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Continuity of entropy for Lorenz maps
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Continuity of entropy for Lorenz maps✩ Z. Cooperbanda , E.P.J. Pearsea , B. Quackenbusha , J. Rowleya , T. Samuelb ,∗, M. Westc a
Mathematics Department, California Polytechnic State University, CA, USA b School of Mathematics, University of Birmingham, UK c Department of Mathematics, University of California: Irvine, CA, USA
Received 15 February 2019; received in revised form 4 September 2019; accepted 28 October 2019 Communicated by S. van Strien
Abstract Let (T p ) p denote a one parameter family of expanding interval maps with two increasing and continuous branches and indexed by their point of discontinuity. Using the pressure formula from thermodynamics, P. Raith (2000) showed that the topological entropy h(T p ) of T p varies continuously as a function of p. Here we provide a new and alternative proof of this result based on Milnor–Thurston kneading theory, as well as some observations on the monotonicity of p ↦→ h(T p ). c 2019 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved. ⃝ Keywords: Kneading sequences; Lorenz maps; Topological entropy; Matching
1. Introduction and main results Since the pioneering work of R´enyi [21] and Parry [16,17], an increasing amount of attention has been paid to interval maps. Their study has provided solutions to practical problems within biology, engineering, information theory and physics. Applications appear in analogue to digital conversion [10], analysis of electroencephalography (EEG) data [12], data storage [13], electronic circuits [3], mechanical systems with impacts and friction [2] and relay systems [23]. ✩ Acknowledgements. The work presented here was carried out with some of our students, notably the first, third, fourth and sixth authors as well E. Esquivel and E. Mihanovich. We are extremely grateful to the Bill and Linda Frost Fund which supported this undergraduate research project. ∗ Corresponding author. E-mail addresses: [email protected] (Z. Cooperband), [email protected] (E.P.J. Pearse), [email protected] (B. Quackenbush), [email protected] (J. Rowley), [email protected] (T. Samuel), [email protected] (M. West).
https://doi.org/10.1016/j.indag.2019.10.002 c 2019 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved. 0019-3577/⃝
Please cite this article as: Z. Cooperband, E.P.J. Pearse, B. Quackenbush et al., Continuity of entropy for Lorenz maps, Indagationes Mathematicae (2019), https://doi.org/10.1016/j.indag.2019.10.002.
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Fig. 1.1. An example of a Lorenz map (right) generated by branch functions f 0 and f 1 (left).
The concept of topological entropy, now ubiquitous in the study of dynamical systems, was introduced in [1] as a measure of the complexity of a dynamical system and is an invariant under a continuous change of coordinates, called topological conjugation. Bowen [6] gave an equivalent definition for a continuous map of a (not necessarily compact) metric space. For our purposes the following formulation, given by Misiurewicz and Szlenk in [15] and consistent with the definition given in [1], serves as a definition of the topological entropy. Let T be a piecewise monotonic interval map, such as a Lorenz map (see Definition 1.1 and Fig. 1.1), the topological entropy h(T ) of T is defined by 1 ln(Var(T n )), (1.1) n where Var( f ) denotes the total variation of the function f and T n denotes the n-fold composition of T with itself. Here we are concerned with how the topological entropy of a Lorenz map changes as we perturb the point of discontinuity. Thus the perturbations being studied here are not additions of C 1 -functions nor are they close in the R0 topology considered in [19]. The simplest example of a Lorenz map is a (normalised) β-transformation, and the topological entropy of such a transformation is equal to ln(β); this was first shown in [18]. However, for a general Lorenz map, the question of determining the topological entropy is more delicate, see for instance [4,11]. Before stating our main results, let us define what is meant by a Lorenz map. Note, in our definition, the branch functions need not be C 1 . h(T ) := lim
n→∞
Definition 1.1. Let 0 < a ≤ p ≤ b < 1. An upper, or lower, Lorenz map is a map T + : [0, 1] → [0, 1], respectively T − : [0, 1] → [0, 1], of the form { { f 0 (x) if 0 ≤ x < p, f 0 (x) if 0 ≤ x ≤ p, + − T (x) := respectively T (x) := f 1 (x) if p ≤ x ≤ 1, f 1 (x) if p < x ≤ 1, where f 0 and f 1 , called the branch functions, satisfy the following conditions. (i) The functions f 0 : [0, b] → [0, 1] and f 1 : [a, 1] → [0, 1] are continuous, strictly increasing and surjective. (ii) There exist constants C, c > 1 with C −1 |x − y| ≤ | f i−1 (x) − f i−1 (y)| ≤ c−1 |x − y| for i ∈ {0, 1} and x ∈ [0, 1]. The region [a, b] is often referred to as the switch region. Please cite this article as: Z. Cooperband, E.P.J. Pearse, B. Quackenbush et al., Continuity of entropy for Lorenz maps, Indagationes Mathematicae (2019), https://doi.org/10.1016/j.indag.2019.10.002.
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When we wish to emphasise the point of discontinuity, we write T p± for T ± . Further, by definition, we have h(T p+ ) = h(T p− ), and hence, for ease of notation, we let h(T p ) denote the common value. Note, a direct consequence of (1.1) is that h(T p ) ≥ ln(c).
(1.2)
In [20], Raith proved the following result. His method of proof uses the pressure formula from thermodynamics as developed by Bowen and Ruelle. Here we give a new alternative proof which utilises Milnor’s and Thurston’s kneading theory and has a more combinatorial flavour. Theorem 1.2.
For a fixed pair of branch functions, p ↦→ h(T p ) is continuous.
With this a hand, a natural and interesting question which arises is how regular is the map p ↦→ h(T p ), namely is it Lipschitz or H¨older continuous? Initial computer simulations indicate that large oscillations can occur on relatively small scales, as depicted in Figs. 3.2 and 3.3, even when the branch functions are affine. Here we would also like to mention that for a parameter class of piecewise affine maps with a single discontinuity and one increasing and one decreasing branch it has been shown that topological entropy displays a semi-regular behaviour. In particular, it is smooth on an open and dense set, see [8]. This feature is due to a property known as matching, see [7,8] and references therein. A similar property also plays a key rˆole in our proof of Theorem 1.2, Lemma 3.2. Outline In Section 2 we provide necessary definitions and preliminary results, and Section 3 is dedicated to our new and alternative proof of Theorem 1.2. We conclude this article with examples which demonstrate, for a fixed pair of branch functions, p ↦→ h(T p ) is not necessarily monotonic even if the branch functions are linear, compare with [9] and reference therein. 2. Preliminaries Throughout we use the convention that ± means either + or − and when we write T p± , we require that both T p+ and T p− are defined using the same branch functions. The set of all infinite words over the alphabet {0, 1} is denoted by Ω and is equipped with the product topology where we place the discrete topology on the set {0, 1} as well as the lexicographic ordering. For n ∈ ⋃ N, define Ωn to be the set of finite words over the alphabet {0, 1} of length n, and set Ω ∗ := n∈N0 Ωn , where by convention Ω0 is the set containing only the empty word ∅. For ω = ω0 · · · ωk and v = v0 · · · vn ∈ Ω ∗ , we set ωv := ω0 · · · ωk v0 · · · vn , that is the concatenation of ω and v, and let v := vvv · · · . Further, for k ∈ N we let v k denote the kth fold concatenation of v with itself. The length of v ∈ Ω ∗ is denoted by |v| with |∅| = 0 and, for a natural number k ≤ |v|, we set v|k := v0 · · · vk−1 . We use the same notations when v is an infinite word. The continuous map S : Ω → Ω defined by S(ω0 ω1 · · · ) := ω1 ω2 · · · is called the left-shift. We also allow for S to act on finite words as follows. For k ∈ N0 and v = v0 · · · vk ∈ Ω ∗ , we set S(v) = v1 · · · vk , if k ≥ 1 and S(v) = ∅ otherwise. The upper and lower itinerary maps τ p± : [0, 1] → Ω encode the orbit of a point x ∈ [0, 1] under T p± and are defined by τ p+ (x) = ω0 ω1 · · · and τ p− (x) = v0 v1 · · · where { { 0 if (T p+ )k (x) < p, 0 if (T p− )k (x) ≤ p, ωk := and v := k 1 if (T p+ )k (x) ≥ p, 1 if (T p− )k (x) > p. Please cite this article as: Z. Cooperband, E.P.J. Pearse, B. Quackenbush et al., Continuity of entropy for Lorenz maps, Indagationes Mathematicae (2019), https://doi.org/10.1016/j.indag.2019.10.002.
