How many inflections are there in the Lyapunov spectrum?

Iommi&Kiwi showed that the Lyapunov spectrum of an expanding map need not be concave, and posed various problems concerning the possible number of inflection points. In this paper we answer a conjecture of Iommi&Kiwi by proving that the Lyapunov spectrum of a two branch piecewise linear map has at most two points of inflection. We then answer a question of Iommi&Kiwi by proving that there exist finite branch piecewise linear maps whose Lyapunov spectra have arbitrarily many points of inflection. This approach is used to exhibit a countable branch piecewise linear map whose Lyapunov spectrum has infinitely many points of inflection.


Introduction
For a differentiable dynamical system T : X → X, where for simplicity X is a subset of the unit interval, the Lyapunov exponent of a point x ∈ X is given by log |(T n ) (x)| whenever this limit exists. Typically the set of all Lyapunov exponents for a given map T is a closed interval of positive length. An investigation into the size of the set of points x corresponding to a given Lyapunov exponent α in this interval leads to the notion, introduced by Eckmann & Procaccia [4], of the associated Lyapunov spectrum L, being a map given by defining L(α) as the Hausdorff dimension of the level set {x ∈ X : λ(x) = α}. The Lyapunov spectrum was studied rigorously by Weiss [21], continuing a broader programme with Pesin (see e.g. [15,16]). In the setting of conformal expanding maps with finitely many branches, Weiss [21] proved the real analyticity of L, and also claimed that L is always concave 1 . By contrast for expanding maps (on a subset of [0, 1], say) with infinitely many branches the Lyapunov spectrum L can never be concave (see e.g. [9,Thm. 4.3]), a simple consequence of the non-negativity of L and the unboundedness of its domain (i.e. the interval of all Lyapunov exponents); the Lyapunov spectrum in the specific case of the Gauss map has been analysed by Kesseböhmer & Stratmann [12], and in the case of the Rényi map by Iommi [9].
Motivated by these examples, Iommi & Kiwi [11] revisited the case of finite branch expanding maps, and discovered that in fact the Lyapunov spectrum is not always concave; indeed even in the simplest possible setting of two-branch piecewise linear maps (see Definition 2.1) there exist examples with non-concave Lyapunov spectra (so such examples have points of inflection, i.e. points at which the second derivative vanishes). For finite branch maps the number of inflection points is necessarily even (cf. [11, p.539]), and all examples of non-concave Lyapunov spectra exhibited in [11] have precisely two points of inflection.
The authors would like to thank Victor Klepsyn for his very helpful suggestions. The second author was partly supported by the ERC Grant 833802-Resonances. The third author was partly supported by EPSRC grant EP/T001674/1. 1 In fact the word convex (rather than concave) is used in the claim [21,Thm. 2.4 (1)], though the interpretation is that of concave in the sense that we use it (see [11, p. 536]); all specific examples of Lyapunov spectra known at the time of [21] were indeed concave.
The natural problem suggested by the work of Iommi & Kiwi is the extent to which it is possible to find Lyapunov spectra with strictly more than two points of inflection. Specifically, the following conjecture and question are contained in [11, p. [11]) The Lyapunov spectrum of a 2-branch expanding map has at most two points of inflection. [11]) Is there an upper bound on the number of inflection points of the Lyapunov spectrum for piecewise expanding maps?
More broadly, the work of Iommi & Kiwi provokes interest in constructing maps whose Lyapunov spectra have more than two inflection points, and in understanding the general properties responsible for producing such inflection points. Henceforth for brevity we shall often use the term Lyapunov inflection (of a map) to denote a point of inflection in the Lyapunov spectrum of that map.
In this article we address both Conjecture 1.1 and Question 1.2, as well as the more general issue of understanding maps with more than two Lyapunov inflections. Specifically, we first give an affirmative answer to Conjecture 1.1 in the setting of 2-branch piecewise linear maps: Theorem 1.3. The Lyapunov spectrum of a two branch piecewise linear map has at most two points of inflection.
