Symplectomorphisms and discrete braid invariants

Area and orientation preserving diffeomorphisms of the standard 2-disc, referred to as symplectomorphisms of $\mathbb{D}^{2}$, allow decompositions in terms of positive twist diffeomorphisms. Using the latter decomposition we utilize the Conley index theory of discrete braid classes as introduced in [Ghrist et al., C. R. Acad. Sci. Paris S\'er. I Math., 331(11), 2000, Invent. Math., 152(2), 2003] in order to obtain a Morse type forcing theory of periodic points: a priori information about periodic points determines a mapping class which may force additional periodic points.

Area and orientation preserving diffeomorphisms of the standard 2-disc, referred to as symplectomorphisms of D 2 , allow decompositions in terms of positive twist diffeomorphisms. Using the latter decomposition we utilize the Conley index theory of discrete braid classes as introduced in [1,2] in order to obtain a Morse type forcing theory of periodic points: a priori information about periodic points determines a mapping class which may force additional periodic points.
Let D 2 ⊂ R 2 be the standard unit 2-disc with coordinates z = (x, y) ∈ R 2 , and let ω = dx ∧ dy be the standard area 2-form on R 2 . A diffeomorphism F : D 2 → D 2 is said to be symplectic if F * ω = ω -area and orientation preserving -and is referred to as a symplectomorphism of D 2 . Symplectomorphisms of the 2-disc form a group which is denoted by Symp(D 2 ). A diffeomorphism F is Hamiltonian if it is given as the time-1 mapping of a Hamiltonian systemẋ = ∂ y H(t, x, y); where H ∈ C ∞ (R × D 2 ) a the Hamiltonian function with the additional property that H(t, ·)| ∂D 2 = const. for all t ∈ R. The set of Hamiltonians satisfying these requirements is denoted by H(D 2 ) and the associated flow of (1) is denoted by ψ t,H . The group Ham(D 2 ) signifies the group of Hamiltonian diffeomorphisms of D 2 . Hamiltonian diffeomorphisms are symplectic by construction. For the 2-disc these notions are equivalent, i.e. Symp(D 2 ) = Ham(D 2 ) and we may therefore study Hamiltonian systems in order to prove properties about symplectomorphisms of D 2 , cf. [3], and Appendix B. A subset B ⊂ D 2 is a invariant set for F if F(B) = B. We are interested in finite invariant sets. Such invariant sets consist of periodic points, i.e. points z ∈ D 2 such that F k (z) = z for some k 1. Since ∂D 2 is also invariant, periodic point are either in int D 2 , or ∂D 2 .
The main result of this paper concerns a forcing problem. Given a finite invariant set B ⊂ int D 2 for F ∈ Symp(D 2 ), do there exist additional periodic points? More generally, does there exist a finite invariant set A ⊂ int D 2 , with A ∩ B = ∅? This is much alike a similar question for the discrete dynamics on an interval, where the famous Sharkovskii theorem establishes a forcing order among periodic points based on their period. In the 2-dimensional situation such an order is much harder to establish, cf. [3]. The main results are based on braiding properties of periodic points and are stated and proved in Section 7. The braid invariants introduced in this paper add additional information to existing invariants in the areapreserving case. For instance, Example 7.3 describes a braid class which forces additional invariant sets solely in the area-preserving case hence extending the non-symplectic methods described in [4].
The theory in this paper can be further genelarized to include symplectomorphisms of bounded subsets of R 2 with smooth boundary, eg. annuli, and symplectomorphisms of R 2 .
A priori knowledge of finite invariant sets B for F categorize mappings in so-called mapping classes. Traditionally mapping class groups are defined for orientation preserving homeomorphisms, cf. [5] for an overview. Denote by Homeo + (D 2 ) the space of orientation preserving homeomorphisms and by Homeo + 0 (D 2 ) the homeomorphisms that leave the boundary point wise invariant. Two homeomorphisms F, G ∈ Homeo + (D 2 ) are isotopic if there exists an isotopy φ t , with φ t ∈ Homeo + (D 2 ) for all t ∈ [0, 1], such that φ 0 = F and φ 1 = G. The equivalence classes in π 0 Homeo + (D 2 ) = Homeo + (D 2 )/∼ are called mapping classes and form a group under composition. The latter is referred to as the mapping class group of the 2-disc and is denoted by Mod(D 2 ). For homeomorphisms that leave the boundary point wise invariant the mapping class group is denoted by Mod 0 (D 2 ) = π 0 Homeo + 0 (D 2 ) . In Appendix A we provide proofs of the relevant facts about mapping class groups. The mapping class groups Mod(D 2 ) and Mod 0 (D 2 ) may also be defined using diffeomorphisms, cf. Appendix A. In Proposition B.1, we show that π 0 Symp(D 2 ) = Mod(D 2 ) and in Propositon B.3 we show that Ham(D 2 ) = Symp(D 2 ), which implies that every homeomorphism, or diffeomorphism is isotopic to a Hamiltonian symplectomorphism. More refined information about mapping classes is obtained by considering finite invariant sets B. This leads to the notion of the relative mapping classes. Two homeomorphisms F, G ∈ Homeo + (D 2 ) are of the same mapping class relative to B if