Markov shifts and topological entropy of families of homoclinic tangles

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Markov shifts and topological entropy of families of homoclinic tangles Bráulio Garcia ∗ , Valentín Mendoza Instituto de Matemática e Computação, Universidade Federal de Itajubá, Av. BPS 1303, Bairro Pinheirinho, CEP 37500-903, Itajubá, MG, Brazil Received 3 May 2018; revised 30 October 2018

Abstract The existence of a homoclinic orbit in dynamical systems implies chaotic behaviour with positive entropy. In this work, we determine explicitly the Markov shifts associated to certain Smale horseshoe homoclinic orbits which allow us to compute a lower bound for the topological entropy that such a system can have. It is done associating a heteroclinic orbit which belongs to the same isotopy class and then determining the Markov partition of the dynamical core of an end periodic mapping. © 2019 Elsevier Inc. All rights reserved. Keywords: Homoclinic and heteroclinic horseshoe orbits; Markov shifts; Entropy

1. Introduction The use of topological methods for giving a qualitative description of dynamics has had a long and important history. Excellent prototype results began with Poincaré and Birkhoff, who were interested in problems from celestial mechanics, specially deterministic systems with chaotic behaviour. It is by now well known that iterate maps of a space, when viewed as dynamical systems, account for some of this irregular behaviour observed in physics. For example, models that comes from differential equations, like the Lorenz system, exhibits return Poincaré maps f with chaotic * Corresponding author.

E-mail addresses: [email protected] (B. Garcia), [email protected] (V. Mendoza). https://doi.org/10.1016/j.jde.2019.01.002 0022-0396/© 2019 Elsevier Inc. All rights reserved.

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dynamics (e.g. Hénon map). A standard measure for the complexity of such systems is the topological entropy. This is an invariant h(f ) of f which gives, besides other things, a quantitative information of the number of orbits that separate in finite time. There are several natural definitions of this invariant, but except in particular cases, for example when there is a symbolic description of the system, it is difficult to compute or estimate it. Furthermore, in those special cases the symbolic dynamics usually acts as a lower bound for the dynamics of f since it is coded in such way that for any orbit in the symbolic description there is at least, via the semiconjugacy, one corresponding orbit of f . In general, except in one dimension dynamics systems (see [22]), the entropy function f → h(f ) are not lower semi-continuous, so is hard to estimate from below. It is also known that are not upper semi-continuous for C r maps with r < ∞, but relative to the C ∞ topology the topological entropy depends continuously on the dynamics (see [23] and [33]). Other famous technique is computed the entropy of an isotopy class, that is, the infimum of all the entropies of the homeomorphisms in the class. If a surface isotopy class has positive entropy, a Thurston’s Theorem [30] guarantees the existence of a canonical map which realizes that infimum and whose dynamics are described by a finite collection of mixing subshift of finite type. Further, a Handel’s theorem [15] states that this dynamics are present up to semiconjugacy in every homeomorphism in that class. For two-dimensional smooth systems, due to its modern form to Smale [28] and Katok [18], existence of transverse homoclinic points implies the existence of horseshoes (non trivial hyperbolic sets), and so diffeomorphisms with transverse homoclinic points have strictly positive topological entropy. Conversely, any C 1+ diffeomorphism of a surface with positive topological entropy has a hyperbolic periodic saddles with transverse homoclinic points. In short, positive entropy surface diffeomorphisms exhibit horseshoes – a very elegant and useful geometric model that capture the essence of the effect of homoclinic and heteroclinic phenomena on dynamical systems. And, knowledge about whether and how the stable and unstable manifolds intersect produces a panorama of many characteristics of the system as illustrated by Poincaré. These are some of the main motivations of our work. We investigate the information about the dynamics of a system essentially looking just a finite number of intersection points of a given homoclinic tangle. These object has been called trellis since Collins [5]. Its safe to mention that Easton [10] used the word trellis for what we will call tangle. The main goal of this paper is to show how to lead with homoclinic and heteroclinic tangles of the Smale horseshoe. To state the techniques we need appropriate notation. Precisely, we study homoclinic orbits P0w of the Smale horseshoe which are coded by symbol sequences of the form ∞ 0101 w 01 10∞ where w is a finite word called decoration. Moreover, since a broad class of these orbits usually appears in physical models, as we shall see in examples, this work may also be useful for both theoretical and experimental physicists. Well known examples include the Van der Pol system and others non-linear oscillators periodically forced. Remarkable in the theory of forcing on homoclinic orbits is the work of Rom-Kedar [24] which introduces symbolic dynamics and estimates the topological entropy for a class of Easton trellises [10] of type l, that are homeomorphic to the tangles defined by the decorations 0l−2 , and of type (l, l, l, 0) that are homeomorphic to the tangles defined by the decorations 0l−2 110l−2 , see [26]. A Handel’s theorem [14] gives conditions for which a family of homoclinic orbits force the existence of a fixed point and them imply, in particular, that ∞ 0101 10∞ forces the existence of the full Smale horseshoe. A generalization of the Bestvina–Handel’s algorithm for obtaining a tight graph for the dynamics relative to a homoclinic orbit was given by Hulme [17]. And in [31,32]

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Yamaguchi and Tanikawa give a procedure to derive the dynamical ordering of homoclinic orbits for maps in the Hénon family. The approach given by Collins in [5,7] in order to obtain a theoretical basis for describing the dynamics forced by a homoclinic tangle allows him to compute the topological entropy of horseshoe homoclinic orbits whose decorations are short (length less than 7), see [6]. His method is based in a fundamental operation called pruning that is closely related to the pruning isotopies introduced by de Carvalho in [8], that in turn are the main tools to be used in our work. 1.1. Results Despite of the relevance of the homoclinic and heteroclinic orbits there are only few cases where their symbolic dynamics is well-known, e.g. [6,31,32]. In [21] it was started a study of forcing from the pruning point of view. Let f be a Smale map with a basic hyperbolic set K on the disc D 2 as, for example, the usual Smale horseshoe. Given a homoclinic orbit P of f one first associates a Smale map f¯ with a heteroclinic orbit also denoted by P , that is isotopic to f and has almost the same dynamics. See Section 2. Thus f¯ restricted to M = D 2 \ P can be considered as an end-periodic map and then the Handel–Miller’s theory, stated in section 2 of [11] due to Fenley and extended by Cantwell and Conlon in [4], can be used for finding the dynamics relative to P . To be precise [4, Theorem 10.13] states that the dynamics that can not be removed in its isotopy class is contained in a set named the dynamical core CP of P which is the intersection of two transversely totally disconnected pseudo-geodesic laminations. Moreover it was proved that CP in [4] has a symbolic description given by a Markov shift of finite type. In [21] it is proved that CP can be found using pruning domains, that is, if D is a region relative to P satisfying pruning conditions and the pruned map of f associated to D has no bigons then CP is formed by the orbits that do not intersect D, CP = K \



f i (Int(D)) = {x ∈ K : f i (x) ∈ / Int(D), ∀i ∈ Z},

(1)

i∈Z

up a topological finite-to-one semi-conjugacy which is injective on the set of non-boundary periodic points. Here we intend to determine a Markov shift for the core CP , where P belongs to some infinite families of horseshoe homoclinic orbits that has been appeared in many recently works. Many of these orbits were studied in [21] where their dynamical cores were found, but their Markov shifts were not determined. Now we fill this gap determining the Markov partition and the topological entropy of some of those families. Our methods rely on pruning techniques for constructing symbol systems. Besides offering an explicit description of the dynamics, the constructed symbol system allows the computation of the characteristic polynomials, whose largest root gives the topological entropy of the Markov shift. Moreover the approach given here can also be used for any particular homoclinic orbit or family of homoclinic orbits, and can also be generalized to Smale maps on other surfaces. Starting with the standard partition of the Smale horseshoe, we define a sequence of (no Markovian) partitions using the iterates of the stable boundary of the pruning domain relative to the orbit and eliminating regions which are disjoint from the core. Thus we obtain a partition that satisfies the Markovian condition on the stable boundaries. Redefining the rectangles by iterations, one can change the unstable boundaries for getting a Markov partition with rectangles containing the core. We state our main theorem in the general setting.