We do not know, however, whether there are nonlinear 2-branch expanding maps with more than two Lyapunov inflections.
Secondly, we construct explicit examples of maps with more than two Lyapunov inflections (see e.g. Figures 1 and 2 below), and resolve Question 1.2 as follows, proving that there is no upper bound on the number of Lyapunov inflections, and that indeed this can be established within the class of piecewise linear maps: Theorem 1.4. For any integer n ≥ 0, there is a piecewise linear map whose Lyapunov spectrum has at least n points of inflection.
A natural corollary of Theorem 1.4 is that there is also no upper bound on the number of zeros of higher order derivatives of the Lyapunov spectrum: Corollary 1.5. For any integers n ≥ 0, k ≥ 2, there is a piecewise linear map whose Lyapunov spectrum has at least n points at which its k th order derivative vanishes.
A natural by-product of our approach to proving Theorem 1.4 is that by moving into the realm of infinite branch maps, the first example of a map with infinitely many Lyapunov inflections can be exhibited: Theorem 1.6. There is an infinite branch piecewise linear map whose Lyapunov spectrum has a countable infinity of inflection points.
As above, this implies a corresponding result for zeros of higher order derivatives of the Lyapunov spectrum: There is an infinite branch piecewise linear map such that for all k ≥ 2, its Lyapunov spectrum has its k th order derivative equal to zero at infinitely many distinct points.
The organisation of this article is as follows. Section 2 consists of various preliminary definitions and results concerning the Lyapunov spectrum and its first two derivatives. Although we work exclusively with piecewise linear maps (see Definition 2.1), much of §2 is valid in the more general setting of expanding maps. While most of §2 is already in the literature in some form, our subsequent focus on inflection points motivates the careful derivation of the formula for the second derivative of L (see §2.4 and §2.5) in a way that is relatively  self-contained. The key ingredients here are a characterisation of L due to Feng, Lau & Wu [7] (see Proposition 2.14), together with the well known formula (8) for the derivative of pressure.
In §3 we prove Theorem 1.3, exploiting an explicit formula for the Lyapunov spectrum in order to show that it has at most two points of inflection as a consequence of a more general result (Theorem 3.3) concerning symmetric functions whose lower order derivatives are of prescribed sign.
In §4 we prove Theorem 1.4, by exhibiting piecewise linear maps together with explicit lower bounds on the number of their Lyapunov inflections. More precisely, we define a sequence of maps T N , where the lower bound on the number of Lyapunov inflections for T N grows linearly with N . It is possible to view the maps T N as evolving from each other, in the sense that each map T N +1 can be described in terms of adjoining additional branches to those of T N . At each stage the adjoined branches have derivatives much larger than the existing branches, a phenomenon reminiscent of the construction of Iommi & Kiwi [11], who showed that the 2-branch piecewise linear maps with Lyapunov inflections are such that the derivative on one branch is much larger than that on the other branch (in a certain precise sense, see [11,Thm. A], and Theorem 3.2). The strategy for bounding from below the number of Lyapunov inflections for T N exploits the characterisation of inflection points as solutions to an explicit equation (namely (17), derived in Proposition 2.27) involving a function related to the pressure of a certain family of potentials. The T N are then constructed so as to facilitate the definition of two interlaced sequences of numbers converging to zero, with the property that one side of the equation is dominant along one sequence, and the other side dominant along the other sequence, up to a certain point (increasing with N ) in the sequences. Consideration of the intervals defined by consecutive points in the two interlaced sequences then yields at least one solution to (17) in each such interval, up to a certain point that grows with N , thereby guaranteeing an increasing number of Lyapunov inflections for the maps T N . In §5 we see that the coherence of the construction of the T N produces, by allowing the process to evolve indefinitely, an infinite branch piecewise linear map T (alternatively, the map T could be considered as the primary object, with the T N viewed as finite branch truncations of T ). Minor modifications to the approach of §4 then yield Theorem 1.6.
