Article Computing the Topological Entropy of Multimodal Maps via Min-Max Sequences

We derive an algorithm to recursively determine the lap number (minimal number of monotonicity segments) of the iterates of twice differentiable l-modal map, enabling to numerically calculate the topological entropy of these maps. The algorithm is obtained by the min-max sequences—symbolic sequences that encode qualitative information about all the local extrema of iterated maps.


Introduction
Entropy is a ubiquitous tool in physics and mathematics.It measures randomness in dynamical systems, uncertainty in information theory and disorder in statistical mechanics.
Topological entropy was introduced in 1965 by Adler, Konheim and McAndrew [1] as an invariant of topological conjugacy for maps of the interval.Along with the Lyapunov exponent, topological entropy is one of the preferred indicators for complexity in topological dynamics.The numerical computation of topological entropy has been and remains an active topic of research, as witnessed by a number of relevant publications in the last decades.
Let I be a compact interval [a, b] ⊂ R and f : I → I be a continuous piecewise monotone map.Such a map is called l-modal if f has precisely l turning points (i.e., points in (a, b) where f has a local extremum).Assume that f has local extrema at c 1 < ... < c l and that f is strictly monotone in each of the l + 1 intervals To such map one can assign a positive or negative shape which describes whether f is increasing or decreasing on its first lap I 1 .In proofs it is occasionally convenient to use the convention c 0 ≡ a and c l+1 ≡ b.Sometimes the additional condition f ({a, b}) ⊂ {a, b} is also required (see for instance [2]); in this case we speak of boundary-anchored maps.We shall also consider boundary-anchored maps below but only as a special case, since the general algorithm for the topological entropy then simplifies quite a bit.The itinerary of x ∈ I under f is the sequence i(x) = (i 0 (x), i 1 (x), ..., i n (x), ...) defined as follows: The itineraries of the critical points, are called the kneading sequences (or invariants) of f .Let h(f ) denote the topological entropy of an l-modal map f : I → I. Then [3,4], where Var(f n ) stands for the variation of f n , and n is shorthand for the lap number of f n (i.e., the number of maximal monotonicity segments of f n ).There are relations similar to (1) and (2), involving the number of fixed points of f n (i.e., the number of periodic points of period n), or the length of the graph of f n .The methods proposed in the literature to compute h(f ), use typically kneading sequences [5][6][7], approximating piecewise linear maps [8] and Markov maps [9], the Ruelle-Perron-Frobenius operator [10], or one of the expressions (1) and (2) [11,12].Their virtues and shortcomings are also discussed in the literature.For instance, some are meant only for unimodal maps [5,7] or bimodal maps [6].Others apply to not necessarily continuous piecewise monotone maps of the interval, however they are not efficient nor even accurate [8].
The method proposed here calculates the lap numbers n , n ≥ 1, and the topological entropy follows from (2).It applies to multimodal maps with or without boundary conditions.The main ingredient of this approach are the so-called min-max sequences-symbolic sequences that encode the coarse-grained information about the extrema of the maps f n , n ≥ 1.It generalizes an approach for unimodal, boundary anchored maps, introduced in [13,14], further developed in [15], and extended for boundary free maps in [16].
The method proposed here is conceptually simple, is direct, is geometrical and is computationally efficient, calculating the lap numbers in a recursive way.The structure of the algorithm is the same for all l-modal maps, independently of the value of l.Regarding computing speed, we shall not provide any sharper bound than the O(1/n) convergence rate derivable on general grounds [12].Nonetheless, numerical simulations confirm the excellent performance of the algorithm-except when h(f ) 0, in which case the convergence is slow.
The rest of the paper is organized as follows.In Section 2 we introduce the min-max sequences of a map f ∈ F l , where F l is the class of twice differentiable l-modal maps.This assumption simplifies the proofs but the results obtained in this paper apply to the class of continuous piecewise monotonous maps.In Section 3, we derive a number of technical lemmas, which are needed in the next two sections.Section 4 is devoted to clarify the connection between the min-max sequences of a map and the structure of its extrema, exploring the geometrical meaning of the min-max sequences.This connection leads in Section 5 to the main result of the paper, Theorem 5.3, which provides a recursive scheme for computing n (hence h(f )) with arbitrary precision.It turns out that the general scheme of Theorem 5.3 simplifies in some special cases, notably for boundary-anchored maps and for unimodal maps; these cases are separately discussed in Section 6.The paper concludes with the logical flow of the algorithm (Section 7), and a summary of numerical simulations with 2and 3-modal maps (Section 8).

