2 Thurston measure

Let ${\mathcal {M}}\mathcal L(S)$ be the set of measured laminations on S. We can consider measured laminations as currents. Indeed, ${\mathcal {M}}\mathcal L(S)$ is nothing other than the set of currents whose support projects to a lamination of S, that is, a compact set foliated by disjoint simple geodesics. As for curves, we will denote by the current associated to the measured lamination . We refer to [Reference Aramayona and Leininger1, Reference Bonahon5, Reference Erlandsson and Souto11] for basic facts about measured laminations and their relation to currents.

What will be very important for us is that the set ${\mathcal {M}}\mathcal L(S)$ of all measured laminations on S is naturally endowed with a measure, the Thurston measure ${\mathfrak m}_{\operatorname {\mathrm {Thu}}}$ . There are different ways of obtaining the Thurston measure. For example, one can get it via the standard symplectic structure on the space of measured laminations, but the more natural way here is as a scaling limit

(2.1) $$ \begin{align} {\mathfrak{m}}_{\operatorname{\mathrm{Thu}}}=\lim_{L\to\infty}\frac 1{L^{6g-6}}\sum_{\gamma\in{{\mathcal{M}}}{\mathcal L}_{\mathbb{Z}}(S)}\delta_{(1/L)\widehat{\gamma}} \end{align} $$

where g is the genus of S, where ${{\mathcal {M}}}{\mathcal L}_{\mathbb {Z}}(S)$ is the set of integrally weighted simple multicurves on S, where we denote by $\widehat {\gamma }$ the counting current associated to the multicurve $\gamma $ , and where the limit is taken with respect to the weak-* topology on the space of currents. See, for example, [Reference Erlandsson and Souto11, Reference Mirzakhani16] for the construction of the Thurston measures and [Reference Monin and Telpukhovskiy18] for the relations between the two possible constructions just mentioned. Anyway, the reason why we care about the Thurston measure is that we have the following theorem.

Theorem 2.1. For every closed geodesic $\gamma _0$ in S there is a positive constant $C(\gamma _0)$ such that

$$ \begin{align*}\lim_{L\to\infty}\frac 1{L^{6g-6}}\sum_{\gamma\sim\gamma_0}\delta_{(1/L)\widehat{\gamma}}=C(\gamma_0)\cdot{\mathfrak m}_{\operatorname{\mathrm{Thu}}}.\end{align*} $$

Here $\delta _x$ stands for the Dirac measure centred at x, the sum is taken over all $\gamma $ of type $\gamma _0$ , and the convergence takes place with respect to the weak-* topology on the space ${\mathcal {C}}(S)$ .

Theorem 2.1 is due to Mirzakhani [Reference Mirzakhani16] when $\gamma _0$ is a simple multicurve, and a complete proof for arbitrary curves appears in [Reference Erlandsson and Souto11, Ch. 8]. Theorem 2.1 also appears as Theorem 5.1 in [Reference Erlandsson, Parlier and Souto8], where it is explained how to deduce it from Mirzakhani’s (1.4) together with [Reference Erlandsson and Souto9, Proposition 4.1]. Now, as the referee reminded us, Mirzakhani never came to completely ironing out [Reference Mirzakhani17] and not everyone agrees whether her proof of (1.4) is complete or not when $\gamma _0$ is not filling. This is not a worry because in [Reference Erlandsson and Souto10] we used results from [Reference Erlandsson and Souto9] and from Mirzakhani’s original paper [Reference Mirzakhani16] (the one about simple curves) to give a very short proof of (1.4) in full generality. We should also point out that in the present paper we could bypass (1.4) altogether. Indeed, since the measures in Theorem 1.1 are normalized to be probability measures, we could make do with a version of Theorem 2.1 which holds only up to taking subsequences—this is Proposition 4.1 in [Reference Erlandsson and Souto9] (see [Reference Erlandsson and Souto11, Ch. 6] for a nicer exposition of this result).

The Thurston measure ${\mathfrak m}_{\operatorname {\mathrm {Thu}}}$ has many similarities with Lebesgue measure on ${\mathbb {R}}^{6g-6}$ . For example, it is homogeneous of degree $6g-6$ , meaning that for all $U\subset {\mathcal {C}}(S)$ measurable we have

$$ \begin{align*}{\mathfrak m}_{\operatorname{\mathrm{Thu}}}(t\cdot U)=t^{6g-6}\cdot{\mathfrak m}_{\operatorname{\mathrm{Thu}}}(U)\end{align*} $$

for all $t\geqslant 0$ . It follows that if $X\subset {\mathcal {C}}(S)$ is a compact set such that for every $\alpha \in {\mathcal {C}}(S)\setminus \{0\}$ there is a unique $t\in {\mathbb {R}}_{>0}$ with $t\cdot \alpha \in X$ , then there is a unique measure $\mathfrak n$ on X such that the map

$$ \begin{align*}({\mathbb{R}}_{>0}\times X, t^{6g-7}\, dt\otimes\mathfrak n)\to({\mathcal{C}}(S)\setminus\{0\},{\mathfrak m}_{\operatorname{\mathrm{Thu}}})\end{align*} $$

is a measure-preserving homeomorphism. The measure $\mathfrak n(U)$ of $U\subset X$ is given by

$$ \begin{align*}\mathfrak n(U)=\lim_{t\searrow 1}\frac{{\mathfrak m}_{\operatorname{\mathrm{Thu}}}([1,t]\times U)}{t-1}.\end{align*} $$

Combining all of this together we have the following lemma.

