Applications of the min-max symbols of multimodal maps

The kneading sequences of a multimodal map f of a closed interval I are symbolic sequences that locate the iterates of its critical values up to the precision set by the partition defined by its critical points [15, 16]. See Sect. 2 for the exact meaning of these concepts in the present work and the general mathematical setting. Since the nth iterate of a critical point of f is a critical value of the nth iterate of f, fⁿ, one may attach to the symbols of each kneading sequence of f a label informing about their minimum/maximum (or “critical”) character. The result is called a min-max sequence, one per critical point, consisting of min-max symbols. These symbols and sequences were introduced in [10, 11] for unimodal maps, and in [3] for multimodal maps. Thus, min-max sequences generalize kneading sequences in that they give additional geometric information about the extrema structure of fⁿ at the critical points for all n ≥ 1. It turns out that the computational cost of a min-max symbol is virtually the same as of a kneading symbol. Indeed, the extra piece of information contained in each symbol of the min-max sequence generated by a critical point, as compared to the corresponding symbol of the kneading sequence generated by the same critical point, can be automatically retrieved from a look-up table once the min-max symbol of the previous iterate of f and the kneading symbol of the current iterate of f have been calculated.

The min-max symbols have been studied in recent years in the papers in [12] for twice-differentiable uni-modal maps, in [3] for twice-differentiable multimodal maps, and in [4,5] for just continuous multimodal maps. Along with theoretical aspects, such as a number of relations between the min-max symbols of a unimodal or multimodal map and certain geometrical properties of the map and its iterates, the practical aspects were also on the fore in those references. In particular, several numerical algorithms for the topological entropy [1, 19] can be found as well in those references, the algorithm in [5] being a variant of the algorithm in [4] and this one a simplification of the algorithm in [3]. The interested reader is also referred to [6–9, 13, 14, 18] for several numerical techniques to compute the topological entropy with various degrees of generality.

In this paper we review two interesting applications of the min-max symbols of a multimodal map f , namely, new expressions for its topological entropy, h(f), and a general algorithm to compute it. In so doing we resort to the well-known expression

h(f)=limn→∞1nloglap(fn),$$\begin{array}{} \displaystyle h(f) = \mathop {\lim }\limits_{n \to \infty } \frac{1}{n}\log {\rm{ lap}}\left( {{f^n}} \right), \end{array}$$(1)

where lap(fⁿ) is shorthand for the lap number of fⁿ (i.e., the number of maximal monotonicity segments of fⁿ) [2, 17]. We will see in Sect. 3 that the min-max symbols allow to relate lap( fⁿ) with the number of ‘transversal’ intersections of fⁿ with the so-called critical lines. Let us also recall at this point other remarkable formulas such as

h(f)=limn→∞1nlog#{x∈I:fn(x)=x}$$\begin{array}{} \displaystyle h(f){\rm{\;}} = \mathop {\lim }\limits_{n \to \infty } \frac{1}{n}\log \# \left\{ {x \in I:{f^n}(x) = x} \right\} \end{array}$$(2)

=limn→∞1nlog+Var(fn),$$\begin{array}{} \displaystyle = \mathop {\lim }\limits_{n \to \infty } \frac{1}{n}{\log ^ + }{\rm{Var}}({f^n}), \end{array}$$(3)

where Var(fⁿ) stands for the variation of fⁿ [17].

The rest of this paper is organized as follows. In order to make the paper self-contained, we summarize in Sect. 2 all the basic concepts, especially the concept of min-max sequences, needed in the subsequent sections, and the concept of ‘bad’ symbols, which are the hallmark of this approach. In Sect. 3 we derive the two new expressions (11) and (19) for h(f ), which add to (1)-(3). A third expression, Eq. (22), is derived in Sect. 4 and, in turn, transformed into a numerical scheme to compute h(f). The resulting algorithm is put to test with bimodal and trimodal maps in Sects. 4.1 and 4.2.

