Core entropy of polynomials with a critical point of maximal order

This paper discusses some properties of the topological entropy of the systems generated by polynomials of degree \begin{document}$ d$\end{document} with two critical points. A partial order in the parameter space is defined. The monotonicity of the topological entropy of postcritically finite polynomials of degree \begin{document}$ d$\end{document} acting on Hubbard tree is generalized.

1. Introduction. One of the biggest issues in holomorphic dynamics is to understand the global dynamics of the polynomials of degree d. The family of quadratic polynomials has been studied by many researchers, hence there are many known results which are relevant, [5,2,9,10]. However, for higher degrees very little is known.
Given a function f defined on a compact set X to itself, the topological entropy h(f ) is the measure of the complexity degree of the dynamics of f and it is a way of measuring the chaos in the system, [13]. More precisely, it measures the growing rate of the number of orbits of f . It is known that given a polynomial P d : C → C of degree d, h(P d ) = log d. Even more, the entropy of P d restricted to its filled Julia set is the same as the entropy of P d restricted to its Julia set, and is equal to log d, [8].
The entropy for real polynomial functions was studied by Milnor, Thurston, Tresser and Douady among others, they studied the quadratic and cubic family, proved the monotonicity of entropy, [12,13,4]. On the other hand, Radulescu studied the behavior of the entropy for a real quartic family obtained by composition of two logistic maps, using symbolic dynamics, [15]. Thurston generalized the entropy concept defined for invariant intervals in the real case to the entropy study of a polynomial restricted to its Hubbard tree, which keeps the dynamics information of P d . This is called the core entropy, [17,18].
The core entropy can be used as a tool to classify the different dynamics of a polynomial family in its parameter space. For the quadratic polynomials it was proved that the core entropy grows through the veins of the Mandelbrot set, M, 116 DOMINGO GONZÁLEZ AND GAMALIEL BLÉ [18,8]. Recently, Tiozzo proved the continuity of the core entropy varying the external argument, [19]. In this paper we calculate the core entropy of a polynomial family P d of degree d with one parameter. The family is ). (1.1) The polynomial function P a (z) has two critical points, 0 that is a fixed point and −a that is a free point. The parameter space of this family was studied by Roesch in [16]. She studies the connectedness locus of this family and gives a description of the closures of the hyperbolic components and the Mandelbrot copies contained in this set. The goal of this work is to generalize the results obtained for quadratic and cubic families in [8]. We give a description of the dynamics in the family P d , showing the behavior of the entropy in Hubbard's trees. For this purpose, we define the following partial order in the parameter space. We say that a ≺ b if the polynomial P a is dynamically included in P b , and we obtain the next result: Main Theorem 1. Let a 1 , a 2 ∈ C such that P a1 , P a2 ∈ P d and their critical points are periodic. If a 2 ≺ a 1 then h(H(a 2 ), P a2 ) ≤ h(H(a 1 ), P a1 ), where H(a 1 ) and H(a 2 ) denote the Hubbard trees of P a1 and P a2 , respectively.
For the proof of this result, we use the external rays and the conjugation of the dynamic of polynomials of degree d with multiplication by d.

2.
Topological entropy. In this section we state the definition and some properties of the topological entropy for a function f defined on a compact set X to itself, where X is a topological compact set. These results and properties can be consulted in [1,4].
If α and β are two open covers of X, the refinement cover of α and β is the union of all sets A ∩ B, A ∈ α and B ∈ β and its denoted by α ∨ β.
where N (α) is the infimum number of open sets in any finite subcover. It can be seen that H(α) ≥ 0, and that the equality is achieved if and only if X ∈ α.
The limit exists and it is called the topological entropy of f relative to the cover α and satisfies The number is called the topological entropy of f , where the supremum is taken over all open covers of X.

2.1.
Properties of the entropy.
1. If f : X → X is a continuous function on a compact metric space X, then 3. If f : X → X and g : Y → Y are topologically conjugate continuous functions, then 3. Polynomial family of degree d. Let us consider a polynomial family of degree d ≥ 3 that has two critical points and one of them is a fixed critical point of maximum multiplicity. It is described modulo affine conjugacy as the polynomial family where the parameter a ∈ C and P a has two critical points, 0 is a critical point of maximum multiplicity and −a is a critical point of simple multiplicity. The topological properties of the closures of the hyperbolic components and Mandelbrot copies in the parameter space for this family are analyzed in [16]. The filled Julia set K a of P a consists of the points whose orbits do not go to infinity and the Julia set J a of P a is the boundary of K a .