Lemma 2.2. Let $X\subset {\mathcal {C}}(S)$ be a compact set so that for every $\alpha \in {\mathcal {C}}(S)\setminus \{0\}$ there is a unique $t\in {\mathbb {R}}_{>0}$ with $t\cdot \alpha \in X$ . There is a measure $\mathfrak n$ on the set X such that the map

$$ \begin{align*}({\mathbb{R}}_{>0}\times X, t^{6g-7}\, dt\otimes\mathfrak n)\to({\mathcal{C}}(S)\setminus\{0\},{\mathfrak m}_{\operatorname{\mathrm{Thu}}})\end{align*} $$

is a measure-preserving homeomorphism.

Continuous homogeneous positive functions $F:{\mathcal {C}}(S)\to {\mathbb {R}}_{\geqslant 0}$ on the space of currents are a particular source of such sets X, where positive means that $F(\alpha )>0$ for all $\alpha \neq 0$ and homogeneous means that $F(t\cdot \alpha )=t\cdot F(\alpha )$ . Indeed, if we are given any such function F then the set $X_F=F^{-1}(1)$ satisfies the condition in Lemma 2.2. Let $\mathfrak n_F$ be the measure on $X_F$ provided by the said lemma. This measure comes in particularly handy when integrating functions on ${\mathcal {C}}(S)$ whose value at $\widehat {\alpha }$ solely depends on $F(\widehat {\alpha })$ . For example, with notation as in Lemma 2.2, we have

(2.2) $$ \begin{align} \int_{\{F(\cdot)\leqslant 1\}}F(\widehat{\unicode{x3bb}})^n\ d{\mathfrak m}_{\operatorname{\mathrm{Thu}}}(\widehat{\unicode{x3bb}})=\frac 1{6g-6+n}\mathfrak n_F(X_F) \end{align} $$

for every $n\geqslant 0$ . Taking $n=0$ , we get in particular that

(2.3) $$ \begin{align} \mathfrak n_F(X_F)=(6g-6)\cdot{\mathfrak m}_{\operatorname{\mathrm{Thu}}}(\{F(\cdot)\leqslant 1\}). \end{align} $$

The reason why we care about continuous, homogeneous and positive functions on ${\mathcal {C}}(S)$ is that there is such a function

$$ \begin{align*}\ell_S:{\mathcal{C}}(S)\to{\mathbb{R}}_{\geqslant 0}\end{align*} $$

satisfying $\ell _S(\widehat {\gamma })=\ell _S(\gamma )$ for every closed geodesic $\gamma $ , where the first $\ell _S(\cdot )$ is the function we are talking about and where the second one is just the length of the geodesic $\gamma $ . The reader can see [Reference Erlandsson, Parlier and Souto8, Reference Martínez-Granado and Thurston14] for other examples of continuous homogeneous functions on the space of currents—in fact we will encounter yet other such functions below.

3 Proof of Theorem 1.1

We are now ready to prove Theorem 1.1. The first step is to compute the total measure of the measures $m_L^{S,\gamma _0}$ introduced above.

Lemma 3.1. Let S be a closed orientable surface of genus g endowed with a negatively curved metric and denote by $\ell _S(\cdot )$ the associated length function. For every closed geodesic $\gamma _0$ , we have

$$ \begin{align*}\lim_{L\to\infty}\frac 1{L^{6g-5}}\Vert\overline m_L^{S,\gamma_0}\Vert=\frac{6g-6}{6g-5}\cdot C(\gamma_0)\cdot{\mathfrak m}_{\operatorname{\mathrm{Thu}}}(\{\ell_S(\cdot)\leqslant 1\})\end{align*} $$

where $\overline m_L^{S,\gamma _0}$ is as in (1.10) and where $C(\gamma _0)$ is as in Theorem 2.1.

Note that the exponents of L in Theorem 2.1 and Lemma 3.1 differ, $6g-6$ versus $6g-5$ .

Proof. Still writing $\gamma \sim \gamma _0$ to indicate that two closed unoriented geodesics are of the same type, we have

$$ \begin{align*} \frac 1{L^{6g-5}}\Vert\overline m_L^{S,\gamma_0}\Vert &=\frac 1{L^{6g-5}}\sum_{\gamma\in\mathcal S_L(S,\gamma_0)}\Vert\overline{\gamma}\Vert=\frac 1{L^{6g-5}}\sum_{\tiny\begin{array}{c}\gamma\sim\gamma_0\\ \ell_S(\gamma)\leqslant L\end{array}}\ell_S(\gamma)\\ &=\frac 1{L^{6g-5}}\int_{\{\ell_S(\cdot)\leqslant L\}}\ell_S(\cdot)\, d\bigg(\sum_{\gamma\sim\gamma_0}\delta_{\widehat{\gamma}}\bigg)\\ &=\int_{\{\ell_S(\cdot)\leqslant L\}}\frac{\ell_S(\cdot)}L\, d\bigg(\frac 1{L^{6g-6}} \sum_{\gamma\sim\gamma_0}\delta_{\widehat{\gamma}}\bigg)\\ &=\int_{\{\ell_S(\cdot)\leqslant 1\}}\ell_S(\cdot)\, d\bigg(\frac 1{L^{6g-6}}\sum_{\gamma\sim\gamma_0}\delta_{(1/L)\widehat{\gamma}}\bigg). \end{align*} $$