there exists an isotopy φ t , with φ t ∈ Homeo + (D 2 ) and φ t (B) = B for all t ∈ [0, 1], such that φ 0 = F and φ 1 = G. The subgroup of such homeomorphisms is denoted by Homeo + (D 2 rel B) and Homeo + 0 (D 2 rel B) in case ∂D 2 is point wise invariant. The associated mapping class groups are denoted by Mod(D 2 rel B) = π 0 Homeo + (D 2 rel B) and Mod 0 (D 2 rel B) = π 0 Homeo + 0 (D 2 rel B) respectively. Let C m D 2 be the configuration space of unordered configurations of m points in D 2 . Geometric braids on m strands on D 2 are closed loops in C m D 2 based at B 0 = {z 1 , · · · , z m }, where the points z i are defined as follows: z i = (x i , 0), x 0 = −1, and x i+1 = x i + 2/(m + 1). The classical braid group on D 2 is the fundamental group π 1 C m D 2 , B 0 and is denote by B m D 2 . The (algebraic) Artin braid group B m is a free group spanned by the m − 1 generators σ i , modulo following relations: Full twists are denoted algebraically by = (σ 1 . . . σ m−1 ) m and generate the center of the braid group B m . Presentation of words consisting only of the σ i 's (not the inverses) and the relations in (2) form a monoid which is called the positive braid monoid B + m . There exists a canonical isomorphism i m : can be understood as follows, cf. [6], [7].
, the composition and scaling of the isotopies defines isotopic braids based at B ∈ C m D 2 . The isomorphism , with β(t) = φ t (B) the geometric braid generated by φ t . The isomorphism d * is given in Appendix A.2 and [β] B denotes the homotopy class in π 1 C m D 2 , B . For Mod(D 2 rel B) we use the same notation for the isomorphism which is given by The above mapping class groups can also be defined using diffeomorphisms and symplectomorphisms.
In Appendix B we show that π 0 Symp(D 2 rel B) = Mod(D 2 rel B) and that Symp(D 2 rel B) = Ham(D 2 rel B) and therefore that every mapping class can be represented by Hamiltonian symplectomorphisms.
Considering free loops in a configuration space as opposed to based loops leads to classes of closed braids, which are the key tool for studying periodic points. .

Discretized braids
From [1] we recall the notion of positive piecewise linear braid diagrams and discretized braids.
Definition 3.1. The space of discretized period d closed braids on n strands, denoted D d m , is the space of all pairs (b, τ) where τ ∈ S m is a permutation on m elements, and b is an unordered set of m strands, b = {b µ } m µ=1 , defined as follows: for all µ = 1, . . . , m; (c) for any pair of distinct strands b µ and b µ such that x µ j = x µ j for some j, the transversality condition Remark 3.2. Two discrete braids (b, τ) and (b , τ ) are close if the strands b ζ(µ) and b µ are close in R md for some permutation ζ such that τ = ζτζ −1 . We suppress the permutation τ from the notation. Presentations via the braid monoid B + m store the permutations. To a configuration b ∈ D d m one can associate a piecewise linear braid diagram B(b). For each strand b µ ∈ b, consider the piecewise-linear (PL) interpolation for t ∈ [0, 1]. The braid diagram B(b) is then defined to be the superimposed graphs of all the functions B µ (t). A braid diagram B(b) is not only a good bookkeeping tool for keeping track of the strands in B(b), but also plays natural the role of a braid diagram projection with only positive intersections, cf. Section 4. The set of t-coordinates of intersection points in B(b) is denoted by {t i }, i = 1, · · · , |b|, where |b| is the total number of intersections in B(b) counted with multiplicity. The latter is also referred to as the word metric and is an invariant for b. A discrete braid b is regular if all points t i and anchor points x µ j are distinct. The regular discrete braids in [b] form a dense subset and every discrete braid is positively isotopic to a regular discrete braid. To a regular discrete braid b one can assign a unique positive word β = β(b) defined as follows: where k i and k i + 1 are the positions that intersect at t i , cf. [8,Def. 1.13].
On the positive braid monoid B + m two positive words β and β are positively equal, notation β . = β , if they represent the same element in B + m using the relations in (2). On B + m we define an equivalence relation which acts as an analogue of conjugacy in the braid group, cf. [9,Sect. 2.2]. For a given word σ i 1 · · · σ i n , define the relation σ i 1 σ i 2 · · · σ i n ≡ σ i 2 · · · σ i n σ i 1 . Definition 3.4. Two positive words β, β ∈ B + m are positively conjugate, notation β . ∼ β , if there exists a sequence of words β 0 , · · · , β ∈ B + m , with β 0 = β and β = β , such that for all k, either β k .
Positive conjugacy is an equivalence relation on B + m and the set of positive conjugacy classes β of the braid monoid B + m is denoted by C B + m . The above defined assignment b → β(b) can be extended to all discrete braids. A discrete braid b is positively isotopic to a regular braid b and the mapping D d m → C B + m , given by b → β(b) , is well-defined by choosing β(b) to be any representative in the positive conjugacy class β(b ) . Observe that for fixed d the mapping D d m → C B + m is not surjective.