Lastly, in §6, we present several explicit examples of piecewise linear maps with a prescribed number of Lyapunov inflections. While these examples are of a rather ad hoc nature, it is noteworthy that (unlike the T N of §4) it is possible to give an exact count for the number of Lyapunov inflections in each case, and moreover the number of branches needed in order to produce a given number of Lyapunov inflections is more economical than in §4. This prompts a natural question (Question 6.5): what is the minimum number of branches needed in order to witness a given number of Lyapunov inflections? 2. Preliminaries 2.1. Piecewise linear maps. We shall be interested in full branch piecewise affine maps defined on subsets of the unit interval; for brevity we call such maps piecewise linear : be a collection of pairwise disjoint closed sub-intervals of [0, 1], with lengths |X i | > 0. An associated piecewise linear map is any of the 2 q maps ∪ q i=1 X i → [0, 1] whose restriction to each X i is an affine homeomorphism onto [0, 1] (necessarily with derivative ±|X i | −1 ). Any restriction of the map to an interval X i is referred to as a branch.
For any piecewise linear map T , the set X := {x ∈ [0, 1] : T n (x) ∈ ∪ q i=1 X i for all n ≥ 0} depends only on the collection {X i } q i=1 , and is such that the restricted piecewise linear map T : X → X is surjective. We refer to X as the associated invariant set, and henceforth always consider piecewise linear maps as dynamical systems T : X → X.

Remark 2.2.
(a) The pairwise disjointness of the X i means the invariant set X is a Cantor set (and is self-similar, cf. [5,Ch. 9]), whose Hausdorff dimension is the unique value s such that q i=1 |X i | s = 1 (a result essentially due to Moran [13], see also e.g. [5,Thm. 9.3]). (b) A minor variant of Definition 2.1 would have been to only insist that the intervals X i have pairwise disjoint interiors (i.e. allow possible intersections at their endpoints); this would have involved choosing the value of T at any such points of intersection, but otherwise the theory would have been identical to that developed here. (c) Since each X i has length strictly smaller than 1, the derivative ±|X i | −1 of the piecewise linear map on X i is in modulus strictly larger than 1, so in particular the map is expanding. It should be noted that the discussion in the following subsections 2.2, 2.3 and 2.4 in fact applies to more general expanding maps (i.e. where the restriction to each X i is a diffeomorphism onto [0, 1] with derivative strictly larger than 1 in modulus), and only later (from §2.5 onwards) do we require the piecewise linear assumption. (d) Our piecewise linear maps were referred to as linear cookie-cutters in [11], following e.g. [1,2,19]. (e) In Section 3 we shall be concerned with the general two-branch case, i.e. q = 2. In Section 4 we shall deal with particularly large values of q in order to guarantee Lyapunov spectra with many points of inflection.

2.2.
Lyapunov exponents and the Lyapunov spectrum.
whenever this limit exists, and for a measure µ ∈ M, its Lyapunov exponent Λ(µ) is defined by Remark 2.5. Naturally there is a relation between the two notions of Lyapunov exponent: if µ ∈ M is ergodic then λ(x) = Λ(µ) for µ-almost every x, by the ergodic theorem, since log |(T n ) (x)| = n−1 i=0 log |T (T i x)|. Definition 2.6. Since log |T | is continuous, and M is both convex and weak- * compact (see e.g. [20]), it follows that the set of all possible Lyapunov exponents is a closed interval, which we shall denote by Remark 2.7. Note that the endpoints α min and α max are, respectively, the minimum and the maximum Lyapunov exponent, and can be characterised as and Notation 2.8. For α ∈ R let us write and for a continuous function ϕ : X → R we write We shall be interested in the Hausdorff dimension (denoted dim H ) of the level sets X α , for α ∈ A. Recall (see e.g. [5,15] Remark 2.10. In the special case that all the intervals X i have equal length, the modulus of the derivative of the piecewise linear map T is constant, so the domain A is a singleton, and the Lyapunov spectrum L : A → R is consequently a constant. The Lyapunov spectrum is real analytic onÅ = (α min , α max ).

