Graph Duality in Surface Dynamics

We compare one-dimensional representations for the isotopy stable dynamics of homeomorphisms in two dimensions. We consider the skeleton graph representative, which captures periodic behaviour, and the homotopy graph representative which captures homo-/heteroclinic behaviour. The main result of this paper is to show that the dual to the skeleton graph representative is the homotopy graph representative of the inverse map. This gives a strong link between different methods for computing the dynamics.


Introduction
Homoclinic tangles were first observed by Poincaré [1] in his treatise on celestial mechanics, from which he concluded that the dynamics was nonintegrable and extremely complicated. For low-dimensional (such as two-dimensional discrete-time) systems, it turns out that detailed information about the behaviour in terms of symbolic dynamics and topological entropy can be calculated. This information is useful in studying properties of fluid mixing [2,3,4,5,6,7], ionization of hydrogen [8,9,10,11], optically-injected semiconductor lasers [12], and many other applications.
In this paper we study the topological dynamics of surface diffeomorphisms with homoclinic and heteroclinic tangles. Since tangles contain an infinite amount of combinatorial information, we study finite pieces of the stable and unstable manifolds, which together form a trellis for the diffeomorphism. Other approaches consider homoclinic orbits as the fundamental starting point, such as [13,14].
The topological approximation method of Rom-Kedar constructed symbolic dynamics in the form of rectangular strips in the trellis [15,16,17,18]. The theory of homotopic lobe dynamics [19,20,21,22], which was developed independently, refines the information obtained by the topological approximation method by obtaining a minimal symbolic description for networks of tangles of arbitrary complexity.
In an alternative approach, the theory of trellises [23,24], finds optimal bounds for the topological entropy and symbolic dynamics by finding periodic orbits using a form of Nielsen periodic-point theory. The main step is to compute a one-dimensional representation of the dynamics, called the skeleton graph map.
The main aim of this paper is to show that the "homotopy dynamics" approach based on curves (or strips) is dual to the approach based on skeleton graphs. The main contribution of the paper is to formalise the duality relationship between the homotopy graph and the skeleton graph of the inverse map. This gives a new algorithm for computing the skeleton graph by first computing homotopy graphs, which avoids the use of the Bestvina-Handel algorithm. We illustrate the concepts by two examples, both of which occur in the Hénon family.
The paper is organised as follows. In Section 2, we introduce trellises, and technical preliminaries on homotopies and surface-embedded graphs. In Section 3 we introduce the homotopy dynamics and skeleton dynamics, two ways of describing the dynamics forced by a trellis. In Section 4, we describe the duality relationships between the skeleton and homotopy dynamics. We give some conclusions in Section 5.

Trellises
Let f be a diffeomorphism of a two-dimensional surface M and P an invariant set of periodic saddle points of f . Then by the Stable Manifold Theorem [25], the unstable and stable manifolds W U and W S of P are each disjoint unions of immersed curves in M . An intersection of W U and W S is a homoclinic or heteroclinic point to P , depending on whether the intersection connects the same or different points in P . Since the curves W U and W S are immersed curves, typically of infinite length, they cannot be numerically computed in their entirety and are difficult to analyse completely.
Instead, we consider finite-length pieces of W U and W S .
Following [23], we say that a pair T = (T U , T S ) with T U ⊂ W U and T S ⊂ W S is a trellis for f if T U and T S are of finite length and if f (T U ) ⊃ T U and f (T S ) ⊂ T S . of one unstable and one stable segment, and a rectangle if its boundary consists of two unstable and two stable segments, with interior angles less than 180 • . We define X to be the set n∈Z f n (T U ∩ T S ). An inner bigon is a bigon B such that #(B ∩ X) = 2.
The vertices of an inner bigon are called pseudoneighbours.
Given a trellis T = (T U , T S ), we can cut along the unstable curve(s) to obtain a new surface C U M . Formally, cutting consists of removing each component of T U in M and replacing it by a topological circle. This circle is topologically partitioned into two arcs, each of which is a copy of the component of T U that was removed.
Informally, the result is what one would obtain by cutting along an arc drawn on a piece of paper. Topological details of the construction are discussed in [23]. The stable curve(s) T S in M lift to the arcs C U T S ⊂ C U M . (An arc is a non-self-intersecting curve that begins and ends on the boundary, but otherwise does not intersect the boundary.) The surface diffeomorphism f lifts to a map C U f : C U M → C U M . Since T S is forward-invariant under f , the lifted stable arcs are invariant under the lift of f , i.e. C U f (C U T S ) ⊂ C U T S . Alternatively, we can cut along the stable curves to obtain C S M with unstable arcs C S T U ⊂ C S M . The inverse map f −1 lifts to the map C S f −1 : C S M → C S M , which leaves the lifted unstable arcs invariant, i.e.
We henceforth make the following assumptions on our trellises, which shall simplify the duality result Theorem 4.4.  (v) The preimage of ∂T U and the image of ∂T S are trellis intersections, equivalently

