Definition 2.1. Let $n \geq 2$ be an integer. Let $p_n: \mathbb {R}^2 \to \mathbb {R}^2$ be the map defined by identifying $\mathbb {R}^2 \cong \mathbb {C}$ and sending z to $z^{{n}/{2}}$ . When n is odd, one has to choose a branch; any choice here would work. Consider the foliations of $\mathbb {R}^2$ by vertical and horizontal lines. Let $l_n^s, l_n^u$ be the singular foliations of $\mathbb {R}^2$ obtained by pulling back these foliations under $p_n$ , respectively. We refer to lifts of the quadrants in $\mathbb {R}^2$ under $p_n$ as quadrants as well.

Let $\unicode{x3bb}>1$ . Consider the map $[\begin {smallmatrix} \unicode{x3bb} & 0\\ 0 & \unicode{x3bb} ^{-1} \end {smallmatrix}]: \mathbb {R}^2 \to \mathbb {R}^2$ . Let $\phi _{n,0,\unicode{x3bb} }:\mathbb {R}^2 \to \mathbb {R}^2$ be the lift of this map over $p_n$ that preserves the quadrants. Let $\phi _{n,k,\unicode{x3bb} }: \mathbb {R}^2 \to \mathbb {R}^2$ be the composition of $\phi _{n,0,\unicode{x3bb} }$ and rotation by ${2\pi k}/{n}$ anticlockwise. Since $[\begin {smallmatrix} \unicode{x3bb} & 0\\ 0 & \unicode{x3bb} ^{-1} \end {smallmatrix}]$ preserves the foliations of $\mathbb {R}^2$ by vertical and horizontal lines, respectively, $l_n^s$ and $l_n^u$ are preserved by $\phi _{n,k,\unicode{x3bb} }$ . We depict the dynamics of $\phi _{n,0,\unicode{x3bb} }$ for $n=3$ in Figure 1.

Figure 1 The dynamics of $\phi _{n,0,\unicode{x3bb} }$ for $n=3$ .

Let $\Phi _{n,k,\unicode{x3bb} }$ be the mapping torus of $\phi _{n,k, \unicode{x3bb} }$ , let $\Lambda ^s, \Lambda ^u$ be the suspensions of $l^s_n, l^u_n$ , respectively, and consider the suspension flow on $\Phi _{n,k, \unicode{x3bb} }$ . Call the suspension of the origin the singular orbit of $\Phi _{n,k,\unicode{x3bb} }$ .

Definition 2.2. A pseudo-Anosov flow on a closed 3-manifold M is a $C^1$ -flow $\phi _t$ satisfying the following.

• • There is a finite collection of closed orbits $\{\gamma _1, \ldots , \gamma _s \}$ , called the singular orbits, such that $\phi _t$ is smooth away from the singular orbits.

• • There is a path metric d on M, which is induced from a Riemannian metric g away from the singular orbits.

• • Away from the singular orbits, there is a splitting of the tangent bundle into three $\phi _t$ -invariant line bundles $TM=E^s \oplus E^u \oplus T\phi _t$ , such that

  $$ \begin{align*}|d\phi_t(v)| < C \unicode{x3bb}^{-t} |v|\end{align*} $$
  for every $v \in E^s, t>0$ , and
  $$ \begin{align*}|d\phi_t(v)| < C \unicode{x3bb}^t |v|\end{align*} $$
  for every $v \in E^u, t<0$ , for some $C, \unicode{x3bb}>1$ .

• • Each singular orbit $\gamma _i$ has a neighborhood $N_i$ and a map $f_i$ sending $N_i$ to a neighborhood of the singular orbit in $\Phi _{n_i, k_i, \unicode{x3bb} }$ , for some $n_i \geq 3$ , such that $f_i$ is bi-Lipschitz on $N_i$ and smooth away from $\gamma _i$ , preserves the orbits, and sends $E^s, E^u$ to line bundles tangent to $\Lambda ^s, \Lambda ^u$ , respectively. In this case, we say that $\gamma _i$ is $n_i$ -pronged. By extension, we also say that a non-singular orbit is $2$ -pronged.

