2.1 Curves

2.2 Crossings

Loosely speaking, an essential crossing is a crossing of a multi-curve that cannot be homotoped away. We make this definition precise as follows. We cover cases where $\gamma $ does not have transverse crossings for convenience of some of the examples.

In case (2)b, the endpoints of $\widetilde {\gamma _p}$ and $\widetilde {\gamma _q}$ are the same.

This definition might be somewhat looser than expected. For instance, in the chain of three crossings

the circled middle crossing is essential iff the other two are, since ‘linking at infinity’ does not see the direction of crossing. This does not matter for our purposes.

Definition 2.7 smoothings

Let $(p,q) \in X(\gamma )$ be an essential crossing of $\gamma $ on S. To make a smoothing $\gamma '$ of $(p,q)$ , cut $X(\gamma )$ at p and q and reglue the resulting four endpoints in one of the two other possible ways, getting a new $1$ -manifold $X(\gamma ')$ . The map $\gamma '$ agrees with $\gamma $ ; this is well-defined since $\gamma (p) = \gamma (q)$ . In pictures we will homotop $\gamma '$ slightly to round out the resulting corners. If $\gamma $ is oriented, then the oriented smoothing is the smoothing that respects the orientation on $X(\gamma )$ :

If we obtain a concrete curve $\gamma '$ from $\gamma $ by a sequence of k smoothings of essential crossings, we will write $\gamma \mathrel {\searrow }_k \gamma '$ . (We check whether the crossings are essential at each stage of this process; this is more restrictive than checking at the beginning.)

We have similar notions for weighted curves.

Using this, we define conditions on a weighted curve functional, extending Definition 1.1.

• • Weighted quasi-smoothing: There is a constant $R>0$ so that $C \mathrel {\searrow }_w C'$ are weighted curves related by a smoothing of weight w,

  $$ \begin{align*} f(C) \ge f(C')-wR. \end{align*} $$

• • Weighted smoothing: Take $R=0$ in the definition of weighted quasi-smoothing.

See Proposition 3.6 for justification for these definitions.

2.3 Space of geodesics

Definition 2.12 Space of oriented geodesics

Let $G^{+}(\Sigma )$ denote the space of oriented geodesics in $\widetilde {\Sigma }$ ; that is,

$$ \begin{align*} G^+(\Sigma) :=\partial_\infty \Sigma \times\partial_\infty\Sigma - \Delta. \end{align*} $$

Since this is independent of the hyperbolic structure, we will also write $G^+(S)$ .

2.4 Geodesic currents

Since the action of $\pi _1(S)$ on $G^+(S)$ is not discrete, this definition is hard to visualise. We give alternate definitions that play a key role in our proofs. For a hyperbolic surface $\Sigma $ , let $UT\Sigma $ be the unit tangent bundle and let $\phi _t$ be the geodesic flow on it.

We can also look at induced measures on cross sections.

The equivalence of the three definitions was known to Bonahon [Reference Bonahon10, Chapter 4]. Details can be found in [Reference Aramayona and Leininger1, Section 3.4]. Briefly, given a measure $\mu $ on $UT\Sigma $ as in Definition 2.14 and a cross section  $\tau $ , there is an induced flux $\mu _\tau $ on $\tau $ , as explained in Definition 7.15; this gives a geodesic current in the sense of Definition 2.15. Lifting to the universal cover then gives a geodesic current in the sense of Definition 2.13. We can also relate Definitions 2.13 and 2.14 directly by connecting both to measures on $UT\mathbb {H}^2$ that are invariant under both $\pi _1(\Sigma )$ and the geodesic flow, as described by Benoist and Oh [Reference Benoist and Oh7, Proposition 8.1].

Because Definitions 2.13 and 2.15 are invariant under the mapping class group, we will also write $G^+(S)$ and $\mathcal {GC}^+(S)$ in the sequel. We will also write $\pi _1(S)$ . On the other hand, we will emphasise the dependence of $UT\Sigma $ on the hyperbolic structure.

2.5 Oriented vs unoriented currents

We will be mostly working in the setting of oriented geodesic currents, but most of our natural examples (like measured laminations) use unoriented currents.

Definition 2.17 Unoriented geodesic currents

To define the subspace $\mathcal {GC}(S) \subset \mathcal {GC}^{+}(S)$ of unoriented geodesics currents, let $\sigma : G^{+}(S) \to G^{+}(S)$ be the flip map that switches the two factors in the definition of $G^+(S)$ , reversing the orientation of the geodesic. This induces a map $\sigma _* \colon \mathcal {GC}^{+}(S) \to \mathcal {GC}^{+}(S)$ . Set

$$ \begin{align*} \mathcal{GC}(\Sigma) := \{ \mu \in \mathcal{GC}^{+}(\Sigma) \mid \sigma_*(\mu)=\mu \}. \end{align*} $$

There is a map $\Pi \colon \mathcal {GC}^{+}(S) \to \mathcal {GC}^{+}(S)$ given by $\Pi (\mu ):= \frac {1}{2}(\mu + \sigma _*(\mu ))$ with image the subset of unoriented currents.

In the proof of the main result, we shall work with oriented currents $\mathcal {GC}^+(S)$ ; oriented currents are more general and just as easy to work with for our proof.

The maps $\sigma $ and $\Pi $ have obvious analogues for curves.

2.6 Curves as currents

For an oriented multi-curve C on a hyperbolic surface $\Sigma $ , we can construct a geodesic current as follows.

For Definition 2.13, consider all lifts of all nontrivial components of C to $\widetilde {\Sigma }$ . Each lift gives a quasi-geodesic in $\widetilde {\Sigma }$ and thus a unique fellow-travelling geodesic in $G^+(S)$ ; we thus get an infinite countable subset of $G^+(S)$ , which is easily seen to be discrete and $\pi _1(S)$ -invariant. Define the geodesic current to be the $\delta $ -function of this subset.

For Definition 2.14, take the geodesic representative $\gamma $ of C and consider the canonical lift $\widetilde {\gamma }$ of $\gamma $ to $UT\Sigma $ ; this is an orbit of $\phi _t$ . Let $\mu _C$ be the length-normalised $\delta $ -function on this orbit. That is, for an open set U we set $\mu _C(U)$ to be the total length of $\widetilde {\gamma } \cap U$ with respect to the natural Riemannian metric on $UT\Sigma $ .

For Definition 2.15 on a cross section $\tau $ , again take $\widetilde {\gamma } \subset UT\Sigma $ , and let $\mu _\tau $ be the $\delta $ -function on the discrete set of points $\widetilde {\gamma } \cap \tau $ . (This is compatible with the length normalisation in the previous paragraph.)

From any of these points of view the inclusion naturally extends to weighted multi-curves.

Weighted closed (multi-)curves are dense in the space of geodesic currents [Reference Bonahon10, Proposition 4.4].

Geometric intersection number extends continuously to geodesic currents, as shown by Bonahon [Reference Bonahon10, Proposition 4.5]. The space of measured laminations (defined by Harer and Penner [Reference Penner and Harer49, Section 1.7]) can be characterised, as in Bonahon [Reference Bonahon8, Proposition 17], as a subset of (unoriented) geodesic currents:

$$ \begin{align*} \mathcal{ML}(S) := \{ \alpha \in \mathcal{GC}(S) \mid \sigma(\alpha)=\alpha, i(\alpha,\alpha)=0 \}. \end{align*} $$

The following square of inclusions is useful to keep in mind:

Here the horizontal inclusions have dense image: Douady and Hubbard showed that weighted simple multi-curves are dense in $\mathcal {ML}$ [Reference Douady and Hubbard19, Theorem]. Soon after, Masur showed that weighted simple curves are also dense [Reference Masur39, Theorem 1].

2.7 Topology on currents and measures

Let $\mathcal {M}(X)$ denote the space of positive Borel measures on a topological space X. $\mathcal {M}_1(X)$ will denote the space of Borel probability measures on X. The topology on $\mathcal {M}(X)$ is the weak $^*$ topology; that is, the smallest topology so that, for all continuous, compactly supported functions f on $G^+(S)$ , the functional

$$ \begin{align*} \mu \mapsto \int_{G^+(S)} fd\mu \end{align*} $$

is continuous.

The topology on $\mathcal {GC}^+(S)$ (in Definition 2.13) is the weak $^*$ topology as a subspace of measures on $G^+(S)$ , We could also look at the weak $^*$ topology on currents as a subspace of measures on $UT\Sigma $ (Definition 2.14); these two points of view give the same topology [Reference Benoist and Oh7, Proposition 8.1]. On the other hand, if we take $\tau $ to be a closed cross section (including the boundary), the map $\mu \mapsto \mu _\tau $ relating Definitions 2.14 and 2.15 is not usually continuous with respect to the weak $^*$ topologies, so it is delicate to use the weak $^*$ topology on $\mathcal {M}(\tau )$ ; see Lemma 10.4 and Remark 10.6.

