Entropy Monotonicity and Superstable Cycles for the Quadratic Family Revisited

The main result of this paper is a proof using real analysis of the monotonicity of the topological entropy for the family of quadratic maps, sometimes called Milnor’s Monotonicity Conjecture. In contrast, the existing proofs rely in one way or another on complex analysis. Our proof is based on tools and algorithms previously developed by the authors and collaborators to compute the topological entropy of multimodal maps. Specifically, we use the number of transverse intersections of the map iterations with the so-called critical line. The approach is technically simple and geometrical. The same approach is also used to briefly revisit the superstable cycles of the quadratic maps, since both topics are closely related.


Introduction
Topological entropy is one of the main quantifiers of complexity in continuous dynamics. First of all, it is a tight upper bound of all measure-preserving dynamics generated by a given continuous self-map of a compact metric space [1]. Furthermore, in metric spaces, topological entropy measures the growth rate of the number of ever longer orbits up to a small error [2]. Its analytical calculation is only feasible in some special cases, though. For one-dimensional dynamics, where transformations can be supposed to be continuous and piecewise monotone (multimodal) for practical purposes, a number of numerical algorithms based on symbolic representations of the orbits have been developed. Examples include kneading invariants [3], min-max symbols [4], ordinal patterns [5], context trees [6] and more. Precisely, this paper is the outgrowth of previous work by the authors and collaborators on the numerical computation of the topological entropy of multimodal maps using min-max symbols [7][8][9]. At the heart of our algorithms is the number of transverse intersections (i.e., "X-crossings") of a multimodal map and its iterates with the so-called critical lines. In this paper we also show the potential of this concept in regard to theoretical issues. To this end, we revisit two well-traveled topics in one-dimensional dynamics: (i) The monotonicity of the topological entropy for the family of quadratic maps; (ii) Some basic properties of the periodic orbits of its critical point (superstable cycles).
Next, we elaborate a bit on these two topics.
The family of quadratic maps (or quadratic family) is composed of the logistic maps f µ (x) = 4µx(1 − x), 0 ≤ µ ≤ 1, or, for that matter, any other dynamically equivalent maps; actually, we will use the maps q t (x) = t − x 2 , 0 ≤ t ≤ 2, because they are algebraically handier. When the topological entropy of multimodal maps was studied in the 1980s, the numerical results indicated The points c 1 , ..., c l are called critical or turning points and their images f (c 1 ), ..., f (c l ) are the critical values of f . These maps are also referred to as multimodal maps (for a general l) and unimodal maps (if l = 1). We denote the set of l-modal maps by M l (I), or just M l if the interval I is clear from the context or unimportant for the argument. f ∈ M l (I) is said to have positive (resp. negative) shape if f (c 1 ) is a maximum (resp. minimum); here and hereafter, all extrema are meant to be local unless stated otherwise. Thus, if f has positive shape, then f is strictly increasing in the intervals with odd subindex (I odd ) and strictly decreasing in the intervals with even subindices (I even ).
