ELEMENTARY PROOF OF THE CONTINUITY OF THE TOPOLOGICAL ENTROPY AT θ = 1001 IN THE MILNOR THURSTON WORLD

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1 REVISTA DE LA UNIÓN MATEMÁTICA ARGENTINA Vol. 58, No., 207, Pages Published online: June 3, 207 ELEMENTARY PROOF OF THE CONTINUITY OF THE TOPOLOGICAL ENTROPY AT θ = 00 IN THE MILNOR THURSTON WORLD ANDRÉS JABLONSKI AND RAFAEL LABARCA Abstract. In 965, Adler, Konheim and McAndrew introduced the topological entropy of a given dynamical system, which consists of a real number that explains part of the complexity of the dynamics of the system. In this context, a good question could be if the topological entropy H top(f) changes continuously with f. For continuous maps this problem was studied by Misisurewicz, Slenk and Urbański. Recently, and related with the lexicographic and the Milnor Thurston worlds, this problem was studied by Labarca and others. In this paper we will prove, by elementary methods, the continuity of the topological entropy in a maximal periodic orbit (θ = 00) in the Milnor Thurston world. Moreover, by using dynamical methods, we obtain interesting relations and results concerning the largest eigenvalue of a sequence of square matrices whose lengths grow up to infinity. Introduction It is well know that one of the purposes of the topological theory of dynamical systems is to find universal models describing the topological dynamics of a large class of systems [6]. One of these models is the shift on n-symbols (Σ n, τ, σ), where Σ n is the set of sequences {θ : N 0 {0,, 2,..., n }} endowed with a certain topology, τ, and σ : Σ n Σ n is the shift map defined by (σ(θ))(i) = θ(i + ) (see [2, 3, 7, 2]). These models have been extensively used to obtain a great amount of information about maps defined in an interval, vector fields on three dimensional manifolds, and other kinds of dynamical systems (see, for instance, [3, 6, 9, 0]). By using a result in [8] which is a slight modification of a result proved by Block, Guckenheimer, Misiurewicz and Young ([5]) and elementary methods we compute the topological entropy of some periodic orbits, and using matrix (linear) algebra we are able to prove the continuity of the topological entropy at the periodic orbit θ = 00 in the Milnor Thurston world (the continuity was proved in [] but 200 Mathematics Subject Classification. 5B36, 37B0. Key words and phrases. Topological entropy, Incidence matrix, A-graph, Milnor Thurston world, Periodic orbit, Shift map. Part of this paper is an outgrowth of research during a visit of the second author to the Collège de France. We thank Collège de France and USACH for their support while preparing the present paper. 63

2 64 A. JABLONSKI AND R. LABARCA here we prove it by elementary methods). To do so, we consider a surjective two to one linear map defined in two subintervals of [0, ] which preserve orientation at the first interval and reverse orientation at the second one (T : I 0 I I = [0, ]). The maximal invariant set of this map can be modeled whit a succession of zeros and ones, i.e., full-shift of two symbols. The dynamics of the restriction of the linear map to this maximal invariant set endowed with the order and the topology induced by the interval is called the Milnor Thurston world (MTW) and it is a representation of the shift of two symbols with the order defined by Milnor and Thurston in [4]. While in [] the continuity of the topological entropy is proved by using the Hausdorff dimension, the novelty of the present paper is related with the techniques implemented: we build a graph associated with the maximal invariant set corresponding to the orbit θ = 00, that describes the dynamics of the system. Then we associate to this graph an incidence matrix; we compute its characteristic polynomial and we compute the largest real eigenvalue. Finally, the entropy h 0 associated with the sub-shift generated by the periodic orbit corresponds to the log of the largest real eigenvalue of the matrix ([8, 5]). To prove the continuity we compute the entropy at a family of periodic orbits h n := (00) n, and we prove that h n h 0 as n.. Preliminaries.. The Milnor Thurston world. Let T : I 0 I I = [0, ] be a linear map defined as follows: T I0 preserves orientation and T I reverses orientation, and both restrictions are surjective. For this map we can consider the set Λ T = k=0 T k (I 0 I ), and for any x Λ T we have: x Λ T x T k (I 0 I ), k N or, equivalently, T k (x) (I 0 I ), k N. Figure represents this map. 0 I 0 I (a) Representation of T 0 I 0 I (b) Representation of T 2 Figure. Graph of the map T. Clearly, we can associate to each x Λ T a succession of 0 s and s using the itinerary function I T : Λ T Σ 2, I T (x) = (I T (x)(i)), where I T (x)(i) = j T i (x) I j. Rev. Un. Mat. Argentina, Vol. 58, No. (207)

3 PROOF OF CONTINUITY OF TOPOLOGICAL ENTROPY AT θ = In Λ T we consider the Euclidean topology and we use the bijection I T to induce a topology in Σ 2 : A set A Σ 2 is open I T (A) is open in Λ T. In the same way, we can define an order induced by the map I T : θ T α in Σ 2 I T (θ) I T (α) in R. For being bijective and by inducing the topology and the order (in Σ 2 ) the map I T is an homeomorphism which preserves the order. Using this property and the fact that the shift function σ : Σ 2 Σ 2 satisfies σ(θ 0 θ θ 2...) = θ θ 2..., it is easy to check that the diagram in Figure 2 is commutative. Λ T Λ I T Σ σ Figure 2. I T Σ 2 Now we can formally define the Milnor Thurston world. A sequence α Σ 0 = {θ Σ 2 ; θ 0 = 0} is called minimal if σ i (α) T α for all i N 0. A sequence β Σ = {θ Σ 2 ; θ 0 = } is called maximal if σ i (β) T β for all i N 0. Let Min (resp. Max) denote the set of minimal (resp. maximal) sequences. For α Σ 0 and β Σ let Σ[α, β] = {θ Σ 2 ; α T σ i (θ) T β for any i N 0 } = j=0 σ j ([α, β]), where [α, β] = {θ Σ 2 ; α T θ T β}. Finally, the Milnor Thurston world (MTW) is the set of pairs (α, β) Min Max such that {α, β} Σ[α, β] (see [2, ]). Remark.. Σ[α, β] is a sub-shift of Σ Definitions of topological entropy. Definition.2. Let M and τ a topology such that (M, τ) is a compact space. Let A τ be a covering by open sets of M. We define the entropy of the covering A as H(A) = log(n(a)), where N(A) = min{card(a ) : A A is a finite subcovering of M}. Definition.3. Let A, B be two coverings of the space M. We define A B = {A B : A A, B B}. Now, let φ : M M be a continuous map, and φ (A) = {φ (U); U A}, the inverse image covering. Definition.4. The topological entropy of φ with respect to the covering A is n h(φ, A) = lim n n H( φ k (A)). k=0 Definition.5. The topological entropy of φ is h top (φ) = sup{h(φ, A)}, where the sup is taken over all the open coverings A of M. Finally, in our case, we define the entropy map H: Definition.6. H : MTW [0, ln(2)], H(α, β) = h top (σ ΛT ([α,β])). A Rev. Un. Mat. Argentina, Vol. 58, No. (207)