In sum, the following pages give a general panorama of the min-max symbols in action. For other applications of the min-max symbols the reader is referred to [3]. Further applications are the subject of current research.

Let I be a compact interval [a,b] ⊂ ℝ and f : I → I a piecewise monotone continuous 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). More precisely, we assume that f has local extrema at c₁ < ... < c_l and that f is strictly monotone on each of the l + 1 intervals

I1=[a,c1),I2=(c1,c2),...,Il=(cl−1,cl),Il+1=(cl,b].$$\begin{array}{} \displaystyle {I_1} = [a,{c_1}),{I_2} = ({c_1},{c_2}),...,{I_l} = ({c_{l - 1}},{c_l}),{I_{l + 1}} = ({c_l},b]{\rm{.}} \end{array}$$(4)

The set of l-modal maps will be denoted hereafter as ℳ_l(I), ℳ_l([a,b]), or just ℳ_l if the interval I = [a,b] is clear from the context or unimportant for the argument. As in the Introduction, sometimes one also speaks of multimodal maps in general, the name unimodal map being reserved for the case l = 1. Furthermore, if f(c₁) is a maximum (resp. minimum), f is said to have a positive (resp. negative) shape.

The itinerary of x ∈ I under f is a symbolic sequence

i(x)=(i0(x),i1(x),...,in(x),...)∈{I1,c1,I2,...,cl,Il+1}ℕ0$$\begin{array}{} \displaystyle {\bf{i}}(x) = ({i_0}(x),{i_1}(x),...,{i_n}(x),...) \in {\left\{ {{I_1},{c_1},{I_2},...,{c_l},{I_{l + 1}}} \right\}^{{_{N_0}}}} \end{array}$$

(ℕ₀ ≡ {0} ∪ ℕ) defined as follows:

in(x)={Ijif fn(x)∈Ij(1≤j≤l+1),ckif fn(x)=ck(1≤k≤l).$$\begin{array}{} \displaystyle {i_n}(x) = \left\{\begin{array}{*{20}{l}} {{I_j}}&{{\rm{if\;}}{f^n}(x) \in {I_j}{\rm{\;}}(1 \le j \le l + 1),}\\ {{c_k}}&{{\rm{if\;}}{f^n}(x) = {c_k}{\rm{\;}}(1 \le k \le l).} \end{array}\right. \end{array}$$

The itineraries of the critical values,

γi=(γ1i,...,γni,...)=i(f(ci)),1≤i≤l,$$\begin{array}{} \displaystyle {\gamma ^i} = (\gamma _1^i,...,\gamma _n^i,...) = {\bf{i}}(f({c_i})),\,1 \le i \le l, \end{array}$$

are called the kneading sequences of f [15].

It is easily shown [3] that the iterates of the critical points, fⁿ(c_i), are local extrema. This information is included in the min-max sequences of an l-modal map f,

ωi=(ω1i,ω2i,...,ωni,...),1≤i≤l,$$\begin{array}{} \displaystyle {\omega ^i} = (\omega _1^i,\omega _2^i,...,\omega _n^i,...),\,1 \le i \le l, \end{array}$$

where

ωni={mγniif fn(ci) is a minimum,Mγniif fn(ci) is a maximum,$$\begin{array}{} \displaystyle \omega _n^i = \left\{\begin{array}{*{20}{l}} {{m^{{\bf{\gamma }}_n^i}}}&{{\rm{if\;}}{f^n}({c_i}){\rm{\;is\;a\;minimum,}}}\\ {{M^{{\bf{\gamma }}_n^i}}}&{{\rm{if\;}}{f^n}({c_i}){\rm{\;is\;a\;maximum,}}} \end{array}\right. \end{array}$$(5)