Geometry of the Itineraries: The Min-Max Sequences for l-Modal Maps
Henceforth we consider the class F l of twice differentiable l-modal maps.Since the results we obtain in Section 5 for the calculation of lap numbers and topological entropy do not depend on the shape of f , we shall assume throughout that the shape of f is positive, that is, where I odd [resp.I even ] denotes any I k with k ∈ {1, ..., l + 1} odd [resp.even], and f (a), f (b) are meant to be the appropriate one-sided derivatives.The chain rule for derivation applied to the nth iterate of f , written f n (f 0 is the identity map), implies trivially which shows that c 1 , ..., c l are critical points of f n for every n ≥ 1.From (5) we conclude also the following.
Therefore, the critical points of f n with n ≥ 1 are the pre-images of the critical points c 1 ,..., c l up to order n − 1.
To locate the extrema of f n in I up to the precision set by the partition P, we introduce a new alphabet where m stands for "minimum", M stands for "maximum", and the superscript σ is the pertaining signature, i.e., if f n (x) is the minimum or maximum considered, then σ = σ(f n (x)).Correspondingly we say that Next we define l sequences ω i = (ω i n ) n≥1 ∈ M N , 1 ≤ i ≤ l, as follows: The sequences ω 1 , ..., ω l are called the min-max sequences of f ∈ F l , or MMSs for short.The geometric meaning of ω i n is clear: . By particularizing Lemma 2.2 to x = c 2k+1 , 0 ≤ k ≤ (l − 1)/2 , and x = c 2k , 1 ≤ k ≤ l/2 , we get the transition rules listed in Table 1.
Table 1.Consecutive symbols in the MMS follow the above transition rules.

Auxiliary Lemmas
As stated before, the generic structure of a signature is (+, ..., +, −, ..., −) or (+, ..., +, 0, −, ..., −) Therefore, when comparing component-wise two signatures, only three cases can happen: (i) all components coincide, (ii) they differ in a single component, or (iii) they differ in a number of consecutive components.Of course, case (ii) can be considered as a "degenerate" subcase of (iii), as we will do in the sequel.
Let f ∈ F l and set, In particular, S 1 = {c 1 , ..., c l }.According to Lemma 2.1, S n contains S 1 and its preimages up to order 2 that all these critical points are local maxima or minima, but not inflexion points.Furthermore, let ξ n [resp.η n ] be the leftmost [resp.rightmost] critical point of f n , i.e., Then, The geometrical interpretation of this lemma in the Cartesian plane (x, y) is clear.
, crosses transversally the "ith critical line" y = c i , and none of the other critical lines (if any) y = c k , k = i.If j 0 > 1, then this curve crosses transversally j 0 successive critical lines, namely, y = c i 0 up to y = c i 0 +j 0 −1 , and none of the remaining ones (if any).In (b) both f n (z n,1 )) and f n (z n,2 )) belong to the same interval I i ∈ P, so y = f n (x) does not cross any critical line when x ∈ (z n,1 , z n,2 ).

Proof. (a) Suppose c
] and the Mean Value Theorem, there exist exactly Therefore, ) is a maximum in the first case, and a minimum in the second.The statement about the relative positions of z n+1,j , 1 ≤ j ≤ j 0 is obvious from the geometrical interpretation.
(b) This assertion is straightforward.
Setting z n,1 = a, z n,2 = ξ n in Lemma 3.1, we conclude the following results.
Proof.(a) is a corollary of Lemma 3.1 (a).The first statement of (b) is a corollary of Lemma 3.1 (b).
As for (b1) and (b2), And setting z n,1 = η n , z n,2 = b in Lemma 3.1, we derive the following results in a way similar to Lemma 3.2.
The results for boundary-anchored maps are simpler.Since we are assuming that f ∈ F l has a positive shape, the boundary conditions of such a map read: f (a) = a for any l, and To prove the next two lemmas, the following weaker boundary conditions are sufficient, though: for n ≥ 1. Maps satisfying the confinement conditions (BC1) and (BC2) at the boundary, will be called quasi boundary-anchored maps for obvious reasons.Lemma 3.4.Let f ∈ F l be a quasi boundary-anchored map such that (H1) f (c 1 ) > c l , and Then, for all n ≥ 1, (c) Suppose l even, f n (b) > c l for all n ≥ 1 (BC2), and f (c l ) < c 1 (H2).Then σ i (f (η 1 )) Lastly, the next lemma is a kind of complementary result to Lemma 3.4.Lemma 3.5.Let f ∈ F l be a quasi boundary-anchored map such that (H1) f (c 1 ) < c 1 , and Then, for all n ≥ 1, (6)] and f (c 1 ) ∈ I 1 .By induction it follows that ξ n = c 1 and (b) Suppose l odd, f n (b) < c 1 for all n ≥ 1 (BC2), and f (c l ) < c 1 (H2).Then f n (c l ) < c 1 for any n ≥ 1, because f is assumed to be strictly increasing in I 1 = [a, c 1 ) and f (c 1 ) < c 1 (H1).Therefore, (c) Suppose l even, f n (b) > c l for all n ≥ 1 (BC2), and f (c l ) > c l (H2).Then f n (c l ) > c l for any n ≥ 1, because f is assumed to be strictly increasing in 6) with l even] and f (c l ) ∈ I l+1 (l +1 odd).By induction it follows that η n = c l and f n (η n ) = f n (c l ) > c l is a minimum for n ≥ 1.