Now, Theorem 2.1 asserts that the measures on the last line converge:

$$ \begin{align*}\lim_{L\to\infty}\frac 1{L^{6g-6}}\sum_{\gamma\sim\gamma_0}\delta_{(1/L)\widehat{\gamma}}=C(\gamma_0)\cdot{\mathfrak m}_{\operatorname{\mathrm{Thu}}}.\end{align*} $$

It follows that

$$ \begin{align*} \lim_{L\to\infty}\frac 1{L^{6g-5}}\Vert\overline m_L^{S,\gamma_0}\Vert &=C(\gamma_0)\cdot\int_{\{\ell_S(\cdot)\leqslant 1\}}\ell_S(\cdot)\, d{\mathfrak m}_{\operatorname{\mathrm{Thu}}}\\ &=\frac{6g-6}{6g-5}\cdot C(\gamma_0)\cdot{\mathfrak m}_{\operatorname{\mathrm{Thu}}}(\{\ell_S(\cdot)\leqslant 1\}) \end{align*} $$

where we have used (2.2) and (2.3) to get the last equality.

Recall at this point the identification (1.8) between the spaces ${\mathcal {M}}_{\text {flip-flow}}(S)$ of flip- and flow-invariant measures and ${\mathcal {C}}(S)$ of currents, and suppose that we are given a continuous function $f\in C^0(T^1S)$ . Then we get a continuous function

$$ \begin{align*}\widehat f:{\mathcal{C}}(S)\to{\mathbb{R}},\ \ \widehat f(\widehat{\alpha})=\int_{T^1S}f(v)\, d\overline\alpha(v)\end{align*} $$

on the space of currents. It is continuous and homogeneous.

Lemma 3.2. We have

$$ \begin{align*} &\lim_{L\to\infty}\int_{T^1S}f\, d\bigg(\frac 1{\Vert\overline m_L^{S,\gamma_0}\Vert}\overline m_L^{S,\gamma_0}\bigg)\\[4pt] &\quad=\frac{6g-5}{(6g-6)\cdot{\mathfrak m}_{\operatorname{\mathrm{Thu}}}(\{\ell_S(\cdot)\leqslant 1\})} \cdot\int_{\{\ell_S(\cdot)\leqslant 1\}}\widehat f\, d{\mathfrak m}_{\operatorname{\mathrm{Thu}}}\end{align*} $$

for every $f\in C^0(T^1S)$ .

Proof. We compute as in Lemma 3.1:

$$ \begin{align*} \int_{T^1S}f\, d\bigg(\frac 1{\Vert\overline m_L^{S,\gamma_0}\Vert}\overline m_L^{S,\gamma_0}\bigg) &=\frac 1{\Vert\overline m_L^{S,\gamma_0}\Vert}\sum_{\tiny\begin{array}{c} \gamma\sim\gamma_0\\ \ell_S(\gamma)\leqslant L\end{array}}\int_{T^1S}f\, d\overline{\gamma}\\ &=\frac 1{\Vert\overline m_L^{S,\gamma_0}\Vert}\sum_{\tiny\begin{array}{c} \gamma\sim\gamma_0\\ \ell_S(\gamma)\leqslant L\end{array}}\widehat f(\widehat{\gamma})\\ &=\frac L{\Vert\overline m_L^{S,\gamma_0}\Vert}\sum_{\tiny\begin{array}{c} \gamma\sim\gamma_0\\ \ell_S(\gamma)\leqslant L\end{array}}\widehat f\,\bigg((1/L)\widehat{\gamma}\bigg)\\ &=\frac L{\Vert\overline m_L^{S,\gamma_0}\Vert}\int_{\{\ell_S(\cdot)\leqslant 1\}}\widehat f\, d\bigg(\sum_{\gamma\sim\gamma_0}\delta_{(1/L)\widehat{\gamma}}\bigg) \end{align*} $$

In light of Lemma 3.1 we get that

(3.1) $$ \begin{align} \int_{T^1S}f\, d\bigg(\frac 1{\Vert\overline m_L^{S,\gamma_0}\Vert}\overline m_L^{S,\gamma_0}\bigg) &\approx \frac K{C(\gamma_0)\cdot L^{6g-6}}\int_{\{\ell_S(\cdot)\leqslant 1\}}\widehat f\, d\bigg(\sum_{\gamma\sim\gamma_0}\delta_{(1/L)\widehat{\gamma}}\bigg)\nonumber\\[4pt] &= \frac K{C(\gamma_0)}\int_{\{\ell_S(\cdot)\leqslant 1\}}\widehat f\, d\bigg(\frac 1{L^{6g-6}}\sum_{\gamma\sim\gamma_0}\delta_{(1/L)\widehat{\gamma}}\bigg) \end{align} $$

where $\approx $ means that the ratio between both sides tends to $1$ when L tends to $\infty $ , and where

$$ \begin{align*}K=\frac{6g-5}{(6g-6)\cdot{\mathfrak m}_{\operatorname{\mathrm{Thu}}}(\{\ell_S(\cdot)\leqslant 1\})}.\end{align*} $$

