Decoration invariants for horseshoe braids

The Decoration Conjecture describes the structure of the set of braid types of Smale's horseshoe map ordered by forcing, providing information about the order in which periodic orbits can appear when a horseshoe is created. A proof of this conjecture is given for the class of so-called lone decorations, and it is explained how to calculate associated braid conjugacy invariants which provide additional information about forcing for horseshoe braids.


Introduction
Forcing relations are a valuable tool in the dynamical study of parameterized families of transformations: they provide information about when the presence of certain dynamical features, such as the existence of periodic orbits of a particular type, imply the presence of other dynamical features.
For surface homeomorphisms, Smale's horseshoe is the paradigmatic map with complicated dynamical behaviour, and understanding how it is created in parameterized families of homeomorphisms is an important problem. In this context, it is fruitful to study the forcing order on the set of braid types of horseshoe periodic orbits [Boy84]: this partial order describes constraints on the order in which periodic orbits can appear during the creation of a horseshoe.
Algorithmic implementations [BH95,FM93,Los93] of Thurston's classification of surface homeomorphisms provide a means of deciding whether or not one given braid type forces another, but such an approach doesn't provide any information about the global structure of the forcing order on the set of all horseshoe braid types.
The decoration conjecture [dCH02a] claims that the set of horseshoe braid types is partitioned into families D w = {β w q }, each parameterized by a rational number q, which are totally ordered by the forcing order, in such a way that β w q forces β w q ′ if and only if q ≤ q ′ . The families are labelled by decorations w, which are finite words in the symbols 0 and 1. Within families this trivializes the computation of forcing: simply compare the rational parameters in the usual order (or, equivalently, compare the symbolic representations of the braids using the unimodal order). Some special cases of this conjecture have been proved, and it is supported by strong intuitive evidence from pruning theory: however, a general proof has so far been elusive. In this paper, the conjecture is proved for a class of decorations called lone. There are many lone decorations: two infinite families of them are described in Sections 6.2 and 6.3 below. Collins [Col05] states that 21 of the 63 decorations of length 5 or less are lone 1 .
The proofs of the main results presented here combine the pruning techniques introduced in [dC99] with unremovability arguments of the type developed in [Hal94]. Although pruning has provided an inspiration for several previous results about forcing, this seems to be the first time that it has been used effectively in proving such results.
The second goal of the paper is to present a new set of braid type invariants. Each totally ordered family D w = {β w q } gives rise to an invariant (or, looking at it from the point of view of the horseshoe braids themselves, a braid conjugacy invariant) r w , defined on the set of all horseshoe braid types β by r w (β) = inf{q : β w q ≤ β}.
These invariants provide much additional information about the forcing order: a periodic orbit of braid type β can only be created once periodic orbits of braid types β w q have been created for all q > r w (β). Notice in this statement that, while β w q is restricted to be a horseshoe braid of lone decoration, β can be any horseshoe braid.
The techniques presented here make it possible, for lone decorations, both to prove that the family D w is totally ordered by forcing, and to calculate the associated decoration invariant r w .
The necessary background material on horseshoe braids and the decoration conjecture is given in Section 2, before the main theorem and the algorithm for computing decoration invariants are presented in Section 3. Some further background material required for the proof, mostly concerning pruning and the Asimov-Franks theorem, is given in Section 4. The proof of the main theorem is given in Section 5. This is followed by some examples and applications in Section 6. The applications include: • a treatment of the so-called "Star" decorations (Section 6.2), which were introduced in [dCH04], including the completion of the main theorem of that paper (Theorem 54 here), thus providing a description of forcing between the families corresponding to different star decorations; • an example of how decoration invariants can be used to prove that certain other decorations are lone, and hence provide their own invariants (Theorem 59); • an example of how decoration invariants can be used to prove that certain horseshoe braids are of pseudo-Anosov type (Theorem 62); and • a discussion of topological entropy bounds arising from decoration invariants (Section 6.4).

Horseshoe braids: height and decoration
This section contains the background material necessary to understand the statement of the main theorem (Theorem 19). Although complete, the treatment is rather terse: the papers [dCH02a,dCH02b] are recommended for readers seeking a more detailed account.
2.1. Smale's horseshoe and the unimodal order. In this paper, the standard model of Smale's horseshoe map [Sma67] F : D 2 → D 2 depicted in Figure 1 is used. The set Λ = {x ∈ D 2 : F n (x) ∈ S for all n ∈ Z} (where S is the square depicted in Figure 1) is a Cantor set, and the itinerary map

2.2.
Notation. Much of the technical part of the paper is concerned with constructing elements of {0, 1} N or {0, 1} Z from words w ∈ {0, 1} j for various j. The following notation will be used.
A word is an element of j≥0 {0, 1} j .
When two of these accents are combined, they are applied 'bottom upwards': thus, for example, ũ =ȗ is the word obtained by changing the final symbol of u and then reversing the result. Denote by u + the word u 0 u 1 . . . u j−1 u j ∈ {0, 1} j+1 , where u j is chosen so that u + is even; and by + u the word u −1 u 0 u 1 . . . u j−1 ∈ {0, 1} j+1 , where u −1 is chosen so that + u is even. A non-empty initial subword of u ∈ {0, 1} j is a word u 0 . . . u i−1 ∈ {0, 1} i for some i with 0 < i ≤ j. Similarly, a non-empty final subword of u is a word u i u i+1 . . . u j−1 for some i with 0 ≤ i < j.
2.3. The horseshoe and its inverse. Recall (see e.g. [dCH03]) that F is conjugate to its inverse: F −1 = φ • F • φ −1 , where φ : D 2 → D 2 is the (orientationreversing) homeomorphism obtained by first reflecting S in its horizonal centre line, and then rotating it anticlockwise about its centre point through an angle π/2. The involution φ restricts to an involution Λ → Λ which corresponds to reversing 2.4. Height. The height function [Hal94] is a (not strictly) decreasing function {0, 1} N → [0, 1/2] which is central to the results and methods of this paper. In order to define it, is is necessary first to introduce, for each rational m/n ∈ (0, 1/2], a word c m/n ∈ {0, 1} n+1 : these words will also play a central rôle throughout the paper. Definition 1. Let m/n be a rational in (0, 1/2]. Let L m/n be the straight line in R 2 from (0, 0) to (n, m). For 0 ≤ i ≤ n, let s i = 1 if L m/n crosses some line y = integer for x ∈ (i − 1, i + 1), and s i = 0 otherwise. Then the word c m/n ∈ {0, 1} n+1 is defined by c m/n = s 0 s 1 . . . s n .
So, for example, c 3/10 = 10011011001 can be read off from Figure 2. The general form of these words can be seen in Table 1, which shows c m/n for all m/n ∈ (0, 1/2) with 1 ≤ m ≤ 4 and 3 ≤ n ≤ 11. Note that the words c q are clearly palindromic, and c m/n is of the form 10 κ1 1 2 0 κ2 1 2 . . . 1 2 0 κm 1 for some integers κ i ≥ 0 (an explicit formula for κ i can be found in [Hal94], but is not needed here: note, however, that each κ i is equal either to κ 1 or to κ 1 − 1). It can also be seen easily from this description (see Lemma 2.7 of [Hal94] with the property that c ≺ (c q 0) ∞ for all rationals q ∈ (0, q(c)), and (c q 0) ∞ ≺ c for all rationals q ∈ (q(c), 1/2].   [Hal94]) provides a practical means of calculating q(c) for all c ∈ {0, 1} N which contain the subword 010 (this is all that will be required in this paper).
Lemma 3. Let c ∈ {0, 1} N , and suppose that c contains the subword 010. Then q(c) is rational, and can be calculated as follows. First, if 10 is not an initial subword of c, then q(c) = 1/2. If 10 is an initial subword, then write c = 10 κ1 1 µ1 0 κ2 1 µ2 . . . , where each κ i ≥ 0, each µ i is either 1 or 2, and µ i = 1 only if κ i+1 > 0 (thus κ i and µ i are uniquely determined by c). For each r ≥ 1, define and let s ≥ 1 be the least integer such that either µ s = 1, or if µ s = 1, or µ s = 2 and w > y for all w ∈ I s+1 (c).