4 66 A. JABLONSKI AND R. LABARCA.3. Some notation and results. Let I R be an interval, f : I R a s continuous map, and A = {I,,..., I s } a partition of I such that I i = I, with i j and each pairwise intersection is at most one point. Definition.7. We say that I f covers n times J if there are subintervals K, K 2,..., K n on I (with disjoint interior) such that f(k i ) = J, i =, 2,..., n. Definition.8 (A-graph). Associated to the partition A of the interval I we define a A-graph oriented with vertices I,,..., I s such that if I i f covers n times I j then there are n arrows from I i to I j. Definition.9 (Incidence matrix). Associated to the A-graph, we define the incidence matrix M = (m ij ) s s where m ij is the number of arrows from I i to I j. Now we define the entropy associated to the A-graph. Definition.0 (Perron s eigenvalue). The entropy of the A-graph is defined as log(r(m)), where r(m) = max{ λ : λ eigenvalue of M}. Definition.. Let M = (m ij ) be an n n matrix. Associated to a finite sequence p = (p j ) k j=0 of elements from {, 2,..., n} we define its width as w(p) = k j= m p j p j. We will say that p is a path in M if w(p) 0. In this case, the number k = l(p) is called the length of the path p. A loop is a path p = (p 0, p,..., p k ) for which p i p i+, i = 0,,..., k and p k = p 0. A rome in M is a subset R {,..., n} for which there is no loop outside of R, i.e., there is no loop p = (p 0,..., p k ) with {p 0, p,..., p k } R =. Given a rome R and a path p = (p 0,..., p k ), we will say the path p is simple regarding R if {p 0, p k } R and {p,..., p k } R =. Finally, given a rome R = {r,..., r k } with r i r j for i j the matrix A R is defined by A R = A R (x) = (a ij ) k i,j= = (a ij(x)) k i,j=, where a ij (x) = p w(p)x l(p) and the sum is over all the simple paths that originate in r i and end at r j. 2. The main result The following proposition has an important role in proving the main result of this article, because we use it to compute the characteristic polynomial for a given matrix M. Proposition 2.. Given a matrix M = (m ij ) n n and a rome R = {I r0, I r,..., I rk } (where r i r j if i j) over M, then For the proof see [8]. p M (x) = det(m xi n ) = ( ) n k x n det(a R (x) I k ). Using this result and Perron s definition.0, we prove in the present paper the following theorem. i= Rev. Un. Mat. Argentina, Vol. 58, No. (207)

5 PROOF OF CONTINUITY OF TOPOLOGICAL ENTROPY AT θ = Theorem 2.2. Let h top be the topological entropy and σ the shift map. Then h top (θ) = H(σ(θ), θ) = h top (σ Λθ ) is continuous at θ = 00, where Λ θ = σ n ([σ(θ), θ]). n=0 In this paper we follow the line of research started in [4]. 3. Proof of the main result To prove the main result we will prove that the map h top is continuous from the right and from the left at the value θ = 00. Lemma 3.. h top is continuous from the right at θ = 00. Proof. In the MTW we have 00 < 000, so first we will prove h top (00) = h top (000). To do that we will compute the topological entropy for both periodic orbits. In order to do that we can use the following methodology: Given a maximal periodical orbit θ Σ 2 of period n, we have its maximal invariant set Λ θ = n=0 σ n ([σ(θ), θ]) (which is a finite sub shift; see Figure 3). Initially we produce a partition of the interval [σ(θ), θ] by using the induced order of the iterations σ i (θ), i {0,, 2,..., n}. I 0 0 Figure 3. Graph of Λ θ. I In the interior of this partition we can find the sequences 0θ and θ (= σ n (θ)). We include both values as part of the partition (one of these two values is part of the orbit of θ) and we compute the associated A-graph. With this we have associated the incidence matrix M = (m ij ) n n. Finally, we choose a rome to compute the polynomial matrix A R (x) and we calculate the maximal real root of the characteristic polynomial. Rev. Un. Mat. Argentina, Vol. 58, No. (207)