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Table 1 Decorations and their characteristic polynomial. Decoration q=m n ∈ Q ∩ (0, 1/2)

w = ωq

k≥1

w = 12k−1

k≥1

w = 10k

Characteristic polynomial Qw (x)   m−1 jmn +1 x n+1 − x n − 2 j =0 x   2k (−1)j x 2k+3−j x 6+2k − 2x 5+2k + x 4+2k − 2 j =0   3k 4+j − 2x 3+2k x 8+4k − 2x 7+4k + x 6+3k − 2x 5+3k + 2 j =2k x

k≥1

w = 0k 10k

x 5k+10 − x 5k+9 − x 4k+8 − x 4k+7 − 2x 2k+4

ωq

Fig. 1. The largest eigenvalue λq of Qωq (x) for q ∈ Q ∩ (0, 1/2). ln λq is the entropy of the homoclinic orbit P0 .

Theorem 1. Let w be a decoration of Table 1. Then there exists a Markov shift w defined by a transition matrix Aw which is topologically conjugated to the dynamics of the core relative to P0w . Moreover, if λw is the largest real root of Qw (x) then the topological entropy of P0w is log λw . The first type of decorations were defined by T. Hall in [13], that is, if q ∈ Q ∩ (0, 1/2) then ωq is palindromic and the orbits 10ωq 01 are in the isotopy class of the rotation of angle αq = 2πq. Note that for such q the polynomial Qωq (x) has a unique larger than 1 root λ(q) which, as function of q, has some Devil’s staircase behaviour. See Fig. 1. By density and monotonicity, λ(.) can be extended to a function λ : (0, 1/2) → R. The function λ = λ(r) has connection with the lower bound function β(r) for the topological entropy of an one-dimensional map of the circle with interval rotation [0, r] given in [1]. As in that case one can conclude that λ(r) is discontinuous from the right and from the left at every rational number but continuous in all irrational number. What we have observed is that λ(r) = β(r) for all r ∈ R \ Q and λ(r) is in the middle of the discontinuity gap for any r ∈ Q. Although neither the Handel–Miller theory for end-periodic maps nor a differentiable pruning for Smale maps can be applied in the irrational case, it is conjectured that the topological entropy relative to the set of two orbits {∞ 010 · ωα , ωα · 010∞ } will be exactly ln λ(α), for all irrational number α. 2. From homoclinic orbits to heteroclinic ones In this section we state the techniques that will be used for finding the topological entropy and the Markov shift associated to a homoclinic or heteroclinic tangle of a hyperbolic map-

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Fig. 2. The classical Smale horseshoe and the homoclinic orbit of p0 = ∞ 010 · 1110∞ . This orbit is represented in the symbol plane in Figure (b).

ping. Since it is ubiquitous in systems with high complexity, our working map is the classical Smale horseshoe but the main argument can be applied to any hyperbolic map. Let D 2 be the disc {(x, y) : x 2 + y 2 < 4}. The Smale horseshoe is a orientation-preserving hyperbolic map F : D 2 → D 2 which has an invariant set K with the following properties: (a) F restricted to K is conjugated to the shift σ defined on the Cantor set 2 := {0, 1}Z , that is, every point x ∈ K has a code given by a symbol sequence ι(x) = (si )i∈Z ∈ 2 such that ι(F (x)) = σ (ι(x)); (b) every point x ∈ K has an unstable manifold W u (x) and a stable manifold W s (x) which are dense in K and K = W u (x) ∩ W s (x). In this work we are particularly interested in homoclinic orbits in W u (0∞ ) ∩ W s (0∞ ), that is, orbits converging to 0∞ by forward and backward iterates. A such orbit is denoted by P0w = {pi } where p0 has code ∞ 0110 w 10 10∞ being w a finite word. A representation of some parts of these invariant manifolds is called a tangle. An example of the homoclinic tangle associated to the orbit P01 containing the point p0 = ∞ 010 · 1110∞ is shown in the Fig. 2. Since M := D 2 \ P0w is not a surface of finite type the Thurston–Nielsen theory can not be applied but its extension done for end periodic homeomorphisms, outlined by Handel and Miller and developed by Cantwell and Conlon in [4], ensures us that if f : M → M is isotopic to an end periodic homeomorphism it has a dynamical core Cw which is the intersection of two geodesic laminations (+ , − ) that only depends of the isotopy class of f . Moreover, due to [4], the dynamics of f restricted to Cw is conjugated to a Markov shift and the topological entropy htop (f |Cw ) is a lower bound of the topological entropy of f : M → M, thus we define the topological entropy of P0w as htop (Pqw ) := htop (f |Cw ).

(2)

In order to obtain an isotopy from the horseshoe F : M → M to an end-periodic map one needs to construct a DS (Derived from Smale) stable diffeotopy {ft } around the point 0 = 0∞ . It is obtained by a pitchfork bifurcation in the stable direction of 0 such that

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Fig. 3. A homoclinic orbit {pi } is converted into a heteroclinic orbit by a DS-diffeotopy. Both orbits are in the same isotopy class.

• 0 becomes an attractor and two new saddle fixed points x1 and x2 are created; • f1 is a Smale map with a hyperbolic set K1 which is a topological extension of F : there exists a semiconjugacy π : K1 → K between f1 and F that is almost one-to-one, i.e., π −1 (x) is single point for every x ∈ K \ W s (0). Since f1 is also Smale, an unstable DS isotopy can be done around the point x2 . As before, by a pitchfork bifurcation in the unstable direction of x2 , an Smale map f2 can be constructed satisfying • x2 becomes a repeller and two new saddle fixed points y1 and y2 are created; • f2 is a Smale map with a hyperbolic set K2 which is a topological extension of F : there exists a semiconjugacy π : K2 → Kbetween f2 and F that is almost one-to-one, i.e., π −1 (x) is single point for every x ∈ K \ W s (0) ∪ W u (0) . Then f2 has an heteroclinic orbit also called {pi } satisfying limi→+∞ pi = x1 and limi→−∞ pi = y2 . See Fig. 3. Finally isotoping x1 and y2 to the boundary of D 2 without changing the dynamics of the Smale horseshoe, we obtain a end-periodic map f¯ which has an infinite orbit {pi } in the isotopy class of {pi }. Thus that map f¯ restricted to D 2 \ {pi } has a dynamical core that actually is Cw up to conjugacy. Now is when differentiable pruning must be applied. In order to find the core Cw one needs to eliminate the superfluous dynamics. It is done by using pruning domains. Definition 2. Let D ⊂ D 2 be a simply connected region. We will say that D is a pruning domain if • there exist periodic points zs and zu such that ∂D = θs ∪ θu where θs ⊂ W s (zs ) and θu ⊂ W u (zu ); • f n (θs ) ∩ Int(D) = f −n (θu ) ∩ Int(D) = ∅, ∀n ≥ 1. A pruning domain relative to the orbit of p0 = ∞ 010 · 1110∞ is shown in Fig. 4(a). If D is a pruning domain, a differentiable version of pruning [8] allows to construct another Smale map g, named pruned map, which has the following properties

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Fig. 4. The pruning domain and the associated pruned map.

Fig. 5. End periodic map associated to the homoclinic tangle of Fig. 4.