For n ≥ 0, f n denotes the nth iterate of f , where f 0 is the identity map. Since f is continuous and piecewise strictly monotone, so is f n for all n ≥ 1. The proof of the following Proposition is direct (see [8], Lemma 2.2).
Proposition 1. Let f ∈ M l (I) with positive shape and n ≥ 1. We have: (i) f n (x) = c odd , (ii) f n (x) ∈ I even and f n (x) is a minimum, or (iii) f n (x) ∈ I odd and f n (x) is a maximum, (1) and (i) f n (x) = c even , (ii) f n (x) ∈ I odd and f n (x) is a minimum, or (iii) f n (x) ∈ I even and f n (x) is a maximum. (2) If f has negative shape, then replace " f n+1 (x) is a maximum if" by " f n+1 (x) is a minimum if" in (1), and the other way around in (2).
Apply Proposition 1 to f n , f n−1 , ..., f to conclude that f n+1 has local extrema at all x ∈ I such that f k (x) = c i for k = 0, 1, .., n and some i. This proves: Proposition 2. Let f ∈ M l (I) and n ≥ 1. Then f n has local extrema at the critical points and their preimages up to order n − 1.
For n ≥ 1, let s n,i = #{x ∈ (a, b) : f n (x) = c i , f k (x) = c j for 0 ≤ k ≤ n − 1, 1 ≤ j ≤ l}, i.e., the number of interior simple zeros of the function f n (x) − c i , and set for the total number of such zeros. For the convenience of notation, definition (3) can be extended to n = 0: s 0,i = #{x ∈ (a, b) : x = c i } = 1, so that s 0 = l.
In the case of differentiable maps (to be considered in Sections 3-5), s n,i amounts geometrically to the number of transverse intersections of y = f n (x) with the ith critical line y = c i . Indeed, by the chain rule of derivation, Therefore, if f k (x) = c j for all 0 ≤ k ≤ n − 1 and 1 ≤ j ≤ l, then d f n (c i )/dx = 0. A solution x * of f n (x) − c i = 0 such that d f n (x * )/dx = 0 corresponds to a tangential intersection of the curve y = f n (x) with the critical line y = c i . Abusing the language, we will speak of transverse and non-transverse intersections in the general case too. Incidentally, Equation (5) proves Proposition 2 for differentiable maps.
Next, let e n be the number of local extrema of f n .
Proof. If n = 0, then e 0 = 0 and s 0 = l, so that e 0 + s 0 gives the right answer e 1 = l. Suppose now that n ≥ 1 and f n+1 has a local extremum at x 0 ∈ I, so that e n+1 is the number of such x 0 's. According to Proposition 1, there are two exclusive possibilities: (a) f n (x 0 ) = c i for some 1 ≤ i ≤ l (Proposition 1(i)); or (b) f n (x 0 ) = c i for all 1 ≤ i ≤ l and f n has a local extremum at x 0 (Proposition 1(ii) and (iii)).
In turn, (a) subdivides according to whether x 0 is a transverse or a tangential intersection of y = f n (x) with the critical line y = c i : Therefore, each x 0 ∈ I that contributes to e n+1 contributes to s n (if case (a1) holds) or, otherwise, to e n (if case (a2) or (b) holds). The bottom line is Equation (6).
In the next two sections we discuss how the transverse and tangential intersections of f n with the critical lines are related to two salient aspects of the dynamics generated by f : topological entropy and superstable periodic orbits.