2.5. Periodic orbits of the horseshoe. Let P be a period n orbit of the horseshoe F : D 2 → D 2 (throughout this paper, "period n" means least period n). Let p be the rightmost point of P : thus k(p) = c P for some word c P of length n, which is called the code of the periodic orbit P . Note that the choice of p as the rightmost point of P means that Recall [Boy84] that the braid type bt(P ; f ) of a period n orbit P of an orientationpreserving homeomorphism f : D 2 → D 2 is a conjugacy class in the mapping class group MCG(D n ) of the n-punctured disk D n , namely the conjugacy class of the isotopy class of h −1 f h : D n → D n , where h : D n → D 2 \ P is any orientationpreserving homeomorphism (if P ⊆ ∂D 2 , then first extend f over an exterior collar). Braid types can thus be classified using the Thurston classification [Thu88] as finite order, reducible, or pseudo-Anosov. The forcing order ≤ on the set BT of all braid types is a partial order defined as follows: if β, γ ∈ BT, then β ≤ γ if and only if every orientation preserving homeomorphism f : D 2 → D 2 which has a periodic orbit P with bt(P ; f ) = γ also has a periodic orbit Q with bt(Q ; f ) = β. If P is a periodic orbit of the horseshoe, then the symbol P will often be used to denote the braid type bt(P ; F ) as well as the periodic orbit itself. In particular, the notation P ≤ Q is used as a shorthand for bt(P ; F ) ≤ bt(Q ; F ).
Most periodic orbits P of the horseshoe are paired, in the sense that c P is also the code of a horseshoe periodic orbit P : in this case bt(P ; F ) = bt( P ; F ). Thus, for example, the two periodic orbits P and P with codes c P = 10010 and cP = 10011 have the same braid type, and it is common to write c P = 1001 0 1 , reflecting the fact that the object of interest is the braid type rather than the periodic orbit itself. The only orbits which are not paired are those of even period 2k, whose codes are of the form c P = ww for some word w of length k.
The height q(P ) ∈ (0, 1/2] of a horseshoe periodic orbit P is defined to be q(c ∞ P ). It is a braid type invariant [Hal94], so in particular q(P ) = q( P ) if P is paired, and (taking the code which ends with 0), q(P ) is rational and can be calculated using the algorithm of Lemma 3. If P is not paired, then again c ∞ P contains the word 010, so the algorithm terminates and q(P ) is rational.
Periodic orbits of the horseshoe can be classified as follows: Orbits of finite order braid type: There are two fixed points, with codes 0 and 1 (and a fixed point outside of S). There is one period two orbit, with code 10. For each rational m/n ∈ (0, 1/2), there is exactly one pair of period n orbits of finite order braid type whose rotation number about the fixed point of code 1 is m/n. The codes of these orbits are d m/n 0 1 , where d m/n is the word consisting of the first n − 1 symbols of c m/n . These periodic orbits have height m/n. There are no other periodic orbits of finite order braid type. (These results are due to Holmes and Williams [HW85].) NBT orbits: For each rational q = m/n ∈ (0, 1/2), the words c m/n 0 1 are the codes of a pair of period n + 2 orbits of pseudo-Anosov braid type, called NBT orbits and denoted P * q [Hal94]. These periodic orbits have height q. There are no other horseshoe periodic orbits of these braid types. For q = 1/2, the word c 1/2 1 = 1011 is the code of a periodic orbit of reducible braid type, denoted P * 1/2 . Orbits described by height and decoration: All other horseshoe periodic orbits P can be described by their height q(P ) ∈ (0, 1/2] ∩ Q and their decoration, which is a word w ∈ {0, 1} k for some k ≥ 0: a periodic orbit of height q has decoration w if and only if it has one of the four codes c q 0 1 w 0 1 . Two periodic orbits of the same height and decoration have the same braid type [dCH03]: the notation P w q can therefore be used for any periodic orbit of height q and decoration w.
Only certain heights are compatible with a given decoration w. Define the scope q w of w to be the height of the horseshoe periodic orbit containing the point with itinerary 10w0, that is, The following result can be found in [dCH02a].
Lemma 6. For q < q w there are four periodic orbits of height q and decoration w (i.e. those with codes c q 0 1 w 0 1 ), while for q > q w there are no periodic orbits of height q and decoration w. (For q = q w , the four words c q 0 1 w 0 1 may or may not be codes of periodic orbits of height q.) It is convenient for many purposes to consider * to be a decoration as well (with scope q * = 1/2). Then every periodic orbit P of the horseshoe is either of finite order braid type, or can be described uniquely (up to braid type) by its decoration w ∈ { * } ∪ k≥0 {0, 1} k and its height q ≤ q w . (If q(P ) = m/n, then the period N of P is either n, or greater than or equal to n + 2. If N = n, then P is of finite order braid type; if N = n + 2, then P has decoration * ; and if N ≥ n + 3, then P has decoration w of length N − (n + 3).) The following lemma, which will be used several times, follows immediately from the definition of q w above, and the fact that q( P ) = q(P ) for any horseshoe periodic orbit P , where P is the periodic orbit containing the point of itinerary c P (Lemma 3.8 of [Hal94]).
The results presented in this paper are related to the Decoration Conjecture [dCH02a].
(Statement c) below is not relevant in this paper, and the definition of topological train track type is therefore not given.) a) There is an equivalence relation ∼ on the set of decorations, with the property that bt(P w q ; F ) = bt(P w ′ q ′ ; F ) if and only if q = q ′ and w ∼ w ′ . b) If q = q w then P w q has pseudo-Anosov braid type. c) For each decoration w, the periodic orbits P w q with 0 < q < q w all have the same topological train track type. d) For each decoration w, the set of braid types of periodic orbits of decoration w is totally ordered by forcing, with P w q ≤ P w q ′ if and only if q ≥ q ′ . e) There is a partial order on the set of equivalence classes of decorations, with the property that if q > q ′ and w w ′ then P w q ≤ P w ′ q ′ .
The main theorem of this paper, Theorem 19 below, concerns d) and e) of Conjecture 8, for a certain class of decorations: Definition 9. A decoration w ∈ k≥0 {0, 1} k is said to be lone if for all q ∈ (0, q w ) ∩ Q, the four horseshoe periodic orbits of height q and decoration w are the only horseshoe periodic orbits of their braid type.
Theorem 19 states that Conjecture 8 d) holds for lone decorations w, when restricted to those P w q which have pseudo-Anosov braid type (so if b) is true, then so is d) for lone decorations). It also elucidates the partial order of e) by providing a means of determining, for any lone decoration w and any horseshoe periodic orbit R, which of the orbits P w q are forced by R.
Remark 10. If Conjecture 8 a) holds, then if there is a single q ∈ (0, q w ) ∩ Q for which the four orbits of height q and decoration w are the only horseshoe periodic orbits of their braid type, then the same is true for all q ∈ (0, q w ) ∩ Q. In this case, the lone decorations are precisely those which are alone in their ∼-equivalence classes.
The results stated in the next lemma can all be found in [Hal94]: Lemma 11. Let P and Q be periodic orbits of the horseshoe. a) Height is a braid type invariant.
b) If q < q(P ) then P * q ≥ P , while if q > q(P ) then P * q ≥ P . c) If P ≥ Q then q(P ) ≤ q(Q).

Statement of results
Let w ∈ k≥0 {0, 1} k be a lone decoration. Let the set of braid types of horseshoe periodic orbits of decoration w. Let K w be the subset of D w consisting of pseudo-Anosov braid types: this contains bt(P w q ; F ) for a dense set of q ∈ (0, q w ] ∩ Q by Lemma 18 below (and, if Conjecture 8 b) holds, for all q except possibly q w ).
The main results of this paper are: • that K w is totally ordered by forcing, with P w q ≤ P w q ′ if and only if q ≥ q ′ ; and • that there exists a practical algorithm to determine, for any horseshoe periodic orbit R, the number r w (R) ∈ (0, q w ] with the property that In particular, for each lone decoration w, r w is a braid type invariant defined on the set of all horseshoe periodic orbits. The algorithm to compute r w (R) is complicated to state, although it is easily implemented and is computationally light 2 . It will therefore be described informally and illustrated by examples first; a formal description which is more suitable for use in later proofs will then be given. The purpose of the algorithm is to decide for which values of q points of R are contained in certain disks bounded by segments of stable and unstable manifold through points of the orbits P w q : this is what determines which of the P w q are forced by R (Theorem 51). Let R be any horseshoe periodic orbit, with code c R : if R is paired, then choose the code ending with 0 (so that the algorithm of Lemma 3 can be used to compute the various heights below). Let w be any word. Then r w (R) is given by where λ w (R), µ w (R), and ν w (R) are elements of (0, q w ] ∩ Q which will now be described. µ w (R). Recall that + w is the word obtained by prepending one symbol to the front of w, in such a way that + w is even. Let v be a non-empty even final subword of + w, and seek all occurrences of the wordsv 0 1 10 in one period of c R (that is, all occurrences of eitherv010 or ofv110: recall thatv is the word obtained from v by changing its initial symbol). For each such occurrence, compute the height of the forward sequence in c R starting at the final symbols 10 in the occurrence. µ w (R) is the minimum of such heights taken over all such occurrences and all non-empty even final subwords v of + w. If there are no such occurrences, or if the minimum of the heights is greater than q w , then µ w (R) = q w .