Curves and homotopies
Our main way of studying trellises is via curves embedded in the surfaces M , C S M , and C U M . A (directed ) curve in a general surface X is a continuous map γ : I → X, where I = [0, 1] is the unit interval. The initial and final endpoints of γ are the points version rel Y of a curve α : I → X is the multicurve {α 1 , . . . , α m }, where each exact α i : I → X rel Y is obtained by cutting α along the set Y , and the concatenation of the α i 's returns the curve α. ‡ In applications M is often a topological disc; assuming the boundary of the disc maps to itself with trivial winding, this case reduces to the case M = S 2 via a one-point compactification.
As usual, assuming Y is simply connected, we define the product of homotopy classes in X rel Y in terms of the catenation of curves. For two curves α : I → X and β : I → X, for which α(1) and β(0) lie on the same pathwise-connected component is the homotopy class [γ] of the curve γ : I → X obtained by concatenating α, δ, and β, where δ is an arbitrary curve within Y joining α(1) to β(0). An explicit parameterisation of γ is given by: In general, the collection Π of all homotopy classes [α] rel Y forms a groupoid [26] under the homotopy product.
An arc α in a surface X is an injective curve with endpoints, but no other points, in ∂X, i.e. α is an injective, exact curve in X rel ∂X. A homotopy between arcs is an isotopy if each curve α t is injective, i.e. is an arc. If arcs α 0 and α 1 are homotopic in S 2 or D 2 , then α 0 and α 1 are isotopic [27].
We recall that an isotopy of a topological space X is a function h : X × I → X such that each h t = h(·, t) is a homeomorphism, and an isotopy of the identity is an isotopy such that h 0 = id . We say arcs α 0 and α 1 in X are ambient isotopic if there is an isotopy of the identity h : X × I → X such that α t = h t • α 0 is a homotopy between α 0 and α 1 . Clearly arcs which are ambient isotopic are isotopic. By the isotopy extension theorem [28], any isotopy α t extends to an ambient isotopy.
A multiarc in X is a list of mutually-disjoint arcs {α 1 , . . . , α m }. The above results on isotopies of arcs extend easily to multiarcs. We now give some results about homotopies and minimal position of multiarcs.
Similar results can be found in [23,29]. The case where X is a topological disc is straightforward, and the case where X has higher genus can be proved by passing to the universal cover. Proof. We directly construct the isotopy h : X × I → X of the identity such that An example of pulling-tight is shown in Fig. 2 The following lemma shows that if the multiarcs {α i } and {β j } are in minimal position, then the number of intersections of any α i with the β j is minimal, and the relative ordering of the intersections is well-defined. Let s i,1 < s i,2 < · · · < s i, denote the parameters of the intersection points, i.e.
for each k = 1, . . . , , α i (s i,k ) ∈ β j k (I), for some j k and α i (s) ∈ β j (I) for all s = s 1 , . . . , s and j = 1, . . . , n. Supposeα i is homotopic to α i . Then there exist parameterss i,1 <s i,2 < · · · <s i, such thatα i (s i,k ) ∈ β j k (I) for all k. That is, there is a subset of the intersections betweenα i and {β 1 , . . . , β n } that occur in the same order and with the same β j 's as α i .
Proof. The proof is a modification of that for Lemma 2. Finally, we show that the minimal position is unique up to topological conjugacy. Proof. We first define a homeomorphism h 1 taking the β j toβ j , for all j = 1, . . . , n.
We then define a homeomorphism h 2 that preserves the multiarcsβ j and that takes intersections of the h 1 • α i with theβ j to those of theα i withβ j , for all i and j; Similar to the case above, we now define the notion of a multiarc being in minimal position, or tight, with respect to a collection of mutually-disjoint trees. We say a multitree is a collection {t 1 , . . . , t n } of mutually-disjoint trees embedded in X with endpoints attached to the boundary of X. Unlike the prior case in Definition 2.2, it is not always possible to pull the arcs α i tight with respect to the t j 's. Figure 3 shows both the cases in which an arc cannot be pulled tight and in which an arc can. However, if each α i can be pulled-tight with respect to each tree t j (which on the disc is equivalent to α i crossing each t j at most once), then the natural extension of Lemmas 2.3, and 2.5 hold.
Lemma 2.7. Let X be a surface with boundary, {t 1 , . . . , t n } a multitree in X, and