Characterisations of the Lyapunov spectrum.
Notation 2.13. For a measure µ ∈ M, let h(µ) denote its entropy. We refer to h : M → R as the entropy map.
The Lyapunov spectrum L admits the following characterisation in terms of entropy: Proof. The identity was established in [7, Thm. 1.1].
(a) The characterisation (1) was implicit in the work of Weiss [21], and appeared explicitly in Kesseböhmer & Stratmann [12] in the setting of the continued fraction map (cf. the discussion in [11, p. 539]). It was generalised in [10, Thm. 1.3] to a wider class of countable branch expanding maps (see also §5).
Notation 2.18. Recall that for a general continuous function ϕ : X → R, the pressure P (ϕ) is defined (see e.g. [14,18,20]) by and admits the well known alternative characterisation The function p is C ω and strictly convex, so its derivative p is an orientation-preserving C ω diffeomorphism onto its image. Clearly p(t) = P (t log |T |) = h(m(t)) + t log |T | dm(t), so in particular p(τ (α)) = h(m(τ (α))) + ατ (α) for all α ∈Å, in other words We deduce the following result (which is well known, see e.g. [11,Eq. (3), p. 539] which differs superficially due to usage of P (−t log |T |) rather than the P (t log |T |) considered here): Proposition 2.19. For a piecewise linear map, on the interiorÅ = (α min , α max ), the Lyapunov spectrum L can be written as Proof. This follows from Proposition 2.17 together with the identity (6).

Formulae for derivatives of the Lyapunov spectrum.
The identity (10) yields the following formula for the derivative of the Lyapunov spectrum: For a piecewise linear map, on the interiorÅ = (α min , α max ), the first derivative L of the Lyapunov spectrum can be expressed as In other words, L (α) = −p((p ) −1 (α))/α 2 for all α ∈Å = (α min , α max ). (10), and differentiating this identity yields as required.
We are now able to derive a formula for the second derivative of the Lyapunov spectrum: Proposition 2.23. For a piecewise linear map, and for α ∈Å = (α min , α max ), the second derivative L of the Lyapunov spectrum can be expressed as where t = (p ) −1 (α) = τ (α).
Corollary 2.24. For a piecewise linear map, the Lyapunov spectrum L has a point of inflection at α ∈Å = (α min , α max ) if and only if Remark 2.25. Formulae for the first and second derivatives of the Lyapunov spectrum in terms of entropy appear in [9, §8]. Proposition 2.23 should be compared to the derivation on [11, p. 544] 3 .

Lyapunov inflections for piecewise linear maps.
While the preceding analysis could have been stated in the more general setting of (nonlinear) expanding maps, henceforth we use the fact that a piecewise linear map T is such that on each interval X i the absolute value of its derivative |T | is equal to |X i | −1 (i.e. the reciprocal of the length of X i ). In this case from (5) we see that the pressure is given by For ease of notation in what follows, we introduce the following function F : Clearly Proposition 2.27. For a piecewise linear map, the corresponding Lyapunov spectrum L has a point of inflection at α ∈ (α min , α max ) if and only if where s = −(p ) −1 (α) = −τ (α), and F is given by (15).

Two branch maps have at most two Lyapunov inflections
Consider a piecewise linear map with two branches, i.e. where q = 2. If |X 1 | = |X 2 | then the Lyapunov spectrum is constant (cf. Remark 2.10), and in particular has no points of inflection. If |X 1 | = |X 2 | then without loss of generality |X 1 | > |X 2 |, and defining a = |X 1 | −1 and b = |X 2 | −1 (so that b > a > 1), the domain is A = [log a, log b]. As noted by Iommi & Kiwi [11, p. 539], the Lyapunov spectrum L has the closed form expression since m(τ (α)) is the Bernoulli measure giving mass log b−α log(b/a) to X 1 and mass α−log a log(b/a) to X 2 , with entropy (see e.g. [20,Thm. 4.26 so that (19) is a consequence of Proposition 2.17.