Topological Entropy
The connection of the recursive Formula (6) with the topological entropy of f ∈ M l (I), h( f ), is readily established through the lap number n of f n , which is defined as the number of maximal monotonicity intervals of f n . First, replace n = e n + 1 in (6) to obtain The initial values 0 = 1 and s 0 = l yield 1 = l + 1, as it should. Second, use the relation [18] h( f ) = lim n→∞ 1 n log n .
Finally, Equations (8) and (9) lead then to the expression which was first derived in [8]. For the general concept of entropy, see [19][20][21]. As a technical remark, the topological entropy of a continuous map (in particular, a multimodal map f : I → I) only depends on its non-wandering set [22]. A point x ∈ I is said to be non-wandering for f if for any neighborhood U of x, there is an integer n ≥ 1 such that f n (U) ∩ U = ∅; otherwise, x is said to be a wandering point for f . The non-wandering set for f consists of all the points that are non-wandering for f .
Equations (9) and (11) add to other similar expressions of h( f ) in terms of e n = n − 1, the number of n-periodic points, the variation of f n ( [14], Theorem 1.1), etc. In this regard, the quantities s k in Equation (11) can be viewed in the following three different ways: (1) Algebraically, s k is by definition (3)-(4) the number of interior simple zeros of the equations (2) Geometrically, s k is the total number of transverse intersections of the iterated map f k with the critical lines.
(3) Dynamically, s k is the total number of preimages of the critical points of minimal order k. Whatever the interpretation, we are going to show that s k is a useful tool to study multimodal maps.
Several numerical algorithms for the topological entropy of multimodal maps based on Equation (11) can be found in [7][8][9], the algorithm in [9] being a variant of the algorithm in [8] and this, in turn, a simplification of the algorithm in [7]. The performance of the algorithm [8] has recently been benchmarked in [23] with favorable results. The computation of s n from the values of s 0 , ..., s n−1 is possible via the so-called min-max sequences [4], which are closely related to the kneading sequences [3,17]. As compared to the kneading symbols, the min-max symbols contain additional information on the minimum/maximum character of the critical values f n (c i ), 1 ≤ i ≤ l, with virtually no extra computational penalty [7,8]. The geometrical properties of the min-max symbols were studied in [24] and [7] for twice-differentiable uni-and multimodal maps, respectively, and in [8,9] for just continuous multimodal maps. A brief overview is given in the Introduction of [9].
Let f t ∈ M l (I t ) be a one-parametric family of l-modal maps whose parameter t ranges in an interval J ⊂ R. Denote by s n (t) the total number of transverse intersections of y = f n t (t) with the critical lines. From (11) and the monotonicity of the logarithmic function it follows: Proposition 4. Let f t ∈ M l (I t ), and t 1 , t 2 ∈ J with t 1 < t 2 . Suppose s n (t 1 ) ≤ s n (t 2 ) for all n ≥ n 0 .
As we will see in Section 4, Proposition 4 provides a handle to prove the monotonicity of the topological entropy for the family of quadratic maps. We mentioned already in the introduction that, according to [14,15], the existing monotonicity proofs [3,[11][12][13] rely in one way or another on complex analysis. Unlike them, our approach uses real analysis. Let us remind at this point that the topological entropy of a family of unimodal maps labeled by some natural parameter (such as its critical value) is not usually monotone, even under very favorable assumptions [25]. More generally, let f v be a polynomial map parametrized by its critical values v = (v 1 , ..., v l ). Then, according to ([14], Theorem 1.1), for l ≥ 2 there exist fixed values of v 2 , ..., v l such that the map v 1 → h( f v ) is not monotone. For multimodal maps, monotonicity of the map is replaced by the connectivity of the isentropes ( [15], Theorem 1.2). See also [15] for related results and open conjectures.