Surface-embedded graphs
We now give a brief overview of the use of graph maps in surface dynamics. Selfmaps of surface-embedded graphs with a differentiable structure (known as traintracks) were used in Thurston [30] to represent the dynamics of psuedo-Anosov surface homeomorphisms, and in the proof of Thurston's classification theorem by Bestvina and Handel [31]. We will view a graph both as a combinatorial object and as a topological object.
Combinatorially, a graph G is a pair (V, E), where V is a finite set of vertices, and E is a finite set of undirected edges, each of which has two directed versions.
The reverse of a directed edge e is denotedē. The initial vertex of a directed edge e is denoted ı(e). An edge-path is a list of directed edges e 1 e 2 · · · e k such that ı(ē i ) = ı(e i+1 ) for all i = 1, . . . , k−1. The reverse of the edge-path is¯ =ē kēk−1 · · ·ē 1 .
The edge-path is a loop if ı(e 1 ) = ı(ē k ). The edge-path is said to back-track if it has a substring · · · eē · · · for some directed edge e.
Topologically, a graph is a 1-dimensional CW (closure-finite in the weak topology) complex, and maps between graphs are continuous functions mapping vertices to vertices. We will be interested in graphs embedded in an oriented surface M , which may have punctures or boundary components. A face of a graph G embedded in a surface M is a component of M \ G.
We say a surface-embedded graph G is a skeleton graph if G is a deformation- For a surface-embedded graph G, a surface-embedding of a self-map g : The embedding structure of a surface-embedded graph is given by the relation describing the (anticlockwise) cyclic ordering of outgoing-directed edges around a vertex. We say that an edge-path = e 1 e 2 · · · e k is peripheral if either for all i, e i e i+1 or for all i, e i+1 ē i . The faces F of a surface-embedded graph correspond to peripheral loops.
The dual of a surface-embedded graph G is a surface-embedded graph G * with one vertex for each face of G, and one edge for each edge of G. For each directed edge e of G, the dual (directed) edge e * crosses e once, transversely and such that (e, e * ) defines a positively oriented frame. The cyclic ordering of outgoing-directed edges around a vertex of G * is given by e * i e * j ifē j e i . The dual G * * of G * is canonically isomorphic to G.
If G is the skeleton graph of a surface M with boundary, then each face of G is an annulus and the vertex of G * corresponding to the face may be identified with that part of the boundary of the annulus formed by ∂M . In this case, the edges of G * may be realized as mutually disjoint arcs in M .
Combinatorially, a graph map is a self-map of G specified by mapping each vertex v to a vertex g(v), and each directed edge e to an edge-path g(e) such that ı(g(e)) = g(ı(e)). Such a combinatorial graph map g is efficient if g n (e) does not back-track for any edge e and any n > 0.
For a graph map g, the transition matrix A of g has components A ij that count the number of times g(e j ) contains the edge e i (in either direction). Since the matrix A consists of positive integer elements, its maximal eigenvalue λ is strictly positive and has positive left and right eigenvectors; log λ is the entropy of g. The left eigenvector l gives the length of each edge, and the right eigenvector w gives the width of each edge.
An edge is infinitesimal if it has zero length or width. We say g is weakly irreducible if w is the only positive (right) eigenvector of A whose eigenvalue is not 1.
We often wish to simplify the representation of a graph map. The most important way of doing this is by combining edges. Suppose there are edges e 1 and e 2 such that e 1 and e 2 are the only incident edges at some vertex v. Suppose further that no other vertex maps to v. Then whenever e 1 or e 2 occur in an edge-path α = g(a), they occur together, either as e 1 e 2 orē 2ē1 . We can then simplify the graph map by eliminating vertex v and combining e 1 and e 2 into a single edge e. We denote this transformation symbolically by e = e 1 e 2 . It is easy to show that if g 1 and g 2 are related by combining edges, then the entropy of g 1 and g 2 are equal. We will also need the inverse transformation, that of splitting edges. Finally, we can also simplify the transition matrix by identifying and relabelling the edges. Suppose g(e 1 ) = g(e 2 ).
Then we can label both e 1 and e 2 by e, which we denote symbolically by e = e 1 = e 2 .