and γni$\begin{array}{} \displaystyle \gamma _n^i \end{array}$ are kneading symbols [3, 10–12]. Hence, the min-max symbolsωni$\begin{array}{} \displaystyle \omega _n^i \end{array}$ are a generalization of the kneading symbols γni$\begin{array}{} \displaystyle \gamma _n^i \end{array}$ in that they also specify whether fⁿ(c_i) is a maximum or a minimum. In the exponential-like notation of (5), the ‘base’ belongs to the alphabet {m,M}, and the ‘exponent’ (also called signature in [3]) belongs to the alphabet {I₁,c₁,I₂,...,c_l,I_l+1}. Therefore, the extra information of a min-max symbol ωni$\begin{array}{} \displaystyle \omega _n^i \end{array}$ as compared to a kneading symbol γni$\begin{array}{} \displaystyle \gamma _n^i \end{array}$ is contained in its base.

In [4, Theorem 1] it is proved that if f ∈ ℳ_l has a positive shape, then the ‘transition rules’ given in Table 1 hold:

Table 1

Transition rules for l-modal maps with a positive shape.

Here “even” and “odd” refer to the parity of the subindices of I_j(1 ≤ j ≤ l + 1) or c_k(1 ≤ i ≤ l) in the exponent of ωni$\begin{array}{} \displaystyle \omega _n^i \end{array}$. If, otherwise, f has a negative shape, then one has to swap m and M on the right column of Table 1 [4, Theorem 1]:

Table 2

Transition rules for l-modal maps with a negative shape.

These transition rules prove our claim in the Introduction that, from the point of view of the computational cost, min-max sequences and kneading sequences are virtually equivalent.

Let the ith critical line, 1 ≤ i ≤ l, be the line y = c_i on the Cartesian plane {(x,y) : x,y ∈ ℝ}. Min-max symbols split into bad and good symbols with respect to the ith critical line. Geometrically, the latter correspond to local maxima strictly above the line y = c_i, and to local minima strictly below the line y = c_i. All other min-max symbols are bad by definition with respect to the ith critical line. We use the notation

Bi={MI1,Mc1,...,MIi,Mci,mci,mIi+1,...,mcl,mIl+1}$$\begin{array}{} \displaystyle {{\mathscr B}^i} = \left\{ {{M^{{I_1}}},{M^{{c_1}}},...,{M^{{I_i}}},{M^{{c_i}}},{m^{{c_i}}},{m^{{I_{i + 1}}}},...,{m^{{c_l}}},{m^{{I_{l + 1}}}}} \right\} \end{array}$$(6)

for the set of bad symbols of f ∈ ℳ_l with respect to the ith critical line. There are 2(l + 1) bad symbols and 2l good symbols with respect to a given critical line. Fig. 1 shows four bad min-max symbols with respect to the critical line y = c_i. Since the concept of bad symbol needs the minimum/maximum character of fⁿ(c_i), it is a distinctive ingredient of properties derived via min-max symbols.

Fig. 1

For further reference, define the sets of pairs of indices

Kni={(k,m),1≤k≤l,1≤m≤n:ωmk∈Bi},$$\begin{array}{} \displaystyle {\mathscr K}_n^i = \left\{ {(k,m),1 \le k \le l,1 \le m \le n:\omega _m^k \in {{\mathscr B}^i}} \right\}, \end{array}$$(7)

(n ≥ 1, 1 ≤ i ≤ l). That is, Kni$\begin{array}{} \displaystyle {\mathscr K}_n^i \end{array}$ collects the upper and lower indices (k,m), respectively, of the bad symbols with respect to the ith critical line in all the initial segments of length n of the min-max sequences of f:

ω11,ω21,...,ωn1;ω12,ω22,...,ωn2;...;ω1l,ω2l,...,ωnl.$$\begin{array}{} \displaystyle \omega _1^1,\omega _2^1,...,\omega _n^1;\,\,\omega _1^2,\omega _2^2,...,\omega _n^2;\,\,...;\,\,\omega _1^l,\omega _2^l,...,\omega _n^l. \end{array}$$