There are in fact two topologies on spaces of measures that are sometimes called the weak $^*$ topology; the one above is also called the wide topology [Reference Minlos41, interalia]. There is also the narrow topology on measures $\mathcal {M}(X)$ on a space X, defined as the smallest topology so that, for all continuous bounded f on X, the functional $\mu \mapsto \int _{X} fd\mu $ is continuous. (That is, replace compactly supported with bounded in the functions considered.)

In general, the weak $^*$ or wide topology is weaker than the narrow topology, but in some particular cases they are equivalent.

A topological space X is called Polish if its topology has a countable base and can be defined by a complete metric.

Thus, $\mathcal {GC}^+(S)$ is second countable completely metrisable and second countable and so in particular sequential continuity is the same as continuity. Although we will be dealing with Radon measures, for Polish spaces it is equivalent to consider the a priori more general class of Borel measures.

The narrow and wide topology agree in certain sequences on locally compact spaces.

In particular, when X is Polish, the two topologies agree for the space $\mathcal {M}_1(X)$ of Borel probability measures.

Remark 2.26. Geodesic currents can also be defined more generally for finite type hyperbolic surfaces. Depending on if we consider ends as cusps or funnels, we get two different spaces, which we will call $\mathcal {GC}_{\mathrm {cusp}}(S)$ and $\mathcal {GC}_{\mathrm {open}}(S)$ , respectively. In the first case, we define the space of geodesic currents analogously to Definition 2.13 for closed surfaces; that is, as invariant measures supported on the space of geodesics of the universal cover, noting that now the space of geodesics contains arcs going from cusp to cusp. In the second case, we consider geodesic currents supported on geodesics projecting to the convex core of the surface. Extending continuously curve functionals to these spaces is more delicate.

In the case of $\mathcal {GC}_{\mathrm {cusp}}(S)$ , let S be a surface with two open ends and let $\Sigma $ be a complete hyperbolic metric of finite area, with respect to which the ends of S are cusps. Let a be an arc going from cusp to cusp. Let $C_n$ be the closed curve going along for some time a, winding n times around one cusp, going along a again and winding n times around the other cusp. Observe that although $C_n \to a$ in the weak $^*$ topology and $i(a,a)=0$ , we have $i(a,C_n)=2n$ , so intersection number is not a continuous function on geodesic currents.

The case of $\mathcal {GC}_{\mathrm {open}}(S)$ is different, since intersection number is continuous. Indeed, let $\overline {\Sigma }$ denote the convex core of the complete hyperbolic surface of infinite area, which is a compact surface with geodesic boundary. We can consider the intersection number on the double $D(\overline {\Sigma })$ of $\overline {\Sigma }$ , which is a closed surface $\overline {\Sigma }$ embeds into. This intersection number on $D(\overline {\Sigma })$ is continuous [Reference Bonahon10, Proposition 4.5]. Restricting this intersection number to $\overline {\Sigma }$ , we obtain continuity of intersection number on $\mathcal {GC}_{\mathrm {open}}(S)$ . However, the conditions of Theorem A alone are not enough to guarantee a continuous extension $f \colon \mathcal {GC}_{\mathrm {open}}(S) \to \mathbb {R}$ . Indeed, let $\ell $ be the restriction to $\overline {\Sigma }$ of the hyperbolic length on $D(\overline {\Sigma })$ and consider the modified curve functional $\ell '$ obtained by setting

$$ \begin{align*} \ell'(C) := \begin{cases} 0 & C\ \text{is a boundary curve}\\ \ell(C) & \text{otherwise.} \end{cases} \end{align*} $$

We note that $\ell '$ satisfies additivity, stability and homogeneity properties because $\ell $ does. Also, it satisfies smoothing because $\ell $ does and nonboundary curves do not intersect boundary curves. However, $\ell '$ does not extend to a continuous function on $\mathcal {GC}_{\mathrm {open}}(S)$ : let $\gamma ,\beta \in \pi _1(S,p)$ be elements based at a point $p \in S$ and denote $C=[\beta ]$ , $D=[\gamma ]$ . Assume that C is a boundary curve and D is not. For each n, define a nonsimple, nonboundary parallel curve by $C_n := [\gamma \beta ^n]$ . Observe that the sequence $\frac {1}{n}C_n$ converges to C in the weak $^*$ topology but

$$ \begin{align*} \ell'(C_n) &\asymp n, \end{align*} $$

whereas

$$ \begin{align*} \ell'(C)&=0, \end{align*} $$

so $\ell '$ cannot be a continuous function on $\mathcal {GC}_{\mathrm {open}}(S)$ . So additional conditions on f are needed to guarantee a continuous extension to $\mathcal {GC}_{\mathrm {open}}(S)$ .

3.1 Convexity on the reals

The curve functionals f we study have some convexity property as a function of the weights, because of the convex union and homogeneity properties. We first review some background on convex functions and their continuity properties.

A function $f\colon \mathbb {R}^n \to \mathbb {R}$ is called $\mathbb {R}$ -convex (respectively $\mathbb {Q}$ -convex) if

$$ \begin{align*} f(ax + (1-a)y) \leq a f(x) + (1-a)f(y) \end{align*} $$

for all $x,y\in \mathbb {R}^n$ and $a \in [0,1]$ (respectively text $a \in [0,1] \cap \mathbb {Q}$ ). A function $f \colon \mathbb {Q}^n \to \mathbb {R}$ might also be $\mathbb {Q}$ -convex, with the same definition. We furthermore say that f is midpoint-convex if

$$ \begin{align*} f\bigl({\textstyle \frac{1}{2}} x + {\textstyle \frac{1}{2}} y \bigr) \le {\textstyle \frac{1}{2}} f(x) + {\textstyle \frac{1}{2}} f(y). \end{align*} $$

It is not true that all $\mathbb {Q}$ -convex functions $f \colon \mathbb {R}^n \to \mathbb {R}$ must be continuous, but all counterexamples are highly pathological. In particular, any measurable $\mathbb {Q}$ -convex function $f \colon \mathbb {R}^n \to \mathbb {R}$ is necessarily continuous (see [Reference Kuczma37, Theorem 9.4.2]).

3.2 Convexity for curve functionals

We now apply the results above to our setting of real-valued functions on curves.

As an immediate consequence of Proposition 3.1 (iv), for a curve functional f satisfying convex union and homogeneity, we can extend f to a weighted curve functional that is convex and therefore continuous for a fixed set of components. This will play a role in the proof of Theorem A, specifically in Proposition 10.3.

First, for any curve functional satisfying homogeneity, we adopt the convention that we extend f to rationally weighted curves $\mathbb {Q}\mathcal {C}^+(S)$ in the usual way by clearing denominators: set

(3.3) $$ \begin{align} f\Bigl(\sum a_i C_i\Bigr) := \frac{1}{d}f\Bigl(\sum d a_i C_i\Bigr) \end{align} $$

for some integer d sufficiently large so all the $d a_i$ are integers. By homogeneity of f, the extension does not depend on d.

Proposition 3.4. Let $C = (C_i)_{i=1,\dots ,n}$ be a finite sequence of multi-curves and consider combinations $\sum _{i=1}^n a_i C_i$ . Let f be a curve functional that satisfies homogeneity and convex union. Define a function $f_{C}\colon \mathbb {Q}^n \to \mathbb {R}$ by

$$ \begin{align*} f_{C}(a_1,\dots,a_n):= f\left( \sum_{i=1}^n a_i C_i \right). \end{align*} $$

Then $f_C$ is $\mathbb {Q}$ -convex and thus continuous.

Proof. We first observe that f, as a function on integrally weighted multi-curves, satisfies weighted quasi-smoothing. If $C=[\gamma ]$ , $\gamma \mathrel {\searrow } \gamma '$ and k is an integer, then $k\gamma \mathrel {\searrow }_k k \gamma '$ , since $k\gamma $ are k disjoint parallel copies of $\gamma $ . Thus,

$$ \begin{align*} f(kC) \geq f(kC')-kR. \end{align*} $$

By the method of clearing denominators and homogeneity, as in equation (3.3), we obtain rational weighted quasi-smoothing.

Finally, by continuity of f as a function of the weights of a fixed multi-curve, we get real weighted quasi-smoothing.

Theorems A and B as stated start from a curve functional of various types. Many curve functionals naturally come as functions on weighted curves (see Section 4). On the other hand, we have seen in Proposition 3.6 that a curve functional satisfying convex union and homogeneity and stability properties yields weighted curve functional satisfying the same properties.