Controlled graphs
In the following, we assume that T is a trellis of a map f : M → M , satisfying Assumptions 2.1. Definition 2.9 (Controlled graph). Let G be a graph embedded in M . Call the edges of G that cross T control edges and those that do not cross T free edges. Then G is a stable (unstable) controlled graph, relative to T , if there is a pairing between the control edges of G and the unstable (stable) edges of T such that (i) Each control edge of G intersects T at exactly one unstable (stable) edge of T . (ii) Each unstable (stable) edge of T intersects exactly one control edge of G. The free edges of G are said to be of stable (unstable) type.
The tree in Fig. 3 and the skeleton graph in Fig. 4 are examples of controlled graphs that we have already encountered.
Recall the construction of the dual of a surface-embedded graph from Section 2.3.
For the case of a controlled graph, such as the homotopy graph of a trellis, we make some adjustments to the construction. Definition 2.10 (Controlled dual graph). Let G U be an unstable controlled graph with control edges C U crossing segments of T S and free edges E U . Then the controlled dual graph (G U ) * is the standard dual graph of the graph whose edges are the segments of T U and the free edges E U and which is embedded in the surface M \ (T S ∪ C U ).
The controlled dual graph is a stable controlled graph whose control edges are those crossing T U and whose free edges are those crossing E U . An analogous definition applies for the dual (G S ) * of a stable controlled graph G S .
An example of a controlled graph and its controlled dual is given in Fig. 12, which will be discussed in detail in Sect. 3.3. The following algorithm explicitly constructs the controlled dual.
Algorithm 2.11. The controlled dual graph (G U ) * of G U is constructed by the following steps. The construction of the dual to a stable controlled graph G S is entirely analogous.
(ii) For every segment U of T U , place a control edge z across U , joining the vertices in the components of M \ (T U ∪ T S ∪ G) separated by U .
(iii) For every free edge e u of G U , place a free edge across e u , joining the vertices in The following lemma shows that the construction merits the name "dual".
Lemma 2.12. If G is a stable or unstable controlled graph, then (G * ) * is canonically isomorphic to G.
Proof. Immediate from the construction, since the control edges of (G * ) * are precisely those of G and the free edges are the usual duals of the dual edges.
We use the terminology spanning graph for a controlled graph all of whose vertices are the endpoints of control edges. The following lemma shows that spanning graphs and skeleton graphs are mutual duals.
Lemma 2.13. Suppose T is a trellis satisfying Assumptions 2.1 and G is either a stable or unstable controlled graph.
(i) If G is a spanning graph, then G * is a skeleton graph.
(ii) If G is a skeleton graph, then G * is a spanning graph. Proof.
(i) Assume G is spanning. In each region R of T , the restriction of G * to R is simplyconnected, and hence a skeleton graph, since every vertex of G has a controlled edge that prevents loops in G * .
(ii) Assume G is a skeleton graph. If G is of unstable-type, then every component contains a segment U of T U on its boundary. So the vertex of G * within F is the endpoint of a control edge crossing U . The case of a stable-type graph is analogous.
Definition 2.14 (Controlled graph map). For an unstable controlled graph G, a surface-embedded graph map g acting on G is called a controlled graph map if each control edge z of G maps to a control edge g(z) such that if z crosses segment S of T S , then g(z) crosses the segment S that contains f (S). A controlled graph map for a stable controlled graph is defined analogously using the inverse map f −1 .
Definition 2.15 (Optimal controlled graph map). A controlled graph map g : G → G is said to be optimal if g is locally injective on the set of free edges.

Homotopy and Skeleton Dynamics
In this section we study trellises based on homotopy classes of curves joining stable segments. This approach corresponds to the homotopic lobe dynamics of [19,20,21,22] and the strips of [18].

Dynamics on the fundamental groupoid
Given a trellis T in M , an unstable-type curve is a curve α in intersects C U T S at the endpoints of α. We say that an unstable-type curve α crosses need not be at a point, but may contain an interval. For simplicity, we shall also say that α crosses T S , where the cutting by T U is implicit.
We say two unstable-type curves α 0 , α 1 : Under this product, the collection of U -homotopy classes Π U forms a groupoid [26]. A curve is trivial if it is homotopic to a curve lying entirely in T S , and a homotopy class is trivial if it contains a trivial curve; trivial classes play the role of identity elements in Π U . We shall denote trivial classes by [•]. As in Section 2.3, we use an overbar to denote the reverseᾱ of a curve α. The reverse of a curve corresponds to the groupoid inverse of the homotopy class, which we also denote by an overbar, i.e., Fig. 6 for an illustration of these concepts.
Since T S is forward invariant and T U is backward invariant, the image of an unstable-type curve α : I → C U M is also an unstable-type curve. We thus define the The entire unstable homotopy formalism for f can be translated into a stable homotopy formalism by using the map f −1 and the trellis (T S , T U ). Thus, a stable-type  Fig. 7(b) map as Example 3.3. Figure 8 shows a more complicated example of the homotopy action.
All the bridge classes are segment classes, except for γ u 5 (Fig. 8a) and γ s 3 (Fig. 8b). Note that γ s 3 ∼ sγ The homotopic approach is summarized by the following algorithm, which generates the concise homotopy action induced by a trellis. (iv) Insert any new nontrivial bridge classes from the above concise products into H U .
(v) Repeat from (iii) until no additional bridge classes are added.
These first five steps construct the full homotopy action. The recurrent homotopy action includes the following additional step.
R Recursively remove any homotopy elements from H U which have no preimages under h u .
Alternatively, the collapsed homotopy action includes the following additional step.
C Remove all inert classes from H U , that is those bridge classes that map to a trivial class. Then, identify those bridge classes that only differ by an inert class, i.e. for a given bridge class a, if either au, va or vau were also a bridge class, for inert classes u and v, then that class would be identified with a.
Either of steps R or C can be applied by itself, or they can be applied in combination, forming the recurrent collapsed homotopy action, or what we call the reduced homotopy action, for short.
Note that step C can be viewed as passing to the groupoid quotient under the kernel of f n , for a sufficiently large n. The induced action of f on the quotient groupoid is then injective.
Step R ensures that the resulting action is surjective. Applying both steps R and C guarantees that the resulting action of f is a groupoid automorphism.
The following proposition summarises the properties of the homotopy action h u . Writing , then for all indices j = 1, . . . , k i (iii) β u i,j (1) lies in the same segment of C U T S as β u i,j+1 (0).
(iv) β u i,j and β u i,j+1 lie in different regions of T .
(v) No β u i,j is trivial.
Proof. Immediate from the construction.
Proof. We recursively construct a sequence of curves β u i such that for all i ≥ 0: . This completes the inductive proof for the existence of the β u i 's. By conditions (ii) and (iii), for any k ≥ 0, any point in β u k ([0, 1]) that does not lie in T S must equal f k (x) for some x ∈ β u 0 ([0, 1]), and further, for any i < k, ∀i k} is nonempty. Since the sets A k are also compact and nested (A k+1 ⊂ A k ), they have a nonempty intersection, which contains a point x such that f i (x) ∈ R i for all i.
Since the regions form a topological partition, and since all points on the stable region-boundaries are asymptotically forward stable to each other and all points on the unstable region-boundaries are asymptotically backward stable, we can deduce the following result on topological entropy (see [18,23]).