Superstable Periodic Orbits
Let x 0 ∈ I and set x k = f k (x 0 ) = f (x k−1 ) for k ≥ 1. Suppose for the time being that f is differentiable and a critical point c i is periodic with prime period p. Then, each point of the orbit In other words, d f p (x j )/dx (whose absolute value quantifies the stability of the fixed points c i , x 1 ,..., x p−1 of f p ) vanishes at each point of the periodic orbit.
On the other hand, f n has local extrema at all critical points for n ≥ 1, so that the periodicity condition f p (c i ) = c i amounts to a tangential intersection of the curve y = f p (x) and the critical line y = c i at x = c i . Therefore, while the transverse intersections of f n with the critical lines are the only input needed to calculate the topological entropy of multimodal maps, the tangential intersections, if any, are the main ingredient of the periodic orbits (cycles) of a critical point. All in all, the intersections of f n with the critical lines, whether transverse or tangential, give information about the dynamical complexity and superstability of the orbits.

Application Case: Quadratic Maps
Quadratic maps have been the workhorse of chaotic dynamics for two good reasons: their dynamic exhibits a mind-boggling complexity despite being algebraically so simple and, precisely because of this simplicity, many of their dynamical properties are amenable to analytical scrutiny. We consider henceforth the family of the real quadratic maps where x ∈ R and 0 ≤ t ≤ 2. The critical point and the critical value of q t are c = 0 and q t (0) = t, respectively, so the critical line y = 0 is the x-axis in the Cartesian plane {(x, y) ∈ R 2 }. The quadratic family has two fixed points, Therefore, an invariant finite interval I t , i.e., q t (I t ) ⊂ I t , where defining a dynamic generated by q t , is It so that the boundary of I t , Since all x / ∈ I t escape to −∞ under iterations of q t , the set I t contains the non-wandering set of q t . See Figure 2 for some instances of the quadratic family. The bifurcation diagram of q t (x) in Figure 3 shows that the asymptotic dynamics of the quadratic family (chaotic attractors, along with stable fixed points and periodic orbits) lives in the interval −2 ≤ x ≤ 2. After the period-doubling cascade, chaos onset occurs at the Feigenbaum point t F = 1.401155..., i.e., the topological entropy of q t is positive for t > t F .
The dynamical systems generated by q t (x), where x ∈ I t and 0 ≤ t ≤ 2, and the more popular or ]. An advantage of the quadratic map (13) is that the transverse (resp. tangential) intersections of y = q n t (x) with the critical line correspond to the simple (resp. multiple) roots of q n t (x), a polynomial of degree 2 n . Since q t (x) is unimodal (l = 1), Equation (4) simplifies to whereI t = I t \∂I t is the interior of I t . Therefore, s n (t) stands for the number of simple zeros of q n t (x) in I t or, equivalently, for the number of transverse intersections of the curve y = q n t (x) with the critical line y = 0. We show in Remark 1 below thatI t contains all zeros of q n t (x), thereforeI t can be safely replaced byI 2 = (−2, 2) (or R, for that matter) in Equation (19).
In the panels of Figure 4 we see that the 2 |σ| root branches φ σ (t), 1 ≤ |σ| ≤ 5, build 2 |σ|−1 parabola-like curves, which we denote φ σ 1 ,...,σ i−1 ,±,σ i+1 ,...,σ n (t) (1 ≤ i ≤ n), this notation meaning that the curves φ σ 1 ,...,σ i =+,...,σ n (t) and φ σ 1 ,...,σ i =−,...,σ n (t) (the branches of the parabola) emerge from a ) with a vertical tangent. The vertex and the abscissa t b will be called indistinctly branching point (geometrical terminology) or bifurcation point (dynamical terminology) of the parabola or any of its branches. Root parabolas with the vertex on the t-axis, φ ±,σ 2 ,...,σ n (t), are sometimes called on-line parabolas, otherwise off-axis parabolas. The branching point t b has also a direct geometrical interpretation in state space: the curve q |σ| t b (x) intersects tangentially the x-axis (the critical line) at the point x = φ σ 1 ,...,σ i−1 ,±,σ i+1 ,...,σ n (t b ). The opening of the branches to the right means that, if the contact occurs from the upper half-plane as t increases, the corresponding local extremum is a minimum, whereas if the contact occurs from the lower half-plane, it is a maximum. In panel (d) of Figure 4 we see that different branches do not cross but touch at the bifurcation points ("T-crossings"). We will show below that all these properties hold in general.

Smoothness Domains of the Root Branches
A crucial issue for our purposes is the distinction between dom φ σ and sdom φ σ , the subset of dom φ σ where φ σ (t) is smooth. As it will turn out in Section 4, sdom φ σ comprises the parametric values t for which the root φ σ (t) is simple-precisely the t's that count for s |σ| (t), Equation (19). Therefore sdom φ σ can be read not only as "smoothness domain" but also as "simplicity domain".
and, in general, so the conditions (C) amount to: therefore, which means that 0 ( . Such a point t b is a branching (or bifurcation) point of φ σ 1 ,...,±σ k+1 ,...,σ n (t) if both branches are defined in a neighborhood of t b ; otherwise, t b is an isolated point of dom φ σ 1 ,...,±σ k+1 ,...,σ n (actually, one can check that the isolated points of φ σ (t), if any, correspond to branching points of some predecessor). Branching points and isolated points are called singular points; the complement are the regular points of the corresponding root parabola or branches. This proves the following result: Proposition 5. The singular points t b of dom φ σ correspond to multiple zeros of q n t b (x). In either case, φ σ k+1 ,...,σ n (t b ) = 0 for some k = 0, 1, ..., n − 1.
Furthermore, if |σ| = s ≥ r = |ρ| and φ σ (t 0 ) = φ ρ (t 0 ), i.e., otherwise, keep squaring the Equation (34) and recursively applying Equation (35) to the resulting equalities to derive: or else By Proposition 5, t 0 is a singular point of φ σ (t). We conclude: Proposition 6. A root branch φ σ (t) can be smoothly extended from the boundary t = 2 to a maximal interval sdom φ σ : In other words, root branches do not cross or touch in their smoothness domains. Table 1, obtained from Figure 4, lists the smoothness domains (t σ , 2] of the 15 root parabolas up to rank 4.