6 68 A. JABLONSKI AND R. LABARCA The topological entropy at θ = 00: First, we apply the shift map to the periodic orbit: 0: θ = 00 (3.) : σ(θ) = 00 (3.2) 2: σ 2 (θ) = 00 (3.3) 3: σ 3 (θ) = 00 (3.4) From (3.) and (3.4) we have θ > σ 3 (θ); (3.2) and (3.3) imply σ 2 (θ) > σ(θ); finally, σ 3 (θ) > σ 2 (θ), then we have θ > σ 3 (θ) > σ 2 (θ) > σ(θ). Defining 0θ = 000 and θ = 00 we have θ = 00 = = 00 = σ 3 (θ); also, 0θ > σ 2 (θ). Expanding the inequality: θ > θ = σ 3 (θ) > 0θ > σ 2 (θ) > σ(θ). Figure 4 shows the relationship between the iterations of the periodical orbit and the associated A-graph. Its incidence matrix is θ I θ θ 2 σ θ) σ θ) σ θ (a) A-graph associated to 00 (b) Graph of the shift map associated to 00 Figure 4. Topological entropy at θ = 00. We consider a rome with a unique element, I, so the closed paths are :, 2 3 and 3, so the A R (x) matrix is given by x + x 2 + x 3. Applying Proposition 2. we have: ( ) 3 x 3 det(a R (x) I) = x 3 (x + x 2 + x 3 ) = + x + x 2 x 3. Finally, we have p(00)(x) = + x + x 2 x 3, whose largest real root is We conclude that h top (00) log( ). From now on, the vector (x, y, z, u, v, w) corresponds to x y z u v w. Rev. Un. Mat. Argentina, Vol. 58, No. (207)

7 PROOF OF CONTINUITY OF TOPOLOGICAL ENTROPY AT θ = The topological entropy at θ = 000: For the iterations of the sequence 000 we have: 0: θ = 000 (3.5) : σ(θ) = 000 (3.6) 2: σ 2 (θ) = 000 (3.7) 3: σ 3 (θ) = 000 (3.8) Defining 0θ = 0000 and θ = 000 we have 0θ = 0000 = = 000 = σ 3 (θ), also 0θ > σ 2 (θ). Expanding the inequalities: θ > θ > 0θ = σ 3 (θ) > σ 2 (θ) > σ(θ). The graph associated to the shift map associated to 000 is the same as the previous one. This implies that we obtain the same value for the topological entropy. Hence, we have h top (00) = h top (000), so for any subsequence η n Σ 2 such that 00 η n 000 we have h top (00) h top (η n ) h top (000). Applying the limit over n we conclude: lim n h top(η n ) = h top (00) = h top (000). Remark 3.2. To complete the proof of Proposition 2.2 we have to prove the continuity of topological entropy at 00 from the left. This is essentially the main difficulty of this paper. Lemma 3.3. h top is continuous from the left at 00. To prove this lemma we will use a subsequence θ n that converges from the left to 00, then we calculate h top for θ n, with n =, 2, 3, 4, 5, to find a pattern on the subsequence of the roots of the characteristic polynomials. Finally, we will show that for any sequence between a subsequence θ k(n) and θ k(n)+ its topological entropy converges to the topological entropy of 00. However, this procedure is not direct and we have to prove it by induction. 3.. Induction procedure. 3 Step. Deduction of the general procedure. For θ = 00 we can order its iterations like 0 > 3 > 4 = θ > 0θ > 2 >. This allows us to define the following intervals: I = [θ = σ 4 (θ ), σ 3 (θ )], = [σ 3 (θ ), θ ], = [σ(θ ), σ 2 (θ )], = [σ 3 (θ ), 0θ ]. By ordering them we get Figure 5. I σ θ 2 σ θ 0θ 4 3 σ (θ)=θ σ (θ) θ Figure 5. Intervals associated to θ. Rev. Un. Mat. Argentina, Vol. 58, No. (207)

8 70 A. JABLONSKI AND R. LABARCA Also, we can check that I and obtain the A-graph of θ by the associated graph of σ θ (Figure 6). I I (a) A-graph associated to θ 0 I 2 3 (b) shift map associated to θ Figure 6. Representation of the dynamics associated to θ. The restriction σ θ has the following dynamics: I σ-cover I and ; σ-cover and ; σ-cover and I ; and σ-cover (Figure 7). I I I Figure 7. Representation of the σ-covering associated to θ. Then We summarize the loops and paths associated to the rome R = {, 4}: a : I I : (, ) and (, 2, 3, ), a 4 : I : (, 2, 4) and (, 2, 3, 4), a 4 : I : (4, 2, 3, ), a 44 : : (4, 2, 4) and (4, 2, 3, 4). Finally, the transition matrix A R (x) is ( ) x + x 3 x 2 + x 3 x 3 x 2 + x 3. p (x) = ( ) 4 2 x 4 det(a R (x) I) = xp(00)(x) +, whose maximal real root is λ = So we have that h top log( ). Rev. Un. Mat. Argentina, Vol. 58, No. (207)

9 PROOF OF CONTINUITY OF TOPOLOGICAL ENTROPY AT θ = 00 7 For θ 2 = (00) 2 = 0000 we can order its iterations like 0 > 4 > 7 > 3 > 8 = θ 2 > 0θ 2 > 6 > 2 > 5 >. This order allows us to define the following intervals: I = [σ 3 (θ 2 ), σ 7 (θ 2 )], = [σ 7 (θ 2 ), σ 4 (θ 2 )], = [σ 5 (θ 2 ), σ 2 (θ 2 )], = [σ 6 (θ 2 ), 0θ 2 ], = [σ 4 (θ 2 ), θ 2 ], = [σ(θ 2 ), σ 5 (θ 2 )], = [σ 2 (θ 2 ), σ 6 (θ 2 )], = [θ 2 = σ 8 (θ 2 ), σ 3 (θ 2 )]. By ordering them we get Figure 8. I ( ) ( ) ( ) ( ) ( )= ( ) ( ) ( ) Figure 8. Intervals associated to θ 2. Also, we can check that I and obtain the A-graph of σ θ2 (Figure 9). 4 7 I 3 I I (a) A-graph associated to θ 2 (b) shift map associated to θ 2 Figure 9. Representation of the dynamics associated to θ 2. The restriction σ θ2 has the following dynamics: I σ-cover I, and ; σ-cover, and ; σ-cover and ; σ-cover and ; σ-cover ; σ-cover ; σ-cover I ; and σ-cover (Figure 0). From this representation we can obtain the loops and paths of the rome R = {, 4}: a : (, ); (, 2, 7, ); (, 8, 5, 6, 7, ); (, 2, 3, 8, 5, 6, 7, ), a 4 : (, 2, 4) and (, 2, 3, 4), a 4 : (4, 2, 7, ); (4, 5, 6, 7, ); (4, 2, 3, 8, 5, 6, 7, ), a 44 : (4, 2, 4) and (4, 2, 3, 4). Rev. Un. Mat. Argentina, Vol. 58, No. (207)

