2.1 Convex projective surfaces

A properly convex set $\Omega $ in $\mathbb {RP}^{2}$ is a bounded open convex subset of an affine chart. A properly convex set whose boundary does not contain open line segments is strictly convex. We will exclusively focus on strictly convex sets in this paper. We equip a strictly convex set $\Omega $ with its Hilbert metric $d_{\Omega }$ . More precisely, if $x,y\in \Omega $ , the projective line $\overline {xy}$ passing through x and y intersects the boundary of $\Omega $ in two points $a,b$ , where $a,x,y,b$ appear in this order along $\overline {xy}$ . The Hilbert distance between x and y is

$$ \begin{align*} d_{\Omega}(x,y)=\tfrac{1}{2}\log [a,x,y,b], \end{align*} $$

where $[a,x,y,b]$ denotes the crossratio of four points on a projective line. With the Hilbert metric, geodesics are segments of a projective line intersecting $\Omega $ . Typically, the Hilbert metric is not Riemannian, but it derives from a Finsler norm. Thus, one can study the unit tangent bundle $T^{1}\Omega $ of a strictly convex set $\Omega $ .

The main objects of interest are representations $\rho \colon \Gamma \to \mathrm {SL}(3,\mathbb {R})$ such that $\rho (\Gamma )$ preserves a properly convex set $\Omega _{\rho }$ on which it acts properly discontinuously with quotient homeomorphic to the closed surface S. In this case, we say that $\rho $ is a (marked real) convex projective structure which divides $\Omega _{\rho }$ and denote by $X_{\rho }$ the surface S equipped with the convex projective structure $\rho $ . Since S is a closed surface of negative Euler characteristic, $\Omega _{\rho }$ is strictly convex and if $\gamma \in \Gamma $ is non-trivial, then the moduli $\unicode{x3bb} _{1}(\rho (\gamma ))>\unicode{x3bb} _{2}(\rho (\gamma ))>\unicode{x3bb} _{3}(\rho (\gamma ))>0$ of the eigenvalues of $\rho (\gamma )$ are distinct. (See for example [Reference Goldman30, Theorem 3.2] and references therein).

The Hilbert distance on $\Omega _{\rho }$ induces the Hilbert length spectrum $\ell _{\rho }^{H}$ for non-trivial conjugacy classes of group elements in $\Gamma $ . Algebraically, if $[\Gamma ]$ is the set of conjugacy classes of non-identity elements in $\Gamma $ , namely $[\gamma ]\in [\Gamma ]$ is the conjugacy class of $\gamma \neq \text {id}$ , then

$$ \begin{align*} \ell_{\rho}^{H}([\gamma])=\frac{1}{2}\log\frac{\unicode{x3bb}_{1}(\rho(\gamma))}{\unicode{x3bb}_{3}(\rho(\gamma))}. \end{align*} $$

Benoist [Reference Benoist7] proved that if $\rho $ is a convex projective structure on S, then $(\Omega _{\rho },d_{\Omega _{\rho }})$ is Gromov hyperbolic, the Gromov boundary and the topological boundary of $\Omega _{\rho }$ coincide, and $\partial \Omega _{\rho }$ is of class $C^{1+\alpha }$ for some $\alpha \in (0,1]$ . For a(ny) point $o\in \Omega _{\rho }$ , the orbit map $\gamma \mapsto \rho (\gamma ).o$ is a quasi-isometric embedding. It follows (see e.g. [Reference Ghys and de la Harpe26, Ch. 7, Proposition 14]) that the orbit map induces a limit map $\xi _{\rho }\colon \partial \Gamma \to \partial \Omega _{\rho }$ between Gromov boundaries which is a $\rho $ -equivariant bi-Hölder homeomorphism.

If $\rho (\Gamma )$ divides a strictly convex set $\Omega _{\rho }$ in $\mathbb {RP}^{2}$ , then the representation $\rho ^{\ast }$ divides a (typically different) strictly convex set $\Omega _{\rho ^{\ast }}$ . Here we recall that the contragredient representation is defined as $\rho ^{\ast }(\gamma )=(\rho (\gamma )^{-1})^{t}$ for all $\gamma \in \Gamma $ . We refer to this operation on the space of convex projective structures as the contragredient involution. The contragredient involution preserves the Hilbert length because for all $[\gamma ]\in [\Gamma ]$ ,

$$ \begin{align*} \ell^{H}_{\rho}([\gamma])=\frac{1}{2}\log\frac{\unicode{x3bb}_{1}(\rho(\gamma))}{\unicode{x3bb}_{3}(\rho(\gamma))}=\frac{1}{2}\log \frac{1/\unicode{x3bb}_{3}(\rho(\gamma))}{{1}/{\unicode{x3bb}_{1}(\rho(\gamma))}}=\ell^{H}_{\rho^{\ast}}([\gamma]). \end{align*} $$

A standard computation using the irreducible representation $\mathrm {PSL}(2,\mathbb {R})\to \mathrm {PSL}(3,\mathbb {R})$ shows that if $\rho $ is a hyperbolic structure, then $\rho =\rho ^{\ast }$ . The converse holds by [Reference Benoist5, Theorem 1.3]: if $\rho $ is a convex projective structure such that $\rho =\rho ^{*}$ , then $\rho $ is a hyperbolic structure.

2.2 The Hilbert geodesic flow and reparametrization function

Suppose that $\rho $ is a convex projective surface. The Hilbert geodesic flow $\Phi ^{\rho }$ is defined on the unit tangent bundle of the surface $T^{1}X_{\rho }$ . The image $\Phi ^{\rho }_{t}(w)$ of a point $w=(x,v)$ is obtained by following the unit speed geodesic for time t leaving x in the direction v. When it is clear from context, we simply write $\Phi ^{\rho }$ as $\Phi $ .

The Hilbert geodesic flow $\Phi $ on $T^{1}X_{\rho }$ is an example of a topologically mixing Anosov flow by [Reference Benoist7, Propositions 3.3 and 5.6]. A standard reference for the theory of Anosov flows is [Reference Katok and Hasselblatt35, §6]. A key property for our discussion is that topologically mixing Anosov flows can be modeled by Markov partitions and symbolic dynamics in the sense of Bowen [Reference Bowen11].

Given a positive Hölder continuous function $f\colon T^{1}X_{\rho }\to \mathbb {R}$ , one can define a Hölder reparameterization of the flow $\Phi $ by time change. We construct the flow $\Phi ^{f}$ following [Reference Sambarino53, §2]. First, we define $\kappa :T^{1} X_{\rho } \times \mathbb {R}\to \mathbb {R} $ as

$$ \begin{align*} \kappa(x,t)=\int_{0}^{t} f(\Phi_{s}(x))\,ds. \end{align*} $$

Given the fact that f is positive and $T^{1}X_{\rho }$ is compact, the function $\kappa (x,\cdot )$ is an increasing homeomorphism of $\mathbb {R}$ . We therefore have an inverse $\alpha : T^{1} X_{\rho } \times \mathbb {R}\to \mathbb {R} $ that verifies

$$ \begin{align*} \alpha(x,\kappa(x,t))=\kappa(x,\alpha(x,t))=t \end{align*} $$

for every $(x,t)\in T^{1}X_{\rho } \times \mathbb {R}$ . The Hölder reparametrization of $\Phi $ by the Hölder continuous function f is given by $\Phi ^{f}_{t}(x)= \Phi _{\alpha (x,t)}(x)$ . We say that f is a reparameterization function for $\Phi ^{f}$ . The new flow $\Phi ^{f}=\{\Phi _{t}^{f}\}_{t\in \mathbb {R}}$ shares the same set of periodic orbits of $\Phi $ . For any periodic orbit $\tau $ of $\Phi $ with period $\unicode{x3bb} (\tau )$ , its period as a $\Phi ^{f}$ periodic orbit is

(2.1) $$ \begin{align} \unicode{x3bb}(f,\tau)=\int_{0}^{\unicode{x3bb}(\tau)} f(\Phi_{s}(x))\,ds. \end{align} $$

Property (2.1) is a simple application of the definitions of $\alpha (x,t)$ and $\kappa (x,t)$ .

Let $\rho _{1}$ and $\rho _{2}$ be two convex projective structures on a surface S. The next lemma, which is essentially due to Sambarino [Reference Sambarino53, §5], states that there exists a positive Hölder continuous reparameterization function $f_{\rho _{1}}^{\rho _{2}}: T^{1}X_{\rho _{1}} \to \mathbb {R}$ encoding the Hilbert length spectrum of $\rho _{2}$ . We include a proof for completeness.

Lemma 2.3. (Sambarino [Reference Sambarino53])

Let $\rho _{1}$ and $\rho _{2}$ be convex projective structures on a surface S. There exists a positive Hölder continuous function $f_{\rho _{1}}^{\rho _{2}}: T^{1}X_{\rho _{1}} \to \mathbb {R}$ such that for every periodic orbit $\tau $ corresponding to $[\gamma ]\in [\Gamma ]$ , one has

$$ \begin{align*} \unicode{x3bb} (f_{\rho_{1}}^{\rho_{2}},\tau)=\ell_{\rho_{2}}^{H}([\gamma]). \end{align*} $$

2.3 Thermodynamic formalism

In this subsection, we will introduce several important concepts from thermodynamic formalism in our context that will be needed later. Standard references for thermodynamic formalism and Markov codings are [Reference Bowen12, Reference Parry and Pollicott47].

For a continuous function $f: T^{1}X_{\rho } \to \mathbb {R}$ , we define its pressure with respect to $\Phi $ as

$$ \begin{align*} P(\Phi,f)=\limsup\limits_{T\longrightarrow \infty} \frac{1}{T}\log\bigg(\sum\limits_{\tau\in R_{T}} e^{\unicode{x3bb}(f,\tau)}\bigg), \end{align*} $$

where $R_{T}:=\{\tau \text { } \text {periodic orbit of }\Phi \mid \unicode{x3bb} (\tau )\in [T-1, T]\}$ . One can check that the topological entropy $h^{H}(\rho )$ is $P(\Phi ,0)$ . For simplicity, we omit the geodesic flow $\Phi $ from the notation and write $P(\cdot )$ for $P(\Phi , \cdot )$ . The pressure can be characterized as follows.

Proposition 2.4. (Variational principle)

The pressure of a continuous function $f:T^{1}X_{\rho }\to \mathbb {R}$ satisfies

$$ \begin{align*}P(f)= \sup\limits_{\mu\in \mathcal{M}^{\Phi}}\bigg(h(\mu)+ \int f\, d\mu\bigg), \end{align*} $$

where $\mathcal M^{\Phi }$ is the space of $\Phi $ -invariant probability measures on $T^{1}X_{\rho }$ and $h(\mu )=h(\Phi ,\mu )$ denotes the measure-theoretic entropy of $\Phi $ with respect to $\mu \in \mathcal {M}^{\Phi }$ .