4.1 Intersection number

Fix a multi-curve D and consider the curve functional $\mathcal {C}(S)$ defined by $f(C) = i(C, D)$ , where $i(C,D)$ is the minimal number of intersection points between representatives of C and D in general position. Then f is homogeneous, additive and stable.

There is a simple geometric argument, which we will use repeatedly, to see that f satisfies smoothing. Fix a minimal representative $\delta $ for D. Take a curve C with an essential self-intersection x and a representative $\gamma $ with minimal intersection with $\delta $ . Then $\gamma $ has a self-intersection point $x'$ of the homotopy type of x. If we consider the curve representative $\gamma ' \in C'$ obtained by smoothing at $x'$ , then, since $i(C',D)$ is an infimum, we have

$$ \begin{align*} i(C',D) \le i(\gamma',\delta) = i(\gamma,\delta) = i(C,D), \end{align*} $$

as desired.

By Theorem A, intersection number with D extends to a continuous function on geodesic currents

$$ \begin{align*} i(\cdot, D) \colon \mathcal{GC}(S) \to \mathbb{R}. \end{align*} $$

We can then fix C and vary D to show that, for $\mu $ a geodesic current, $i(\cdot ,\mu )$ is a continuous function on $\mathcal {GC}(S)$ . In [Reference Bonahon10, Proposition 4.5], Bonahon shows that the geometric intersection number $i \colon \mathbb {R}\mathcal {C}(S) \times \mathbb {R}\mathcal {C}(S) \to \mathbb {R}_{\geq 0}$ between two weighted multi-curves extends to a continuous two-variable function

$$ \begin{align*}i \colon \mathcal{GC}(S) \times \mathcal{GC}(S) \to \mathbb{R}_{\geq 0}.\end{align*} $$

Following Example 1.12, proving that there is a continuous extension of $\sqrt {i(C,C)}$ to a function on $\mathcal {GC}(S)$ is equivalent to proving continuity of i as a two-variable function, by a simple polarisation argument:

$$ \begin{align*} i(C,D) = \frac{i(C\cup D, C \cup D) - i(C,C) - i(D,D)}{2}. \end{align*} $$

4.2 Hyperbolic length

We continue with the original motivating example for geodesic currents in Bonahon’s paper [Reference Bonahon8, Proposition 14]. Fix a hyperbolic metric g on S and denote the hyperbolic structure by $\Sigma $ . Then, for any closed curve C on S (not necessarily simple), we can consider its hyperbolic length with respect to the Riemannian metric. In terms of the holonomy representation $\rho _g: \pi _1(S) \to \operatorname {\mathrm {\mathit {PSL}}}_2(\mathbb {R})$ , this is given by

$$ \begin{align*} \ell_g(C) = 2\cosh^{-1}\left({\textstyle \frac{1}{2}} \operatorname{\mathrm{tr}}(\rho(c))\right) \end{align*} $$

where $c \in \pi _1(S)$ is a representative of C. We extend $\ell _g$ to a weighted curve functional by additivity and homogeneity:

$$ \begin{align*} \ell_g(t_1C_1 \cup \dots \cup t_nC_n) = \sum_{i=1}^n t_i\ell_g(C_i). \end{align*} $$

By definition, $\ell _g$ is additive and homogeneous. Stability follows from properties of the trace of $2\times 2$ matrices or geometrically from the length. Smoothing follows by the argument for intersection number. Thus, by Theorem A, $\ell _g$ extends to a continuous function on geodesic currents.

We recall that Bonahon shows that

$$ \begin{align*} \ell_g(C)=i(\mathcal{L}_{\Sigma},C) \end{align*} $$

where $\ell _g(C)$ denotes the hyperbolic length of C – that is, the length of the g-geodesic representative – and $\mathcal {L}_{\Sigma }$ denotes the Liouville current, a geodesic current induced by the volume form on $UT\Sigma $ . (For an equivalent formulation in terms of Definition 2.13, see Bonahon [Reference Bonahon8, Section 2].)

4.3 Length with respect to arbitrary metrics

The argument from Subsection 4.2 applies equally well to show that for any Riemannian or, more generally, length metric g on S, length $\ell _g$ with respect to g satisfies smoothing. For completeness and later use, we prove that these curve functionals are stable. (Freedman–Hass–Scott give a proof in the Riemannian case [Reference Freedman, Hass and Scott27, Lemma 1.3].)

4.4 Length with respect to embedded graphs

We can generalise further beyond length metrics. Let $\iota \colon \Gamma \hookrightarrow S$ be an embedding of a finite graph in S that is filling, in the sense that the complementary regions are disks or, equivalently, $\iota _*$ is surjective on $\pi _1$ . Endow $\Gamma $ with a length metric g. Then any closed multi-curve C on S can be homotoped so that it factors through $\Gamma $ , in many different ways. Let $\ell _\Gamma (C)$ be the length of the smallest multi-curve D on $\Gamma $ so that $\iota (D)$ is homotopic to C. It is easy to see that this length is realised and is positive. (In fact, we can see $\ell _\Gamma $ as a limit of lengths with respect to Riemannian metrics, by fixing an embedding of $\Gamma $ and making the metric on the complement of a regular neighbourhood of $\Gamma $ be very large, following Shepard [Reference Rodin53].)

As before, $\ell _\Gamma $ is clearly additive and homogeneous and is stable by the argument of Lemma 4.2. To see that $\ell _\Gamma $ satisfies smoothing at an essential crossing, take a minimal-length concrete representative $\delta $ of D on $\Gamma $ . Since the image $\iota \circ \delta $ has a corresponding crossing by Lemma 2.8 and $\iota $ is an embedding, there is a corresponding crossing of $\delta $ that can be smoothed and then tightened to get the desired inequality.

As a special case, we can consider the case when $\Gamma $ is a rose graph with only one vertex $\ast $ and edges of length $1$ . Since $\iota $ is filling, the image of the edges of $\Gamma $ give generators for $\pi _1(S,\iota (\ast ))$ . Then the length $\ell _\Gamma (C)$ of a curve C is the length of C as a conjugacy class in $\pi _1(S)$ with respect to these generators. This is a simple generating set in the sense of Erlandsson [Reference Erlandsson22], who proved this continuity and constructed an explicit multi-curve K so that $\ell _\Gamma (C) = i(C,K)$ .

4.5 Stable lengths

Generalising the previous example, let $\iota \colon \Gamma \to S$ be an immersion from a finite graph to S so that $\iota _* \colon \pi _1(\Gamma ) \to \pi _1(S)$ is surjective, and again give a length metric on $\Gamma $ . For instance, if $\Gamma $ has a single vertex and all edges have length $1$ , this is equivalent to giving an arbitrary generating set for $\pi _1(S)$ . We can define $\ell _\Gamma (C)$ as before, as the minimum length of any multi-curve D on $\Gamma $ so that $\iota _*(D) = C$ .

The curve functional $\ell _\Gamma $ is still additive, but unlike the previous examples it is not stable (see Example 4.10). Thus, we cannot hope to extend $\ell _\Gamma $ to currents but rather extend the stable curve functional ${\lVert } \ell _\Gamma {\rVert }$ (defined in Section 13). We do have quasi-smoothing.

This is a special case of a more general result. Let V be a connected length space, with a continuous, $\pi _1$ -surjective map $\iota \colon V \to S$ . Then for C a multi-curve on S, a lift of C is a multi-curve $\tilde {C}$ in V so that $\iota _* \tilde {C} = C$ . (Both $\tilde {C}$ and C are defined up to homotopy.) Define $\ell _{\iota ,V}(C)$ to be the infimum, over all lifts $\tilde {C}$ of C, of the length of $\tilde {C}$ in V.

As a corollary of Proposition 4.5 and Theorems B and A, ${\lVert } \ell _V {\rVert }$ extends continuously to a function on $\mathcal {GC}(S)$ . This continuous extension was first proved by Bonahon [Reference Bonahon9, Proposition 10] in the context of a hyperbolic group acting discretely and cocompactly on a length space (replacing $\pi _1(S)$ acting on $\widetilde {V}$ ), with an additional technical assumption that the space is uniquely geodesic at infinity. Later, Erlandsson, Parlier, and Souto [Reference Erlandsson, Parlier and Souto23, Theorem 1.5] lifted this assumption.

We remark that given a properly discontinuous action of $\pi _1(S)$ on X a CW-complex, we can construct a $\pi _1(S)$ -surjective map $\iota \colon X/\pi _1(S) \to S$ . In fact, we will construct a $\pi _1(S)$ -equivariant map $\iota \colon X \to \widetilde {S}$ . First, we define $\iota \colon X_0 \to \tilde {S}$ by picking a value on each $\pi _1(S)$ orbit of the 0-skeleton arbitrarily and extending equivariantly. Similarly, on the 1-skeleton $X_1$ , for each $\pi _1(S)$ orbit on $X_1$ , pick a path in $\widetilde {S}$ between the images of the endpoints. Continue the construction inductively. This construction works because $\pi _1(S)$ acts freely – since $\pi _1(S)$ is torsion-free and acts properly discontinuously – and $\widetilde {S}$ is contractible.