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The infinite words α := τ p− ( p) and β := τ p+ ( p) are called the kneading sequences of T p± , and play an important rˆole in what is to follow. We say that τ p± ( p) is periodic if there exists n ∈ N such that (T p± )n ( p) = p, and the period of τ p± ( p) is the smallest n ∈ N for which this holds. If τ p± ( p) are periodic, then there exist v, ω ∈ Ω ∗ such that τ p+ ( p) = v and τ p− ( p) = ω. Lemma 2.1 ([4,5]). The maps x ↦→ τx± (x) are strictly increasing. Additionally, x ↦→ τx+ (x) is right-continuous and x ↦→ τx− (x) is left-continuous. The following lemma extends this result. Lemma 2.2. If p ̸= a and β is non-periodic, then x ↦→ τx+ (x) is continuous at p and if p ̸= b and α is non-periodic, then x ↦→ τx− (x) is continuous at p. Proof. We prove the first statement, as the proof of the second statement is analogous. By definition and Lemma 2.1, it is sufficient to show, for a fixed N ∈ N, that there exists + ′ + ′ δ ∈ (0, p − a) such that τ p−δ ′ ( p − δ )| N = τ p ( p)| N for all δ ∈ (0, δ). This means we require ′ δ > 0 so that for n ∈ {0, 1, . . . , N − 1} and δ ∈ (0, δ) either + n ′ ′ + n (T p−δ ′ ) ( p − δ ) < p − δ and (T p ) ( p) < p
or + n (T p−δ ′) (p
(2.1)
− δ ) > p − δ and ′
′
(T p+ )n ( p)
> p.
To this end, let c and C be as in Definition 1.1 and choose δ ∈ (0, p − a) such that } { |(T p+ )k ( p) − p| : k ∈ {1, 2, . . . , N − 1} . 0 < δ < min Ck
(2.2)
Note, since p ̸= a and since β is not periodic, the value on the right hand side of (2.2) is positive. We claim, for all n ∈ {0, 1, . . . , N − 1} and δ ′ ∈ (0, δ), that + n ′ ′ n δ ′ cn ≤ (T p+ )n ( p) − (T p−δ ′) (p − δ ) ≤ δ C .
(2.3)
The case, n = 0, is immediate. Assume (2.3) holds for some n ∈ {0, 1, . . . , N − 1}. + n ′ + n ′ n If (T p+ )n ( p) < p, then (2.3) implies (T p−δ < p − δ ′ . If ′ ) ( p − δ ) ≤ (T p ) ( p) − δ c + n + n ′ n (T p ) ( p) > p, then (2.2) implies (T p ) ( p) − p > δ C ; combining this with our hypothesis + n ′ + n ′ n ′ yields (T p−δ ′ ) ( p − δ ) > (T p ) ( p) − δ C > p > p − δ . Hence, we have (2.3) for n + 1. To complete the proof, notice (2.3) implies (2.1). □ In our proof of Theorem 1.2, we use the following Laurent series which can be thought of as a generating function of the kneading sequences α = τ p− ( p) = α1 α2 · · · and β = τ p+ ( p) = β1 β2 · · · and is related to the function P(·, ·) defined in [11] by ξ p (z) = 2P(z −1 , z −1 ). For z ∈ C \ {0}, set ξ p (z) :=
∞ ∑ (βk − αk )z −k .
(2.4)
k=0
Observe that the interval (1, 2) belongs to the domain of convergence of ξ p . Further, we have the following result, which identifies the maximal zero of ξ p and the value γ = γ p := exp(h(T p )). Please cite this article as: Z. Cooperband, E.P.J. Pearse, B. Quackenbush et al., Continuity of entropy for Lorenz maps, Indagationes Mathematicae (2019), https://doi.org/10.1016/j.indag.2019.10.002.
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Fig. 3.1. The graph of ξ p together with a neighbourhood in which the graph of ξ p−δ belongs.
Theorem 2.3 ([4,11]). The topological entropy of T p± is equal to ln(r ), where r is the maximal positive real zero of ξ p . Additionally, if r is not simple, then it is the only zero of ξ p in the interval (1, 2] and has multiplicity two. 3. Proof of Theorem 1.2 In Sections 3.1 and 3.2 for a fixed pair of branch functions we prove that the map p ↦→ h(T p ) is left-continuous; right-continuity follows by an identical argument, see Section 3.3 for further details. The proof of left-continuity is subdivided into two sub-cases: when the kneading sequences of T p± are not periodic, and when they are periodic. For each sub-case we use the same approach. (i) Fix p ∈ (a, b) and show the existence of an ε > 0 with (γ − ε, γ + ε) ⊆ (1, 2), so that τ ± are continuous on (γ − ε, γ + ε). (ii) Show there exists δ > 0 such that ξ p−δ (x) has a zero r ∈ (γ − ε, γ + ε). (iii) Show there are no zeros larger than r . With this at hand, Theorem 2.3 allows us to conclude that ln(r ) = h(T p−δ ). Note, in Step (ii) we must take into account the multiplicity of γ ; see Fig. 3.1. By Theorem 2.3, if γ is simple, one can appeal to the intermediate value theorem, but more care is required in the case when γ is not simple. 3.1. Left continuity: β non-periodic Fix ε > 0 with (γ −ε, γ +ε) ⊆ (1, 2). By Lemma 2.2, the assumption that β is non-periodic ensures x ↦→ τx± (x) are left-continuous at p; this completes the proof of (i). To prove (ii) and (iii), that is there exists δ > 0 such that ξ p−δ (x) has a maximal zero r ∈ (γ − ε, γ + ε), we replace the infinite sum ξ p (x) with a partial sum (a polynomial) and approximate γ p−δ as the root of this polynomial. To this end, for n ∈ N, set R p,n (x) :=
∞ ∑
(βk − αk )x −k
k=n
Lemma 3.1.
so that
ξ p (x) =
n−1 ∑ (βk − αk )x −k + R p,n (x). k=0
For n ∈ N, there exists δ = δ(n) > 0 so that, for δ ′ ∈ (0, δ) and x ∈ (1, 2),
|ξ p−δ′ (x) − ξ p (x)| ≤
2x −n . 1 − x −1
(3.1)
Please cite this article as: Z. Cooperband, E.P.J. Pearse, B. Quackenbush et al., Continuity of entropy for Lorenz maps, Indagationes Mathematicae (2019), https://doi.org/10.1016/j.indag.2019.10.002.