A $\Phi $ -invariant probability measure $ \mu $ on $T^{1}X_{\rho }$ is called an equilibrium state for f if the supremum is attained at $\mu $ .

The following lemma, derived from Abramov’s formula [Reference Abramov1], allows us to rescale a reparameterization function to be pressure zero.

Lemma 2.6. (Sambarino [Reference Sambarino53, Lemma 2.4], Bowen and Ruelle [Reference Bowen and Ruelle13, Proposition 3.1])

For a positive Hölder reparameterization function f on $T^{1}X_{\rho }$ and $h\in \mathbb {R}$ , the pressure function satisfies

$$ \begin{align*}P(-hf)=0\end{align*} $$

if and only if $h=h(\Phi ^{f})$ , where

$$ \begin{align*} h(\Phi^{f})=\limsup_{T\to\infty} \frac{1}{T}\log\#\{\tau \text{ periodic orbit} \mid \unicode{x3bb}(f,\tau)\leq T\}. \end{align*} $$

By definition, $h(\Phi ^{f})$ is the topological entropy of the reparameterized flow $\Phi ^{f}$ .

We will use Lemma 2.6 in the proofs of Theorems 1.1, 1.3, and 1.5.

Finally, we introduce (Livšic) cohomology. We say two Hölder continuous functions f and g are (Livšic) cohomologous if there exists a Hölder continuous function $V\colon T^{1}X_{\rho }\to \mathbb {R}$ that is differentiable in the flow’s direction such that for all $x\in T^{1}X_{\rho }$ ,

$$ \begin{align*} f(x)-g(x)= \frac{\partial}{\partial t}\bigg|_{t=0}V(\Phi_{t}(x)). \end{align*} $$

2.4 Independence of convex projective surfaces

We start by recalling the notion of (topologically) weakly mixing flows which motivates the concept of independence of representations. A flow $\varphi $ on $T^{1}X_{\rho }$ is weakly mixing if its periods do not generate a discrete subgroup of $\mathbb {R}$ . In particular, we can ask whether the Hilbert geodesic flow $\Phi $ is weakly mixing. Equivalently, we ask whether there exists some non-zero real number $a\in \mathbb {R}$ such that $a\ell _{\rho }^{H}(\gamma )\in \mathbb {Z}$ for all $[\gamma ]\in \Gamma $ . In the proof of Theorem 1.1, we will need a strengthening of this property for a pair of representations.

Two convex projective structures $\rho _{1}$ and $\rho _{2}$ are dependent if there exist $a_{1},a_{2}\in \mathbb R$ , not both equal to zero, such that $a_{1}\ell ^{H}_{\rho _{1}}([\gamma ])+a_{2}\ell ^{H}_{\rho _{2}}([\gamma ])\in \mathbb Z$ for all $[\gamma ]\in [\Gamma ]$ . Otherwise, $\rho _{1}$ and $\rho _{2}$ are independent over $\mathbb Z$ . The next definition clarifies that this notion of independence is of a dynamical nature.

We now prove that convex projective surfaces with distinct Hilbert length spectra are independent.

Proof. Our proof follows from combining the results of Benoist [Reference Benoist5, Reference Benoist6] and an argument of Glorieux [Reference Glorieux28].

We prove this statement by contradiction. Consider the product representation $\eta =\rho _{1}\times \rho _{2}\colon \Gamma \to \mathrm {G}_{1}\times \mathrm {G}_{2}$ , where $\mathrm {G}_{i}$ denotes the Zariski closure of $\rho _{i}$ . Benoist [Reference Benoist5, Theorem 1.3] proved that $\mathrm {G}_{i}$ is $\mathrm {PSL}(3,\mathbb {R})$ or isomorphic to $\mathrm {PSO}(1,2)$ . Either way, this Zariski closure is connected, center-free, and simple, so $\mathrm {G}_{1}\times \mathrm {G}_{2}$ is semi-simple. Choose a Cartan subspace of $\mathrm {G}_{1}\times \mathrm {G}_{2}$ such that

$$ \begin{align*} \mathfrak{a}\subseteq\{(\vec x,\vec y)\in\mathbb{R}^{3}\times\mathbb{R}^{3}\mid x_{1}+x_{2}+x_{3}=0=y_{1}+y_{2}+y_{3}\} \end{align*} $$

and a positive Weyl chamber $\mathfrak {a}^{+}$ contained in $\{(\vec x,\vec y)\in \mathfrak {a}\mid x_{1}\geq x_{2}\geq x_{3}\text { and }y_{1}\geq y_{2}\geq y_{3}\}$ . Denote by $\unicode{x3bb} \colon \mathrm {G}_{1}\times \mathrm {G}_{2}\to \mathfrak {a}^{+}$ the corresponding Jordan projection and by $\phi ^{H}_{a_{1},a_{2}}$ the non-zero linear functional $\phi ^{H}_{a_{1},a_{2}}(\vec x,\vec y)=a_{1}(x_{1}-x_{3})+a_{2}(y_{1}-y_{3})$ so that $\phi ^{H}_{a_{1},a_{2}}(\unicode{x3bb} (\eta (\gamma )))=a_{1}\ell ^{H}_{\rho _{1}}([\gamma ])+a_{2}\ell ^{H}_{\rho _{2}}([\gamma ])$ .

Denote by $\mathrm {H}\subseteq \mathrm {G}_{1}\times \mathrm {G}_{2}$ the Zariski closure of $\eta (\Gamma )$ . As a first step, observe that $\mathrm {H}\neq \mathrm {G}_{1}\times \mathrm {G}_{2}$ . Otherwise, Benoist [Reference Benoist6, Proposition on p. 2] shows that the additive group generated by $\unicode{x3bb} (\eta (\Gamma ))$ is dense (in the standard topology) in $\mathfrak {a}$ . We obtain a contradiction as we assumed $\phi ^{H}_{a_{1},a_{2}}(\unicode{x3bb} (\eta (\gamma )))\in \mathbb {Z}$ for all $\gamma \in \Gamma $ and $\phi ^{H}_{a_{1},a_{2}}$ is continuous.

Let $\pi _{i}\colon \mathrm {H}\to \mathrm {G}_{i}$ for $i=1,2$ denote the projection maps. It follows from [Reference Dal’bo and Kim24, Lemma 3.1] that $\pi _{i}$ is surjective for $i=1,2$ . Denote by $\mathrm {N}_{1}=\pi _{2}^{-1}(\text {id})$ (respectively $\mathrm {N}_{2}=\pi _{1}^{-1}(\text {id})$ ) the kernel of $\pi _{2}$ (respectively $\pi _{1}$ ) which is naturally identified with a normal subgroup of $\mathrm {G}_{1}$ (respectively $\mathrm {G}_{2}$ ) (note the indices in the definition of $\mathrm {N}_{i}$ ). Then, Goursat’s lemma [Reference Hall32, Theorem 5.5.1] states that the image of $\mathrm {H}$ in $\mathrm {G}_{1}/\mathrm {N}_{1}\times \mathrm {G}_{2}/\mathrm {N}_{2}$ is the graph of an isomorphism $\mathrm {G}_{1}/\mathrm {N}_{1}\cong \mathrm {G}_{2}/\mathrm {N}_{2}$ . Since $\mathrm {G}_{1}$ is simple, connected, and center-free, then $\mathrm {N}_{1}=\{e\}$ or $\mathrm {G}_{1}$ .

Case 1: Suppose $\mathrm {N}_{1}=\mathrm {G}_{1}$ . Then $\mathrm {N}_{2}=\mathrm {G}_{2}$ and $\mathrm {H}$ is the direct product $\mathrm {G}_{1}\times \mathrm {G}_{2}$ , which is a contradiction.

Case 2: Suppose $\mathrm {N}_{1}=\{e\}$ . Since $\mathrm {G}_{2}$ is simple, $\mathrm {N}_{2}=\{e\}$ and $\mathrm {G}_{1}\cong \mathrm {G}_{2}$ . In other words, $\mathrm {H}$ is the graph of an automorphism $\iota \colon \mathrm {G}_{1}\to \mathrm {G}_{2}$ . This induces an automorphism of the corresponding Lie algebras and, by the classification of their outer automorphisms given in [Reference Gündoğan31] (see also [Reference Bridgeman, Canary, Labourie and Sambarino14, Theorem 11.9]), we deduce that $\rho _{2}$ is conjugated to either $\rho _{1}$ or $\rho _{1}^{\ast }$ , which contradicts our hypothesis.

Observe that Lemma 2.11 readily implies that the geodesic flow of a convex projective structure is weakly mixing. Otherwise there exist $a\in \mathbb R$ , $a\neq 0$ such that $a\ell ^{H}_{\rho }([\gamma ])\in \mathbb Z$ which directly contradicts Lemma 2.11 with $a_{1}=a$ , $a_{2}=0$ , $\rho _{1}=\rho $ , and $\rho _{2}\in \mathfrak {C}(S)$ different from $\rho _{1}$ and $\rho _{1}^{\ast }$ .

2.5 Geodesic currents for convex projective surfaces

Fix an auxiliary hyperbolic structure m on S. A geodesic current is a Borel, locally finite, $\pi _{1}(S)$ -invariant measure on the set of complete geodesics of the universal cover $\widetilde {S}$ . An important example is the geodesic current $\delta _{\gamma }$ given by Dirac measures on the axes of the lifts of a closed geodesic $\gamma $ in S.

The space $\mathcal {C}(S)$ of geodesic currents is a convex cone in an infinite dimensional vector space. Bonahon [Reference Bonahon8] extended the intersection pairing on closed curves to the space of geodesic currents, that is, there exists a continuous, positive, symmetric, bilinear pairing

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

such that $i(\delta _{c},\delta _{d})$ equals the intersection number of the closed geodesics c and d.

Extending the work of Bonahon [Reference Bonahon9], in [Reference Bridgeman, Canary, Labourie and Sambarino15, Reference Martone and Zhang45], it was shown that for each convex projective surface $\rho $ , there exists a Hilbert geodesic current $\nu _{\rho }$ such that for every $[\gamma ]\in [\Gamma ]$ ,

$$ \begin{align*} i(\nu_{\rho},\delta_{\gamma})=\ell_{\rho}^{H}([\gamma]), \end{align*} $$

where $\gamma $ denotes the unique closed geodesic in its free homotopy class $[\gamma ]$ . Bonahon [Reference Bonahon9, Proposition 15] proves that the geodesic current $\nu _{\rho }$ of a hyperbolic structure $\rho $ has self-intersection $i(\nu _{\rho },\nu _{\rho })=-\pi ^{2}\chi (S)$ . However, if $\rho \in \mathfrak {C}(S)$ is not in the Fuchsian locus, then $i(\nu _{\rho },\nu _{\rho })>-\pi ^{2}\chi (S)$ by Corollary 5.3 in [Reference Bridgeman, Canary, Labourie and Sambarino15].