Counting Laps
Given the kneading sequences of a map f ∈ F l , it is possible to draw qualitatively the graph of f n for any n ≥ 1.The procedure to be explained shortly is based on the geometrical meaning of the MMSs, Lemma 2.1, and the auxiliary lemmas 3.1-3.3;see Example 4.2 below for an illustration.
(A) Fix n ≥ 1 and using the transition rules in Table 1, determine the first n terms of the min-max sequences ω i , 1 ≤ i ≤ l, from the seeds ω i 1 = M σ(γ i 1 ) , if i is odd, and ω i 1 = m σ(γ i 1 ) , if i is even.For the exposition it is convenient to introduce the notation  2 and 3).These columns will be called the c 0 -, ..., c l+1 -column, respectively.Leave ample space between these columns to insert further columns as we proceed with the present construction.
(C) Proceed now row-wise, say left to right, starting with the ν = 1 row.We are going to compare pair-wise the signatures of neighboring symbols.
In order to bring clarity into the notation, we stick in the sequel to the above usage: n and the Greek letters ν, μ, κ, τ (mostly as subindices, and belonging to N or N 0 ) will refer to map iterations, while the Latin letters i, j, k, p, q (mostly as upper indices, and belonging to {1, ..., l}) will refer to the critical points.
We call the MM-table of f a table constructed following the rules (A)-(D), as exemplified in Tables 2 and 3.This construction provides the basic tools to derive our algorithm for the lap number n (Theorem 5.3 below).