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Proof. For m ∈ N, we have βm − αm ∈ {−1, 0, 1}, whence, for n ∈ N and x ∈ (1, 2), ∞ ∑ x −n . |R p,n (x)| ≤ x −k = 1 − x −1 k=n
(3.2)
Let n ∈ N be fixed. Since the maps x ↦→ τx± (x) are both left-continuous at p, there exists δ > 0 ± ′ ± such that, if δ ′ ∈ (0, δ), then τ p−δ ′ ( p − δ )|n = τ p ( p)|n . As (3.2) also holds for R p−δ ′ ,n (x), we have established (3.1). □ Proof of Theorem 1.2 (Left-Continuity with β Non-Periodic). Note that γ is an isolated zero of ξ p . Indeed, in its domain of convergence, the function ξ p is holomorphic. Consequently, the existence of a sequence of zeros of ξ p converging to γ would imply that ξ p is the constant zero function; a contradiction. Let ε > 0 be fixed such that (γ − ε, γ + ε) ⊆ (1, 2) and such that ξ p has a single root in this interval, namely at γ . Let c > 1 be as in Definition 1.1 and fix u ∈ (1, c). The specific value of u is not important, so for convenience set u = (1 + c)/2, and by (1.2) and Theorem 2.3 the maximal real zero of ξq is greater than u for all q ∈ [a, b]. Assume that γ is simple. Let n ∈ N be such that 2u −n (3.3) 1 − u −1 and let δ be chosen in accordance with Lemma 3.1. In that case, for all δ ′ ∈ (0, δ), 2u −n (3.4) |ξ p−δ′ (γ ± ε) − ξ p (γ ± ε)| < 1 − u −1 which ensures sgn(ξ p−δ′ (γ ± ε)) = sgn(ξ p (γ ± ε)), respectively. This together with the fact that ξ p is smooth and has a single root in (γ − ε, γ + ε) and an application of the intermediate value theorem yields that ξ p−δ′ has a zero in (γ − ε, γ + ε) for all δ ′ ∈ (0, δ); see Fig. 3.1. Assume that γ has multiplicity two. { By Theorem 2.3 and since ξ p (2) ≥ 0, } we have ξ p (x) > 0 for all x ∈ [u, 2] \ {γ }. Let ρ := inf ξ p (x)/2 : x ∈ [u, γ − ϵ] ∪ [γ + ϵ, 2] , let n ∈ N be such that 2u −n /(1 − u −1 ) < ρ, and let δ be chosen in accordance with Lemma 3.1. In that case, for all δ ′ ∈ (0, δ) and x ∈ [u, γ − ϵ] ∪ [γ + ϵ, 2], 2u −n ξ p−δ′ (x) ≥ ξ p (x) − > ξ p (x) − ρ > 0 1 − u −1 which ensures that ξ p−δ′ (x) has no zeros in [u, γ − ϵ] ∪ [γ + ϵ, 2]. Therefore, by Theorem 2.3 and Eq. (1.2), namely that the maximal real zero of ξq belongs to (u, 2] for q ∈ [a, b], we have ξ p−δ′ necessarily has a zero in the interval (γ − ε, γ + ε) Therefore, regardless of the multiplicity of γ , it is necessarily the case that ξ p− δ′ has a zero in (γ − ε, γ + ε) for all δ ′ ∈ (0, δ), whence Theorem 2.3 implies that |h(T p ) − h(T p−δ′ )| ≤ ε for all δ ′ ∈ (0, δ), as required. □ |ξ p (γ ± ε)| ≥
3.2. Left continuity: β periodic Throughout this section, we assume β is periodic with period N , for some N ∈ N. Note, in this case p ̸= a, since when p = a we have that β = 10 which is not periodic. Lemma 3.2. For n ∈ N with n ≥ N , there exists δ = δ(n) > 0 so that, for δ ′ ∈ (0, δ), the + ′ concatenation of τ p+ ( p)| N and τ p− ( p)|n−N equals τ p−δ ′ ( p − δ )|n , namely + ′ + − τ p−δ ′ ( p − δ )|n = (τ p ( p)| N )(τ p ( p)|n−N ). Please cite this article as: Z. Cooperband, E.P.J. Pearse, B. Quackenbush et al., Continuity of entropy for Lorenz maps, Indagationes Mathematicae (2019), https://doi.org/10.1016/j.indag.2019.10.002.
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Proof. Let n ≥ N denote a fixed integer. By construction, there exists η > 0 such that, + j ′ ′ if η′ ∈ (0, η), then (T p−η ′ ) ( p − η ) ̸ = p − η for all j ∈ {1, 2, . . . , N + n − 1}. Using the same arguments as in the proof of Lemma 2.2, we may choose η small enough so that, + j ′ ′ j in addition to this, if η′ ∈ (0, η), then η′ c j ≤ (T p+ ) j ( p) − (T p−η for ′) (p − η ) ≤ η C all j ∈ {0, 1, . . . , N }; here c and C are as in Definition 1.1. If j = N , then this yields + N ′ ′ N η′ c N < p − (T p−η for η′ ∈ (0, η). Since c > 1, this implies that ′) (p − η ) < η C + ′ N ′ ′ N 0 < ( p − η ) − (T p−η′ ) ( p − η ) < η (C − 1). Utilising both the left-continuity and monotonicity of x ↦→ τx− (x) at p, given in Lemma 2.1, there exists λ ∈ (0, 1) so that, for q ∈ [a, b] with 0 < p − q < λ, we have τ p− ( p)|n = τq− (q)|n . Setting δ := min{λC −N /2, η}, if δ ′ ∈ (0, δ), then + N ′ 0 < ( p − δ ′ ) − (T p−δ ′ ) ( p − δ ) < λ/2,
which implies − − + ′ N ′ τ p−δ ′ ( p − δ )|n = τ p−δ ′ ((T p−δ ′ ) ( p − δ ))|n + + N ′ N + ′ = τ p−δ ′ ((T p−δ ′ ) ( p − δ ))|n = S (τ p−δ ′ ( p − δ ))|n .
(3.5)
+ + − ′ ′ ′ Therefore, τ p−δ ′ ( p − δ )|n = (τ p−δ ′ ( p − δ )| N )(τ p−δ ′ ( p − δ )|n−N ). By the same arguments as in + the proof of Lemma 2.2, if δ is positive and suitably small, then τ p−δ ( p − δ)| N = τ p+ ( p)| N , and as above, utilising both the left-continuity and monotonicity of x ↦→ τx− (x) at p, guaranteed − by Lemma 2.1, for δ > 0 suitably small, τ p−δ ( p − δ)|n−N = τ p− ( p)|n−N . This in tandem with (3.5) completes the proof. □
The previous result is closely related to the property known as matching, which has been extensively studied, see for instance [7,8] and reference therein. Lemma 3.3. If ε > 0 and (γ − ε, γ + ε) ⊆ (1, 2), then there exists n ∈ N such that for the δ = δ(n) guaranteed by Lemma 3.2, we have |ξ p (x) − ξ p−δ′ (x)| < ε + x −N |ξ p (x)|, for all x ∈ [u, 2] and δ ′ ∈ (0, δ), where, as in Section 3.1, we set u = (1 + c)/2. Proof. Since β is periodic with period N , for all x ∈ (1, 2], N −1 ∞ ∑ ∑ 1 −k β x − αk x −k . (3.6) k −N 1 − x k=0 k=0 k=0 k=0 ∑ −k Let v ∈ N be fixed. For x ∈ (1, 2) set η1,v (x) := ∞ and observe that k=N (v−1) (βk − αk )x
ξ p (x) =
ξ p (x) =
∞ ∑
βk x −k −
∞ ∑
αk x −k =
N (v−1)−1 N −1 ∑ 1 − x −N v ∑ −k β x − αk x −k + η1,v (x). k −N 1−x k=0 k=0
(3.7)
Note, η1,v (x) is bounded by the tail of a geometric series, namely we have that |η1,v (x)| ≤ x −N (v−1) /(1 − x −1 ). In Lemma 3.2, letting n ≥ N v, there exists δ = δ(N v) > 0 so that, for all δ ′ ∈ (0, δ), − ′ τ p−δ ′ ( p − δ )| N v−1 = α0 · · · α N v−2 + τ p−δ ′(p
and
− δ )| N v−1 = β0 · · · β N −1 α0 · · · α N (v−1)−1 . ′
From this, we obtain an expansion of ξ p−δ′ (x) similar to (3.7), namely ξ p−δ′ (x) =
N −1 ∑ k=0
βk x −k −
N∑ v−1 k=0
αk x −k +
N∑ v−1
αk−N x −k + η2,v (x, δ ′ ).
(3.8)
k=N
Please cite this article as: Z. Cooperband, E.P.J. Pearse, B. Quackenbush et al., Continuity of entropy for Lorenz maps, Indagationes Mathematicae (2019), https://doi.org/10.1016/j.indag.2019.10.002.