In this construction, proper discontinuity of the action is crucial. For example, it was shown by Bonahon in [Reference Bonahon9, Proposition 11] that if W is a finite graph which is a deformation retract of S, the action of $\pi _1(S)$ on the universal cover $\widetilde {W}$ of W is cocompact but not properly discontinuous and translation length of conjugacy classes of $\pi _1(S)$ acting on X does not extend continuously to $\mathcal {GC}(S)$ .

Proof of Proposition 4.5. Let $\widetilde {\iota } \colon \widetilde {V} \to \widetilde {S}$ be the pull-back of $\iota $ along the universal cover $\pi _S$ of S, part of the pullback square

(4.6)

where $\widetilde {V}=V \times _S \widetilde {S}$ . Since $\iota $ is $\pi _1$ -surjective, $\widetilde {V}$ is a connected covering space of V. For $\widetilde {x} \in \widetilde {S}$ , let $\operatorname {\mathrm {diam}}_\iota (\widetilde {x})$ be the diameter of $\widetilde {\iota }^{-1}(\widetilde {x}) \subset \widetilde {V}$ . (Set $\operatorname {\mathrm {diam}}_\iota (\widetilde {x}) = 0$ if $\widetilde {x}$ is not in the image of $\widetilde {\iota }$ .) Set

$$ \begin{align*} \operatorname{\mathrm{diam}}_\iota(V) := \mathop{\mathrm{sup}}\limits_{\widetilde{x} \in \widetilde{S}} \operatorname{\mathrm{diam}}_\iota(\widetilde{x}). \end{align*} $$

We wish to see that $\operatorname {\mathrm {diam}}_\iota (V)$ is finite. First, since $\iota $ is $\pi _1$ -surjective, for every $\gamma \in \pi _1(S,x)$ there exists $\delta _{\gamma } \in \pi _1(V,\tilde x)$ so that $\iota _* \delta _\gamma = \gamma $ . This implies

$$ \begin{align*} \delta_{\gamma} \cdot \tilde{\iota}^{-1}(\tilde{x}) = \tilde{\iota}^{-1}(\gamma \cdot \tilde{x}). \end{align*} $$

But $\delta _{\gamma }$ is a deck transformation and thus acts as an isometry on $\widetilde {V}$ , so $\operatorname {\mathrm {diam}}(\tilde {\iota }^{-1}(\gamma \cdot \tilde {x}))=\operatorname {\mathrm {diam}}(\tilde {\iota }^{-1}(\tilde {x}))$ and so we can define

(4.7) $$ \begin{align} \operatorname{\mathrm{diam}}_\iota(x) = \operatorname{\mathrm{diam}}_\iota(\widetilde{x}) \end{align} $$

for $x \in S$ and any lift $\widetilde {x}$ of x. (Note that $\operatorname {\mathrm {diam}}_\iota (x)$ is not in general the diameter of $\iota ^{-1}(x)$ ; rather than looking at the length of a shortest path connecting two points in $\iota ^{-1}(x)$ , we restrict to paths that map to null-homotopic loops.)

Proof. For each $x_0 \in S$ , consider an evenly covered neighbourhood U of $x_0$ and fix a lift $\tilde {x}_0 \in \widetilde {S}$ . We want to show that for all sequences $\{ x_i \} \subset U$ with $x_i \to x$ and for all $\varepsilon>0$ , there exists $i_0$ so that for all $i \geq i_0$

$$ \begin{align*} \operatorname{\mathrm{diam}}(\tilde{\iota}^{-1}(\tilde{x_i})) < \operatorname{\mathrm{diam}}(\tilde{\iota}^{-1}(\tilde{x})) + \varepsilon. \end{align*} $$

Now, $\tilde {\iota }$ is a pullback of a proper map, so it is a closed map.

Lemma 4.9 [Reference Shepard54, Theorem 005R]

Let X be a metric space and $f \colon X \to Y$ a proper map. For any continuous map $g \colon Z \to Y$ , the pullback map $X \times _Y Z \to Z$ is closed.

By definition of diameter, and since the fibres $\tilde {\iota }^{-1}(\tilde {x})$ are compact for any $\tilde {x}_i$ , we can find points $p_i,q_i \in \tilde {\iota }^{-1}(\tilde {x_i})$ so that $\operatorname {\mathrm {diam}}(\tilde {\iota }^{-1}(\tilde {x_i}))=d(p_i,q_i)$ . Also,by closedness of $\tilde {\iota }$ , a subsequence of the $p_i$ and $q_i$ converges to points $p,q \in \tilde {\iota }^{-1}(\tilde {x})$ . Furthermore, by continuity of distance, for any $\varepsilon>0$ , we have, for i large enough,

$$ \begin{align*} \operatorname{\mathrm{diam}}(\tilde{\iota}^{-1}(\tilde{x_i}))=d(p_i,q_i) <d(p,q) + \varepsilon \leq \operatorname{\mathrm{diam}}(\tilde{\iota}^{-1}(\tilde{x})) + \varepsilon, \end{align*} $$

finishing the proof of Lemma 4.8.

As a result of Lemma 4.8, the function $\operatorname {\mathrm {diam}}_\iota (x)$ is bounded on S; let $R(V)$ be this global bound.

We now finish the proof of Proposition 4.5. We are given a curve C with an essential crossing p and corresponding smoothing $C'$ . Pick a concrete curve $\tilde \gamma $ on V that comes within $\varepsilon $ of realising $\ell _{\iota ,V}(C)$ ; in particular, $\iota \circ \tilde \gamma $ represents C. By Lemma 2.8, there are points $x,y \in X(\tilde \gamma )$ so that $\iota (\tilde \gamma (x)) = \iota (\tilde \gamma (y))$ is a crossing corresponding to p. We wish to find another curve $\tilde \gamma '$ on V, with length not too much longer, so that $\iota \circ \tilde \gamma '$ represents $C'$ . We can do this by cutting $\widetilde {\gamma }$ at x and y, yielding endpoints $x_1$ , $x_2$ and $y_1$ , $y_2$ and reconnecting $x_1$ to $y_2$ and $y_1$ to $x_2$ by paths in V that project to the identity in $\pi _1(S)$ .

But the maximal length of a path connecting any two points $x,y \in V$ with $\iota (x) = \iota (y)$ that projects to a null-homotopic path is exactly $\operatorname {\mathrm {diam}}_\iota (\iota (x))$ . We can therefore construct a desired representative $\tilde \gamma '$ with

$$ \begin{align*} \ell_V(\tilde\gamma') \le \ell_V(\tilde\gamma) + 2R(V) \le \ell_{\iota,V}(C) + \varepsilon + 2R(V). \end{align*} $$

Since $\varepsilon $ was arbitrary, we have proved the result with quasi-smoothing constant $2R(V)$ .

We show now an example of a curve functional that satisfies quasi-smoothing but not strict smoothing.

Example 4.10. Consider the torus with one puncture with fundamental group generated by the usual horizontal loop a and vertical loop b, as in Figure 4.1. (This does not, strictly speaking, fit in the context of closed surfaces considered in this article, but we can embed this punctured torus in a larger surface without essential change.) Its fundamental group is the free group $F_2 = \langle a, b\rangle $ . We will consider word length f with respect to the generating set $(a, a^2, b)$ . Word length satisfies quasi-smoothing and additive union but not stability. (For instance, $f(a^2) = 1 \ne f(2a) = 2$ .) The stable word length $\| f\|$ satisfies stability and still satisfies quasi-smoothing, but it does not satisfy strict smoothing. We will show it behaves more erratically than word length with respect to embedded generating sets.

Figure 4.1 A punctured torus with, in blue, a train track carrying a slice of measured laminations on the punctured torus depending on a parameter $x\in [0,1]$ . In red, the parallel loop a. In green, the meridian loop b.

Consider, for example, the collection of weighted curves $C(x)$ carried by the train track in Figure 4.1, with weights depending on a rational parameter $x \in [0,1]$ . For instance, for $x = 2/5$ , the curve is $1/5[aabab]$ , with stable length $(1/5) \cdot 4$ . If we plot the stable word length of $C(x)$ multiplied by the weight, we obtain the saw-tooth graph in Figure 4.2. We note the erratic behavior as a function of x. In particular, it is far from convex. If $\|f\|$ satisfied the smoothing property, then it would be a convex function of the train track weights, since if $w_1$ and $w_2$ are two rational weights on a train track T, the weighted multi-curve $T(w_1) \cup T(w_2)$ can be smoothed to $T(w_1 + w_2)$ (observed by Mirzakhani in [Reference Mirzakhani42, Appendix A] and the second author in [55, Subsection 3.2]).