Let f ∈ ℳ_l(I). Since f is continuous and piecewise strictly monotone, so is fⁿ for all n ≥ 2 as well. We say that x₀ ∈ I is an interior simple zero of fⁿ(x) − c_i = 0, n ≥ 0, if a < x₀ < b, fⁿ(x₀) = c_i, and fⁿ(x₀) is not a local extremum, thus fⁿ(x) −c_i takes both signs in every neighborhood of x₀. In the case of (everywhere) differentiable maps, an interior simple zero x₀ of fⁿ(x) − c_i = 0 amounts geometrically to a transversal intersection on the two-dimensional interval I × I = {(x,y) : x,y ∈ I} of the curve y = fⁿ(x) and the critical line y = c_i.

Let sni$\begin{array}{} \displaystyle s_n^i \end{array}$, 1 ≤ i ≤ l, stand for the number of interior simple zeros of fⁿ(x) − c_i = 0, n ≥ 0. In order to simplify the notation, set

sn=∑i=1lsni,$$\begin{array}{} \displaystyle {s_n} = \mathop \sum\limits^l_{i = 1} s_n^i, \end{array}$$(8)

n ≥ 0. In particular,

s0=∑i=1ls0i=∑i=1l1=l.$$\begin{array}{} \displaystyle {s_0} = \mathop \sum\limits^l_{i = 1} s_0^i = \mathop \sum\limits^l_{i = 1} 1 = l. \end{array}$$(9)

According to [3, Eqn. (31)], lap(fⁿ) satisfies

lap(fn)=1+∑ν=0n−1sν.$$\begin{array}{} \displaystyle {\rm{lap}}({f^n}) = {\rm{1 + }}\mathop \sum\limits^{n - 1}_{\nu = 0} {s_\nu}. \end{array}$$(10)

Plug (10) into (1) to obtain

h(f)=limn→∞1nlog(1+∑ν=0n−1sν),$$\begin{array}{} \displaystyle h(f) = \mathop {\lim }\limits_{n \to \infty } \frac{1}{n}\log \left( {1 + \mathop \sum\limits^{n - 1}_{\nu = 0} {s_\nu}} \right), \end{array}$$(11)

which expresses the topological entropy of a multimodal map f by means of the number of interior simple zeros of fⁿ(x) − c_i = 0, 1 ≤ i ≤ l, or, in more geometrical terms, via the number of ‘transversal’ intersections of fⁿ with the critical lines, for n ≥ 1.

For (11) to provide also a practical tool for computing h(f), a procedure is needed to go from s₀,s₁,...,s_n−1 to s_n. Here is where the min-max sequences of f enter the scene.

We say that an l-modal map f of the interval I = [a,b] is boundary-anchored if f{a,b} ⊂ {a,b}, i.e., if

f(a)=a,and f(b)={aif l is odd,bif l is even,$$\begin{array}{} \displaystyle f(a) = a,\,{\rm{and\;}}f(b) = \left\{\begin{array}{*{20}{l}} a&{{\rm{if\;}}l{\rm{\;is\;odd,}}}\\ b&{{\rm{if\;}}l{\rm{\;is\;even,}}} \end{array}\right. \end{array}$$

in case that f has a positive shape, or

f(a)=b,and f(b)={bif l is odd,aif l is even,$$\begin{array}{} \displaystyle f(a) = b,\,{\rm{and\;}}f(b) = \left\{\begin{array}{*{20}{l}} b&{{\rm{if\;}}l{\rm{\;is\;odd,}}}\\ a&{{\rm{if\;}}l{\rm{\;is\;even,}}} \end{array}\right. \end{array}$$

in case that f has a negative shape.