Now, again invoking Theorem 2.1, we get that the measure at the very end of (3.1) converges to $C(\gamma _0)\cdot {\mathfrak m}_{\operatorname {\mathrm {Thu}}}$ , meaning that

$$ \begin{align*}\lim_{L\to\infty}\int_{\{\ell_S(\cdot)\leqslant 1\}}\widehat f\, d\bigg(\frac 1{L^{6g-6}}\sum_{\gamma\sim\gamma_0}\delta_{(1/L)\widehat{\gamma}}\bigg)=C(\gamma_0)\int_{\{l_S(\cdot)\leqslant 1\}}\widehat f\, d{\mathfrak m}_{\operatorname{\mathrm{Thu}}}.\end{align*} $$

Putting all of this together, we get that

$$ \begin{align*}\lim_{L\to\infty}\int_{T^1S}f\, d\bigg(\frac 1{\Vert\overline m_L^{S,\gamma_0}\Vert}\overline m_L^{S,\gamma_0}\bigg)= K\cdot\int_{\{\ell_S(\cdot)\leqslant 1\}}\widehat f\, d{\mathfrak m}_{\operatorname{\mathrm{Thu}}}\end{align*} $$

as claimed.

We are basically home and dry.

Proof of Theorem 1.1

nf your party affiliation, from the very definition of measure) that there is a unique measure ${\mathfrak m}$ on $T^1S$ with

$$ \begin{align*}\int_{T^1S}f\, d{\mathfrak m}=\int_{\{\ell_S(\cdot)\leqslant 1\}}\bigg(\int_{T^1S} f\, d\overline{\unicode{x3bb}}\bigg)\,d{\mathfrak m}_{\operatorname{\mathrm{Thu}}}({\unicode{x3bb}})\end{align*} $$

where, as usual, we denote by the flip- and flow-invariant measure associated to the measured lamination .

In those terms, Lemma 3.2 says that

$$ \begin{align*}\lim_{L\to\infty}\int_{T^1S}f\, d\bigg(\frac 1{\Vert\overline m_L^{S,\gamma_0}\Vert}\overline m_L^{S,\gamma_0}\bigg)=\frac{6g-5}{(6g-6)\cdot{\mathfrak m}_{\operatorname{\mathrm{Thu}}}(\{\ell_S(\cdot)\leqslant 1\})}\cdot\int_{T^1S}f\, d{\mathfrak m}.\end{align*} $$

This just means that the measures $ (1/{\Vert \overline m_L^{S,\gamma _0}\Vert })\overline m_L^{S,\gamma _0}$ converge, when L tends to $\infty $ , to the measure

$$ \begin{align*}{\mathfrak m}^S=\frac{6g-5}{(6g-6)\cdot{\mathfrak m}_{\operatorname{\mathrm{Thu}}}(\{\ell_S(\cdot)\leqslant 1\})}\cdot{\mathfrak m}.\end{align*} $$

Since, by (1.11), we have

$$ \begin{align*}\frac 1{\Vert m_L^{S,\gamma_0}\Vert}m_L^{S,\gamma_0}=\frac 1{\Vert\overline m_L^{S,\gamma_0}\Vert}\overline m_L^{S,\gamma_0},\end{align*} $$

we are done.

4 Proof of Theorem 1.2

We are now ready to prove Theorem 1.2. We recall the statement for the convenience of the reader.

Proof. We start by proving that the measure ${\mathfrak m}^S$ is not ergodic. To begin with, recall that the set

$$ \begin{align*}\mathcal U\mathcal E=\{{\unicode{x3bb}}\in{\mathcal{M}}\mathcal L(S)\text{ with }{\unicode{x3bb}}\text{ uniquely ergodic}\}\end{align*} $$

of uniquely ergodic measured laminations has full Thurston measure by a result of Masur [Reference Masur15]. Decompose the associated projective space

$$ \begin{align*}P\mathcal U\mathcal E=\mathcal U\mathcal E/{\mathbb{R}}_{>0}\end{align*} $$

into two disjoint measurable sets

$$ \begin{align*}P\mathcal U\mathcal E=A_1\sqcup A_2,\end{align*} $$

such that the associated subsets of ${\mathcal {M}}\mathcal L(S)$ have positive Thurston measure

(4.1) $$ \begin{align} {\mathfrak m}_{\operatorname{\mathrm{Thu}}}(\{{\unicode{x3bb}}\in{\mathcal{M}}\mathcal L(S),\ [{\unicode{x3bb}}]\in A_i\})>0\quad\text{for }i=1,2. \end{align} $$

The sets $\mathcal A_1,\mathcal A_2\subset T^1S$ given by

$$ \begin{align*}\mathcal A_i=\{v\in T^1S\text{ tangent to a leaf of the support of some }{\unicode{x3bb}}\in A_i\}\end{align*} $$

are then disjoint and invariant under the geodesic flow. In turn, (4.1) implies that both have positive ${\mathfrak m}^S$ -measure:

$$ \begin{align*}{\mathfrak m}^S(\mathcal A_1),{\mathfrak m}^S(\mathcal A_2)>0.\end{align*} $$

This proves that ${\mathfrak m}^S$ is not ergodic.