For d fixed, the connectedness locus of the family P d is defined as The parameter space C is divided into two regions, C and W ∞ , where W ∞ is the set of all the parameters for which the free critical point −a is attracted to ∞. In the Figure 1 it is shown the connectedness locus for d = 4 and d = 5. In the connectedness locus, there is a special set of parameters, the set of parameters a ∈ C for which P a is post-critically finite, that is, the critical orbits are finite. This set is denoted by C 0 .
Note that K a = J a because K a contains the basin of attraction of 0, B a = {z ∈ C : P n a (z) → 0}. Let us denote by B a the inmediate basin of attraction of zero, this is the component ofB a containing 0. By the maximum modulus principle, B a is topologically a disk.
By Böttcher's Theorem, for the super-attracting fixed points p = 0, ∞ there are neighborhoods V p a , W p a of p such that P a (V p a ) ⊂ V p a , and conformal isomorphisms ϕ p a : V p a → W p a , such that the following diagrams are commutative, That is a tangent to the identity, in a neighborhood of ∞ and ϕ 0 a tangent to z → λ(a)z around zero, where λ(a) is a (d − 2)-th root of da d−1 , [5]. On the other hand, if a ∈ C 0 then J a is locally connected, and then the set V ∞ a is the set C \ K a , [16]. Definition 3.1. Set ψ = (ϕ ∞ a ) −1 . For θ ∈ T = R/Z, we define the external ray of argument θ by R a (θ) = ψ({re 2πiθ : r > 1}).
3.1. The parameter space. Since P τ a (τ z) = τ (P a (z)), where τ = e πi d−1 , the rotation is the only conformal conjugation between two polynomials P a , P a ∈ P d . Moreover, P a is conjugated to the polynomial P a by the complex conjugation σ(z) = z, [16]. In this way, a fundamental domain for the study of the polynomial family P a is According to Milnor, there are four possible types of bounded hyperbolic components in the polynomials family with two critical points [10]. For the family P d , we have three component types, the main component W 0 is the only adjacent component and there are other components of the capture type or disjoint type. In particular, we can classify the capture components in terms of the iterated to reach the basin of zero. The connectedness locus C is compact and connected. Moreover, W ∞ is isomorphic to C \ D, [16]. In fact, the isomorphism , where ϕ ∞ a is the Böttcher coordinate of P a at infinity, for a ∈ C \ C. From ϕ ∞ a we can define the external rays R a (θ) to K a and with φ ∞ we can define the external rays R C (θ) to C. In particular, if α ∈ Q the external ray R C (α) lands in the parameter a ∈ C or in the root of hyperbolic component containing a. In the dynamic plane of P a , the external ray R a (α) lands in P a (−a) or in the root of the Fatou component containing P a (−a), [16]. 4. Hubbard trees. Let a ∈ C, since P a has a super-attracting fixed point at 0, the interior int(K a ) of the filled Julia set is not empty. Every component U of int(K a ) is a bounded Fatou component, with closure U homeomorphic to the closed disc D. Also, by Sullivan's Theorem every component U ⊂ int(K a ) is eventually periodic, [2]. Therefore if −a is pre-periodic then U is mapped in the fixed Fatou component Since φ is unique, except for a rotation of D, the radial arcs are well defined.
An embedded arc in K a is any subset of K a which is homeomorphic to the closed interval [0, 1] ⊂ R. An embedded arc I is regulated if for every bounded Fatou component U , the intersection I ∩ U is either empty, a point, or it consists of radial arcs in U .
The following Lemma is shown in [5,20] for quadratic polynomials but the argument is valid for polynomials of degree d.
Lemma 4.1. Given any two points x, y ∈ K a , there exists a unique regulated arc I ∈ K with endpoints x, y. Thus, if η is any embedded arc in K a , which connects x with y, then I ∩ J a ⊂ η ∩ J a .
We denote the regulated arc I from previous Lemma by [x, y]. The open arc (x, y) is defined as [x, y]\{x, y}. Similarly we can define a semi open arc [x, y). In general, given a finite set of points {x 1 , x 2 , . . . , x n } ⊂ K a there exists a unique connected smallest set [x 1 , x 2 , . . . , x n ] ⊂ K a which consists of regulated arcs containing these points. This set is a topological tree and it is called the regulated tree generated by {x 1 , x 2 , . . . , x n }, [5].