• g is isotopic to f¯, • the hyperbolic basic set Kg of g is an incomplete horseshoe: there exists a semiconjugacy πp : Kg → KD , where KD = {x ∈ K : f n (x) ∈ / Int(D), ∀n ∈ Z}. The pruned map for the current example is shown in Fig. 4(b). Thus the dynamics of g is semiconjugated to the open system obtained from F and associated to Int(D). What pruning does is uncross the invariant manifolds lying in Int(D) and in all its iterates. See Fig. 4(b). If KD is exteriorly situated, i.e. if there are no null-homotopic loops formed by a piece of an stable manifold and a piece of an unstable manifold relative to P0w , then the results in [21] ensure us that the dynamical core Cw agrees with KD up to a topological finite-to-one semi-conjugacy which is injective on the set of non-boundary periodic points. More precisely, every leaf of + can be associated to a leaf of W u (KD ) and every leaf of − can be associated to a leaf of W s (KD ). Finally, the end-periodic map associated to P0w with w = 1 is shown in Fig. 5. 3. Markov shifts and a formula for the topological entropy In this section we will be concerned with the proof of Theorem 1, i.e., we will find the Markov partition and the topological entropy for some of the orbits introduced in [21]. Recall that a rectangle of a Smale map f , having a basic set K, is a region R such that there exist an homeomorphism h : [0, 1] × [0, 1] → R and two closed sets Fs and Fu included in [0, 1] satisfying

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• {0, 1} ⊂ Fs and {0, 1} ⊂ Fu , • R ∩ W s (K) = h(Fs × [0, 1]) and R ∩ W u (K) = h([0, 1] × Fu ). Definition 3. Let z1 , z2 , · · · , zn be points of K where n is even. We will denote [z1 , z2 , · · · , zn ] to the region bounded by a stable segment going from z1 to z2 , an unstable segment going from z2 to z3 , · · · , an unstable segment going from zn to z1 . Using the orientation of D 2 , if R = [z1 , z2 , z3 , z4 ], the points zi are chosen in such way that [z1 , z2 ]s is the left stable boundary of R, [z3 , z4 ]s is the right stable boundary of R, [z2 , z3 ]u is the top unstable boundary of R and [z1 , z4 ]u is the bottom unstable boundary of R. 3.1. Star homoclinic orbits The first type of orbits are named star. Given a rational number q = define the exponents {κi }m i=1 by the expressions:  κi =

1/q − 1

if i = 1

i/q − (i − 1)/q − 2

if 2 ≤ i ≤ m.

m n

 := (0, 1 ) ∩ Q, ∈Q 2

(3)

Then define the word ωq = 0κ1 −1 12 0κ2 · · · 12 0κm −1 which has been very useful for understanding the forcing relation. For example, by a result due to Hall in [13], the isotopy class of the periodic orbit 10ωq 01 is of finite order. Observe that for q = n1 for n ≥ 3, the formula (3) gives ω1/n = 0n−3 (using the second condition). Another result found in [13] is the following proposition. Proposition 4. Let {κi } be the exponents of ωq . Then: (1) for mi = 1, · · · , m, κi = κm−i+1 ; and (2) i=1 (κi + 2) = n + 1.  the star homoclinic orbit of rotation q is the orbit P ωq which contains Definition 5. Given q ∈ Q, 0 the point pq = ∞ 010 · ωq 010∞ .  Lemma 6. Let q = m n ∈ Q. Then F restricted to Cωq is conjugated to a Markov shift (σA , A ) where A ∈ M(n+1)×(n+1) is a transition matrix given by the relations (see Fig. 6): J1 → {J1 , J2 , · · · , Jm+1 } J2 → Jm+2 , J3 → Jm+3 , J4 → Jm+4 , · · · , Jn−m → Jn Jn−m+1 → {J1 , Jn+1 }, Jn−m+2 → {J1 , J2 }

(4)

Jn−m+3 → J3 , Jn−m+4 → J4 , · · · , Jn+1 → Jm+1 .

Proof of Lemma 6. Let 0 and 1 be the rectangles of the standard partition of the Smale horseω shoe F . From [21] it is known that the pruning domain associated to P0 q is a domain Dq such that ∂Dq = θu ∪ θs where θs ⊂ W s (∞ 010 · ωq 010∞ ) and θu ⊂ W u (1∞ ), i.e. Dq = [X, Y ] where

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Fig. 6. Markov partition associated to a star homoclinic orbit.

X = ∞ 1 · ωq 010∞ and Y = ∞ 10 · ωq 010∞ . See Fig. 7(a). Note that Cw ⊂ Int(D) ∪ F −1 (Int(D)), then one can subdivide the rectangles labelled by 0 and 1 in 5 sub-rectangles: R11 , R21 , R31 ⊂ 0 and R41 , R51 ⊂ 1 such that Cw ⊂ ∪5i=1 Ri1 in such a way that R11 = [0∞ , ∞ 1 · 0∞ , ∞ 1 · 0ωq 010∞ , ∞ 0 · 0ωq 010∞ ], R21 = [∞ 0 · 0ωq 010∞ , ∞ 1 · 0ωq 010∞ , X, ∞ 0 · ωq 010∞ ], R31 = [∞ 0 · ωq 010∞ , Y, ∞ 10 · 010∞ , ∞ 0 · 010∞ ], R41 = [∞ 0 · 110∞ , ∞ 10 · 110∞ , ∞ 10 · 10∞ , ∞ 0 · 10∞ ], R51 = [∞ 1 · 1ωq 010∞ , ∞ 01 · 1ωq 010∞ , ∞ 01 · 10∞ , ∞ 1 · 10∞ ]. These rectangles for the case ω3/11 = 01100110 are displayed in Fig. 7(a). Now observe that iterates F i (θs ) ⊂ R31 for i = 1, · · · , κ1 − 2. Thus one can apply procedure 1 that consists in subdividing R31 along the stable segment passing through F i (θs ) and define (κ1 − 1) new rectangles j −1 where {B3 }jκ1=1 B3 = [∞ 0 · 0κ1 −j 11 · · · 0κm 10∞ , F j −1 (Y ), F j (X), ∞ 0 · 0κ1 −j −1 11 · · · 0κm 10∞ ] j

for i = 1, · · · , κ1 − 2 and B3κ1 −1 = [∞ 0 · 0110κ2 · · · 0κm 10∞ , F κ1 −2 (Y ), ∞ 10κ1 −1 · 010∞ , ∞ 0 · 010∞ ]. See Fig. 8. Then one applies procedure 2 associated to R41 and the segment F κ1 −1 (θs ) that consists in subdividing R41 in two new rectangles C41 = [∞ 0 ·10∞ , ∞ 10κ1 −1 ·10∞ , F κ1 −1 (X), ∞ 0 · 110κ2 · · · 0κm 10∞ ] and C42 = [∞ 0 · 110κ2 · · · 0κm 10∞ , F κ1 −1 (Y ), ∞ 10κ1 · 10∞ , ∞ 0 · 10∞ ]. See Fig. 9. Now one can apply procedure 3 associated to R51 and the segment F κ1 (θs ) (see Fig. 10). It consists in subdividing R51 in two new rectangles D51 and D52 such that D51 = [∞ 10κ1 1 · 1ωq 010∞ , ∞ 01 · 1ωq 010∞ , ∞ 01 · 10κ2 11 · · · 0κm 10∞ , F κ1 (Y )] and D52 = [F κ1 (X), ∞ 01 · 10κ2 11 · · · 0κm 10∞ , ∞ 01 · 10∞ , ∞ 10κ1 −1 1 · 10∞ ].

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Fig. 7. Steps of the construction of a Markov partition the star orbit with rotation q = 3/11.