Note that the support of ${\mathfrak m}^S$ is the set ${\mathcal {K}}$ of vectors tangent to a leaf in the support of some measured lamination. The projection of ${\mathcal {K}}$ to the surface S has Hausdorff dimension $1$ by the well-known Birman–Series theorem [Reference Birman and Series3]. That ${\mathcal {K}}$ itself has Hausdorff dimension $1$ was proved by Fathi [Reference Fathi12, Theorem 3.1], who also obtained in Corollary 3.4 of the same paper that the topological entropy of the restriction of the geodesic flow to ${\mathcal {K}}$ vanishes—this implies that every measure supported by ${\mathcal {K}}$ , and in particular ${\mathfrak m}^S$ , has vanishing entropy.

5 Proof of Theorem 1.3

We will now prove Theorem 1.3. First recall that, as in (1.6), every homeomorphism

$$ \begin{align*}\phi:S\to S'\end{align*} $$

between negatively curved surfaces induces a homeomorphism

$$ \begin{align*}\phi_*:{\mathcal{C}}(S)\to{\mathcal{C}}(S')\end{align*} $$

between the spaces of currents on the domain and target. Since we also have the homeomorphism (1.8) between the space of currents and the space of flip- and flow-invariant measures on the unit tangent bundle, we get an induced homeomorphism

(5.1) $$ \begin{align} \phi_*:{\mathcal{M}}_{\text{flip-flow}}(S)\to{\mathcal{M}}_{\text{flip-flow}}(S'). \end{align} $$

Our first goal is to figure out the image of ${\mathfrak m}^S$ under this map.

Lemma 5.1. For every $f\in C^0(T^1S')$ we have

$$ \begin{align*}\int_{T^1S'}f\, d\phi_*({\mathfrak m}^{S})=\frac{6g-5}{6g-6}\cdot{\mathfrak m}_{\operatorname{\mathrm{Thu}}}(\{\ell_S(\cdot)\leqslant 1\})\cdot\int_{\{\ell_S(\phi^{-1}(\cdot))\leqslant 1\}}\widehat f\, d{\mathfrak m}_{\operatorname{\mathrm{Thu}}},\end{align*} $$

where the second integral is taken over the set

(5.2) $$ \begin{align} \{\ell_S(\phi^{-1}(\cdot))\leqslant 1\}=\{{\unicode{x3bb}}\in{\mathcal{M}}\mathcal L(S')\vert\ell_S(\phi^{-1}({\unicode{x3bb}}))\leqslant 1\} \end{align} $$

of those measured laminations in ${\mathcal {M}}\mathcal L(S')$ which have length at most $1$ in the surface S.

Proof. We start by tracking what the map $\phi _*$ does to the flip- and flow-invariant measure $\overline {\gamma }$ on $T^1S$ associated to a free homotopy class $\gamma $ :

$$ \begin{align*}\matrix{ {\mathcal{M}}_{\text{flip-flow}}(S) \rightarrow & {\mathcal{C}}(S) \rightarrow & {\mathcal{C}}(S') \rightarrow & {\mathcal{M}}_{\text{flip-flow}}(S')\\ \overline{\gamma}\rightarrow & \widehat{\gamma} \rightarrow & \widehat{\phi(\gamma)} \rightarrow & \overline{\phi(\gamma)}}\end{align*} $$

Now, linearity of (5.1) implies that

$$ \begin{align*}\phi_*(m^{S,\gamma_0}_L)=\sum_{{\tiny\begin{array}{c} \gamma\sim\gamma_0\\ \ell_S(\gamma)\leqslant L\end{array}}}\overline{\phi(\gamma)}\end{align*} $$

where $\gamma _0$ is some arbitrary but otherwise fixed closed geodesic in S.

Then, exactly the same computation as we used in the first part of the proof of Lemma 3.2 shows that for any continuous function $f\in C^0(T^1S')$ we have

$$ \begin{align*}\int_{T^1S'}f\, d\phi_*\bigg(\frac 1{\Vert\overline m_L^{S,\gamma_0}\Vert}\overline m_L^{S,\gamma_0}\bigg)=\frac L{\Vert\overline m_L^{S,\gamma_0}\Vert}\int_{\{\ell_S(\phi^{-1}(\cdot))\leqslant 1\}}\widehat f\, d\bigg(\sum_{\gamma\sim\phi(\gamma_0)}\delta_{(1/L)\widehat{\gamma}}\bigg)\end{align*} $$

where the integral is taken over the set (5.2). Still as in the proof of Lemma 3.2, we get that

$$ \begin{align*}\int_{T^1S'}f\, d\phi_*\bigg(\frac 1{\Vert\overline m_L^{S,\gamma_0}\Vert}\overline m_L^{S,\gamma_0}\bigg)\approx\frac{K(S)}{C(\gamma_0)}\int_{\{\ell_S(\phi^{-1}(\cdot))\leqslant 1\}}\widehat f\, d\bigg(\frac 1{L^{6g-6}}\sum_{\gamma\sim\phi(\gamma_0)}\delta_{(1/L)\widehat{\gamma}}\bigg)\end{align*} $$

where

$$ \begin{align*}K(S)=\frac{6g-5}{6g-6}\cdot{\mathfrak m}_{\operatorname{\mathrm{Thu}}}(\{\ell_S(\cdot)\leqslant 1\}).\end{align*} $$

At this point, still as in Lemma 3.2, we get that

$$ \begin{align*}\lim_{L\to\infty}\int_{T^1S'}f\, d\phi_*\bigg(\frac 1{\Vert\overline m_L^{S,\gamma_0}\Vert}\overline m_L^{S,\gamma_0}\bigg)=K(S)\cdot\int_{\{\ell_S(\phi^{-1}(\cdot))\leqslant 1\}}\widehat f\, d{\mathfrak m}_{\operatorname{\mathrm{Thu}}}.\end{align*} $$