In general, given a geodesic current $\nu $ , we can use the intersection number to define its length spectrum $\ell _{\nu }\colon [\Gamma ]\to \mathbb {R}^{+}$ as $\ell _{\nu }([\gamma ])=i(\nu ,\delta _{\gamma })$ . The systole of $\nu $ is then $ \mathrm {Sys}(\nu ):=\inf _{[\gamma ]\in [\Gamma ]}\ell _{\nu }([\gamma ])$ . Corollary 1.5 in [Reference Burger, Iozzi, Parreau and Pozzetti18] shows that $\mathrm {Sys}\colon \mathcal {C}(S)\to [0,\infty )$ is a continuous function.

A geodesic current is period minimizing if for all $T>0$ , the set $ \#\{[\gamma ]\in [\Gamma ]\mid \ell _{\nu }([\gamma ])<T\}$ is finite. We define the exponential growth rate of a period minimizing geodesic current $\nu $ by

$$ \begin{align*} h(\nu)=\lim_{T\to \infty}\frac{1}{T}\log \#\{[\gamma]\in[\Gamma]\mid \ell_{\nu}([\gamma])<T\}. \end{align*} $$

The notation is motivated by the fact that if $\rho $ is a convex projective structure and $\nu _{\rho }$ is the corresponding Hilbert geodesic current, then $h(\nu _{\rho })$ is equal to the topological entropy $h^{H}(\rho )$ of $\rho $ .

The systole and the exponential growth rate of a geodesic current are related by the following inequality, which will play an important role in the proof of Theorem 1.3.

Theorem 2.12. (Corollary 7.6 in [Reference Martone and Zhang45])

Let S be a closed, connected, oriented surface of genus $g\geq 2$ . There exists a constant $C>0$ depending only on g such that for every period minimizing geodesic current $\nu \in \mathcal C(S)$ ,

$$ \begin{align*} \text{Sys}(\nu)h(\nu)\leq C. \end{align*} $$

4.1 Manhattan curve and correlation number

Let $\rho _{1}$ and $\rho _{2}$ be two convex projective structures. The Manhattan curve of $\rho _{1},\rho _{2}$ is the curve $\mathcal {C}(\rho _{1},\rho _{2})$ that bounds the convex set

$$ \begin{align*} \bigg\{(a,b) \in \mathbb{R}^{2}: \sum_{[\gamma]\in[\Gamma]} e^{-(a\ell^{H}_{\rho_{1}}([\gamma])+b\ell^{H}_{\rho_{2}}([\gamma]))}< +\infty\bigg\}. \end{align*} $$

Equivalently (see [Reference Sharp58]), the Manhattan curve can be defined in terms of the pressure function and the reparameterization function $f=f_{\rho _{1}}^{\rho _{2}}: T^{1}X_{\rho _{1}} \to \mathbb {R}$ from Lemma 2.3 as

$$ \begin{align*} \mathcal{C}(\rho_{1},\rho_{2})=\{(a,b)\in\mathbb{R}^{2}\colon P(-a-bf)=0\}=\{(a,b)\in\mathbb{R}^{2}\colon P(-bf)=a\}. \end{align*} $$

The next theorem collects the properties of the Manhattan curve we will need which were first discussed in the setting of representations into isometry groups of rank one symmetric spaces (see [Reference Burger17, Theorem 1]). We recall that a convex curve is a simple curve in the Euclidean plane which lies completely on one side of any of its tangent lines. The boundary of a convex set is always a convex curve. A strictly convex curve is a convex curve that does not contain any line segments.

Proof. For item (1), convexity of the Manhattan curve follows from its definition since it bounds a convex set. However, because the Manhattan curve is the graph of a function q defined implicitly as $P(-q(s)f)=s$ , the implicit function theorem and real analyticity of the pressure function yield real analyticity of the Manhattan curve.

For item (2), we notice $P(0)=h^{H}(\rho _{1})$ and $P(-h^{H}(\rho _{2})f)=0$ (recall Lemma 2.6). Therefore, $\mathcal C(\rho _{1},\rho _{2})$ passes through $(h^{H}(\rho _{1}),0)$ and $(0,h^{H}(\rho _{2}))$ . To see that the normal to the Manhattan curve at $(h^{H}(\rho _{1}),0)$ has slope ${\mathbf {I}}(\rho _{1},\rho _{2})$ , we differentiate the equality $P(-q(s)f)=s$ using equation (3.1) at $(h^{H}(\rho _{1}),0)$ to obtain

$$ \begin{align*} 1=\frac{d}{ds}\bigg|_{s=h^{H}(\rho_{1})}P(-q(s)f)=\bigg(-\int f\,{d}\mu_{\Phi^{\rho_{1}}}\bigg)q^{\prime}(h^{H}(\rho_{1})). \end{align*} $$

Therefore, the normal to the Manhattan curve at $(h^{H}(\rho _{1}),0)$ has slope ${\mathbf {I}}(\rho _{1},\rho _{2})=\int \! f\,{d}\mu _{\Phi ^{\rho _{1}}}$ . The rest of item (2) follows by symmetry.

For item (3), if $\mathcal C(\rho _{1},\rho _{2})$ is not strictly convex, then by real analyticity, it must be a line segment. From item (2), we have ${\mathbf {I}}(\rho _{1},\rho _{2})=1/\mathbf {I}(\rho _{2}, \rho _{1})$ . Therefore, we conclude ${\mathbf {J}}(\rho _{1},\rho _{2})=1/\mathbf {J}(\rho _{2}, \rho _{1})$ . By Proposition 3.4, we conclude ${\mathbf {J}}(\rho _{1},\rho _{2})={\mathbf {J}}(\rho _{2},\rho _{1})=1$ and therefore $L^{H}_{\rho _{1}}=L^{H}_{\rho _{2}}$ . Then Remark 1.2 gives the results.

Sharp [Reference Sharp58] expressed the correlation number $M(\rho _{1},\rho _{2})$ in terms of the Manhattan curve for hyperbolic structures. We establish an analogous result for convex real projective structures.

Theorem 4.2. Let $\rho _{1}$ and $\rho _{2}$ be convex projective structures with distinct renormalized Hilbert length spectra. Then, their correlation number can be written as

$$ \begin{align*} M(\rho_{1},\rho_{2})=\frac{a}{h^{H}(\rho_{1})}+\frac{b}{h^{H}(\rho_{2})}, \end{align*} $$

where $(a,b)\in \mathcal {C}(\rho _{1},\rho _{2})$ is the unique point on the Manhattan curve at which the tangent line is parallel to the line passing through $(h^{H}(\rho _{1}),0)$ and $(0,h^{H}(\rho _{2}))$ . See Figure 1.

Figure 1

Manhattan curve and the point $(a,b)$ described in Theorem 4.2.

Proof. By the proof of Theorem 1.1, the correlation number is such that $h^{H}(\rho _{1}) M(\rho _{1},\rho _{2})=h(\mu _{a_{0}})$ , where $a_{0}=\int f\, d\mu _{a_{0}}= ({h^{H}(\rho _{1})}/{h^{H}(\rho _{2})})$ . By definition,

$$ \begin{align*} h(\mu_{a_{0}})=P(t_{a_{0}}f)-t_{a_{0}}\frac{h^{H}(\rho_{1})}{h^{H}(\rho_{2})}. \end{align*} $$

Note that $\mathcal {C}(\rho _{1},\rho _{2})$ is the graph of a real analytic function q defined implicitly as $P(-q(s)f)=s$ . Setting $q(s_{0})=-t_{a_{0}}$ , it follows that

$$ \begin{align*} M(\rho_{1},\rho_{2})=\frac{h(\mu_{a_{0}})}{h^{H}(\rho_{1})}=\frac{P(-q(s_{0})f)}{h^{H}(\rho_{1})}+\frac{q(s_{0})}{h^{H}(\rho_{2})}= \frac{s_{0}}{h^{H}(\rho_{1})}+\frac{q(s_{0})}{h^{H}(\rho_{2})}. \end{align*} $$

Moreover, observe that

$$ \begin{align*} 1=\frac{d}{ds}P(-q(s)f)=\bigg(-\int f\,d\mu_{-q(s)f}\bigg)q^{\prime}(s). \end{align*} $$

We conclude by recalling that $\int \!f\, d\mu _{t_{a_{0}}f}= ({h^{H}(\rho _{1})}/{h^{H}(\rho _{2})})$ and that the line passing through $(h^{H}(\rho _{1}),0)$ and $(0,h^{H}(\rho _{2}))$ has slope $- ({h^{H}(\rho _{2})}/{h^{H}(\rho _{1})})$ .

4.2 Decay of correlation number

This section is dedicated to the proof of Theorem 1.3 from the introduction, which we restate here for the convenience of the reader.

Theorem 1.3.

There exist sequences $(\rho _{n})_{n=1}^{\infty }$ and $(\eta _{n})_{n=1}^{\infty }$ in the Teichmüller space $\mathcal {T}(S)$ such that the correlation numbers $M(\rho _{n},\eta _{n})$ satisfy

$$ \begin{align*} \lim_{n\to\infty} M(\rho_{n},\eta_{n})=0. \end{align*} $$

Proof. We construct two special families of hyperbolic structures $\rho _{n}$ , $\eta _{n}$ and consider their corresponding geodesic currents $\nu _{\rho _{n}}$ , $\nu _{\eta _{n}}$ as in §2.5. Our proof proceeds in two steps. First, we take geodesic currents $\nu _{\rho _{n}}+\nu _{\eta _{n}}$ given by the sum of the currents of $\rho _{n}$ and $\eta _{n}$ and show that their exponential growth rates satisfy $\lim \nolimits _{n\to \infty } h(\nu _{\rho _{n}}+\nu _{\eta _{n}})=0$ . Then we show that this condition implies that the correlation number $M(\rho _{n},\eta _{n})$ goes to zero as well.