10 72 A. JABLONSKI AND R. LABARCA I ; Figure 0. Representation of the σ-covering associated to θ 2. I Finally, the transition matrix A R (x) is: ( ) x + x 3 + x 5 + x 7 x 2 + x 3 x 3 + x 4 + x 7 x 2 + x 3. Then, p 2 (x) = ( ) 8 2 x 8 det(a R (x) I) = x 4 p (x) (x 3 + x 2 + x ). For the next steps (3, 4 and 5) we just show the main stage of each iteration. Intervals associated to θ i, i = 3, 4, 5 (Figures 3): I 0 I I σ θ σ θ σ (θ) σ (θ) σ (θ) σ (θ) θ θ σ (θ) σ (θ) σ (θ) σ (θ) σ (θ) θ Figure. Intervals associated to θ 3. I 0 I I ( ) σ (θ) σ (θ) σ (θ) σ (θ) σ (θ) σ (θ) σ (θ) θ θ σ (θ) σ (θ) σ (θ) σ (θ) σ (θ) σ (θ) σ (θ) θ Figure 2. Intervals associated to θ 4. I I I I 3 4 σ θ σ 5 θ σ 6 θ 7 8 σ θ σ θ σ θ σ θ σ θ σ θ σ θ 0θ θ σ θ σ θ σ θ σ θ σ θ σ θ σ θ σ θ σ θ θ Figure 3. Intervals associated to θ 5. Rev. Un. Mat. Argentina, Vol. 58, No. (207)

11 PROOF OF CONTINUITY OF TOPOLOGICAL ENTROPY AT θ = Representation of the A-graphs and shift maps associated to θ i, i = 3, 4, 5 (Figures 4 6): I I I 0 (a) A-graph associated to θ I I 0 I I (b) shift map associated to θ 3 I I 0 Figure 4. Representation of the dynamics asssociated to θ 3. Rev. Un. Mat. Argentina, Vol. 58, No. (207)

12 74 A. JABLONSKI AND R. LABARCA I I2 I I 0 I9 (a) A-graph associated to θ 4 θ θ I 0θ θ θ θ I 0 I I I I 0 (b) shift map associated to θ 4 Figure 5. Representation of θ 4. Rev. Un. Mat. Argentina, Vol. 58, No. (207)

13 PROOF OF CONTINUITY OF TOPOLOGICAL ENTROPY AT θ = I I5 I I0 (a) A-graph associated to θ 5 θ θ 0θ I θ θ θ I 0 I 0 I (b) shift map associated to θ 5 I 0 I I 0 Figure 6. Representation of θ 5. Rev. Un. Mat. Argentina, Vol. 58, No. (207)

14 76 A. JABLONSKI AND R. LABARCA σ-coverings associated to θ i, i = 3, 4, 5 (Figures 7 9): I I 2 I 0 I I ; Figure 7. σ-covering associated to θ 3. I I I I 0 I I Figure 8. σ-covering associated to θ 4. I I I 0 0 I 0 I I ; Figure 9. σ-covering associated to θ 5. Rev. Un. Mat. Argentina, Vol. 58, No. (207)

15 PROOF OF CONTINUITY OF TOPOLOGICAL ENTROPY AT θ = A R (x) matrices associated to θ i, i = 3, 4, 5: For θ 3 : ( ) x + x 3 + x 5 + x 7 + x 9 + x x 2 + x 3 x 3 + x 4 + x 7 + x 8 + x x 2 + x 3, p 3 (x) = ( ) 2 2 x 2 det(a R (x) I) = x 4 p 2 (x) (x 3 + x 2 + x ). For θ 4 : ( ) x + x 3 + x 5 + x 7 + x 9 + x + x 3 + x 5 x 2 + x 3 x 3 + x 4 + x 7 + x 8 + x + x 2 + x 5 x 2 + x 3, p 4 (x) = ( ) 6 2 x 6 det(a R (x) I) = x 4 p 3 (x) (x 3 + x 2 + x ). For θ 5 : ( ) x + x 3 + x 5 + x 7 + x 9 + x + x 3 + x 5 + x 7 + x 9 x 2 + x 3 x 3 + x 4 + x 7 + x 8 + x + x 2 + x 5 + x 6 + x 9 x 2 + x 3, p 5 (x) = ( ) 20 2 x 20 det(a R (x) I) = x 4 p 5 (x) (x 3 + x 2 + x ) Step 2. Generalization. Considering the previous constructions, we can assume (inductively) the following order for θ = θ k = (00) k and its iterations: θ > 4 > 8 > 2 > 6 > 20 > > 4(k ) > 4k > 4(k ) > 4(k 2) > > 23 > 9 > 5 > > 7 > 3 > 4k = θ k > 0θ k > 4k 2 > 4(k ) 2 > 4(k 2) 2 > > 22 > 8 > 4 > 0 > 6 > 2 > 4k 3 > 4(k ) 3 > 4(k 2) 3 > > 2 > 7 > 3 > 9 > 5 >. This order of the iterations allows us to define the intervals I = [σ 4k 5 (θ k ), σ 4k (θ k )] = [σ 4k (θ k ), σ 4(k ) (θ k )] = [σ 4k 3 (θ k ), σ 2 (θ k )] = [σ 4k 2 (θ k ), 0θ k ] = [σ 4 (θ k ), θ k ] = [σ(θ k ), σ 5 (θ k )] = [σ 2 (θ k ), σ 6 (θ k )] = [σ 3 (θ k ), σ 7 (θ k )] = [σ 8 (θ k ), σ 4 (θ k )] I 0 = [σ 5 (θ k ), σ 9 (θ k )] I = [σ 6 (θ k ), σ 0(θ k )] = [σ 7 (θ k ), σ (θ k )]. k 2 = [σ 4k 7 (θ k ), σ 4k 3 (θ k )] k = [σ 4k 6 (θ k ), σ 4k 2 (θ k )] k = [θ k = σ 4k (θ k ), σ 3 (θ k )], whose representation, over the Milnor Thurston world, is given by Figure 20:. I 0 k-3 I k- k k-4 I k ( ) 4k k σ (θ) σ (θ) σ (θ) σ (θ) σ (θ) σ (θ) σ (θ) σ (θ) σ (θ) 0 σ (θ) σ (θ) σ (θ) 4k-4 σ (θ) σ (θ) σ (θ) σ (θ) 4k- Figure 20. Order of the intervals associated to θ k. Rev. Un. Mat. Argentina, Vol. 58, No. (207)