Since ${\mathfrak m}^{S}$ arises by Theorem 1.1 as the limit of the measures $( 1/{\Vert \overline m_L^{S,\gamma _0}\Vert })\overline m_L^{S,\gamma _0}$ and since $\phi _*$ is continuous, we get

$$ \begin{align*}\int_{T^1S'}f\, d\phi_*({\mathfrak m}^{S})=K(S)\cdot\int_{\{\ell_S(\phi^{-1}(\cdot))\leqslant 1\}}\widehat f\, d{\mathfrak m}_{\operatorname{\mathrm{Thu}}},\end{align*} $$

as claimed.

We will now rewrite the integral on the right in Lemma 5.1 using polar coordinates in ${\mathcal {M}}\mathcal L(S')$ . To that end we choose a compact set $X\subset {\mathcal {C}}(S')$ such that for every $\alpha \in {\mathcal {C}}(S')\setminus \{0\}$ there is a uniquely determined $t>0$ with $t\cdot \alpha \in X$ . Let $\mathfrak n$ be the measure on X such that the map

$$ \begin{align*}({\mathbb{R}}_{>0}\times X,t^{6g-7}dt\otimes\mathfrak n)\to({\mathcal{C}}(S')\setminus\{0\},{\mathfrak m}_{\operatorname{\mathrm{Thu}}})\end{align*} $$

is a measure-preserving homeomorphism. Note that this induces a homeomorphism between

$$ \begin{align*}\{(t,{\unicode{x3bb}})\in{\mathbb{R}}_{>0}\times X\text{ with }0<t\leqslant \ell_S(\phi^{-1}({\unicode{x3bb}}))^{-1}\}\to\{\ell_S(\cdot)\leqslant 1\}\end{align*} $$

We then have for every $f\in C^0(T^1S')$ that

$$ \begin{align*} \int_{\{\ell_S(\phi^{-1}(\cdot))\}}\widehat f\, d{\mathfrak m}_{\operatorname{\mathrm{Thu}}} &=\int_{\{\ell_S(\phi^{-1}(\cdot))\}}\bigg(\int_{T^1S'}f\, d\overline{\unicode{x3bb}}\bigg)\, d{\mathfrak m}_{\operatorname{\mathrm{Thu}}}({\unicode{x3bb}})\\ &=\int_X\int_0^{ 1/{\ell_S(\phi^{-1}({\unicode{x3bb}}))}}\bigg(\int_{T^1S'}f\, d(\overline{t\cdot{\unicode{x3bb}}})\bigg) t^{6g-7}\cdot dt\cdot d\mathfrak n({\unicode{x3bb}})\\ &=\int_X\int_0^{1/{\ell_S(\phi^{-1}({\unicode{x3bb}}))}}\bigg(\int_{T^1S'}f\, d\overline{\unicode{x3bb}}\bigg) t^{6g-6}\cdot dt\cdot d\mathfrak n({\unicode{x3bb}})\\ &=\int_X \frac 1{(6g-5)\cdot\ell_S(\phi^{-1}({\unicode{x3bb}}))^{6g-5}}\bigg(\int_{T^1S'}f\, d\overline{\unicode{x3bb}}\bigg)\, d\mathfrak n({\unicode{x3bb}}). \end{align*} $$

In light of Lemma 5.1, we get

$$ \begin{align*}\int_{T^1S'}f\, d\phi_*({\mathfrak m}^S)=\frac{{\mathfrak m}_{\operatorname{\mathrm{Thu}}}(\{\ell_S(\cdot)\leqslant 1\})}{6g-6}\int_X \frac 1{\ell_S(\phi^{-1}({\unicode{x3bb}}))^{6g-5}}\bigg(\int_{T^1S'}f\, d\overline{\unicode{x3bb}}\bigg)\, d\mathfrak n({\unicode{x3bb}}).\end{align*} $$

Applying this computation to $S=S'$ and $\phi =\operatorname {\mathrm {Id}}$ , we also get

$$ \begin{align*}\int_{T^1S'}f\, d{\mathfrak m}^{S'}=\frac{{\mathfrak m}_{\operatorname{\mathrm{Thu}}}(\{\ell_{S'}(\cdot)\leqslant 1\})}{6g-6}\int_X \frac 1{\ell_{S'}({\unicode{x3bb}})^{6g-5}}\bigg(\int_{T^1S'}f\, d\overline{\unicode{x3bb}}\bigg)\, d\mathfrak n({\unicode{x3bb}}).\end{align*} $$

Let us now denote by $\mathcal U\mathcal E\subset {\mathcal {M}}\mathcal L(S')$ the set of uniquely ergodic laminations and consider the function

$$ \begin{align*}\xi^{\phi,S}_{S'}:T^1S'\to{\mathbb{R}}\end{align*} $$

given by

$$ \begin{align*}\xi^{\phi,S}_{S'}(v)=\left\{\begin{array}{ll} \bigg(\frac{\ell_{S'}({\unicode{x3bb}})}{\ell_S(\phi^{-1}({\unicode{x3bb}}))}\bigg)^{6g-5} & \text{ if }v\text{ is tangent to }{\unicode{x3bb}}\in\mathcal U\mathcal E,\\ 0 & \text{ otherwise}. \end{array}\right.\end{align*} $$