We consider two pair-of-pants decompositions $(\alpha _{i})$ and $(\beta _{i})$ whose union is filling on a hyperbolic structure $\rho _{0}$ . See Figure 2 for an example. A family of simple closed curves is filling if the complement of their union consists of topological discs. We take $(\rho _{n})_{n=1}^{\infty }$ to be a sequence of hyperbolic structures obtained by pinching all $\alpha _{i}$ on $\rho _{0}$ so that the hyperbolic length $\ell _{\rho _{n}}(\alpha _{i})=\epsilon _{n}$ with $\epsilon _{n} \to 0$ when $n\to \infty $ . Similarly, we take $(\eta _{n})$ to be another sequence of hyperbolic structures obtained by pinching all $\beta _{i}$ on $\rho _{0}$ so that the hyperbolic length $\ell _{\eta _{n}}(\beta _{i})=\epsilon _{n}$ . Note in such cases, we have that $\ell ^{H}_{\rho _{n}}=\ell _{\rho _{n}}$ and $\ell ^{H}_{\eta _{n}}=\ell _{\eta _{n}}$ and that the topological entropy $h^{H}(\rho _{n})=h^{H}(\eta _{n})=1$ for all n. We now proceed to prove $\lim \nolimits _{n\to \infty } h(\nu _{\rho _{n}}+\nu _{\eta _{n}})=0$ .

Figure 2

Closed curves $\alpha _{1}$ , $\alpha _{2}$ , and $\alpha _{3}$ and $\beta _{1}$ , $\beta _{2}$ , and $\beta _{3}$ give two pair-of-pants decompositions whose union fills the surface of genus two.

By definition, the systole of $\nu _{\rho _{n}}+\nu _{\eta _{n}}$ is equal to $L_{n}=\inf \nolimits _{[\gamma ]\in [\Gamma ]}\{\ell ^{H}_{\rho _{n}}([\gamma ])+\ell ^{H}_{\eta _{n}}([\gamma ])\}$ . Note that for a fixed n, $\#\{[\gamma ]\in [\Gamma ]\mid \ell ^{H}_{\rho _{n}}([\gamma ])+\ell ^{H}_{\eta _{n}}([\gamma ])<T\}<\infty $ for all $T>0$ . In other words, the geodesic current $\nu _{\rho _{n}}+\nu _{\eta _{n}}$ is period minimizing and so we can apply Theorem 2.12 to find a constant C depending only on the topology of S such that

$$ \begin{align*} h(\nu_{\rho_{n}}+\nu_{\eta_{n}})\leq \frac{C}{L_{n}}. \end{align*} $$

Therefore, to show $\lim \nolimits _{n\to \infty }h(\nu _{\rho _{n}}+\nu _{\eta _{n}})=0$ , it suffices to show that $\lim \nolimits _{n\to \infty } L_{n}=\infty $ . Because of the filling condition, the geodesic representative of any $[\gamma ]\in [\Gamma ]$ must intersect transversally either curves in $(\alpha _{i})$ or curves in $(\beta _{i})$ . By the collar lemma (see [Reference Buser19, Ch. 4]), each geodesic representative of $\alpha _{i}$ (respectively $\beta _{i}$ ) for $\rho _{n}$ (respectively $\eta _{n}$ ) is enclosed in a standard collar neighborhood of width approximately $\log ({1}/{\epsilon _{n}})$ . In particular, every closed curve traverses a collar neighborhood of width approximately $\log ({1}/{\epsilon _{n}})$ for the hyperbolic metric $\rho _{n}$ or the hyperbolic metric $\eta _{n}$ which implies that $\lim \nolimits _{n\to \infty }L_{n}=\infty $ .

We now show that $\lim \nolimits _{n\to \infty } h(\nu _{\rho _{n}}+\nu _{\eta _{n}})=0$ implies that the correlation number goes to zero as well. Consider the reparameterization functions $f^{\eta _{n}}_{\rho _{n}}$ as in Lemma 2.3 and notice that $\unicode{x3bb} (1+f^{\eta _{n}}_{\rho _{n}},\tau )=\ell ^{H}_{\rho _{n}}([\gamma ])+\ell ^{H}_{\eta _{n}}([\gamma ])$ for every periodic orbit corresponding to $[\gamma ]\in [\Gamma ]$ . By Lemma 2.6, for all n,

$$ \begin{align*} P(-h(\nu_{\rho_{n}}+\nu_{\eta_{n}})-h(\nu_{\rho_{n}}+\nu_{\eta_{n}})f^{\eta_{n}}_{\rho_{n}})=P(-h(\nu_{\rho_{n}}+\nu_{\eta_{n}})(1+f^{\eta_{n}}_{\rho_{n}}))=0. \end{align*} $$

We have that $(h(\nu _{\rho _{n}}+\nu _{\eta _{n}}),h(\nu _{\rho _{n}}+\nu _{\eta _{n}}))\in \mathcal {C}(\rho _{n},\eta _{n})$ thanks to the characterization of the Manhattan curve in terms of the pressure function. If the line $y+x=2h(\nu _{\rho _{n}}+\nu _{\eta _{n}})$ is tangent to the Manhattan curve $\mathcal {C}(\rho _{n},\eta _{n})$ , then by Theorem 4.2, we obtain $M(\rho _{n},\eta _{n})=2h(\nu _{\rho _{n}}+\nu _{\eta _{n}})$ . If not, then the line $y+x=2h(\nu _{\rho _{n}}+\nu _{\eta _{n}})$ must intersect transversally $\mathcal {C}(\rho _{n},\eta _{n})$ at two points. See Figure 3. Now, by the mean value theorem and the convexity of the Manhattan curve, we have from Theorem 4.2 that there exists $0<a_{n},b_{n}< 2h(\nu _{\rho _{n}}+\nu _{\eta _{n}})$ such that

$$ \begin{align*} 0<M(\rho_{n},\eta_{n})=a_{n}+b_{n}<2h(\nu_{\rho_{n}}+\nu_{\eta_{n}}). \end{align*} $$

This immediately implies that $M(\rho _{n},\eta _{n})$ goes to zero as n goes to infinity.

Figure 3

Manhattan curve and the point $(h(\nu _{\rho _{n}}+\nu _{\eta _{n}}),h(\nu _{\rho _{n}}+\nu _{\eta _{n}}))$ described in the proof of Theorem 1.3.

5.1 Cubic rays and affine spheres

Building on work of Hitchin [Reference Hitchin33], Labourie [Reference Labourie38], and Loftin [Reference Loftin43] have independently parameterized $\mathfrak {C}(S)$ as the vector bundle of holomorphic cubic differentials over $\mathcal {T}(S)$ . This is called the Labourie–Loftin’s parameterization. We briefly recall this parameterization since we will use it in this subsection to study the correlation number. The reader can refer to [Reference Labourie38, Reference Loftin43] for a detailed construction. This parameterization shows that the space of pairs formed by a complex structure J on S and a J-holomorphic cubic differential q is in one-to-one correspondence with the space of convex real projective structures on S.

Because of the classical correspondence between hyperbolic structures and complex structures, we will sometimes blur the difference between a Riemann surface X and a hyperbolic structure X. We recall that a holomorphic cubic differential q is a holomorphic section of $K\otimes K \otimes K$ , where K is the canonical line bundle of a Riemann surface X. Locally, a holomorphic cubic differential can be written in complex coordinate charts as $q=q(z)dz\otimes dz \otimes dz$ (often written as $q(z)dz^{3}$ ) with $q(z)$ holomorphic. The cubic differential q is invariant under change of coordinates, meaning that if we pick a different coordinate chart w, then $q(w)dw^{3}=q(z)dz^{3}$ . The hyperbolic metric $\sigma $ that corresponds to the complex structure of X induces a pointwise Hermitian metric $\langle \cdot ,\cdot \rangle $ for holomorphic cubic differentials on X defined as follows:

$$ \begin{align*}\langle q_{1},q_{2} \rangle= \frac{q_{1} \overline{q_{2}}}{\sigma^{3}}.\end{align*} $$

The $L^{2}$ -norm of a cubic differential q is defined as

$$ \begin{align*}\lVert q\rVert {}^{2}_{X}=\int_{X}\langle q,q \rangle\, {d}\sigma.\end{align*} $$

The correspondence between the space of pairs of complex structures on S and holomorphic cubic differentials on associated Riemann surfaces with the set of convex real projective structures on S goes through a geometrical object invariant under affine transformations, called an affine sphere. We briefly describe the relation. Consider a pair $(J, q)$ , where J is a complex structure and q is a holomorphic cubic differential on the Riemann surface $X_{J}$ associated to J. Then one can obtain an affine sphere, which is a hypersurface in $\mathbb {R}^{3}$ that is invariant under affine transformations, by solving a second-order elliptic partial differential equation (PDE) called Wang’s equation (see [Reference Loftin43, §4]). The affine sphere can be projected to $\mathbb {RP}^{2}$ so as to obtain the developing image $\Omega _{\rho }$ of some convex projective structure $\rho $ . Conversely, given a convex projective structure $\rho $ and its developing image $\Omega _{\rho }$ , we can construct an affine sphere by solving a Monge–Ampére equation (see [Reference Loftin43, §7]). The affine sphere will provide a complex structure J and a holomorphic cubic differential q on $X_{J}$ .

Since hyperbolic structures correspond to complex structures, the above identification provides a way to parameterize the space of convex projective structures $\mathfrak {C}(S)$ as the vector bundle of holomorphic cubic differentials over $\mathcal {T}(S)$ . In particular, we can fix a non-zero cubic differential q and consider the associated family of representations $(\rho _{t})_{\scriptscriptstyle t\in \mathbb {R}}$ in $\mathfrak {C}(S)$ parameterized by $(tq)_{t\in \mathbb {R}}$ . When $t\geq 0$ , we call such a family a cubic ray. In this section, we study the properties of the renormalized Hilbert length spectrum along cubic rays in $\mathfrak {C}(S)$ .

Note that we can use the holomorphic data introduced above to describe when two convex projective structures $\rho _{1}$ and $\rho _{2}$ have the same renormalized Hilbert length spectra (see Remark 1.2). Fixing a Riemmann surface $X_{J}$ , the case $\rho _{1}=\rho _{2}$ happens when their corresponding holomorphic cubic differentials $q_{1}$ and $q_{2}$ with respect to $X_{J}$ are equal in the Labourie–Loftin’s parameterization. However, if $\rho _{1}$ is the contragredient of $\rho _{2}$ , then their corresponding holomorphic objects must be $(X_{J}, q_{1})$ and $(X_{J}, q_{2})=(X_{J}, -q_{1})$ , respectively. For a detailed explanation of this fact, we refer the reader to [Reference Loftin42, §8].

For the reminder of this section, when there is no ambiguity, we ease notation by writing $h_{t}=h^{H}(\rho _{t})$ for the topological entropy, and $L^{H}_{t}=L^{H}_{\rho _{t}}$ for the renormalized Hilbert length spectrum of a cubic ray $(\rho _{t})_{t\geq 0}$ . With these notation, $\rho _{0}$ is the hyperbolic structure associated to $X_{0}$ . The entropy $h_{0}$ equals $1$ and the renormalized Hilbert length spectrum $L^{H}_{0}$ is the $X_{0}$ -length spectrum.