Iommi & Kiwi [11, p. 539] conjectured that, in the setting of a two branch expanding map, the Lyapunov spectrum has at most two points of inflection. In fact the number of such inflections is necessarily even, since L is concave on some neighbourhood [log a, γ] of the left endpoint of A, and some neighbourhood [δ, log b]] of the right endpoint of A (see [11, p. 539]). In the case that the map is piecewise linear, we are able to answer Iommi & Kiwi's conjecture in the affirmative: Theorem 3.1. For a 2-branch piecewise linear map, the Lyapunov spectrum L is either concave or has precisely two inflection points.
The following more precise characterisation follows from Theorem 3.1 and [11, Thm. A]: Theorem 3.2. For a 2-branch piecewise linear map, where without loss of generality |X 1 | ≥ |X 2 |, the Lyapunov spectrum L is concave if and otherwise has precisely two inflection points. To see that Theorem 3.1 follows from Theorem 3.3, note that if we introduce then the Lyapunov spectrum L in (19) can be written as and if we define and it is worth recording the following easy lemma: The fact that Theorem 3.1 follows from Theorem 3.3 is then a consequence of the following properties of the function ϕ defined in (20).

Now we prove Theorem 3.3, concerning the inflection points of the function
Note that hence necessarily either or If we now define functions Φ + and Φ − by and then we see that (21) and (22) are respectively equivalent to and That is, if x 0 is an inflection point of M then necessarily either (25) or (26) holds. We now show that in fact (25) can never hold: Proof. In view of the above discussion it suffices to show that (25) can never hold. For this, note that an assumption of Theorem 3.3 is that c > 1, so that −c < −1. We claim that the image of Φ + is disjoint from (−∞, −1), so that no x 0 can satisfy Φ + (x 0 ) = −c.
Having eliminated the need to consider the function Φ + , we now simplify our notation by defining Φ := Φ − , in other words we set and from Lemma 3.6 we know that if x 0 is an inflection point of M then necessarily To conclude the proof of Theorem 3.3, it now suffices to show (in Lemma 3.7 below) that Φ is strictly decreasing on (0, 1/2), and strictly increasing on (1/2, 1), since from the above discussion it follows that any point −c ∈ (−∞, −1) has at most two Φ-preimages, and hence that M has at most two points of inflection. Proof. It will be shown that Φ is strictly negative on (0, 1/2), and strictly positive on (1/2, 1).

The number of Lyapunov inflections is unbounded
In this section we define a particular sequence of piecewise linear maps T N , with the property that the number of Lyapunov inflections of T N tends to infinity as N → ∞.
which is strictly smaller than 1 for all N ≥ 6 (note that N j=6 2 j 2 −2 j < 10 −8 ), so it is certainly possible to choose the X i to be pairwise disjoint and contained in [0, 1]. (b) We prescribe the lengths of the intervals in the collection X N , but need no further information about the intervals themselves (beyond the fact that they are pairwise disjoint, and contained in [0, 1]), since translating various of the X i does not change the Lyapunov spectrum. Clearly it could be arranged that X N ⊂ X N +1 for all N ≥ 6, which would lend the interpretation of T N +1 evolving from the preceding map T N (as described in §1) by adjoining 2 (N +1) 2 new branches. (c) The number of branches q N of T N is large. For example T 6 has 2 36 = 68, 719, 476, 736 branches, with |T 6 | = 2 64 = 18, 446, 744, 073, 709, 551, 616 on each branch. In Theorem 4.2, the smallest value of N yielding more than two Lyapunov inflections is N = 28, and the map T 28 has q 28 > 10 236 branches. Notation 4.4. Following 4 the notation of (15), define F N : R → R by Define U j (s) := j 2 − 2 j s , so that For j ≥ 1 define s j := 2j + 1 2 j , and for j ≥ 2 define the midpoint m j := s j + s j−1 2 = 6j − 1 2 j+1 , so in particular s 1 > m 2 > s 2 > m 3 > s 3 > . . . 4 The fact that various of the intervals in XN have identical lengths allows the representation (31) as a sum over the range 6 ≤ j ≤ N , rather than over 6 ≤ j ≤ qN . and lim j→∞ s j = lim j→∞ m j = 0 .