Since the set of uniquely ergodic measured laminations has full Thurston measure ${\mathfrak m}_{\operatorname {\mathrm {Thu}}}$ [Reference Masur15] we get that

$$ \begin{align*}\int_X \frac 1{\ell_S(\phi^{-1}({\unicode{x3bb}}))^{6g-5}}\bigg(\int_{T^1S'}f\, d\overline{\unicode{x3bb}}\bigg)\, d\mathfrak n({\unicode{x3bb}})=\int_X \frac 1{\ell_{S'}({\unicode{x3bb}})^{6g-5}}\bigg(\int_{T^1S'}f\xi^{\phi,S}_{S'}\, d\overline{\unicode{x3bb}}\bigg)\, d\mathfrak n({\unicode{x3bb}}).\end{align*} $$

Taking all of this together, we get

$$ \begin{align*}\frac 1{{\mathfrak m}_{\operatorname{\mathrm{Thu}}}(\{\ell_S(\cdot)\leqslant 1\})}\cdot\int_{T^1S'}f\, d\phi_*({\mathfrak m}^S)=\frac 1{{\mathfrak m}_{\operatorname{\mathrm{Thu}}}(\{\ell_{S'}(\cdot)\leqslant 1\})}\cdot\int_{T^1S'}f\xi^{\phi,S}_{S'}\, d{\mathfrak m}^{S'}.\end{align*} $$

Since this holds true for all continuous functions $f\in C^0(T^1S')$ , we get that indeed $\phi _*({\mathfrak m}^S)$ is absolutely continuous with respect to ${\mathfrak m}^{S'}$ with Radon–Nikodym derivative

(5.3) $$ \begin{align} \frac{d\phi_*({\mathfrak m}^S)}{d{\mathfrak m}^{S'}}=\frac{{\mathfrak m}_{\operatorname{\mathrm{Thu}}}(\{\ell_{S}(\cdot)\leqslant 1\})}{{\mathfrak m}_{\operatorname{\mathrm{Thu}}}(\{\ell_{S'}(\cdot)\leqslant 1\})}\cdot\xi^{\phi,S}_{S'}. \end{align} $$

Since the Radon–Nikodym derivative is essentially positive on the support of ${\mathfrak m}^{S'}$ , we get that indeed both measures are in the same measure class. We have proved Theorem 1.3, which we restate for the convenience of the reader.

All that is left to do is to prove Theorem 1.4, which we do in the next section.

6 Proof of Theorem 1.4

Let us remind the reader what we need to prove.

Proof. One direction is clear: if $\phi $ is an isometry, then $\phi _*({\mathfrak m}^S)={\mathfrak m}^{S'}$ . Suppose conversely that S and $S'$ are hyperbolic surfaces and that $\phi _*({\mathfrak m}^S)={\mathfrak m}^{S'}$ . This means that the Radon–Nikodym derivative (5.3) is identically $1$ . This means that we have

(6.1) $$ \begin{align} \ell_S({\unicode{x3bb}})=\frac{{\mathfrak m}_{\operatorname{\mathrm{Thu}}}(\{\ell_{S}(\cdot)\leqslant 1\})}{{\mathfrak m}_{\operatorname{\mathrm{Thu}}}(\{\ell_{S'}(\cdot)\leqslant 1\})}\cdot\ell_{S'}(\phi({\unicode{x3bb}})) \end{align} $$

for almost every uniquely ergodic . The density of the set of uniquely ergodic laminations and continuity of the length function imply that (6.1) holds for all . The following claim implies that this cannot happen unless $\phi $ is homotopic, and hence isotopic, to an isometry.

Claim. If $\phi :S\to S'$ is not homotopic to an isometry then there are $\alpha ,\beta \in {\mathcal {M}}{\mathcal {L}}(S)$ with $\ell _S(\alpha )<\ell _{S'}(\phi (\alpha ))$ and $\ell _S(\beta )>\ell _{S'}(\phi (\beta ))$ .

Proof. Let $\operatorname {\mathrm {Lip}}(\phi )$ denote the infimum of the Lipschitz constants over all maps homotopic to $\phi $ . It is due to Thurston [Reference Thurston20] that $\operatorname {\mathrm {Lip}}(\phi )>1$ unless $\phi $ is homotopic to an isometry and that

$$ \begin{align*}\operatorname{\mathrm{Lip}}(\phi)=\max_{{\unicode{x3bb}}\in{\mathcal{M}}\mathcal L(S)\setminus\{0\}}\frac{\ell_{S'}(\phi({\unicode{x3bb}}))}{\ell_S({\unicode{x3bb}})}.\end{align*} $$

The existence of $\alpha $ follows. The existence of $\beta $ follows as well once we repeat the argument replacing $\phi $ by $\phi ^{-1}$ .

Having proved the claim, we have proved that if S and $S'$ are hyperbolic and if $\phi ({\mathfrak m}^S)={\mathfrak m}^{S'}$ then $\phi $ is homotopic to an isometry.