Given a polynomial P a with a ∈ C 0 we define the Hubbard tree of P a as the smallest regulated tree that contains the orbits of the critical points and it is denoted by H(a). Some important points that characterize a Hubbard tree are the following: Directly of definition we have the following results, see [14].

Remark 4.2. The tree H is P-invariant.
Lemma 4.3. Let η be a regulated arc that does not contain critical points, except in its endpoints, then P a | η is injective and P a (η) is a regulated arc. Proof. Assume that a = 0 and that the free critical point −a is an endpoint of H(a) that is not fixed. Since any non extreme point that is mapped to an end point must be a critical point, and the critical point 0 is fixed, then there is no non extreme point that can be mapped to −a. Thus we conclude that the preimage of −a is an end point and therefore −a is periodic and its orbit is exactly the set of end points. Now assume that a = 0 and the free critical point −a is not an end point of H(a). Since P a is locally injective except in the critical points and the critical point 0 is a fixed point, then the only non extreme point that is mapped to an end point is the critical point −a. Therefore, the end points of H(a) are generated by the first iterations of the critical point −a.

5.
Entropy on the Hubbard tree. By the Remark 4.2, the Hubbard tree H(a) of a polynomial P a with postcritical finite set is invariant. Consequently, we can talk about the entropy of P a restricted to H(a) and we will denote it by h(H(a), P a ) 5.1. Small copies of the Mandelbrot set. Starting from the main component W 0 , across any direction t ∈ T periodic under the multiplication by d − 1, there is a small copy of the Mandelbrot set directly attached to W 0 , [16]. We denote the center of this copy by a(t), then the small copy of the Mandelbrot set can be represented by P a(t) * M.
Let us assume that the angle α ∈ Q and the critical point −a(α) has period k, then the Hubbard tree H(a(α)) consists of k radial edges coming out from a central vertex located in the zero, connecting a point of the orbit to the free critical point −a(α). Since the dynamics of polynomial P a(α) over H(a(α)) is conjugated to multiplication by d − 1 in these edges, h(H(a(α)), P a(α) ) = 0.
Any polynomial P b with a ∈ P a(t) * M has the form P a(t) ⊥ P c where P c is a quadratic polynomial with c ∈ M and ⊥ denote the tuning, [3]. The filled Julia set K b of P b can be obtained from the filled Julia set K a(t) as follows: the closure of each component U from the interior of K a(t) is replaced by a copy of K c , [3].
where A are k copies of the Hubbard tree H(c) and B is the Hubbard tree of P a removing the k regulated arcs containing the points of the periodic orbit. Since P k b is conjugated to P c , in each copy of H(c), then kh(A, To complete the proof, we must show that h(B, P a ) = h(H(a), P a ). Applying a similar idea to the previous one about the tree, we can write H(a), H(a) = A ∪ B, where A is the set of the regulated arcs containing the critical orbit of P a and B is the Hubbard tree of P a taking out the k regulated arcs of the critical orbit. Since the k-th iteration of P a restricted to any of these regulated arcs is conjugated to z d in the interval and this one has entropy zero, then h(A , P a ) = 0. Hence h(B, P a ) = h(H(a), P a ). Therefore Any copy of the Mandelbrot set P a(t) * M has in its tips a capture component C directly attached to P a(t) * M, [16].
Let b be any tip of P a(t) * M, Proof. Given a parameter b ∈ ∂C such that the free critical point −b is strictly pre-periodic for P b , there exists the smallest integer n such that P n b (−b) is periodic and P n b (−a) ∈ ∂B b . We denote by k the period of P n b (−b), let l ≤ n be the smallest integer such that P That is, the orbit of P l b (−a) consists of n + k − l points and all of them are in the boundary of B b . Let T be the tree generated by the regulated arcs connecting the orbit of P l b (−b), which has n + k − l edges with a common vertex, the super attracting critical point 0. Let b i be the edge that connects 0 and P

5.3.
External rays in the Hubbard tree. The entropy on the Hubbard tree for a given polynomial P a can be studied in a similar way as Douady in [4]. Using the semi-conjugation given by Böttcher's theorem we have that the entropy of P a in the Hubbard tree is the same that the entropy of the map z → dz restricted to a T J ⊂ T. This set can be obtained by the following diagram, where γ a : T → J a is the surjective and continuous map given by the Carathéodory's Theorem. Thus, h(H(a), P a ) = h(γ −1 a (H(a) ∩ J a ), ρ). To determine the set T J , we will analyze the external rays properties at J a , [5,6].