Root Parabolas
Sdom The ordering of the branching points t σ is related to the ordering of the root branches. Due to the strictly increasing/decreasing monotonicity of the positive/negative square root function, Thus, attaching a sign "+" (resp. "−") in front the signature preserves (resp. reverses) the ordering. This generalizes to the following signed lexicographical order for root branches.

Remark 1. According to Equation
and lim are the optimal upper and lower bounds, respectively, of all root branches for t > 0. From Equations (15) and (18) we see that where µ = 1 4 (1 +
The "⊂" bifurcation at "time" t 1 corresponds to a local minimum (resp. local maximum) of q n t (x) crossing the x-axis from above (resp. below) at the point x 1 = φ σ 1 ,...,σ n (t 1 ) = 0 if k 1 = 0 or Figure 5b,c and Equation (33)). Bifurcation points with branches opening to the right occur at the left endpoint of the smoothness domains, in particular at t = 0, so they are certainly allowed.
The "⊃" bifurcation at "time" t 2 corresponds to local a minimum (resp. maximum) of q n t (x) crossing the x-axis from below (resp. above) at the point Figure 5b,c and Equation (33)). To show that bifurcation points with branches opening to the left, however, are not allowed, we are going to exploit the following Fact derived from the hypothetical existence of ⊃ bifurcations.
It is even more true: said change of sign occurs both in (x 2 − ε, x 2 ) due to the left branch, and in (x 2 , x 2 + ε) due to the right branch. Note that the above Fact does not hold for ⊂ bifurcations.
Therefore, we will consider only the first case (Figure 5b with x = x 2 and t = t 2 ). There are several subcases.
(a) If ∂q n t (x 2 )/∂t| t=t 2 > 0, then where 0 < x ≤ ε, 0 < t ≤ τ, and we wrote ∂ t q n t 2 (x 2 ) ≡ ∂ t q n t (x 2 )| t=t 2 for brevity. From Equation (45) it follows q n t i − t (x) < 0 in (x 2 − ε, x 2 + ε) for all t, which contradicts the above Fact. This excludes the possibility of having a ⊃ bifurcation at t 2 if the "velocity" of q n t (x 2 ) at t = t 2 is positive. (b) Suppose now ∂q n t 2 (x 2 )/∂t = 0, so where ∂ xx q n t 2 (x 2 ) ≡ ∂ xx q n t 2 (x) x=x 2 ≥ 0 (because q n t 2 (x 2 ) is a minimum in the case we are considering), ∂ tt q n t 2 (x 2 ) ≡ ∂ tt q n t (x 2 )| t=t 2 and similarly for the mixed term. (b1) If ∂ xx q n t 2 (x 2 ) > 0 and some of the other O(2) terms is not zero, let t → 0 while x is fixed to conclude from Equation (46) that q n t 2 −∆t (x 2 ± x) does not change sign for sufficiently small t, 0 < t x, in contradiction to the above Fact. The same contraction follows, of course, if all O(2) terms in Equation (46) except ∂ xx q n t 2 (x 2 ) vanish. (b2) If all terms O(2) vanish at x = x 2 and t = t 2 , repeat the same argument with the terms O(3). Since q n t 2 (x 2 ) is a minimum and q n t (x) is a polynomial of degree 2 n , it holds ∂ m x m q n t 2 (x 2 ) > 0 for some 2 ≤ m ≤ 2 n .