The following observation is important and follows immediately from the work of Tholozan [Reference Tholozan59].

Lemma 5.1. Let $(\rho _{t})_{t> 0}$ be a cubic ray. There exists $D>1$ such that for all $t\geq 0$ ,

$$ \begin{align*} \frac{1}{D}L^{H}_{0} \leq L^{H}_{t}\leq DL^{H}_{0}. \end{align*} $$

Proof. Tholozan [Reference Tholozan59, Theorem 3.9] proves that there exists $B>1$ such that for all $t>0$ ,

(5.1) $$ \begin{align} \frac{1}{B}t^{1/3}\ell^{H}_{0}\leq\ell^{H}_{t}\leq Bt^{1/3}\ell^{H}_{0}. \end{align} $$

In turn, this implies that for all $T\geq 0$ and $t> 0$ ,

$$ \begin{align*} \#\bigg\{[\gamma] \mid \ell^{H}_{0}([\gamma])\leq \frac{1}{Bt^{1/3}}T\bigg\}\leq \#\{[\gamma]\mid \ell^{H}_{t}([\gamma])\leq T\}\leq \#\bigg\{[\gamma] \mid \ell^{H}_{0}([\gamma])\leq \frac{B}{t^{1/3}}T \bigg\}. \end{align*} $$

Since $h_{0}=1$ , by definition of topological entropy, the above inequalities imply that for all $t> 0$ ,

(5.2) $$ \begin{align} \frac{1}{Bt^{1/3}}\leq h_{t}\leq \frac{B}{t^{1/3}}. \end{align} $$

The result follows by taking $D=B^{2}$ and combining the inequalities (5.1) and (5.2). The case for $t=0$ is obvious.

The following example points out a problem with comparing un-renormalized length spectra and partially motivates our study of the renormalized Hilbert length spectrum. In particular, equation (5.3) below should be compared to the correlation theorem.

Example 5.2. Fix $\epsilon>0$ . Let $\rho _{0}$ be a Fuchsian representation in $\mathfrak C(S)$ and consider a cubic ray $(\rho _{t})_{t\geq 0}\subset \mathfrak C(S)$ . Consider any $t>0$ . Then,

(5.3) $$ \begin{align} \lim_{x\to\infty}\# \{[\gamma]\in[\Gamma] \mid \ell_{0}^{H}([\gamma]) \in (x,x+\varepsilon),\ \ell_{t}^{H}([\gamma]) \in (x, x+\varepsilon)\}=0. \end{align} $$

Proof. For $t>0$ , consider $\delta>0$ small enough so that $h_{t}<1< D-\delta $ , where D is as in Lemma 5.1. Here we recall that $h^{H}(\rho )$ is strictly less than one when $\rho \in \mathfrak {C}(S)$ is not in the Fuchsian locus (see [Reference Crampon22]). There exists M sufficiently large such that $({Dx}/{x+\varepsilon }) \geq D-\delta $ for all $x>M$ . Therefore, for all $x>M$ , if $x<\ell _{0}^{H}([\gamma ])<x+\epsilon $ , then

$$ \begin{align*} \ell^{H}_{t}([\gamma])\geq\frac{D}{h_{t}}\ell^{H}_{0}([\gamma])>\frac{D}{D-\delta}x\geq x+\varepsilon. \end{align*} $$

This shows that $\{[\gamma ]\in [\Gamma ] \mid \ell _{0}^{H}([\gamma ]) \in (x,x+\varepsilon ),\ \ell _{t}^{H}([\gamma ]) \in (x, x+\varepsilon )\}=\emptyset $ for all $x>M$ , which implies that equation (5.3) holds.

5.2 Cubic rays and correlation numbers

In contrast with Theorem 1.3, we show some instances in which the correlation numbers $M(\rho _{t}, \eta _{t})$ arising from two different cubic rays are uniformly bounded away from zero. We first introduce a convenient lemma that will be used in the proof of Theorems 1.4 and 1.5.

Lemma 5.3. Suppose $\rho $ and $\eta $ are different convex real projective structures such that $\rho ^{*} \neq \eta $ . Let $\upsilon _{\rho }$ and $\upsilon _{\eta }$ be the renormalized Hilbert currents associated to $\rho $ and $\eta $ , respectively. For $s\in [0,1]$ , let $u_{s}$ denote the geodesic current $s\upsilon _{\rho }+(1-s)\upsilon _{\eta }$ . Then there exists a unique $s_{0}\in (0,1)$ such that

$$ \begin{align*} M(\rho,\eta)=h(u_{s_{0}}). \end{align*} $$

Proof. Since $\eta \neq \rho ,\rho ^{*}$ , the Manhattan curve $\mathcal {C}(\rho ,\eta )$ is strictly convex. The line $ ({x}/{h^{H}(\rho )}) + ({y}/{h^{H}(\eta )}) =1$ intersects the Manhattan curve in the first quadrant only at the points $(h^{H}(\rho ),0)$ and $(0,h^{H}(\eta ))$ . It follows that for every $s\in [0,1]$ , the straight line connecting $(sh^{H}(\rho ),(1-s)h^{H}(\eta ))$ and the origin must intersect $\mathcal {C}(\rho ,\eta )$ . By Lemma 2.6,

$$ \begin{align*} &P(-h(u_{s})sh^{H}(\rho)-h(u_{s})(1-s)h^{H}(\eta)f^{\eta}_{\rho})\\&\quad= P(-h(u_{s})(sh^{H}(\rho)+(1-s)h^{H}(\eta)f^{\eta}_{\rho}))= 0, \end{align*} $$

where $f^{\eta }_{\rho }$ is the reparameterization function from Lemma 2.3. It follows from the definition of the Manhattan curve that every point on $\mathcal {C}(\rho ,\eta )$ in the first quadrant can be written in the form of $(h(u_{s})sh^{H}(\rho ),h(u_{s})(1-s)h^{H}(\eta ))$ for some $s\in [0,1]$ . By Theorem 4.2, there exists $s_{0}\in (0,1)$ such that

$$ \begin{align*} M(\rho,\eta)=\frac{s_{0}h(u_{s_{0}})h^{H}(\rho)}{h^{H}(\rho)}+\frac{(1-s_{0})h(u_{s_{0}})h^{H}(\eta)}{h^{H}(\eta)}=h(u_{s_{0}}).\\[-3.6pc] \end{align*} $$

We first study the case of two cubic rays which lie in two different fibers of the vector bundle $\mathfrak {C}(S)\to \mathcal {T}(S)$ given by the Labourie–Loftin parameterization of the space of convex projective surfaces.

Theorem 1.4.

Let $(\rho _{t})_{t\geq 0}$ , $(\eta _{r})_{r\geq 0}$ be two cubic rays associated to two different hyperbolic structures $\rho _{0}\neq \eta _{0}$ . Then, there exists a constant $C>0$ such that for all $t,r \geq 0$ ,

$$ \begin{align*} M(\rho_{t},\eta_{r})\geq CM(\rho_{0},\eta_{0}). \end{align*} $$

Proof. Fix $t, r\geq 0$ and write $\rho =\rho _{t}$ and $\eta =\eta _{r}$ , for simplicity. Note that by hypothesis, $L^{H}_{\rho }\neq L^{H}_{\eta }$ and $L^{H}_{\rho _{0}}\neq L^{H}_{\eta _{0}}$ . This is because $L_{\rho }^{H}=L_{\eta }^{H}$ if and only if $\rho $ equals $\eta $ or its contragredient involution $\eta ^{*}$ . Clearly, $\rho \neq \eta $ . Moreover, $\rho =\eta ^{*}$ only happens if their associated pairs of complex structures and holomorphic cubic differentials by the above construction are $(X_{0},q)$ and $(X_{0}, -q)$ , respectively (see [Reference Loftin42, §8]), which is excluded as well. For $s\in [0,1]$ , let $u_{s}$ denote the geodesic current given by $ s \upsilon _{\rho }+(1-s)\upsilon _{\eta }$ , where $\upsilon _{\rho }$ and $\upsilon _{\eta }$ are the renormalized Hilbert geodesic currents of $\rho $ and $\eta $ , respectively. Denote by $h(u_{s})$ the exponential growth rate for the geodesic current $u_{s}$ .

By Lemma 5.1, there exist constants $D_{1},D_{2}$ depending on $\rho _{0}$ and $\eta _{0}$ , respectively, such that

$$ \begin{align*} sL^{H}_{\rho}+(1-s)L^{H}_{\eta}\leq sD_{1} L^{H}_{\rho_{0}}+(1-s)D_{2} L^{H}_{\eta_{0}}\leq \max\{D_{1},D_{2}\}( s L^{H}_{\rho_{0}}+(1-s) L^{H}_{\eta_{0}}). \end{align*} $$

Set $C={1}/{\max \{D_{1},D_{2}\}}$ and $w_{s}=s\upsilon _{\rho _{0}}+(1-s)\upsilon _{\eta _{0}}$ , where $\upsilon _{\rho _{0}}$ and $\upsilon _{\eta _{0}}$ are the renormalized Hilbert geodesic currents of $\rho _{0}$ and $\eta _{0}$ , respectively. Then,

$$ \begin{align*} h(u_{s})\geq C h(w_{s}). \end{align*} $$

By Lemma 2.6, the intersection between the Manhattan curve $\mathcal {C}(\rho _{0}, \eta _{0})$ and the line passing through the origin and the point $(s,(1-s))$ has coordinates $(s h(w_{s}), (1-s)h(w_{s}))$ and lies on the line $y+x=h(w_{s})$ . Then, by Theorem 4.2,

$$ \begin{align*} h(w_{s})\geq M(\rho_{0},\eta_{0}). \end{align*} $$

See Figure 4. Finally, by Lemma 5.3, there exists $s_{0}\in (0,1)$ such that

$$ \begin{align*} M(\rho,\eta)=h(u_{s_{0}})\geq C h(w_{s_{0}})\geq C M(\rho_{0},\eta_{0}), \end{align*} $$

which concludes the proof.

Figure 4

The correlation number of $\rho _{0}$ and $\eta _{0}$ is less or equal to the exponential growth rate of the geodesic current $w_{s}$ .

Finally, we prove Theorem 1.5 from the introduction which can be seen as a ‘limiting case’ of Theorem 1.4 if we avoid pairs of sequences related by the contragredient involution.