The homotopy graph
To clarify the organization of the (unstable) bridge classes, we seek to represent these homotopy classes as edges of a surface-embedded graph. Ideally, we would like a graph whose edges correspond to the bridge classes and whose vertices correspond to stable segments. In the simplest case (e.g. Fig. 7(a)), we may select the edges so that they share common endpoints; if edge α and β both terminate on the same component of C U T S , then they can be chosen to have the same endpoint. However, this procedure fails for more complex situations, such as the local topologies in Figs. 9(a1) and 9(a2).
Consider the edges (thick, red lines) shown in Fig. 9(b1). We would like to distort the three on the right side so that they share a common endpoint with the edge on the left, but this is impossible without at least one of the curves passing through the cut along T U (thin, red lines). A similar problem is evident in Fig. 9(b2).
To solve the above problem, we use the idea of a controlled graph, and in particular the concept of control edges, introduced in Section 2.4 (Definition 2.9). Control edges are "infinitesimally-short" edges crossing stable segments. We see how they can be used to connect up bridge classes in Figs. 9(c1) and 9(c2).
Example 3.8. An example of a homotopy graph is shown in Fig. 10(b). The control edges are denoted ζ i , where the i subscript orders them by their distance from the fixed-point p along T S . The free edges correspond to the bridge classes, with the edge corresponding to exact curve α u in Fig. 10(a) being labelled α, etc.
The control edges are mapped to each other, with the image of the control edge ζ i crossing segment S i being the control edge ζ j crossing segment S j ⊃ f (S i ). The image of the edge corresponding to an exact curve α u is based on the homotopy action, but must now also include a control edge for every essential intersection of [f • α u ] with T S . The homotopy dynamics therefore induces the following action on the edges of the homotopy graph.
where • indicates that the edge maps to a single vertex.
The general construction proceeds as follows. The α u are chosen to be mutually disjoint from one another.
The homotopy graph map h u G acts on H U G as follows.
(iv) If ζ is the control edge crossing stable segment S, then h u G (ζ) is the control edge crossing the segment S containing f (S).
(v) The image of the homotopy edge α u G corresponding to [α u ] is the edge-path in H U G corresponding to the concise homotopy action h u ([α u ]), including a control edge whenever h u ([α u ]) has an essential intersection with T S .
Note that the G subscript distinguishes H U G , which is a graph, from H U , which is simply a set of homotopy classes, the elements of which are identified with the free edges of H U G . The first five steps construct the full homotopy graph representative. The recurrent homotopy graph representative includes the following additional step. R Recursively remove any control or free edge that has no preimage. Note that any remaining homotopy edges must connect two non-removed control edges.
Alternatively, the collapsed homotopy graph representative includes the following additional step.
C Collapse all inert homotopy edges, i.e. those homotopy edges that eventually map to a single vertex. Then, in the event that multiple homotopy edges connect the same two vertices, keep only one of these edges.
Either of steps R or C can be applied by itself, or they can be applied in combination, forming the recurrent collapsed homotopy graph representatives or simply the reduced homotopy graph representative, for short.
The recurrent and reduced homotopy graphs of the trellis T 3 are shown in Figs. 10c and 10d. A somewhat more involved example of the transition from the recurrent homotopy graph to the reduced homotopy graph, by applying step C, is shown in Fig. 11 for the trellis T 3e .
It is clear from the construction and the properties of the homotopy action that the image of any edge of H U G is an edge-path which contains no two consecutive edges in the same region.
Theorem 3.10. The homotopy graph constructed in Algorithm 3.9 is an unstable controlled graph, which is also spanning, i.e. all vertices are endpoints of control edges. The corresponding homotopy graph map is an optimal controlled graph map.
Proof. Straightforward from the construction. The conciseness of the dynamics ensures that the graph map is optimal.
We can also construct the homotopy graph representative of (f −1 , (T S , T U )), which we call the stable homotopy graph representative of (f, (T U , T S )). The stable homotopy graph and graph map are denoted H S G and h s G . Remark. It will be noted that each 2n-gon in the complement of the trellis generically is crossed by 2n − 3 unstable bridge classes. The complements of these edges form either strips along an unstable segment, or triangles bounded by three homotopy classes. For a bigon however, this formula suggests −1 edges, and indeed, the case of a bigon is rather degenerate. For this paper, we consider a bigon to contain no nontrivial homotopy element.

