2.1 Planar topology

Let $\mathbb R^2$ be the Euclidean plane, equipped with the topology induced by the Euclidean metric d. An open ball of radius $r>0$ centered at $x\in \mathbb R^2$ is denoted by $B(x,r)$ .

A path $\gamma : I\longrightarrow \mathbb R^2$ is a continuous map defined on an interval $I\subset \mathbb R$ . A path $\gamma $ is said to be a reparameterization of another path $\gamma ':I'\longrightarrow \mathbb R^2$ if there exists a strictly increasing map $\varphi :\mathbb R\longrightarrow \mathbb R$ that satisfies $\varphi (I) = I'$ and $\gamma = \gamma ' \circ \varphi \vert _I$ . A loop $\gamma :\mathbb {S}^1\longrightarrow \mathbb R^2$ is a continuous map defined on the unit circle $\mathbb {S}^1$ . A path or a loop is said to be simple if it is injective. By abuse of notation, if there is no ambiguity, we may consider paths and loops as subset of the plane by identifying them with their images.

A topological line is a path $\ell : \mathbb R \longrightarrow \mathbb R^2$ that is injective and proper. Observe that any topological line admits a natural orientation induced by the positive orientation on $\mathbb R$ . Similarly, we define a half-line as a path $\ell ^+: [0,\infty )\longrightarrow \mathbb R^2$ that is injective and proper. Often enough, it is convenient to consider half-lines defined on $(-\infty ,0]$ as well.

In the sphere compactification of the plane $\mathbb {S}^2 = \mathbb R^2 \sqcup \{\infty \}$ , every topological line $\ell $ can be extended to a simple loop on $\mathbb S^2$ by adding the point at infinity. According to the Jordan curve theorem [Reference JordanJor87], the complement of any simple loop on the sphere has exactly two connected components. By the Schoenflies theorem (see [Reference CairnsCai51] for an elementary proof), each of these connected components is homeomorphic to $\mathbb R^2$ . In our context, this means that every topological line $\ell $ separates the plane into two sets, each of which is homeomorphic to the plane. These sets are called the left and right sides of $\ell $ , denoted $ L(\ell ) $ and $ R(\ell ) $ , according to the orientation of $\ell $ .

Now, we present a fundamental generalization of the Schoenflies theorem, which provides a criterion for extending homeomorphisms defined on subsets of the sphere to the entire sphere. This theorem is particularly useful in the context of low-dimensional topology because it allows us to handle ‘nice’ subsets of $\mathbb{S}^2$ with more flexibility.

Here, by arc, we mean a simple path $\gamma : [0,1] \longrightarrow \mathbb S^2$ . A Y-set is a subset of $\mathbb S^2$ , which is the union of three arcs intersecting at a single point that is an endpoint of each arc. Note that the orientation of $\mathbb S^2$ induces a cyclic order on the arcs constituting a Y-set in a neighborhood of the intersection point.

Applying the Homma–Schoenflies theorem is not necessarily straightforward, as it requires checking the preservation of the cyclic order of Y-sets. However, there are many classic examples of applications of this theorem (see [Reference Le RouxLR06, Appendix]). Along this preliminary section, we will guide the reader through the process of applying the Homma–Schoenflies theorem as we introduce the new concepts and tools.

A family $\{A_i\}_{i \in I}$ of subsets of $\mathbb R^2$ is called locally finite if, for every compact set ${K\subset \mathbb R^2}$ , we have that $\{i \in I \mid A_i \cap K \neq \varnothing \}$ is a finite set.

Proof. For each $i \in I$ , consider the set $Y_i:=\bigcup _{j \in I} X_j \setminus X_i$ . First, we show that $Y_i$ is closed. For a given $i \in I$ , let $(x_n)_{n\in \mathbb N}$ be a sequence of points in $Y_i$ that converges to a point $x\in \mathbb R^2$ . Observe that, since the family $\{X_j\}_{j \in I}$ is locally finite, the set $\{j \in I \mid d(x, X_j) < r\}$ is finite for any $r>0$ . Consequently, there exists an index $j_0 \in I\setminus \{i\}$ and a subsequence $(x_{n_k})_{k\in \mathbb N}$ that satisfies $x_{n_k} \in X_{j_0}$ for every $k\in \mathbb N$ . Since $X_{j_0}$ is a closed set, we have that $x \in X_{j_0}$ . Since $X_{j_0}$ is disjoint from $X_i$ , we have that $x \not \in X_i$ . This allows us to conclude that $x \in Y_i$ , which proves that the set $Y_i$ is closed.

We can now define, for each $i \in I$ , the set

$$ \begin{align*} U_i := \{z \in \mathbb R^2 \mid d(z, X_i)<{\min}( d(z, Y_i), 1)\}.\end{align*} $$

Since $X_i$ and $Y_i$ are disjoint sets, we observe that

$$ \begin{align*} d(z, X_i) = 0 < {\min}( d(z, Y_i), 1) \quad \text{for all } z \in X_i.\end{align*} $$

This proves that $X_i \subset U_i$ for every $i \in I$ . Next, we remark that

$$ \begin{align*} d(z, X_i) <{\min}( d(z, Y_i), 1) \leq d(z, X_j) \quad \text{for all } j\neq i,\ \text{ for all } z \in U_i.\end{align*} $$

This implies that $U_i$ is disjoint from $U_j$ for every $j \in I\setminus \{i\}$ . Finally, note that the function

$$ \begin{align*}g_i:z \longmapsto d(z, X_i) - {\min}( d(z, Y_i), 1)\end{align*} $$

is continuous. Thus, by writing $U_i = g_i^{-1}((-\infty ,0))$ , we conclude that $U_i$ is an open set.

To conclude the proof, we need to show that the family $\{U_i\}_{i \in I}$ is locally finite. For that, we consider for each $i \in I$ , the closed ball

$$ \begin{align*}B(X_i,1):=\{z \in \mathbb R^2 \mid d(z, X_i) \leq 1\}.\end{align*} $$

Since the family $\{X_i\}_{i \in I}$ is locally finite, the family of closed balls $\{B(X_i,1)\}_{i \in I}$ is also locally finite. From the definition of the set $U_i$ , we can observe that

$$ \begin{align*} U_i\subset B(X_i,1) \quad \text{for all } i \in I.\end{align*} $$

Therefore, we conclude that the family of open sets $\{U_i\}_{i \in I}$ also is locally finite.

Here, we provide a first application of the Homma–Schoenflies theorem.

Let $\{\ell _{n}\}_{n\in \mathbb Z}$ be a locally finite family of pairwise disjoint topological lines such that

$$ \begin{align*} L(\ell_{n_1})\subset L(\ell_{n_2}) \subset L(\ell_{n_3})\quad \text{for all } n_1<n_2<n_3\in \mathbb Z.\end{align*} $$

In this context, the union of the lines in the family $\{\ell _n\}_{n \in \mathbb Z}$ and the point at infinity forms a compact, connected, and locally connected subset of $\mathbb S^2$ . It is worth noting that this would not be the case if the family $\{\ell _n\}_{n \in \mathbb Z}$ was not locally finite. Using the Homma–Schoenflies theorem, we can obtain an orientation-preserving homeomorphism on the plane that maps each line $\ell _n$ to the vertical line $\{n\}\times \mathbb R$ oriented downwards.

For more applications, we refer the reader to [Reference Le RouxLR17, Appendix B].

2.2 Topological planar foliations

A topological planar foliation $\mathcal {F}$ is a partition of the plane into disjoint topological lines, called leaves, that admits a local parallel structure. In other words, for any point $x \in \mathbb {R}^2$ , there exists a closed neighborhood $U\subset \mathbb R^2 $ of x and a homeomorphism $\varphi : U \to [0,1] \times [0,1]$ such that, for every leaf $\phi \in {\mathcal {F}}$ , each connected component of the set $\phi \cap U$ is of the form $\varphi^{-1}(\{t\} \times [0,1]).$

In the particular case of foliations of $\mathbb R^2$ , the condition above can be exchanged for the condition that the entire set $\phi \cap U$ is of the form $\varphi ^{-1}(\{t\} \times [0,1])$ for some $t \in [0,1] $ .

A cross-section of ${\mathcal {F}}$ over a leaf $\phi \in {\mathcal {F}}$ is a compact arc $\sigma \subset \mathbb {R}^2$ that intersects $\phi $ and is transverse to ${\mathcal {F}}$ , meaning that the arc $\sigma $ intersects each leaf of ${\mathcal {F}}$ at most once.

Let $\phi \in {\mathcal {F}}$ be a leaf and let $\{\sigma _k\}_{k \in \mathbb Z^*}$ be a locally-finite collection of pairwise disjoint compact cross-sections of ${\mathcal {F}}$ over the leaf $\phi $ . Assume that, for every $k>0$ , it holds that:

• • the cross-sections $\sigma _{-k}$ and $\sigma _{k}$ intersect the same leaves in ${\mathcal {F}}$ ;

• • the sub-arc $u_k\subset \phi $ joining $\sigma _{-k}$ and $\sigma _{k}$ satisfies $u_k \subset u_{k+1}$ ;

• • the only leaf of ${\mathcal {F}}$ intersecting all cross-sections in $\{\sigma _k\}_{k \in \mathbb N^*}$ is the leaf $\phi $ ;

• • every leaf of ${\mathcal {F}}$ that intersects $\sigma _{k+1}$ also intersects $\sigma _k$ .

The closed set $V\subset \mathbb R^2$ , obtained as the union of all compact sub-arcs of leaves in ${\mathcal {F}}$ joining the cross-sections $\sigma _{-k}$ and $\sigma _{k}$ , for some $k>0$ , is a trivial neighborhood of the leaf $\phi $ . Observe that, for any open neighborhood $U\subset \mathbb R^2$ of $\phi $ , we can consider sufficiently small cross-sections $\{\sigma _k\}_{k \in \mathbb Z^*}$ so that the neighborhood V is contained in U.

It is worth noting that trivial neighborhoods of leaves can be taken to be arbitrarily small. More precisely, for any leaf $\phi \in {\mathcal {F}}$ and any open neighborhood $U\subset \mathbb R^2$ of $\phi $ , there exists a trivial neighborhood V of $\phi $ such that $V\subset U$ . Such a trivial neighborhood V can be obtained by taking the cross-sections in the collection $\{\sigma _k\}_{k \in \mathbb Z^*}$ to be sufficiently small.