The presentation of discrete braids via words in B + m yields the following alternative equivalence relation. Taking d sufficiently large is a sufficient condition to ensure free braid classes, but this condition is not a necessary condition. are two different path components of D 2 3 . The positive conjugacy class of σ 1 σ 2 2 σ 1 is given by σ 1 σ 2 2 σ 1 = {σ 1 σ 2 2 σ 1 , σ 2 2 σ 2 1 , σ 2 σ 2 1 σ 2 , σ 2 1 σ 2 2 }. The words σ 2 2 σ 2 1 and σ 2 1 σ 2 1 are not represented in D 2 3 . If we consider b ∈ D 3 3 given by b , then the associated braid class [b ] is free, which confirms that the condition in Proposition 3.7 is not a necessary condition, see Figure 1[right].

Discrete 2-colored braid classes
On closed configuration spaces we define the following product: The strand labels in a range from µ = 1, · · · , n and the strand labels in b range from µ = n + 1, · · · , n + m. The associated permutation τ a,b = τ a ⊕ τ b ∈ S n+m , where τ a ∈ S n and τ b ∈ S m , and τ a acts on the labels {1, · · · , n} and τ b acts on the labels {n + 1, · · · , n + m}. The strands a = {x µ j }, µ = 1, · · · , n are the red, or free strands and the strands b = {x µ j }, µ = n + 1, · · · , n + m are the black, or skeletal strands. A path component [a rel b] in D d n,m is called a 2-colored discretized braid class. The canonical projections are given by The mapping yields a fibration The pre-images −1 (b) = [a] rel b ⊂ D d n , are called the relative discretized braid class fibers.
There exists a natural embedding D d n,m → D d n+m , defined by a rel b → a b. Via the embedding we define the notion of topological equivalence of two 2-colored discretized braids: a rel b .
∼ a b . The associated equivalence classes are denoted by [a rel b] . ∼ , which are not The set of collapsed singular braids inD d n,m is given by: is not proper it is called improper. In [9] properness is considered in a more general setting. The notion of properness in this paper coincides with weak properness in [9]. .

Algebraic presentations
Discretized braid classes are presented via the positive conjugacy classes of the positive braid monoid B + m . For 2-colored discretized braids we seek a similar presentation.
In order to keep track of colors we define coloring on words in B + n+m . Words in B + n+m define associated permutations τ and the permutations τ yield partitions of the set {1, · · · , n + m}. Let γ ∈ B + n+m be a word for which the induced partition contains a union of equivalence classes a ⊂ {1, · · · , n + m} consisting of n elements. The set a is the red coloring of length n and the remaining partitions are colored black, denoted by b. The pair (γ, a) is called a 2-colored positive braid word, see Figure 2. For a given coloring a ⊂ {1, · · · , n + m} of length n the set of all words (γ, a) forms a monoid which is denoted by B + n,m,a and is referred as the 2-colored braid monoid with coloring a.
Two pairs (γ, a) and (γ , a ) are positively conjugate if γ . ∼ γ and a = ζ −1 (a), where ζ is a permutation conjugating the induced permutations τ γ and τ γ , i.e. τ γ = ζτ γ ζ −1 . If ξ is another permutation such that τ γ = ξτ γ ξ −1 , then ζτ γ ζ −1 = ξτ γ ξ −1 . This implies that τ γ = ζ −1 ξτ γ ξ −1 ζ and thus ξ −1 ζ(a) = a, which is equivalent to ζ −1 (a) = ξ −1 (a). This shows that the conjugacy relation in well-defined. Positive conjugacy for 2-colored braid words is again denoted by (γ, a) . ∼ (γ , a ) and a conjugacy class is denoted by γ, a . The set of 2-colored positive conjugacy classes with red colorings of length n is denoted by C B + n,m . The words corresponding to the different colors (γ, a) can be derived from the information in (γ, a). Let a 0 ⊂ a be a cycle of length n and let k ∈ a 0 . If γ = σ i 1 · · · σ i d , then we define an -periodic sequence {k j }, by considering the word γ . Now use the following rule: Moreover, σ i j is replaced by σ i j −1 , if k j = k j−1 < i j and σ i j remains unchanged otherwise. We repeat this procedure for all cycles in a and we obtain the mapping (γ, a) → β ∈ B + m denoted by By considering the complementary color b we construct a mapping (γ, b) → α ∈ B + n using the same scheme. With the notion of coloring braid words we can encode the information of a 2-colored discretized braid a rel b in a 2-colored word (γ, a). Given cf. (4) and the coloring a = c −1 ({1, · · · , n}), where the permutation c is defined as follows. Order the coordinates x µ 1 0 < · · · < x µ n+m 0 and define The permutations τ γ and τ a,b are conjugated: .
∼ via the mapping γ → E q (γ) ∈ D d+q n+m , with d the number of generators in γ and q 0, cf. Figure 2[right]. The representation a rel b ∈ D d+q n,m is obtained from the coloring a.