Fig. 8. Procedure 1.

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Fig. 9. Procedure 2.

Fig. 10. Procedure 3.

Applying procedure 1, 2 and 3 one obtains Fig. 7(b) for the current example. Thus a partition P = {R11 , R21 , B31 , B32 , · · · , B3κ1 −1 , C41 , C42 , D51 , D52 } is obtained and, since the eliminated regions are contained some in F k (Int(D)) with k ∈ Z, it follows that Cw ⊂ ∪R∈P R. This partition is not still a Markov partition since F (∂D52 ) is not included in the boundary of a rectangle. To solve it firstly we construct a partition that satisfies the condition If F (Ri ) ∩ Rj = ∅ =⇒ F (∂s Ri ) ⊂ ∂s Rj .

(5)

It is done by induction. For j = κ1 + 1, · · · , n − 1, let R be the rectangle of P that contains F j (θs ). Two cases can happen • if F j (pq ) = · · · 0κi −l · 0l 11 · · · for some l ≥ 0 then apply procedure 2 associated to R and F j (θs ), creating two new rectangles R1 and R2 , and substitute R in P by R1 and R2 ; and • if F j (pq ) = · · · 0κi 1 · 1 · · · then apply procedure 3 associated to R and F j (θs ), creating two new rectangles R1 and R2 , and substitute R in P by R1 and R2 . This creates a partition P = {J1 , · · · , Jn+1 } with n + 1-rectangles that can be indexed in the counterclockwise direction starting from J1 = R11 , which satisfies condition (5) and the transitions given in (4). See Fig. 7(c) and 7(b) for the current example. Unfortunately it does not satisfy the condition on their unstable boundaries: If F −1 (Ri ) ∩ Rj = ∅ =⇒ F −1 (∂u Ri ) ⊂ ∂u Rj .

(6)

From P we must construct a partition satisfying condition (6). Iterating once F n−4 (X) one sees that F n−3 (X) = ∞ 10κ1 −1 11 · · · 0κm −1 · 010∞ is the right top vertex of Jn−2m+1 . Since

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Fig. 11. Details of the construction of Jm+1 .

Jn−2m+1 → Jn−m+1 one have to decrease the top unstable boundary of Jn−m+1 until the unstable leaf of F n−2 (X). See Fig. 11. Since Jn−m+1 → Jn+1 one have to decrease the bottom unstable boundary of Jn+1 until the unstable leaf of F n−1 (X). Since Jn+1 → Jm+1 one have to decrease the top unstable boundary of Jm+1 until the unstable leaf containing the point F n (X). That is the top right vertex of Jm+1 now is Q2 = ∞ 10κ1 −1 11 · · · 0κm 11 · ωq 010∞ . Let Q1 , Q2 , Q3 and Q4 be the vertexes of Jm+1 . Since there is a path of length n from Jm+1 to Jm+1 , Jm+1 → J2m+1 → · · · → Jn−r1 +1 → Jm−r1 +1 → J2m−r1 +1 → · · · → Jn−r2 +1 

 

 κ1 + 1 terms

κ2 + 2 terms

→ Jm−r2 +1 → J2m−r2 +1 → · · · → Jn−r3 +1 

 κ3 + 2 terms

.. . → Jm−rm−1 +1 → J2m−rm−1 +1 → · · · → Jn−m+1 

 → Jn+1 → Jm+1

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κm + 1 terms

where in = m(κ1 + 1 + κ2 + 2 + · · · + κi + 2) + ri for i = 1, · · · , m − 1, one can apply the same process to obtain a new rectangle Jm+1 for which the new vertex Q2 is ∞ 1(0κ1 −1 11 · · · 0κm 11)2 · ω 010∞ . See Fig. 12. Repeating by induction one can obtain a rectanq gle Jm+1 whose vertex Q2 is ∞ 1(0κ1 −1 11 · · · 0κm 11)k · ωq 010∞ . In the limit Jm+1 has as vertexes ∞ ∞ κ1 −1 11 · · · 0κm 11) · ω 010∞ , Q and Q implying that J Q∞ q 3 4 m+1 has the periodic 1 , Q2 = (0 κ −1 κ ∞ m 1 11 · · · 0 11) in its top unstable boundary. Extending by iterations using the point xq = (0 path (7) it follows that the rectangles J2 , · · · , Jn+1 have an unstable boundary contained in the unstable manifold of a point of the periodic orbit of xq . The current example is shown in Fig. 12 and Fig. 7(e). Due to a consequence of [9, Theorem 40], the orbit xq = (ωq 011)∞ is of finite order. Thus the partition satisfies that Cωq ⊂ ∪n+1 i=1 Ji . Let (σA , A ) be the Markov shift defined by the rectangles. If y ∈ A then Orb(y) ∩ Int(Dq ) = ∅ for all i = 1, · · · , n + 1. Thus y ∈ Cωq . 2

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Fig. 12. Reduction of the unstable boundary of Jm+1 .

Lemma 7. The characteristic polynomial of the Markov shift given in Lemma 6 is  m−1

Qωq (x) = x n+1 − x n − 2

 jn x m +1 .

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j =0

Proof of Lemma 7. In order to compute the characteristic polynomial for the transition matrix given by relations (4) we take as a rome R = {J1 , J2 }, in the language of [2]: that is to say, there is no path Ji1 → Ji2 → · · · → Jil → Ji1 which is disjoint from  R. Now, we compute the elements of the reduced matrix AR (x) = (Aij )2i,j =1 , where Aij (x) = p x −l(p) and the summation is over all the simple paths starting in Ji and ending in Jj and l(p) is the length of the path p. Now, from [2, Theorem 1.7] it follows that the characteristic polynomial of the transition matrix A is equal to (−1)n−1 x n+1 det(AR (x) − I2 ), where I2 is the identity matrix of size 2. j Define a0 = 1 and for 1 ≤ j ≤ m − 1, aj = jmn + 1. Then aj = i=1 (κi + 2). There exist three ways for arising J1 : using J1 → J1 , Jn−m+1 → J1 or Jn−m+2 → J1 . In the following path will refer to a simple path. (a) If n = m(κ1 + 1) + m − 1 then n − m + 1 = m(κ1 + 1), and κi = κ1 for all i = 2, · · · , m. Noting that there exist transitions from J1 → {J2 , J3 , · · · , Jm+1 }, Jn−m+1 → {J1 , Jn+1 }, Jn−m+2 → {J1 , J2 }, Jn+1 → Jm+1 and the path J1 → Jm+1 → · · · → Jn−m+2 → J2 → J2+m → · · · → Jn−m+3 

 

 κ1 + 2 terms

κ2 + 2 terms

→ J3 → J3+m → · · · → Jn−m+4 

 κ3 + 2 terms

.. . → Jm → J2m → · · · → Jn−m+1 → Jn+1 

 κm + 1 terms

one can computed the reduced matrix,

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AR (x) =

 x −1 + x −(κ1 +2) + L + x −(κ1 +2) L x −1 + x −(κ1 +2) + x −(κ1 +2) L , x −n x −(am−1 −1) + x −n

where L = x −(κm +2) + x −(κm +κm−1 ) + · · · + x −(κm +2+κm−1 +2+···+κ3 +2) By a straightforward calculation it is easy to check that x

n+1

det(AR (x) − I2 ) = x

n+1

−x −2 n

 m−1

 jn x m +1 .

j =0

(b) For i = 1, · · · , m − 1, let ri ∈ {1, · · · , m − 1} be the natural number satisfying in = m(κ1 + 1 + κ2 + 2 + · · · + κi + 2) + ri . Let p ∈ {2, · · · , m − 1} be the smallest number such that rp = m − 1 implying that Jm−rp +1 = J2 and Jn−rp +1 = Jn−m+2 . Thus one divides the set V = {J3 , · · · , Jm+1 } = W ∪ Z with W = {Jm+1 , Jm−r1 +1 , · · · , Jm−rp−1 +1 } and Z = {Jm−rp+1 +1 , Jm−rp+2 +1 , · · · , Jm−rm−1 +1 }. Note that J1 → {J3 , · · · , Jm+1 } and Jm+1 → J2m+1 · · · → Jn−r1 +1 → Jm−r1 +1 → J2m−r1 +1 · · · → Jn−r2 +1 

 

 κ1 + 1 terms

κ2 + 2 terms

→ Jm−r2 +1 → J2m−r2 +1 · · · → Jn−r3 +1 

 κ3 + 2 terms

.. .