We call the (possibly singular) foliation which is tangent to $E^s \oplus T\phi _t$ away from the singular orbits and given by the image of $\Lambda ^s \subset \Phi _{n_i,k_i,\unicode{x3bb} }$ under $f_i$ near the singular orbits the stable foliation $\Lambda ^s$ . We define the unstable foliation $\Lambda ^u$ similarly.

An Anosov flow is a pseudo-Anosov flow without singular orbits.

Definition 2.3. A flow on a closed 3-manifold M is said to be transitive if it has an orbit that is dense in M.

Definition 2.4. Let $\phi _i$ be a flow on a 3-manifold $M_i$ for $i=1,2$ . We say that $\phi _1$ is orbit equivalent to $\phi _2$ if there is a homeomorphism $h:M_1 \to M_2$ sending orbits of $\phi _1$ to orbits of $\phi _2$ in an orientation preserving way (but not necessarily preserving the parameterizations by the flows). In this case, we say that h is an orbit equivalence.

Definition 2.5. Let $\phi $ be a pseudo-Anosov flow on a closed 3-manifold M, and let $\mathcal {C}$ be a non-empty finite collection of closed orbits of $\phi $ which includes all the singular orbits of $\phi $ . Lift $\phi $ up to a flow $\widetilde {\phi }$ on the universal cover $\widetilde {M}$ . Let $\widetilde {\mathcal {C}}$ be the set of orbits of $\widetilde {\phi }$ which cover the orbits in $\mathcal {C}$ .

Let $\mathcal {O}$ be the space of orbits of $\widetilde {\phi }$ , endowed with the quotient topology. We refer to $\mathcal {O}$ as the orbit space of $\phi $ . It is shown in [Reference Fenley and MosherFM01, Proposition 4.2] that $\mathcal {O}$ is homeomorphic to $\mathbb {R}^2$ , and the images of $\Lambda ^s, \Lambda ^u$ under the projection $\widetilde {M} \to \mathcal {O}$ are two (possibly singular) one-dimensional foliations $\mathcal {O}^s, \mathcal {O}^u$ , respectively.

A perfect fit rectangle is a rectangle-with-one-ideal-vertex properly embedded in $\mathcal {O}$ such that the restrictions of $\mathcal {O}^s$ and $\mathcal {O}^u$ to the rectangle foliate it as a product that is conjugate to the foliations of $[0,1]^2 \backslash \{(1,1)\}$ by vertical and horizontal lines. See Figure 2.

Figure 2 A perfect fit rectangle.

We say that $\phi $ has no perfect fits relative to $\mathcal {C}$ if every perfect fit rectangle in $\mathcal {O}$ intersects $\widetilde {\mathcal {C}}$ .

If $\phi $ has no perfect fits relative to the collection of singular orbits, then we say that $\phi $ has no perfect fits.

Definition 2.14. Let $\phi $ be a pseudo-Anosov flow on a closed 3-manifold M. An immersed Birkhoff section is an immersed cooriented compact surface with boundary S such that the following hold.

• • The interior of S is positively transverse to the orbits of $\phi $ .

• • The boundary of S is a union of closed orbits of $\phi $ .

• • Every orbit of $\phi $ intersects S in finite foward and finite backward time, that is, for every $x \in M$ , there exists $t_1, t_2> 0$ such that $\phi _{t_1}(x) \in S$ and $\phi _{-t_2}(x) \in S$ .

When M is oriented, we orient the boundary components of S using the induced orientation on S. We say that a boundary component of S is positive if its orientation agrees with the flow direction, otherwise it is negative. See Figure 4.

Figure 4 A Birkhoff section near a positive/negative boundary component.