Since the strands in b have labels µ = 2, 3, the permutation is given by τ a,b = (23) and γ = β(a b) = σ 2 σ 1 σ 2 . The coloring permutation is given as follows: Figure 2[left]. The topological type of a rel b is given by the ∼ is a proper and free 2-colored discrete braid class, see Figure 2[right]. In order to compute the skeletal braid word β we consider the following sequence: This yields the differences k 1 − k 0 = 1, k 2 − k 1 = 0 and k 3 − k 2 = −1, and therefore both letters σ 2 are removed from γ, which gives γ → β = σ 1 .  We summarize the construction of a topological invariant for 2-colored relative braid classes as described in [1,2] for all j ∈ Z and for all ν = 1, · · · , m. The recurrence relation R may regarded as vector field and is integrated via the equations There is an intrinsic way to define h(a rel b) without using parabolic recurrence relations. We define N − ⊂ ∂N to be the set of boundary points for which the word metric is locally maximal. The pair (N, N − ) is an index pair for any parabolic system R such that R(b) = 0, and thus by the  Via the projection : Define the following extension mapping E : D d m → D d+1 m , cf. [1], via concatenation with the trivial braid of period one: Properness remains unchanged under the extension mapping E, however boundedness may not be preserved. Define the skeletal augmentation: for all k and therefore H(a rel b) = H(a rel b * ). One can define second skeletal augmentation: The independence of H on the skeleton b can be derived from the Stabilization Theorem. Since a 2-colored discretized braid class is free when d is sufficiently large, we have that [E p a rel E p b * ] is free for some p > 0 sufficiently large, and by stabilization H(a rel which shows that the index H only depends on the topological type γ, a , with γ = β(a rel b).
The braid Conley index H may be computed using any representative a rel b * for any sufficiently large d and any associated recurrence relation R.
Finally, we mention that besides the extension E, we also have a half twist extension operator T: Every discretized braid can be dualized via the mapping n,m be proper, then where the wedge is the 2n-suspension of the Conley index. From the singular homology H * (H(a rel b * )) the Poincaré polynomial is denoted by P t (a rel b * ), or P t γ, a in terms of the topological type. This yields an important invariant: |P t (a rel b * )| = |P t γ, a |, which is the number of monomial term in the Poincaré polynomial.
For a given symplectomorphism F ∈ Symp(D 2 ) the problem of finding periodic points can be reformulated in terms of parabolic recurrence relations. .

Twist symplectomorphisms
Let F(x, y) = f(x, y), g(x, y) be a symplectomorphism of R 2 , with f, g smooth functions on R 2 . Recall that F ∈ Symp(R 2 ) is a positive twist symplectomorphism if ∂f(x, y) ∂y > 0.
For twist symplectomorphisms there exists a variational principle for finding periodic points, cf. [11], [12]. Such a variational principle also applies to symplectomorphisms that are given as a composition: with F j ∈ Symp(R 2 ) positive twist symplectomorphisms for all j. It is important to point out that F itself is not twist in general. An important question is whether every mapping F ∈ Symp(R 2 ) can be written as a composition of (positive) twist symplectomorphisms, cf. [11]. Suppose F ∈ Ham(R 2 ), and F allows a Hamiltonian isotopy ψ t,H with appropriate asymptotic conditions near infinity, such that ψ t i ,H • ψ −1 t i−1 ,H is close to the identity mapping in the C 1 -norm for sufficiently small time steps t i − t i−1 . Then, define G i = ψ t i ,H • ψ −1 t i−1 ,H , i = 1, · · · , k, and F = G k • · · · • G 1 . We remark that in this construction the individual mappings G i are not twist necessarily. The following observation provides a decomposition consisting solely of positive twist symplectomorphisms. Consider the 90 o degree clockwise rotation ψ(x, y) = (y, −x), ψ 4 = id, which is positive twist symplectomorphism. This yields the decomposition: where F 4i = G i • ψ and F j = ψ for j = 4i for some i and d = 4k. Since the mappings G i are close to the identity, the compositions G i • ψ are positive twist symplectomorphisms. The above procedure intertwines symplectomorphisms with k full rotations. As we will see later on this results in positive braid representations of mapping classes. The choice of ψ is arbitrary since other rational rotations also yield twist symplectomorphisms. For symplectomorphisms F ∈ Symp(D 2 ) we establish a similar decomposition in terms of positive twist symplectomorphisms, with the additional property that the decomposition can be extended to symplectomorphisms of R 2 , which is necessary to apply the variational techniques in [1].

. Interpolation
A symplectomorphism F ∈ Symp(R 2 ) satisfies the uniform twist condition if the there exists a δ > 0 such that The subset of such symplectomorphism is denoted by SV(R 2 ), cf. [11]. A result by Moser implies that all symplectomorphisms of R 2 with a uniform twist condition are Hamiltonian.
For completeness we give a self-contained proof of Proposition 5.1, which is the same as the proof in [12] modulo a few alterations.