By the Homma–Schoenflies theorem, for any trivial neighborhood V of a leaf $\phi \in {\mathcal {F}}$ , we can find a homeomorphism $\varphi : \mathbb R^2\longrightarrow \mathbb R^2$ that satisfies the following properties:

• • the leaf $\phi $ is mapped by $\varphi $ into the vertical line $\{0\}\times \mathbb R$ ;

• • the set V is mapped by $\varphi $ into the set $\bigcup _{n \in \mathbb N^*} [-{1}/{n}, {1}/{n}]\times [ -n,n]$ ;

• • for each leaf $\phi ' \in {\mathcal {F}}$ , the set $\varphi (\phi '\cap V)$ is a vertical line $\{t\}\times \mathbb R$ for some $t\in \mathbb R$ .

As a direct application of Lemma 2.1, we have the following lemma, which will be of main importance in the developments of §3.

We can endow any foliation ${\mathcal {F}}$ with the quotient topology, which corresponds to the finest topology on ${\mathcal {F}}$ for which the quotient map $\pi _{_{\mathcal {F}}}:\mathbb R^2\longrightarrow {\mathcal {F}}$ that sends each point to the leaf passing through it is continuous. In this topology, the open sets of the foliation ${\mathcal {F}}$ are those whose preimages under $\pi _{_{\mathcal {F}}}$ are open in $\mathbb R^2$ . Throughout this text, we will always consider the quotient topology on foliations, unless otherwise stated.

A subset $A\subset \mathbb R^2$ is called saturated (with respect to ${\mathcal {F}}$ ) if it contains all leaves of ${\mathcal {F}}$ that intersect it. The restriction of a foliation ${\mathcal {F}}$ to a saturated subset $A\subset \mathbb R^2$ is denoted by ${\mathcal {F}}\vert _A$ . Note that, in the case of saturated sets, we have ${\mathcal {F}}\vert _A = \pi _{_{\mathcal {F}}}(A)$ . By a slight abuse of notation, we often treat saturated sets $A\subset \mathbb R^2$ as subsets of the foliation ${\mathcal {F}}$ by identifying $A = {\mathcal {F}}\vert _A$ . This is the abuse that allows us to denote saturated sets as subsets $A\subset {\mathcal {F}}$ or to refer to leaves as elements $\phi \in A$ . This convention introduces no topological ambiguity when referring to an open saturated set or to its boundary. Indeed, from the definition of the quotient topology, a saturated set $A\subset \mathbb R^2$ is open if, and only if, the restriction ${\mathcal {F}}\vert _A$ is an open subset of ${\mathcal {F}}$ . Additionally, the boundary $\partial A$ of a saturated set A may be understood equivalently as the boundary of A inside $\mathbb {R}^2$ or as the boundary of ${\mathcal {F}}|_A$ inside ${\mathcal {F}}$ , with no risk of confusion.

This next lemma states that the connectedness is also preserved through this notation.

Equipped with the quotient topology, the foliation ${\mathcal {F}}$ becomes a simply connected and non-Hausdorff one-dimensional manifold, often called the leaf space of ${\mathcal {F}}$ . As a matter of fact, the classical result due to Haefliger and Reeb [Reference Haefliger and ReebHR57] proves that every simply connected, non-Hausdorff 1-manifold can be realized as the leaf space of a planar foliation. Note that, any simply connected (non-Hausdorff) manifold is orientable. In the one-dimensional context, this corresponds to the existence of an atlas of charts mapping the local orientation at each point of the manifold to the positive orientation of $\mathbb R$ . By choosing an orientation, we fix a local notion of left and right sides around each point. Since, by removing a point, we separate any simply connected non-Hausdorff 1-manifold into two connected components, this local notion of left and right sides is extended globally to each connected component.

In the context of the leaf space, this means that for each leaf $\phi \in {\mathcal {F}}$ , the set ${\mathcal {F}}\setminus \{\phi \}$ has exactly two connected components, one labeled as the left side $L(\phi )$ and the other as the right side $R(\phi )$ . Each leaf of $\mathcal {F}$ thereby inherits an orientation and this assignment is locally coherent, since it is determined by the local orientation at each point of the leaf space. In terms of local trivializing maps of the foliation ${\mathcal {F}}$ , this means that one can choose each trivializing map $\varphi : U \to [0,1] \times [0,1]$ so that the orientation of every leaf $\phi \in {\mathcal {F}}$ is mapped to the positive direction of the second coordinate on the set $\varphi (U \cap \phi ) = \{t\} \times [0,1]$ . This is the classical definition of an oriented foliation.

Once the foliation ${\mathcal {F}}$ is equipped with an orientation, it inherits a naturally defined partial order, where $\phi \leq \phi '$ if and only if $L(\phi ') \subset L(\phi )$ . Note that pairs of leaves $\phi ,\phi '\in {\mathcal {F}}$ which are not Hausdorff-separable must satisfy either $R(\phi ) \cap R(\phi ') = \varnothing $ or $L(\phi ) \cap L(\phi ') = \varnothing $ and, consequently, they cannot be comparable under this partial order relation. Recall that, an interval of the foliation ${\mathcal {F}}$ with respect to this partial order consists in a subset of ${\mathcal {F}}$ that contains all leaves between any two of its leaves. Observe that leaf intervals are not necessarily homeomorphic to an interval of the real line $\mathbb R$ , as the leaf space of ${\mathcal {F}}$ is not necessarily Hausdorff.

A saturated set $A\subset \mathbb R^2$ is said to be non-separating if for every leaf $\phi \in A$ , one of the connected components of $\mathbb R^2\setminus \phi $ contains all leaves in $A\setminus \{\phi \}$ .

A saturated set $A\subset \mathbb R^2$ is said to be trivially foliated if the restriction ${\mathcal {F}}\vert _A$ is totally ordered by the natural order on the foliation ${\mathcal {F}}$ .

Following Lemmas 2.3 and 2.4, we observe that non-empty leaf domains of ${\mathcal {F}}$ correspond exactly to the subsets of ${\mathcal {F}}$ that are homeomorphic to $\mathbb R$ . Moreover, the boundary of a leaf domain is formed by a locally finite and non-separating collection of leaves in ${\mathcal {F}}$ .

The boundary of a leaf domain $A\subset {\mathcal {F}}$ admits two important subsets, defined as

$$ \begin{align*} \partial_R A := \partial A \cap \biggl(\ \bigcap_{\phi \in A} R(\phi)\biggr) \quad \text{and} \quad \partial_L A := \partial A \cap \biggl(\ \bigcap_{\phi \in A} L(\phi)\biggr).\end{align*} $$

These sets are disjoint by definition and their union may not cover the entire boundary $\partial A$ . The set A lies on the left side of each leaf in $\partial _R A$ and on the right side of every leaf in $\partial _L A$ .

The sets $\partial _R A$ and $\partial _L A$ are each totally ordered by a relation $\prec $ , defined as follows.

• • Fix a leaf of reference $\phi _* \in A$ . For any two leaves $\phi , \phi ' \in \partial _L A$ (respectively $\partial _R A$ ), we define $\phi \prec \phi '$ if and only if there exist two disjoint paths $\gamma ,\gamma ':[0,1]\longrightarrow \mathbb R^2$ with endpoints $\gamma (1) \in \phi $ , $\gamma '(1) \in \phi '$ , and $\gamma (0), \gamma '(0) \in \phi _*$ , ordered $\gamma (0)< \gamma '(0) $ along the leaf $\phi _*$ .

We remark that $\prec $ is well defined and independent of the choice of reference leaf $\phi _*\in A$ . Later, in §3.3, we provide a more formal alternative definition of this order relation.

A path $\gamma :I\longrightarrow \mathbb R^2$ defined on an interval $I \subset \mathbb R$ is said to be positively transverse to an oriented foliation ${\mathcal {F}}$ if the map $\pi _{_{\mathcal {F}}}\circ \gamma :I\longrightarrow {\mathcal {F}}$ is strictly increasing with respect to the natural order on ${\mathcal {F}}$ . In other words, a positively transverse path intersects each leaf of ${\mathcal {F}}$ at most once and it does so crossing each leaf from right to left.

Each positively transverse path $\gamma $ defines a saturated set $C_\gamma \subset \mathbb R^2$ , which is the union of all leaves in ${\mathcal {F}}$ that are intersected by $\gamma $ . From the definition of positively transverse paths, it follows that the set $C_\gamma $ (seen as a subset of the foliation ${\mathcal {F}}$ ) is homeomorphic to an interval of the real line $\mathbb R$ . In particular, this proves the following elementary lemma.

Conversely, for any leaf domain $A\subset {\mathcal {F}}$ and any pair of points $x,y \in A$ that do not lie on the same leaf, there exists a positively transverse path $\gamma :I\longrightarrow \mathbb R^2$ that satisfies either $\gamma (0)=x$ and $\gamma (1)=y$ or $\gamma (0)=y$ and $\gamma (1)=x$ .

We present two examples of applications of the Homma–Schoenflies theorem.

(1) For any path $\gamma :[0,1]\longrightarrow \mathbb R^2$ positively transverse to a foliation ${\mathcal {F}}$ , there exists an orientation-preserving homeomorphism $\varphi :\mathbb R^2\longrightarrow \mathbb R^2$ that satisfies the following properties:

• • the path $\gamma $ is mapped by $\varphi $ into the horizontal arc $[0,1]\times \{0\}$ ;

• • for $i\in \{0,1\}$ , the leaf $\phi _{\gamma (i)}\in {\mathcal {F}}$ passing through $\gamma (i)$ is mapped by $\varphi $ into the vertical line $\{i\}\times \mathbb R$ oriented downwards;

• • there exists a closed set $V\subset \mathbb R^2$ containing $\gamma $ that is mapped by $\varphi $ into the square $[0,1]\times [ -1,1]$ such that, for any leaf $\phi \in {\mathcal {F}}$ intersecting V, the set $\phi \cap V$ is mapped by $\varphi $ into a vertical arc of the form $\{t\}\times [ -1,1]$ for some $t\in [0,1]$ .

(2) For any topological line $\Gamma :\mathbb R\longrightarrow \mathbb R^2$ positively transverse to ${\mathcal {F}}$ , there exists an orientation-preserving homeomorphism $\varphi :\mathbb R^2\longrightarrow \mathbb R^2$ that satisfies the following properties:

• • the topological line $\Gamma $ is mapped by $\varphi $ into the horizontal line $\mathbb R\times \{0\}$ ;

• • there exists a closed set $V\subset \mathbb R^2$ containing $\Gamma $ that is mapped by $\varphi $ into the horizontal strip $\mathbb R\times [ -1,1]$ and such that, for any leaf $\phi \in {\mathcal {F}}$ intersecting V, the set $\phi \cap V$ is mapped by $\varphi $ into a vertical arc of the form $\{t\}\times [ -1,1]$ oriented downwards.