Let $\mathcal {C}$ be the set of closed orbits for which some element of $\partial S$ lies along. The complexity of S is defined to be $\mathfrak {c}(S) = -\chi _{{\mathrm {top}}} (S \backslash \mathcal {C})$ , where $\chi _{{\mathrm {top}}}$ denotes the Euler characteristic of the underlying space. In other words, we puncture the immersed surface at the points where it intersects its boundary and compute the negative of its Euler characteristic.

A Birkhoff section is an immersed Birkhoff section that is embedded in its interior. Note that the complexity of a Birkhoff section is equal to the negative of its Euler characteristic.

Definition 2.17. A surface with corners is a surface with boundary S along with a finite collection of points on $\partial S$ , which we call the corners. The complementary regions of the corners in $\partial S$ are called the sides. A punctured surface with corners is a surface with corners S with a finite collection of interior points and corners removed.

The index of a surface with corners S is defined to be ${\mathrm {ind}}(S) = \chi _{{\mathrm {top}}}(S) - \tfrac 14 \text {# corners}$ . If we remove x interior points and y corners from S to get a punctured surface with corners $S'$ , then the index of $S'$ is defined to be ${\mathrm {ind}}(S') = {\mathrm {ind}}(S)-x-\tfrac 14 y$ .

Definition 2.19. Let $\phi $ be a pseudo-Anosov flow on a closed 3-manifold M. A broken transverse surface is an immersed cooriented surface with corners S such that the following hold.

• • The interior of S is positively transverse to the orbits of $\phi $ .

• • Every boundary component $\alpha $ of S has an even number of corners. The sides along $\alpha $ lie along closed orbits or are transverse to the orbits of $\phi $ alternatingly.

We denote the union of sides of S that lie along closed orbits by $\partial _v S$ and call it the vertical boundary of S. We denote the union of sides of S that are transverse to orbits of $\phi $ by $\partial _h S$ and call it the horizontal boundary of S.

When M is oriented, we orient the sides of S using the induced orientation on S. A side in $\partial _v S$ is said to be positive if its orientation agrees with the flow direction, otherwise it is negative.

Definition 2.20. An ideal tetrahedron is a tetrahedron with its four vertices removed. The removed vertices are called the ideal vertices. An ideal triangulation of a 3-manifold M is a decomposition of M into ideal tetrahedra glued along pairs of faces.

A taut structure on an ideal triangulation is a labeling of the dihedral angles by $0$ or $\pi $ , such that the following hold.

• • Each tetrahedron has exactly two dihedral angles labeled $\pi $ , and they are opposite to each other.

• • The angle sum around each edge in the triangulation is $2\pi $ .

A transverse taut structure is a taut structure along with a coorientation on each face, such that for any 0-labeled edge in a tetrahedron, exactly one of the faces adjacent to it is cooriented inwards. A transverse taut ideal triangulation is an ideal triangulation with a transverse taut structure.

In this paper, we will take the convention that the coorientations are always pointing upward.

Definition 2.21. A veering structure on a transverse taut ideal triangulation of an oriented 3-manifold is a coloring of the edges by red or blue, so that if we look at each tetrahedron with a $\pi $ -labeled edge in front, the four outer $0$ -labeled edges, starting from an end of the front edge and going counter-clockwise, are colored red, blue, red, blue, respectively. See Figure 6.

Figure 6 A tetrahedron in a veering triangulation. There are no restrictions on the colors of the top and bottom edges.

A veering triangulation is a transverse taut ideal triangulation with a veering structure.

Definition 2.23. A tetrahedron in $\Delta $ is called a toggle tetrahedron if the colors on its top and bottom edges differ. It is called a red/blue fan tetrahedron if both its top and bottom edges are red/blue, respectively.

Note that some authors call toggle and fan tetrahedra hinge and non-hinge, respectively.

Definition 2.25. Let t be a tetrahedron in $\Delta $ . An equatorial square (or square in short) in t is a quadrilateral-with-4-ideal-vertices properly embedded in t with its four sides along the side edges of t.