Proof. Following [12] we consider action integral 1 0 L(t, x(t),ẋ(t))dt for functions x(t) with x(0) = x 0 and x(1) = x 1 . We require that extremals are affine lines, i.e.ẍ(t) = 0. For extremals the action is given by S(x 0 , x 1 ) = 1 0 L(t, x(t),ẋ(t))dt and we seek Lagrangians such that S = h, where h is the generating function for F. For Lagrangians this implies where p =ẋ. Solving the first order partial differential equation yields and m = m(t, x) to be specified later, cf. see [12] for details. The extremals x(t) are also extremals for L 0 . Let S 0 (x 0 , and hence Differentiating S yields and for the mixed derivate Then, S 0 (x 0 , implies S = h. Differentiating the relation y = −∂ x h(x, x 1 ) with respect to y and using the fact that x 1 = f(x, y), yields
From the above analysis we obtain the following expression for the isotopy ψ t,H : Let π x denote the projection onto the x-coordinate. Then, ∂ y π x ψ t,H (x, y) = ∂ y λ(x, y)t = ∂ 2 yy Ht, which proves that ψ t,H is positive twist for all t ∈ (0, 1]. Using Proposition 5.1 we obtain a decomposition of symplectomorphisms F ∈ Symp(R 2 ) as given in (15) and which satisfy additional properties such that the discrete braid invariants in [1] are applicable.

The decomposition
is a generalization of the decomposition given in (15).
The isotopy constructed in Proposition 5.2 is called a chained Moser isotopy. Before proving Proposition 5.2 we construct analogues of the rotation mapping used in (15).
In order to extend chained Moser isotopies yet another type of Hamiltonian twist diffeomorphism is needed.
Proof of Proposition 5.2. Consider the subgroup Symp c (R 2 ) formed by compactly supported symplectomorphisms of the plane. 2 Recall that due to the uniform twist property the set SV(R 2 ) is open in the topology given by C 1 -convergence on compact sets, cf. [11]. Let Ψ ∈ Ham(R 2 ) be given by Lemma 5.3 for some 3. Then, there exists an open neighborhood W ⊂ Symp c (R 2 ) of the identity, such that ϕ • Ψ ∈ SV(R 2 ) for all ϕ ∈ W .
For F ∈ Symp(D 2 ), Proposition 2.2 provides a Hamiltonian H ∈ H(D 2 ) such that F = ψ 1,H . Let H † be a smooth extension to R × R 2 and U (D 2 ) = {z ∈ R 2 | |z| < 1 + } and let α : R 2 → R be a smooth bump function satisfying α| D 2 = 1, α = 0 on R 2 \ U (D 2 ). Take ∈ (0, 1/2) and define H = αH † with H ∈ H(R 2 ). The associated Hamiltonian isotopy is denoted by ψ t, H and F = ψ 1, H ∈ Ham(R 2 ). Moreover, ψ t, H equals the identity on Fix 3 and choose k > 0 sufficiently large such that the symplectomorphisms are elements of W . Each G i restricted to D 2 can be decomposed as follows: where Ψ and Ψ are obtained from Lemma 5.3 by choosing rotation angles 2π/ and 2π/κ respectively. Observe that Ψ • Ψ κ( −1) | D 2 = id. From F we define the mapping F ∈ Symp(R 2 ): By construction we have F| D 2 = F. Let κ = κ( − 1) + 1 and d = κ k and put with F j ∈ SV(R 2 ) for j ∈ {1, . . . , d} and F j = F j | D 2 . Using the latter we obtain a decomposition of F as given in (15), and with the additional property that the mappings F j extend to twist symplectomorphisms of the R 2 , which proves (25). Each symplectomorphism F j can be connected to identity by a Hamiltonian path. Let H j be the Hamiltonian given by Proposition 5.1, which connects F j to the identity via the Moser isotopy ψ s,H j , s ∈ [0, 1]. Let t j = j/d for all j ∈ {0, . . . , d} and define with s j (t) = d(t − t j ) and F 0 = id. Observe that, by construction, φ t j • φ −1 t j−1 = F j , for all j = 1, · · · , d and (i) -(iv) is satisfied. Condition (v) follows from (iv) and from the fact that each F j leaves the disc D 2 invariant.
All the symplectomorphisms in the decomposition are supported in the disc U (D 2 ), hence Conditions (ii) and (iii) of Lemma 5.3 imply Properties (vi) and (vii).
Remark 5.5. The chained Moser isotopies in Proposition 5.1 can be extended with two more parameters r 0 and ρ 0. Consider the decoposition where Ψ r r | D 2 = id and r 3, and Υ ρ ρ | D 2 = id and ρ 3. We can again define an Moser isotopy as in (33) with d replaced by d + r r + ρ ρ . The isotopy is again called a chained Moser isotopy and denoted by φ t , and the extended period will again be denoted by d. The strands φ t (z ± ) link with the cylinder [0, 1] × D 2 and with each other with linking number 2ρ. .