Figure 4.2 We consider the curve $C(x)$ carried by a train track depending on a parameter x and plot the stable word length of $C(x)$ as a function of x. The graph has vertices at $\big (\frac {1}{2n+1},\frac {n+1}{2n+1}\big )$ and $ \big (\frac {1}{2n},\frac {n+1}{2n}\big )$ .

4.6 Asymmetric lengths

The arguments in Subsection 4.5 apply equally well to cases where distances may be zero or not symmetric. For instance, we can take a directed graph $\Gamma $ with a nonnegative length on each edge, together with a map $\iota \colon \Gamma \to S$ so that the corresponding cover $\widetilde {\Gamma }$ is strongly connected (every vertex can be reached from any other vertex). The same arguments apply to show that $\ell _\Gamma (\vec C)$ satisfies the oriented quasi-smoothing property and so its stabilisation ${\lVert } \ell _\Gamma {\rVert }$ extends to a continuous function $\mathcal {GC}^+(S) \to \mathbb {R}_{\ge 0}$ .

One example would be to take a generating set for $\pi _1(S)$ as a monoid. This corresponds to taking $\Gamma $ to be a graph with a single vertex and one edge for each monoid generator.

4.7 Generalised translation lengths from higher representations

Let G be a real, connected, noncompact, semi-simple, linear Lie group. Let K denote a maximal compact subgroup of G, so that $X=G/K$ is the Riemannian symmetric space of G. Let $[P]$ be the conjugacy class of a parabolic subgroup $P \subset G$ . Then there is a natural notion of $[P]$ -Anosov representation $\rho \colon \pi _1(S) \to G$ ; see, for example, Kassel’s notes [Reference Kassel34, Section 4]. When $\operatorname {rank}_{\mathbb {R}}(G)=1$ there is essentially one class $[P]$ , so we can simply refer to them as Anosov representations, and they can be defined as those injective representations $\rho \colon \pi _1(S) \to G$ where $\Gamma := \rho (\pi _1(S))$ preserves and acts cocompactly on some nonempty convex subset V of X.

Rank $1$ Anosov representations include two familiar examples:

1. (1) Fuchsian representations into $G=\operatorname {\mathrm {\mathit {PSL}}}(2,\mathbb {R})$ . Here, $K=\operatorname {\mathrm {\mathit {SO}}}(2)$ and $X=\mathbb {H}^2$ . The convex set V in this case is the lift of the convex core of the hyperbolic surface.

2. (2) Quasi-Fuchsian representations of surface groups into $G=\operatorname {\mathrm {\mathit {PSL}}}(2,\mathbb {C})$ . In this case, $K=\operatorname {\mathrm {\mathit {SU}}}(2)$ , $X=\mathbb {H}^3$ , and V is the lift of the convex core of the hyperbolic quasi-Fuchsian manifold.

In general, the conjugacy classes of parabolic subgroups of G correspond to subsets $\theta $ of the set of restricted simple roots $\Delta $ of G. For a given $[P]$ -Anosov representation and each $\alpha \in \theta $ , Martone and Zhang define [Reference Martone and Zhang38, Definition 2.21] a curve functional

$$ \begin{align*} l_{\alpha}^{\rho}\colon \mathcal{C}(S) \to \mathbb{R}_{\geq 0} \end{align*} $$

and show that for a certain subset of Anosov representations, these can be extended to geodesic currents as intersection numbers with some fixed geodesic current.

For the two rank 1 examples above, this length $l_{\alpha }^{\rho }(C)$ corresponds to hyperbolic length of the closed geodesic in the homotopy class C in the quotient hyperbolic manifold $\mathbb {H}^2/\rho (\pi _1(S))$ or $\mathbb {H}^3/\rho (\pi _1(S))$ .

Bonahon showed the length in the Fuchsian case extends to geodesic currents [Reference Bonahon8, Proposition 14]. Our techniques give an extension to geodesic currents of the quasi-fuchsian length; that is, the hyperbolic length of the geodesic representatives in the quasi-Fuchsian manifold.

4.8 Extremal length

We now turn to curve functionals that satisfy only convex union and not additive union, starting with the original motivation for this work, extremal length.

Definition 4.15. Fix $\Sigma $ a Riemann surface with a metric g. Let $C = \bigcup t_iC_i$ be a weighted multi-curve on $\Sigma $ . For $\rho \colon \Sigma \to \mathbb {R}_{\ge 0}$ a measurable rescaling function, the area of $\rho $ is

$$ \begin{align*} \operatorname{\mathrm{Area}}(\rho g) := \int_{x \in \Sigma} \rho(x)^2 \mu_g(x), \end{align*} $$

where $\mu _g$ is the Lebesgue measure of g. The length of C is

$$ \begin{align*} \ell_{\rho g}(C) := \inf_{\gamma \in C} \sum_i t_i \int_{x\in\gamma_i} \rho(x)\,dx, \end{align*} $$

where $dx$ is measured with respect to g arc-length and the infimum runs overall all representatives $\gamma = \bigcup _i \gamma _i$ of C, where $\gamma _i$ is a representative of the component $C_i$ of C. When $\rho $ is continuous, $\ell _\rho (C)$ is the length with respect to the metric g rescaled by $\rho $ . The square root of the extremal length of C is

$$ \begin{align*} \sqrt{\operatorname{\mathrm{EL}}}(C) := \mathop{\mathrm{sup}}\limits_\rho \frac{\ell_{\rho g}(C)}{\sqrt{\operatorname{\mathrm{Area}}(\rho g)}}. \end{align*} $$

Observe that the supremand is unchanged under multiplying $\rho $ by a positive constant. It is a standard result that the supremum is realised by some generalised metric (not necessarily Riemannian) [Reference Tyrell Rockafellar52, Theorem 12] and that, when C is a simple multi-curve, the optimum metric $\rho g$ is the cone Euclidean metric associated to a quadratic differential, as shown by Jenkins [Reference Jenkins33]. Very little is known about the optimum metric when C is not simple, except in special cases [Reference Wolf and Zwiebach62, Reference Calabi14, Reference Headrick and Zwiebach31, Reference Naseer and Zwiebach46].

Proof. This follows since $\ell _{\rho g}$ satisfies these properties for each $\rho $ ; here is the argument for smoothing.

Let C be a multi-curve with an essential crossing and let $C'$ be the curve obtained by smoothing at the crossing. Then, for any scaling function $\rho $ ,

$$ \begin{align*} \frac{\ell_{\rho g}(C')}{\sqrt{\operatorname{\mathrm{Area}}(\rho g)}} \le \frac{\ell_{\rho g}(C)}{\sqrt{\operatorname{\mathrm{Area}}(\rho g)}}. \end{align*} $$

Since $\sqrt {\operatorname {\mathrm {EL}}}(C')$ and $\sqrt {\operatorname {\mathrm {EL}}(C)}$ are the suprema of such terms, the result follows.

Convex union is different, since on one side of the inequality we have a sum of values of $\sqrt {\operatorname {\mathrm {EL}}}$ . (Extremal length does not satisfy additivity.)

Proof. Fix a curve split as a union $C = C_1 \cup C_2$ , and let $\rho \colon \Sigma \to \mathbb {R}$ be the function realising the supremum in the definition of extremal length for C. Then

$$ \begin{align*} \sqrt{\operatorname{\mathrm{EL}}}(C_1 \cup C_2) &= \frac{\ell_{\rho g}(C_1)}{\sqrt{\operatorname{\mathrm{Area}}(\rho g)}} + \frac{\ell_{\rho g}(C_2)}{\sqrt{\operatorname{\mathrm{Area}}(\rho g)}}\\ &\le \sqrt{\operatorname{\mathrm{EL}}}(C_1) + \sqrt{\operatorname{\mathrm{EL}}}(C_2), \end{align*} $$

where the last inequality holds by the supremum in the definition of $\operatorname {\mathrm {EL}}$ .

Thus, by Theorem A, $\sqrt {\operatorname {\mathrm {EL}}}$ (and $\operatorname {\mathrm {EL}}$ ) extend uniquely to continuous functions on geodesic currents. With this extension, we propose the following conjecture.

Conjecture 4.18. For some universal constant C,

$$ \begin{align*}\operatorname{\mathrm{EL}}_{\Sigma}(\mathcal{L}_{\Sigma})=C\operatorname{\mathrm{Area}}(\Sigma),\end{align*} $$

where $\mathcal {L}_{\Sigma }$ is the Liouville current (compare Subsection 4.2).