Example: Let R have code c R = 100010111001010, and let w = 1. Then + w = 11, which has only one non-empty even final subword, namely v = 11. Hencev = 01, and occurrences of 01010 and 01110 are sought. The three occurrences of such words in one period of c R are shown in Figure 3. The corresponding forwards sequences have heights q(10010 . . .) = 1/3, q(1010 . . .) = 1/2, and q(100010 . . .) = 1/4. Hence µ w (R) = 1/4. ν w (R). The computation of ν w (R) is similar: let v be a non-empty even initial subword of w + , and seek all occurrences of the words 01 0 1ṽ in one period of c R . For each such occurrence, compute the height of the backward sequence in c R starting at the initial symbols 01 in the occurrence. ν w (R) is the minimum of such heights taken over all such occurrences and all non-empty even initial subwords v of w + . If there are no such occurrences, or if the minimum of the heights is greater than q w , then ν w (R) = q w .
Example: Continuing with the above example, w + = 11 which has only one non-empty even initial subword, namely v = 11. Henceṽ = 10, and occurrences of 01010 and 01110 are sought. This gives ν w (R) = 1/3 (see Figure 4). Seek all occurrences of the four words 01 0 1 w 0 1 10 in one period of c R . For each such occurrence, compute max(q(b), q(f )), where b is the backward sequence starting at the initial symbols 01 in the occurrence, and f is the forward sequence starting at the final symbols 10. Then λ w (R) is the minimum value of this quantity taken over all such occurrences. If there are no such occurrences, or if the minimum is greater than q w , then λ w (R) = q w . Example: Continuing with the above example, occurrences of 0101010, 0111010, 0101110, and 0111110 are sought. There are two such occurrences, one with q(b) = 1/3 and q(f ) = 1/4, and the other with q(b) = 1/4 and q(f ) = 1/3 (see Figure 5).
Theorem 13. Let w be a decoration and q ∈ (0, q w ). Let p n denote the proportion of period n horseshoe orbits R with r w (R) < q. Then p n → 1 as n → ∞.
As a further illustration of the use of these invariants, let R = P 10 1/6 be the periodic orbit of code c R = 10000011100. Computing r w (R) for the same lone decorations w as in Table 2 gives r * (R) = 1/3, r · (R) = 1/3, r 0 (R) = 1/6, r 1 (R) = 1/3, r 00 (R) = 1/6, r 11 (R) = 1/3, r 000 (R) = 1/6, r 101 (R) = 1/3, and r 111 (R) = 1/3. The periodic orbits of these decorations are depicted in Figure 6: for each decoration w, there are periodic orbits P w q for each rational q in (0, q w ) (represented by points on the vertical lines), but not for rationals q > q w . The decoration invariants show that all orbits on the thicker parts of the lines are forced by R, but that none of the orbits on the thinner parts of the lines are forced by R (the theorem doesn't state whether or not the orbits represented by points at the transition from thin to thick are forced by R). Notice that since q(R) = 1/6, R cannot force any orbit P w q with q < 1/6 by Lemma 11 c). The decoration 10 of R itself is not lone, and is not required to be by Theorem 19.  Figure 6. Some orbits forced, and not forced, by the orbit of code 10000011100 A formal statement of the algorithm for computing r w (R) will now be given. Some preliminary definitions are useful.

Definitions 14.
Let v and c be words, with |c| ≥ 1.
Finally, if w is a word and R is a horseshoe periodic orbit, define : v is a non-empty even initial subword of w + }, and The pieces are now all in place to describe the invariant r w : Definition 16. Let w ∈ k≥0 {0, 1} k be a decoration, and R be a horseshoe peri- A number of the results presented below require that the braid type of P w q be pseudo-Anosov, hence the following definition: Definition 17. Let w be a decoration. Define Q w = {q ∈ (0, q w ) ∩ Q : P w q has pseudo-Anosov braid type}.
Conjecture 8 b) states that Q w = (0, q w ) ∩ Q for all decorations w, and this can be proved in a number of special cases. However, in this paper all that is needed is the following straightforward lemma: Lemma 18. For any decoration w, Q w is dense in (0, q w ).
Proof. Let w be a decoration of length k ≥ 0, and let q = m/n ∈ (0, q w ). Since P w q does not have finite order braid type, it must have pseudo-Anosov braid type if its period n + k + 3 is prime [Boy84]. Hence However, A w is dense in (0, q w ). Indeed, for any N ∈ N, the set Q N = {m/n : (m, n) = 1 and n + N is prime} is dense in Q. For let r/s ∈ Q. For each K ≥ 1, let r K = K(N − 1)r + 1 and s K = K(N − 1)s + 1, so that r K /s K → r/s as K → ∞, and it suffices to show that each r K /s K can be approximated arbitrarily closely by elements of Q N . Since r K s K is coprime to N − 1, Dirichlet's theorem gives a strictly increasing sequence The main theorem of this paper can now be stated: Theorem 19. Let w be a lone decoration. Then a) The braid types of the orbits P w q with q ∈ Q w are totally ordered by forcing, with P w q ≥ P w q ′ if and only if q ≤ q ′ . b) For any horseshoe periodic orbit R, and hence r w is a braid type invariant.
Remark 20. There is a corresponding result for the family {P * q : q ∈ (0, 1/2] ∩ Q} of NBT orbits [Hal94]. This family is totally ordered by forcing (with P * q ≥ P * q ′ if and only if q ≤ q ′ ), and the corresponding braid type invariant r * can be calculated by Remark 21. r w (R) gives information about which periodic orbits of decoration w are forced by R. Invariants q w (R) which give information about which periodic orbits of decoration w force R can be defined similarly: . However, the authors know of no means of computing these invariants, except for the NBT decoration w = * , for which q * (R) = q(R) by Lemma 11 b).

Linking numbers, the Asimov-Franks theorem, and pruning
The main technique used in this paper to show that the braid type of one periodic orbit P forces the braid type of a second periodic orbit Q is to apply the Asimov-Franks theorem [AF83,Hal91b] to show that Q is unremovable in D 2 \ P . In order to show that the conditions of the Asimov-Franks theorem hold, linking number considerations will be used. The necessary definitions and results are presented in Sections 4.1 -4.3.
On the other hand, the technique used to show that the braid type of one periodic orbit does not force the braid type of a second is to show that all orbits of the second braid type can be pruned away by an isotopy supported in the complement of the first orbit. The pruning theorem used to do this [dC99] is stated in Section 4.4. 4.1. Linking numbers. Let f : D 2 → D 2 be an orientation-preserving homeomorphism, and {f t } : id ≃ f be an isotopy (which will be referred to as a suspension of f ). Recall that, given two distinct periodic orbits P and Q of f , the linking number L(Q, P ) of Q about P with respect to the suspension {f t } is an integer which determines the homology class of the suspension π( If the suspension {f t } is changed, then the linking number L(Q, P ) changes by a multiple of the period n of Q: thus the linking number is well-defined modulo n.
The following two straightforward lemmas can be found in [Hal91a], where they appear as Lemma 1.23 and Corollary 1.25 respectively. The first can be used to show that two periodic orbits have distinct linking numbers about a third periodic orbit P : intuitively, this means that they cannot move together and annihilate under an isotopy relative to P .
Lemma 22. Let f : D 2 → D 2 be an orientation preserving homeomorphism which has periodic points q 0 and q 1 of least period n lying on orbits Q 0 and Q 1 , and let α : [0, 1] → D 2 be a path from q 0 to q 1 . Let γ be the closed curve α · (f n • α) −1 .
Suppose that P is a periodic orbit of f with no points lying on α, and that γ has winding number w γ (P ) about P . Then where the linking numbers are calculated with respect to any fixed suspension of f .
The second lemma can be used to exclude the possibility of a given periodic orbit collapsing onto an orbit of lower period under isotopy.
Lemma 23. Let f i : D 2 → D 2 be a sequence of orientation-preserving homeomorphisms, converging in the C 0 topology to a homeomorphism f : D 2 → D 2 . Let p i → p and q i → q be sequences in D 2 with the property that each p i lies on a period m orbit P i of f i and each q i lies on a period n orbit Q i of f i . If q is a period n/l point of f which does not lie on the f -orbit of p, then L(Q i , P i ) is a multiple of l for sufficiently large i (with respect to any suspension of f i ).