Remark 2.
The images of the critical point build a sequence of polynomials P n (t) := q n t (0), that is, P n (t) is the nth image of 0 under q t . Alternatively, one can define polynomial maps P n : [0, 2] → [−2, 2] by the recursion P 0 (t) = 0, P n (t) = t − P n−1 (t) 2 for n ≥ 1.
The first polynomials are: If, as in the proof of Proposition 8, we interpret the parameter t as time, then the time of passage of q n t (0) through the x-axis is given by the zeros of P n (t) = 0. Note that P n (0) = 0 for n ≥ 1, while P 1 (2) = 2, P n (2) = −2 for n ≥ 2 (51) (see Figure 2d for n = 4). In physical terms, q t (0) moves upwards from x = 0 (t = 0) to x = 2 (t = 2) at constant speedṖ 1 (t) = 1 (the dot denotes time derivative), while, for n ≥ 2, q n t (0) moves from x = 0 (t = 0) to x = −2 (t = 2), reversing the speed whenṖ n (t) = 0 and crossing the x-axis when P n (t) = 0. In Section 5 we will come back to these polynomials from a different perspective.
At this point we have already cleared our way to the monotonicity of the topological entropy for the quadratic family, where s k (t) is the number of simple zeros of q k t (x) or, equivalently, the number of transverse intersections of the curve y = q k t (x) with the critical line y = 0 (the x-axis); see Equation (19). According to Equation (41) and Remark 1, all zeros of q k It follows from Propositions 5 and 9, that, for each signature σ with |σ| = n, dom φ σ \sdom φ σ comprises multiple roots of q n t (t) (isolated points and the branching point t σ ), while the roots φ σ (t) are simple for t σ < t ≤ 2 by Proposition 6. The bottom line is: The smoothness domain sdom φ σ comprises the values of t for which the root φ σ (t) of q |σ| t (x) is simple.
For this reason we anticipated at the beginning of Section 3.2 that sdom φ σ may be called the simplicity domain of φ σ as well. This being the case, each root φ σ (t) contributes 0 or 1 to s n (t), the number of simple zeros of q n t (x), depending on whether 0 < t ≤ t σ or t σ < t ≤ 2, respectively. We conclude that s n (t) = ∑ σ∈{+,−} n for n ≥ 1, where we used t −σ 1 ,σ 2 ,...,σ n = t σ 1 ,σ 2 ,...,σ n , and χ (t  (19), is piecewise constant and monotonically increasing for every n ≥ 1. Its discontinuities occur at the branching points t σ of the root branches φ σ (t) with |σ| = n, where s n (t) is lower semicontinuous. Figure 6 shows the function s 4 (t) based on Figure 4c. Apply now Proposition 4 to prove Milnor's Monotonicity Conjecture for the quadratic family: Theorem 2. The topological entropy of q t is a monotonically increasing function of t. Figure 7 shows the topological entropy of q t superimposed on the bifurcation diagram ( Figure 3); in particular, h(q t ) > 0 (i.e., q t is chaotic) for t > t F = 1.4011551... (Feigenbaum point) and h(q 2 ) = log 2, the highest value that the topological entropy of a unimodal map can take, see Equation (10). It can be proved that the function t → h(q t ) is a Devil's staircase, meaning that it is continuous, monotonically increasing (Theorem 2), but there is no interval of parameters where it is strictly increasing [27,28]. The plateaus where h(q t ) is constant correspond to intervals containing a periodic attractor and the subsequent period doubling cascade (e.g., the period-3 window, clearly visible in Figure 7). This shows that periodic orbits do not disappear as t increases, however, the new periodic orbits that are created do not necessarily increase h(q t ). The topological entropy in Figure 7 was computed using the general algorithm presented in [9], but see [24] for a simpler and quicker algorithm adapted to unimodal maps. The small but positive values of h(q t ) to the left of t F are due to the slow convergence of the algorithm.

Application II: Superstable Period Orbits
In Section 4 we studied the solutions of the equation q n t (x) = 0, where the parameter t was thought to be fixed. In other words, we were interested in the zeros of a polynomial function of the variable x and, more particularly, in the values of t for which those zeros were simple. If we fix x instead, then the solutions of q n t (x) = 0 are the parametric values t such that x is an n-order preimage of 0, which is the critical point of q t . If, moreover, x = 0, then the solutions are the parametric values t for which the critical value is periodic with period n. As explained at the beginning of Section 2.3, these orbits are called superstable because then the derivative of q n t at each point of the periodic orbit is 0 (see Equation (12)). For the quadratic maps, q t (0) = t, so 0 is not a fixed point for t > 0.