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Here, η2,v (·, δ ′ ) consists of remaining terms in the expansion of ξ p−δ′ (x). Note the remainder term η2,v (·, δ ′ ) is bounded by the tail of a geometric series, namely |η2,v (x)| ≤ x −N v /(1 − x −1 ). Combining (3.6) and (3.8), we have N −1 N∑ v−1 x −N ∑ −k βk x − η3,v (x) − αk−N x −k − η2,v (x, δ ′ ), (3.9) ξ p (x)−ξ (x) = 1 − x −N k=0 k=N ∑ −k ′ where η3,v (x) := ∞ k=N v αk x . As with η1,v and η2,v (·, δ ), observe that η3,v is bounded by −N v the tail of a geometric series, namely |η3,v (x)| ≤ x /(1 − x −1 ). Reindexing the second series in (3.7) and rearranging yields ) ( N −1 N∑ v−1 1 − x −N v ∑ −k −k −N βk x − ξ p (x) + η1,v (x) . αk−N x = x 1 − x −N k=0 k=N p−δ ′
Substituting this into (3.9) gives, for δ ′ ∈ (0, δ), ξ p (x) − ξ p−δ′ (x) =
N −1 x −N v x −N ∑ βk x −k − η3,v (x) + x −N ξ p (x) − x −N η1,v (x) − η2,v (x, δ ′ ). 1 − x −N k=0
(3.10)
Taking absolute values and using for η1,v , η2,v (·, δ ′ ) and η3,v gives, for a ⏐ ⏐ the bounds obtained −N ⏐ ⏐ suitable constant K > 0, that ξ p (x) − ξ p−δ′ (x) ≤ K x v + x −N |ξ p (x)|, for all δ ′ ∈ (0, δ) and x ∈ [u, 2], where we note that δ is dependent on N v. Since this latter statement holds for all v, the required result follows. □ Proof of Theorem 1.2 (Left-Continuity with β Periodic). The proof proceeds as in the proof of left-continuity of Theorem 1.2 for β non-periodic, with the following modification. Instead of choosing δ = δ(n) in accordance with Lemma 3.1, we use Lemmas 3.2 and 3.3 to choose δ = δ(n); we replace (3.3) by |ξ p (γ ± ε)| ≥
1 2u −n 1 − u −N 1 − u −1
and (3.4) by 2u −n + u −N |ξ p (γ ± ε)| ≤ |ξ p (γ ± ε)|; 1 − u −1 the remainder of the proof follows identically. □ |ξ p−δ′ (γ ± ε) − ξ p (γ ± ε)| <
3.3. Right continuity In Section 3.1 and 3.2, the point p was shifted to the left by subtracting some δ > 0 from p. By adding δ > 0 to p instead of subtracting it, right continuity can be shown by repeating Section 3.1 and 3.2 with the following substitutions: swap τ p+ with τ p− , T p+ with T p− , and p − δ with p + δ. Then, proceeding by cases (whether or not α is periodic). Therefore, it follows that x ↦→ h(Tx ) is continuous at every point p ∈ (a, b). This concludes the proof of Theorem 1.2. 4. Monotonicity of topological entropy An affine Lorenz map T p is a Lorenz map where the branch functions f 0 and f 1 are affine, namely f 0 (x) = b0 x and f 1 (x) = 1 − b1 + b1 x where 0 < b0 ≤ p ≤ b1 , b0 , b1 > 1 and Please cite this article as: Z. Cooperband, E.P.J. Pearse, B. Quackenbush et al., Continuity of entropy for Lorenz maps, Indagationes Mathematicae (2019), https://doi.org/10.1016/j.indag.2019.10.002.
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Fig. 3.2. Numerical approximation of the topological entropy of the affine Lorenz map T p , with first branch f 0 (x) := 1.1x and second branch f 1 (x) := 1.9x − 0.9, using the algorithm developed in [22] – truncation term: n = 500; tolerance: ε = 10−7 .
Fig. 3.3. Agreement of separate numerical methods to compute topological entropy on the highlighted nonmonotonic feature of Fig. 3.2. The magenta curve uses the algorithm developed in [22] and the blue curve computes the lap number of (T p± )50 . (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
b0 +b1 > b0 b1 . In the following we take a first step to address Milnor’s monotonicity conjecture in the setting of affine Lorenz maps: that is to say, in our setting, given h 0 ∈ (1, 2), the set of p such that h(T p ) = ln(h 0 ) is either empty or connected. Indeed, numerical experiments show that there exist affine Lorenz maps T p± (x) where x ↦→ h(Tx ) is non-monotonic and non-constant. This phenomenon is demonstrated using the algorithm developed in [22] and is corroborated by a second algorithm that gives an approximation of topological entropy by computing lap numbers, see for instance [14]. Let (T p ) p∈[9/19,10/11] denote the family affine Lorenz maps with branch functions f 0 and f 1 given by f 0 (x) := 1.1x and f 1 (x) := 1.9x − 0.9. The graph of the map p ↦→ h(T p ) is shown in Fig. 3.2; here h(T p ) has been computed by using the algorithm given in [22] with Please cite this article as: Z. Cooperband, E.P.J. Pearse, B. Quackenbush et al., Continuity of entropy for Lorenz maps, Indagationes Mathematicae (2019), https://doi.org/10.1016/j.indag.2019.10.002.
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truncation term n = 500 and tolerance ε = 10−7 . We see that there are many instances of nonmonotonicity in the plot that exceed the algorithm convergence tolerance. Fig. 3.2 captures a significant non-monotonic feature. Indeed, there are many non-monotonic sections of the graph that exceed the algorithm’s error tolerance. References [1] R.L. Adler, A.G. Konheim, M.H. McAndrew, Topological entropy, Trans. Amer. Math. Soc. 114 (1965) 309–319. [2] J. Awrejcewicz, C.-H. Lamarque, Bifurcation and chaos in nonsmooth mechanical systems, in: World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises, vol. 45, World Scientific Publishing Co., 2003. [3] S. Banerjee, G.C. Verfghese, Nonlinear Phenomena in Power Electronics: Attractors, Bifurcations, Chaos and Nonlinear Control, Wiley-IEEE Press, 2001. [4] M. Barnsley, B. Harding, A. Vince, The entropy of a special overlapping dynamical system, Ergodic Theory Dynam. Systems 34 (2) (2014) 483–500. [5] M. Barnsley, N. Mihalache, Symmetric itinerary sets, 2011, Pre-print: arXiv:1110.2817. [6] R. Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, in: Lecture Notes in Mathematics, vol. 470, Springer-Verlag, Berlin-New York, 1975. [7] H. Bruin, C. Carminati, C. Kalle, Matching for generalised β-transformations, Indag. Math. 28 (1) (2017) 55–73. [8] H. Bruin, C. Carminati, S. Marmi, A. Profeti, Matching in a family of piecewise affine maps, Nonlinearity 32 (1) (2019). [9] H. Bruin, S. van Strien, Monotonicity of entropy for real multimodal maps, J. Amer. Math. Soc. 28 (1) (2015) 1–61. [10] I. Daubechies, R. DeVore, C.S. Gunturk, V.A. Vaishampayan, Beta expansions: a new approach to digitally corrected a/d conversion, in: IEEE International Symposium on Circuits and Systems. Proceedings (Cat. No. 02CH37353), vol. 2, 2002, pp. II–784–II–787. [11] P. Glendinning, T. Hall, Zeros of the kneading invariant and topological entropy for Lorenz maps, Nonlinearity 9 (4) (1996) 999–1014. [12] K. Keller, T. Mangold, I. Stolz, J. Werner, Permutation entropy: New ideas and challenges, Entropy 19 (3) (2017). [13] D. Lind, B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, 1995. [14] J. Milnor, C. Tresser, On entropy and monotonicity for real cubic maps, Comm. Math. Phys. 209 (1) (2000) 123–178. [15] M. Misiurewicz, W. Szlenk, Entropy of piecewise monotone mappings, Studia Math. 67 (1) (1980) 45–63. [16] W. Parry, On the β-expansions of real numbers, Acta Math. Hungar. 11 (1960) 401–416. [17] W. Parry, Representations for real numbers, Acta Math. Hungar. 15 (1964) 95–105. [18] W. Parry, Symbolic dynamics and transformations of the unit interval, Trans. Amer. Math. Soc. 122 (1966) 368–378. [19] P. Raith, Continuity of the entropy for monotonic mod one transformations, Acta Math. Hungar. 77 (3) (1997) 247–262. [20] P. Raith, On the continuity of the pressure for monotonic mod one transformations, Comment. Math. Univ. Carolin. 41 (1) (2000) 61–78. [21] A. Rényi, Representations for real numbers and their ergodic properties, Acta Math. Hungar. 8 (1957) 477–493. [22] T. Samuel, N. Snigireva, A. Vince, Embedding the symbolic dynamics of Lorenz maps, Math. Proc. Cambridge Philos. Soc. 156 (3) (2014) 505–519. [23] Z. Zhusubaliyev, E. Mosekilde, Bifurcations and chaos in piecewise-smooth dynamical systems, in: World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises, vol. 44, World Scientific Publishing Co., 2003.
Please cite this article as: Z. Cooperband, E.P.J. Pearse, B. Quackenbush et al., Continuity of entropy for Lorenz maps, Indagationes Mathematicae (2019), https://doi.org/10.1016/j.indag.2019.10.002.