Remark 4.5. The s j are defined so that U j (s j ) and U j+1 (s j ) are equal, more precisely and each m j is the mid-point of s j and s j−1 , with In light of Proposition 2.27, to prove Theorem 4.2 it suffices to establish the following result: Proposition 4.6. For N ≥ 27, the equation has at least 2(N − 26) distinct solutions.
Proof. Introducing the auxiliary functions To prove (34) and (35), first note that the derivative of F N can be written as and its second derivative as where we set V j (s) := 2j + U j (s) = 2j + j 2 − 2 j s . First we prove (34). The strategy for this will be to approximate each of F N (m k ), F N (m k ), F N (m k ) by the single j = k term in the respective sums (32), (36), (37). In particular, we claim that if c = 16/5 then, for all 26 ≤ k ≤ N , and and combine these with the obvious lower bound to give whereas (38) yields log F N (m k ) < (log 2)U k (m k ) + log(1 + c2 −k/2 ) < (log 2)(k 2 − 3k + 1/2) + c2 −k/2 and therefore Since c = 16/5, from (41) and (42) it is readily verified that G N (m k ) > H N (m k ) for 5 26 ≤ k ≤ N , as required.
To prove (52), note that so combining with (50) gives We now establish an upper bound on by combining the exact evaluation of the two-term sum with upper bounds on the head and tail To estimate (57), write j = k − i to give where and Clearly D(2) = 1 and D(k) > 1. We claim that To establish (61) note that if 2 ≤ i ≤ k − 1 then n k (i) ≤ n k (2), therefore 2 n k (i) ≤ 2 n k (2) , so The righthand side of (62) is decreasing in k, and we are assuming that k ≥ 26, so therefore combining (63) with (60) easily gives the claimed bound (61). Combining (61) with (59) then gives which is our desired bound on the head (57). We now wish to estimate the tail (58), and for this write j = k + i so that and then write Now k ≥ 1, and 2 + i − 2 i < 0 for i ≥ 3, so 2k so using this estimate in (66) gives and substituting into (65) yields which is our desired bound on the tail (58). Substituting into (55) the identity (56), together with the bounds (64) and (67), gives Combining (54) and (68) gives Note that the righthand side of (69) is increasing in k, and since k ≥ 26 then and therefore (52) follows, as required.
Although the map T N is defined so that its derivative is identical on various branches, it should be apparent that this is a convenience rather than a strict condition. The dependence of the Lyapunov spectrum (with respect to e.g. some C k topology) on the underlying map is continuous, so that for example a sufficiently small perturbation of T N to a piecewise linear mapT N on a slightly different collection of intervalsX N = {X i } q N i=1 , such that the lengths |X i | (and hence the derivatives of the various branches) are all distinct, will not change the number of Lyapunov inflections.