Remark 3.
If in Section 4 our main concern were the transverse intersections of q n t (x) with the x-axis, in this section it will be the transverse intersection of the bisector with the positive root branches (if any).
In this regard, note that the bisector can intersect a positive root branch φ +,σ (t) at a regular point t 0 (i.e., t 0 ∈ (t +,σ , 2] = sdom φ +,σ ) only once and transversally, from above to below. Otherwise, the root parabola φ ±,−,σ (t) = ± t − φ +,σ (t) would have multiple branching points, contradicting Proposition 9. Singular points are isolated or at the boundary of smoothness domains (branching points), so the concept of transversal intersection do not apply to them.

Symbolic Sequences
To study the superstable periodic orbits of q t , it suffices to consider symbolic orbits. As an advantage, the results hold also under order-preserving conjugacies, as happens with q t | [−2µ,2µ] and f µ [0,1] under the affine transformation (18). We come back to this point below.
Consider a superstable periodic orbit (0, x 1 , x 2 , ..., x p−1 ) ∞ of prime period p ≥ 2, so that x k = 0 for k = 1, ..., p − 1. For the time being, we drop the exponent "∞" and rearrange the cycle as (x 1 , x 2 , ..., x p−1 , 0) so that the first point is the critical value (also the greatest value) q t (0) = t > 0. In this case, Σ k ∈ {+, −} for 1 ≤ k ≤ p − 1, so that we will fittingly use σ's instead of Σ's and write the pertaining symbolic sequence as (+, σ 2 , ..., σ p−1 , C). Therefore, by writing Σ = (+, σ 2 , ..., σ p−1 , C) we do not need to specify that Σ is a superstable cycle of prime period p. The parameter values for which q t has superstable cycles are discrete because q n t (0) = 0 is a polynomial equation in t for each n (see Remark 2); we will see below that there are infinitely many such values and that they accumulate at the right endpoint of the parametric interval, t max = 2.

Remark 4.
The main ingredient in the proof of Theorem 3 is the fact that sdom φ σ is a half-interval (t σ , 2] (Proposition 9), from which Milnor's Monotonicity Conjecture (Theorem 2) was derived. Reciprocally, from Theorem 3 it follows that the bisector can transversally intersect a positive root branch φ +,σ (t) only once. In turn, it recursively follows from this that the simplicity domains of the root branches are half-intervals and, hence, Milnor's Monotonicity Conjecture.
Two maps of the interval f (x) and g(y) are called combinatorially equivalent if they are conjugate via an order-preserving transformation ϕ(x). For instance, q t (x) and cq 1/c (y) = 1 − cy 2 are combinatorially equivalent via ϕ(x) = 1 t x and c = t, whereas q t (x) and −q −c (y) = c + y 2 are conjugate via ϕ(x) = −x and c = −t, but they are not combinatorially equivalent because ϕ(x) reverses the order in this case. It is plain that combinatorially equivalent maps have the same symbolic sequences for corresponding initial conditions x 0 and ϕ(x 0 ). Theorem 4 (Thurston's Rigidity [15]). Consider q t 1 and q t 2 for which their critical points c = 0 have finite orbits O and O . If q t 1 and q t 2 are combinatorially equivalent, then t 1 = t 2 .
Proof. Suppose that q t 1 and q t 2 are combinatorially equivalent via an order-preserving conjugacy ϕ. Then, the symbolic sequence Σ of O and the symbolic sequence Σ of ϕ(O) = O are equal. Apply now Theorem 3(a) to conclude t 1 = t 2 .
As mentioned in the Introduction, Thurston's Rigidity implies Milnor's Monotonicity Conjecture for the quadratic maps [15]. In Remark 4 we sketched how this derivation could be done using Theorem 3, which is a sort of symbolic version of Thurston's Rigidity.