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Continuity of entropy for Lorenz maps✩ Z. Cooperbanda , E.P.J. Pearsea , B. Quackenbusha , J. Rowleya , T. Samuelb ,∗, M. Westc a
Mathematics Department, California Polytechnic State University, CA, USA b School of Mathematics, University of Birmingham, UK c Department of Mathematics, University of California: Irvine, CA, USA
Received 15 February 2019; received in revised form 4 September 2019; accepted 28 October 2019 Communicated by S. van Strien
Abstract Let (T p ) p denote a one parameter family of expanding interval maps with two increasing and continuous branches and indexed by their point of discontinuity. Using the pressure formula from thermodynamics, P. Raith (2000) showed that the topological entropy h(T p ) of T p varies continuously as a function of p. Here we provide a new and alternative proof of this result based on Milnor–Thurston kneading theory, as well as some observations on the monotonicity of p ↦→ h(T p ). c 2019 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved. ⃝ Keywords: Kneading sequences; Lorenz maps; Topological entropy; Matching
1. Introduction and main results Since the pioneering work of R´enyi [21] and Parry [16,17], an increasing amount of attention has been paid to interval maps. Their study has provided solutions to practical problems within biology, engineering, information theory and physics. Applications appear in analogue to digital conversion [10], analysis of electroencephalography (EEG) data [12], data storage [13], electronic circuits [3], mechanical systems with impacts and friction [2] and relay systems [23]. ✩ Acknowledgements. The work presented here was carried out with some of our students, notably the first, third, fourth and sixth authors as well E. Esquivel and E. Mihanovich. We are extremely grateful to the Bill and Linda Frost Fund which supported this undergraduate research project. ∗ Corresponding author. E-mail addresses: [email protected] (Z. Cooperband), [email protected] (E.P.J. Pearse), [email protected] (B. Quackenbush), [email protected] (J. Rowley), [email protected] (T. Samuel), [email protected] (M. West).
https://doi.org/10.1016/j.indag.2019.10.002 c 2019 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved. 0019-3577/⃝
Please cite this article as: Z. Cooperband, E.P.J. Pearse, B. Quackenbush et al., Continuity of entropy for Lorenz maps, Indagationes Mathematicae (2019), https://doi.org/10.1016/j.indag.2019.10.002.
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Fig. 1.1. An example of a Lorenz map (right) generated by branch functions f 0 and f 1 (left).
The concept of topological entropy, now ubiquitous in the study of dynamical systems, was introduced in [1] as a measure of the complexity of a dynamical system and is an invariant under a continuous change of coordinates, called topological conjugation. Bowen [6] gave an equivalent definition for a continuous map of a (not necessarily compact) metric space. For our purposes the following formulation, given by Misiurewicz and Szlenk in [15] and consistent with the definition given in [1], serves as a definition of the topological entropy. Let T be a piecewise monotonic interval map, such as a Lorenz map (see Definition 1.1 and Fig. 1.1), the topological entropy h(T ) of T is defined by 1 ln(Var(T n )), (1.1) n where Var( f ) denotes the total variation of the function f and T n denotes the n-fold composition of T with itself. Here we are concerned with how the topological entropy of a Lorenz map changes as we perturb the point of discontinuity. Thus the perturbations being studied here are not additions of C 1 -functions nor are they close in the R0 topology considered in [19]. The simplest example of a Lorenz map is a (normalised) β-transformation, and the topological entropy of such a transformation is equal to ln(β); this was first shown in [18]. However, for a general Lorenz map, the question of determining the topological entropy is more delicate, see for instance [4,11]. Before stating our main results, let us define what is meant by a Lorenz map. Note, in our definition, the branch functions need not be C 1 . h(T ) := lim
n→∞
Definition 1.1. Let 0 < a ≤ p ≤ b < 1. An upper, or lower, Lorenz map is a map T + : [0, 1] → [0, 1], respectively T − : [0, 1] → [0, 1], of the form { { f 0 (x) if 0 ≤ x < p, f 0 (x) if 0 ≤ x ≤ p, + − T (x) := respectively T (x) := f 1 (x) if p ≤ x ≤ 1, f 1 (x) if p < x ≤ 1, where f 0 and f 1 , called the branch functions, satisfy the following conditions. (i) The functions f 0 : [0, b] → [0, 1] and f 1 : [a, 1] → [0, 1] are continuous, strictly increasing and surjective. (ii) There exist constants C, c > 1 with C −1 |x − y| ≤ | f i−1 (x) − f i−1 (y)| ≤ c−1 |x − y| for i ∈ {0, 1} and x ∈ [0, 1]. The region [a, b] is often referred to as the switch region. Please cite this article as: Z. Cooperband, E.P.J. Pearse, B. Quackenbush et al., Continuity of entropy for Lorenz maps, Indagationes Mathematicae (2019), https://doi.org/10.1016/j.indag.2019.10.002.
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When we wish to emphasise the point of discontinuity, we write T p± for T ± . Further, by definition, we have h(T p+ ) = h(T p− ), and hence, for ease of notation, we let h(T p ) denote the common value. Note, a direct consequence of (1.1) is that h(T p ) ≥ ln(c).
(1.2)
In [20], Raith proved the following result. His method of proof uses the pressure formula from thermodynamics as developed by Bowen and Ruelle. Here we give a new alternative proof which utilises Milnor’s and Thurston’s kneading theory and has a more combinatorial flavour. Theorem 1.2.
For a fixed pair of branch functions, p ↦→ h(T p ) is continuous.
With this a hand, a natural and interesting question which arises is how regular is the map p ↦→ h(T p ), namely is it Lipschitz or H¨older continuous? Initial computer simulations indicate that large oscillations can occur on relatively small scales, as depicted in Figs. 3.2 and 3.3, even when the branch functions are affine. Here we would also like to mention that for a parameter class of piecewise affine maps with a single discontinuity and one increasing and one decreasing branch it has been shown that topological entropy displays a semi-regular behaviour. In particular, it is smooth on an open and dense set, see [8]. This feature is due to a property known as matching, see [7,8] and references therein. A similar property also plays a key rˆole in our proof of Theorem 1.2, Lemma 3.2. Outline In Section 2 we provide necessary definitions and preliminary results, and Section 3 is dedicated to our new and alternative proof of Theorem 1.2. We conclude this article with examples which demonstrate, for a fixed pair of branch functions, p ↦→ h(T p ) is not necessarily monotonic even if the branch functions are linear, compare with [9] and reference therein. 2. Preliminaries Throughout we use the convention that ± means either + or − and when we write T p± , we require that both T p+ and T p− are defined using the same branch functions. The set of all infinite words over the alphabet {0, 1} is denoted by Ω and is equipped with the product topology where we place the discrete topology on the set {0, 1} as well as the lexicographic ordering. For n ∈ ⋃ N, define Ωn to be the set of finite words over the alphabet {0, 1} of length n, and set Ω ∗ := n∈N0 Ωn , where by convention Ω0 is the set containing only the empty word ∅. For ω = ω0 · · · ωk and v = v0 · · · vn ∈ Ω ∗ , we set ωv := ω0 · · · ωk v0 · · · vn , that is the concatenation of ω and v, and let v := vvv · · · . Further, for k ∈ N we let v k denote the kth fold concatenation of v with itself. The length of v ∈ Ω ∗ is denoted by |v| with |∅| = 0 and, for a natural number k ≤ |v|, we set v|k := v0 · · · vk−1 . We use the same notations when v is an infinite word. The continuous map S : Ω → Ω defined by S(ω0 ω1 · · · ) := ω1 ω2 · · · is called the left-shift. We also allow for S to act on finite words as follows. For k ∈ N0 and v = v0 · · · vk ∈ Ω ∗ , we set S(v) = v1 · · · vk , if k ≥ 1 and S(v) = ∅ otherwise. The upper and lower itinerary maps τ p± : [0, 1] → Ω encode the orbit of a point x ∈ [0, 1] under T p± and are defined by τ p+ (x) = ω0 ω1 · · · and τ p− (x) = v0 v1 · · · where { { 0 if (T p+ )k (x) < p, 0 if (T p− )k (x) ≤ p, ωk := and v := k 1 if (T p+ )k (x) ≥ p, 1 if (T p− )k (x) > p. Please cite this article as: Z. Cooperband, E.P.J. Pearse, B. Quackenbush et al., Continuity of entropy for Lorenz maps, Indagationes Mathematicae (2019), https://doi.org/10.1016/j.indag.2019.10.002.