2.3 Foliated Brouwer theory

Let $ f:\mathbb {R}^2 \longrightarrow \mathbb {R}^2 $ be a Brouwer homeomorphism, that is, an orientation-preserving and fixed-point free homeomorphism of the plane.

The orbit of a point $ x \in \mathbb {R}^2 $ under $ f $ is defined as the set

$$ \begin{align*} \mathcal{O}(f, x) = \{f^n(x) \mid n \in \mathbb{Z}\} \subset \mathbb{R}^2.\end{align*} $$

Sometimes, it may be useful to denote the set of orbits of $ f $ by $\mathrm {Orb}(f)$ . As proved in [Reference BrouwerBro12], every orbit of f is wandering and, thus, it has an empty limit set. This means that every orbit of f is the image of a proper embedding of $\mathbb {Z}$ into the plane.

A Brouwer line for $ f $ is an oriented topological line $ \unicode{x3bb} $ on $ \mathbb {R}^2 $ that satisfies

$$ \begin{align*} f(L(\unicode{x3bb})) \subset L(\unicode{x3bb}) \quad \text{and} \quad f^{-1}(R(\unicode{x3bb})) \subset R(\unicode{x3bb}),\end{align*} $$

where $ L(\unicode{x3bb} ) $ and $ R(\unicode{x3bb} ) $ denote the left and right sides of $ \unicode{x3bb} $ . The classical Brouwer translation theorem [Reference BrouwerBro12] asserts that through every point of the plane passes a Brouwer line for $ f $ .

Any foliation of the plane by Brouwer lines of $ f $ is called a transverse foliation of $ f $ . It is worth noting that a Brouwer homeomorphism admits several transverse foliations, which may exhibit different structures and non-homeomorphic leaf spaces, for instance.

Let us fix a transverse foliation ${\mathcal {F}}$ of a Brouwer homeomorphism $ f $ . In this context, one can show that every point $ x \in \mathbb {R}^2 $ may be connected to its image $ f(x) $ by a path $\gamma _x$ that is positively transverse to the foliation ${\mathcal {F}}$ . By concatenating the paths $\gamma _{f^n(x)}$ for all $n\in \mathbb Z$ , we obtain

$$ \begin{align*} \Gamma := \prod_{n\in \mathbb Z} \gamma_{f^n(x)},\end{align*} $$

which is a positively transverse path connecting the entire orbit $\mathcal {O}(f,x)$ , called a transverse trajectory of ${O}(f,x)$ (with respect to the foliation ${\mathcal {F}}$ ). Since ${\mathcal {F}}$ is fixed, we often omit its mention when referring to transverse trajectories. Note that each orbit ${O}(f,x)$ admits uncountably many transverse trajectories, depending on the choice of $\{\gamma _{f^n(x)}\}_{n \in \mathbb Z}$ .

Independently of the choice of transverse paths $\{\gamma _{f^n(x)}\}_{n \in \mathbb Z}$ , the leaves that are intersected by the transverse trajectory $\Gamma _{O}$ of ${O}={O}(f,x)$ are determined by the orbit itself. In fact, the set $C_{O} \subset {\mathcal {F}}$ of leaves crossed by the orbit ${O}$ can be equivalently defined as

$$ \begin{align*} C_{O} = \{\phi \in {\mathcal{F}} \mid \phi \cap \Gamma_{O} \neq \varnothing\} = \{\phi \in {\mathcal{F}} \mid {O} \text{ intersects } R(\phi) \text{ and } L(\phi)\}.\end{align*} $$

We remark that, if the choice of paths $\{\gamma _{f^n(x)}\}_{n \in \mathbb Z}$ is locally finite on the plane, then the transverse trajectory $\Gamma _{O}$ is a topological line and it is called a proper transverse trajectory of the orbit ${O}$ . The converse is also true.

Let ${\mathcal {O}}=\{{O}_1, \ldots ,{O}_r\}$ be a finite collection of orbits of f. Throughout this text, we may treat ${\mathcal {O}}$ as a subset of $\mathbb R^2$ by identifying it with ${\mathcal {O}} = \bigcup _{i=1}^{ r} {O}_i$ . A transverse foliation ${\mathcal {F}}$ of $ f $ is said to be generic for the collection ${\mathcal {O}}$ if each leaf $\phi \in {\mathcal {F}}$ intersects at most one orbit in ${\mathcal {O}}$ . This condition is indeed generic, as we explain in the following.

Assume that a leaf $\phi \in {\mathcal {F}}$ intersects two orbits in ${\mathcal {O}}$ , at points $x,x^{\prime }\in \phi $ . Let B be a closed ball centered at x that is sufficiently small so that $B \cap {\mathcal {O}} = \{x\}$ and $f(B) \cap B = \varnothing $ . We can perturb ${\mathcal {F}}$ inside the ball B in such a way that x no longer lies on the leaf passing by $x^{\prime }$ . Since B satisfies $f(B) \cap B = \varnothing $ , the leaves of ${\mathcal {F}}$ continue to be Brouwer lines of $ f $ after the perturbation. Now, since ${\mathcal {O}}$ is a finite collection of orbits of f and each orbit of f is a closed discrete subset of $\mathbb R^2$ , we have that for any compact set $K\subset \mathbb R^2$ , the set $K\cap {\mathcal {O}}$ is finite. Therefore, by repeating the perturbation process described above on a locally finite family of disjoint balls covering ${\mathcal {O}}$ , we can obtain a transverse foliation ${\mathcal {F}}'$ of $ f $ that is generic for ${\mathcal {O}}$ .

3.1 Limitation of the framework

Let $x \in \mathbb R^2$ be a point in the plane and let $O=O(f,x)$ . For each $n\in \mathbb Z$ , let $\phi _n \in {\mathcal {F}}$ be the leaf ${\mathcal {F}}$ containing the point $f^n(x)$ . Note that $(\phi _n)_{n\in \mathbb Z}$ is a bi-infinite sequence of leaves in $C_{O}$ that is increasing with respect to the natural order of the foliation ${\mathcal {F}}$ , meaning, $L(\phi _{n+1})\subset L(\phi _n)$ for all ${n\in \mathbb Z}$ . Although ${O}$ must escape to infinity, the sequence $(\phi _n)_{n\in \mathbb Z}$ may accumulate at certain leaves of ${\mathcal {F}}$ . Since $(\phi _n)_{n\in \mathbb Z}$ is increasing, there can be two types of leaves accumulated by the sequence $(\phi _n)_{n\in \mathbb Z}$ . Either a leaf $\phi _L\in {\mathcal {F}}$ , which is a limit leaf of the forward sequence $(\phi _n)_{n\geq 0}$ , or a leaf $\phi _R \in {\mathcal {F}}$ , which is a limit leaf of the backward sequence $(\phi _{n})_{n\leq 0}$ . It is worth noting, since ${\mathcal {F}}$ is not necessarily Hausdorff, each of these sequences may admit multiple limit leaves.

A priori, each point $f^n(x) \in {O}$ can be located anywhere along its underlying leaf $\phi _n$ . Hence, in case where the sequence $(\phi_n)_{n\geq0}$ converges to a leaf $\phi _L$ , the orbit ${O}$ could possibly exhibit a weird asymptotic behavior with respect to $\phi _L$ (see Figure 7) which impedes the existence of properly embedded transverse trajectories of ${O}$ . Such behavior would pose a significant limitation to the control of topological properties of transverse trajectories.

Figure 7

Weird asymptotic behavior that would make the existence of proper transverse trajectories impossible.

Figure 7 Long description

The diagram illustrates a mathematical vector field using grey lines with downward-pointing arrows.

On the left side, the grey flow lines are nearly vertical. A dark blue line, labeled with a capital O, moves horizontally across these lines in a jagged, oscillating pattern marked by blue circular nodes at each peak and trough.

In the center, the grey flow lines begin to curve around a parabolic region. The blue trajectory, now labeled Gamma sub O, becomes more compressed and follows the curvature of the flow.

On the right side, a smooth red curve labeled phi sub L forms a U-shape opening to the right. The blue trajectory lines become extremely dense and tightly packed as they approach the exterior of this red curve, demonstrating asymptotic behavior where the paths converge but do not cross the red boundary. The grey flow lines in this region follow the outer contour of the red and blue structures.

In this section, we overcome this limitation by proving that every orbit of f indeed admits proper transverse trajectories (see Theorem A). Moreover, we show that a proper transverse trajectory $\Gamma _{O}$ associated to an orbit ${O}$ encodes additional information about the asymptotic behavior of ${O}$ that is not captured by $C_{O}$ alone, but rather by $C_{O}$ and four distinguished subsets of $\partial C_{O}$ , namely $\partial _L^{\mathrm {top}} {O}$ , $\partial _L^{\mathrm {bot}} {O}$ , $\partial _R^{\mathrm {top}} {O}$ , and $\partial _R^{\mathrm {bot}} {O}$ (see Figure 8).

Figure 8

Description of the information encoded by proper transverse trajectories.

Figure 8 Long description

A vector field diagram with numerous thin grey streamlines flowing horizontally. On the left side, there are two blue U-shaped boundaries. The top blue boundary is labeled partial sub R super top O. The bottom blue boundary is labeled partial sub R super bot O. On the right side, there is a single red U-shaped boundary labeled partial sub L super bot O. A thick black line with a central arrow, labeled Gamma sub O, originates from the left between the two blue boundaries and curves upward to pass between the top blue boundary and the red boundary. The grey streamlines follow a similar flow, curving around the boundaries and converging through the narrow channels between the blue and red regions.

3.2 Crossed leaves and limit leaves

For each orbit ${O} \in \mathrm {Orb}(f)$ , we define the set

$$ \begin{align*} C_{O}:= C({\mathcal{F}}, {O}) = \{\phi \in {\mathcal{F}} \mid \phi \cap \Gamma_{O} \neq \varnothing\},\end{align*} $$

where $\Gamma _{O}$ represents any transverse trajectory of ${O}$ . By abuse of notation, we treat $C_{O}$ as a saturated subset of $\mathbb R^2$ (see §2.2). According to Lemmas 2.7 and 2.4, the saturated set $C_{O} \subset \mathbb R^2$ is a leaf domain of ${\mathcal {F}}$ and, consequently, it is homeomorphic to a trivially foliated plane, and its boundary $\partial C_{O}$ is a locally finite and non-separating collection of leaves in ${\mathcal {F}}$ . The set $C_{O}$ is often referred to as the leaf domain of the orbit ${O}$ leaves in $C_{O}$ are said to be crossed by the orbit ${O}$ .