→ Jm−rp−1 +1 → J2m−rp−1 +1 · · · → Jn−rp +1 

 κp + 2 terms

.. . → Jm−rm−1 +1 → J2m−rm−1 +1 · · · → Jn−m+1 

 κm + 1 terms

and Jn−m+2 → {J1 , J2 }, Jn−m+1 → {J1 , Jn+1 } and Jn+1 → Jm+1 . Concatenating the paths one obtain

AR (x) =

L+B C + x −n

 L , x −n

where L = T + x −a B B = x −(1+κm +1) + x −(1+κm−1 +2+κm +1) + · · · + x −(1+κp+2 +2···+κm−1 +2+κm +1) C = x −(κp+1 +2+κp+2 +2+···+κm +1)

(9)

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ωq

Fig. 13. Tangle which is topologically equivalent to the end-periodic map of P0 , the orbit of pq = ∞ 010 · 010∞ , with q = 1/3.

where a = κ1 + 2 + · · · + κp + 2 and T = x −1 + x −(1+κp +2) + x −(1+κp−1 +2+κp +2) + · · · + x −(1+κ1 +1+κ2 +2+···+κp +2) . After a long calculation x n+1 det(AR (x) − I2 ) = x n+1 − x − x n−a+1 B − x n+1 B − x a+1 T − x n+1 T Now using Proposition 4 and the equalities (3) for i = 1, · · · , m − p, κp+i = κi , and (4) κp = κ1 − 1, one have that

x n+1 det(AR (x) − I2 ) = x n+1 − x n − 2

 m−1

 x aj .

2

j =0

Example 8. Star homoclinic orbits appear frequently in applications. For example the type l ω tangle, introduced by Rom-Kedar is [24] and [26], corresponds to the homoclinic orbit P0 q ω 1 with q = l+1 as was showed in Huaraca and Mendoza [16]. The orbit P0 q with q = 13 appears in its heteroclinic form (see Fig. 13) in Fig. 12 of Camassa and Wiggins [3] and is associated to a chaotic fluid-particle motion, and the orbits of Fig. 11 of [12] due Guantes, Borondo and 1 Miret-Artés correspond to the star orbits P ωq with q = 18 , · · · , 12 which appears in atom-surface chaotic scattering. Moreover, Fig. 10(a) and Fig. 13 of [27] by Rückerl and Jung correspond ω to star homoclinic orbits with q = 14 and q = 25 , respectively. It seems that P0 q , with q = 14 , emerges in the dynamics of a time-periodic incompressible fluid as in Fig. 48(a) of [29]. 3.2. The topological entropy formula for decoration w = 12k−1 Without loss of generality, we can suppose that P0w is the orbit of pw = ∞ 010 · 12k−1 110∞ , where k ≥ 1. Lemma 9. Let w = 12k−1 be a finite word with k ≥ 1. The Markov shift associated to the homoclinic orbit P0w is given by the relations (see Fig. 14):

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Fig. 14. Markov partition associated to homoclinic orbit 12k−1 .

J1 → {J1 , J3 , J4 , · · · , J4+k }, J2 → Jk+5 , J3 → J5+2k , J4 → J6+2k J5 → J4+2k , J6 → J3+2k , · · · , J3+k → J6+k , J4+k → {J4+k , J5+k } J5+k → J3+k , J6+k → J2+k , · · · , J4+2k → J4

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J5+2k → {J1 , J2 }, J6+2k → {J1 , J3 } Proof of Lemma 9. As in the case of star orbits one needs to define an initial partition. In this case the pruning domain is the disc D = [X, Y ] with X = ∞ (10)11.12k−1 110∞ , Y = ∞ (10).12k−1 110∞ , that is ∂D = θs ∪ θu where θs ⊂ W s (∞ 010.12k−1 110∞ ) and θu ⊂ W u ((10)∞ ). Using the pre-images F −1 (D) and F −2 (D) one obtains a partition P = {R11 , · · · , R71 }, containing the core Cw , that consists of 7 rectangles R11 = [0∞ , ∞ 01 · 0∞ , ∞ 01 · 012k−1 110∞ , ∞ 0 · 012k−1 110∞ ], R21 = [0∞ · 012k−1 110∞ , ∞ (01) · 012k−1 110∞ , ∞ (01) · 010∞ , ∞ 0 · 010∞ ], R31 = [∞ 0 · 010∞ , ∞ (01) · 010∞ , ∞ (01) · 12k 110∞ , ∞ 0 · 12k 110∞ ], R41 = [∞ 0 · 12k 110∞ , ∞ 01 · 12k 110∞ , ∞ 01 · 12k+1 110∞ , ∞ 0 · 12k+1 110∞ ], R51 = [∞ (10) · 12k+1 110∞ , ∞ 01 · 12k+1 110∞ , ∞ 01 · 12k−1 110∞ , ∞ (10) · 12k−1 110∞ ], R61 = [∞ 0 · 1012k−1 110∞ , ∞ (10) · 1012k−1 110∞ , ∞ (10) · 10∞ , ∞ 0 · 10∞ ], R71 = [∞ (10)11 · 12k−1 110∞ , ∞ 01 · 12k−1 110∞ , ∞ 01 · 10∞ , ∞ (10)11 · 10∞ ]. See Fig. 15(a) for an example with k = 2. Iterating until F −3 (D) and eliminating its intersections with R11 and R21 one obtain three {B11 , B21 , B31 } rectangles from them as in Fig. 15(b). There are the followings B11 = [0∞ , ∞ 01 · 0∞ , ∞ 01 · 012k+1 110∞ , ∞ 0 · 012k+1 110∞ ], B21 = [∞ (01) · 012k+1 110∞ , ∞ 01 · 012k+1 110∞ , ∞ 01 · 012k−1 110∞ , ∞ (01) · 012k−1 110∞ ],

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Fig. 15. Steps of the construction for 12k−1 with k = 2.

B31 = [∞ 0 · 01012k−1 110∞ , ∞ (01) · 01012k−1 110∞ , ∞ (01) · 010∞ , ∞ 0 · 010∞ ]. See Fig. 15(b). For i = 1, · · · , k − 1 we will consider the iterates F 2i−1 (θs ) and F 2i (θs ). Observe that F 1 (θs ) intersects R31 and thus one can divide it in two new rectangles C13 and C23 as it was done in procedure 2 for star orbits, and since F (D) also intersects B31 one can reduce one its top unstable boundary until the unstable leaf passing through F (X). See Fig. 15(c). Moreover applying the same procedure to F 2 (X) it follows that R71 is divided in two rectangles such that the unstable leaf of one of them is a segment of the unstable manifold of X and the other has an unstable leaf which is a piece of the unstable manifold of F 2(X).