Therefore, assume for the time being that f ∈ ℳ_l is boundary-anchored. Then it is proved in [4, Theorem 2] that for any n ≥ 1, 1 ≤ i ≤ l,

sni=1+∑ν=0n−1sν−Sni,$$\begin{array}{} \displaystyle s_n^i = 1 + \mathop \sum\limits_{\nu = 0}^{n - 1} {s_\nu} - S_n^i, \end{array}$$(12)

where

Sni=2∑(k,m)∈Knisn−mk$$\begin{array}{} \displaystyle S_n^i = 2\mathop \sum\limits_{(k,m) \in {\mathscr K}_n^i} s_{n - m}^k \end{array}$$(13)

with Sni=0$\begin{array}{} \displaystyle S_n^i = 0 \end{array}$ if Kni=∅$\begin{array}{} \displaystyle {\mathscr K}_n^i = \emptyset \end{array}$. Thus, see (8),

sn=l(1+∑ν=0n−1sν)−Sn,$$\begin{array}{} \displaystyle {s_n} = l\left( {1 + \mathop \sum\limits_{\nu = 0}^{n - 1} {s_\nu}} \right) - {S_n}, \end{array}$$(14)

where

Sn=∑i=1lSni.$$\begin{array}{} \displaystyle {S_n} = \mathop \sum\limits_{i = 1}^l S_n^i. \end{array}$$(15)

Note that the right hand of (12) only contains the numbers s_ν with 0 ≤ ν ≤ n − 1, what allows to calculate s_n in a fully recursive way from the seeds s01=...=s0l=1$\begin{array}{} \displaystyle s_0^1 = ... = s_0^l = 1 \end{array}$. This fact and Remark 1 render Eq. (11) a formula for actually computing the topological entropy of any (i.e., not necessarily boundary-anchored) l-modal map f. Since the calculation of Sni$\begin{array}{} \displaystyle S_n^i \end{array}$ involves the bad symbol set Kni$\begin{array}{} \displaystyle {\mathscr K}_n^i \end{array}$, the corresponding algorithm presupposes the knowledge of the min-max symbols ωmk$\begin{array}{} \displaystyle \omega _m^k \end{array}$ with 1 ≤ k ≤ l, and 1 ≤ m ≤ n.

The relation (14) not only promotes (11) to the basis of an algorithm for the computation of h(f) via minmax symbols, but also leads to a further expression for h(f). For this insert (10) in (14) to write the lap number lap(fⁿ) of a boundary-anchored l-modal map as

lap(fn)=1l(sn+Sn).$$\begin{array}{} \displaystyle {\rm{lap}}({f^n}) = \frac{{\rm{1}}}{l}({s_n} + {S_n}). \end{array}$$(16)

Next we derive from (16) a formula for h(f) which involves the min-max symbols of f in an explicit manner. To this end we are going to express s_n in terms of S₁,...,S_n.

From (16) and (17) it follows

lap(fn)=(l+1)n(1−∑ν=1n−1Sν(l+1)ν+1)$$\begin{array}{} \displaystyle {\rm{lap}}({f^n}) = {(l + 1)^n}\left( {{\rm{1}} - \mathop \sum\limits_{\nu{\rm{ = 1}}}^{n - 1} \frac{{{S_\nu}}}{{{{(l + 1)}^{\nu + 1}}}}} \right) \end{array}$$(18)

for boundary-anchored l-modal maps. Apply now (1) to (18) and derive the following formula for the topological entropy of any l-modal map in virtue of Remark 1.

According to (19) h(f) ≤ log(l+1) =: h(f)_max, a well-known result for l-modal maps. Therefore, (19) expresses the difference h(f)_max − h(f) by means of the sequence (S_n)_n≥1. Note that the convergence is monotonic, i.e.,

log(l+1)−1n|log(1−∑ν=1n−1Sν(l+1)ν+1)|↘h(f)$$\begin{array}{} \displaystyle \log (l + 1) - \frac{1}{n}\left| {\log \left( {1 - \mathop \sum\limits_{\nu = 1}^{n - 1} \frac{{{S_\nu}}}{{{{(l + 1)}^{\nu + 1}}}}} \right)} \right| \searrow h(f) \end{array}$$(20)

as n → ∞. See [5, Sect. 5] for details of the convergence (20). The application of (19) to the logistic family,

qν(x)=4νx(1−x),$$\begin{array}{} \displaystyle {q_\nu }(x) = 4\nu x(1 - x), \end{array}$$(21)

0 ≤ x ≤ 1, 0 < ν ≤ 1, was extensively studied in [5, Sect. 6].