Since the set $C_{O}$ is a leaf domain, its boundary $\partial C_{O}$ admits two distinguished subsets, namely the left-limit and right-limit leaves of $C_{O}$ , which are defined as follows:

$$ \begin{align*} \partial_L C_{O} := \partial C_{O} \cap \biggl(\ \bigcap_{\phi \in C_{O}} L(\phi)\biggr) \quad \text{and} \quad \partial_R C_{O} := \partial C_{O} \cap \biggl(\ \bigcap_{\phi \in C_{O}} R(\phi)\biggr).\end{align*} $$

By fixing a basepoint $x \in O$ and considering the bi-infinite sequence $(\phi _n)_{n\in \mathbb Z}$ formed by the leaves $\phi _n \in {\mathcal {F}}$ passing through the points $f^n(x)$ , we note that the forward sequence $(\phi _n)_{n\geq 0}$ converges simultaneously to all leaves in $\partial _L C_{O}$ (with respect to the quotient topology on ${\mathcal {F}}$ ), while the backward sequence $(\phi _n)_{n\leq 0}$ converges to all leaves in $\partial _R C_{O}$ . This can be deduced from the fact that, for any leaf $\phi \in C_{O}$ , there exists $n \geq 0$ such that $L(\phi_{-n}) \subset L(\phi) \subset L(\phi_n)$ . Observe that the sets $\partial _L C_{O}$ and $\partial _R C_{O}$ are disjoint by definition, and their union may not cover the entire boundary set $\partial C_{O}$ (see Figure 1).

3.3 Definition of order $\prec $ on the sets $\partial _L C_{O}$ and $\partial _R C_{O}$

Fix ${O} \in \mathrm {Orb}(f)$ and a basepoint $x \in {O}$ . Since $\partial C_{O}$ is a locally finite collection of leaves disjoint from the closed set $\phi _0 \cup {O}$ , we can apply Lemma 2.2 to affirm that there exists a locally finite family $\{V_\phi \}_{\phi \in \partial C_{O}}$ of pairwise disjoint closed sets such that, for each $\phi \in \partial C_{O}$ , the set $V_\phi $ is a trivial neighborhood of $\phi $ disjoint from $\phi _0 \cup {O}$ .

For each $n\in \mathbb Z$ , let $\phi _n \in {\mathcal {F}}$ be the leaf passing through the point $f^n(x)$ and define

$$ \begin{align*} J_n := \{\phi \in \partial_L C_{O} \cup \partial_R C_{O} \mid V_\phi \cap \phi_n \neq \varnothing\}\end{align*} $$

(see Figure 12).

Figure 12

Definition of the sets $J_n$ .

Figure 12 Long description

A mathematical diagram illustrating flow lines and regions. On the left, vertical wavy lines represent a vector field. On the right, three horizontal, finger-like lobes extend toward the left.

Top Lobe: Outlined in blue and shaded light blue. A black arrowed line follows the outer curve, labeled V sub phi. Inside the lobe, a red arrowed line loops around the center. The text inside the lobe reads phi is an element of J sub n in red.

Middle Lobe: Outlined in blue and shaded light blue. A black arrowed line follows the outer curve, labeled V sub phi double prime. Inside the lobe, a gray arrowed line loops around the center. The text inside the lobe reads phi double prime is not an element of J sub n in gray.

Bottom Lobe: Outlined in blue and shaded light blue. A black arrowed line follows the outer curve, labeled V sub phi prime. Inside the lobe, a red arrowed line loops around the center. The text inside the lobe reads phi prime is an element of J sub n in red.

A single black line labeled phi sub n at the top center traces the boundary between the vertical flow lines and the tips of the three lobes.

Observe that, by construction, $J_n \subset \partial _L C_{O}$ when $n>0$ and $J_n \subset \partial _R C_{O}$ when $n<0$ .

Proof. We prove the lemma for the sequence $(J_n)_{n\geq 0}$ ; the proof for $(J_n)_{n\leq 0}$ is analogous.

Since the leaf $\phi _0$ is disjoint from each set in the family $\{V_\phi \}_{\phi \in \partial _L C_{O}}$ , it holds that $J_0 = \varnothing $ . Next, as the sequence of leaves $(\phi _n)_{n\geq 0}$ is increasing in ${\mathcal {F}}$ , if some leaf $\phi _n$ intersects $V_\phi $ for a given integer $n \geq 0$ , then the leaf $\phi _{n+1}$ also intersects $V_\phi $ . In other words, it holds that

$$ \begin{align*} J_n \subset J_{n+1} \quad \text{for all } n\geq0. \end{align*} $$

Finally, note that for each leaf $\phi \in C_{O}$ , there exists some $n>0$ such that $L(\phi _n) \subset L(\phi )$ . Therefore, for each $\phi \in \partial _L C_{O}$ , there exists the smallest positive integer $N_\phi>0$ satisfying

$$ \begin{align*}\phi_{N_\phi} \cap V_\phi \neq \varnothing.\end{align*} $$

This implies that $\partial _L C_{O} = \bigcup _{n\geq 0} J^{n}$ and, thus, we conclude the proof of the lemma.

First, note that for each $n \in \mathbb Z$ and $\phi \in J_n$ , the set $\alpha _n(\phi ):=\phi _n \cap V_\phi $ is a non-empty compact sub-arc of the leaf $\phi _n$ . This gives us, for each $n \in \mathbb Z$ , a family $A_n:=\{\alpha _n(\phi )\}_{\phi \in J_n}$ of pairwise disjoint sub-arcs of the leaf $\phi _n$ . Next, note that the orientation of $\phi _n$ induces a natural total order on the set $A_n$ , which is defined as follows. For any two arcs $\alpha ,\alpha '\in A_n$ , we write $\alpha \prec _n \alpha '$ if, along the oriented leaf $\phi _n$ , the arc $\alpha '$ is positioned after $\alpha $ . Through the bijection $\phi \longmapsto \alpha _n(\phi )$ , we import this order to $J_n$ . Given two leaves $\phi ,\phi '\in J_n$ , we say that

$$ \begin{align*} \phi \prec_n \phi' \quad \text{if } \alpha_n(\phi) \prec_n \alpha_n(\phi') \end{align*} $$

(see Figure 13).

Figure 13

Definition of the order $\prec_n$ on the set $J_n$ .

Figure 13 Long description

On the far left, the text alpha sub n (phi) followed by a curly less-than symbol with a subscript n and alpha sub n (phi prime) establishes the ordering rule. To the right, a series of gray contour lines flow vertically and curve around three horizontal, finger-like structures.

* The top structure is shaded light blue and outlined in dark blue. It contains a black U-shaped arrow pointing left then right. Inside the white center of this structure is the label phi is an element of J sub n. A red label alpha sub n (phi) is positioned at the tip of this structure, associated with a red curved arrow.

* The middle structure is similar but lacks specific labels, showing only the black U-shaped directional arrow and gray contour lines.

* The bottom structure is also shaded light blue and contains the label phi prime is an element of J sub n. A red label alpha sub n (phi prime) is at its tip with a corresponding red arrow.

A black line labeled phi sub n at the top right traces the boundary of these structures, separating the blue-shaded regions from the surrounding gray field.

Observe that $\prec _n$ and $\prec _{n+1}$ are compatible with the inclusion $J_n \subset J_{n+1}$ for any $n\geq 0$ . This means that, if two leaves $\phi ,\phi '\in J_n$ satisfy $\phi \prec _n \phi '$ , then they also satisfy $\phi \prec _{n+1} \phi '$ . This ensures that the set $ \partial _L C_{O} = \bigcup _{n>0} J_{n} $ inherits a total order $\prec $ that extends the sets in the filtration $(J_n)_{n\geq 0}$ , in the sense that, for any $\phi ,\phi '\in \partial _L C_{O}$ , we define the relation

$$ \begin{align*} \phi \prec \phi'\quad \text{if there exists } n>0 \text{ such that } \phi \prec_n \phi'. \end{align*} $$

Note that the set $\partial _R C_{O}$ inherits an analogous total order $\prec $ from the filtration $(J_n)_{n\leq 0}$ .

3.4 Orbit induced cuts

Let $\text {Cut}_\prec (\partial _L C_{O})$ be the set of cuts of $(\partial _L C_{O},\prec )$ , that is, the set of all pairs $(A,B)$ such that $A, B \subset \partial _L C_{O}$ are disjoint subsets of $\partial _L C_{O}$ satisfying:

• • $A \cup B = \partial _L C_{O}$ ;

• • for any leaf $\phi \in A$ and any leaf $\phi ' \in B$ , it holds that $\phi \prec \phi '$ .

Similarly, let $\text {Cut}_\prec (\partial _R C_{O})$ denote the set of all cuts of the totally ordered set $(\partial _R C_{O},\prec )$ .

Proposition 3.4. Every orbit ${O} \in \mathrm {Orb}(f)$ admits two canonically defined cuts

$$ \begin{align*} (\partial_L^{\mathrm{top}} {O},\partial_L^{\mathrm{bot}} {O}) \in \text{Cut}_\prec(\partial_L C_{O}) \quad \text{and} \quad ( \partial_R^{\mathrm{top}} {O},\partial_R^{\mathrm{bot}} {O}) \in \text{Cut}_\prec(\partial_R C_{O}). \end{align*} $$

Figure 14

The cuts $(\partial_L^{\mathrm{top}} O,\partial_L^{\mathrm{bot}} O)$ and $(\partial_R^{\mathrm{top}} O,\partial_R^{\mathrm{bot}} O)$ canonically associated to the orbit $O$ .

Figure 14 Long description

A line diagram depicting a mathematical vector field with various flow lines and a central orbit.

* On the left side, three blue U-shaped curves with arrows pointing inward are stacked vertically. They are labeled from top to bottom as partial R super top O, partial R super top O, and partial R super bot O.

* In the center, a series of light gray vertical and wavy flow lines run from top to bottom. A diagonal path of black dots, labeled O, crosses from the lower left to the upper right. A gray label C sub O is positioned to the right of the central vertical axis.

* On the right side, three red U-shaped curves with arrows pointing outward are stacked vertically. They are labeled from top to bottom as partial L super top O, partial L super bot O, and partial L super bot O.

* The background consists of thin, light gray contour lines that follow the general flow of the vector field, curving around the blue and red boundary structures.

It is worth remarking that the sets $\partial _L^{\mathrm {top}} {O}$ , $\partial _L^{\mathrm {bot}} {O}$ , $\partial _R^{\mathrm {top}} {O}$ , and $\partial _R^{\mathrm {bot}} {O}$ depend not only on ${O}$ , but also on the foliation ${\mathcal {F}}$ . However, we often omit the dependence on ${\mathcal {F}}$ in the notation.