4.2. The Asimov-Franks theorem. The Asimov-Franks theorem [AF83] gives conditions under which periodic orbits of homeomorphisms f : M → M persist (or are unremovable) under arbitrary isotopy of f . The version given here is from [Hal91b], and is restricted to the case of interest in this paper (unremovability of single periodic orbits of orientation-preserving homeomorphisms of D 2 under isotopy relative to some other periodic orbit).
Definitions 24. Let f : D 2 → D 2 and g : D 2 → D 2 be orientation-preserving homeomorphisms having period n points p and q respectively; and let R be a periodic orbit of f in Int(D 2 ). Then (p ; f ) and (q ; g) are connected by isotopy rel. R (denoted (p ; f ) ∼ (q ; g)) if there exists an isotopy {f t } : f ≃ g relative to R and a path α in D 2 from p to q, such that α(t) is a period n point of f t for all t.
(p ; f ) is said to be unremovable in (D 2 , R) if every homeomorphism g : which is isotopic to f rel. R has a period n point q with (p ; f ) ∼ (q ; g).
Notice that being connected by isotopy rel. R is clearly an equivalence relation on the set of all pairs (p ; f ), where f : D 2 → D 2 is an orientation-preserving homeomorphism and p is a periodic point of f . The Asimov-Franks theorem gives three conditions which together ensure the unremovability of (p ; f ). The first condition prevents the orbit of p from collapsing onto an orbit of lower period under isotopy.
Definition 25. (p ; f ) is said to be uncollapsible if, given any sequence of homeomorphisms g j : D 2 → D 2 which converge in the C 0 topology to g : D 2 → D 2 , and a sequence q j → q in D 2 such that q j is a period n point of g j with (q j ; g j ) ∼ (p ; f ) for all j, then q is a period n point of g.
The second condition prevents the orbit of p from falling into the periodic orbit R.
The final condition is that the total fixed point index of periodic points which could interact with p under isotopy is non-zero. Recall (see for example [Jia83]) that if f : X → X is a continuous self-map of a compact manifold, then the fixed point index index(S, f ) of a subset S of Fix(f ) can be defined, generalising the familiar notion of the index of an isolated fixed point, provided that S is compact and is open in Fix(f ).
Definition 27. Let p be a period n point of f . The strong Nielsen class snc(p ; f ) of (p ; f ) is the set of all period n points q of f with (p ; f ) ∼ (q ; f ).
If (p ; f ) is uncollapsible and separated from R, then snc(p ; f ) is compact and open in the set of fixed points of f n , so the following definitions can be made: Theorem 29 (Asimov-Franks). If (p ; f ) is uncollapsible, separated from R, and essential, then it is unremovable.
The following result can be found in [Hal91b]: it says that relevant topological information is preserved under connection by isotopy. The following trivial result will also be useful: Lemma 32. Let p and q be period n points of f and g. Then (p ; f ) ∼ (q ; g) if and only if (f (p) ; f ) ∼ (g(q) ; g).

4.3.
A method for showing that two periodic points are connected by isotopy.
Definition 33. Let f : D 2 → D 2 be an orientation-preserving homeomorphism with distinct period n points p and q. Suppose that α : [0, 1] → D 2 is an arc from p to q with the property that b) Condition a) is ponderous, but the reason for it is simply explained. In the applications in this paper, where f is the horseshoe, the arc α will be taken to be an arc of the stable manifold of p followed by an arc of the unstable manifold of q. Suppose, for example, that p is a periodic point of negative index (so that f n sends each branch of its stable manifold to itself), while q is a periodic Thus α i has endpoints on P ∪ Q and is disjoint from ∂∆ for 1 ≤ i < n, hence by Definition 33 c) α i ∩ ∆ = ∅ for 1 ≤ i < n. Let D be a disk which contains α 0 ∪ α n in its interior, but which is disjoint from R and from α i for 1 ≤ i < n; and let C be a simple closed curve bounding a disk containing α 0 in its interior such that both C and f n (C) are contained in Int D, while f i (C) is disjoint from D for 1 ≤ i < n.
Then f n (C) is isotopic to C rel. P ∪ Q ∪ R, since both are simple closed curves contained in D, surrounding the only two points of P ∪ Q ∪ R which lie in D.
Hence (by a theorem of Epstein [Eps66]) there is a homeomorphism h : D 2 → D 2 , supported in D and isotopic to the identity rel. P ∪ Q ∪ R, with h(f n (C)) = C. Let F = h • f , so that F n (C) = C and f ≃ F rel. P ∪ Q ∪ R. Clearly (p ; f ) ∼ (p ; F ) and (q ; f ) ∼ (q ; F ) (using the constant paths from p to p and from q to q, and the isotopy f ≃ F rel. P ∪ Q ∪ R), so it remains to show that (p ; F ) ∼ (q ; F ).
Let E 0 be the closed disk bounded by C, write E i = F i (E 0 ) for 1 ≤ i < n, However G agrees with F on P and outside of E: thus applying the Alexander trick to all of the components of E (which each contain a single point of P ), there is an isotopy from G to F relative to P ∪ R, which provides an isotopy connection (p ; F ) ∼ (p ; G). Hence (p ; F ) ∼ (q ; F ) as required.
4.4. Pruning theory. Pruning theory provides a means of destroying some of the dynamics of a surface homeomorphism by an isotopy with controlled support. The following definition and theorem are from [dC99], simplified in accordance with the requirements of this paper.
Definitions 36. A pruning disk for the horseshoe map F : D 2 → D 2 is a closed topological disk ∆ ⊆ D 2 whose boundary is the union of an arc C of stable manifold and an arc E of unstable manifold, intersecting only at their endpoints, which satisfy The common endpoints of the two arcs are called the vertices of ∆.
Theorem 37. Let ∆ be a pruning disk for F . Then there exists an isotopy, supported in n∈Z F n (Int(∆)), from F to a homeomorphism F ∆ for which all points of Int(∆) are wandering.
The pruning isotopy therefore destroys all of the dynamics in Int(∆), while leaving untouched any orbits which do not enter Int(∆). In this paper the pruning theorem will be applied in the form of the following corollary: Corollary 38. Let w be a lone decoration and q ∈ (0, q w ). Suppose that there is a pruning disk ∆ containing points of all four periodic orbits P w q in its interior. If R is a horseshoe periodic orbit disjoint from ∆, then R ≥ P w q .
Proof. The set of braid types of the pruned homeomorphism F ∆ is precisely the set of braid types of periodic orbits of F which are disjoint from ∆. In particular, F ∆ has a periodic orbit of the braid type of R, but none of the braid type of P w q .

Proof of the main theorem
Let w be a lone decoration of length k. This decoration will be fixed throughout the section, and therefore the dependence of many objects on w will be suppressed: on the few occasions when it is temporarily important to indicate this dependence, this will be done by means of a superfix w. For the sake of clarity, it is assumed at first that w is even: the modifications necessary in the case of odd w are described in Section 5.5. 5.1. Iterated arcs. The proof of the theorem depends on the details of the configuration of the F -images of a collection of arcs joining points of the four periodic orbits of height q and decoration w for each q ∈ (0, q w )∩Q. For the remainder of this subsection, let q = m/n be a fixed element of (0, q w ). For the sake of notational clarity, arcs α : [0, 1] → D 2 and their images α([0, 1]) will not be distinguished carefully; and points x ∈ Ω(F ) will often be identified with k(x) ∈ {0, 1} Z .
Let p 1 = c q 0w0, p 2 = c q 0w1, p 3 = c q 1w1, and p 4 = c q 1w0 be the rightmost points on each of the four periodic orbits of height q and decoration w (which have period n + k + 3). Note that index(p i , F n+k+3 ) = (−1) i , since the words c q 0w0 and c q 1w1 are even and the words c q 0w1 and c q 1w0 are odd.
Let γ and δ be the following arcs connecting these points: a) γ goes from p 1 to p 2 . It is the concatenation of the vertical arc from p 1 to the point ∞ (c q 0w1) · (c q 0w0) ∞ and the horizontal arc from ∞ (c q 0w1) · (c q 0w0) ∞ to p 2 . b) δ goes from p 3 to p 4 . It is the concatenation of the vertical arc from p 3 to the point ∞ (c q 1w0) · (c q 1w1) ∞ and the horizontal arc from ∞ (c q 1w0) · (c q 1w1) ∞ to p 4 .
These arcs are depicted schematically in Figure 7. The points p i are shown with their correct relative horizontal and vertical orderings, calculated using the fact that w is even.