Proof. This proof is similar to the proof of Theorem 1.4. We note that $\rho _{0}=\eta _{0}$ and $L_{\rho _{0}}^{H}=L_{\eta _{0}}^{H}=L_{0}^{H}$ . Recall from Lemma 5.1, there exists $D_{1},D_{2}>1$ such that for all $t,r\geq 0$ ,

$$ \begin{align*} \frac{1}{D_{1}}L^{H}_{0} \leq L^{H}_{\rho_{t}}\leq D_{1}L^{H}_{0}\quad\text{and}\quad \frac{1}{D_{2}}L^{H}_{0} \leq L^{H}_{\eta_{r}}\leq D_{2}L^{H}_{0}. \end{align*} $$

Consider the renormalized Hilbert currents $\upsilon _{\rho _{t}}$ and $\upsilon _{\eta _{r}}$ for $\rho _{t}$ and $\eta _{r}$ , respectively. Let $s\in [0,1]$ and set the geodesic current $u_{t,r,s}=s\upsilon _{\rho _{t}}+(1-s)\upsilon _{\eta _{r}}$ . As a first step, we show that the entropy of the geodesic current $u_{t,r,s}=s\upsilon _{\rho _{t}}+(1-s)\upsilon _{\eta _{r}}$ is uniformly bounded away from zero. We have from Lemma 5.1, for all $[\gamma ]\in [\Gamma ]$ ,

$$ \begin{align*} s L_{\rho_{t}}^{H}([\gamma])+(1-s)L^{H}_{\eta_{r}}([\gamma])\leq (sD_{1}+(1-s)D_{2})L_{0}^{H}([\gamma])\leq \max\{D_{1}, D_{2}\} L^{H}_{0}([\gamma]). \end{align*} $$

Thus, we obtain $h(u_{t,r,s}) \geq {1}/{D}$ for every $s\in (0,1)$ , where $D=\max \{D_{1},D_{2}\}$ .

Next we want to show this implies the correlation number $M(\rho _{t},\eta _{r})$ is bounded away from zero. Because $q_{1}$ and $q_{2}$ both have unit $L^{2}$ -norm and $q_{2}\neq -q_{1}$ , we have $L_{\rho _{t}}^{H}\neq L_{\eta _{t}}^{H}$ as ${\rho _{t}}^{*} \neq \eta _{t}$ when $t=r>0$ . Again, here we use the fact that $\rho _{t}=\eta _{t}^{*}$ only if their associated pairs of complex structures and holomorphic cubic differentials (via affine spheres) are $(X_{0},tq_{1})$ and $(X_{0}, -tq_{1})$ , respectively. For each $t>0$ and $r>0$ , by Lemma 5.3, we can find some $s_{0}\in (0,1)$ such that

$$ \begin{align*} M(\rho_{t},\eta_{r})=h(u_{t,r,s_{0}}). \end{align*} $$

We conclude that $M(\rho _{t},\eta _{r}) \geq {1}/{D}$ for all $t>0$ and $r>0$ , as desired.

5.3 Cubic rays and geodesic currents

In this section, we observe two properties of renormalized Hilbert currents $\upsilon _{\rho _{t}}=h_{t}\nu _{\rho _{t}}$ along cubic rays $(\rho _{t})_{t\geq 0}$ which readily follow from Lemma 5.1. Let us start by showing that the self-intersection of the renormalized Hilbert geodesic current is uniformly bounded along cubic rays.

Proposition 5.4. There exists $C>1$ such that for all $t\geq 0$ ,

$$ \begin{align*} \frac{1}{C}\leq i(\upsilon_{\rho_{t}},\upsilon_{\rho_{t}})\leq C, \end{align*} $$

where $\upsilon _{\rho _{t}}$ is the renormalized Hilbert geodesic current of $\rho _{t}$ .

Proof. Recall from §2.5 that Bonahon proved that $i(\upsilon _{\rho _{0}},\upsilon _{\rho _{0}})=-\pi ^{2}\chi (S)$ , where $\chi (S)$ is the Euler characteristic of S. Corollary 5.2 in [Reference Bridgeman, Canary, Labourie and Sambarino15] states that

$$ \begin{align*} \bigg(\inf_{[\gamma]\in[\Gamma]}\frac{L^{H}_{t}([\gamma])}{L^{H}_{0}([\gamma])}\bigg)^{2}\leq \frac{i(\upsilon_{\rho_{t}},\upsilon_{\rho_{t}})}{i(\upsilon_{\rho_{0}},\upsilon_{\rho_{0}})}\leq \bigg(\sup_{[\gamma]\in[\Gamma]}\frac{L^{H}_{t}([\gamma])}{L^{H}_{0}([\gamma])}\bigg)^{2}, \end{align*} $$

so

$$ \begin{align*} -\pi^{2}\chi(S)\bigg(\inf_{[\gamma]\in[\Gamma]}\frac{L^{H}_{t}([\gamma])}{L^{H}_{0}([\gamma])}\bigg)^{2}\leq i(\upsilon_{\rho_{t}},\upsilon_{\rho_{t}})\leq-\pi^{2}\chi(S) \bigg(\sup_{[\gamma]\in[\Gamma]}\frac{L^{H}_{t}([\gamma])}{L^{H}_{0}([\gamma])}\bigg)^{2}. \end{align*} $$

We conclude by applying Lemma 5.1.

Bonahon [Reference Bonahon9] showed that Thurston’s compactification of the Teichmüller space can be understood via geodesic currents. Explicitly, given a diverging sequence of hyperbolic structures $m_{t}$ , there exists $\unicode{x3bb} _{t}>0$ and a geodesic current $\alpha $ with $i(\alpha ,\alpha )=0$ , a measured lamination, such that $\unicode{x3bb} _{t}\nu _{m_{t}}$ converges up to subsequences to $\alpha $ . In this case, the systole of $\alpha $ vanishes, that is,

$$ \begin{align*} \mathrm{Sys}(\alpha)=\inf_{[\gamma]\in[\Gamma]}i(\alpha,\delta_{\gamma})=0. \end{align*} $$

Burger et al [Reference Burger, Iozzi, Parreau and Pozzetti18] show that there exist diverging sequences of Hilbert geodesic currents which converge projectively to a current $\alpha $ with $\mathrm {Sys}(\alpha )>0$ . We use Lemma 5.1 to show that this happens along cubic rays.

Proof. Suppose $(\alpha _{i})$ and $(\beta _{j})$ are pair-of-pants decompositions which fill the surface, that is, they are pants decompositions such that the complement of their union is a collection of topological discs. Set $u=\sum _{\gamma \in (\alpha _{i})\cup (\beta _{j})}\delta _{\gamma }$ and let $M=\max _{\gamma \in (\alpha _{i})\cup (\beta _{j})}L^{H}_{0}([\gamma ])$ . For every $t\in \mathbb {R}$ , we have

$$ \begin{align*} i(\upsilon_{\rho_{t}},u)=\sum_{\gamma\in(\alpha_{i})\cup(\beta_{j})}L^{H}_{t}([\gamma]). \end{align*} $$

By Lemma 5.1, $\upsilon _{t}$ lies in $ \{\nu \in \mathcal {C}(S)\mid i(\nu ,u)\leq (6g-6)DM \}$ for some $D>1$ . This set is compact by [Reference Bonahon9, Proposition 4] and by linearity of the intersection number. Thus, $\upsilon _{t}$ converges, up to passing to a subsequence, to $\upsilon \in \mathcal {C}(S)$ . Applying Lemma 5.1 again, we see that $\mathrm {Sys}(\upsilon _{\rho _{t}})$ is greater or equal to $D^{-1}\mathrm {Sys}(\upsilon _{\rho _{0}})>0$ . By continuity, the systole $\mathrm {Sys}(\upsilon )$ is strictly positive.

6.1 Length functions, Busemann cocycles, and entropy

In this section, we define length functions for Hitchin representations and the Busemann cocycles associated to them. We refer to [Reference Quint51, Reference Sambarino53] for a more detailed construction.

We need to recall some Lie theory. Let $\mathrm {G}=\mathrm {PSL}(d,\mathbb {R})$ and consider the standard choices of Cartan subspace

$$ \begin{align*} \mathfrak{a}=\{\vec x\in\mathbb{R}^{d}\mid x_{1}+\cdots+x_{d}=0\} \end{align*} $$

and positive Weyl chamber $\mathfrak {a}^{+}=\{\vec x\in \mathfrak {a}\mid x_{1}\geq \cdots \geq x_{d}\}$ . Let $\unicode{x3bb} \colon \mathrm {G}\to \mathfrak {a}^{+}$ be the Jordan projection

$$ \begin{align*} \unicode{x3bb}(g)=(\log \unicode{x3bb}_{1}(g),\ldots, \log \unicode{x3bb}_{d}(g)) \end{align*} $$

consisting of the logarithms of the moduli of the eigenvalues of g in non-increasing order. We will use the following fundamental property of the Jordan projection of the image of a Hitchin representation.

Theorem 6.1. (Fock–Goncharov [Reference Fock and Goncharov25], Labourie [Reference Labourie37])

Every Hitchin representation $\rho \in \mathcal H_{d}(S)$ is discrete and faithful and for every $[\gamma ]\in [\Gamma ]$ ,

$$ \begin{align*} \unicode{x3bb}_{1}(\rho(\gamma))>\cdots>\unicode{x3bb}_{d}(\rho(\gamma))>0. \end{align*} $$

The limit cone $\mathcal {L}_{\rho }$ of a Hitchin representation $\rho \in \mathcal H_{d}(S)$ , introduced in [Reference Benoist6], is the closed cone of $\mathfrak {a}^{+}$ generated by $\unicode{x3bb} (\rho (\gamma ))$ . This cone contains all the rays spanned by positive multiples of $\unicode{x3bb} (\rho (\gamma ))$ for all $\gamma \in \Gamma $ . The (positive) dual cone of $\mathcal {L}_{\rho }$ is defined as $\mathcal {L}_{\rho }^{*}=\{\phi \in \mathfrak {a}^{*}: \phi |_{\mathcal {L}_{\rho }} \geq 0\}$ . For every $i=1,\ldots , d-1$ , the ith simple root $\alpha _{i}\colon \mathfrak {a}\to \mathbb {R}$ , defined by $\alpha _{i}(\vec x)=x_{i}-x_{i+1}$ . We will focus on linear functionals in the set

$$ \begin{align*} \Delta=\bigg\{c_{1}\alpha_{1}+\cdots+c_{d-1}\alpha_{d-1} \, \bigg\vert \, c_{i}\geq 0,\ \sum_{i}c_{i}>0\bigg\}, \end{align*} $$

which is contained in the interior of $\mathcal {L}_{\rho }^{*}$ by a result of Potrie–Sambarino [Reference Potrie and Sambarino50, Theorem B and Lemma 2.7].