The leaf domain $C_{O}$ and the cuts $(\partial _L^{\mathrm {top}} {O},\partial _L^{\mathrm {bot}} {O})$ and $(\partial _R^{\mathrm {top}} {O},\partial _R^{\mathrm {bot}} {O})$ describe the asymptotic behavior of the orbit ${O}$ with respect to ${\mathcal {F}}$ (Figure 14). The qualitative data associated with these objects are strictly greater than those given by the set $C_{O}$ alone. Throughout this work, we refer to the qualitative data as the ${\mathcal {F}}$ -asymptotic behavior of ${O}$ .

Later, in §3.5, we show that the cuts $(\partial _L^{\mathrm {top}} {O},\partial _L^{\mathrm {bot}} {O})$ and $(\partial _R^{\mathrm {top}} {O},\partial _R^{\mathrm {bot}} {O})$ can be interpreted by proper transverse trajectories of ${O}$ (see Theorem A). These proper transverse trajectories completely describe the ${\mathcal {F}}$ -asymptotic behavior of the orbit ${O}$ , differently from arbitrary (non-proper) trajectories, which only encode the information carried by $C_{O}$ .

Proof of Proposition 3.4.

Fix an orbit ${O} \in \mathrm {Orb}(f)$ , a basepoint $x \in {O}$ , a family $\{V_\phi \}_{\phi \in \partial C_{O}}$ of trivial-neighborhoods, the leaves $(\phi _n)_{n\in \mathbb Z}$ , and the sets $(J_n)_{n\in \mathbb Z}$ as defined in §3.3.

Observe that, by cutting each leaf $\phi _n$ at the point $f^n(x)$ , we obtain two half-lines

$$ \begin{align*} \phi^+_n:[0,\infty)\longrightarrow\mathbb R^2 \quad \text{and} \quad \phi^-_n:(-\infty,0]\longrightarrow\mathbb R^2, \end{align*} $$

with endpoints $\phi ^+_n(0) = \phi ^-_n(0) = f^n(x)$ and satisfying $\phi ^+_n, \phi ^-_n \subset \phi _n$ . Recall that each set in the family $\{V_\phi \}_{\phi \in \partial C_{O}}$ is disjoint from the orbit ${O}$ . Consequently, for each $n \in \mathbb Z$ and $\phi \in J^n$ , the arc $\alpha _n(\phi ) := \phi _n \cap V_\phi $ must lie entirely within one of the two half-lines, either $\phi _n^+$ or $\phi _n^-$ . This allows us to define, for any $n \in \mathbb Z$ , a cut $(J_n^-,J_n^+) \in \text {Cut}_\prec (J_n)$ given by the sets

$$ \begin{align*} J_n^-:=\{\phi \in J_n \mid \alpha_n(\phi) \subset \phi_n^-\}\quad \text{and} \quad J_n^+:=\{\phi \in J_n \mid \alpha_n(\phi) \subset \phi_n^+\}. \end{align*} $$

Claim 3.5. For each leaf $\phi \in \partial _L C_{O}$ , there exists $M_\phi>0$ which is the smallest positive integer such that, for any $n \geq M_\phi $ , the following two equivalences hold

$$ \begin{align*} \phi \in J_n^+ \iff \phi \in J_{n+1}^+, \\ \phi \in J_n^- \iff \phi \in J_{n+1}^-. \end{align*} $$

Similarly, for each leaf $\phi \in \partial _R C_{O}$ , there exists $M_\phi <0$ satisfying an analogous property.

Proof of Claim 3.5.

Fix a leaf $\phi \in \partial _L C_{O}$ and denote by $\phi _R \in C_{O}$ the right-most leaf that intersects the trivial-neighborhood $V_\phi $ . The open set $L(\phi _R) \cap R(\phi )\setminus V_\phi $ has exactly two connected components, named $U^+$ and $U^-$ according to the following equivalences:

$$ \begin{align*} \phi\in J_n^+ \iff f^n(x) \in U^+;\\ \phi\in J_n^- \iff f^n(x) \in U^-. \end{align*} $$

To simplify the argument, we can profit from Homma–Schoenflies theorem to suppose, up to conjugacy, that the topological lines $\phi _R$ , $\phi $ , and $f(\phi )$ are verticals on the plane oriented downwards; the set $U^-$ lies on the upper half-plane; the set $U^+$ lies on the lower half-plane; and that f maps the points of $\phi $ into $f(\phi )$ as a horizontal (see Figure 15). We represent $V_\phi $ disjoint from $f(\phi )$ because $V_\phi $ is considered small enough; however, our proof does not depend on this property and it is possible that $V_\phi $ intersects $f(\phi )$ .

Figure 15

Setting in the proof of Claim 3.5 after applying the Homma–Schoenflies theorem.

Now, we suppose by contradiction that the claim of the lemma does not hold. More precisely, we suppose that there exists an increasing sequence of integers $(n_k)_{k \in \mathbb N}$ satisfying

$$ \begin{align*} \phi \in J_{n_k}^- \bigcap J_{n_k + 1}^+. \end{align*} $$

According to the equivalence that defines $U^-$ and $U^+$ , for every $k \in \mathbb N$ , it holds that

$$ \begin{align*} z_k:=f^{n_k}(x) \in U^+ \quad \text{and} \quad f(z_k) \in U^-.\end{align*} $$

Since f is a Brouwer homeomorphism, $(z_k)_{k\in \mathbb N}$ has no accumulation points on the plane. As the set $U^+$ is bounded in the horizontal coordinate and contained in the lower half-plane, the sequence $(z_k)_{k\in \mathbb N}$ is unbounded from below on the vertical coordinate. Up to taking a subsequence, we may suppose that $(z_k)_{k\in \mathbb N}$ is strictly decreasing on the vertical coordinate. Consequently, the family $\{\alpha _k\}_{k \in \mathbb N}$ of horizontal arcs $\alpha _k$ that join $z_k$ to a point $y_k \in \phi $ is, by construction, locally finite (see Figure 16).

Figure 16

Illustration of the family $\{\alpha_k\}_{k \in \mathbb{N}}$ of horizontal arcs in the proof of Claim 3.5.

Let $K\subset \mathbb R^2$ be the compact arc of zero vertical coordinate connecting $\phi _R$ and $f(\phi )$ . From the hypothesis given by the Homma–Schoenflies theorem, for any $k \in \mathbb N$ , we have that the point $y_k \in \phi $ and its image $f(y_k) \in f(\phi )$ have exactly the same horizontal coordinate. Consequently, the sequence $(f(y_k))_{k \in \mathbb N}$ lies on the lower half-plane, while $(f(z_k))_{k \in \mathbb N}$ lies on the upper half-plane. Since the arcs in the family $\{\alpha _k\}_{k\in \mathbb N}$ are contained in the set $\overline {L(\phi _R) \cap R(\phi )}$ , their images $\{f(\alpha _k)\}_{k \in \mathbb N}$ are contained in $\overline {L(\phi _R) \cap R(f(\phi ))}$ . Note that, since $f(\alpha _k)$ joins the point $f(y_k)$ to $f(z_k)$ , which are respectively in the lower and upper half-planes, each arc in the family $\{f(\alpha _k)\}_{k \in \mathbb N}$ must intersect the compact set K. In other words, $\{f(\alpha _k)\}_{k \in \mathbb N}$ is not a locally finite family of arcs (see Figure 17).

Figure 17

Illustration of the family $\{f(\alpha_k)\}_{k \in \mathbb{N}}$ of horizontal arcs in the proof of Claim 3.5.

Figure 17 Long description

The diagram is composed of three vertical lines intersected by a central horizontal line.

At the top left, a series of blue dots curves upward, labeled f (z sub k) sub k. A solid blue curve, labeled f (alpha sub k), originates from one of these dots, crosses the central vertical line, and descends to meet the rightmost vertical line.

A light gray shaded region shaped like a stepped cross is centered on the intersection of the middle vertical and horizontal lines, labeled with a capital K.

The horizontal axis is labeled phi sub R on the left and f (phi) on the right.

In the lower-left quadrant, a family of nine thin red horizontal arcs extends from the leftmost vertical line toward the center. Each arc terminates in a small red dot, with the lengths of the arcs decreasing as they move lower.

On the rightmost vertical line, a series of black dots corresponds to the vertical positions of the red arcs and the blue curve, labeled at the bottom as (f (y sub k)) sub k.

However, since f is a homeomorphism and $\{\alpha _k\}_{k \in \mathbb N}$ is locally finite, $\{f(\alpha _k)\}_{k \in \mathbb N}$ should be, in fact, locally finite. This gives us a contradiction.

Due to Claim 3.5, we can now define the following sets:

$$ \begin{align*} \partial_L^{\mathrm{top}} {O} := \{\phi \in \partial_L C_{O} \mid \text{ there exists } M>0 \text{ for all } n>M, \phi \in J_n^+ \};\\ \partial_L^{\mathrm{bot}} {O}:=\{\phi \in \partial_L C_{O} \mid \text{ there exists } M>0 \text{ for all } n>M, \phi \in J_n^- \}. \end{align*} $$

By construction, we have that $(\partial _L^{\mathrm {top}} {O},\partial _L^{\mathrm {bot}} {O}) \in \text {Cut}_\prec (\partial _L C_{O})$ . We can define the other two sets $\partial _R^{\mathrm {top}} {O}$ and $\partial _R^{\mathrm {bot}} {O}$ in a similar way, which thus concludes the proof of the proposition.

3.5 Proper transverse trajectories

We now prove Theorem A, which we restate.

Theorem A. Every orbit ${O} \in \mathrm {Orb}(f)$ admits a proper transverse trajectory. Moreover, any proper transverse trajectory $\Gamma _{O}$ associated to the orbit ${O}$ satisfies the equalities

$$ \begin{align*} \partial_L^{\mathrm{top}} {O} = \partial_L C_{O} \cap L(\Gamma_{O}), \quad \partial_R^{\mathrm{top}} {O} = \partial_R C_{O} \cap L(\Gamma_{O}), \\[0.1ex]\partial_L^{\mathrm{bot}} {O} = \partial_L C_{O} \cap R(\Gamma_{O}), \quad \partial_R^{\mathrm{bot}} {O} = \partial_R C_{O} \cap R(\Gamma_{O}) \end{align*} $$

(see Figure 18).

Figure 18

Depiction of how a proper transverse trajectory $\Gamma_O$ of an orbit $O$ recovers the cannonically associated cuts $(\partial_L^{\mathrm{top}} O, \partial_L^{\mathrm{bot}} O)$ and $(\partial_R^{\mathrm{top}} O,\partial_R^{\mathrm{bot}} O)$ , as stated in Theorem A.