Proof. Denote by γ j the arc F j • γ, for 0 ≤ j ≤ n + k + 3. A straightforward induction shows that for 0 ≤ j ≤ n + k + 2, the arc γ j is contained in S, and is the concatenation of the vertical arc from σ j (c q 0w0) to σ j ( ∞ (c q 0w1) · (c q 0w0) ∞ ) and the horizontal arc from σ j ( ∞ (c q 0w1) · (c q 0w0) ∞ ) to σ j (c q 0w1). This is true by definition when j = 0, and follows for 1 ≤ j ≤ n + k + 2 since γ j−1 does not cross the vertical centre line C of S. Thus γ n+k+2 is the concatenation of the vertical arc from 0c q 0w to ∞ (1c q 0w) · (0c q 0w) ∞ and the horizontal arc from ∞ (1c q 0w)·(0c q 0w) ∞ to 1c q 0w (which crosses C).
Hence (see Figure 8) γ n+k+3 is the concatenation of i) the vertical arc from p 1 = c q 0w0 to ∞ (1c q 0w) 0 · (c q 0w0) ∞ ; ii) the horizontal arc from ∞ (1c q 0w) 0 · (c q 0w0) ∞ to the right hand edge of S; iii) a semicircular arc outside of S; and iv) the horizontal arc from the right hand edge of S to c q 0w1 = p 2 .
Condition a) of Definition 33 thus follows (with α(b) = ∞ (1c q 0w) 0 · (c q 0w0) ∞ ) and conditions b) and c) are satisfied because p 1 and p 2 lie to the right of all other points in both of their orbits, and the arcs γ j are contained in S for 1 ≤ The corresponding result for the arc δ is analogous, and its proof is omitted. For the purposes of the current argument, denote the arc γ, which depends on q and on the decoration w, by γ w q . Now q < q w = qŵ (Lemma 7), and hence, applying Lemma 39 with decorationŵ, the arc γŵ q (which joins c q 0ŵ0 to c q 0ŵ1) defines a disk Cŵ q under F , and a point x = b · f of Ω(F ) lies in Int Cŵ q if and only if Applying the involution φ of Section 2.3, the arc φ(γŵ q ) (which joins 0w0c q to 1w0c q ) defines a disk φ(Cŵ q ) under F −1 , and a point x = b · f of Ω(F ) lies in Write α = α w q = φ(γŵ q ), and similarly β = β w q = φ(δŵ q ) (see Figure 9). Thus α joins the highest point r 1 = 0w0c q on the orbit of p 1 to the highest point r 4 = 1w0c q on the orbit of p 4 ; while β joins the highest point r 2 = 0w1c q on the orbit of p 2 to the highest point r 3 = 1w1c q on the orbit of p 3 , and the argument above gives: Figure 10 depicts the disks A 1/3 , B 1/3 , C 1/3 , and D 1/3 for w = 11, together with the four periodic orbits of height 1/3 and decoration w. This figure is drawn to scale, and short segments of stable and unstable manifolds cannot be discerned on it. The disks A 1/3 and C 1/3 are shaded lightly, and the disks B 1/3 and D 1/3 are shaded heavily. Figure 10. The disks A 1/3 , B 1/3 , C 1/3 , and D 1/3 for w = 11 This figure motivates the following straightforward lemma.
Lemma 43. For any decoration w and any q ∈ (0, q w ), D q ⊆ C q and B q ⊆ A q .
Proof. Let x = b·f ∈ Ω(F ). It suffices to show that if x ∈ Int(D q ) then x ∈ Int(C q ).
Since both c q andŵ are even words, it follows that f ≻ (c q 0w0) ∞ and σ(b) ≻ (ŵ0c q 1) ∞ , so that x ∈ Int(C q ) by Lemma 39.

Linking properties.
Let q ∈ (0, q w ) ∩ Q, and for 1 ≤ i ≤ 4 denote by P i,q the periodic orbit containing the point p i of Section 5.1. Fix a suspension {f t } of the horseshoe map F . Given a horseshoe periodic orbit R distinct from each P i,q , denote by L i,q (R) the linking number of P i,q about R with respect to the suspension {f t }.
Proof. Note first that the boundaries of the disks A q , B q , C q , and D q are composed of segments of the stable and unstable manifolds of points of the orbits P i,q , and so cannot intersect the orbit R.
Corollary 45. Let R be any horseshoe periodic orbit distinct from each P i,q . Then Proof. Both are equal to L 3,q (R) − L 1,q (R).

5.3.
Forcing conditions. The first main result is that {P w q : q ∈ Q w } is totally ordered by the forcing relation. The following lemma will be used: Proof. Observe first that a point of R lies in one of the disks if and only if it lies in its interior, since R is distinct from the orbits P i,q .
a) The rightmost point ∞ . Now q ′ = q(f ) < q = q((c q 0w0) ∞ ) and hence f ≻ (c q 0w0) ∞ , so condition a) of Lemma 39 (for a point to lie in C q ) is satisfied.
Since q w = q b w (Lemma 7) and q ′ < q < q w , the words c q ′ 0ŵ0 and c q 1ŵ0 are codes of periodic orbits of heights q ′ and q by Lemma 6. Therefore (c q ′ 0ŵ0) ∞ ≻ (c q 1ŵ0) ∞ .
Theorem 19 a) follows immediately from the following result.
Suppose first that L 1,q (R) is not divisible by n + k + 3. It will be shown that (p 1 ; F ) is unremovable in (D 2 , R), which will establish the result. Note first that (p 1 ; F ) is certainly separated from R, since R and P 1,q have different braid types (Remark 31). To show that (p 1 ; F ) is uncollapsible, suppose that g j → g is a sequence of homeomorphisms D 2 → D 2 , and r j → r in D 2 is such that r j is a period n + k + 3 point of g j with (r j ; g j ) ∼ (p 1 ; F ) for each j. The fact that n+k +3 |L 1,q means (by Lemma 23) that r cannot be a fixed point of g. If r were a period (n+k+3)/ℓ point of g for some ℓ with 1 < ℓ < n+k+3, then for j sufficiently large the braid type of the g j -orbit of r j would be reducible with (n + k + 3)/ℓ reducing curves, contradicting the fact that P 1,q has pseudo-Anosov braid type.
Thus r is a period n + k + 3 point of g, establishing the uncollapsibility of (p 1 ; F ).
Finally, to show that (p 1 ; F ) is essential: if r ∈ snc(p 1 ; F ), then r lies on a periodic orbit of braid type bt(P 1,q ) and has linking number L 1,q (R) about R with respect to some suspension of F by Lemma 30. Since w is a lone decoration and L 2,q (R) = L 3,q (R) = L 4,q (R) = L 1,q (R) + 1, it follows that r ∈ P 1,q , and hence On the other hand, if L 1,q (R) is divisible by n+ k + 3, then L 2,q (R), L 3,q (R), and L 4,q (R) are not. The above argument can then be repeated with p 2 , the only part which is any different being the proof that I(p 2 ; F ) = 0. For this part, observe first that by Lemma 30 the only points which can lie in snc(p 2 ; F ) are points of the form F i (p j ) for j ∈ {2, 3, 4}, since these are the only periodic points of F which lie on periodic orbits of braid type bt(P 2,q ) and have linking number L 2,q (R) about R.
Remark 48. A cleaner approach to this proof would be to start by using the fact that R ∩ D q = ∅ to prune away the periodic orbits P 3,q and P 4,q by an isotopy which leaves R, P 1,q , and P 2,q untouched (a suitable pruning disk can be constructed by analogy with Lemma 50 below). There would then remain only two periodic orbits of the braid type of P 1,q , whose linking numbers about R would differ by 1. However, in order to take this approach it is necessary to ensure that the indices of p 1 and p 2 are unchanged after the pruning isotopy, which requires more careful control of the support of this isotopy. The technical details involved in doing this are more complicated than the approach taken in the proof above.
The next lemma describes how the disk C q can be enlarged to a pruning disk ∆ q which contains all of the points p i . (See Figure 11.) ∆ q is "approximately the same" as C q in the sense that if q has large denominator then the boundary of ∆ q is close to that of C q , and in particular ∆ q \ C q cannot contain periodic points of low period. This idea is encapsulated in the following definition:  Figure 11. The pruning disk ∆ q Lemma 50. Let v 0 and v 1 be the points ∞ 0c q 0w 0 1 · (c q 0w011) ∞ . Then v 0 and v 1 are the vertices of a pruning disk ∆ q which contains {p 1 , p 2 , p 3 , p 4 }. Moreover Let C and E be the segments of stable and unstable manifold constituting the boundary of ∆ q . Observe that, for n ≥ 1, F n (C) (respectively F −n (E)) is a segment of stable (respectively unstable) manifold, with endpoints F n (v i ) (respectively F −n (v i )), which is contained in the central square S of the horseshoe. Thus to show that ∆ q is a pruning disk (Definition 36), it is enough to show that F n (v i ) ∈ Int(∆ q ) for all n ∈ Z. Now q w01 = min 0≤i≤k+4 q(σ i ((10w010) ∞ )) = min 0≤i≤k+2 q(σ i ((10w0) ∞ )) = q w , and hence q < q w01 . Thus (Lemma 6) c q 0w011 is the code of a periodic orbit, whose rightmost point is on the stable leaf containing C. It follows that F n (v i ) ∈ Int(∆ q ) for all n ≥ 0. If n < 0, then h(σ n (v i )) ≻ (c q 0w011) ∞ could only occur if h(σ n (v i )) = 10 κr 1 2 . . . 1 2 0 κm 10w 0 1 (c q 0w011) ∞ for some 1 ≤ r ≤ m (recall the horiztonal and vertical coordinate functions h and v from Section 2.1). If r > 1 then this is not possible (as q(10 κr 1 2 . . . 1 2 0 κm 10 . . .) > q), while if r = 1 then σ(v(σ n (v i ))) = 0 ∞ ≻ŵ0c q 0 ∞ . Thus ∆ q is a pruning disk as required.