Given $\phi $ in the interior of $\mathcal {L}_{\rho }^{*}$ , the $\phi $ -length of $[\gamma ]\in [\Gamma ]$ is defined as

$$ \begin{align*} \ell^{\phi}_{\rho}([\gamma])=\phi(\unicode{x3bb}(\rho(\gamma)))>0, \end{align*} $$

and its exponential growth rate is

$$ \begin{align*} h^{\phi}(\rho)={\limsup_{T\to\infty} \frac{1}{T}\#\log\{[\gamma]\in[\Gamma]\mid \ell^{\phi}_{\rho}([\gamma])\leq T\}.} \end{align*} $$

Sambarino [Reference Sambarino54] established the following important property of linear functionals in $\Delta $ . We include the proof for completeness.

The $\phi $ -length function can be realized via the period of the $\phi $ -Busemann cocycle. To define $\phi $ -Busemann cocycles, we need to introduce the Frenet curve $\xi _{\rho }$ for a Hitchin representation $\rho $ . A complete flag F in $\mathbb {R}^{d}$ is a maximal nested sequence of subspaces of $\mathbb {R}^{d}$ . Explicitly, F is a collection of subspaces $F^{(i)}$ of $\mathbb {R}^{d}$ , with the properties that $F^{(i)}\subset F^{(i+1)}$ and $\dim F^{(i)}=i$ for all $i=0,1,\ldots ,d$ . Denote by $\textsf {F}_{d}$ the space of complete flags in $\mathbb {R}^{d}$ . Fock and Goncharov [Reference Fock and Goncharov25] and Labourie [Reference Labourie37] show that for every Hitchin representation $\rho $ , there exists a unique (up to $\mathrm {PSL}(d,\mathbb {R})$ ) $\rho $ -equivariant Frenet curve $\xi _{\rho }\colon \partial \Gamma \to \textsf {F}_{d}$ which is bi-Hölder continuous onto its image, transverse, positive, and it satisfies certain contraction properties. Moreover, the Frenet curve is dynamics preserving: if $\gamma ^{+}\in \partial \Gamma $ is the attracting fixed point of $\gamma \in \Gamma $ , then $\xi _{\rho }(\gamma ^{+})$ is the attracting eigenflag of $\rho (\gamma )$ . We refer to [Reference Fock and Goncharov25, Theorem 1.15] and [Reference Labourie37, Theorem 4.1] for precise statements.

We are now ready to define $\phi $ -Busemann cocycles for $\phi \in \Delta $ . Fix $F_{0}\in \textsf {F}_{d}$ and observe that for every $F\in \textsf {F}_{d}$ , there exists $k\in \mathrm {SO}(d)$ such that $F=kF_{0}$ . Every $M\in \mathrm {SL}(d,\mathbb {R})$ can be written as $M=L(\exp {\sigma (M)})U$ , for some $L\in \mathrm {SO}(d)$ , $\sigma (M)\in \mathfrak {a}$ , and U is unipotent and upper triangular. This is known as the Iwasawa decomposition of M. The Iwasawa cocycle $B\colon \mathrm {SL}(d,\mathbb {R})\times \textsf {F}_{d}\to \mathfrak {a}$ is defined as $B(A,F)=\sigma (Ak)$ .

The vector valued Busemann cocycle $B_{\rho }:\Gamma \times \partial \Gamma \to \mathfrak {a}$ of a Hitchin representation $\rho $ with Frenet curve $\xi _{\rho }$ is

$$ \begin{align*} B_{\rho}(\gamma,x)=B(\rho(\gamma),\xi_{\rho}(x)). \end{align*} $$

For every linear functional $\phi \in \Delta $ , we set $B^{\phi }_{\rho }(\gamma ,x)=\phi (B_{\rho }(\gamma ,x))$ . Lemma 7.5 in [Reference Sambarino53] directly shows that the cocycle $B^{\phi }_{\rho }$ encodes the $\phi $ -length via the equality $B_{\rho }^{\phi }(\gamma ,\gamma ^{+})=\ell ^{\phi }_{\rho }([\gamma ])$ , for every $\gamma \in \Gamma $ .

The $\phi $ -Busemann cocycle is an example of a Hölder cocycle.

Definition 6.3. A Hölder cocycle is a map $c\colon \Gamma \times \partial \Gamma \to \mathbb {R}$ such that

$$ \begin{align*} c(\gamma\eta,x)=c(\gamma,\eta x)+c(\eta,x) \end{align*} $$

for any $\gamma ,\eta \in \Gamma $ and $x\in \partial \Gamma $ , and there exists $\alpha \in (0,1)$ such that $c(\gamma ,\cdot )$ is $\alpha $ -Hölder for all $\gamma \in \Gamma $ .

For a general Hölder cocycle, let $\ell _{c}(\gamma )=c(\gamma ,\gamma ^{+})$ denote the c-length (also known as period) of $\gamma \in \Gamma $ . Two Hölder cocycles are cohomologous if they have the same periods. We can define the exponential growth rate $h_{c}$ of a general Hölder cocycle $c:\Gamma \times \partial \Gamma \to \mathbb {R}$ as

$$ \begin{align*} h_{c}=\limsup_{T\to\infty} \frac{1}{T}\#\log\{[\gamma]\in[\Gamma]\mid \ell_{c}(\gamma)\leq T\}. \end{align*} $$

Note that the exponential growth rate $h^{\phi }(\rho )$ of the linear functional $\phi \in \Delta $ is the same as the exponential growth rate $h_{B^{\phi }_{\rho }}$ of the cocycle $B^{\phi }_{\rho }$ . The number $h^{\phi }(\rho )$ is also called the topological entropy of a Hitchin representation $\rho $ with respect to the $\phi $ -length. Indeed, $h^{\phi }(\rho )$ is the topological entropy of a translation flow, as we recall in the next section.

6.2 Translation flows and metric Anosov flows

In this subsection, we recall how to equip Hitchin representations with translation flows which will allow us to use thermodynamic formalism tools to study them. We start by recalling the construction of Sambarino’s translation flow [Reference Sambarino53], which is analogous to Hopf’s parameterization of the geodesic flow of a negatively curved manifold.

Let $\partial ^{2}\Gamma =\{(x,y)\in \partial \Gamma \times \partial \Gamma : x\neq y\}$ . The translation flow $\{\varphi _{t}\}_{t\in \mathbb {R}}$ on $\partial ^{2}\Gamma \times \mathbb {R}$ is

$$ \begin{align*}\varphi_{t}(x,y,s)=(x,y,s-t).\end{align*} $$

Given a Hölder cocycle $c\colon \Gamma \times \partial \Gamma \to \mathbb {R}$ , the group $\Gamma $ acts on $\partial ^{2}\Gamma \times \mathbb {R}$ by

$$ \begin{align*}\gamma(x,y,t)=(\gamma x, \gamma y, t-c(\gamma, y) ).\end{align*} $$

The translation flow $\{\varphi _{t}\}_{t\in \mathbb {R}}$ then descends to the quotient $M_{c}$ of $\partial ^{2} \Gamma \times \mathbb {R}$ by the $\Gamma $ -action via c. Sambarino proves a reparameterization theorem for the translation flow $\varphi _{t}$ .

First, recall that two (Hölder) continuous flows $\psi _{t}$ and $\psi _{t}^{\prime }$ on compact metric spaces X and Y, respectively, are (Hölder) conjugated if there exists a (bi-Hölder) homeomorphism $h: X\to Y$ such that $\psi _{t}^{\prime }=h\circ \psi _{t} \circ h^{-1}$ for every $t\in \mathbb R$ .

From now on, we will always assume that the cocycle c is of the form $B^{\phi }_{\rho }$ for some Hitchin representation $\rho $ and $\phi \in \Delta $ . In this case, the hypotheses of Theorem 6.4 are satisfied. We will still denote the translation flow associated to $B^{\phi }_{\rho }$ by $\varphi _{t}$ and write $M_{\rho }^{\phi }=M_{B^{\phi }_{\rho }}$ .

We want to use thermodynamic formalism methods for the translation flow $\varphi _{t}$ and we briefly explain why we can do so in this setting. While it is well known that a Hölder reparameterization of an Anosov flow is an Anosov flow [Reference Anosov and Sinai3, p. 122], Hölder conjugacy does not necessarily preserve the Anosov property [Reference Gogolev29]. To construct symbolic codings and use results from the thermodynamic formalism for Hitchin representations, we will work in the more general setting of metric Anosov flows. Metric Anosov flows (or Smale flows) were introduced by Pollicott in [Reference Pollicott48] to generalize classical results for Anosov flows and Axiom A flows. In particular, he constructed a symbolic coding for metric Anosov flows [Reference Pollicott48].

Our definition of metric Anosov flow, which is better suited to our purposes, differs from the original definition in [Reference Pollicott48]. The main difference is that we allow an $\alpha $ -power of the distance $d(x,y)$ for the exponential diverging and expanding properties in part (1) of Definition 6.5 below. This property guarantees that Hölder conjugacy preserves our notion of metric Anosov flow (see Proposition 6.7) and is useful in our setting, for example, in Proposition 6.8.

Recall for any continuous flow $\psi _{t}$ on a compact metric space X, the local stable set of a point $x\in X$ is defined for $\epsilon>0$ as

$$ \begin{align*} W^{s}_{\epsilon}(x)=\{y\in X: d(\psi_{t} x , \psi_{t} y) \leq \epsilon, \text{ for all } t \geq 0 \textrm{ and } d(\psi_{t} x, \psi_{t} y) \to 0 \textrm{ as } t \to \infty\}. \end{align*} $$

The local unstable set of a point x for $\epsilon>0$ is

$$ \begin{align*} W^{u}_{\epsilon}(x)=\{y\in X: d(\psi_{-t} x , \psi_{-t} y) \leq \epsilon, \text{ for all } t \geq 0 \textrm{ and } d(\psi_{-t} x, \psi_{-t} y) \to 0 \textrm{ as } t \to \infty\}. \end{align*} $$

Proof. We show here that a Hölder conjugacy preserves metric Anosov properties. By hypothesis, there exists a bi-Hölder homeomorphism $h:X\to Y$ with Hölder exponent $\alpha _{0}\in (0,1]$ (for both h and $h^{-1}$ ) such that $h\circ \psi _{t}^{1}= \psi _{t}^{2}\circ h$ . We want to show that $\psi _{t}^{2}$ also satisfies the metric Anosov property.