Similarly, T N could be perturbed in a nonlinear way, yielding a nearby nonlinear (full branch) expanding map with the same number of Lyapunov inflections. More precisely, given X N := {X i } q N i=1 as in Definition 4.1, for any integer k ≥ 1 let C k N denote the set of maps ∪ q N i=1 X i → [0, 1] such that the restriction to each X i is a C k diffeomorphism onto [0, 1], equipped with the C k norm T k = sup{|T (i) (x)| : where T (i) denotes the i-th order derivative of T . As noted in Remark 2.2 (c), the definitions and basic properties of Lyapunov spectra detailed in subsections 2.2, 2.3 and 2.4 are valid for expanding members of C k N , in particular those that are sufficiently C k -close to T N , and we deduce:

An infinite branch map with infinitely many Lyapunov inflections
Using essentially the same strategy as in §4, we can prove the following: There is a countably infinite branch piecewise linear map whose Lyapunov spectrum has a countable infinity of inflection points.
To prove Theorem 5.1 we can define X = {X i } ∞ i=1 to be a countable collection of pairwise disjoint closed sub-intervals of [0, 1], consisting of 2 j 2 intervals of length 2 −2 j for all j ≥ 6. By analogy with Definition 2.1 we then define T to be a full branch affine homeomorphism onto [0, 1] on each interval in X , with the analogous definition of its invariant set. Such a T is in particular an expanding Markov-Rényi map (see e.g. [10] for multifractal analysis in this general context), and the collection of its inverse branches is a conformal iterated function system (see e.g. [8] for multifractal analysis in this general context).
The theory of §2 holds almost verbatim in this countable branch setting, the only significant difference being that is defined only for t > 0, and The approach of Proposition 4.6 can then be used to establish that has infinitely many solutions, by again simply proving the analogues of (34) and (35) for all k ≥ 26, and this yields Theorem 5.1. We also deduce the following: There is a countably infinite branch piecewise linear map such that for all k ≥ 2, its Lyapunov spectrum has its k th order derivative equal to zero at infinitely many distinct points.

Examples with a specified number of Lyapunov inflections
In this section we consider some specific piecewise linear maps with strictly more than two points of inflection in their Lyapunov spectra. By way of a complement to the explicit examples of Figures 1 and 2, and the maps T N of §4, the cases considered here show that a given number of Lyapunov inflections (namely 4, 6, and 8) can be attained for maps with relatively few branches (namely 7, 62, and 821), an issue we return to in Question 6.5 below. The maps are defined as follows. Proof. In view of Proposition 2.27 it suffices to show that there are precisely n solutions s 1 < s 2 < . . . < s n to equation (17). With F (t) = q i=1 |X i | t and p(t) = log F (−t), the corresponding n points of inflection of L are (by Proposition 2.27) given by In each of the three cases the corresponding function I, which we denote respectively by I 4 , I 6 , I 8 , can be graphed (see Figures 3,4,5, together with graphs of the corresponding second derivative of the Lyapunov spectrum), and its zeros located (to arbitrary precision, using e.g. the bisection method). For the map S 4 , four zeros of I 4 are located, at s 1 ≈ −0.3975,  2n points s − 1 < s + 1 < s − 2 < s + 2 < . . . < s − n < s + n can be chosen, where s + n < dim H (X), such that I(s − i ) and I(s + i ) have opposite sign, and I can be shown to be strictly monotone on (s − i , s + i ) (by bounding I away from zero). Moreover I can be bounded away from zero on the complementary intervals (−∞, s − 1 ), (s + 1 , s − 2 ), (s + 2 , s − 3 ),. . . , (s + n−1 , s − n ), while on (s + n , ∞) the function I is strictly positive on (s + n , dim H (X)), has a singularity at dim H (X) (since F (dim H (X)) = 1), and is strictly negative on (dim H (X), ∞).
In view of Theorem 1.3, and the 7-branch map S 4 with four Lyapunov inflections, it is natural to wonder if there is a 3-branch piecewise linear map with four Lyapunov inflections, or failing that whether four Lyapunov inflections occurs for any piecewise linear map with strictly fewer than 7 branches. More generally: Question 6.5. For any (even) natural number n, what is the smallest number Q n such that there exists a Q n -branch piecewise linear map whose Lyapunov spectrum has n points of inflection?