Dark Lines and the Misiurewicz Points
To wrap up our excursion into the superstable cycles of the quadratic family, let us remind that the "dark lines" in the bifurcation diagram ( Figure 3) that go through the chaotic regions or build their boundaries are determined by the superstable periodic orbits. To briefly study those dark lines, we resort again to the polynomials P n (t) ≡ q n t (0) introduced in Equations (47) and (49). We have already discussed in Section 5.1 how to pinpoint superstable cycles (0, P 1 (t), ..., P p−1 (t)) ∞ in the parametric interval using symbolic sequences and root branches. More generally, consider orbits of 0 that are eventually periodic, that is: Such parametric values are called Misiurewicz points [31] and denoted as M h,T , where we assume that h ≥ 1 is the minimal length of the preperiodic "tail" (the preperiod) and T ≥ 1 is the prime period of the periodic cycle. Therefore, if M h,T is a Misiurewicz point, then so that the curves P h+kT (t), k ≥ 0, meet at t = M h,T in the (t, x)-plane. For example, (P n (2)) ∞ n=0 = (0, 2, (−2) ∞ ), i.e., the orbit of 0 hits a (repelling) fixed point after only two iterations. Comparison of Equations (59) and (57) shows that 2 = M 2,1 , therefore, all curves P n (t) with n ≥ 2 meet at t = 2 (see Equation (51)). The graphs of P 0 (t), ..., P 7 (t) are shown in Figure 10. As a first observation, one can recognize the main features of the chaotic bands in the bifurcation diagram of the quadratic family, in particular band merging. We also see that the curves x = P n (t) intersect transversally or tangentially; all these intersections are related to important aspects of the dynamic. Chaos bands merge where those curves intersect transversally, while periodic windows open where they intersect tangentially the upper and lower edges. Moreover, the functions P n (t) intersect the t-axis precisely at the parameter values for which 0 is periodic: P n (t 0 ) ≡ q n t 0 (0) = 0 ⇔ φ σ 1 ,...,σ n (t 0 ) = 0.

Conclusions
In the previous sections we have revisited two classical topics of the continuous dynamics of interval maps: entropy monotonicity (Section 4) and superstable cycles (Section 5) for the quadratic family q t (x) (Section 3). The novelty consists in the starting point: we use Equation (52) for h(q t ), the topological entropy of q t , where s n (t) is the number of transversal intersections of the polynomial curves q n t (x) with the x-axis. Equation (52) and several numerical schemes for its computation were derived in [7][8][9]. This approach leads directly to the root functions (φ σ (t)), bifurcation points (t σ ) and smoothness domains (sdom φ σ ) studied in Sections 3.1 and 3.2. It is precisely the structure of the smoothness domains, sdom φ σ = (t σ , 2] (Proposition 9), which implies that s n (t) is a nondecreasing staircase function for each n ≥ 1 (Theorem 1) and, in turn, that the function t → h(q t ) is monotone (Theorem 2). Unlike existing proofs [3,[11][12][13], Theorem 2 proves Milnor's Monotonicity Conjecture via real analysis. This also shows that the transversal intersections of a multimodal map and its iterates with the critical lines is a useful tool in one-dimensional dynamics. Sections 2.2 and 4 contains further details on Milnor's Monotonicity Conjecture and its generalization to multimodal maps.
In Section 5.1 we derived some basic results on the superstable cycles of the quadratic family, in particular Theorem 3, which is a sort of symbolic version of Thurston Rigidity (Theorem 4). The commonalities between entropy monotonicity and the superstable cycles of the quadratic maps go beyond the techniques used, namely, root branches, bifurcation points, transversality, and a geometrical language. There is also a flow of ideas in both directions. We started with the topological entropy and worked our way towards the superstable cycles, but the other direction works too, although we only indicated this possibility in Remark 4. We also made a brief excursion into the preperiodic orbits of the critical point in Section 5.2 (Misiurewicz points). In conclusion, both topics complement and intertwine in remarkable ways, as well as being interesting on their own. Funding: This work was financially supported by the Spanish Ministry of Science and Innovation, grant PID2019-108654GB-I00.