The discrete action functional
Let F ∈ Symp(D 2 ) be the given symplectomorphism of the 2-disc and let {φ t } t∈R and {t j } j∈{0,...,d} be the associated continuous isotopy and sequence of discretization times as given in Proposition 5.2 for the extension F. The isotopy is extended periodically, that is φ t+s = φ t • φ s 1 and t j+sd = s + t j for all s ∈ Z. The decomposition of F given by Proposition

yields a periodic sequence of positive twist symplectomorphisms
If (x j+n , y j+d ) = (x j , y j ) for all j, then {(x j , y j )} j∈Z is called an d-periodic sequence for the system { F j }.
For every twist symplectomorphism F j ∈ SV(R 2 ) we assign a generating function h j = h j (x j , x j+1 ) on the x-coordinates, which implies that y j = −∂ 1 h j and y j+1 = ∂ 2 h j . From the twist property it follows that Note that the sequence {h j } is d-periodic.
Define the action functional W d : for all j ∈ Z. The y-coordinates satisfy y j = ∂ 2 h j−1 x j−1 , x j .
Periodicity and exactness of R j is immediate. The monotonicity follows directly from inequality (35). A periodic point z, i.e. F d (z) = z, is equivalent to the periodic sequence {(x j , y j )} j∈Z , with z = (x 0 , y 0 ) ∈ D 2 . Since z = (x 0 , y 0 ) ∈ D 2 , the invariance of D 2 under F implies that (x j , y j ) ∈ D 2 for all j. The above considerations yield the following variational principle.
The idea of periodic sequences can be generalized to periodic config- For a d-periodic sequence {B j }, the x-projection yields a discretized braid b = {b µ } = {x µ j }, cf. Definition 3.1. The above action functional can be extended to the space of discretized braids D d m : where W d (b µ ) is given by (36). This yields the following extension of the variational principle.
A discretized braid b that is stationary for W d if it satisfies the parabolic recurrence relations in (37) for all µ and the periodicity condition in Definition 3.1(b). In Section 6 we show that d-periodic sequences of configurations {B j } for the system { F j } yields geometric braids.
For symplectomorphisms F ∈ Symp(D 2 ), with a finite invariant set B ⊂ int D 2 , the mapping class can be identified via a chained Moser isotopy.  Let z, z ∈ R 2 be distinct points with the property that F n (z) = z and F n (z ) = z , for some n 1, and where F = φ 1 and φ t a chained Moser isotopy constructed in Proposition 5.2. Define the continuous functions z(t) = φ t (z) and z (t) = φ t (z ) and let x(t) and x (t) the x-projection of z(t) and z (t) respectively. By Proposition 5.2, x(t) and x (t) are (continuous) piecewise linear functions that are uniquely determined by the sequence {t j } nd j=0 , t j = j/d. Lemma 6.4 (cf. [1]). The two x-projections x(t) and x (t) form a (piecewise linear) braid diagram, i.e. no tangencies. The intersection number ι x(t), x (t) , given as the total number of intersections of the graphs of x(t) and x (t) on the interval t ∈ [0, n], is well-defined and even.
Proof. Let x j = x(t j ) and x j = x (t j ), j = 0, · · · , nd and by the theory in Section 5.3 the sequences satisfy the parabolic recurrence rela- Suppose the sequences {x j } and {x j } have a tangency at x j = x j (but are not identically equal). Then, either x j−1 < x j−1 and x j+1 x j+1 , or x j−1 x j−1 and x j+1 < x j+1 , and similar with the role of {x j } and {x j } reversed. Since functions R j are strictly increasing in the first and third variables and since x j = x j both evaluations of R j cannot be zero simultaneously, which contradicts the existence of tangencies. All intersections of x(t) and x (t) are therefore transverse in the sense of Definition 3.1(c) and thus ι x(t), x (t) is well-defined and even.
The curves z(t) and z (t) may be regarded as 3-dimensional (continuous) curves z, z : R/Z → R 3 , t → (t, z(t)) and t → (t, z (t)). Due to the special properties of the chained Moser isotopy φ t we have: Lemma 6.5. The graphs t → (t, z(t)) and t → (t, z (t)) form a positive 2-strand braid. Intersections in the x, x -braid diagram correspond to positive crossings in the z, z -braid. The linking number is given by Link z(t), z (t) = 1 2 ι x(t), x (t) .
Proof. By Lemma 6.4 the projection graphs t → x(t) and t → x (t) form a braid diagram. In order to show that the graphs t → (t, z(t)) and t → (t, z (t)) form a positive 2-strand braid we examine the intersections in the x, x -braid diagram.
(i) Consider an intersection on the interval [t j , t j+1 ] ⊂ [0, n] for which x j < x j and x j+1 > x j+1 . Let τ ∈ [t j , t j+1 ] be the intersection point and x(τ) = x (τ) = x * . After rescaling and shifting to the interval [0, 1] we have x(s(t)) = x j + (x j+1 − x j )s(t), s(t) = d(t − t j ) ∈ [0, 1] and the same for x (s(t)). Recall that φ t is given by (33) and therefore by (23), where L j are the Lagrangians for the Moser isotopies ψ t,H j in Proposition 5.1. Since ∂ 2 pp L j δ > 0 and x j+1 − x j > x j+1 − x j , we conclude that y(s(τ)) > y (s(τ)). By reversing the role of x and x , i.e. x j > x j and x j+1 < x j+1 , we obtain y(s(τ)) < y (s(τ)), which shows that an intersection in the x, x -diagram corresponds to a positive crossing in the z, z -braid.