Remark 4.19. For the extremal length without the square root, we instead have inequalities

$$ \begin{align*} \operatorname{\mathrm{EL}}(C_1) + \operatorname{\mathrm{EL}}(C_2) \le \operatorname{\mathrm{EL}}(C_1 \cup C_2) \le 2(\operatorname{\mathrm{EL}}(C_1) + \operatorname{\mathrm{EL}}(C_2)). \end{align*} $$

The second inequality is a simple consequence of Lemma 4.17. To see the first inequality, take optimal rescaling functions $\rho _i$ for $\operatorname {\mathrm {EL}}(C_i)$ , normalised so that $\ell _{\rho _i g}(C_i) = \operatorname {\mathrm {Area}}(\rho _i g) = \operatorname {\mathrm {EL}}(C_i)$ . Then using $\rho _1 + \rho _2$ as the test function for $\operatorname {\mathrm {EL}}(C_1 \cup C_2)$ gives the desired inequality after elementary manipulations.

4.9 Extremal length with respect to elastic graphs

There is a parallel notion of extremal length with respect to elastic graphs, as introduced by the second author [Reference Thurston56], just as there is for ordinary lengths (Subsection 4.4).

An elastic graph $(\Gamma ,\alpha )$ is a 1-dimensional CW complex $\Gamma $ (i.e., allowing multiple edges and loops) together with an assignment of positive real numbers $\alpha (e)$ for each $e \in \operatorname {Edge}(\Gamma )$ , where the edges are the 1-dimensional cells of $\Gamma $ .

By a concrete multi-curve $\gamma $ on $\Gamma $ we mean a 1-manifold $X(\gamma )$ and a PL map $\gamma \colon X(\gamma ) \to \Gamma $ . Given a scaling function $\rho \colon \operatorname {Edge}(\Gamma ) \to \mathbb {R}_{\geq 0}$ , the length metric $\rho \alpha $ on $\Gamma $ gives edge e the length $\rho (e)\alpha (e)$ . We define the length of $\gamma $ as

$$ \begin{align*} \ell_{\rho \alpha}(\gamma) := \sum_{e \in \operatorname{Edge}(\Gamma)} n_\gamma(e)\rho(e)\alpha(e), \end{align*} $$

where $n_\gamma (e)$ is the weighted number of times that $\gamma $ runs over e. We can likewise define the length of a multi-curve D on $\Gamma $ as the infimum over of concrete multi-curves in D.

The area of $\Gamma $ with respect to $\rho \alpha $ is defined to be

$$ \begin{align*} \operatorname{\mathrm{Area}}_{\rho}(\Gamma,\alpha) := \sum_{e \in \operatorname{Edge}(\Gamma)} \rho(e)^2 \alpha(e). \end{align*} $$

Intuitively, each edge is turned into a rectangle of width $\rho (e)$ , aspect ratio $\alpha (e)$ and thus area $\rho (e)^2\alpha (e)$ .

As for extremal length for surfaces, we define the square root of extremal length of a multi-curve on $\Gamma $ by

(4.20) $$ \begin{align} \sqrt{\operatorname{\mathrm{EL}}}(D; \Gamma,\alpha) := \mathop{\mathrm{sup}}\limits_{ \rho : \operatorname{Edge}(\Gamma) \to \mathbb{R}_{\geq 0}} \frac{\ell_{\rho \alpha}(D)}{\sqrt{\operatorname{\mathrm{Area}}_{\rho}(\Gamma,\alpha)}}. \end{align} $$

It is easy to do this optimisation. We get a more interesting quantity by incorporating a filling embedding $\iota \colon \Gamma \hookrightarrow S$ of $\Gamma $ in a surface S; that is, an embedding $\iota $ that is $\pi _1$ -surjective. Then, for a multi-curve C on S and scaling $\rho $ , the length is defined as in Subsection 4.4:

$$ \begin{align*} \ell_{\rho\alpha;\iota}(C) := \inf_{\substack{D\text{ on }\Gamma\\\iota_*(D) = C}} \ell_{\rho\alpha}(D). \end{align*} $$

(The ‘filling’ condition guarantees that there are such multi-curves D with $\iota _*(D) = C$ .) We can then define a version of extremal length, following equation (4.20), with respect to $\iota $ :

(4.21) $$ \begin{align} \sqrt{\operatorname{\mathrm{EL}}}(C; \Gamma,\alpha,\iota) := \mathop{\mathrm{sup}}\limits_{ \rho : \operatorname{Edge}(\Gamma) \to \mathbb{R}_{\geq 0}} \frac{\ell_{\rho \alpha;\iota}(D)}{\sqrt{\operatorname{\mathrm{Area}}_{\rho}(\Gamma,\alpha)}}. \end{align} $$

We can also consider extremal length with respect to an immersion $\iota $ (rather than an embedding), defined in the same way.

Proof. Convex union and homogeneity still hold by the same argument. We need an extra argument for quasi-smoothing. Instead of taking the supremum over all $\rho $ , rewrite equation (4.21) as

$$ \begin{align*} \sqrt{\operatorname{\mathrm{EL}}}(C; \Gamma, \alpha, \iota) = \mathop{\mathrm{sup}}\limits_{\substack{\rho : \operatorname{Edge}(\Gamma) \to \mathbb{R}_{\ge 0}\\ \operatorname{Area}_\rho(\Gamma,\alpha) = 1}} \ell_{\rho\alpha}(C). \end{align*} $$

The immersed graph $\iota (\Gamma )$ has finitely many self-intersections. For each self-intersection x, take the supremum over the compact set of metrics $\{\rho \mid \operatorname {\mathrm {Area}}_\rho (\Gamma ,\alpha ) = 1\}$ of the diameter $\operatorname {\mathrm {diam}}_\iota (x)$ defined in equation (4.7). By the replacement argument in Proposition 4.5, all $\ell _{\rho \alpha }$ for $\rho $ in this set satisfy quasi-smoothing with a uniform quasi-smoothing constant. It follows that $\operatorname {\mathrm {EL}}(C; \Gamma ,\alpha ,\iota )$ also satisfies quasi-smoothing with the same constant.

4.10 p-Extremal-length with respect to immersed graphs

Extremal length fits into a family of energies for graphs, as explored by the second author in [Reference Thurston56, Appendix A]. For $\Gamma $ a metric graph with metric g, a constant p with $1 \le p \le \infty $ and C a curve on $\Gamma $ , define

(4.25) $$ \begin{align} E_p(C;\Gamma,g) :=\! \mathop{\mathrm{sup}}\limits_{\sigma:\operatorname{Edge}(\Gamma) \to \mathbb{R}_{\ge 0}} \frac{\ell(C; \sigma g)}{{\lVert} \sigma {\rVert}_p} \end{align} $$

where the $L^p$ norm ${\lVert } \sigma {\rVert }_p$ is taken with respect to the metric g. As in the previous section, we can also consider a $\pi _1$ -surjective immersion $\iota \colon \Gamma \to S$ and consider C to be a curve on S rather than on $\Gamma $ .

For $p=\infty $ , $E_\infty (C)$ is in fact just the length with respect to g (as in Subsection 4.5). Indeed, let $\sigma $ be any scaling factor, and let $\gamma $ be the shortest representative of C on $\Gamma $ with respect to g (not with respect to $\sigma g$ ). Then

$$ \begin{align*} \ell(C; \sigma g) \le \ell(\gamma; \sigma g) \le {\lVert} \sigma {\rVert}_\infty \ell(\gamma; g), \end{align*} $$

from which the result easily follows.

7.1 Smeared first return map

Let Y be a smooth closed manifold with a smooth flow $\phi _t$ . For us, a cross section is a compact smooth codimension $1$ submanifold-with-boundary $\tau $ that is smoothly transverse to the foliation of Y given by $\phi _t$ . A global cross section is a cross section $\tau $ so that its interior $\tau ^\circ $ intersects all forward and backward orbits: for all $x \in Y$ , there exists $s<0$ and $t> 0$ with $\phi _s(x), \phi _t(x) \in \tau ^\circ $ . Any flow on a compact manifold has global cross section consisting of a union of finitely many disks, although not all flows on compact manifolds admit a global cross section consisting of a single connected component. (For instance, the Reeb foliation on $\mathbb {T}^2$ has no connected global cross section.) By the implicit function theorem, for any cross section $\tau $ there is a larger cross section $\tau '$ with $\partial \tau \subset \tau '{}^\circ $ , which we also write $\tau \Subset \tau '$ .

Let $t_\tau \colon Y \to \mathbb {R}$ be the first return time defined by $t_{\tau }(x):=\min \{ t>0 \mid \phi _t(x) \in \tau \}$ , and let $p_{\tau }(x):=\phi _{t_{\tau }(x)}(x)$ . Then $p_{\tau }$ (restricted to $\tau $ ) is the first return map associated to the cross section $\tau $ . We will omit the subscript on $p_\tau $ if it is clear from context. We also have the first return time to the interior, denoted $t_\tau ^\circ $ . (Recall we assume $t_\tau ^\circ (x)$ is finite.)