In this section we are going to review a general algorithm to compute h(f) [4]. This algorithm is based on the formula for h(f) which follows readily from (1) and (16),

h(f)=limn→∞1nlog1l(sn+Sn)=limn→∞1nlog1l∑i=1l(sni+2∑(k,m)∈Knisn−mk),$$\begin{array}{} \displaystyle \begin{array}{*{20}{l}} {h(f)}&{ = \mathop {\lim }\limits_{n \to \infty } \frac{1}{n}\log \frac{1}{l}({s_n} + {S_n})}\\ {}&{ = \mathop {\lim }\limits_{n \to \infty } \frac{1}{n}\log \frac{1}{l}\mathop \sum\limits_{i = 1}^l \left( {s_n^i + 2\mathop \sum\limits_{(k,m) \in {\mathscr K}_n^i} s_{n - m}^k} \right),} \end{array} \end{array}$$(22)

again for any l-modal map f (Remark 1). All we need is a recursive scheme to compute the right hand side of (22) for ever larger n’s.

The core of the algorithm consists of a loop over n. Each time the algorithm enters the loop, the values of s_n−1 and S_n−1 are updated to s_n and S_n, and the current estimation of h(f) is compared to the previous one. Note that the computation of Sni$\begin{array}{} \displaystyle S_n^i \end{array}$, 1 ≤ i ≤ l, requires s0i=1$\begin{array}{} \displaystyle s_0^i = 1 \end{array}$, s1i,...,sn−1i$\begin{array}{} \displaystyle s_1^i,...,s_{n - 1}^i \end{array}$, see (13), while the computation of sni$\begin{array}{} \displaystyle s_n^i \end{array}$, 1 ≤ i ≤ l, requires s0i,s1i,...,sn−1i$\begin{array}{} \displaystyle s_0^i,s_1^i,...,s_{n - 1}^i \end{array}$, and Sni$\begin{array}{} \displaystyle S_n^i \end{array}$, see (12). Note also that Kni=Kn−1i∪(Kni\Kn−1i)$\begin{array}{} \displaystyle {\mathscr K}_n^i = {\mathscr K}_{n - 1}^i \cup ({\mathscr K}_n^i\backslash {\mathscr K}_{n - 1}^i) \end{array}$, where

Kni\Kn−1i={(k,n),1≤k≤l:ωnk∈Bi}.$$\begin{array}{} \displaystyle {\mathscr K}_n^i\backslash {\mathscr K}_{n - 1}^i = \left\{ {(k,n),1 \le k \le l:\omega _n^k \in {{\mathscr B}^i}} \right\}. \end{array}$$(23)

We summarize next the algorithm resulting from (22) in the following scheme (“A → B” stands for “B is computed by means of A”).

(A1) Parameters:l ≥ 1 (number of critical points), ε > 0 (dynamic halt criterion), and n_max ≥ 2 (maximum number of loops).

As a matter of course, the parameter ε does not bound the error |h(f)−1nlogsn+Snl|$\begin{array}{} \displaystyle \left| {h(f) - \frac{1}{n}\log \frac{{{s_n} + {S_n}}}{l}} \right| \end{array}$ but the difference between two consecutive estimations, see (25). The number of exact decimal positions of h(f) can be found out by taking different ε’s, as we will see in the numerical simulations below. Equivalently, one can control how successive decimal positions of 1nlogsn+Snl$\begin{array}{} \displaystyle \frac{1}{n}\log \frac{{{s_n} + {S_n}}}{l} \end{array}$ stabilize with growing n. Moreover, the smaller h(f), the smaller ε has to be chosen to achieve a given numerical precision.