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Since F 2 (X) also intersects R61 , one have to decrease the top unstable leaf of R61 until the leaf of the symmetric of F 2 (X) in relation to the central horizontal line, that, is at the leaf of Q1 = ∞ (10)1110 · 10∞ . Repeating the process with the iterates F 2i−1 (θs ) and F 2i (θs ) with i = 2, · · · , k − 1, one obtains a partition P = {J1 , J2 , · · · , J6+2k } that satisfies the transitions given in (10) and the condition If F (Ji ) ∩ Jj = ∅ =⇒ F (∂s Ji ) ⊂ ∂s Jj .

(11)

See Fig. 15(d). To obtain a partition satisfying the condition If F −1 (Ji ) ∩ Jj = ∅ =⇒ F −1 (∂u Ji ) ⊂ ∂u Jj

(12)

we will proceed as follows. Note that J1 = B11 , J2 = B21 , J5+k = R51 and J5+2k = R61 . The top unstable boundary of J5+2k is included in the unstable manifold of ∞ (10)12k+1 0 · 10∞ , that is J5+2k = [∞ 0 · 1012k−1 110∞ , ∞ (10)12k+1 0 · 1012k−1 110∞ , ∞ (10)12k+1 0 · 10∞ , ∞ 0 · 10∞ ]. Considering the transition J5+2k → J2 → J5+k one sees that the rectangles J2 and J5+k have to be redefined as follows J2 = [∞ (01)12k+1 01 · 012k+1 110∞ , ∞ 01 · 012k+1 110∞ , ∞ 01 · 012k−1 110∞ , ∞

(01)12k+1 01 · 012k−1 110∞ ]

J5+k = [∞ (01)12k+1 010 · 12k+1 110∞ , ∞ 01 · 12k+1 110∞ , ∞ 01 · 12k−1 110∞ , ∞

(01)12k+1 010 · 12k−1 110∞ ].

See Fig. 15(e). Repeating the process by iterating J5+k we can get a rectangle J5+2k with its top unstable boundary in the unstable manifold of ∞ (10)(1012k+1 0)2 · 10∞ , that is, J5+2k = [∞ 0 · 1012k−1 110∞ , ∞ (10)(1012k+1 0)2 · 1012k−1 110∞ , ∞

(10)(1012k+1 0)2 · 10∞ , ∞ 0 · 10∞ ]

Repeating by induction and taking the limit of this process one gets that J5+2k = [∞ 0 · 1012k−1 110∞ , ∞ (1012k+1 0) · 1012k−1 110∞ , ∞ (1012k+1 0) · 10∞ , ∞ 0 · 10∞ ], that is J5+2k has a periodic point x with code 1012k+1 0 at its top unstable boundary, and extending it by iterations one conclude that each rectangle J2 , · · · , J6+2k has a periodic point at their unstable boundaries. In Fig. 15(f) there is the final partition for the current example. Using the notation of [20] it follows that 1012k+1 0 = (10) ∗ (10k 1) and thus x is a renormalizable point. 2 Lemma 10. Let w = 12k−1 be a decoration with k ≥ 1. Then, the topological entropy of P0w is equal to the logarithm of the largest positive root of the polynomial 2k   (−1)j x 2k+3−j .

x 6+2k − 2x 5+2k + x 4+2k − 2

j =0

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Fig. 16. Topologically equivalent heteroclinic tangles to the homoclinic tangle defined by w = 1.

Proof of Lemma 10. We use the method described in the Lemma 6. Let A be the transition matrix defined by relations (10). We find that R = {J1 , J3 , Jk+4 } is a rome for A. Thus the reduced matrix is ⎡ ⎢ AR (x) = ⎣

x −1 + x −3 + · · · + x −(2k+1)

x −1 + x −3 + · · · + x −(2k+1)

x −2 + x −(2k+4)

x −(2k+4)

x −(2k+2)

x −(2k+2)

x −1

⎤

⎥ 0 ⎦.

x −1

A calculation shows that the characteristic polynomial x 6+2k det(AR (x) − I3 ) is the desired expression. 2 Example 11. In Fig. 4(b) we have represented the tangle associated to the decoration w = 1, that is k = 1. This orbit appears with different shapes in many applications. For example it is the orbit if Fig. 10(b) of [27] due to Rückerl and Jung, and it is topologically equivalent to the tangles found in the works of Mitchell and Steck [19] (see Fig. 16(a)), Rom-Kedar, Leonard and Wiggins [25] (see Fig. 16(b)), and Camassa and Wiggins [3] (see Fig. 16(c)). Thus the topological entropy is the log of the largest root of the polynomial x 8 − 2x 7 + x 6 − 2(x 5 − x 4 + x 3 ) = 0 that is, log 1.89110302 = 0.63716 with five decimal round-off coordinates. 3.3. The Markov shift associated to the decoration w = 10k with k ≥ 1 Let us consider P0w as the orbit of the point pw = ∞ 010 · 10k 110∞ . Lemma 12. Let w = 10k with k ≥ 1. Then the Markov shift associated to the dynamics relative to P0w is given by the relations (see Fig. 17): J1 → {J1 , J3 , J4 , J5 } J2 → J6 , J3 → J7 , · · · , J3+4k → J7+4k , J4+4k → J8+4k J5+4k → {J5 , J6 , J9 , J10 , · · · , J5+4k , J6+4k } J6+4k → J4 , J7+4k → {J1 , J2 }, J8+4k → {J1 , J3 }

(13)

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Fig. 17. Markov partition associated to homoclinic orbit 10k .

Proof of Lemma 12. Since w = 10k is a maximal decoration in the notation of [21], it follows that the pruning domain associated is D = [X, Y ] with X = ∞ (10k 0)10k 1 · 10k 110∞ and Y = ∞ (10k 0) · 10k 110∞ , that is, ∂D = θs ∪ θu with θs ⊂ W s (∞ 010 · 10k 110∞ ) and θu ⊂ W u ((10k 0)∞ ). Using the inverse F −1 , eliminating the region between F −i (θu ) and the top unstable leaf for i = 1, · · · , k + 1 (see Fig. 18(a)), and the region between F −i (θ ) and the bottom unstable leaf for i = k + 2, · · · , 2k + 3 (see Fig. 18(b)), F −i (D) allows to define 4(k + 2) rectangles {B11 , B21 , B31 , B41 , · · · , B1k+2 , B2k+2 , B3k+2 , B4k+2 } with the following coordinates: B11 = [0∞ , ∞ 0 · 10∞ , ∞ 01 · 0k 010k 110k 110∞ , ∞ 0 · 0k 010k 110k 110∞ ] B21 = [∞ (10k 0)1 · 0k 010k 010k 110∞ , ∞ 01 · 0k 010k 110k 110∞ , ∞

01 · 0k 010k 110∞ , ∞ (10k 0)1 · 0k 010k 110∞ ]

B31 = [∞ 0 · 0k 010k 010k 110∞ , ∞ (10k 0)1 · 0k 010k 010k 110∞ , ∞

(10k 0)1 · 0k 010∞ , ∞ 0 · 0k 010∞ ]

B41 = [∞ 0 · 0k 110∞ , ∞ (10k 0)1 · 0k 110∞ , ∞ (10k 0)1 · 0k 110k 110∞ , ∞ 0 · 0k 110k 110∞ ] and for i = 2, · · · , k + 1, B1i = [∞ 0 · 0k−i+2 110k 110∞ , ∞ 01 · 0k−i+2 110k 110∞ , ∞ 01 · 0k−i+1 010k 110k 110∞ , ∞ 0 · 0k−i+1 010k 110k 110∞ ]

B2i = [∞ (10k 0)10i−1 · 0k−i+1 010k 110k 110∞ , ∞ 01 · 0k−i+1 010k 110k 110∞ , ∞ 01 · 0k−i+1 110k 110∞ , ∞ (10k 0)10i−1

· 0k−i+1 010k 110∞ ]

B3i = [∞ 0 · 0k−i+1 010k 010k 110∞ , ∞ (10k 0)10i−1 · 0k−i+1 010k 010k 110∞ , ∞ (10k 0)10i−1

· 0k−i+1 010∞ , ∞ 0 · 0k−i+1 010∞ ]

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Fig. 18. The construction for w = 10k with k = 2.