(ii) Consider an intersection at x j , for which x j−1 < x j−1 , x j = x j = x * and x j+1 > x j+1 . As in the previous case and since x * − x j−1 > x * − x j−1 (and x j+1 − x * > x j+1 − x * ) we conclude that y(s(τ)) > y (s(τ)). Reversing the role of x and x yields y(s(τ)) < y (s(τ)). In this case we also conclude that an intersection in the x, x -diagram corresponds to a positive crossing in the z, z -braid, which concludes the proof.
Proof of Proposition 6.1. Since φ t is an isotopy and F(B) = φ 1 (B) = B, the path t → φ t (B) in C m D 2 represents a geometric braid β(t). By Lemma 6.5 all crossings in β(t) are positive. Indeed, if we consider the m-fold cover , then all pairs of strands satisfy the hypotheses of Lemma 6.5. Consequently, the presentation of β ∈ B m of β is unique and consists of only positive letters.
Our decomposition automatically selects a braid word in B + m which is positive and which may be represented as a positive piecewise linear braid diagram. This allows us to use the theory of parabolic recurrence relations for finding additional periodic points for F. In the following lemma F is an extension of F to R 2 given by Proposition 5.2. Lemma 6.6. Let A, B ⊂ R 2 be finite, disjoint sets and B ⊂ int D 2 . Let F ∈ Symp(D 2 ) with F(B) = B and let φ t be a chained Moser isotopy given by Proposition 5.2 with φ 1 = F. Suppose A is an invariant set for F and Proof. The set A = α(0) is an invariant set for F. Assume without loss of generality that α(t) is a single component braid. Let α † (t) be the n-fold cover of α, with t ∈ [0, n]. If α † (t j ) ∈ R 2 \ int D 2 , t j = j/d, for some j, then α † (t j ) ∈ R 2 \ int D 2 for all j, since the set φ t (D 2 ) separates the points inside and outside ∂D 2 under the isotopy φ t . Therefore, α † (t) ∈ R 2 \ int φ t (D 2 ) for all t ∈ [0, n] and thus α(t) ∈ R 2 \ int φ t (D 2 ) for all t ∈ [0, 1]. By assumption β(t) ∈ φ t (D 2 ) for all t ∈ [0, 1] and therefore that α can be contracted onto ∂φ t (D 2 ), which contradicts the assumption that α rel β is acylindrical. Consequently, α(t) ∈ φ t (D 2 ) for all t ∈ [0, 1]. The latter implies A ⊂ int D 2 , which completes the proof.
Given F ∈ Symp(D 2 ) and a finite invariant set B ⊂ int D 2 , then The braid word β can chosen positive by adding full twists and can be used to force additional invariant sets as quantified by the following result. (ii) the number of distinct invariants sets A in (i) is bounded from below by |P t γ, a |.
Proof. Recall from Section 3.3 that γ, a determines a 2-colored discretized braid class [a rel b] . ∼ via the mapping γ → E 0 (γ) ∈ D d 0 n+m , with d 0 the number of generators in γ. The representative a rel b ∈ D d 0 n,m is obtained from the coloring a.
for some λ 1. The choices of , k and κ determine d.
Case I: The integers q 0 and κ are chosen large enough such that Figure 5, then by the invariance of the discrete braid invariant we have that   Figure 6: Two representatives for Case II. Numbers of discretization points are linked by d + r r = d 0 + q.
Case II: By Remark 5.5 we use a(n) (extended) chained Moser isotopy, denoted by φ t , with r = λ, r 3 and ρ = 0 and we denote the associated discrete braid by b B := π x φ t j (B) . By construction b B .
The integers q 0 and κ are chosen large enough such that Figure 6, such that This proves the theorem in Case II. Case III: By Remark 5.5 we use a(n) (extended) chained Moser isotopy, denoted by φ t , with r = 0 and ρ = 1 and ρ = 2λ + 2 4. We denote the associated discrete braid by b B := π x φ t j (B) . By construc- . From (14) and the fact that E q (a rel b) = E q a rel E q b and T 2λ+2 E q a rel(E q b * ) # ∼ a B rel b * # B , cf. Figure 7, we derive that which implies that H a B rel b * # B = 0. As in the previous case the nontriviality of the braid invariant proves the theorem in Case III since the braid class [a B rel b * # B ] is free, bounded and proper and stationary braids of Figure 7: Two representatives for Case III. Numbers of discretization points are linked by d + ρ = d 0 + q + 2λ + 2. The dashed lines indicate the oscillating strands due to the action of T and the augmentation of the skeleta by # .