The algorithm (A1)-(A5) was benchmarked in [4] against (27) for unimodal maps and against the algorithm of [3] (also based on (22)) for l-modal maps with 2 ≤ l ≤ 5. The result was that (27) is faster for unimodal maps, otherwise the algorithm (A1)-(A5) is faster. Let us mention in passing that (19) was used in [5] to derive a similar though slower computational scheme.

For the sake of completeness, we show next numerical results obtained with the above algorithm (A1)-(A5) applied to bimodal and trimodal maps. The algorithm was coded for arbitrary l with PYTHON and run on an Intel(R) Core(TM)2 Duo CPU. The base of the logarithm function is 2.

4.1

Bimodal maps

Consider the bimodal maps f_ν1,ν2: [0,1] → [0,1] defined as

fν1,ν2(x)=(2ν2−ν1)−2(ν2−ν1)sin(π6(5−46x+6x2)),$$\begin{array}{} \displaystyle {f_{{\nu_1},{\nu_2}}}(x) = \left( {2{\nu_2} - {\nu_1}} \right) - 2\left( {{\nu_2} - {\nu_1}} \right)\sin \left( {\frac{\pi }{6}\left( {5 - 4\sqrt 6 x + 6{x^2}} \right)} \right), \end{array}$$

where 0 ≤ ν₁ ≠ ν₂ ≤ 1. The critical points of f_ν1,ν2 on the interval [0,1] are

c1=13(2−1)=0.2391...,c2=23=0.8165....$$\begin{array}{} \displaystyle {c_1} = \frac{1}{{\sqrt 3 }}(\sqrt 2 - 1) = 0.2391...,\,\,{c_2} = \sqrt {\frac{2}{3}} = 0.8165.... \end{array}$$

Moreover,

fν1,ν2(0)=ν2,fν1,ν2(c1)=ν1,fν1,ν2(c2)=ν2,fν1,ν2(1){>ν2if ν1>ν2<ν2if ν1<ν2.$$\begin{array}{} \displaystyle {f_{{\nu_1},{\nu_2}}}(0) = {\nu_2},\quad {f_{{\nu_1},{\nu_2}}}({c_1}) = {\nu_1},\quad {f_{{\nu_1},{\nu_2}}}({c_2}) = {\nu_2},\quad {f_{{\nu_1},{\nu_2}}}(1)\left\{\begin{array}{*{20}{l}} { \gt {\nu_2}}&{{\rm{if\;}}{\nu_1} \gt {\nu_2}}\\ { \lt {\nu_2}}&{{\rm{if\;}}{\nu_1} \lt {\nu_2}} \end{array}\right. . \end{array}$$

Therefore, if we choose ν₁ > ν₂, then f_ν1,ν2 has a positive shape, while ν₁ < ν₂ entails a negative shape. Fig. 2 shows the graphs of the full range maps f_1,0 and f_0,1, together with f_0.75,0.25 and f_0.25,0.75.

Fig. 2

Fig. 3 shows the plot of h(f_0,ν2) vs. ν₂, 0 < ν₂ ≤ 1, computed with ε = 10⁻⁴ and Δν₂ = 0.001. Fig. 4 depicts the values of h(f_ν1,ν2) vs. ν₁ and ν₂, with ε = 10⁻⁴ and Δν₁ = Δν₂ = 0.01. Finally, Table 3 illustrates the convergence of the algorithm (as measured by the number of computation loops) when calculating h(f_0.1,0.9). We see that ε = 10⁻⁷ is necessary to fix the first two decimal positions.