B4i = [∞ 0 · 0k−i+1 110∞ , ∞ (10k 0)10i−1 · 0k−i+1 110∞ , ∞ (10k 0)10i1 · 0k−i+1 110k 110∞ , ∞ 0 · 0k−i+1 110k 110∞ ]

and for i = k + 2, B1k+2 = [∞ 0 · 110k 110∞ , ∞ 01 · 110k 110∞ , ∞ 01 · 10k 110k 110∞ , ∞ 0 · 10k 110k 110∞ ] B2k+2 = [∞ (10k 0) · 10k 110k 110∞ , ∞ 01 · 10k 110k 110∞ , ∞ 01 · 10k 110∞ , ∞ (10k 0) · 10k 110∞ ] B3k+2 = [∞ 0 · 10k 010k 110∞ , ∞ (10k 0) · 10k 010k 110∞ , ∞ (10k 0) · 10∞ , ∞ 0 · 10∞ ] B4k+2 = [∞ (10k 0)10k 1 · 10k 110∞ , ∞ 01 · 10k 110∞ , ∞ 01 · 10∞ , ∞ (10k 0)10k 1 · 10∞ ]

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Iterating F i (D) for i = 1, · · · , k + 2 one see that the intersection between F i (D) and B3i can be eliminated and top unstable leaf of B3i can be reduced to leaf of F i (X), that is, now B3i will be B31 = [∞ 0 · 0k 010k 010k 110∞ , ∞ (10k 0)10k 11 · 0k 010k 010k 110∞ , ∞ (10k 0)10k 11 · 0k 010∞ , ∞ 0 · 0k 010∞ ]

B3i = [∞ 0 · 0k−i+1 010k 010k 110∞ , ∞ (10k 0)10k 110i−1 · 0k−i+1 010k 010k 110∞ , ∞ (10k 0)10k 110i−1

· 0k−i+1 010∞ , ∞ 0 · 0k−i+1 010∞ ]

B3k+2 = [∞ 0 · 10k 010k 110∞ , ∞ (10k 0)10k 110k 0 · 10k 010k 110∞ , ∞ (10k 0)10k 110k 0 · 10∞ , ∞ 0 · 10∞ ].

See Fig. 18(c). Since there is a path B3k+2 → B21 → · · · → B2k+2 → B41 → · · · B4k+2 → B31 → B3k+1 → B3k+2 one has to reduce the bottom unstable boundaries of B21 , B22 , · · · , B2k+2 , the top unstable boundaries of B41 , · · · , B4k+1 , the bottom unstable boundary of B4k+2 and the top unstable boundaries of B31 , B32 , · · · , B3k+2 . See Fig. 18(d) and (e). Thus the new B3k+2 is B3k+2 = [∞ 0 · 10k 010k 110∞ , ∞ (10k 0)(10k 010k 110k 0)2 · 10k 010k 110∞ , ∞ (10k 0)(10k 010k 110k 0)2

· 10∞ , ∞ 0 · 10∞ ]

Repeating the process n times one obtains that B3k+2 = [∞ 0 · 10k 010k 110∞ , ∞ (10k 0)(10k 010k 110k 0)n · 10k 010k 110∞ , ∞ (10k 0)(10k 010k 110k 0)n

· 10∞ , ∞ 0 · 10∞ ]

Taking the limit when n → +∞ one obtain the coordinates of B3k+2 B3k+2 = [∞ 0 · 10k 010k 110∞ , ∞ (10k 010k 110k 0) · 10k 010k 110∞ , ∞ (10k 010k 110k 0) · 10∞ , ∞ 0 · 10∞ ]

By iterations, all the rectangles, except B11 , has one unstable boundary included in the unstable manifold of point of the orbit of x = 10k 010k 110k 0. In the notations of [20], x = 10k 0 ∗ 101 is renormalizable. See Fig. 18(e). Then we redefine J4i+l = Bli+1 for i = 0, · · · , k − 1, l = 1, · · · , 4. 2 Lemma 13. The topological entropy of the Markov shift associated to the decoration w = 10k is the largest root of the equation x 8+4k − 2x 7+4k + x 6+3k − 2x 5+3k + 2

3k  j =2k

 x 4+j − 2x 3+2k = 0.

(14)

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Fig. 19. Markov partition associated to homoclinic orbit with decoration 0k 10k .

Proof of Lemma 13. It is not difficult to see that the rome is {J1 , J3 , J5+4k }. Then the reduced matrix is ⎡

x −1 + x −(k+3)

⎢ −(k+2) + x −(3k+6) AR (x) = ⎢ ⎣x E

x −1 + x −(k+3)

x −(k+1)

x −(3k+6)

0

E

x −1 + x −2 + · · · + x −(k+1)

⎤ ⎥ ⎥, ⎦

with E = x −(2k+4) + x −(2k+3) + · · · + x −(k+5) + x −(k+4) . By the same reasons of Lemma 6, one that the characteristic polynomial is x 8+4k det(AR (x) − I3 ) which gives us the expression (14). 2 3.4. The Markov shift associated to the decoration w = 0k 10k Consider P as the orbit of pw = ∞ 010 · 0k 10k 110∞ . Lemma 14. Let w = 0k 10k be a decoration with k ≥ 1. See Fig. 19. Then the dynamics relative to P0w is given by the Markov shift defined by the relations: J1 → {J1 , J3 , J4 }, J2 → J5 , J3 → {J7 , J8 } J4 → J9 , J5 → J10 , · · · , J5(k+1) → J5(k+2) J5k+6 → J3 , J5k+7 → J2 , J5k+8 → J1

(15)

J5k+9 → {J1 , J3 , J4 , J5 }, J5k+10 → {J6 , J8 } Proof of Lemma 14. In this case one needs two pruning domains for describing the forced dynamics:

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Fig. 20. Construction of the Markov partition for w = 0k 10k with k = 2.

• D1 = [∞ (0k 10)0k 11 · 0k 10k 110∞ , ∞ (0k 10) · 0k 10k 110∞ ], that is, ∂D1 = θs ∪ θu with θs ⊂ W s (∞ 010 · 0k 10k 110∞ ) and θu ⊂ W u ((0n 10)∞ ); and • D2 = [∞ 1 · 0k−1 110∞ , ∞ 10 · 0k−1 110∞ ], that is, ∂D2 = θs ∪ θu with θs ⊂ W s (∞ 0100k 10 · 0k−1 110∞ ) and θu ⊂ W u (1∞ ). See Fig. 20(a) for the case 02 102 . Since the point ∞ 0100k 10 · 0k−1 110∞ has the same horiω zontal coordinate as the star homoclinic orbit P0 q = ∞ 010 · 0k−1 010∞ with q = 1/(k + 2), one can use the disc D2 and the procedures in Lemma 6 for getting a Markov partition P = {B1 , B2 , · · · , Bk+3 } where B1 = [0∞ , ∞ 01 · 0∞ , ∞ 01 · 0k 010∞ , ∞ 0 · 0k 010∞ ];

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for i = 2, · · · , k + 2 Bi = [∞ 0 · 0k−i+2 110∞ , ∞ (0k 11)∞ 0i−2 · 0k−i+2 110∞ , ∞ (0k 11)∞ 0i−2