Figure 18 Long description

The diagram features a central black curve labeled Gamma sub O with an arrow pointing right, representing a transverse trajectory. This curve passes through a dense field of light gray flow lines that curve vertically in the center and horizontally toward the edges.

On the left side, three blue U-shaped curves represent cuts. They are labeled from top to bottom as:

* partial R super top O

* partial R super top O

* partial R super bot O

On the right side, three red U-shaped curves represent corresponding cuts. They are labeled from top to bottom as:

* partial L super top O

* partial L super bot O

* partial L super bot O

A gray label C sub O is positioned in the lower-right quadrant among the gray flow lines. The flow lines on the far left and far right have small arrows indicating a direction toward the center along the horizontal axis.

Proof. Consider an orbit ${O} \in \mathrm {Orb}(f)$ , a basepoint $x \in {O}$ , a family $\{V_\phi \}_{\phi \in \partial C_{O}}$ of trivial neighborhoods, the leaves $(\phi _n)_{n\in \mathbb Z}$ , and the sets $(J_n)_{n\in \mathbb Z}$ as defined in §3.3. Recall that in the proof of Lemma 3.2 and Proposition 3.4, we defined, for each leaf $\phi \in \partial _L C_{O}$ , two integers $N_\phi>0$ and $M_\phi>0$ , which are the smallest positive integers satisfying

$$ \begin{align*} \phi_{n} \cap V_\phi \neq \varnothing \quad \text{for all } n\geq N_\phi, \\ \phi \in J_n^+ \iff \phi \in J_{n+1}^+ \quad \text{for all } n\geq M_\phi. \end{align*} $$

Observe that, for $n \geq M_\phi $ , the following two scenarios cannot occur (see Figure 19):

$$ \begin{align*} \text{ (a) } \alpha_n(\phi) \subset \phi_n^+ \text{ and } \alpha_{n+1}(\phi) \subset \phi_{n+1}^-;\quad \text{ (b) } \alpha_n(\phi) \subset \phi_n^- \text{ and } \alpha_{n+1}(\phi) \subset \phi_{n+1}^+. \end{align*} $$

This means that, for each $n\geq M_\phi $ , there exists a path $\gamma _{n,\phi }:[0,1]\longrightarrow \mathbb R^2$ positively transverse to ${\mathcal {F}}$ that joins $f^n(x)$ to $f^{n+1}(x)$ and is disjoint from the neighborhood $V_\phi $ .

Figure 19

Illustration of what cannot occur (on the left) and what actually occurs (on the right).

Figure 19 Long description

The diagram consists of two panels side-by-side.

Left Panel: Two vertical black lines are labeled phi sub n on the left and phi sub n plus 1 on the right. A red dot on the left line is labeled f super n (x). A dashed red curve labeled gamma sub n descends from this dot, passing through a central gray shaded region labeled V sub phi, and ends at a red dot on the right line labeled f super n plus 1 (x). Below the panel is the expression phi is an element of J sub n super minus intersection J sub n plus 1 super plus.

Right Panel: Similar vertical lines phi sub n and phi sub n plus 1. A red dot on the left line is labeled f super n (x). A solid red curve labeled gamma sub n with an arrow pointing right connects this dot to a red dot on the right line labeled f super n plus 1 (x). This path stays above the central gray shaded region V sub phi. Below the panel is the expression phi is an element of J sub n super minus intersection J sub n plus 1 super minus.

The first step of the proof is to show that, by modifying the trivial neighborhoods in the family $\{V_\phi \}_{\phi \in \partial C_{O}}$ , we can ensure that $N_\phi = M_\phi $ for all $\phi \in \partial C_{O}$ . Indeed, to show this, one should simply observe that each leaf $\phi \in \partial _L C_{O}$ also admits the trivial neighborhood

(3.1) $$ \begin{align} V^{\prime}_\phi:= V_\phi \cap \overline{L(\phi^{M_\phi})}. \end{align} $$

Thus, by redefining the original $V_\phi $ to be instead the newly defined trivial neighborhood $V^{\prime }_\phi $ , we automatically get the property that $N_\phi = M_\phi $ . Therefore, we conclude that after this redefinition, we know that for each $n\geq 0$ and $\phi \in \partial _L C_{O}$ , the points $f^n(x)$ and $f^{n+1}(x)$ can be connected by a path $\gamma _{n,\phi }$ positively transverse to ${\mathcal {F}}$ that is disjoint from $V_\phi $ .

Now, the second step of the proof is to show that, for each $n\geq 0$ , we can always construct the paths $\gamma _{n,\phi }$ so that they are the same among all leaves $\phi \in \partial _L C_{O}$ .

For every $n \geq 0$ , we define a partition of the set $A^n=\{\alpha _n(\phi )\}_{\phi \in J_n}$ given by

$$ \begin{align*} A_n^+ : = \{\alpha_n(\phi) \in A^n \mid \phi \in J_n^+\} \quad \text{and} \quad A_n^- : = \{\alpha_n(\phi) \in A^n \mid \phi \in J_n^-\}. \end{align*} $$

The set $A^n$ is ordered according to the orientation of the leaf $\phi _n$ . Observe that any element in $A_n^+$ is greater than any element in $A_n^-$ . Now, since the family $\{V_\phi \}_{\phi \in \partial _L C_{O}}$ is locally finite, any compact sub-arc of $\phi _n$ intersects finitely many arcs of $A^n$ . Consequently, if $A_n^+\neq \varnothing $ , then there exists a minimal element $\alpha _n^{{\min }}$ in $A_n^+$ . Similarly, if $ A_n^-\neq \varnothing $ , then there exists a maximal element $\alpha _n^{\mathrm {max}}$ in $A_n^-$ . In the case where $A_n^+ = \varnothing $ or $A_n^- = \varnothing $ , such arcs are considered empty. Note that the connected component of $\phi _n\setminus (\alpha _n^{\mathrm {max}} \cup \alpha _n^{\mathrm {min}})$ that contains $f^n(x)$ intersects no trivial neighborhood in $\{V_\phi \}_{\phi \in J^n}$ . For each $n \geq 0$ , let $V_n^{\mathrm {max}}$ and $V_n^{\mathrm {min}}$ be the trivial neighborhoods in the family $\{V_\phi \}_{\phi \in J^n}$ containing $\alpha _n^{\mathrm {max}}$ and $\alpha _n^{\mathrm {min}}$ , respectively.

For each $n\geq 0$ , we define the set $ D_n := \overline {L(\phi _n) \cap R(\phi _{n+1})}.$ After the redefinition of the trivial neighborhoods as $V^{\prime }_\phi := V_\phi \cap \overline {L(\phi ^{M_\phi })}$ in (3.1), only two scenarios are possible:

1. (1) either $V_n^{\mathrm {max}} = V_{n+1}^{\mathrm {max}}$ ;

2. (2) or the intersection $V_{n+1}^{\mathrm {max}} \cap D_n$ is contained in $\phi _{n+1}$ .

These scenarios are respectively illustrated in Figure 20. Observe that, if the trivial neighborhood $V_{n+1}^{\mathrm {max}}$ is distinct from $V_n^{\mathrm {max}}$ , then it must stop at the leaf $\phi _{n+1}$ . This is also a consequence of the redefinition in (3.1), as it implies that $V_\phi \subset \overline {L(\phi ^{M_\phi })}$ .

Figure 20

Illustration of scenarios (1) and (2) after redefining $V'_\phi:= V_\phi \cap \overline {L(\phi ^{M_\phi })}$ .

We have completely analogous alternatives for the sets $V_n^{\mathrm {min}}$ for all $n \geq 0$ .

Either way, for each $n\geq 0$ , we have that the points $f^n(x)$ and $f^{n+1}(x)$ are contained in the same connected component, namely $D^*_n$ (see Figure 21), of the set

$$ \begin{align*}D_n\setminus(V_n^{\mathrm{max}} \cup V_n^{\mathrm{min}} ).\end{align*} $$

This further implies that, for each $n \geq 0$ , the points $f^n(x)$ and $f^{n+1}(x)$ can be connected via a path $\gamma _n:[0,1]\longrightarrow \mathbb R^2$ that is positively transverse to ${\mathcal {F}}$ and contained in $D^*_n$ . Therefore, since $\gamma _n \subset D^*_n$ , it follows that $\gamma _n$ is disjoint from every set in the family $\{V_\phi \}_{\phi \in \partial _L C_{O}}$ .

Figure 21

Illustration describing the properties of the set $D^*_n$ .

Figure 21 Long description

Two parallel vertical lines serve as the primary axes. The left axis is labeled phi sub n at the top and the right axis is labeled phi sub n plus 1 at the top.

Between these axes, three distinct regions are defined vertically.

1. The top region is a light gray shaded area labeled V sub n super max.

2. The middle region is a white area filled with thin vertical gray lines, labeled D sub n asterisk.

3. The bottom region is a light gray shaded area labeled V sub n super min.

A red curved path, labeled gamma sub n, originates from a red dot on the left axis labeled f super n (x). The path curves downward and to the right, ending at a red dot on the right axis labeled f super n plus 1 (x).

On the right axis, a specific vertical segment within the middle region is marked with tick marks and labeled V sub n plus 1 super max. Both vertical axes feature downward-pointing arrowheads indicating a directional flow.

By repeating the same construction with respect to the set $\partial _R C_{O}$ , we end up with a family of positively transverse paths $\{\gamma _n\}_{n\in \mathbb Z}$ , where each $\gamma _n$ connects $ f^n(x)$ to $ f^{n+1}(x)$ , and is disjoint from $V_{\phi }$ for any leaf $\phi \in \partial _L C_{O} \cup \partial _R C_{O}$ .