Given $x_{2}\in Y$ , we want to find suitable parameters so that conditions (1) and (2) in Definition 6.5 hold. We show condition (1) for local stable sets. Suppose $x_{2}=h(x_{1})$ . Because $\psi _{t}^{1}$ is metric Anosov, we can find $\varepsilon _{1}, C_{1}, \unicode{x3bb} _{1}, \delta _{1}$ positive and $\alpha _{1}\in (0,1]$ so that

$$ \begin{align*} W^{s}_{\epsilon_{1}}(x_{1})=\{&y_{1}\in X: d_{X}(\psi^{1} {}_{t} x_{1},\psi^{1}_{t} y_{1})\leq \epsilon_{1} \text{ and } d_{X}(\psi^{1}_{t} x_{1} , \psi^{1}_{t} y_{1}) \\[3pt] &{\leq}\, C_{1} e^{-\unicode{x3bb}_{1} t} d_{X}(x_{1},y_{1})^{\alpha_{1}} \text{ for all } t \geq 0 \}. \end{align*} $$

Note,

$$ \begin{align*} d_{Y}(\psi^{2}_{t}h(x_{1}), \psi^{2}_{t} h(y_{1}))&=d_{Y}(h(\psi_{t}^{1}x_{1}), h(\psi_{t}^{1} y_{1}))\leq C_{h} d_{X}(\psi^{1}_{t}x_{1},\psi^{1}_{t}y_{1})^{\alpha_{0}}\\[3pt] &\leq C_{h}(C_{1} e^{-\unicode{x3bb}_{1}t} d_{X}(x_{1},y_{1})^{\alpha_{1}})^{\alpha_{0}} \leq C_{2} e^{-\unicode{x3bb}_{1} \alpha_{0} t}d_{Y}(h (x_{1}), h (y_{1}))^{\alpha_{0}^{2}\alpha_{1}}. \end{align*} $$

Here we have used the Hölder properties of both h and $h^{-1}$ , and $C_{h}$ is a constant from the Hölder property of h. This implies that we can find $\varepsilon _{2}>0$ so that $W^{s}_{\varepsilon _{2}}(x_{2})$ satisfies condition (1) in Definition 6.5 for $\psi ^{2}_{t}$ with positive $C_{2}, \unicode{x3bb} _{2}=\unicode{x3bb} _{1} \alpha _{0}$ and $\alpha _{2}= \alpha _{0}^{2}\alpha _{1}$ . The argument is similar for local unstable sets. Furthermore, given $x_{2}, y_{2}\in Y$ close enough, condition (2) in Definition 6.5 can be verified by taking $\langle x_{2},y_{2}\rangle = h(\langle h^{-1} (x_{2}), h^{-1} (y_{2}) \rangle )$ . We conclude that $\psi ^{2}_{t}$ is metric Anosov.

The translation flow has been introduced and extensively studied by Sambarino in [Reference Sambarino53]. For completeness, we provide a proof that the translation flow is metric Anosov.

6.3 Reparameterization functions

After constructing flows for Hitchin representations, the next thing that we want to do is to define the reparameterization function with domain $M_{\rho }^{\phi }$ for $\rho $ a Hitchin representation. This construction is well known and it is contained in the work of Sambarino [Reference Sambarino53]. We include this discussion to make our exposition self-contained.

In §2.2, given a positive Hölder continuous function f on the unit tangent bundle $T^{1}X_{\rho }$ of a convex real projective structure, we defined a reparameterization of the flow by time change. This can be done more generally for any Hölder continuous flow on a compact metric space X. In particular, given two Hitchin representations $\rho _{1}$ and $\rho _{2}$ in $\mathcal {H}_{d}(S)$ , we will show the existence of a positive Hölder continuous reparameterization function $f_{\rho _{1}}^{\rho _{2}}: M^{\phi }_{\rho _{1}} \to \mathbb {R}_{> 0}$ that encodes the $\phi $ -length spectrum of $\rho _{2}$ .

We start with a lemma that relates the positivity of the Hölder reparameterization function to the positivity of its entropy. The entropy of a Hölder function f on a compact metric space X equipped with a metric Anosov flow is defined as

$$ \begin{align*} h_{f}=\limsup_{T\to\infty} \frac{1}{T}\#\log\bigg\{\tau \text{ periodic} \bigg| \int_{\tau} f\leq T\bigg\}. \end{align*} $$

We also need the following theorem of Ledrappier [Reference Ledrappier40] which establishes the correspondence between cohomologous Hölder cocycles and cohomologous Hölder continuous functions on $T^{1}X_{0}$ . We will state this theorem for vector valued Hölder cocycles. Their definition is analogous to Definition 6.3.

Theorem 6.10. (Ledrappier [Reference Ledrappier40, p. 105])

Let V be a finite-dimensional vector space. For each Hölder cocycle $c:\Gamma \times \partial \Gamma \to V$ , there exists a Hölder continuous map $F_{c}: T^{1}X_{0} \to V$ , such that for every $\gamma \in \Gamma -\{e\}$ ,

$$ \begin{align*} \ell_{c}(\gamma)=\int_{[\gamma]}F_{c}. \end{align*} $$

This map $c\to F_{c}$ induces a bijection between the set of cohomology classes of V-valued Hölder cocycles and the set of cohomology classes of Hölder maps from $T^{1}X_{0}$ to V.

For a Hitchin representation $\rho $ , this tells us that we can find a Hölder continuous map $g_{\rho }:T^{1}X_{0}\to \mathfrak {a}$ with periods equal to the periods of the vector valued Busemann cocycle $B_{\rho }$ . Since $h_{\phi \circ g_{\rho }}=h_{B^{\phi }_{\rho }}=h^{\phi }(\rho )$ is finite and positive, by Lemma 6.9, the reparameterization $\phi \circ g_{\rho }$ on $T^{1}X_{0}$ is cohomologous to a positive Hölder continuous function.

Now we are ready to state our lemma about the existence of positive reparameterization functions, which should be compared to Lemma 2.3.

Lemma 6.11. Let $\rho _{1}\,$ and $\rho _{2}$ be two different representations in $\mathcal {H}_{d}(S)$ . There exists a positive Hölder continuous function $f_{\rho _{1}}^{\rho _{2}}: M^{\phi }_{\rho _{1}} \to \mathbb {R}_{>0}$ such that for every periodic orbit $\tau $ corresponding to $[\gamma ]\in [\Gamma ]$ ,

$$ \begin{align*} \unicode{x3bb}(f_{\rho_{1}}^{\rho_{2}},\tau)=\ell_{\rho_{2}}^{\phi}([\gamma]). \end{align*} $$

Once we have defined reparameterization functions, we see that all definitions, remarks, and results in §2.3 regarding thermodynamic formalism readily generalize to this context by simply changing the domain from $T^{1}X_{\rho }$ to $M_{\rho }^{\phi }$ .

6.4 Correlation and Manhattan curve theorem revisited

First, we state the independence lemma for Hitchin representations which generalizes Lemma 2.11.

Recall that a flow is weakly mixing if its periods do not generate a discrete subgroup of $\mathbb {R}$ . In particular, the independence lemma implies that the translation flow $\varphi _{t}$ is weakly mixing.

We are now ready to state our correlation theorem for Hitchin representations. Recall that we denote the renormalized $\phi $ length spectrum of $\rho $ by $L^{\phi }_{\rho }=h^{\phi }(\rho )\ell ^{\phi }_{\rho }$ .

Theorem 1.7.

Given a linear functional $\phi \in \Delta $ and a fixed precision $\varepsilon>0$ , for any two different Hitchin representations $\rho _{1}, \rho _{2}\colon \Gamma \to \mathrm {PSL}(d,\mathbb {R})$ such that $\rho _{2}\neq \rho _{1}^{*}$ , there exist constants $C=C(\varepsilon , \rho _{1},\rho _{2},\phi )>0$ and $M=M(\rho _{1},\rho _{2}, \phi ) \in (0,1)$ such that

$$ \begin{align*} \# \{[\gamma]\in[\Gamma]\, \mid\, L_{\rho_{1}}^{\phi}([\gamma]) \in (x,x+h^{\phi}(\rho_{1})\varepsilon),\ L_{\rho_{2}}^{\phi}([\gamma]) \in (x, x+h^{\phi}(\rho_{2})\varepsilon)\} \sim C \frac{e^{Mx}}{x^{3/2}}. \end{align*} $$

For two Hitchin representations $\rho _{1}$ , $\rho _{2}$ with $\rho _{2}\neq \rho _{1},\rho _{1}^{*}$ and $\phi \in \Delta $ , consider the Manhattan curve

$$ \begin{align*} \mathcal{C}^{\phi}(\rho_{1},\rho_{2})=\{(a,b)\in\mathbb{R}^{2}\colon P(-a-bf)=0\}, \end{align*} $$

where f is the reparameterization function from Lemma 6.11. We obtain a characterization of the correlation number analogous to Theorem 4.2.

Theorem 6.14. Fix $\phi \in \Delta $ , and let $\rho _{1}$ and $\rho _{2}$ be Hitchin representations in $\mathcal {H}_{d}(S)$ such that $\rho _{2} \neq \rho _{1}, \rho _{1}^{*}$ . Their correlation number can be written as

$$ \begin{align*} M(\rho_{1},\rho_{2}, \phi)=\frac{a}{h^{\phi}(\rho_{1})}+\frac{b}{h^{\phi}(\rho_{2})}, \end{align*} $$

where $(a,b)\in \mathcal {C}^{\phi }(\rho _{1},\rho _{2})$ is the unique point on the Manhattan curve at which the tangent line is parallel to the line passing through $(h^{\phi }(\rho _{1}),0)$ and $(0,h^{\phi }(\rho _{2}))$ .

We conclude by raising two questions which are motivated by Theorems 1.3 and 1.5.

Recall that Potrie and Sambarino [Reference Potrie and Sambarino50, Theorem B] showed that for every Hitchin representation $\rho $ , the simple root lengths are such that $h^{\alpha _{i}}(\rho )=1$ for $i=1,\ldots ,d-1$ . Moreover, the simple root lengths restrict to the hyperbolic length on the Fuchsian locus $\mathcal {T}(S)\subset \mathcal {H}_{d}(S)$ . Thus, Theorem 1.7 in this case is a particularly natural generalization of the correlation theorem for hyperbolic surfaces [Reference Schwartz and Sharp56]. Theorem 1.3 exhibits examples of sequences for which the $\alpha _{i}$ -correlation number decays. We ask whether there exist similar examples which lie outside the Fuchsian locus.

Finally, we raise a conjecture motivated by Theorem 1.5. Similarly to cubic rays introduced in §5, in general for the Hitchin component $\mathcal {H}_{d}(S)$ , one can consider a dth-order holomorphic differential q over $X_{0}$ and its associated family of representations $(\rho _{t})_{\scriptscriptstyle t\in \mathbb {R}}$ in $\mathcal {H}_{d}(S)$ parameterized by $(tq)_{t\in \mathbb {R}}$ . These representations are given by holonomies of cyclic Higgs bundles. We refer to [Reference Baraglia4] for a detailed definition and discussion. When $t\geq 0$ , we say this family of representations $(\rho _{t})_{t\geq 0}$ is a ray associated to q.