· 0k−i+2 10∞ , ∞ 0 · 0k−i+2 10∞ ];

and Bk+3 = [∞ (10k 1) ·10k−1 010∞ ,∞ 01 ·10k−1 010∞ , ∞ 01 ·10∞ ,∞ (10k 1) ·10∞ ]. See Fig. 20(b). The transitions are the following: B1 → {B1 , B2 }; Bi → Bi+1 , for i = 2, · · · , k + 1; Bk+2 → {B1 , Bk+3 }; Bk+3 → {B1 , B2 }. (16) Now consider the region E = [1∞ · 0k 110∞ , 1∞ · 10k−1 110∞ ]. Note that E ∩ Cw = ∅ then by 1 2 and Bk+3−i for each backward iterations F −i (E) divides Bk+3−i in two new rectangles Bk+3−i 1 2 1 2 i = 1, · · · , k + 2. See Fig. 20(c). Thus P = {B1 , B1 , · · · , Bk+2 , Bk+2 , Bk+3 } is a Markov partition that contains the core. Now let us consider the disk D1 . It is not difficult to see that the region E = [∞ (0k+1 1) · 00k 110k−1 110∞ , ∞ (0k+1 1) · 10k 110k−1 110∞ ] contains F −1 (D1 ) and is disjoint from the core. Then the upper unstable leaf of B12 has to be changed to the leaf of (0k+1 1)∞ . 1 2 By backward iterations, for i = 1, · · · , k + 1, F −i (E ) divides Bk+3−i and Bk+3−i in two new 1 −(k+2)

(E ) divides B1 in two rectangles and decrease the left starectangles, respectively; and F ble boundary of B12 . See Fig. 20(d). Iterating one more time, for i = 1, · · · , k + 2, F −(k+2+i) (E ) 2 divides the two rectangles in Bk+3−i in three new rectangles as in Fig. 20(e). Therefore we have obtained 5(k + 2) rectangles {J1 , · · · , J5k+10 } which satisfies the stable condition for Markov partitions. Adjusting the unstable boundaries as in the previous sections, one obtains a Markov partition for the core which still has the transitions given in (15). See Fig. 20(f). 2 Lemma 15. The entropy of the Markov shift given by relations (15) is the largest root of the polynomial equation x 5k+10 − x 5k+9 − x 4k+8 − x 4k+7 − 2x 2k+4 = 0. Proof of Lemma 15. Note that a rome for the relations (15) is the set {J1 , J3 , J4 }. Counting the simple loops starting in elements of the rome, we get the following reduced matrix: ⎡

x −1 −(k+2) ⎣ AR (x) = x + x −(3k+6) −(k+2) x + x −(3k+5)

x −1

x −(3k+6) + x −(3k+5)

x −(k+2)

x −1 0

x −(k+2)

⎤ ⎦.

Thus the characteristic polynomial is x 5(k+2) det(AR (x) − I3 ) which gives the desired expression. 2 4. Concluding comments It follows from Theorem 1 that the dynamics implied by a homoclinic tangle can be explicitly described by a Markov shift associated to the orbits, namely the dynamics on the dynamical core of an end periodic mapping. A lower bound for the topological entropy, given in terms of

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families of their symbolic codes, was determined from the characteristic polynomial, computed by the rome of a transition matrix. The families of homoclinic tangles studied here are well known in many physical models, where lower bounds for the entropy are calculated only for some cases. The method outlined in those families of homoclinic tangles is a general algorithm, being also applicable to heteroclinic orbits; in fact the procedure is started in a heteroclinic orbit that belongs in the same isotopy class of the homoclinic one. The reader is encouraged to apply it, for example, for orbits with decoration w = (10)k . References [1] Ll. Alsedà, J. Llibre, M. Misiurewicz, C. Simó, Twist periodic orbits and topological entropy for continuous maps of the circle of degree one which have a fixed point, Ergodic Theory Dynam. Systems 5 (4) (1985) 501–517. [2] L. Block, J. Guckenheimer, M. Misiurewicz, L.S. Young, Periodic points and topological entropy of onedimensional maps, in: Global Theory of Dynamical Systems, Proc. Internat. Conf., Northwestern Univ., Evanston, Ill., 1979, in: Lecture Notes in Math., vol. 819, Springer, Berlin, 1980, pp. 18–34. [3] R. Camassa, S. Wigging, Chaotic advection in a Rayleigh–Bénard flow, Phys. Rev. A 43 (2) (1991) 774–797. [4] J. Cantwell, L. Conlon, Endperiodic automorphisms of surfaces and foliations, arXiv:1006.4525v6, 2017. [5] P. Collins, Dynamics of surface diffeomorphisms relative to homoclinic and heteroclinic orbits, Dyn. Syst. 19 (1) (2004) 1–39. [6] P. Collins, Forcing relations for homoclinic orbits of the Smale horseshoe map, Exp. Math. 14 (1) (2005) 75–86. [7] P. Collins, Entropy-minimizing models of surface diffeomorphisms relative to homoclinic and heteroclinic orbits, Dyn. Syst. 20 (4) (2005) 369–400. [8] A. de Carvalho, Pruning fronts and the formation of horseshoes, Ergodic Theory Dynam. Systems 19 (4) (1999) 851–894. [9] A. de Carvalho, T. Hall, Unimodal generalized pseudo-Anosov maps, Geom. Topol. 8 (2004) 1127–1188. [10] R. Easton, Trellises formed by stable an unstable manifolds in the plane, Trans. Amer. Math. Soc. 294 (2) (1986) 719–732. [11] S. Fenley, End periodic surface homeomorphisms and 3-manifolds, Math. Z. 224 (1997) 1–24. [12] R. Guantes, F. Borondo, S. Miret-Artés, Periodic orbits and the homoclinic tangle in atom-surface chaotic scattering, Phys. Rev. E 56 (1) (1997) 378–389. [13] T. Hall, The creation of horseshoes, Nonlinearity 7 (1994) 861–924. [14] M. Handel, A fixed-point theorem for planar homeomorphisms, Topology 38 (2) (1999) 235–264. [15] M. Handel, Global shadowing of pseudo-Anosov homeomorphisms, Ergodic Theory Dynam. Systems 5 (1985) 373–377. [16] W. Huaraca, V. Mendoza, Minimal topological chaos coexisting with a finite set of homoclinic and periodic orbits, Phys. D 315 (2016) 83–89. [17] H.M. Hulme, Finite and Infinite Braids: a Dynamical Systems Approach, Ph.D. thesis, Univ. of Liverpool, 2000. [18] A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math. 51 (1980) 137–173. [19] K. Mitchell, D. Steck, Fractal templates in the escape dynamics of trapped ultracold atoms, Phys. Rev. A 76 (2007), 031403(R). [20] V. Mendoza, Renormalization and forcing of horseshoe orbits, Topology Appl. 173 (2014) 234–239. [21] V. Mendoza, Dynamics forced by homoclinic orbits, preprint, 2014. [22] M. Misiurewicz, Horseshoes for mappings of an interval, Bull. Acad. Pol. Sci., Sér. Sci. Math. 27 (1979) 167–169. [23] S. Newhouse, Continuity properties of entropy, Ann. of Math. (2) 129 (1990) 215–235, Errata in Ann. of Math. 131 (1990) 409–410. [24] V. Rom-Kedar, Transport rates of a class of two-dimensional maps and flows, Phys. D 43 (1990) 229–268. [25] V. Rom-Kedar, A. Leonard, S. Wiggins, An analytical study of transport, mixing and chaos in an unsteady vortical flow, J. Fluid Mech. 214 (1990) 347–394. [26] V. Rom-Kedar, Homoclinic tangles – classification and applications, Nonlinearity 7 (1994) 441–473. [27] B. Rückerl, C. Jung, Scaling properties of a scattering system with an incomplete horseshoe, J. Phys. A 27 (1994) 55–77. [28] S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967) 747–817.

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