Since 0 is a super attracting fixed point, we give a description of the external rays that land on the Hubbard tree using [8]. Let P a be a polynomial with a ∈ C 0 . Given x ∈ H(a), we define θ(x) as the angles of the external rays that land on x, if x ∈ J a , or the angles of the external rays that land in the root of the Fatou component containing x, if x / ∈ J a . We denote by R a (ϑ(x)) the external ray with angle ϑ and landing on H(a). If there is more than one external ray that lands on x or in the root of Fatou component that contains x we write it as For a fixed a ∈ C 0 , we denoted by H = H(a) the Hubbard tree corresponding to P a . Also, H 1 = P −1 a (H) and H n+1 = P −1 a (H n ). Note that H ⊂ H 1 ⊂ H 2 ⊂ . . . . Definition 5.5. Let a ∈ C 0 and [x, y] be a regulated arc such that [x, y]∩[−a, 0] = ∅. We say that x ≺ y if x ∈ (0, y) and θ(x) = θ(y).
Let P a be a polynomial of degree d with a ∈ C 0 and let [x, y] be a regulated arc that does not contain critical points. Let {θ(x) − , θ(x) + , θ(y) − , θ(y) + } be the angles of the four external rays, such that [θ(x) − , θ(y) − ] and [θ(x) + , θ(y) + ] are the smallest intervals that contain all the external rays that land on [x, y]. The set is called the external rays associated to [x, y]. When only one ray lands in y, we have that θ(y) − = θ(y) + = θ(y), in this case the external rays associated to [x, y] are denoted by {R a (θ(x) − ), R a (θ(x) + ), R a (θ(y))}.
From the definition of the associated external rays, we have the next result.
Lemma 5.6. Let a ∈ C 0 and x, y ∈ H(a). If x ≺ y, then are in a positive cyclic order. Thus, ≺ is a partial order.

5.4.
Intervals of characteristic angles. Let P a be a post-critically finite polynomial and let H be the Hubbard tree associated to P a . Suppose that the free critical point −a is not an endpoint of H. We can describe the set H 1 \ H in a unique way as the finite union of semi open regulated arcs. Explicitly On the other hand, let H ∩ ∂B a = {z 1 , z 2 , . . . , z m } be the set of points on H that intersects the boundary of the inmediate basin B a of zero. The external rays that land on these points z l are denoted by R a (ε l ) ± for 1 ≤ l ≤ m, see the Figure 2. We define the subset of the circle We define the interval of the characteristic angles U(a) associated to the Hubbard tree H as U(a) = ν(a) ( n k=1 U k ).  Lemma 5.9. Given a polynomial P a with a ∈ C 0 , we have that an external ray R a (θ) lands on H if and only if the orbit of θ under ρ(θ) = dθ does not get inside U.

Remark 5.7. Any two regulated arcs of H 1 \ H have empty intersection or if
Proof. First we are going to prove the necessary condition. Since H is invariant, P a (H) = H. If R a (θ) lands on H then R a (ρ(θ)) also lands on H, this ensures that the orbit of θ never intersects U(a).
6. Monotonicity of the entropy. In this section we are going to analyze and compare the entropy of polynomials restricted to Hubbard trees (core entropy) for polynomials with periodic critical point −a. To obtain this, we propose a way to compare Hubbard trees following the ideas in [7].
We start introducing an equivalence relation on Hubbard trees, such that any equivalence class is characterized by the dynamics in the vertex set. Definition 6.1. Let H be the Hubbard tree associated to the polynomial P a , let P be the partition of H that has five elements {c 1 }, {c 2 }, T 0 , T 1 , T 2 (some of them might be empty), such that P a | Ti is a homeomorphism for all i. Thus c 1 , c 2 ∈ T 0 and for i = 1, 2, c i ∈ T i unless T i = ∅. P is called the partition associated to H, where c 1 = 0 and c 2 = −a.
With this partition we can define the itinerary for each point of H, as follows: Using this definition we will give a characterization of the Hubbard trees, which allows us to define an equivalence relation.
Using the previous definition we define a partial order in C 0 as follows: Let a 1 , a 2 ∈ C 0 , we will say that a 2 ≺ a 1 if the Hubbard tree H(a 2 ) is inside dynamically in the Hubbard tree H(a 1 ), and the Hubbard trees H(a 1 ) and H(a 2 ) are not in the same equivalence class. Remark 6.6. If a 2 ≺ a 1 then N (a 2 ) ≤ N (a 1 ).