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The infinite words α := τ p− ( p) and β := τ p+ ( p) are called the kneading sequences of T p± , and play an important rˆole in what is to follow. We say that τ p± ( p) is periodic if there exists n ∈ N such that (T p± )n ( p) = p, and the period of τ p± ( p) is the smallest n ∈ N for which this holds. If τ p± ( p) are periodic, then there exist v, ω ∈ Ω ∗ such that τ p+ ( p) = v and τ p− ( p) = ω. Lemma 2.1 ([4,5]). The maps x ↦→ τx± (x) are strictly increasing. Additionally, x ↦→ τx+ (x) is right-continuous and x ↦→ τx− (x) is left-continuous. The following lemma extends this result. Lemma 2.2. If p ̸= a and β is non-periodic, then x ↦→ τx+ (x) is continuous at p and if p ̸= b and α is non-periodic, then x ↦→ τx− (x) is continuous at p. Proof. We prove the first statement, as the proof of the second statement is analogous. By definition and Lemma 2.1, it is sufficient to show, for a fixed N ∈ N, that there exists + ′ + ′ δ ∈ (0, p − a) such that τ p−δ ′ ( p − δ )| N = τ p ( p)| N for all δ ∈ (0, δ). This means we require ′ δ > 0 so that for n ∈ {0, 1, . . . , N − 1} and δ ∈ (0, δ) either + n ′ ′ + n (T p−δ ′ ) ( p − δ ) < p − δ and (T p ) ( p) < p
or + n (T p−δ ′) (p
(2.1)
− δ ) > p − δ and ′
′
(T p+ )n ( p)
> p.
To this end, let c and C be as in Definition 1.1 and choose δ ∈ (0, p − a) such that } { |(T p+ )k ( p) − p| : k ∈ {1, 2, . . . , N − 1} . 0 < δ < min Ck
(2.2)
Note, since p ̸= a and since β is not periodic, the value on the right hand side of (2.2) is positive. We claim, for all n ∈ {0, 1, . . . , N − 1} and δ ′ ∈ (0, δ), that + n ′ ′ n δ ′ cn ≤ (T p+ )n ( p) − (T p−δ ′) (p − δ ) ≤ δ C .
(2.3)
The case, n = 0, is immediate. Assume (2.3) holds for some n ∈ {0, 1, . . . , N − 1}. + n ′ + n ′ n If (T p+ )n ( p) < p, then (2.3) implies (T p−δ < p − δ ′ . If ′ ) ( p − δ ) ≤ (T p ) ( p) − δ c + n + n ′ n (T p ) ( p) > p, then (2.2) implies (T p ) ( p) − p > δ C ; combining this with our hypothesis + n ′ + n ′ n ′ yields (T p−δ ′ ) ( p − δ ) > (T p ) ( p) − δ C > p > p − δ . Hence, we have (2.3) for n + 1. To complete the proof, notice (2.3) implies (2.1). □ In our proof of Theorem 1.2, we use the following Laurent series which can be thought of as a generating function of the kneading sequences α = τ p− ( p) = α1 α2 · · · and β = τ p+ ( p) = β1 β2 · · · and is related to the function P(·, ·) defined in [11] by ξ p (z) = 2P(z −1 , z −1 ). For z ∈ C \ {0}, set ξ p (z) :=
∞ ∑ (βk − αk )z −k .
(2.4)
k=0
Observe that the interval (1, 2) belongs to the domain of convergence of ξ p . Further, we have the following result, which identifies the maximal zero of ξ p and the value γ = γ p := exp(h(T p )). Please cite this article as: Z. Cooperband, E.P.J. Pearse, B. Quackenbush et al., Continuity of entropy for Lorenz maps, Indagationes Mathematicae (2019), https://doi.org/10.1016/j.indag.2019.10.002.
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Fig. 3.1. The graph of ξ p together with a neighbourhood in which the graph of ξ p−δ belongs.
Theorem 2.3 ([4,11]). The topological entropy of T p± is equal to ln(r ), where r is the maximal positive real zero of ξ p . Additionally, if r is not simple, then it is the only zero of ξ p in the interval (1, 2] and has multiplicity two. 3. Proof of Theorem 1.2 In Sections 3.1 and 3.2 for a fixed pair of branch functions we prove that the map p ↦→ h(T p ) is left-continuous; right-continuity follows by an identical argument, see Section 3.3 for further details. The proof of left-continuity is subdivided into two sub-cases: when the kneading sequences of T p± are not periodic, and when they are periodic. For each sub-case we use the same approach. (i) Fix p ∈ (a, b) and show the existence of an ε > 0 with (γ − ε, γ + ε) ⊆ (1, 2), so that τ ± are continuous on (γ − ε, γ + ε). (ii) Show there exists δ > 0 such that ξ p−δ (x) has a zero r ∈ (γ − ε, γ + ε). (iii) Show there are no zeros larger than r . With this at hand, Theorem 2.3 allows us to conclude that ln(r ) = h(T p−δ ). Note, in Step (ii) we must take into account the multiplicity of γ ; see Fig. 3.1. By Theorem 2.3, if γ is simple, one can appeal to the intermediate value theorem, but more care is required in the case when γ is not simple. 3.1. Left continuity: β non-periodic Fix ε > 0 with (γ −ε, γ +ε) ⊆ (1, 2). By Lemma 2.2, the assumption that β is non-periodic ensures x ↦→ τx± (x) are left-continuous at p; this completes the proof of (i). To prove (ii) and (iii), that is there exists δ > 0 such that ξ p−δ (x) has a maximal zero r ∈ (γ − ε, γ + ε), we replace the infinite sum ξ p (x) with a partial sum (a polynomial) and approximate γ p−δ as the root of this polynomial. To this end, for n ∈ N, set R p,n (x) :=
∞ ∑
(βk − αk )x −k
k=n
Lemma 3.1.
so that
ξ p (x) =
n−1 ∑ (βk − αk )x −k + R p,n (x). k=0
For n ∈ N, there exists δ = δ(n) > 0 so that, for δ ′ ∈ (0, δ) and x ∈ (1, 2),
|ξ p−δ′ (x) − ξ p (x)| ≤
2x −n . 1 − x −1
(3.1)
Please cite this article as: Z. Cooperband, E.P.J. Pearse, B. Quackenbush et al., Continuity of entropy for Lorenz maps, Indagationes Mathematicae (2019), https://doi.org/10.1016/j.indag.2019.10.002.