The Main Result
Given f ∈ F l , let ν denote the lap number of f ν , and e ν the number of local extrema (or critical points) of f ν , with ν ≥ 1.Since f ν is continuous and piecewise monotone, the laps are separated by critical points, hence the relation, holds.In particular, e 0 = 0, and 0 = 1 (17) since f 0 , the identity, is monotonically strictly increasing, and e 1 = l, and ν is the number of transversal intersections in the Cartesian plane (x, y) of the curve y = f ν (x) and the straight line y = c i , over the interval (a, b).Note that for all i.
Consider fixed but otherwise arbitrary indices i ∈ {1, ..., l} and ν ≥ 1.The following two observations are trivial: (i) the upper bound s i ν,max ≡ e ν + 1 of s i ν corresponds to the case in which the graph of f ν crosses the ith critical line y = c i on every lap; (ii) the row ν of the MM-table of f ∈ F l contains alternating maxima and minima, i.e., alternating symbols m σ and M σ corresponding to the graph points, say, (x r,s , f ν (x r,s )) and (x r ,s , f ν (x r ,s )), respectively.If σ i • σ i < 0, then the curve y = f ν (x) joining the corresponding extrema crosses the critical line y = c i .If, otherwise, σ i • σ i ≥ 0, then one of the two symbols involved is necessarily a "bad" symbol, to wit: (i) m σ with σ i ∈ {0, +}, so as the curve y = f ν (x) does not cross the ith critical line on the lap (x r,s , x r ,s ), or (ii) M σ with σ i ∈ {−, 0}, so as the curve y = f ν (x) does not cross either the ith critical line on the same lap.Call B i = {M (...,σ i =−,...) , M (...,σ i =0,...) , m (...,σ i =0,...) , m (...,σ i =+,...) } (25) the set of bad symbols or types with respect to the ith critical line.Moreover note that if a bad symbol appears on a column other than the ξ ν -or η ν -column, then the same conclusion concerning the zeros of f ν (x) − c i applies to the two laps of y = f ν (x) left and right of corresponding extremum.And if a bad symbol appears on the ξ ν -and/or η ν -column, then there is no zero of f ν (x) − c i in (a, ξ ν ) and/or (η ν , b). Figure 2 illustrates the geometrical meaning of a bad symbol ω i ν ∈ B i : The branches of the parabolic approximation to a local extrema f ν (x) whose type is a bad symbol point away from the critical line y = c i .It is easy to check that For example, for l = 2 B 1 = {M (−,−) , M (0,−) , m (0,−) , m (+,−) , m (+,0) , m (+,+) } B 2 = {M (−,−) , M (0,−) , M (+,−) , M (+,0) , m (+,0) , m (+,+) } If ω i ν / ∈ B i we say that ω i ν is a good symbol with respect to the ith critical line.Since there are 2l + 1 symbols m σ and 2l + 1 symbols M σ , the number of good symbols with respect to the ith critical line is (4l + 2) − (2l + 2) = 2l.M Therefore, the equality s i ν = e ν + 1 ≡ s i ν,max is only possible if the row ν in the MM-table of f ∈ F l contains only good symbols with respect to the critical line y = c i .Indeed, we have just seen that each bad symbol on row ν subtracts two simple zeros (solutions of f ν (x) − c i = 0, f ν (x) = 0) from s i ν,max .But that condition is not sufficient.It could also happen that the leftmost extremum f ν (ξ ν ) is of good type, but f ν (x) − c i has no zero in the interval (a, ξ ν ) because σ i (f ν (a)) • σ i (f ν (ξ ν )) ≥ 0, i.e., the graph points (ξ ν , f ν (ξ ν )) and (a, f ν (a)) are both above or both below the critical line y = c i .Of course, a similar consideration holds for the rightmost extremum f ν (η ν ) and f ν (b) too.
All these facts can be encapsulated in the relation where b i ν is the number of symbols from the bad set B i and Before using the previous results to formulate a recursive procedure to calculate the lap number n , we need to relate the symbols ω i n on the ξ ν -and η ν -columns to the critical values f ν (ξ ν ) and f ν (η ν ).Remember that in the construction of the MM-table of f , we may encounter two situations in the intervals (a, ξ ν ) (a similar discussion holds for the intervals (η ν , b)).
Figure 3 (left) shows the graphs of the full range map f 1,0 , together with f 0.9,0.1 and f 0.8,0.2 .The convergence rate of ( n ) 1/n to 2 h(f ) for these three maps when n increases is shown in Figure 3 (right).The precision obtained for h(f v 1 ,v 2 ) in this range of parameters and n = 200, is the following: For n = 500, the estimation of h(f 0.8,0.2 ) has four exact decimal digits.
Figure 3. Left, graphs of the maps f 1,0 , f 0.9,0.1 , f 0.8,0.2 .Right, the corresponding convergence plots of ( n ) 1/n to 2 h(f ) as a function of n (top to bottom).A typical benchmark for estimators of the topological entropy consists in determining the entropy as a function of the control parameter(s).Since f v 1 ,v 2 depends on two control parameters, we have calculated that dependence both on one parameter (while keeping fixed the other one), and on the two of them.Figure 4 is a plot of h(f 1,v 2 ) vs. v 2 .As h(f 1,v 2 ) gets smaller, the number of iterations needed to get the entropy with a given precision grows higher.In Figure 4, the mesh constant used was Δv 2 = 10 −3 , and the precision ε = 10 −4 .
Figure 5 is the same kind of plot, this time for h(f v 1 ,v 2 ) as a function of both control parameters, with 0.5 ≤ v 1 ≤ 1, and 0 ≤ v 2 ≤ v 1 − 0.5.This figure depicts also some level sets, just to illustrate the monotonicity of the topological entropy in the parametric space.This property, first conjectured by Milnor and Thurston [17], was later proved for quadratic maps in [18,19].Only recently did H.Bruin and S. van Strien succeed in proving it also for multimodal maps [20].The computation parameters were set as follows: This family verifies f v 2 ,v 3 (0) = 0, f v 2 ,v 3 (c 2 ) = v 2 , f (c 3 ) = v 3 , and Thus, f v 2 ,v 3 has two fixed critical points (c 2 and c 3 ), while the critical point c 1 depends on the control parameters v 2 , v 3 .And again, v 2 , v 3 coincide with the critical values at c 2 and c 3 , respectively.The restriction v 2 < v 3 postulated above relates to v 2 being a local minimum and v 3 a local maximum.Moreover, the left endpoint, x = 0, is a fixed point.
In particular, the choice v 2 = 0 and v 3 = 1 produces a full range quartic, Figure 6, with equation

Conclusions
We have given an algorithm to efficiently calculate the lap number n (hence, the topological entropy) for the iterates of a twice differentiable l-modal map f .The algorithm is based on l + 1 symbolic sequences (ω i ν ) ν∈N , 0 ≤ i ≤ l + 1, -the min-max sequences of f -that contains qualitative information about the structure of maxima and minima of the map iterates f n and the orbits of the endpoints.Theorem 6.3 shows that n is determined by the initial segments (ω i n ) 1≤ν≤n−1 , hence by the itineraries of the critical and boundary points up to order n − 1.This approach builds on previous results for unimodal, boundary-anchored maps obtained in [13] and [15] (Corollary 6.4) and [16].To test if the topological entropy is positive, we test if the kneading sequences are similar or differ from the kneading sequences associated with the Feigenbaum period doubling cascade ( [16], Section 5).If the kneading sequences are similar, than the map has zero topological entropy.Finally, we would like to add that the counting techniques developed here can be extended to maps with jump discontinuities and to piecewise continuous and monotonous maps.However in this case, the kneading sequence calculus must be substantially changed.