16 78 A. JABLONSKI AND R. LABARCA Also, we can check that I and that the associated graph of σ θk in Figure 2; its diagram of σ-covering is given by Figure 22. is as shown θ 2 4k-2 4k-8 4k-4 4k- 4k-5 4k θ 0θ 4k-2 4k-6 0 k-7 k-3 I k-4 k-8 k k- k k-3 4k-7 I k-2 k k-9 4k-5 4k- 4k-4 4k-8 5 4k-7 4k k-6 4k-2 0θ θ θ 4k-2 k-6 I k-2 k-5 k- k I I k-8 k-4 2 k-3 k-7 Figure 2. Shift map associated to θ k. I I k-5 k (k-) k-7 k k- k-3 k k-3 k-2 k- I Figure 22. Diagram of σ-covering associated to θ k. Rev. Un. Mat. Argentina, Vol. 58, No. (207)

17 PROOF OF CONTINUITY OF TOPOLOGICAL ENTROPY AT θ = From that diagram we can deduce the loops and paths associated to the rome R = {, 4}. With this we can obtain its associated A R (x) matrix ( ) A k a (x) x R(x) = 2 + x 3 a 2 (x) x 2 + x 3, where a (x) = x + x x (4(k ) 3) + x (4(k ) ) + x (4k 3) + x (4k ) a 2 (x) = x 3 + x x (4(k ) 4) + x (4(k ) ) + x (4k 4) + x (4k ). Finally, for the characteristic polinomial associated to the matrix defined by the A-graph, we have: p k (x) = ( ) 4k 2 x 4k det(a k R(x) I) = x 4 p k (x) (x 3 + x 2 + x ) Step 3. Proof for the case n = k +. Now, let us take θ k+ = (00) k+. It is not hard to see that θ k+ and its iterations satisfy: (σ j (θ k+ ), j = 0,, 2,..., 4k+ 4): θ > 4 > 8 > 2 > 6 > > 4(k 2) > 4(k ) > 4k > 4k + 3 > 4k > 4(k ) > > 9 > 5 > > 7 > 3 > 4k = θ k+ > 0θ k+ > 4k + 2 > 4k 2 > 4(k ) 2 > > 22 > 8 > 4 > 0 > 6 > 2 > 4k + > 4k 3 > 4(k ) 3 > > 2 > 7 > 3 > 9 > 5 >. (A proof of this fact will be given later in this section). This order allows us to define the following intervals: I = [σ 4k (θ k+ ), σ 4k+3 (θ k+ )] = [σ 4k+3 (θ k+ ), σ 4k (θ k+ )] = [σ 4k+ (θ k+ ), σ 2 (θ k+ )] = [σ 4k+2 (θ k+ ), 0θ k+ ] = [σ 4 (θ k+ ), θ k+ ] = [σ(θ k+ ), σ 5 (θ k+ )] = [σ 2 (θ k+ ), σ 6 (θ k+ )] = [σ 3 (θ k+ ), σ 7 (θ k+ )] = [σ 8 (θ k+ ), σ 4 (θ k+ )] I 0 = [σ 5 (θ k+ ), σ 9 (θ k+ )]. k+ = [σ 4k 4 (θ k+ ), σ 4k (θ k+ )] k+2 = [σ 4k 3 (θ k+ ), σ 4k+ (θ k+ )] k+3 = [σ 4k 2 (θ k+ ), σ 4k+2 (θ k+ )] k+4 = [θ k+ = σ 4k+3 (θ k+ ), σ 3 (θ k+a )], whose representation, over the Milnor Thurston world, is given by Figure 23:. I 0 I 4 I I I I 5 k+4 I I I 4k+ I I I k-3 4k- 4k+3 4 4k-4 2 I k 4k ( ) 4k-4 4k σ (θ) σ (θ) σ (θ) σ (θ) σ (θ) σ (θ) σ (θ) σ (θ) σ 4k-2 4k+2 (θ) 0 σ (θ) σ (θ) σ (θ) 4k- σ 4k+3 (θ) 4k σ (θ) σ (θ) σ (θ) σ (θ) σ (θ) σ (θ) Figure 23. Order of the intervals associated to θ k+. Also, we can check that I and that the graph associated to σ θk+ shown in Figure 24; its diagram of σ-covering is given by Figure 25. is as Rev. Un. Mat. Argentina, Vol. 58, No. (207)

18 80 A. JABLONSKI AND R. LABARCA θ 4k-8 4k-4 4k 4k+3 4k- 4k-5 k-3 k+ I k k θ 0θ 4k+2 4k k+ 4k k-4 4k k-2 4k+2 0θ θ 3 7 4k-5 4k- 4k+3 4k 4k-4 4k-8 θ I 6 k-3 I I 4k+ k- k+3 k+4 I 8 I I I 2 k-4 k 2 k+ k-3 k+4 k+3 k- I k+2 k-2 Figure 24. Shift map associated to θ k+. I I k k k-7 k- (k+) k+3 k-3 k- k... k+3 I (k+) Figure 25. Diagram of σ-covering associated to θ k+. Rev. Un. Mat. Argentina, Vol. 58, No. (207)