Remark 7.4. One can also consider 2-colored words (γ, a), γ ∈ B n+m . Braid words can be expressed in normal form. The fundamental element, or Garside element in B + m is denoted by := (σ 1 · · · σ m−1 )(σ 1 · · · σ m−2 ) · · · (σ 1 σ 2 )σ 1 , and is also referred to as a half twist. The element is a factor of a positive word β, if β = β β , β , β positive (possibly trivial) words. If not, β is said to be prime to . If we use the lexicographical order on the presentations of a positive word β, then the smallest positive word β , positively equal to β, is called the base of β.
For positive words β, prime to , the base is denoted byβ. Following [6] and [13], every word β ∈ B m is uniquely presented by a word ρδ , with ρ ∈ Z andδ ∈ B + m , which is called left Garside normal form. Via the relation β = r(β) we obtain the right Garside normal form β = r(δ) ρ . For the 2-colored braid words the Garside normal form is not the appropriate normal form, since odd powers of do not represent trivial permutations. In the case that the Garside power of γ is even, then 2 = defines the identity permutation and therefore the associated base (δ, a), withδ ∈ B + n+m , is a positive 2-colored braid word. In this case the left and right normal form are the same since is at the center of the braid group B n+m . When the Garside power is odd, we argue as follows. Let ρ = 2λ + 1, then where¯ = δ = r(δ) is the base and¯ is prime to . The power λ is related to the Garside power via λ = λ(γ) = ρ/2 and is referred to as the symmetric Garside power of γ. For 2-colored braid words (γ, a) this yields the following symmetric normal form: γ = λ¯ =¯ λ , with λ ∈ Z and (¯ , a) a positive 2-colored braid word. Since mapping classes of F are given modulo full twists the latter normal form suggests that we capture all forcing via positive braid words. If conjugacy is incorporated, the power λ can be optimized with respect to different conjugate braid words. with γ = σ 4 σ 1 σ 2 σ 3 σ 2 2 σ 3 σ 2 σ 1 σ 4 and a = {3}.
We give overview various known and less known facts about mapping class groups of the 2-disc and the 2-disc with a finite number of marked points. .
For a treatment of mapping class groups via symplectic and Hamiltonian diffeomorphisms see, Appendix B. .
Write ω t = a t (x, y)dx ∧ dy, with a t (x, y) > 0 on [0, 1] × D 2 and a 0 = a 1 = 1. In order to construct a symplectic isotopy we invoke Moser's stability argument, cf. [20], [21], Sect. 3.2. Consider potential functions Φ t : D 2 × [0, 1] → R and define the vector fields X t = 1 a t (x,y) ∇Φ t , with X t (x), n = 0 for x ∈ ∂D 2 , n the outward pointing normal. The boundary condition guarantees that D 2 is invariant for the associated flow χ t . Furthermore, define 1-forms θ t = −ι X t ω t . In order to apply Moser stability we seek potentials Φ t such that dω t dt = dθ t , which is equivalent to the Neumann problem Since the forms ω t are cohomologous, Stokes' theorem implies that which shows that the Neumann problem is well-posed and has a unique solution (up to an additive constant), which depends smoothly on t.
By construction χ * t ω t = ω and the desired symplectic isotopy is given by ξ t • χ t . The symplectic isotopy φ t := F • ξ t • χ t is an isotopy between F and G, which proves that F and G are symplectically isotopic, and thus π 0 Symp(D 2 ) = Mod(D 2 ).
As for homeomorphisms we can also consider relative symplectic mapping classes. Two symplectomorphisms F, G ∈ Symp(D 2 ) are of the same relative symplectic mapping class if there exists an isotopy φ t , with φ t ∈ Symp(D 2 ) and φ t (B) = B for all t ∈ [0, 1], such that φ 0 = F and φ 1 = G. The subgroup of such symplectomorphisms is denoted by Symp(D 2 rel B).
Lemma B.2. Let F, G ∈ Symp(D 2 ) be isotopic in Diff + (D 2 rel B), then they are isotopic in Symp(D 2 rel B).
Proof. Let F, G ∈ Symp(D 2 ) be isotopic in Diff + (D 2 rel B), then F −1 G is isotopic to the identity via a smooth isotopy ξ t with the additional condition that ξ t (B) = B for all t ∈ [0, 1]. In order to find a symplectic isotopy we repeat the proof of Proposition B.1 with a few modifications.
Let X t = 1 a t (x,y) ∇Φ t + 1 a t (x,y) J∇λ t (x), where λ t : [0, 1] × D 2 → R is a smooth function compactly supported in int D 2 , where t = Jn is a unit tangent such that {n, t} is positively oriented. In order for the associated flow χ t to restrict to a flow on D 2 we need that ∂λ t ∂t + ∂Φ t ∂n = 0 at ∂D 2 . Since λ t is compactly supported in int D 2 we use the Neumann boundary condition for Φ t . For the 1-forms we obtain As in the proof of Proposition B.1 the potential Φ t is determined by the Neumann problem in (41). Since λ t can be chosen arbitrarily we define λ t (x) = J∇Φ t (z 0 ), x on neighborhoods of z 0 ∈ B and a smooth extension outside, compactly supported in int D 2 . This guarantees that the isotopies χ t preserve B point wise, which completes the proof.