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Proof. For m ∈ N, we have βm − αm ∈ {−1, 0, 1}, whence, for n ∈ N and x ∈ (1, 2), ∞ ∑ x −n . |R p,n (x)| ≤ x −k = 1 − x −1 k=n
(3.2)
Let n ∈ N be fixed. Since the maps x ↦→ τx± (x) are both left-continuous at p, there exists δ > 0 ± ′ ± such that, if δ ′ ∈ (0, δ), then τ p−δ ′ ( p − δ )|n = τ p ( p)|n . As (3.2) also holds for R p−δ ′ ,n (x), we have established (3.1). □ Proof of Theorem 1.2 (Left-Continuity with β Non-Periodic). Note that γ is an isolated zero of ξ p . Indeed, in its domain of convergence, the function ξ p is holomorphic. Consequently, the existence of a sequence of zeros of ξ p converging to γ would imply that ξ p is the constant zero function; a contradiction. Let ε > 0 be fixed such that (γ − ε, γ + ε) ⊆ (1, 2) and such that ξ p has a single root in this interval, namely at γ . Let c > 1 be as in Definition 1.1 and fix u ∈ (1, c). The specific value of u is not important, so for convenience set u = (1 + c)/2, and by (1.2) and Theorem 2.3 the maximal real zero of ξq is greater than u for all q ∈ [a, b]. Assume that γ is simple. Let n ∈ N be such that 2u −n (3.3) 1 − u −1 and let δ be chosen in accordance with Lemma 3.1. In that case, for all δ ′ ∈ (0, δ), 2u −n (3.4) |ξ p−δ′ (γ ± ε) − ξ p (γ ± ε)| < 1 − u −1 which ensures sgn(ξ p−δ′ (γ ± ε)) = sgn(ξ p (γ ± ε)), respectively. This together with the fact that ξ p is smooth and has a single root in (γ − ε, γ + ε) and an application of the intermediate value theorem yields that ξ p−δ′ has a zero in (γ − ε, γ + ε) for all δ ′ ∈ (0, δ); see Fig. 3.1. Assume that γ has multiplicity two. { By Theorem 2.3 and since ξ p (2) ≥ 0, } we have ξ p (x) > 0 for all x ∈ [u, 2] \ {γ }. Let ρ := inf ξ p (x)/2 : x ∈ [u, γ − ϵ] ∪ [γ + ϵ, 2] , let n ∈ N be such that 2u −n /(1 − u −1 ) < ρ, and let δ be chosen in accordance with Lemma 3.1. In that case, for all δ ′ ∈ (0, δ) and x ∈ [u, γ − ϵ] ∪ [γ + ϵ, 2], 2u −n ξ p−δ′ (x) ≥ ξ p (x) − > ξ p (x) − ρ > 0 1 − u −1 which ensures that ξ p−δ′ (x) has no zeros in [u, γ − ϵ] ∪ [γ + ϵ, 2]. Therefore, by Theorem 2.3 and Eq. (1.2), namely that the maximal real zero of ξq belongs to (u, 2] for q ∈ [a, b], we have ξ p−δ′ necessarily has a zero in the interval (γ − ε, γ + ε) Therefore, regardless of the multiplicity of γ , it is necessarily the case that ξ p− δ′ has a zero in (γ − ε, γ + ε) for all δ ′ ∈ (0, δ), whence Theorem 2.3 implies that |h(T p ) − h(T p−δ′ )| ≤ ε for all δ ′ ∈ (0, δ), as required. □ |ξ p (γ ± ε)| ≥
3.2. Left continuity: β periodic Throughout this section, we assume β is periodic with period N , for some N ∈ N. Note, in this case p ̸= a, since when p = a we have that β = 10 which is not periodic. Lemma 3.2. For n ∈ N with n ≥ N , there exists δ = δ(n) > 0 so that, for δ ′ ∈ (0, δ), the + ′ concatenation of τ p+ ( p)| N and τ p− ( p)|n−N equals τ p−δ ′ ( p − δ )|n , namely + ′ + − τ p−δ ′ ( p − δ )|n = (τ p ( p)| N )(τ p ( p)|n−N ). Please cite this article as: Z. Cooperband, E.P.J. Pearse, B. Quackenbush et al., Continuity of entropy for Lorenz maps, Indagationes Mathematicae (2019), https://doi.org/10.1016/j.indag.2019.10.002.
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Proof. Let n ≥ N denote a fixed integer. By construction, there exists η > 0 such that, + j ′ ′ if η′ ∈ (0, η), then (T p−η ′ ) ( p − η ) ̸ = p − η for all j ∈ {1, 2, . . . , N + n − 1}. Using the same arguments as in the proof of Lemma 2.2, we may choose η small enough so that, + j ′ ′ j in addition to this, if η′ ∈ (0, η), then η′ c j ≤ (T p+ ) j ( p) − (T p−η for ′) (p − η ) ≤ η C all j ∈ {0, 1, . . . , N }; here c and C are as in Definition 1.1. If j = N , then this yields + N ′ ′ N η′ c N < p − (T p−η for η′ ∈ (0, η). Since c > 1, this implies that ′) (p − η ) < η C + ′ N ′ ′ N 0 < ( p − η ) − (T p−η′ ) ( p − η ) < η (C − 1). Utilising both the left-continuity and monotonicity of x ↦→ τx− (x) at p, given in Lemma 2.1, there exists λ ∈ (0, 1) so that, for q ∈ [a, b] with 0 < p − q < λ, we have τ p− ( p)|n = τq− (q)|n . Setting δ := min{λC −N /2, η}, if δ ′ ∈ (0, δ), then + N ′ 0 < ( p − δ ′ ) − (T p−δ ′ ) ( p − δ ) < λ/2,
which implies − − + ′ N ′ τ p−δ ′ ( p − δ )|n = τ p−δ ′ ((T p−δ ′ ) ( p − δ ))|n + + N ′ N + ′ = τ p−δ ′ ((T p−δ ′ ) ( p − δ ))|n = S (τ p−δ ′ ( p − δ ))|n .
(3.5)
+ + − ′ ′ ′ Therefore, τ p−δ ′ ( p − δ )|n = (τ p−δ ′ ( p − δ )| N )(τ p−δ ′ ( p − δ )|n−N ). By the same arguments as in + the proof of Lemma 2.2, if δ is positive and suitably small, then τ p−δ ( p − δ)| N = τ p+ ( p)| N , and as above, utilising both the left-continuity and monotonicity of x ↦→ τx− (x) at p, guaranteed − by Lemma 2.1, for δ > 0 suitably small, τ p−δ ( p − δ)|n−N = τ p− ( p)|n−N . This in tandem with (3.5) completes the proof. □
The previous result is closely related to the property known as matching, which has been extensively studied, see for instance [7,8] and reference therein. Lemma 3.3. If ε > 0 and (γ − ε, γ + ε) ⊆ (1, 2), then there exists n ∈ N such that for the δ = δ(n) guaranteed by Lemma 3.2, we have |ξ p (x) − ξ p−δ′ (x)| < ε + x −N |ξ p (x)|, for all x ∈ [u, 2] and δ ′ ∈ (0, δ), where, as in Section 3.1, we set u = (1 + c)/2. Proof. Since β is periodic with period N , for all x ∈ (1, 2], N −1 ∞ ∑ ∑ 1 −k β x − αk x −k . (3.6) k −N 1 − x k=0 k=0 k=0 k=0 ∑ −k Let v ∈ N be fixed. For x ∈ (1, 2) set η1,v (x) := ∞ and observe that k=N (v−1) (βk − αk )x
ξ p (x) =
ξ p (x) =
∞ ∑
βk x −k −
∞ ∑
αk x −k =
N (v−1)−1 N −1 ∑ 1 − x −N v ∑ −k β x − αk x −k + η1,v (x). k −N 1−x k=0 k=0
(3.7)
Note, η1,v (x) is bounded by the tail of a geometric series, namely we have that |η1,v (x)| ≤ x −N (v−1) /(1 − x −1 ). In Lemma 3.2, letting n ≥ N v, there exists δ = δ(N v) > 0 so that, for all δ ′ ∈ (0, δ), − ′ τ p−δ ′ ( p − δ )| N v−1 = α0 · · · α N v−2 + τ p−δ ′(p
and
− δ )| N v−1 = β0 · · · β N −1 α0 · · · α N (v−1)−1 . ′
From this, we obtain an expansion of ξ p−δ′ (x) similar to (3.7), namely ξ p−δ′ (x) =
N −1 ∑ k=0
βk x −k −
N∑ v−1 k=0
αk x −k +
N∑ v−1
αk−N x −k + η2,v (x, δ ′ ).
(3.8)
k=N
Please cite this article as: Z. Cooperband, E.P.J. Pearse, B. Quackenbush et al., Continuity of entropy for Lorenz maps, Indagationes Mathematicae (2019), https://doi.org/10.1016/j.indag.2019.10.002.