19 PROOF OF CONTINUITY OF TOPOLOGICAL ENTROPY AT θ = 00 8 We can obtain the loops and paths associated to the rome R = {, 4} and their associated A R (x) matrix, ( ) A k a (x) x R(x) = 2 + x 3 a 2 (x) x 2 + x 3, where a (x) = x + x x (4k 3) + x (4k ) + x (4k+) + x (4k+3) a 2 (x) = x 3 + x x (4k 4) + x (4k ) + x 4k + x (4k+3). Then, if we write ( A k a R(x) = we can see that Then, k a 4 k a 4 k a 44 k ) (, A k+ a R (x) = k+ a 4 k+ a 4 k+ a 44 k+ a k+ = a k + x (4k+) + x (4k+3), a 4 k+ = a 4 k, a 4 k+ = a 4 k + x (4k+3) + x 4k, a 44 k+ = a 44 k. p k+ (x) = ( ) 4(k+) 2 x 4(k+) det(a k+ R (x) I) = x 4k+4 a k+ a k+ 4 a k+ 4 a k+ 44 = x 4k+4 a k + x (4k+) + x (4k+3) a k 4 a k+ 4 + x (4k+3) + x 4k a k 44 = x 4(k+)[ ((a k ) + x (4k+) + x (4k+3) )(a k 44 ) (a k 4 + x (4k+3) + x 4k )(a k 4) ] = x 4(k+)[ {(a k )(a k 44 ) a k 4a k 4} Since a k 44 = a k 4 we have: ), + (a k 44 )(x (4k+) + x (4k+3) ) a4 k (x (4k+3) + x 4k ) ]. = x 4 p k (x) + x 4k+4 [a k 44x (4k+) x (4k+) x (4k+3) a k 4x 4k ] = x 4 p k (x) + x 4k+4 [x (4k+3) + x (4k+4) x (4k+) x (4k+3) x (4k+2) x (4k+3) ] = x 4 p k (x) + x 4k+4 [x (4k+4) x (4k+3) x (4k+2) x (4k+) ] = x 4 p k (x) + ( x x 2 x 3 ) = x 4 p k (x) (x 3 + x 2 + x ). Therefore p k+ (x) = x 4 p k (x) (x 3 + x 2 + x ). Rev. Un. Mat. Argentina, Vol. 58, No. (207)

20 82 A. JABLONSKI AND R. LABARCA So, let us prove that iterations of the sequence θ k+ appear in the given order. For that we consider: σ(θ k+ ) = σ((00) k+ ) = (00) k+ 2 σ() = σ( ) = σ(0(00) k 0) = (00) k 00 3 σ(2) = σ(0(00) k 0) = (00) k 00 4 σ(3) = σ((00) k 00) = (00) k (00) 5 σ(4) = σ((00) k (00)) = (00) k (00) 6 σ(5) = σ(0(00) k 000) = (00) k 0(00)0 7 σ(6) = σ(0(00) k (00)00) = (00) k (00)00 8 σ(7) = σ((00) k (00)00) = (00) k (00) 2 9 σ(8) = σ((00) k (00) 2 ) = (00) k (00) 2 0 σ(9) = σ(0(00) k 2 0(00) 2 ) = (00) k 2 0(00) 2 0 σ(0) = σ(0(00) k 2 (00) 2 0) = (00) k 2 (00) σ() = σ((00) k 2 (00) 2 00) = (00) k 2 (00) 3 3 σ(2) = σ((00) k 2 (00) 3 ) = (00) k 2 (00) 3 4 σ(3) = σ(0(00) k 3 0(00) 3 ) = (00) k 3 0(00) σ(4) = σ(0(00) k 3 (00) 3 0) = (00) k 3 (00) σ(5) = σ((00) k 3 (00) 3 00) = (00) k 3 (00) 4 So, 0 = (00) k+ ; 4 = (00) k (00); 8 = (00) k (00) 2 ; 2 = (00) k 2 (00) 3 ; 6 = (00) k 3 (00) 4. = 4j 4 = (00) k (j 2) (00) j, j. On the other hand, = (00) k+ ; 5 = (00) k (00); 9 = (00) k (00) 2 ; 3 = (00) k 2 (00) 3 = 4j 3 = (00) k (j 2) (00) j, j. Also, 2 = (00) k 00; 6 = (00) k 0(00)0; 0 = (00) k 2 0(00) 2 0; 4 = (00) k 3 0(00) 3 0 = 4j 2 = (00) k (j ) 0(00) j 0, j. Finally, 3 = (00) k 00; 7 = (00) k (00)00; = (00) k 2 (00) 2 00; 5 = (00) k 3 (00) 3 00 = 4j = (00) k (j ) (00) j 00, j. Rev. Un. Mat. Argentina, Vol. 58, No. (207)

21 PROOF OF CONTINUITY OF TOPOLOGICAL ENTROPY AT θ = Taking j = k we have 4k 4 = (00) 2 (00) k 4k 3 = (00) 2 (00) k 4k 2 = (00)0(00) k 0 4k = (00)(00) k 00. Iterating the previous part by σ 4, we get: 4(k + ) 4 = (00)(00) k 4(k + ) 3 = (00)(00) k 4(k + ) 2 = 0(00) k 0 4(k + ) = (00) k 00. So, from the expressions of the iterations of θ k+ = (00) k+ we conclude that θ > 4 > 8 > 2 > 6 > > 4(k ) > 4k > 4k +3 > 4k > 4(k ) > > > 9 > 5 > > 7 > 3 > 4k = θ k > 0θ k > 4k + 2 > 4k 2 > 4(k ) 2 > > 22 > 8 > 4 > 0 > 6 > 2 > 4k + > 4k 3 > 4(k ) 3 > > 2 > 7 > 3 > 9 > 5 >. Which completes the inductive procedure and we have the characteristic polynomials. Let us now show the existence of the largest real root of the polynomials p((00) k )(x), k =, 2, 3,... ; later we will see that this sequence of roots is increasing and tends to the largest root of the polynomial p(00)(x). Let x = be the largest real root of the polynomial p(00)(x). We have: and p((00))(x) = xp(00)(x) + p((00))(x) = > 0 p((00))() = p(00)() + = ( ) + < 0. So, x ], x[ such that p((00))(x ) = 0. We have < x < x. From p((00))(x) = x( + x + x 2 x 3 ) + = x 4 x 3 x 2 x +, we obtain for x x, p((00))(x) p((00))(x) =, i.e., x is the largest real root of p((00))(x). For k = 2, Therefore, p((00) 2 )(x) = x 4 p((00))(x) (x 3 + x 2 + x ) = x 4 ( xp(00)(x) + ) (x 3 + x 2 + x ) = p((00) 2 )(x) = x 4 () (x 3 + x 2 + x ) = x ( x 3 + x 2 + x + ) +. }{{} 0 p((00) 2 )(x) = > 0. Rev. Un. Mat. Argentina, Vol. 58, No. (207)

