Hence, to prove Theorem 1, it remains to show that I is σ-porous.

For each pair r_{1} < r_{2} ∈ Q set

I_{r_{1},r_{2}} = {a | (r_{1}, r_{2}) ⊂ [ξ_{2}(a), ξ_{1}(a)], ξ_{n}(a) ∉ (r_{1}, r_{2}) for all n, and lim_{k→∞} Q_{a}(k) = ∞}.

We will show that each nonempty I_{r_{1},r_{2}} is porous and therefore that I is σ-porous. Theorem 1 will then immediately follow.

Fix r_{1} < r_{2} ∈ Q such that I_{r_{1},r_{2}} ≠ ∅ and fix a ∈ I_{r_{1},r_{2}}. Let ϵ > 0 be as defined in Section 6 for the fixed a. Let {δ_{i} > 0} be such that lim_{i→∞} δ_{i} = 0. For each i apply Proposition 4 with the fixed a, ϵ, and δ = δ_{i}, to generate sets J′_{i} and positive integers k_{i} and l_{i} which satisfy for each i the conditions (1)-(4) of Proposition 4. Since η_{1} from Proposition 4 is independent of the choice of δ (depending only on a and ϵ) we have a common η_{1} for all i.

Let P be a partition of [0, 1] into subintervals of length \frac{ϵ}{4}, with perhaps the partition element containing 0 having a smaller length. Set r′_{1} = r_{1} + \frac{(r_{2}−r_{1})}{4} and r′_{2} = r_{2} − \frac{(r_{2}−r_{1})}{4}. Since T_{a} is leo, for each I ∈ P there exists a closed interval U_{I} ⊂ I and k_{I} ∈ N such that T_{a}^{k_{I}} (U_{I}) ⊂ (r′_{1}, r′_{2}). Let γ > 0 be such that if a′ ∈ (a − γ, a), then for each I ∈ P we have T_{a′}^{k_{I}} (U_{I}) ⊂ (r_{1}, r_{2}). Passing to a subsequence if needed, assume that

J′_{i} ⊂ (a − γ, a)  (39)

for all i.

From P a finite partition, |ξ_{S_{k_{i}} +l_{i}}(J′_{i})| > \frac{ϵ}{2} for all i, and |I| ≤ \frac{ϵ}{4} for all I ∈ P, we may choose I^{∗} ∈ P such that I^{∗} ⊂ ξ_{S_{k_{i}} +l_{i}}(J′_{i}) for infinitely many i. Again, passing to a subsequence if needed, assume that I^{∗} ⊂ ξ_{S_{k_{i}} +l_{i}}(J′_{i}) for all i. Next for each i set

J_{i}^{∗} = ξ_{S_{k_{i}} +l_{i}}^{−1}(U_{I^{∗}}).

Then for all i, J_{i}^{∗} ⊂ J′_{i} and

ξ_{S_{k_{i}} +l_{i}+k_{I^{∗}}} (J_{i}^{∗}) = ξ_{k_{I^{∗}}} (U_{I^{∗}}).  (40)

It follows from (39) and (40) that for all i, ξ_{S_{k_{i}} +l_{i}+k_{I^{∗}}} (J_{i}^{∗}) ⊂ (r_{1}, r_{2}). Hence,

J_{i}^{∗} ⊄ I_{r_{1},r_{2}}  (41)

for all i.

Claim 1: For each i,

|J′_{i}| ≤ \frac{(1 + ϵ_{0})}{|U_{I^{∗}}|}|J_{i}^{∗}|.

Proof: For each i set H_{i} = ξ_{S_{k_{i}} +l_{i}}(J′_{i}). Set M = max_{x∈J′_{i}}{|ξ′_{S_{k_{i}} +l_{i}}(x)|} and m = min_{x∈J′_{i}}{|ξ′_{S_{k_{i}} +l_{i}}(x)|}. Then,

|U_{I^{∗}}| = ∫_{J_{i}^{∗}} |ξ′_{S_{k_{i}} +l_{i}}(x)| dx ≤ M|J_{i}^{∗}|,  (42)

and

|H_{i}| = ∫_{J′_{i}} |ξ′_{S_{k_{i}} +l_{i}}(x)| dx ≥ m|J′_{i}|.  (43)

Combining (42) and (43) and using Lemma 2 we have

\frac{|U_{I^{∗}}|}{|H_{i}|} ≤ \frac{M|J_{i}^{∗}|}{m|J′_{i}|} ≤ (1 + ϵ_{0})\frac{|J_{i}^{∗}|}{|J′_{i}|}.

Hence,

\frac{|J_{i}^{∗}|}{|J′_{i}|} ≥ \frac{1}{(1 + ϵ_{0})} \frac{|U_{I^{∗}}|}{|H_{i}|} ≥ \frac{|U_{I^{∗}}|}{(1 + ϵ_{0})};

use |H_{i}| ≤ 1. This completes Claim 1. □

For each i let Δ_{i} be the length of the smallest closed interval containing a and J_{i}^{∗}. We need to find λ > 0 such that

\frac{|J_{i}^{∗}|}{Δ_{i}} ≥ λ

for all i. Write

\frac{|J_{i}^{∗}|}{Δ_{i}} = \frac{|J_{i}^{∗}|}{|J′_{i}|} \frac{|J′_{i}|}{Δ_{i}}.  (44)

From Proposition 4 we have for each i that \frac{|J′_{i}|}{E_{J′_{i}}} ≥ η_{1}, and by definition Δ_{i} ≤ E_{J′_{i}} for all

i; therefore

\frac{|J′_{i}|}{Δ_{i}} ≥ η_{1}  (45)

for all i. Joining (44) and (45), we see that to get a lower bound for \frac{|J_{i}^{∗}|}{Δ_{i}} it suffices to

have a lower bound for \frac{|J_{i}^{∗}|}{|J′_{i}|}; but this is precisely Claim 1. Thus Claim 1, (44), and (45) give that

\frac{|J_{i}^{∗}|}{Δ_{i}} ≥ \frac{|U_{I^{∗}}|}{(1 + ϵ_{0})}η_{1} ≡ λ,

for all i; note that λ is independent of i. Recalling (41) we now have that for each i

\frac{|J_{i}^{∗}|}{Δ_{i}} ≥ λ and J_{i}^{∗} ∩ I_{r_{1},r_{2}} = ∅.

Thus I_{r_{1},r_{2}} is porous. □

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