Definition 3.2. A rectangle R in $\mathcal {P}$ is a rectangle embedded in $\mathcal {P}$ such that the restrictions of $\mathcal {P}^s$ and $\mathcal {P}^u$ foliate the rectangle as a product that is conjugate to the foliations of $[0,1]^2$ by vertical and horizontal lines, and such that no element of $\mathcal {S}$ lies in the interior of R.

Figure 8 From left to right: an edge rectangle, a face rectangle, and a tetrahedron rectangle.

Let $R_1$ and $R_2$ be rectangles in $\mathcal {P}$ . Here, $R_1$ is said to be taller than $R_2$ if every leaf of $\mathcal {P}^u$ that intersects $R_2$ intersects $R_1$ . Additionally, $R_1$ is said to be wider than $R_2$ if every leaf of $\mathcal {P}^s$ that intersects $R_2$ intersects $R_1$ .

An edge rectangle in $\mathcal {P}$ is a rectangle with two opposite corners on $\mathcal {S}$ . A face rectangle in $\mathcal {P}$ is a rectangle with one corner on $\mathcal {S}$ and the two opposite sides to the corner containing elements of $\mathcal {S}$ in their interior. A tetrahedron rectangle in $\mathcal {P}$ is a rectangle all of whose sides contain elements of $\mathcal {S}$ in their interior. See Figure 8.

In this paper, we will take the convention of drawing leaves of $\mathcal {P}^s$ as vertical lines and leaves of $\mathcal {P}^u$ as horizontal lines. As such, we will call the sides of a rectangle R that lie along leaves of $\mathcal {P}^s$ as the vertical sides of R, and the sides that lie along leaves of $\mathcal {P}^u$ as the horizontal sides of R.

Note that by our assumption of no perfect fits, we could have alternatively defined a face rectangle to be a rectangle maximal with respect to the property that one corner lies on $\mathcal {S}$ , and a tetrahedron rectangle to be a maximal rectangle.

Definition 3.3. Let R be a rectangle in $\mathcal {P}$ and let p be a corner of R. Let s be the vertical side of R which contains p. Orient s to point inward at p. Similarly, let u be the horizontal side of R which contains p and orient u to point inward at p. Here, p is said to be the SW or NE corner of R if (orientation of u, orientation of s) agrees with the orientation of $\mathcal {P}$ at $\gamma $ . Otherwise, p is said to be the NW or SE corner of R.

Let R be an edge rectangle. We say that R is red if its SW and NE corners lie on $\mathcal {S}$ . We say that R is blue if its NW and SE corners lie on $\mathcal {S}$ .

Definition 3.4. Let R be an edge rectangle in $\mathcal {P}$ , let p and q be the two corners of R that lie in $\mathcal {S}$ . A veering diagonal is an arc in R between p and q that is transverse to $\mathcal {P}^s$ and $\mathcal {P}^u$ .

Definition 3.5. Let $\gamma $ be a point on $\mathcal {O}$ . Recall Definition 2.1. There exists a neighborhood of $\gamma $ and a map from the neighborhood to $\mathbb {R}^2$ sending $\gamma $ to $0$ and sending the foliations $\mathcal {O}^s$ and $\mathcal {O}^u$ to $l^s_n$ and $l^u_n$ , respectively. We call the preimage of a quadrant under the map a quadrant at $\gamma $ .

Let p be a point on $\mathcal {P}$ . Suppose p maps to $\gamma $ in $\mathcal {O}$ . Then we call the lift of a quadrant at $\gamma $ to p a quadrant at p.

Recall again that $\mathcal {P}$ is oriented. If $q_1$ and $q_2$ are quadrants at p, we say that $q_1$ is on the left of $q_2$ if there exists an orientation preserving local homeomorphism $\mathbb {R} \times [0,\infty )$ mapping $(0,0)$ to p, $(-\infty ,0] \times [0,\infty )$ to $q_1$ , and $[0,\infty ) \times [0,\infty )$ to $q_2$ . If there exists quadrants $q_1,\ldots ,q_{n+1}$ at p such that $q_i$ is on the left of $q_{i+1}$ for each i, then we say that $q_1$ is n quadrants to the left of $q_{n+1}$ .