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Here, η2,v (·, δ ′ ) consists of remaining terms in the expansion of ξ p−δ′ (x). Note the remainder term η2,v (·, δ ′ ) is bounded by the tail of a geometric series, namely |η2,v (x)| ≤ x −N v /(1 − x −1 ). Combining (3.6) and (3.8), we have N −1 N∑ v−1 x −N ∑ −k βk x − η3,v (x) − αk−N x −k − η2,v (x, δ ′ ), (3.9) ξ p (x)−ξ (x) = 1 − x −N k=0 k=N ∑ −k ′ where η3,v (x) := ∞ k=N v αk x . As with η1,v and η2,v (·, δ ), observe that η3,v is bounded by −N v the tail of a geometric series, namely |η3,v (x)| ≤ x /(1 − x −1 ). Reindexing the second series in (3.7) and rearranging yields ) ( N −1 N∑ v−1 1 − x −N v ∑ −k −k −N βk x − ξ p (x) + η1,v (x) . αk−N x = x 1 − x −N k=0 k=N p−δ ′
Substituting this into (3.9) gives, for δ ′ ∈ (0, δ), ξ p (x) − ξ p−δ′ (x) =
N −1 x −N v x −N ∑ βk x −k − η3,v (x) + x −N ξ p (x) − x −N η1,v (x) − η2,v (x, δ ′ ). 1 − x −N k=0
(3.10)
Taking absolute values and using for η1,v , η2,v (·, δ ′ ) and η3,v gives, for a ⏐ ⏐ the bounds obtained −N ⏐ ⏐ suitable constant K > 0, that ξ p (x) − ξ p−δ′ (x) ≤ K x v + x −N |ξ p (x)|, for all δ ′ ∈ (0, δ) and x ∈ [u, 2], where we note that δ is dependent on N v. Since this latter statement holds for all v, the required result follows. □ Proof of Theorem 1.2 (Left-Continuity with β Periodic). The proof proceeds as in the proof of left-continuity of Theorem 1.2 for β non-periodic, with the following modification. Instead of choosing δ = δ(n) in accordance with Lemma 3.1, we use Lemmas 3.2 and 3.3 to choose δ = δ(n); we replace (3.3) by |ξ p (γ ± ε)| ≥
1 2u −n 1 − u −N 1 − u −1
and (3.4) by 2u −n + u −N |ξ p (γ ± ε)| ≤ |ξ p (γ ± ε)|; 1 − u −1 the remainder of the proof follows identically. □ |ξ p−δ′ (γ ± ε) − ξ p (γ ± ε)| <
3.3. Right continuity In Section 3.1 and 3.2, the point p was shifted to the left by subtracting some δ > 0 from p. By adding δ > 0 to p instead of subtracting it, right continuity can be shown by repeating Section 3.1 and 3.2 with the following substitutions: swap τ p+ with τ p− , T p+ with T p− , and p − δ with p + δ. Then, proceeding by cases (whether or not α is periodic). Therefore, it follows that x ↦→ h(Tx ) is continuous at every point p ∈ (a, b). This concludes the proof of Theorem 1.2. 4. Monotonicity of topological entropy An affine Lorenz map T p is a Lorenz map where the branch functions f 0 and f 1 are affine, namely f 0 (x) = b0 x and f 1 (x) = 1 − b1 + b1 x where 0 < b0 ≤ p ≤ b1 , b0 , b1 > 1 and Please cite this article as: Z. Cooperband, E.P.J. Pearse, B. Quackenbush et al., Continuity of entropy for Lorenz maps, Indagationes Mathematicae (2019), https://doi.org/10.1016/j.indag.2019.10.002.
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Fig. 3.2. Numerical approximation of the topological entropy of the affine Lorenz map T p , with first branch f 0 (x) := 1.1x and second branch f 1 (x) := 1.9x − 0.9, using the algorithm developed in [22] – truncation term: n = 500; tolerance: ε = 10−7 .
Fig. 3.3. Agreement of separate numerical methods to compute topological entropy on the highlighted nonmonotonic feature of Fig. 3.2. The magenta curve uses the algorithm developed in [22] and the blue curve computes the lap number of (T p± )50 . (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
b0 +b1 > b0 b1 . In the following we take a first step to address Milnor’s monotonicity conjecture in the setting of affine Lorenz maps: that is to say, in our setting, given h 0 ∈ (1, 2), the set of p such that h(T p ) = ln(h 0 ) is either empty or connected. Indeed, numerical experiments show that there exist affine Lorenz maps T p± (x) where x ↦→ h(Tx ) is non-monotonic and non-constant. This phenomenon is demonstrated using the algorithm developed in [22] and is corroborated by a second algorithm that gives an approximation of topological entropy by computing lap numbers, see for instance [14]. Let (T p ) p∈[9/19,10/11] denote the family affine Lorenz maps with branch functions f 0 and f 1 given by f 0 (x) := 1.1x and f 1 (x) := 1.9x − 0.9. The graph of the map p ↦→ h(T p ) is shown in Fig. 3.2; here h(T p ) has been computed by using the algorithm given in [22] with Please cite this article as: Z. Cooperband, E.P.J. Pearse, B. Quackenbush et al., Continuity of entropy for Lorenz maps, Indagationes Mathematicae (2019), https://doi.org/10.1016/j.indag.2019.10.002.
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truncation term n = 500 and tolerance ε = 10−7 . We see that there are many instances of nonmonotonicity in the plot that exceed the algorithm convergence tolerance. Fig. 3.2 captures a significant non-monotonic feature. Indeed, there are many non-monotonic sections of the graph that exceed the algorithm’s error tolerance. References [1] R.L. Adler, A.G. Konheim, M.H. McAndrew, Topological entropy, Trans. Amer. Math. Soc. 114 (1965) 309–319. [2] J. Awrejcewicz, C.-H. Lamarque, Bifurcation and chaos in nonsmooth mechanical systems, in: World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises, vol. 45, World Scientific Publishing Co., 2003. [3] S. Banerjee, G.C. Verfghese, Nonlinear Phenomena in Power Electronics: Attractors, Bifurcations, Chaos and Nonlinear Control, Wiley-IEEE Press, 2001. [4] M. Barnsley, B. Harding, A. Vince, The entropy of a special overlapping dynamical system, Ergodic Theory Dynam. Systems 34 (2) (2014) 483–500. [5] M. Barnsley, N. Mihalache, Symmetric itinerary sets, 2011, Pre-print: arXiv:1110.2817. [6] R. Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, in: Lecture Notes in Mathematics, vol. 470, Springer-Verlag, Berlin-New York, 1975. [7] H. Bruin, C. Carminati, C. Kalle, Matching for generalised β-transformations, Indag. Math. 28 (1) (2017) 55–73. [8] H. Bruin, C. Carminati, S. Marmi, A. Profeti, Matching in a family of piecewise affine maps, Nonlinearity 32 (1) (2019). [9] H. Bruin, S. van Strien, Monotonicity of entropy for real multimodal maps, J. Amer. Math. Soc. 28 (1) (2015) 1–61. [10] I. Daubechies, R. DeVore, C.S. Gunturk, V.A. Vaishampayan, Beta expansions: a new approach to digitally corrected a/d conversion, in: IEEE International Symposium on Circuits and Systems. Proceedings (Cat. No. 02CH37353), vol. 2, 2002, pp. II–784–II–787. [11] P. Glendinning, T. Hall, Zeros of the kneading invariant and topological entropy for Lorenz maps, Nonlinearity 9 (4) (1996) 999–1014. [12] K. Keller, T. Mangold, I. Stolz, J. Werner, Permutation entropy: New ideas and challenges, Entropy 19 (3) (2017). [13] D. Lind, B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, 1995. [14] J. Milnor, C. Tresser, On entropy and monotonicity for real cubic maps, Comm. Math. Phys. 209 (1) (2000) 123–178. [15] M. Misiurewicz, W. Szlenk, Entropy of piecewise monotone mappings, Studia Math. 67 (1) (1980) 45–63. [16] W. Parry, On the β-expansions of real numbers, Acta Math. Hungar. 11 (1960) 401–416. [17] W. Parry, Representations for real numbers, Acta Math. Hungar. 15 (1964) 95–105. [18] W. Parry, Symbolic dynamics and transformations of the unit interval, Trans. Amer. Math. Soc. 122 (1966) 368–378. [19] P. Raith, Continuity of the entropy for monotonic mod one transformations, Acta Math. Hungar. 77 (3) (1997) 247–262. [20] P. Raith, On the continuity of the pressure for monotonic mod one transformations, Comment. Math. Univ. Carolin. 41 (1) (2000) 61–78. [21] A. Rényi, Representations for real numbers and their ergodic properties, Acta Math. Hungar. 8 (1957) 477–493. [22] T. Samuel, N. Snigireva, A. Vince, Embedding the symbolic dynamics of Lorenz maps, Math. Proc. Cambridge Philos. Soc. 156 (3) (2014) 505–519. [23] Z. Zhusubaliyev, E. Mosekilde, Bifurcations and chaos in piecewise-smooth dynamical systems, in: World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises, vol. 44, World Scientific Publishing Co., 2003.
Please cite this article as: Z. Cooperband, E.P.J. Pearse, B. Quackenbush et al., Continuity of entropy for Lorenz maps, Indagationes Mathematicae (2019), https://doi.org/10.1016/j.indag.2019.10.002.