It is straightforward that ∆ q contains all of the p i , since c q For the final part of the lemma, suppose that x = c is a point of a period N = |c| < n/2 orbit R of F . It is required to show that if x ∈ Int(∆ q ) then x ∈ Int(C q ). Suppose for a contradiction, then, that x ∈ Int(∆ q ) \ Int(C q ): that is, by Lemma 39, either If the former inequalities hold, then q(c ∞ ) = q. Since |c| < n/2, both c and cc must be initial words of c q , and so c = 10 κ1 1 2 . . . 1 2 0 κr 1 for some r < m/2. Now the code c R of R is some cyclic permutation of c, so c R = 10 κi 1 2 . . . 1 2 0 κr 1 2 0 κ1 . . . 1 2 0 κi−1 1 for some i between 1 and r. Removing the even initial subword 10 κ1 1 2 . . . 1 2 0 κi−1 1 (of length s, say) from each term in the inequalities gives and hence q(R) = q(c R ∞ ) ≥ q. On the other hand, q(R) ≤ q(c ∞ ) = q, and hence q(R) = q = m/n, and so R has period at least n. This is the required contradiction.
If the latter inequalities hold, then (removing the even initial subwordŵ0 from each term) there is some point d ∞ of R such that and the proof proceeds in exactly the same way.
The description of the invariant r w follows relatively easily from the following theorem: Theorem 51. Let R be a period N horseshoe orbit, and let q = m/n ∈ Q w be such that n > 2N . Then R ≥ P w q if and only if R ∩ A q = ∅ and R ∩ C q = ∅.
Proof. Notice that R cannot intersect the boundary of any of the disks A q , B q , C q , D q , since its period is less than the period n + k + 3 of P w q . If R ∩ C q = ∅, then R ≥ P w q by Lemma 50 and Corollary 38. Similarly, if R ∩ A q = ∅ then φ(R) ∩ φ(A q ) = φ(R) ∩ Cŵ q = ∅, and φ(∆ŵ q ) is a pruning disk which is disjoint from R but contains the highest points {r 1 , r 2 , r 3 , r 4 } of the orbits P w q . Suppose, then, that R ∩ A q = ∅ and R ∩ C q = ∅. It is required to prove that R ≥ P w q . By Lemma 44, the linking number L 1 is strictly smaller than the linking numbers L 2 , L 3 , and L 4 . If L 1 is not divisible by n + k + 3, then an identical argument to that in the proof of Theorem 47 shows that (p 1 ; F ) is unremovable in (D 2 , R), which establishes the result. On the other hand, if L 1 is divisible by n+k+3 then L 2 , L 3 , and L 4 are not. Either one of these linking numbers is distinct from the other two, or all three are equal, and in either case the proof proceeds identically to that of Theorem 47. 5.4. The invariant r w . The next result completes the proof of Theorem 19 for even decorations.
Theorem 52. Let R be any horseshoe periodic orbit, and r w (R) ∈ (0, q w ] ∩ Q be as given in Definition 16. Then for q ∈ Q w 0 < q < r w (R) =⇒ R ≥ P w q and r w (R) < q < q w =⇒ R ≥ P w q .
In particular, r w is a braid type invariant.
Proof. Let q = m/n ∈ Q w . Suppose first that 0 < q < r w (R): it is required to show that R ≥ P w q . Without loss of generality, assume that n/2 is greater than the period of R, so that Theorem 51 applies (if not, replace q with q ′ ∈ (q, r w (R)) ∩ Q w having sufficiently large denominator. Then P w q ≥ P w q ′ by Theorem 47, so showing that R ≥ P w q ′ shows that R ≥ P w q ). Assume for a contradiction that R ≥ P w q , so that R ∩ A q = ∅ and R ∩ C q = ∅ by Theorem 51. Let . Thus there is a word 010w 0 1 10 in c R which forces λ w (R) ≤ q.
To summarize: the existence of a point of R ∩ C q means that either λ w (R) ≤ q or µ w (R) ≤ q.
Now let x = b · f ∈ R ∩ A q (to simplify the notation the same symbols x, b, and f are used although x must be a different point of R). Thus φ(x) = f · b ∈ R ∩ Cŵ q , so that either λŵ( R) ≤ q or µŵ( R) ≤ q. Thus either λ w (R) ≤ q or ν w (R) ≤ q by Remark 15.
So the fact that both R ∩ C q and R ∩ A q are non-empty means that either λ w (R) ≤ q, or both µ w (R) and ν w (R) are less than or equal to q. This is precisely to say that r w (R) ≤ q, which is the required contradiction.
For the converse, suppose that r w (R) < q < q w : it is required to show that R ≥ P w q . Once again, assume without loss of generality that n/2 is greater than the period of R (if not, replace q with an element of (r w (R), q) having sufficiently large denominator).
Since q > r w (R) = min(λ w (R), max(µ w (R), ν w (R))), at least one of the following two possibilities holds: that q > λ w (R), or that q is greater than both µ w (R) and ν w (R). It will be shown that in either case R contains points of both A q and C q , so that R ≥ P w q by Theorem 51 as required.
a) Suppose that q > λ w (R). Thus there is a point In particular, f ≻ (c q 0w0) ∞ and σ(b) =ŵ 0 1 b ′ ≻ (ŵ0c q 1) ∞ , so x ∈ C q by Lemma 39. Similarly, q > λŵ( R) = λ w (R), so there is a point of R in Cŵ q , and hence a point of R in A q . b) Suppose that q > µ w (R) and q > ν w (R). The fact that q > µ w (R) means that there is a point where v is a nonempty even final subword of 0w. Thus f ≻ (c q 0w0) ∞ and σ(b) ≻ (ŵ0c q 1) ∞ , so x ∈ C q by Lemma 39. Similarly, q > ν w (R) = µŵ( R) means that there is a point of R in Cŵ q , and hence a point of R in A q .
5.5. The case of odd decoration. The case where w is an odd decoration works similarly, the only substantial changes being the exchange of the rôles of C q and D q (and similarly of A q and B q ), and the replacement of the inclusions of Lemma 43 with "approximate inclusions". Define the points p i and the arcs γ, δ, α, and β exactly as in Section 5.1: observe that now index(p i , F n+k+3 ) = (−1) i+1 rather than (−1) i as before. Lemmas 39 -42 (describing the disks defined by these arcs) and Lemma 44 and Corollary 45 (describing the linking numbers of the orbits P i,q about a periodic orbit R) are unchanged. However Lemma 43 is false, as can clearly be seen from Figure 12, which is a schematic depiction of the points p i (with their correct relative horizontal and vertical ordering), the arcs γ and δ, and the images of these arcs under F n+k+3 . What is true, though, is that the disk C q is approximately contained in D q in the sense of Definition 49. Lemma 43 (odd version). -For any odd decoration w and any q ∈ (0, q w ), C q ⊆ q D q and A q ⊆ q B q .
Proof. Let q = m/n. Let x = c be a point of a period N = |c| < n/2 orbit R of F .
It suffices to show that if x ∈ Int(C q ) then x ∈ Int(D q ). The result will then follow p1 γ S δ p3 p4 p2 Figure 12. The disks C q and D q when w is odd since A q = φ(Cŵ q ) and B q = φ(Dŵ q ). The proof works in exactly the same way as the final part of the proof of Lemma 50.
The proof works in exactly the same way, the only small difference being the need to show that points of R other than the rightmost point lie in neither C q nor D q : this uses the fact that if w is odd then q(10w010 . . .) = q(10w110 . . .).
Theorem 47 is true as stated for odd decorations, and the proof is identical except that L 3,q (R) plays the rôle of L 1,q (R), since the revised Lemma 46 gives that L 1,q (R) = L 2,q (R) = L 4,q (R) = L 3,q (R) − 1.