Finally, the last step is to prove that the family $\{\gamma _n\}_{n\in \mathbb Z}$ is locally finite. For that, define

$$ \begin{align*}W:=\bigcup_{n \geq 0} \overline{L(\phi_{-n}) \cap R(\phi_{n})} \end{align*} $$

and observe that every point $z \in W$ has an open neighborhood $U_z\subset \mathbb R^2$ that intersects finitely many sets in the family $\{D_n\}_{n \in \mathbb Z}$ . In particular, each neighborhood $U_z$ intersects finitely many paths in the family $\{\gamma _n\}_{n\in \mathbb Z}$ . Let $K \subset \mathbb R^2$ be a compact set and denote

$$ \begin{align*}K^*:=K \setminus \bigcup_{\phi \in \partial_L C_{O} \cup \partial_R C_{O}} \text{int}{(V_\phi)}.\end{align*} $$

Note that $K^*$ is a compact set contained in W satisfying $\gamma _n \cap K = \gamma _n \cap K^*$ for all $n \in \mathbb Z$ . Consider a cover of $K^*$ of the form $\{U_z\}_{z \in K^*}$ . By taking a finite sub-cover, we conclude that $K^*$ intersects at most finitely many transverse paths in $\{\gamma _n\}_{n\in \mathbb Z}$ . Hence, K intersects finitely many paths in $\{\gamma _n\}_{n\in \mathbb Z}$ . This shows that $\{\gamma _n\}_{n\in \mathbb Z}$ is a locally finite family and thus

$$ \begin{align*}\Gamma_{O} := \prod_{n \in \mathbb Z} \gamma_n\end{align*} $$

is a properly embedded transverse trajectory associated to ${O}$ .

The fact that $\Gamma _{O}$ satisfies $\partial _L^{\mathrm {top}} {O} = \partial _L C_{O} \cap L(\Gamma _{O})$ and $\partial _L^{\mathrm {bot}} {O} = \partial _L C_{O} \cap R(\Gamma _{O})$ follows directly from the construction. The analogous properties for $\partial _R^{\mathrm {top}} {O}$ and $\partial _R^{\mathrm {bot}} {O}$ hold as well. To see that these equalities hold for any proper transverse trajectory $\Gamma _{O}$ , it suffices to note that they hold automatically if the trivial neighborhoods in the family $\{V_\phi \}_{\phi \in \partial C_{O}}$ are disjoint from $\Gamma _{O}$ . Since, in the proper case, $\Gamma _{O}\subset \mathbb R^2$ is a closed set disjoint from $\partial C_{O}$ , Lemma 2.2 ensures that such a family $\{V_\phi \}_{\phi \in \partial C_{O}}$ exists. This concludes the proof.

3.6 Equivalence classes of ${\mathcal {F}}$ -asymptotic orbits

As presented in Definition 1.2, we can relate orbits exhibiting similar ${\mathcal {F}}$ -asymptotic behavior via equivalence relations $\stackrel {+}{\sim }$ and $\stackrel {-}{\sim }$ on the set $\mathrm{orb}$ . Below, we interpret these relations in terms of proper transverse trajectories.

Figure 22

Existence of ‘forever intersecting’ proper trajectories for pairs of $\mathcal{F}$ -asymptotic orbits.

Figure 22 Long description

A line graph depicting mathematical trajectories in a flow field. On the left, two primary paths are labeled. A red line, Gamma sub O prime, starts at the top left and curves downward. A blue line, Gamma sub O, starts at the bottom left and curves upward. Both lines have directional arrows pointing toward the center.

In the middle, the red and blue lines intersect and begin to oscillate around each other. They are surrounded by numerous thin grey vertical lines that curve inward toward the center, representing a vector field or flow.

On the right, the oscillating red and blue lines enter a narrow horizontal corridor defined by two thick black parabolic curves.

- The upper curve is labeled partial sub L super top C sub O equals partial sub L super top C sub O prime.

- The lower curve is labeled partial sub L super bot C sub O equals partial sub L super bot C sub O prime.

The red and blue trajectories continue to intertwine in a ‘forever intersecting’ pattern as they move toward the right edge of the frame within these boundaries.

Lemma 3.6 admits a dual version which is slightly stronger, presented in Lemma 3.7.

Figure 23

Forced divergence for any pair of proper transverse trajectories associated to non- $\mathcal{F}$ -asymptotic orbits.

Figure 23 Long description

A two-panel diagram illustrating dynamical systems.

Left Panel: Labeled O not-asymptotic to O prime. A background of light gray streamline curves flows horizontally. A thick black vertical wavy line labeled phi sub R divides the space. Two colored trajectories enter from the left. A red line labeled Gamma sub O starts straight, crosses phi sub R, and then oscillates with decreasing amplitude. A blue line labeled Gamma sub O prime starts below the red line, crosses phi sub R, and then oscillates above the red line before converging toward it.

Right Panel: Labeled O not-asymptotic to O prime with a plus superscript. This panel is a mirrored version of the first. A thick black vertical wavy line is labeled phi sub L. The red trajectory Gamma sub O and blue trajectory Gamma sub O prime enter from the left as oscillating lines that cross the phi sub L boundary and then flatten into straight horizontal lines on the right side.

In both panels, arrows on the colored lines indicate the direction of flow from left to right, and arrows on the black boundary lines point downward.

Remark 3.8. It is important to remark that the results of Lemmas 3.6 and 3.7 are no longer true if we consider non-proper transverse trajectories as in the following example.

We suppose that ${O},{O}^{ \prime } \in \mathrm {Orb}(f)$ are two orbits admitting a leaf $\phi \in \partial _L^{\mathrm {top}} {O} \cap \partial _L^{\mathrm {bot}} {{O}^{ \prime }}$ . These orbits are clearly not forward ${\mathcal {F}}$ -asymptotic. However, as one can see in Figure 24, there still exist non-proper transverse trajectories $\Gamma _{O}$ and $\Gamma _{{O}^{ \prime }}$ associated to ${O}$ and ${O}^{\prime }$ , respectively, that satisfy the property $ \Gamma _{O}([ t, \infty )) \cap \Gamma _{{O}^{ \prime }}([ t, \infty )) \neq \varnothing $ for any $t \in \mathbb R$ .

Figure 24

Illustration of the example described in Remark 3.8.

Figure 24 Long description

The diagram illustrates a mathematical flow field.

* Background: Vertical grey lines with downward-pointing arrows represent a steady flow field across the entire frame.

* Central Feature: A large parabolic black outline on the right side contains the mathematical expression partial sub L super top O intersection partial sub L super bot O prime.

* Red Boundary: A smooth red curve labeled Gamma sub O prime slopes upward from the left toward the top right. It features several red circular nodes where it intersects with blue lines.

* Blue Boundary: A jagged blue curve labeled Gamma sub O runs generally below the red curve. It consists of sharp peaks that touch the red curve at the red nodes and deep troughs marked by blue circular nodes.

* Interaction: Between the red curve and the parabolic black boundary, the blue lines become highly concentrated and compressed, following the contour of the parabola before branching out toward the bottom right.

* Points of Interest: On the far left, the blue curve oscillates independently with blue nodes at its base. As it moves right, the peaks of the blue curve align perfectly with the red nodes on the Gamma sub O prime curve.

Proof of Lemma 3.6.

(i) $\implies $ (ii): From statement (i), we have $C_{O} \cap C_{{O}^{ \prime }}\neq \varnothing $ and $\partial _L C_{O} = \partial _L C_{{O}^{ \prime }}$ . Let $\phi _* \in C_{O} \cap C_{{O}^{ \prime }}$ and let $(\phi _n)_{n\geq 0}$ be the increasing sequence of leaves in ${\mathcal {F}}$ formed by the leaves intersecting the set $({O}\cup {O}^{ \prime })\cap \overline {L(\phi _*)}$ . Note that some leaves of $(\phi _n)_{n\geq 0}$ may intersect simultaneously both orbits ${O}$ and ${O}^{ \prime }$ . Next, according to Lemma 2.2, there exists a family of sufficiently small trivial neighborhoods $\{V_\phi \}_{\phi \in \partial _L C_{O}}$ of the leaves in $\partial _L C_{O}$ that is pairwise disjoint and disjoint from the closed set $\phi _0 \cup {O} \cup {O}^{ \prime }$ .

Similarly to §3.3, we have a filtration $(J_n)_{n\geq 0}$ of the set $\partial _L C_{O}$ by sets of the form $J_n=\{\phi \in \partial _L C_{O} \mid V_\phi \cap \phi _n\}$ . Moreover, for each $n\geq 0$ such that ${O} \cap \phi _n \neq \varnothing $ , we have a cut

$$ \begin{align*}(J_n^-({O}), J_n^+({O})) \in \text{Cut}_\prec(J_n).\end{align*} $$

However, for the values of $n\geq 0$ such that ${O} \cap \phi _n = \varnothing $ , we define $m(n)>n$ as the smallest integer such that ${O} \cap \phi _{m(n)} \neq \varnothing $ , and we set

$$ \begin{align*} J_n^-({O}) := J_n \cap J_{m(n)}^-({O}) \quad\text{and}\quad J_n^+({O}):= J_n \cap J_{m(n)}^+({O}). \end{align*} $$

Performing the same adaptation for the cuts induced by the orbit ${O}^{ \prime }$ , we end up with two cuts $(J_n^-({O}), J_n^+({O}))$ and $(J_n^-({O}^{ \prime }), J_n^+({O}^{ \prime }))$ of the ordered set $(J_n,\prec )$ for every $n\geq 0$ .

Since ${O}\stackrel {+}{\sim }{O}^{ \prime }$ for each leaf $\phi \in \partial _L C_{O}$ , there exists $M_\phi>0$ such that

$$ \begin{align*} \phi \in \partial_L^{\mathrm{top}} {O} \implies \phi\in J_n^-({O}) \cap J_n^-({O}^{ \prime}) \quad \text{for all } n\geq M_\phi,\\ \phi \in \partial_L^{\mathrm{bot}} {O} \implies \phi\in J_n^+({O}) \cap J_n^+({O}^{ \prime}) \quad \text{for all } n\geq M_\phi. \end{align*} $$

By the same redefinition of the family $\{V_\phi \}_{\phi \in \partial _L C_{O}}$ presented in (3.1), we can ensure that each leaf $\phi \in \partial _L C_{O}$ satisfies $\phi \not \in J_n$ for all $n<M_\phi $ and, consequently, for any $n\geq 0$ , we have

$$ \begin{align*} \partial_L^{\mathrm{top}} {O} = J_n^- ({O}) = J_n^- ({O}^{ \prime}) \quad \text{and} \quad \partial_L^{\mathrm{bot}} {O} = J_n^+ ({O}) = J_n^+ ({O}^{ \prime}). \end{align*} $$