22 84 A. JABLONSKI AND R. LABARCA Also, p((00) 2 )(x ) = (x 3 + x 2 + x ) < 0. Therefore x 2 ]x, x[ such that p((00) 2 )(x 2 ) = 0. We have < x < x 2 < x. Now, p((00) 2 )(x) = x 4 (x 4 x 3 x 2 x + ) (x 3 + x 2 + x ) = x 5 (x 3 x 2 x ) + x(x 3 x 2 x ) +. Therefore, if x x, p((00) 2 )(x) p((00) 2 )(x) =, i.e., x 2 is the largest real root of p((00) 2 )(x). For k = 3, p((00) 3 )(x) = x 4 p((00) 2 )(x) (x 3 + x 2 + x ) = p((00) 3 )(x) = x 4 (x 3 + x 2 + x ) = x ( x 3 + x 2 + x + ) + =. }{{} 0 Therefore, p((00) 3 )(x) = > 0. Also, p((00) 3 )(x 2 ) = (x x x 2 ) < 0. Then, x 3 ]x 2, x[ such that p((00) 3 )(x 3 ) = 0. We have < x < x 2 < x 3 < x. Now, p((00) 3 )(x) = x 4 (x(x 4 + )(x 3 + x 2 + x ) + ) (x 3 + x 2 + x ). Therefore, if x x, p((00) 3 )(x) p((00) 3 )(x) =, i.e., x 3 is the largest real root of p((00) 3 )(x). Let us now assume that x k ]x k, x[ such that p((00) k )(x k ) = 0. Also, that we have < x < x 2 < x 3 < < x k < x. For k + we have p((00) k+ )(x) = x 4 p((00) k )(x) (x 3 + x 2 + x ) = p((00) k+ )(x) = x 4 (x 3 + x 2 + x ) = x ( x 3 + x 2 + x + ) + =. }{{} 0 Therefore p((00) k+ )(x) = > 0. Also, p((00) k+ )(x k ) = (x 3 k + x 2 k + x k ) < 0. Therefore x k+ ]x k, x[ such that p((00) k+ )(x k+ ) = 0. We have < x < x 2 < x 3 < < x k < x k+ < x. In the same way, if x x, p((00) k+ )(x) p((00) k+ )(x) =, i.e., x k+ is the largest real root of p((00) k+ )(x). With these arguments, we have shown that there is a sequence of maximal real roots, monotone increasing and bounded above by x. Affirmation. lim x k = x. k Rev. Un. Mat. Argentina, Vol. 58, No. (207)

23 PROOF OF CONTINUITY OF TOPOLOGICAL ENTROPY AT θ = Proof. We have: p((00) k+ )(x) = x 4 p((00) k )(x) (x 3 + x 2 + x ) = x 4 [x 4 p((00) k )(x) (x 3 + x 2 + x )] (x 3 + x 2 + x ) = x 8 p((00) k )(x) (x 3 + x 2 + x )( + x 4 ) = x 8 [x 4 p((00) k 2 )(x) (x 3 + x 2 + x )] (x 3 + x 2 + x )( + x 4 ) = x 2 p((00) k 2 )(x) (x 3 + x 2 + x )( + x 4 + x 8 ). k = x 4k p((00))(x) (x 3 + x 2 + x ) x 4i k = x 4k ( xp(00)(x) + ) (x 3 + x 2 + x ) k = x 4k+ p(00)(x) + x 4k k Defining Z k (x) = x 4k k x 4i+3 k x 4i+3 k x 4i+2 Z (x) = x 4 + x 3 + x 2 + x = p((00))(x) x 4i k x 4i+2 Z 2 (x) = x 8 x 3 x 7 x 2 x 6 x x x 4 k x 4i+ + k x 4i+ + x 4i. x 4i we have = x 8 x 7 x 6 x 5 + x 4 + x 3 + x 2 + x = p((00) 2 )(x) Z 3 (x) = x 2 x 3 x 7 x x 2 x 6 x 0 x x 5 x x 4 + x 8 = x 2 x x 0 x 9 + x 8 x 7 x 6 x 5 + x 4 + x 3 + x 2 + x = p((00) 3 )(x). Inductively, we may conclude that Z k (x) = p((00) k )(x). So, p((00) k+ )(x) = x 4k+ p(00)(x) + p((00) k )(x) = x 4k+ p(00)(x) x 4(k )+ p(00)(x) + p((00) k )(x) = p(00)(x)(x 4k+ + x 4(k )+ ) + p((00) k )(x). = p(00)(x)(x 4k+ + x 4(k ) x 4j+ ) + p((00) k j )(x) Rev. Un. Mat. Argentina, Vol. 58, No. (207)