The pruning disk which contains all of the points p i is different from that used in the case of even decoration. Lemma 50 (odd version). -Let v 0 and v 1 be the points ∞ 0c q 1w 0 1 · (c q 0w10) ∞ if w ends with the word 01 2i for some i ≥ 0, and the points ∞ 0c q 1w 0 1 · (c q 0w110) ∞ otherwise. Then v 0 and v 1 are the vertices of a pruning disk ∆ q which con- The only change to Theorem 51 is the replacement of A q and C q with B q and D q . The proof works identically.
Theorem 51 (odd version). -Let R be a period N horseshoe orbit, and suppose that q = m/n ∈ Q w is such that n > 2N . Then R ≥ P w q if and only if R ∩ B q = ∅ and R ∩ D q = ∅.
Finally, the statement of Theorem 52 is unchanged, and its proof works in just the same way. Using B q and D q in place of A q and C q is exactly what is required to compensate for the changes introduced because w is odd, and because + w = 1w and w + = w1, rather than 0w and w0 as in the even case.
6. Examples and applications 6.1. Generalities. The following straightforward lemma will be useful in this section.
Lemma 53. Let w be a lone decoration. a) If R is a horseshoe periodic orbit of decoration w, then r w (R) = q(R). b) If R is any horseshoe periodic orbit and r w (R) = q w , then r w (R) ≥ q(R). c) If R is any horseshoe periodic orbit, then r w (R) = rŵ( R).
Proof. a) is immediate from Theorem 47, Theorem 52, and Lemma 11 c), while c) is immediate from Remark 15.
These decorations were considered in [dCH04]: their name is due to the fact that the train tracks for periodic orbits with decoration w m/n are all star-shaped, with n branches: the pseudo-Anosov has an n-pronged singularity corresponding to the vertex of the star, whose prongs are rotated by m/n.
The scope q w m/n of the decoration w m/n is m/n. In order to simplify the notation, the periodic orbits P w m/n q of decoration w m/n and height q ∈ (0, m/n) will be denoted P m/n q and the invariants r w m/n will be denoted r m/n throughout this subsection.
It is shown in Lemma 17 and Corollary 18 of [dCH04] that each w m/n is a lone decoration, and that the periodic orbits P The aim in this section is to use Theorem 52 to determine the forcing between braid types in distinct families D m/n : that is, for each m/n, m ′ /n ′ , q, and q ′ , to determine whether or not P This completes the proof of Theorem 15 d) of [dCH04].
Consider the point of R with itinerary σ n ′ +1 (c R ) = 0w m ′ /n ′ 0c q ′ = b · f , so that f = 0w m ′ /n ′ 0c q ′ ∞ and b = c q ′ 0w m ′ /n ′ 0 ∞ (the latter using the fact that w m ′ /n ′ and c q ′ are palindromic). It will be shown that where v is a non-empty even initial subword of w + m/n . Thus for v a non-empty even initial subword of w + m/n , so ν w m/n (P Then Thus it only remains to establish the claim, that f = 0w m ′ /n ′ 0c q ′ ∞ is of the form 0ṽf ′ , where v is a non-empty even initial subword of w + m/n = w m/n 0. Observe first that the words w m ′ /n ′ 0c q ′ = 0 ℓ1−1 1 2 0 ℓ2 1 2 . . . 1 2 0 ℓ m ′ −1 1 2 0 ℓ m ′ 10 . . . and w m/n 0 = 0 κ1−1 1 2 0 κ2 1 2 . . . 1 2 0 κm−1 1 2 0 κm must disagree before the shorter of their lengths. For a) There is a subword of the form 010 in w m ′ /n ′ 0c q ′ but no such subword in w m/n 0.
Hence if |w m/n 0| ≥ |w m ′ /n ′ 0c q ′ | then the two words must disagree.
b) On the other hand, if |w m/n 0| < |w m ′ /n ′ 0c q ′ | and the words do not disagree, then w m ′ /n ′ 0c q ′ = w m/n 0 . . ., and so 10w m ′ /n ′ 0c q ′ = 10w m/n 0 . . . and (since 10w m/n 0 is an odd word) Taking the height of both sides gives m ′ /n ′ ≤ m/n, a contradiction.
Since m ′ /n ′ = q(10w m ′ /n ′ 0c q ′ . . .) > q(10w m/n 010 . . .) = m/n, the word w m ′ /n ′ 0c q ′ is greater than w m/n 0 in the unimodal order. Let u be the longest initial word on which they agree, and v be the length |u| + 1 initial subword of w m/n 0. Then either u is even and v = u0, or u is odd and v = u1. In either case, v is a non-trivial even initial subword of w + m/n = w m/n 0 and 0ṽ is an initial subword of 0w m ′ /n ′ 0c q ′ as required.
6.3. Decorations of the form 1 2i+1 . For each i ≥ 0, consider the decoration w i = 1 2i+1 , with scope q wi = q 101 2i+1 0 ∞ = 1/2. It will be shown that w i is lone for all i, yielding corresponding braid type invariants r wi by Theorem 19. These invariants will then be used to show that P wi q has pseudo-Anosov braid type for all i and all q ∈ (0, 1/2).
In order to simplify the notation, the periodic orbits P wi q will be denoted P i q and the invariants r wi will be denoted r i throughout this subsection. Similarly, λ wi , µ wi and ν wi will be abbreviated to λ i , µ i , and ν i .
The proof that the decorations w i are lone is by induction on i: the fact that w i is lone will be established by using the fact that w i−1 is lone, and hence that r i−1 is a braid type invariant. The proof will also make use of the following theorem and lemma, which can be found in [Hal94], Theorem 56 appearing as Theorems 3.11 and 3.15, and Lemma 57 appearing as Lemma 3.3.
Theorem 56. The rotation interval of any horseshoe periodic orbit R is of the form where rhe(R) ∈ [q(R), 1/2] ∩ Q is equal to 1/2 if and only if c R contains either the word 01010 or the word 01 2m+1 0 for some m ≥ 1.
The following lemma will also be used in the proof.
Lemma 58. Let i ≥ 0, and let R be a horseshoe periodic orbit with r i (R) < 1/2.
Then c R contains a word of the form 01 2k+1 0 for some k ≤ i + 2.
Similarly, if ν i (R) < 1/2 then there is some non-empty even final subword v = 1 2j (1 ≤ j ≤ i + 1) of w i+ = 1 2i+2 with the property that some shift of c R is of the where q(b) < 1/2 and so b = 10 . . .. Hence c R contains a block of 1s of odd length either 2j − 1 or 2j + 1 ≤ 2i + 3 as required.
Proof. The proof is by induction on i. For i = 0, it is required to show that for each q = m/n ∈ (0, 1/2)∩Q, the four horseshoe periodic orbits P w0 q of codes c q Now let i > 0. By the inductive hypothesis, r i−1 is a braid type invariant.
Thus P i q is a period n + 2i + 4 orbit with rotation interval [q, 1/2] and r i−1 (P i q ) = 1/2. It will be shown that any horseshoe orbit with these properties must be P i q , which will complete the proof.
Thus r i is a braid type invariant for all i ≥ 0. These invariants will now be used to prove that P i q has pseudo-Anosov braid type for all i ≥ 0 and q ∈ (0, 1/2). The proof will make use of a more general result (Theorem 62 below) for showing that horseshoe periodic orbits have pseudo-Anosov braid type.
The first step is to give a lower bound on the period of horseshoe orbits R for which r i (R) < r i−1 (R).