The skeleton graph
The skeleton graph representative of a trellis was defined in [23], and gives an alternative representative to the homotopy graph representative. It can be defined by the axioms given in Definition 3.12 below, and is the unique graph map satisfying these axioms. The direct computation of the skeleton graph representative relies on an algorithm similar to that of Bestvina and Handel [31] for computing train-tracks.
In Section 4.2, we shall show that the dual of the recurrent homotopy graph map is precisely the skeleton graph representative, yielding an alternative computation. Definition 3.11. For a trellis T of the map f , satisfying Assumptions 2.1, a compatible skeleton graph K U G and skeleton graph map k u G for (T, f ) satisfy: (i) K U G is an unstable controlled graph.
(ii) K U G is a skeleton graph of M \ T U , i.e., K U G restricted to any region is a tree.
(iii) k u G is a controlled graph map (Definition 2.14) such that the image of any unstable arc α embedded in K U G lies in the homotopy class of f (α).
There are many possible skeleton graphs compatible with a given trellis, as (for example) a region with 4 unstable boundary segments may contain a valence-4 vertex or two valence-3 vertices (in two possible configurations). Further, the image under k u G of an arc in K U G joining two stable segments need not be concise. (iv) The map k u G is locally injective when restricted to the free edges.
In [23], the following result was shown. The topological entropy of f is at least that of k u G .
From this point forward, whenever we mention the skeleton graph representative, we mean the optimal compatible skeleton graph and map (K U G , k u G ). Just as for the homotopy graph representative, we can remove and collapse edges. A recurrent compatible skeleton graph is be computed from a skeleton graph representative by the following additional step. R Recursively remove any skeleton edge that has no preimage, and then remove any control edge that does not thereby share a vertex with any skeleton edge.
Alternatively, a collapsed skeleton graph representative includes the following additional step.
C Collapse all inert skeleton edges, i.e. those skeleton edges that eventually map to a single vertex, and, in the event that multiple skeleton edges connect the same two vertices, keep only one of these edges. Either of steps R or C can be applied by itself, or they can be applied in combination, forming a recurrent collapsed skeleton graph representative.
Note that conditions (i) are equivalent to saying that the topological pair (G, C) is homotopy-equivalent to the pair (M \ T U , T S \ T U ). The dynamical conditions (iii) are equivalent to saying that the map g : (G, C) → (G, C) is homotopy-equivalent to The optimality condition implies that the graph map is efficient in the sense of Bestvina and Handel [31,23].
We note that the image of the skeleton graph restricted to any component of This means that no image of an edge back-tracks, and the incident edges at any vertex map to distinct edge-paths with distinct initial edges. E.g. in Example 4.2, at the vertex with incident edgesc 1 ,c 2 ,c 3 , we havē c 1 → c 3c2 ,c 2 → c 4 andc 3 → c 5z20 z 18 d 1 , beginning, respectively, with edgesc 3 ,c 4 ,c 5 .
In particular, any vertex maps to a vertex of equal or higher valence.

Duality
In this section, we consider duality relationships involving the homotopy dynamics and the skeleton dynamics. We first define a duality relation on controlled graph maps, and show the main result of this paper, that the dual of the (backward) homotopy graph map is the skeleton graph map. We also show a duality relation between the unstable and stable homotopy dynamics. We also show how the forward homotopy graph can be projected onto the skeleton graph.