If $\tau $ has no boundary, then p is a homeomorphism. On the other hand, if the cross section has a noninvariant boundary (i.e., $p(\partial \tau ) \ne \partial \tau $ ), then $t_\tau $ and p will have discontinuities. This necessarily happens for the geodesic flow on the unit tangent bundle of a hyperbolic surface. See Cossarini–Dehornoy [Reference Cossarini and Dehornoy15, Sec. 1] for a justification and examples of global cross sections with boundary for this flow; we construct our own cross section in Section 8. However, by the continuity of $\phi _t$ with respect to initial parameters, we have the following ‘local continuity’ claim.

This is presumably standard (Basener gives this as ‘a useful technical lemma, the proof of which is trivial’ [Reference Basener3, Lem. 1]), but we give a proof for completeness.

Proof. Pick an initial neighbourhood $U_1'$ of $x_1$ in $\tau $ , and let $\varepsilon $ be small enough so that $V_1 := \phi _{(-\varepsilon ,\varepsilon )}(U_1')$ is a 3-dimensional flow-box neighbourhood of $x_1$ in Y. Then the restriction of $\phi _t$ to $V_1$ is a homeomorphism to a neighbourhood $V_2$ of $x_2$ in Y. Set $U_2':= V_2 \cap \tau _2$ . Now consider the composition $\psi := \pi _1 \circ \phi _{-t} \circ \iota _2 \colon U_2' \to U_1'$ , where $\iota _i \colon U_i' \hookrightarrow V_i$ is the inclusion and $\pi _1 \colon V_1 \to U_1$ is the flow projection:

Then $\psi $ is a map from $U_2'$ to $U_1'$ , taking $x_2$ to $x_1$ . By transversality of $\tau _1$ and $\tau _2$ , the differential of $\psi $ at $x_2$ is invertible. Thus, by the inverse function theorem, there is a neighbourhood $U_2$ of $x_2$ and $U_1$ of $x_1$ so that the restriction of $\psi $ is a diffeomorphism from $U_2$ to $U_1$ . For $x \in U_1$ , set $t^1_2(x) := t + \pi _t(\phi _{-t}(\psi ^{-1}(x)))$ , where $\pi _t \colon V_1 \to (-\varepsilon ,\varepsilon )$ is the projection onto the time coordinate of the flow box. We have $\phi _{t^1_2}(x) = \psi ^{-1}(x) \in U_1$ , as desired for the first claim.

For the second claim, by hypothesis, the compact sets $\phi _{[0,t]}(x_1)$ and $\tau _2$ do not intersect, so $\phi _{[0,t]}(x_1)$ has an open neighbourhood that does not intersect $\tau _2$ . It follows that we can shrink $U_1$ and $U_2$ so that $\psi ^{-1}$ restricted to $U_1$ is the first return map to $\tau _2$ .

In the $C^1$ setting, Basener showed that a global cross section can be perturbed slightly so that the first return map is piecewise continuous with a cellular structure [Reference Basener4]. However, we want a continuous version of the first return map, so we proceed in a different direction.

Definition 7.4. Fix a nested pair of global cross sections $\tau _0 \Subset \tau $ . A bump function $\psi $ for this pair is a continuous function $\psi \colon \tau \to [0,1]$ so that $\psi $ is $1$ on $\tau _0$ and $0$ on an open neighbourhood of $\partial \tau $ . Set $\bar {\psi }(x) = 1-\psi (x)$ . Let $p\colon \tau \to \tau $ be the first return map with respect to $\tau $ . Then the smeared first return map of $\psi $ is a function $P_\psi \colon \tau \to \mathbb {R}_1\tau $ defined by

$$ \begin{align*} P_{\psi}(x):= \begin{cases} p(x) & p(x) \in \tau_0\\ \psi(p(x))\cdot p(x) + \bar\psi(p(x))\cdot P_{\psi}(p(x)) & p(x) \in \tau - \tau_0. \end{cases} \end{align*} $$

Here, $\mathbb {R}_1\tau \subset \mathcal {M}_1(\tau )$ is the subspace of measures with finite support and total mass $1$ ; see Convention 2.24.

The convention here is that a smeared map takes values in finite linear combinations of the target space (or maybe in measures). We use capital letters for smeared maps.

Intuitively, we iterate x forward, stopping at each iterate with probability given by $\psi $ . More visually, imagine the original cross section as a disk. As we look along the flow lines, we see an overlapping set of disks, with hard edges between them. To find the smeared first return map, we ‘feather’ the edges by giving the disks partially transparent boundaries made out of cellophane. If we continuously increase the transparency towards the boundary, the resulting image will have soft edges. See Figure 7.1.

Figure 7.1 Smeared first return map, illustrating the proof of continuity in the case $n=3$ (before shrinking $\tau $ ). The bump function $\psi $ is indicated by the density of red.

We will usually omit $\psi $ from the notation and denote the smeared first return map by P.

Since $\tau _0$ is a global cross section, in the definition of P we eventually take the first choice, and so $P(x)$ is a finite sum of elements of $\tau _1$ as claimed. A little more is true.

As a consequence of Lemma 7.5, we can rewrite P directly. Let N be the bound from Lemma 7.5. Then

(7.6) $$ \begin{align} \begin{aligned} P(x) &= \psi(p(x)) \cdot p(x) + \bar\psi(p(x))\psi(p^2(x)) \cdot p^2(x) + \cdots \\ &= \sum_{k=1}^N \bar\psi(p(x))\cdots\bar\psi(p^{k-1}(x)) \psi(p^k(x)) \cdot p^k(x). \end{aligned} \end{align} $$

If we extend the upper limit of the sum beyond N, the additional terms will be $0$ .

Proof. We wish to show that P is continuous at $x \in \tau $ . There is some first $n> 0$ such that $p^n(x) \in \tau _0^\circ $ . If there is any i between $1$ and n so that $p^i(x) \in \partial \tau $ , we first find a smaller cross section $\tau '$ with $\tau _0 \Subset \tau ' \Subset \tau $ without this problem, as follows. Recall that we assumed that $\psi $ vanishes in a neighbourhood of $\partial \tau $ . Since there are finitely many points $p^i(x)$ in the open set $\tau \setminus \operatorname {\mathrm {supp}}(\psi )$ , we can pick $\tau '$ containing $\operatorname {\mathrm {supp}}(\psi )$ so that its boundary avoids those finitely many $p^i(x)$ . Since $\psi $ vanishes on $\tau \setminus \tau '$ , the smeared first return map defined with respect to $\tau '$ agrees with that defined with respect to $\tau $ . By replacing $\tau $ by $\tau '$ , we may thus assume that $p^i(x) \notin \partial \tau $ . Similarly, shrink $\tau _0$ so that $p^i(x) \notin \partial \tau _0$ for $0 < i < n$ .

We proceed by induction on n. If $n=1$ – that is, if $p(x) \in \tau _0^\circ $ – then p is continuous at x by Lemma 7.2, and P is continuous since the map taking a point x to the delta function $\delta _x$ is continuous. Otherwise, note that $p_1$ (the return map for $\tau $ ) is continuous at x (again by Lemma 7.2, since $p_1(x) \in \tau ^\circ $ ). By induction, P is continuous at $p_1(x)$ , and therefore

$$ \begin{align*} P(x) = \psi(p_1(x)) \cdot p_1(x) + \bar\psi(p_1(x)) \cdot P(p_1(x)) \end{align*} $$

is continuous at x.

To define iterates of P, we first extend P and other functions to act on measures.