24 86 A. JABLONSKI AND R. LABARCA = xp(00)(x) j x 4i + p((00) k j )(x) k = xp(00)(x) x 4i + p((00))(x) k = xp(00)(x) x 4i + xp(00)(x) + = xp(00)(x) ( k ) x 4i + +. As for x = x k+ we have p((00) k+ )(x k+ ) = 0. Then ( k ) x k+ p(00)(x k+ ) x 4i + = = x k+ p(00)(x k+ ) = = x k+ p(00)(x k+ ) = x k+ p(00)(x k+ ) = ) x 4i + ( k x 4k k+ x 4 k+ + x 4 k+ x 4k k+ + x4 k+ 2 = p(00)(x k+ ) = x k+ x 4 k+ x 4k k+ + x4 k+ 2. As x k+ >, letting k we have (x 4k k+ +x4 k+ 2), then p(00)(x k+) 0. However, lim p(00)(x k+) = 0 lim x k+ = x. k k Then, we have shown that this sequence of maximal real roots converges to x. Finally we are in a position to show the continuity of the topological entropy at θ = Continuity of the topological entropy at θ = 00 Let us see that Applying limits: h top ((00) n ) = ln x n. lim n h top((00) n ) = lim n ln x n = ln(x) = h top (00) = h( lim n (00)n ). Proposition 4.. The map h top = h top (σ Σ[σ(θ),θ] ) is continuous at θ = 00. Rev. Un. Mat. Argentina, Vol. 58, No. (207)

25 PROOF OF CONTINUITY OF TOPOLOGICAL ENTROPY AT θ = Proof. We have h top (00) = h top (000), so for each θ such that 00 < θ < 000 we have h top (00) h top (θ) h top (000). Hence, h top (θ) = h top (00). Thus, for any sequence (ζ n ) n N Σ such that ζ n+ < ζ n and lim ζ n = 00 we have h top (ζ n ) = h top (00) (as long as 00 < n ζ n < 000). On the other hand, if (ζ n ) n N Σ is a sequence such that ζ n < ζ n+ and lim ζ n = 00, then for each n big enough, there exists k(n) such that n with lim k(n) =. Then n (00) k(n) ζ n (00) k(n)+ x k(n) = h top ((00) k(n) ) h top (ζ n ) h top ((00) k(n)+ ) = x k(n)+. As lim ζ n = lim n n (00)k(n) = 00, we conclude that lim h top(ζ n ) = h top (00). n So, we have proved that h top (θ) is continuous at θ = 00, as we announced in our main result. References [] R. L. Adler, A. G. Konheim, M. H. McAndrew, Topological entropy. Trans. Amer. Math. Soc. 4 (965), MR [2] L. Alsedà, J. Llibre, M. Misiurewicz, Combinatorial dynamics and entropy in dimension one. Second edition. Advanced Series in Nonlinear Dynamics, 5. World Scientific, River Edge, NJ, MR [3] R. Bamón, R. Labarca, R. Mañé, M. J. Pacífico, The explosion of singular cycles. Inst. Hautes Études Sci. Publ. Math. No. 78 (993), (994). MR [4] S. Aranzubía, R. Labarca, Combinatorial dynamics and an elementary proof of the continuity of the topological entropy at θ = 0, in the Milnor Thurston world. Progress and challenges in dynamical systems, Springer Proc. Math. Stat., 54, Springer, Heidelberg, 203. MR 377. [5] L. Block, J. Guckenheimer, M. Misiurewicz, L. S. Young, Periodic points and topological entropy of one-dimensional maps. Global theory of dynamical systems (Proc. Internat. Conf., Northwestern Univ., Evanston, Ill., 979), pp. 8 34, Lecture Notes in Math., 89, Springer, Berlin, 980. [6] J. Guckenheimer, R. F. Williams, Structural stability of Lorenz attractors. Inst. Hautes Études Sci. Publ. Math. No. 50 (979), MR [7] R. Labarca, A note on the topological classification of Lorenz maps on the interval. Topics in symbolic dynamics and applications (Temuco, 997), , London Math. Soc. Lecture Note Ser., 279, Cambridge Univ. Press, Cambridge, MR [8] R. Labarca, La entropía topológica, propiedades generales y algunos cálculos en el caso del shift de Milnor Thurston. Con la colaboración de S. Aranzubia y E. Inda. Serie de las Escuelas Venezolanas de Matemáticas, Instituto Venezolano de Investigaciones Científicas (IVIC), 20. Rev. Un. Mat. Argentina, Vol. 58, No. (207)

26 88 A. JABLONSKI AND R. LABARCA [9] R. Labarca, C. Moreira, Essential dynamics for Lorenz maps on the real line and the lexicographical world. Ann. Inst. H. Poincaré Anal. Non Linéaire 23 (2006), MR [0] R. Labarca, C. Moreira, Bifurcation of the essential dynamics of Lorenz maps on the real line and the bifurcation scenario for Lorenz like flows: the contracting case. Proyecciones 29 (200), MR [] R. Labarca, C. Moreira, A. Pumariño, J. A. Rodríguez, On bifurcation set for symbolic dynamics in the Milnor Thurston world. Commun. Contemp. Math. 4 (202) , 6 pp. MR [2] R. Labarca, A. Pumariño, J. A. Rodríguez, On the boundary of topological chaos for the Milnor Thurston world. Commun. Contemp. Math. (2009), MR [3] D. Lind, B. Marcus, An introduction to symbolic dynamics and coding. Cambridge University Press, Cambridge, 995. MR [4] J. Milnor, W. Thurston, On iterated maps of the interval. Dynamical systems (College Park, MD, ), , Lectures Notes in Math. 342, Springer, Berlin, 988. MR [5] M. Misiurewicz, W. Szlenk, Entropy of piecewise monotone mappings. Studia Math. 67 (980), MR [6] S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (967), MR [7] M. Urbański On Hausdorff dimension of invariant sets for expanding maps of a circle. Ergodic Theory Dynam. Systems 6 (986), MR A. Jablonski Departamento de Matemática y Ciencia de la Computación, Universidad de Santiago de Chile, Santiago, Chile andres.jablonski@gmail.com R. Labarca Departamento de Matemática y Ciencia de la Computación, Universidad de Santiago de Chile, Santiago, Chile rafael.labarca@usach.cl Received: May 9, 205 Accepted: October 2, 206 Rev. Un. Mat. Argentina, Vol. 58, No. (207)