Since the forcing orbit lemma is also valid in a copy of the Mandelbrot set (see [11]), we have the following: Lemma 6.7. Let M 0 ⊂ C be a small Mandelbrot set copy intersecting W 0 , a 1 , a 2 ∈ M 0 and P a1 , P a2 polynomials with periodic critical points. If the hyperbolic component containing a 2 split a 1 of W 0 then a 2 ≺ a 1 .
Lemma 6.8. Let a 1 , a 2 ∈ C 0 . If a 2 ≺ a 1 , N (a 1 ) = N (a 2 ) and z ∈ H(a 1 ) is the characteristic point given by Definition 6.5, then , where ∼ = means they are topologically homeomorphic.
Proof. We denote the number of endpoints by N (a 1 ) = N (a 2 ) = N and P a1 by P . By the property iv) from Definition 6.5, the external rays that land on z in the tree H(a 1 ) are the same external rays that land on P a2 (−a 2 ) in the tree H a2 . Thus z, P (z), P 2 (z), . . . , P N −1 (z) are exactly the endpoints of [z, P (z), P 2 (z), . . . , P N −1 (z)] and therefore H(a 2 ) ∼ = [P (z), P (z), . . . , P N (z)]. Now let us prove by induction that there are not branching points and critical points in the intervals (P k (z), P k (P (−a))) for k = 0, 1, . . . N − 1. Note that, 0, −a / ∈ (z, P (−a)) and there are not branching points in (x, 0), otherwise N (a 1 ) > N (a 2 ). Now, assume that there are not branching points or critical points in (P k (z), P k (P (−a))) for all k ≤ k 0 ≤ N − 1. Since P is injective in each regulated arc that does not contain a critical point, it follows that (P k0+1 (z), P k0+1 (P (−a))) does not contain any branching point.
Since, there are not branching points in (P k (z), P k (P (−a))) for k = 1, 2, . . . , N we have that H(a 1 ) ∼ = [P (z), P 2 (z), . . . , P N (z)]. Lemma 6.9. Let P = P a be as in the previous Lemma, let x, y ∈ H ∩ J a . Assume that P i (x) ≺ P i (y) for 0 ≤ i ≤ k then the following affirmations are valid: 1) If P k (y) ≺ y in H 1 , then P k (x) ≺ x in H 1 ; 2) If y ≺ P k (y) in H 1 , then x ≺ P k (x) in H 1 , where y is any preimage of P (y) different from y and x is the preimage of P (x) such that [x , y ] does not contain any critical points.
Proof. The set H 1 is the union of the Hubbard tree H and its preimages: Let us prove the first affirmation by contradiction. Assume that P k (y) ≺ y and x ≺ P k (x) in H 1 (a), then we have from this, it follows that P k (x), P k (y) ⊂ [x, y ], then P k restricted to [x , y ] is a homeomorphism onto itself, which is a contradiction with the expansion of P | Ja since x , y ∈ J a .
The proof of 2) is obtained by the same argument.
Proof. Note that ν(a 1 ) = ν(a 2 ). Furthermore, U(a 1 ) and U(a 2 ) are disjoint unions of intervals of angles. Each one of these intervals is associated with a regulated arc of the trees H 1 (a 1 ), H 1 (a 2 ), respectively. We will prove that for each interval of angles U ⊂ U(a 1 ) limited by the external rays not landing on B a , there exists an interval of angles U ⊂ U(a 2 ), such that U ⊂ U . We will denote the number of endpoints as N (a 1 ) = N (a 2 ) = N and P a1 by P .
From 1) and 2) we conclude the proof.
Next we show the result that compare the topological entropy in the Hubbard tree, of two comparable polynomials in C 0 . Theorem 6.11. Let a 2 and a 1 be two parameters that represents polynomials of degree d in C 0 each with periodic critical points, if a 2 ≺ a 1 and N (a 2 ) = N (a 1 ) then h(H(a 2 ), P a2 ) ≤ h(H(a 1 ), P a1 ).
Proof. By Theorem 6.10 we know that U(a 1 ) ⊂ U(a i ) and then Therefore, we have the following inequality of the entropy, Since we have the following conjugation between P ai and the function ρ
In the case of the polynomials with parameters a 2 ≺ a 1 whose Hubbard trees have different number of endpoints, we will calculate the core entropy using the Markov matrix associated to the Hubbard tree.