The dual graph map
Example 4.1. For the trellis T 3 we have the stable homotopy graph H S G shown in Fig. 12(a). The controlled dual graph (Definition 2.10), denoted (H S G ) * , is then as shown in Fig. 12(b). The control edges of the dual graph cross the stable segments of T 3 . The free edges of (H S G ) * cross the edges of H S G representing bridge classes. In the region with edges β 1 , β 2 and β 3 , there are therefore three edges of (H S G ) * , labelled b 1 , b 2 and b 3 , meeting at a valence-3 vertex. Note that (H S G ) * is a skeleton graph, as defined in Sect. 2.3.
Having constructed a dual skeleton graph (H S G ) * to the (stable) homotopy graph H S G , we now consider a dual map (h s G ) * to the homotopy action h s G . This graph map will be determined by the order in which the images of skeleton edges are forced to cross stable bridge edges.
Example 4.2. For the trellis T 4 in Fig. 8, we obtain the dual skeleton graph in Fig. 13 if we only focus on the "interior" region of the trellis. (Technically, Fig. 13 shows the reduced skeleton graph; details will be given later.) The skeleton edge a is homotopic to the curve α u (Fig. 8), The skeleton edge c 1 is not in any unstable bridge class. To compute its image, we instead compute essential crossings of c 1 with preimages of stable arcs. We note that It can be seen that the images so constructed yield a consistent image for the end vertices of c 3 .
In summary, (h s G ) * yields the following action on the skeleton graph, a → az 24 z 14 b, b → c 1c2 , e → az 24 z 14 b, This construction of the graph map on the skeleton graph given in Example 4. ) with no back-tracking by Lemma 2.8. Finally, we define the dual graph map (h s G ) * applied to this edge e u to be the edge-path a u 1 , . . . , a u l , where a u i is the edge of (H S G ) * dual to α s i . This edge-path also contains the necessary control edges z u j between the free edges a u i . The dual graph map (h s G ) * applied to a control edge z u of (H S G ) * maps to a control edge z u such that if z u crosses segment S of T U , then z u crosses the segment S that contains f −1 (S). We have therefore shown:

Duality between the homotopy and skeleton dynamics
We now state and prove the main theorem of the paper, which gives the relationship between the homotopy dynamics and the skeleton dynamics. Proof. The basic structure of this proof is to show that the dual to the homotopy graph representative ((H S G ) * , (h s G ) * ) satisfies Definition 3.12 for the skeleton graph representative. First, by construction of the dual (Definition 2.10), (H S G ) * is an unstable controlled graph, which proves property (i) of Definition 3.11.
Second, by Lemma 2.13 we have already seen that the dual of any stable spanning graph, such as H S G , is a skeleton graph, thereby yielding property (ii). Third, by the construction of the dual map (Theorem 4.3), (h s G ) * is a controlled graph map. Next suppose a free edge e maps to an edge path b u zc u , where z ∈ (h s G ) * (C) is a control edge and b u and c u are edge paths. Then there is a control edge z ∈ C such that z = (h s G ) * (z ). But this means that e must have a transverse intersection with T S , which is not true since e lies in a single region. Hence, e does not map to any control edge in (h s G ) * (C). This completes the proof of property (iii).
Finally, we show that (h s G ) * is locally injective on the free edges. Note, for an arbitrary bridge class [α s ], f −1 • α s is homotopic to a tight edge-path that passes through each region at most once (since C S M is simply connected), and hence crosses any edge of the skeleton graph at most once. By the construction of the dual map in Theorem 4.3, (h s G ) * applied to any edge of (H S G ) * cannot map to two copies of the same edge. Hence, (h s G ) * restricted to any edge is injective. By a similar argument, we show that (h s G ) * is locally injective at a vertex connecting free edges in (H S G ) * . Suppose (h s G ) * were not locally injective at a vertex, then (H S G ) * would have two free edges a u 1 and a u 2 beginning at a common vertex such that (h s G ) * applied to either a u 1 or a u 2 would begin with the same initial edge b u . Then, for a homotopy edge [α s ], the concise description of h s G ([α s ]) would contain two homotopy elements in the same region, one each intersecting a u 1 and a u 2 . But we have already determined that there is a single homotopy element in each region for the concise description. Hence (h s G ) * is locally injective at a vertex of free edges. Combining this with the local injectivity along each free edge, allows us to conclude that (h s G ) * is locally injective when restricted to the free edges, condition (iv) of Definition 4.4 By step R of Algorithm 3.9, every free edge in the recurrent homotopy graph H S G has a preimage under h s G . Thus, none of the dual edges in (H S G ) * maps to a trivial edge-path under (h s G ) * .
We illustrate the construction by another example.
Example 4.5. Consider the trellis T 3 in Fig. 12. The edges α, β 1 , β 2 , β 3 , γ, δ of the stable homotopy graph H S are dual to (i.e. cross) the edges a, b 1 , b 2 , b 3 , c, d of the skeleton graph K U G . These homotopy elements map where we have ignored the trivial dynamics on the free edges 1 , 2 , 3 . The corresponding transition matrix (Sect.