Definition 3.6. Let $R_1,\ldots ,R_k$ be edge rectangles such that:

• • there are elements $s_0,\ldots ,s_k \in \mathcal {S}$ such that $R_i$ has corners at $s_{i-1}$ and $s_i$ , for each i;

• • the quadrant determined by $R_{i+1}$ at $s_i$ is $2$ quadrants to the left of that determined by $R_i$ , for each i.

Consider the set $S_k=\bigcup _{i=1}^k [i-1,i] \times [0,i]$ in $\mathbb {R}^2$ . If there exists an orientation-preserving embedding of $S_k$ into $\mathcal {P}$ sending $[i-1,i] \times [i-1,i]$ to $R_i$ for each i and sending the foliations of $\mathbb {R}^2$ by vertical and horizontal lines to $\mathcal {P}^s$ and $\mathcal {P}^u$ , we call the image of the embedding a staircase for $R_1,\ldots ,R_k$ . See Figure 10 left.

Figure 10 Left: A staircase for $R_1,\ldots ,R_k$ . Right: A slope for $R_1,\ldots ,R_k$ .

Note that if $k=1$ , then the edge rectangle $R_1$ is a staircase for itself.

Extending the definition for rectangles, for staircases $S_1$ and $S_2$ , $S_1$ is said to be taller than $S_2$ if every leaf of $\mathcal {P}^u$ that intersects $S_2$ intersects $S_1$ ; $S_1$ is said to be wider than $S_2$ if every leaf of $\mathcal {P}^s$ that intersects $S_2$ intersects $S_1$ .

Suppose we have chosen a veering diagonal $e_i$ for each $R_i$ , then the complementary region of $\bigcup _i e_i$ in the staircase which contains the image of $(k,0)$ is called a slope for $R_1,\ldots ,R_k$ with respect to the choice of veering diagonals. See Figure 10 right.

Note that if $k=1$ , then there are two slopes for $R_1$ . We will specify to which one we are referring in such a case.

Definition 3.7. Let p be a point on $\mathcal {P}$ that is fixed by $g \in \pi _1(M^\circ )$ . We define a red g-edge path to be an infinite sequence of red edge rectangles $(R_i)_{i \in \mathbb {Z}}$ satisfying the following.

• • There is a collection of elements $(s_i)_{i \in \mathbb {Z}}$ of $\mathcal {S}$ such that each $R_i$ has corners at $s_{i-1}$ and $s_i$ .

• • For each i, we have one of the following two cases:

  1. (1) the quadrants determined by $R_i$ and $R_{i+1}$ at $s_i$ are the same and $R_{i+1}$ is taller than $R_i$ ;

  2. (2) the quadrant determined by $R_{i+1}$ at $s_i$ is two quadrants to the left of the quadrant determined by $R_i$

  and case (1) occurs for at least one value of i.

• • If $R_j, \ldots , R_k$ is a maximal subsequence such that $R_i$ and $R_{i+1}$ do not determine the same quadrant at $s_i$ for $j \leq i < k$ , then there is a staircase $S_{j,k}$ for $R_j, \ldots , R_k$ , such that p lies in $S_{j,k}$ .

• • There exists some positive integer P such that $g \cdot R_i=R_{i+P}$ for every i.

See Figure 11. We call P the period of the red g-edge path.

Figure 11 A red g-edge path.

A blue g-edge path is defined similarly. When g and the color red/blue is clear from context, we will abbreviate these as edge paths.

Intuitively, one can think of an edge path as a directed path obtained by transversing some choice of veering diagonals $e_i$ in $R_i$ in the direction of increasing i. Such a directed path is g-invariant and ‘winds around’ p with increasing height and decreasing width. The desirable case is when this intuitive picture holds.