Let us now construct an example showing that Theorem 1.4 fails if, say, $S'$ is allowed to have variable curvature, even after we normalize the area to be equal to that of S. Let us start by choosing S hyperbolic and recall that, by Theorem 1.2, the support ${\mathcal {K}}\subset T^1S$ of ${\mathfrak m}^S$ has Hausdorff dimension $1$ . Its projection ${\mathcal {K}}'\subset S$ to the surface is closed and also has Hausdorff dimension $1$ . Denoting by $\sigma _0$ the hyperbolic metric on S, let $\sigma $ be another metric (with same area) obtained by very slightly perturbing $\sigma _0$ on some open set whose closure does not meet ${\mathcal {K}}'$ . Let $S'=(S,\sigma )$ and $\phi =\operatorname {\mathrm {Id}}$ . Then the map

$$ \begin{align*}\phi:(S,\sigma_0)\to(S,\sigma)\end{align*} $$

is an isometry on the set ${\mathcal {K}}'$ and hence satisfies that for all . It follows that the Radon–Nikodym derivative (5.3) of $\phi _*({\mathfrak m}^S)$ with respect to ${\mathfrak m}^{S'}$ is identically $1$ , meaning that $\phi _*({\mathfrak m}^S)={\mathfrak m}^{S'}$ . On the other hand, there is no isometry from a hyperbolic surface to one with non-constant curvature.

7 Surfaces with cusps

So far we have been considering closed surfaces. We explain briefly how to modify the proofs of the theorems in the introduction so that they apply to surfaces with cusps. Actually, we will just focus on Theorem 1.1, leaving to the reader the pleasure of modifying the other proofs.

Let us suppose that S is a complete hyperbolic surface of finite area and let $\gamma _0$ be a closed primitive geodesic in S. We also suppose that S is not a thrice punctured sphere—all results are trivial in that case.

The first basic fact (see, for example, [Reference Erlandsson, Parlier and Souto8, Lemma 5.2], [Reference Erlandsson and Souto11, Lemma 2.8], or [Reference Bell2, Lemma 4]) is that there is a compact subsurface $S_0\subset S$ which contains every closed geodesic of the same type as $\gamma _0$ , or indeed any closed geodesic $\gamma $ with $\iota (\gamma ,\gamma )\leqslant \iota (\gamma _0,\gamma _0)$ . It follows that the measures $m_L^{S,\gamma _0}$ and $\overline m_L^{S,\gamma _0}$ are supported by $S_0$ for all L.

This means that in the proofs one does not need to worry about the measure wandering off to infinity. But this is only one of the issues one faces when the surface is non-compact. The point is that for technical reasons, when working with currents (and wanting to have a continuous and homogeneous extension of $\iota (\cdot ,\cdot )$ to the space of currents) one is forced by nature to work on compact surfaces. However, while the surfaces have to be compact, they can have a boundary.

Let $S'\subset S$ be a compact subsurface such that

• • $S_0\subset S'\setminus \partial S'$ , and

• • $S\setminus S'$ is a union of annular components, one for each cusps of S.

Endow $S'$ with a negatively curved metric with totally geodesic boundary and which agrees with the metric of S near $S_0$ .

By definition, a current on $S'$ is a $\pi _1(S')$ -invariant Radon measure on the space ${\mathcal {G}}({\tilde {S}}')$ of complete geodesics on the universal cover of $S'$ . Denote by ${\mathcal {C}}_{S_0}(S')$ be the set of currents on $S'$ supported by geodesics which project into $S_0$ . For example, the current $\widehat {\gamma }$ belongs to ${\mathcal {C}}_{S_0}(S')$ for every $\gamma $ of the same type as $\gamma _0$ .

Note also that $S'$ -geodesics contained in $S_0$ are, by the choice of the metric on S, also S-geodesics. It follows that we can consider currents $\widehat {\mu }\in {\mathcal {C}}_{S_0}(S')$ as $\pi _1(S)$ -invariant measures on the set of geodesics on S. We thus get a continuous map

(7.1) $$ \begin{align} {\mathcal{C}}_{S_0}(S')\to{\mathcal{M}}_{\text{flip-flow}}(S),\quad\widehat{\mu}\mapsto\overline\mu=dt\otimes\widehat{\mu} \end{align} $$

where the measure $\overline \mu $ is still given by (1.7). This maps is also proper.

Now, Theorem 2.1 is proved in [Reference Erlandsson and Souto11] not only for closed but also for general compact connected orientable surfaces of negative Euler characteristic (other than the pair of pants). Applying it to $\gamma _0$ when seen as a curve in $S'$ , we get that

(7.2) $$ \begin{align} \lim_{L\to\infty}\frac 1{L^{6g-6+2r}}\sum_{\gamma\sim\gamma_0}\delta_{(1/L)\widehat{\gamma}}=C(\gamma_0)\cdot{\mathfrak m}_{\operatorname{\mathrm{Thu}}}. \end{align} $$

Here g and r are the genus and number of cusps of S and the convergence takes place with respect to the weak-* topology on the space of Radon measures on ${\mathcal {C}}_{S_0}(S')$ .

Armed with (7.1) and (7.2), we can repeat word by word the proof of Lemmas 3.1 and 3.2. Theorem 1.1 follows.

The reader might be wondering if this is actually true. Earlier we were making a big deal of the fact that the space of currents is homeomorphic to that of flip- and flow-invariant measures, and the map (7.1) is obviously not a homeomorphism. If the reader is worried about this, then they can really go back to the proof of Lemmas 3.1 and 3.2 and check that we are only using the continuity and properness of (1.8).