Fig. 3

Fig. 4

Table 3

Performances when computing h(f_0.1,0.9) in bits with the bimodal map.

precision	h	n
ε = 10−4	0.655591287672	179
ε = 10−5	0.643433302022	565
ε = 10−6	0.639578859603	1786
ε = 10−7	0.638359574751	5645

4.2

Trimodal maps

Consider the trimodal maps g_ν1,ν2 : [0,1] → [0,1] defined as

gν1,ν2(x)=ν1cosπ8−ν2+(ν2−ν1)sin(π8(5−8x+8x2))cosπ8−1,$$\begin{array}{} \displaystyle {g_{{\nu_1},{\nu_2}}}(x) = \frac{{{\nu_1}\cos \frac{\pi }{8} - {\nu_2} + \left( {{\nu_2}\ - {\nu_1}} \right)\sin \left( {\frac{\pi }{8}\left( {5 - 8x + 8{x^2}} \right)} \right)}}{{\cos \frac{\pi }{8} - 1}}, \end{array}$$

where 0 ≤ ν₁ ≠ ν₂ ≤ 1. The critical points of g_ν1,ν2 on the interval [0,1] are

c1=12(1−12)=0.1464...,c2=12,c3=12(1+12)=0.8535...$$\begin{array}{} \displaystyle {c_1} = \frac{1}{2}\left( {1 - \frac{1}{{\sqrt 2 }}} \right) = 0.1464...,\,\,{c_2} = \frac{1}{2},\,\,{c_3} = \frac{1}{2}\left( {1 + \frac{1}{{\sqrt 2 }}} \right) = 0.8535... \end{array}$$

Moreover,

gν1,ν2(0)=ν2,gν1,ν2(c1)=ν1,gν1,ν2(c2)=ν2,gν1,ν2(c3)=ν1,gν1,ν2(1)=ν2.$$\begin{array}{} \displaystyle {g_{{\nu_1},{\nu_2}}}(0) = {\nu_2},\quad {g_{{\nu_1},{\nu_2}}}({c_1}) = {\nu_1},\quad {g_{{\nu_1},{\nu_2}}}({c_2}) = {\nu_2},\quad {g_{{\nu_1},{\nu_2}}}({c_3}) = {\nu_1},\quad {g_{{\nu_1},{\nu_2}}}(1) = {\nu_2}. \end{array}$$

As before, if we choose ν₁ > ν₂, then g_ν1,ν2 has a positive shape, while ν₁ < ν₂ entails a negative shape. Fig. 5 shows the graphs of the full range maps g_1,0 and g_0,1, together with g_0.75,0.25 and g_0.25,0.75.

Fig. 5

Fig. 6 shows the plot of h(g_0,ν2) vs. ν₂, 0 < ν₂ ≤ 1, computed with ε = 10⁻⁴ and Δν₂ = 0.001. Fig. 7 depicts the values of h(g_ν1,ν2) vs. ν₁ and ν₂, with ε = 10⁻⁴ and Δν₁ = Δν₂ = 0.01. Finally, Table 4 illustrates the convergence of the algorithm when calculating h(g_0.1,0.9). Here ε = 10⁻⁶ fixes the first two decimal positions.

Fig. 6

Fig. 7

Table 4

Performances when computing h(g_0.1,0.9) in bits with the trimodal map.

precision	h	n
ε = 10−4	1.02013528493	203
ε = 10−5	1.00638666069	640
ε = 10−6	1.00202049572	2023
ε = 10−7	1.00063926538	6394

In this paper we have reviewed a few applications of the min-max symbols to both theoretical and computational aspects of the topological entropy of multimodal maps. Among the first, see Eqs. (19) and (22); among the latter, see the fast and numerically stable algorithm for h(f) presented in Sect. 4, which is based on Eq. (22). The application of this algorithm to bimodal and trimodal maps was illustrated in Sect. 4 as well. The expressions (19), (22), and also (11) add to other well-known ones for h(f) such as (1)-(3).