Suppose we have chosen a veering diagonal $e_i$ for each $R_i$ . Then the red g-edge path $(R_i)$ is said to be winding with respect to the choice of veering diagonals if for every maximal subsequence $R_j, \ldots , R_k$ such that $R_i$ and $R_{i+1}$ do not determine the same quadrant at $s_i$ for $j \leq i < k$ , p lies in the slope for $R_j, \ldots , R_k$ on the side of $R_{j-1}$ and $R_{k+1}$ .

See Figure 12 left for a choice of veering diagonals for which the edge path in Figure 11 is winding, and Figure 12 right for a choice of veering diagonals for which the edge path is not winding.

Figure 12 Left: A choice of veering diagonals for which the edge path in Figure 11 is winding. Right: A choice of veering diagonals for which the edge path in Figure 11 is not winding.

Notice that the condition is automatically true when $k-j \geq 1$ , since p must lie outside of $R_j,\ldots ,R_k$ in this case. So one only needs to check the condition for $R_i$ where $R_{i-1}$ and $R_i$ determine the same quadrant at $s_{i-1}$ and $R_i$ and $R_{i+1}$ determine the same quadrant at $s_i$ .

Definition 3.8. [Reference Landry, Minsky and TaylorLMT22, §5.1.2]

Let R be an edge rectangle. Set $R_0=R$ . We define $R_i$ for $i \geq 1$ inductively in the following way: There exists a unique tetrahedron rectangle $Q_i$ whose two elements of $\mathcal {S}$ on its vertical sides are the corners of $R_{i-1}$ . Consider the sub-rectangle of $Q_i$ which has corners at the elements of $\mathcal {S}$ on its horizontal sides. This is an edge rectangle which we call $R_i$ . Similarly, for each $i \leq -1$ , there exists a unique tetrahedron rectangle $Q_{i+1}$ whose two elements of $\mathcal {S}$ on its horizontal sides are corners of $R_{i+1}$ . Consider the sub-rectangle of $Q_{i+1}$ which has corners at the elements of $\mathcal {S}$ on its vertical sides. This is an edge rectangle which we call $R_i$ .

We call the bi-infinite sequence $(R_i)$ the core sequence of R. The intersection $\bigcap _{i=-\infty }^\infty R_i$ is a single point, which we call the core point of R and denote as $c(R)$ . See Figure 13.

Figure 13 The core sequence and core point of an edge rectangle R.

Definition 3.9. A red g-edge path is said to be nice if whenever $R_{i-1}$ and $R_i$ determine the same quadrant at $s_{i-1}$ , and $R_i$ and $R_{i+1}$ determine the same quadrant at $s_i$ , p lies closer to the vertical side of $R_i$ containing $s_i$ than the core point of $R_i$ .

A nice blue g-edge path is defined similarly.

Definition 3.11. An anchor system is a pair $(A,\alpha )$ where $\alpha $ is a bijection from the set of edge rectangles onto a subset $A \subset \mathcal {P}$ satisfying:

• • for each edge rectangle R, $\alpha (R)$ lies in the interior of R;

• • $g \cdot \alpha (R) = \alpha (g \cdot R)$ for each edge rectangle R and each $g \in \pi _1(M^\circ )$ ;

• • for edge rectangles $R_1$ and $R_2$ sharing a corner $s \in \mathcal {S}$ and determining the same quadrant at s, if $R_1$ is wider than $R_2$ , then the rectangle with corners at s and $\alpha (R_1)$ is wider and no taller than the rectangle with corners at s and $\alpha (R_2)$ .

Let $(A,\alpha )$ be an anchor system. Let F be a face rectangle. Let s be the corner of F that lies in $\mathcal {S}$ , let x be the element of $\mathcal {S}$ that lies in the interior of a vertical side of F, and let y be the element of $\mathcal {S}$ that lies in the interior of a horizontal side of F. Let $a_x$ be the image of the edge rectangle with corners at s and x under $\alpha $ , and $a_y$ be the image of the edge rectangle with corners at s and y under $\alpha $ . If the rectangle R with corners at x and $a_x$ intersects the rectangle Q of F with corners at y and $a_y$ , then F is said to be busy. If F is busy, let $R'$ be the maximal sub-rectangle of R with the property that the stable and unstable leaves through each point in $R'$ do not intersect the interior of Q. An F-buoy is a point in $R'$ which corresponds to an orbit of $\widehat {\phi }$ covering a closed orbit of $\phi $ .