Definition 7.8. When $X, Y$ are measure spaces and $f \colon X \to Y$ is a measurable function, by convention we extend f to a function $\mathcal {M}_1(X) \to \mathcal {M}_1(Y)$ acting on measures, denoted $f_*$ (or simply f), by setting

$$ \begin{align*} f_*(\mu)(S) := \mu(f^{-1}(S)) \end{align*} $$

for $\mu \in \mathcal {M}_1(X)$ and $S \subset Y$ a measurable set. If f is continuous, then this extension is continuous with respect to the weak $^*$ topology on $\mathcal {M}_1(X)$ and $\mathcal {M}_1(Y)$ . (This uses Proposition 2.23 and the fact that if $g \colon Y \to \mathbb {R}$ is bounded, then $g \circ f \colon X \to \mathbb {R}$ is also bounded.) If f is invertible and $\psi \colon X \to \mathbb {R}_{\ge 0}$ is a scaling factor, then

(7.9) $$ \begin{align} f(\psi \cdot \mu) = (\psi \circ f^{-1}) \cdot f(\mu). \end{align} $$

In practice, we will often be interested in the subspace of finitely supported measures, in which case the extension $f\colon \mathbb {R}_1 X \to \mathbb {R}_1 Y$ is given by

$$ \begin{align*} f\Bigl(\sum a_i x_i\Bigr) := \sum a_i f(x_i). \end{align*} $$

For $F: X \to \mathcal {M}_1(Y)$ a smeared function, we extend F to a function $\mathcal {M}_1(X) \to \mathcal {M}_1(Y)$ from measures to measures, denoted $\tilde F$ (or simply F), by setting, for any measurable function $\varphi \colon Y \to \mathbb {R}_{\geq 0}$ ,

$$ \begin{align*} F_\varphi(x) &:= \int_{y \in Y}\varphi(y) \bigl(F(x)\bigr)(y) \\ \int_{y \in Y} \varphi(y)\tilde{F}(\mu)(y) &:= \int_{x \in X} F_\varphi(x)\mu(x), \end{align*} $$

where $F_\varphi \colon X \to \mathbb {R}_{\ge 0}$ is an auxiliary function. See also equations (7.11) and (7.12).

Proof. Let $(x_i)_{i=0}^\infty $ be a sequence approaching $x\in X$ . By assumption, $F(x_i)$ approaches $F(x)$ in the weak $^*$ topology. By Proposition 2.23, this is equivalent to saying that for all continuous bounded functions $\varphi \colon Y \to \mathbb {R}_{\geq 0}$ the function $F_{\varphi }$ above is continuous. Furthermore, $F_{\varphi }$ is bounded since F takes values in probability measures. We now show that $\tilde {F}$ is continuous. Let $\mu _i \to \mu \in \mathcal {M}(X)$ . We want to show that $\tilde F(\mu _i) \to \tilde F(\mu )\in \mathcal {M}_1(Y)$ ; that is, for any continuous bounded function $\varphi \colon Y \to \mathbb {R}_{\geq 0}$ ,

$$ \begin{align*} \int_{x \in X} F_{\varphi}(x)\mu_i(x) \to \int_{x \in X} F_{\varphi}(x)\mu(x). \end{align*} $$

This is true by definition of the weak $^*$ topology in $\mathcal {M}_1(X)$ and Proposition 2.23, since $F_{\varphi }$ is continuous and bounded.

In our applications, F takes values in finitely supported measures, with a bound on the size of the support. Concretely, if $F: X \to \mathbb {R} Y$ can be written as a finite sum

$$ \begin{align*} F(x) = \sum_{i=1}^N \psi_i(x) f_i(x) \end{align*} $$

for real-valued functions $\psi _i$ and invertible Y-valued functions $f_i$ , then, by equation (7.9), the extension is defined by

(7.11) $$ \begin{align} F(\mu) = \sum_{i=1}^N f_i(\psi_i\cdot\mu) = \sum_{i=1}^N (\psi_i\circ f_i^{-1}) \cdot f_i(\mu) \end{align} $$

where the middle expression is a sum of pushforwards of scaled measures, and in the last expression we have pulled the scaling factors out. If $\mu $ is also finitely supported, we have

(7.12) $$ \begin{align} F\Bigl(\sum_i a_i x_i \Bigr) = \sum_{i,j} a_i \psi_j(x_i) f_j(x_i). \end{align} $$

Definition 7.13. With the above extension of notation, the iterates of the smeared return map P are defined by

(7.14) $$ \begin{align} \begin{aligned} P^0(x) &:= x \\ P^n(x) &:= P(P^{n-1}(x)). \end{aligned} \end{align} $$

We have the following invariance property.

For motivation for the factor of $\psi $ in the proposition statement, think about extending the definitions to allow $\psi $ to be the (noncontinuous) characteristic function of $\tau _0 \Subset \tau $ ; then P is the ordinary first return map to $\tau _0$ , and $\psi \nu _\tau = \nu _{\tau _0}$ is the flux of $\tau _0$ . See also Example 9.3.

Proof. The first part is standard. For the second part, let $\mu = \nu _\tau $ . Then we have

$$ \begin{align*} P\bigl(\psi\cdot\mu\bigr) &= p\bigl(\psi\cdot(\psi\circ p)\cdot\mu\bigr) + p^2\bigl(\psi\cdot (\bar\psi\circ p)\cdot (\psi\circ p^2)\cdot\mu\bigr) + \dots\\ &= \sum_{k=1}^N p^k\bigl(\psi\cdot(\bar\psi \circ p)\cdots (\bar\psi\circ p^{k-1})\cdot(\psi \circ p^k)\cdot \mu\bigr)\\ &= \sum_{k=1}^N (\psi\circ p^{-k})\cdot(\bar\psi \circ p^{-k+1})\cdots (\bar\psi \circ p^{-1}) \cdot \psi\cdot p^k(\mu)\\ &= \sum_{k=1}^N (\psi\circ p^{-k})\cdot(\bar\psi \circ p^{-k+1})\cdots (\bar\psi \circ p^{-1}) \cdot \psi\cdot \mu \end{align*} $$

using the definition of P (in the form of equation (7.6)), rewriting as a sum, equation (7.11) and invariance of $\mu $ under p. Since $\psi + \bar \psi = 1$ , this sum telescopes, and the result is $\psi \mu $ .

We will sometimes blur the distinction between a geodesic current and its flux and write, for instance, $\nu (\tau )$ for the total mass of $\nu _\tau $ on $\tau $ , or $\nu (\psi \tau )$ for $\int _{x \in \tau } \psi (x) \nu _\tau (x)$ .

7.2 Homotopy type of return

We will additionally need to track how a point returns to the cross section. For this, we suppose that we have a global cross section $\tau $ contained in a simply connected cross section $\tau '$ . (For a $C^1$ flow on a manifold of dimension at least 3, there is always a simply connected cross section [Reference Basener4].) For such a cross section, from the first return for $x \in \tau $ , we can extract another piece of information: the homotopy class of the return trajectory.

To get the return map for iterates, we also incorporate the point of first return.

Definition 7.19. The homotopy return map is the map $q \colon \tau \to \tau \times \pi _1(Y,*)$ defined by

$$ \begin{align*} q(x) := (p(x), m(x)). \end{align*} $$

We can iterate q by inductively defining $q^{n+1}$ to be the composition

$$ \begin{align*} \tau \overset{q^n}{\longrightarrow} \tau \times \pi_1(Y) \xrightarrow{q \times \operatorname{\mathrm{id}}} \tau \times \pi_1(Y) \times \pi_1(Y) \xrightarrow{(x,g,h)\mapsto(x,hg)} \tau \times \pi_1(Y). \end{align*} $$

Define $m^n(x)\in \pi _1(Y,*)$ to be the second component of $q^n(x)$ .

Definition 7.21. For $\tau $ a global cross section with basepoint $*$ , $\tau _0 \Subset \tau $ a smaller global cross section, $\psi $ a bump function for this pair and $\tau ' \supset \tau $ a simply connected cross section, the smeared homotopy return map $Q\colon \tau \to \mathbb {R}_1(\tau \times \pi _1(Y,*))$ is defined by

$$ \begin{align*} Q(x) &:= \begin{cases} q(x) & p(x) \in \tau_0 \\ \psi(p(x)) \cdot q(x) + \bar\psi(p(x))\cdot L_{m(x)}Q(p(x)) & p(x) \in \tau - \tau_0 \end{cases} \end{align*} $$

where $L_g$ is left translation by $g \in \pi _1(Y,*)$ :

$$ \begin{align*} L_{g}\Bigl(\sum_ia_i(x_i,h_i)\Bigr) := \sum_i a_i (x_i, gh_i). \end{align*} $$

There is once again a natural notion of iteration, defined by inductively setting $Q^{n+1}$ to be the composition

$$ \begin{align*} \tau \overset{Q^n}{\longrightarrow} \mathbb{R}_1(\tau \times \pi_1(Y)) \xrightarrow{\mathbb{R}_1(Q \times \operatorname{\mathrm{id}})} \mathbb{R}_1(\mathbb{R}_1(\tau \times \pi_1)\times\pi_1(Y)) \xrightarrow{\operatorname{join}} \mathbb{R}_1(\tau \times \pi_1(Y)), \end{align*} $$

where $\operatorname {join}$ is the somewhat more involved operation

$$ \begin{align*} \operatorname{join}\Biggl(\sum_i a_i \Bigl(\Bigl(\sum_j b_{ij} (x_{ij}, g_{ij})\Bigr), h_i\Bigr)\Biggr) := \sum_{i,j} a_i b_{ij}(x_{ij},h_ig_{ij}). \end{align*} $$

(The terminology comes from the theory of monads [Reference Moggi45, Reference Wadler and Broy60]. See equation (7.12).)