For each $n\geq 0$ , let $p_n$ and $p_n'$ be the points where the leaf $\phi _n$ intersects ${O}$ and ${O}^{ \prime }$ , respectively. In the case where $\phi _n$ intersects only one of these orbits, we consider $p_n = p_n'$ . Following arguments similar to those in the proof of Theorem A, we obtain for each $n\geq 0$ , two positively transverse paths $\gamma _n, \gamma _n':[0,1]\longrightarrow \mathbb R^2$ that are disjoint from all sets in the family $\{V_\phi \}_{\phi \in \partial _L C_{O}}$ and such that $\gamma ^n_{O}(0) = p_n$ , $\gamma ^n_{{O}^{ \prime }}(0) = p_n'$ , $\gamma ^n_{O}(1) = p_{n+1}$ , and $\gamma ^n_{{O}^{ \prime }}(1) = p_{n+1}'$ . Observe that $p_n$ , $p_n'$ , $p_{n+1}$ , and $p_{n+1}'$ belong to the same connected component of the set

$$ \begin{align*} \overline{L(\phi_n) \cap R(\phi_{n+1})} \ \bigg\backslash \ \bigcup_{\phi \in \partial_L C_{O}} \text{int}(V_\phi).\end{align*} $$

Therefore, for any pair of open sets $U_n, U_{n+1}\subset \mathbb R^2$ satisfying $\phi _n \subset U_n$ and $\phi _{n+1} \subset U_{n+1}$ , the same arguments of §3.5 allow us to construct $\gamma _n$ and $\gamma _n'$ in such a way that they coincide outside the neighborhoods $U_n$ and $U_{n+1}$ . Moreover, in the case where $p_n = p_n'$ and $p_{n+1} = p_{n+1}'$ , we can assume that $\gamma _n = \gamma _n'$ . In all cases, we have $\gamma _n \cap \gamma _n' \neq \varnothing $ for all $n\geq 0$ . The argument of §3.5 proves that the families $\{\gamma _n\}_{n\geq 0}$ and $\{\gamma _n'\}_{n\geq 0}$ are both locally finite. Thus, they can be concatenated, and then extended, into proper transverse trajectories $\Gamma $ and $\Gamma '$ associated respectively to the orbits ${O}$ and ${O}^{ \prime }$ . These proper transverse trajectories satisfy statement (ii).

(ii) $\implies $ (i). Since $\Gamma $ and $\Gamma '$ intersect each other, there exists a leaf $\phi _0 \in C_{O} \cap C_{{O}^{ \prime }}$ . Let $(\phi _n)_{n\geq 0}$ be the increasing sequence of leaves in ${\mathcal {F}}$ formed by the leaves meeting the set $({O}\cup {O}^{ \prime })\cap \overline {L(\phi _0)}$ . Since $\Gamma $ and $\Gamma '$ are proper lines, the set $\Gamma \cup \Gamma ' \subset \mathbb R^2$ is closed. Thus, by applying Lemma 2.2, we know there exists a family $\{V_\phi \}_{\phi \in \partial _L C_{O}}$ of pairwise disjoint trivial neighborhoods of $\partial _L C_{O}$ that satisfies $V_\phi \cap (\phi _0 \cup \Gamma \cup \Gamma ') = \varnothing $ for any $\phi \in \partial _L C_{O}$ .

For each $n\geq 0$ , we consider $J_n=\{\phi \in \partial _L C_{O} \mid V_\phi \cap \phi _n\}$ . As in (i) $\implies $ (ii), consider the sets $J_n^- ({O})$ , $J_n^+ ({O})$ , $J_n^-({O}^{ \prime })$ , and $J_n^+({O}^{ \prime })$ . Note that, as each $V_\phi $ is disjoint from ${\Gamma \cup \Gamma '}$ , hypothesis (ii) implies that, $\text {for all } n\geq 0$ , we have $J_n^+ ({O}) = J_n^+ ({O}^{ \prime }) $ and $J_n^- ({O}) = J_n^- ({O}^{ \prime })$ . This shows that the orbits ${O}$ and ${O}^{ \prime }$ induce the same cut $(\partial _L^{\mathrm {top}} {O}, \partial _L^{\mathrm {bot}}{O})$ on $\partial _L C_{O}$ and, consequently, we conclude that ${O}$ and ${O}^{ \prime }$ are forward ${\mathcal {F}}$ -asymptotic.

Remark 3.9. The argument (i) $\implies $ (ii) within the proof of Lemma 3.6 can be adapted to any finite subset ${\mathcal {O}}\subset \mathrm {Orb}(f)$ satisfying

$$ \begin{align*} {O} \stackrel{+}{\sim} {O}^{ \prime} \quad \text{for all } {O},{O}^{\prime}\in {\mathcal{O}}.\end{align*} $$

If each leaf of ${\mathcal {F}}$ intersects at most one orbit in ${\mathcal {O}}$ , this adaptation proves Lemma 7.2.

Proof of Lemma 3.7.

(i) $\implies $ (ii). Suppose that ${O} \not\stackrel{+}{\sim} {O}^{ \prime }$ . According to Lemma 3.6, for any pair of proper transverse trajectories $\Gamma _{O}$ and $\Gamma _{{O}^{ \prime }}$ of ${O}$ and ${O}^{ \prime }$ , there exists $M>0$ such that

$$ \begin{align*} \Gamma_{O}([ M, \infty)) \cap \Gamma_{{O}^{ \prime}}([ M, \infty)) = \varnothing.\end{align*} $$

Since $\Gamma _{O}$ and $\Gamma _{{O}^{ \prime }}$ are topological lines, the intersection $\Gamma _{O} \cap \Gamma _{{O}^{ \prime }}$ is a closed subset of $\mathbb R^2$ . Therefore, the set $ {\Gamma _{O}}^{-1}(\Gamma _{O} \cap \Gamma _{{O}^{ \prime }})$ is a closed subset of $\mathbb R$ bounded from above by M and, consequently, it admits a maximal element $t_{\mathrm {max}}$ .

Note that the leaf $\phi _{\mathrm {max}}\in {\mathcal {F}}$ passing through the point $\Gamma _{O}(t_{\mathrm {max}})$ belongs to $C_{O} \cap C_{{O}^{ \prime }}$ . Since $C_{O} \cap C_{{O}^{ \prime }}$ is an open set, there exists a leaf $\phi _L\in {\mathcal {F}}$ satisfying $L(\phi _L) \subset L(\phi _{\mathrm {max}})$ and sufficiently close to $\phi _{\mathrm {max}}$ so that it also belongs to $C_{O} \cap C_{{O}^{ \prime }}$ . This leaf proves (i) $\implies $ (ii).

(ii) $\implies $ (i). This implication follows trivially by Lemma 3.6.

4.1 Defining $\lesssim _L$ and $\lesssim _R$

In this section, we will define two relations $\lesssim _L$ and $\lesssim _R$ on the set of orbits $\mathrm {Orb}(f)$ that satisfy the following properties.

• • For each $\phi \in {\mathcal {F}}$ , the relations $\lesssim _L$ and $\lesssim _R$ are total preorder on the set

  $$ \begin{align*} \mathrm{Orb}(\phi) :=\{O \in \mathrm{orb} \mid \phi \in C_{O} \}. \end{align*} $$

• • For any pair of orbits ${O}, {O}^{\prime } \in {\mathrm {Orb}}(f)$ , the following equivalences hold:

  $$ \begin{align*} {O}\stackrel{+}{\sim} {O}^{\prime} \iff {O} \lesssim_L {O}^{\prime} \quad \text{and}\quad {O}^{\prime} \lesssim_L {O}, \\ {O}\stackrel{-}{\sim} {O}^{\prime} \iff {O} \lesssim_R {O}^{\prime} \quad \text{and}\quad {O}^{\prime} \lesssim_R {O}. \end{align*} $$

Before, we note that $C_{O} \cap C_{{O}^{\prime }}$ is a (possibly empty) leaf domain of ${\mathcal {F}}$ (see Lemma 2.6). Moreover, limit sets $\partial _L(C_{O} \cap C_{{O}^{\prime }})$ and $\partial _R(C_{O} \cap C_{{O}^{\prime }})$ are totally ordered by a relation $\prec $ , just as for $\partial _L C_{O}$ and $\partial _R C_{O}$ (see §3.3). This allows us to introduce the next definition.

Figure 25

Two possible configurations at infinity for pairs of orbits $O$ and ${O}^{\prime }$ that cross a common leaf but are neither forward $\mathcal{F}$ -asymptotic nor backward $\mathcal{F}$ -asymptotic.

Figure 25 Long description

The figure consists of two panels, a and b, each displaying a dense field of light gray flow lines representing a foliation. Two vertical, wavy black lines divide each panel into three regions.

In panel a, two specific orbits are highlighted. A red orbit labeled Gamma sub O prime enters from the top-left, crosses the first vertical line, and exits at the top-right. A blue orbit labeled Gamma sub O enters from the bottom-left, crosses the first vertical line, and exits at the bottom-right. The orbits are roughly parallel and do not converge.

In panel b, the configuration changes. The red orbit Gamma sub O prime enters from the top-left but, after crossing the vertical lines, it exits at the bottom-right. Conversely, the blue orbit Gamma sub O enters from the bottom-left and exits at the top-right. This creates a crossing pattern where the orbits swap their vertical positions relative to each other as they pass through the central region. Small arrows on the colored lines indicate the direction of the flow from left to right.

Figure 26

Definition of $\lesssim_{R}$ and $\lesssim_{L}$ .

Figure 26 Long description

The diagram consists of eight panels arranged in two columns. The left column is headed by the relation O less-than-or-equal-to sub R O prime, and the right column by O less-than-or-equal-to sub L O prime. Each panel contains a mathematical definition followed by a vector field diagram.

* Left Column (R-series):

- (R1): Defines conditions for phi and phi prime in the boundary of the intersection of C sub O and C sub O prime. The diagram shows two black U-shaped boundaries on the left with flow lines converging toward a central crossing of a red line Gamma sub O and a blue line Gamma sub O prime.

- (R2): Defines a leaf phi in the boundary. The diagram shows a single black U-shaped boundary on the left with the red line Gamma sub O passing above it and the blue line Gamma sub O prime below it.

- (R3): Similar to R2 but focuses on leaf phi prime. The diagram shows the black U-shaped boundary on the left with the blue line Gamma sub O prime passing through its center.

- (R4): Defines boundary equality and subset relations. The diagram shows a single black U-shaped boundary on the left with flow lines spreading outward toward the right.

* Right Column (L-series):

- (L1): Mirror image of R1. The black U-shaped boundaries are on the right side of the panel. The red line Gamma sub O and blue line Gamma sub O prime cross in the center.

- (L2): Mirror image of R2. The black U-shaped boundary is on the right. The red line Gamma sub O originates from the right boundary area.

- (L3): Mirror image of R3. The black U-shaped boundary is on the right with the blue line Gamma sub O prime originating from it.

- (L4): Mirror image of R4. The black U-shaped boundary is on the right with flow lines converging into it from the left.

In all diagrams, light gray lines represent the background vector field flow, while red and blue lines represent specific orbits Gamma sub O and Gamma sub O prime respectively.