Definition 3.12. Let s be an element of $\mathcal {S}$ , and let q be a quadrant at $\gamma $ . An s-staircase is the union of all edge rectangles that have a corner at s and determine the quadrant q (see Figure 14). Generally, an $\mathcal {S}$ -staircase is an s-staircase for some $s \in \mathcal {S}$ . Notice that these were simply called staircases in [Reference Landry, Minsky and TaylorLMT22] but in this paper, we need to distinguish them from the objects in Definition 3.6.

Figure 14 An s-staircase.

An edge rectangle R is pinched if there exists another edge rectangle Q such that R and Q lie in the same $\mathcal {S}$ -staircase and $c(R)=c(Q)$ . A core point c is pinched if it is the core point of a pinched edge rectangle.

The preimage of a core point c is the union of edge rectangles whose core point is c.

Definition 3.13. A family of rectangles $\{b(R):\text {edge rectangles}\ R\}$ is a choice of core boxes if it satisfies the following properties.

1. (1) For every edge rectangle R, $b(R)$ lies in the interior of R and contains the core point of R.

2. (2) If $R_1$ and $R_2$ are edge rectangles that lie in the same $\mathcal {S}$ -staircase with distinct core points and $R_1$ is taller than $R_2$ , then for any points $x_i \in b(R_i)$ , the rectangle $R^{\prime }_1$ with corners at s and $x_1$ is strictly taller than the rectangle $R^{\prime }_2$ with corners at s and $x_2$ , and $R^{\prime }_2$ is strictly wider than $R^{\prime }_1$ .

3. (3) $b(g \cdot R) = g \cdot b(R)$ for all edge rectangles R and $g \in \pi _1(M^\circ )$ .

Definition 4.1. Let $\Delta $ be a veering triangulation on a 3-manifold M. We define the stable branched surface B of $\Delta $ : as a $2$ -complex, B is the dual cell complex to $\Delta $ . The branched surface structure is then determined by declaring that within each tetrahedra, B contains a smooth quadrilateral with vertices on the top and bottom edges and the two side edges of the same color as the top edge. See Figure 16 left.

Figure 16 Left: The portion of the stable branched surface and the flow graph within each tetrahedron. Right: The portion of the flow graph on each sector of the stable branched surface.

The $2$ -cells of B are known as the sectors, while the $1$ -skeleton of B is known as the branch locus. We orient the 1-cells of B to be positively transverse to the faces of $\Delta $ . Then the 1-skeleton of B becomes a directed graph embedded in M, which we call the dual graph of $\Delta $ and denote by $\Gamma $ .

Suppose c is a directed path of $\Gamma $ , then at a vertex v of c, we say that c takes a branching turn at v if it can be realized by a smooth arc on B near v, otherwise we say that c takes an anti-branching turn at v. A cycle of $\Gamma $ that only takes branching turns is called a branch cycle. A cycle of $\Gamma $ that only takes anti-branching turns is called an AB cycle.

Definition 4.2. [Reference Landry, Minsky and TaylorLMT20]

Let $\Delta $ be a veering triangulation on a 3-manifold M. Define the flow graph $\Phi $ to be a directed graph with the set of vertices equal to the set of edges of $\Delta $ , and adding three edges for each tetrahedron, going from the bottom edge to the top edge and the two side edges of opposite color to the top edge.

Here, $\Phi $ can be naturally embedded in the stable branched surface B, and hence in M, by placing each vertex at the top corner of the sector of B that its corresponding edge of $\Delta $ meets, and placing the edges that enter that vertex within that sector of B. See Figure 16 right. We will always consider the flow graph to be embedded in M in this way.