2.3) is
where we have only recorded transitions between the free edges.
The skeleton edges map with corresponding transition matrix We see that A H = A T S , illustrating that the stable homotopy graph map is dual to the unstable skeleton graph map.
If we reduce the homotopy and skeleton graphs by recursively deleting edges with no preimages (Step R), and also collapsing edges with trivial image (Step C), we see that the edge c is collapsed and its dual counterpart γ is removed, whereas the edge d is removed and its dual counterpart δ is collapsed. The resulting graphs are shown in Fig. 15, and are again dual to each other, as are the graph maps α → αζ 8 ζ 6 β 3 , β 1 → αζ 8 ζ 6 β 3 , β 2 → β 1ζ6 ζ 8ᾱ , β 3 → β 2 and a → az The following result shows that applying the collapsing process applied to the dual of the full stable homotopy graph map also yields the skeleton graph representative.
Proposition 4.6. The skeleton graph representative is obtained from the dual of the full stable homotopy graph map by collapsing all inert skeleton edges.
Proof. The collapsed edges are precisely those which cross elemental homotopy arcs which are removed when constructing the recurrent homotopy graph. Removing an edge from the homotopy graph induces a collapsing of the corresponding edge in the dual graph.

Time-reversal symmetry of the homotopy dynamics
We consider the intersections of unstable homotopy elements with stable homotopy elements. We first construct a geometric figure such that the representatives of the homotopy classes of H U and H S are tight, such as shown in Fig. 16. The homotopy representative α u crosses α s , and β u 1 crosses β s 3 andβ s 2 in that order.
, so it has essential intersections with α s , γ s , β s 3 andβ s 2 in that order. The following results directly from Lemma 2.3. Proof. First, note that if (T U , T S ) is a trellis for f , and h is a homeomorphism such Now, suppose f • α u is homotopic to a path φ u which intersects β s once. By In particular, if f • α u is homotopic to a curve with n intersections with β s , then f −1 • β s is homotopic to a curve with n intersections with α u .
Note that if each component of M \(T U ∪f (T S )) is simply-connected, then f ([α u ]) has at most one essential intersection with β s for any α u and β s .

Projection of the homotopy graph onto the skeleton graph
We notice that there is also a relationship between the unstable skeleton graph and the unstable homotopy graph. Each homotopy graph element is homotopic to an edge-path in the skeleton graph, and the concise description of the homotopy type of the image of the homotopy graph element is equivalent to the image edge-path in the skeleton graph. For example, comparing Fig. 8a to Fig. 13 We now describe this relationship between the (unstable) homotopy and skeleton graphs.
Lemma 4.8. Let [α u ] u be a homotopy element and a 1 · · · a k the edge-path in the skeleton graph which is homotopic to α u . Then the concise description of [f • α u ] u crosses the same stable homotopy arcs whose preimages contain α s 1 , . . . , α s k , where each α s i is dual to a i , for all i = 1, . . . , k.
Proof. The reduced skeleton graph map k u G is locally-injective on each region, so the representative of any nontrivial homotopy element maps to an edge-path that does not back-track. This edge-path therefore crosses each stable homotopy arc at most once, and hence crosses the same arcs as the concise description of [f • α u ] u .
We now show that the forward homotopy graph and skeleton graph can each be easily computed from the other. Theorem 4.9. The homotopy graph H U G and map h u G are related to the skeleton graph K U G and map k u G by an edge-splitting followed by an edge-identification.
Proof. Each edge in the homotopy graph embeds as an edge-path in the skeleton graph. Since each edge in the homotopy graph lies in a single region, the image of the edge in the skeleton graph does not back-track, so equals the projected image under the skeleton graph map. In other words, there is a projection p u G : ) for all homotopy elements α. We next construct the split graph H U G of the homotopy graph H U G . Since a given edge α in H U G projects to an edge-path a 1 . . . a l of length l in K U G , we split α into l by duality, as opposed to constructing an initial skeleton graph simplifying by folding and pulling-tight. In both cases, we have shown how to represent the dynamics via a combinatorial graph embedded in the surface. The two approaches are complementary in the sense that the homotopy dynamics more directly captures the structure of the homoclinic orbits forced by the trellis, whereas the skeleton graph provides a direct proof of existence of periodic orbits via the Nielsen fixed-point theory.
We note that a similar theory could be developed for periodic orbits, with the skeleton graph corresponding to train-tracks, and the homotopy dynamics corresponding to iteration of arcs joining periodic points. However, the theory of finite developments of homoclinic tangles is in some sense actually easier than the theory for periodic orbits. In particular, the skeleton graph and homotopy graphs are unique, while a train-track for a pseudo-Anosov mapping class is not unique, but is related by zipping.
Although in the exposition we have used examples of planar horseshoe trellises, the results generalise in a straightforward way to arbitrary irreducible trellis types in surfaces of higher genus.
An interesting project for further work would be to automatically extract the trellis topology from a computation of a geometric trellis, and hence compute the homotopy and skeleton graph representatives.