If P a is a hyperbolic polynomial, the restriction of P a to any edge e j of H(a) is injective. In addition P a (e j ) is the union of edges e 1 , . . . , e l of H(a). If H(a) has edges e 1 , . . . , e n , we define the Markov matrix M (a) associated to the H(a) as M ij = 1 if e j ⊂ P a (e i ) and 0 otherwise. By definition, the matrix M has dimension n × n and has entries 0 and 1. By Perron-Frobenius' Theorem, the spectral radius λ of M is greater or equal than 1, [1]. Proposition 6.12. Let M 0 ⊂ C be a Mandelbrot copy and let a ∈ M 0 be a parameter such that P a has periodic critical points. If H(a) is its Hubbard tree, then h(H(a), P a ) = log λ.
Proof. Let P be the partition of H(a) whose elements are the l edges e i of H(a). P satisfies that e i ∩e j is either an empty set or has only one point, for all 1 ≤ i < j ≤ k. The proof of this affirmation is obtained using the same arguments as in Lemma 1 in [4].
By the construction of P k a * S, we have that ∂S ⊂ ∂(P a * S) ⊂ · · · ⊂ ∂(P n−1 a * S) and . . , v n ), such that v 1 > 0, it is known that there exist constants 0 < c 1 < c 2 , such that c 1 · λ n ≤ (M T ) n · v ≤ c 2 · λ n .
Since v S = cardS, we have c 1 · λ n ≤ card n * S ≤ c 2 · λ n . Taking logarithm and dividing by n we have logc 1 · λ n n ≤ log card n * S n ≤ log c 2 · λ n n .
If we let n tends to ∞ then we obtain that h(H(a), P a , S) = log λ.
In a general setting, let M be a n × n matrix with entries 0 and 1, and spectral radius λ ≥ 1. Let X be a compact connected space and f : X → X be a continuous map. The set A = {X 1 , X 2 , . . . , X n } of compact subsets of X, such that int(X i ) is not empty and the X i 's mutually disjoint, it is called an over-Markov packing with matrix M , if X j ⊂ f (X i ) when M ij = 1. Proposition 6.14. If A = {X 1 , X 2 , . . . , X n } is an over-Markov packing with matrix M , then h(X, f ) ≥ log λ, where λ is the spectral radius of M .
Main Theorem 2. Let a 2 ≺ a 1 be parameters in a Mandelbrot copy M 0 intersecting W 0 , such that P a1 and P a2 are hyperbolic polynomials and postcritically finite, then h(H(a 2 ), P a2 ) ≤ h(H(a 1 ), P a1 ).
Proof. If N (a 1 ) = N (a 2 ) then we have the result by Theorem 6.11.
Hence we can assume that N (a 2 ) < N (a 1 ). By Remark 6.6 the number of endpoints is not decreasing. Moreover, without loss of generality, we can assume that does not exist a hyperbolic parameter c ∈ C 0 , such that c 2 ≺ c ≺ c 1 and N (c 2 ) < N (c) < N (c 1 ). Let H(a 1 ), H(a 2 ) be the Hubbard trees and let z be the characteristic point given by Definition 6.5.
Let H(z) be the regulated tree generated by the orbit of z. By definition H(z) is a subtree of H(a 1 ) and is not invariant under P a1 . We take as vertices of H(z), the union of orb(z) and the branch points of H(z). There is a bijection between the vertices of H(z) and the vertices of H(a 2 ). Moreover, by Definition 6.5, the tree H(z) is topologically homeomorphic to the Hubbard tree H(a 2 ).
Let e 1 , e 2 , . . . , e n be edges of H(a 2 ) and M (a 2 ) the Markov matrix associated to H(a 2 ). Then h(H(a 2 ), P a2 ) = log λ, where λ is the spectral radius of M . Since, H(z) is homeomorphic to H(a 2 ), we can label the edges of H(z) by e z 1 , e z 2 , . . . , e z n . These edges satisfy the condition that P a1 (e z i ) ⊃ e z j , whenever M ij = 1. Hence the set E z = {e z 1 , e z 2 , . . . , e z n } is an over-Markov packing with matrix M (a 2 ). Then h(H(a 1 ), P a1 ) ≥ log λ. Therefore h(H(a 1 ), P a1 ) ≥ h(H(a 2 ), P a2 ).