Definition 4.3. Let $\Delta $ be a veering triangulation on a 3-manifold M. Lift the triangulation, its stable branched surface B, its dual graph $\Gamma $ , and its flow graph $\Phi $ to the universal cover $\widetilde {M}$ to get $\widetilde {\Delta }$ , $\widetilde {B}$ , $\widetilde {\Gamma }$ , and $\widetilde {\Phi }$ , respectively.

A descending path on $\widetilde {B}$ is a path that intersects the branch locus of $\widetilde {B}$ transversely and induces the maw coorientation at each intersection, that is, it goes from a side with more sectors to a side with less sectors. Let x be a point on $\widetilde {B}$ . The descending set of x, denoted by $\Delta (x)$ , is the set of points on $\widetilde {B}$ that can be reached from x via a descending path. By [Reference Landry, Minsky and TaylorLMT22, Lemma 3.1], each descending set is a union of sectors that forms a quarter-plane.

Now let c be a cycle of the dual graph $\Gamma $ that is not a branch cycle. Lift c to a bi-infinite path $\widetilde {c}$ of $\widetilde {\Gamma }$ . The dynamic plane associated to $\widetilde {c}$ , denoted by $D(\widetilde {c})$ , is the set of points on $\widetilde {B}$ that can be reached from a point on $\widetilde {c}$ via a descending path. A dynamic plane is a union of sectors that forms a plane. In fact, if the vertices of $\widetilde {c}$ are $(v_i)_{i \in \mathbb {Z}}$ , then $D(\gamma ) = \bigcup _i \Delta (v_i)$ .

Consider the restriction of $\widetilde {\Phi }$ to a dynamic plane D. This is an oriented train track with only converging switches. In particular, the forward $\widetilde {\Phi }$ -path starting at any given point $x \in \widetilde {\Phi }$ is well defined.

In Figure 17, which is a reproduction of [Reference Landry, Minsky and TaylorLMT22, Figure 6], we illustrate a descending set in a dynamic plane, and the restriction of $\widetilde {\Phi }$ to it.

Figure 17 A descending set in a dynamic plane and the restriction of $\widetilde {\Phi }$ (green). This figure is a reproduction of [Reference Landry, Minsky and TaylorLMT22, Figure 6].

Consider the case when c is an AB cycle. Let $\widetilde {c}$ be a lift of c. Then there are two $\widetilde {\Phi }$ -paths on the dynamic plane $D(\widetilde {c})$ for which every vertex of $\widetilde {c}$ lies along. The region bounded by the two $\widetilde {\Phi }$ -paths is called an AB strip. By [Reference Landry, Minsky and TaylorLMT22, Proposition 3.10], for a fixed dynamic plane D, all the AB strips, if any, must be adjacent. We call the union of them the AB region of D.

In Figure 18, which is a reproduction of [Reference Landry, Minsky and TaylorLMT22, Figure 6], we illustrate a case where the AB region of a dynamic plane contains two AB strips.

Figure 18 A dynamic plane containing two AB strips in its AB region. The $\widetilde {\Phi }$ -paths starting at $x_1$ and $x_2$ never converge. This figure is a reproduction of [Reference Landry, Minsky and TaylorLMT22, Figure 10].

Definition 4.7. Let $\gamma $ be a closed orbit of $\phi $ which is not an element of $\mathcal {C}$ . Take $c \in F_\Phi ^{-1}(\gamma ^2)$ , which exists by Proposition 4.5. We define the flow graph complexity of $\gamma $ with respect to $\mathcal {C}$ , denoted by $\mathfrak {c}_{\mathcal {C}}(\gamma )$ , to be $\tfrac 12$ times the length of c. By Proposition 4.5, $\mathfrak {c}_{\mathcal {C}}(\gamma )$ is well defined.

When the collection $\mathcal {C}$ is clear from context, we will just write $\mathfrak {c}(\gamma )$ and call it the flow graph complexity of $\gamma $ .