Proof. a) Suppose first that λ i (R) = q. It follows from Definitions 14 that for some s σ s (c R ) = b 0 1 1 2i+10 1 · f, where q(b) ≤ q and q(f ) ≤ q, with equality in one of the two cases. Suppose that q(f ) = q: the case where q(b) = q works identically. Hence (Lemma 60) either f = 10w q 110 . . . or f = 10w q 01 . . .. In the former case, c R contains both the word 01 0 1 1 2i+10 1 10 (which has length 2i + 7 and contains only isolated 0s and blocks of 1s of odd length) and the word 0w q 110 (which has length n + 1 and contains only blocks of 1s of even length). Thus R has period at least (2i + 7) + (n + 1) − 2 > n + 2i + 4 as required. In the latter case, c R again contains the word 01 0 1 1 2i+10 1 10, and also contains the word 0w q 0 (which has length n − 1 and contains only blocks of 1s of even length). Thus R has period at least (2i + 7) + (n − 1) − 2 = n + 2i + 4 as required. (Note that if R has period n + 2i + 4, then σ s (c R ) = 10w q 01 0 1 1 2i+10 1 = c q 0 1 1 2i+10 1 , i.e. R = P i q .) b) Suppose, then, that λ i (R) = q. Since r i (R) = q, it follows from Definitions 14 that λ i (R) > q, and that µ i (R) ≤ q and ν i (R) ≤ q, with equality in one of the two cases. Suppose that µ i (R) = q: the case where ν i (R) = q works identically. Hence, by Definitions 14, there is some s such that where 1 ≤ k ≤ i+1 and q(f ) = q (here 01 2k−1 =v, where v = 1 2k is a non-empty even final subword of + w i = 1 2i+2 ).
i) If k = i + 1, then c R contains the word 01 2i+10 1 10 (of length 2i + 5, with only isolated 0s and blocks of 1s of odd length). It also contains either the word 0w q 110 (length n + 1) or 0w q 0 (length n − 1), which contain only blocks of 1s of even length. So if R has period less than n + 2i + 4, c R must contain the words 01 2i+10 1 10 and 0w q 0, and these words must overlap at either one or both of their endpoints: that is, σ s (c R ) is either 10w q 01 2i+1 0 1 , or 10w q 001 2i+1 0 1 , or 100w q 01 2i+1 0 1 . In the first case, R = P i−1 q , and hence r i−1 (R) = q, contradicting the hypothesis that q < r i−1 (R). In the second case, f = 10w q 00 . . . and in the third case, f = 100w q 0 . . ., each contradicting q(f ) = q (since by Lemma 60, if q(f ) = q then the number of 1s in the first n + 1 symbols of f is either 2m or 2m + 1).
Let ν i (R) = r ≤ q. Then c R contains the word w r 01 0 1 1 2i+1 0 (the block of 1s here can't be shorter, since ν i−1 (R) > r). In particular, it contains the word 01 0 1 1 2i+1 0 (which has length 2i + 5 and contains only isolated 0s and blocks of 1s of odd length). Since µ i (R) = q, c R also contains either the word 0w q 110 (length n + 1) or 0w q 0 (length n − 1). So if R has period less than n + 2i + 4, c R must contain the words 01 0 1 1 2i+1 0 and 0w q 0, and these words must overlap at either one or both of their endpoints: that is, σ s (c R ) is either 10w q 01 0 1 1 2i , or 10w q 001 0 1 1 2i , or 100w q 01 0 1 1 2i . As before, in the first case R = P i−1 q , contradicting r i−1 (R) < q, while the other two cases contradict q(f ) = q.
Note that there is no restriction on the decoration of the horseshoe periodic orbit R in the following result, which thus provides a general test for pseudo-Anosov braid type of horseshoe periodic orbits.
Theorem 62. Let i ≥ 1 and let R be a period N horseshoe orbit with r i (R) = m/n < r i−1 (R). Let d be the largest divisor of N other than N itself. If d < n + 2i + 4, then R has pseudo-Anosov braid type.
Proof. The fact that r i (R) < r i−1 (R) ≤ 1/2 means that R forces infinitely many periodic orbits of decoration w i , so cannot be of finite order braid type.
If R had reducible braid type, then it would force the braid type of the outermost component in its Nielsen-Thurston canonical representative g: in particular, this is the braid type of some horseshoe periodic orbit S. Since R ≥ S, it follows that r i−1 (S) ≥ r i−1 (R). Now R ≥ P i q for all q > r i (R) with q ∈ Q wi , so g has periodic orbits of each of these braid types in its outermost component, and hence S ≥ P i q for all q > r i (R) with q ∈ Q wi . So r i (S) = r i (R) < r i−1 (R) ≤ r i−1 (S), and hence S has period at least n + 2i + 4 by Lemma 61. This contradicts the fact that the period of S is at most d, which is less than n + 2i + 4.
Proof. Let q = m/n. Suppose first that i > 0. Then r i−1 (P i q ) = 1/2, as established in the proof of Theorem 59; and r i (P i q ) = q by Lemma 53 a). Since P i q has period n + 2i + 4, the result follows from Theorem 62.
For the case i = 0, suppose for a contradiction that R has reducible braid type. As in the proof of Theorem 62, let S be a horseshoe periodic orbit whose braid type is that of the outermost component in the Nielsen-Thurston canonical representative of the braid type of R. Then ρi(S) = ρi(R), so in particular q(S) = q(R) = m/n, and hence S has period at least n. Since R has period n + 4, the period of S is at most n/2 + 2: thus n ≤ 4. A direct check verifies that the orbits P 0 1/3 and P 0 1/4 have pseudo-Anosov braid type.
It follows from Theorem 19 that P i q ≥ P i q ′ for all i and all q, q ′ ∈ (0, 1/2) with q ≤ q ′ . The forcing between families with different i can also be determined easily using the invariants r i . The next result says that P i q forces none of the P j q ′ with j < i, while if j ≥ i it forces all those with q ′ > q: in the language of Conjecture 8, this means that w j w i if and only if j ≥ i.
Corollary 65. Let R be a period N horseshoe orbit. Then (r i (R)) is a decreasing sequence, with r i (R) = r i ′ (R) if i, i ′ ≥ ⌊(N − 7)/2⌋.
Proof. Let q > r i (R) and pick q ′ ∈ (r i (R), q). Then R ≥ P i q ′ by Theorem 19, and P i q ′ ≥ P i+1 q by Theorem 64. That is, R ≥ P i+1 q for all q > r i (R), and so r i+1 (R) ≤ r i (R) as required.
That the sequence (r i (R)) stabilises after i = ⌊(N − 7)/2⌋ is immediate from Lemma 61. 6.4. Topological entropy bounds. Recall that the topological entropy h(β) of a braid type β is the minimum topological entropy of orientation-preserving homeomorphisms of the disk having a periodic orbit of braid type β: it is realised by the Nielsen-Thurston canonical representative of the braid type.
Let w be a lone decoration, and let h w (q) = h(P w q ) denote the topological entropy of the braid type of the periodic orbits with height q and decoration w (0 < q < q w ).
It is clear that, for any horseshoe periodic orbit R, h(R) ≥ h w (q) for all q > r w (R) and, in particular, that h(R) ≥h w (r w (R)), wherē It is often possible to calculateh w (q) explicitly using train track techniques, providing a convenient means to compute topological entropy bounds. The approach for the decorations w i of Section 6.3 will be outlined in this section. A similar calculation could in principle be carried out for the star decorations of Section 6.2, using the explicit train track maps described in [dCH04].
An explicit train track and train track map for the periodic orbits P wi q is depicted in Figure 13. Writing q = m/n, the n + 2i + 4 points of the orbit are depicted with solid circles. There are two valence i + 3 vertices, depicted with unfilled circles. The strings of edges denoted A, B, and C contain respectively n − 2m + 1, m, and m points of the orbit (the remaining 2i + 3 points comprising i at valence 1 vertices around the left hand valence i + 3 vertex, i + 1 at valence 1 vertices around the right hand valence i + 3 vertex, and 2 between these two vertices).
A routine but long calculation using this train track map shows that h wi (m/n) is the logarithm of the largest real root of the polynomial x j + 2(x + 1)(x 2i+4 + 1)(1 + f m/n (x)).
Since n > 2m, every term of 2(x+ 1)(x 2i+4 + 1) m−1 j=0 x ⌊jn/m⌋ is cancelled by terms of (x 2i+4 + x 2i+3 + 2) n−1 j=0 x j , so that K(x) is of the form where all of the coefficients a j are non-negative. It follows that all of its positive roots occur with positive derivative, so that K(x) has a unique positive root, and hence H i m/n has a unique root in (1, ∞) as required.
Nowh wi (m/n) is the limit as k → ∞ of the increasing sequence µ k = h wi m 2 k mnk−1 . Pick a rational m ′ /n ′ ∈ (0, 1/2) which is greater than m/n, and let µ = h wi (m ′ /n ′ ), so that µ < µ k for all sufficiently large k. Restrict to such large k, and work with values of x in the interval [e µ , 2]. Then Example 67. Note that entropy bounds obtained in this way depend only on local features of the code of the periodic orbit under consideration. For example, let R be any horseshoe periodic orbit for which c R contains the word 001 0 1 111 0 1 100. Then some shift of c R is of the form bv · f , where v = 0 1 111 0 1 = 0 1 w 1 0 1 , and q(b) ≤ 1/3, q(f ) ≤ 1/3. Thus r w1 (R) ≤ 1/3, and hence h(R) ≥h w1 (1/3), which is the logarithm of the unique root in x > 1 of the polynomial H 1 1/3 (x).
Compare this to the topological entropy of P w1 1/3 itself, which is given by the largest positive root of the polynomial giving h w1 1/3 ≃ log(1.56294).