2.1 Markov shifts

A countable state Markov shift is the data of a countable alphabet $\mathcal A$ , a transition matrix $\mathbb T=(t_{ab})\in \{0,1\}^{\mathcal A\times \mathcal A}$ , the set of words

$$ \begin{align*} \Sigma^+=\{\underline{x}=(x_n)\in\mathcal A^{\mathbb Z_{>0}}\colon t_{x_nx_{n+1}}=1\}, \end{align*} $$

and the left-shift $\sigma \colon \Sigma ^+\to \Sigma ^+$ defined by $\sigma ((x_n))=(x_{n+1})$ . The countable state Markov shift is: (i) topologically mixing if for any letters $a,b\in \mathcal A$ , there exists $N=N(a,b)$ such that for any $n> N$ , there exists a word $\underline {x}\in \Sigma ^+$ with $x_1=a$ and $x_n=b$ ; (ii) has BIP, short for big images and pre-images property, if there exists a finite subset $\mathcal B$ of the alphabet $\mathcal A$ such that for any $a\in \mathcal A$ , there exist letters $b_p,b_s\in \mathcal B$ such that $t_{b_pa}=t_{ab_s}=1$ . For the rest of this section, we fix a topologically mixing, countable state Markov shift with BIP which, with a slight abuse of notation, we denote by $\Sigma ^+$ . Furthermore, we equip $\Sigma ^+$ with the metric $d(\underline {x},\underline {y})=e^{-\inf \{n:\ x_n\neq y_n\}}$ .

For any $k\in \mathbb Z_{>0}$ , a k-cylinder is a non-empty subset $p\subset \Sigma ^+$ defined by the property

$$ \begin{align*} \underline{x},\underline{y}\in p \iff x_i=y_i\quad \text{for all } i=1,\ldots,k. \end{align*} $$

Denote by $\Lambda _k$ the set of k-cylinders.

2.2 Potentials, entropy gaps, and pressure

A potential is a continuous function $f\colon \Sigma ^+\to \mathbb K$ , where $\mathbb K=\mathbb R$ or $\mathbb C$ equipped with the norm $|\cdot |$ . The nth ergodic sum of f is the potential

$$ \begin{align*} S_nf(\underline{x})=\sum_{i=0}^{n-1}f(\sigma^i(\underline{x})). \end{align*} $$

A potential $f\colon \Sigma ^+\to \mathbb K$ is locally Hölder continuous (or $\beta $ -locally Hölder continuous with constant A) if there exist $A,\beta>0$ such that

$$ \begin{align*} |f(\underline{x})-f(\underline{y})|<Ad(\underline{x},\underline{y})^\beta \end{align*} $$

whenever $x_1=y_1$ . We denote by $\mathcal F_\beta $ (respectively $\mathcal F_\beta (\mathbb C)$ ) the space of $\beta $ -locally Hölder continuous potentials valued in $\mathbb R$ (respectively $\mathbb C$ ) equipped with the norm

$$ \begin{align*} \|\phi\|_{\beta}=\|\phi\|_{\infty}+\text{Lip}(\phi)\quad \text{where Lip }(\phi):=\sup \bigg\{\frac{|\phi(\underline{x})-\phi(\underline{y})|}{d(\underline{x},\underline{y})^\beta} \colon\ \underline{x}\neq \underline{y},\ x_1=y_1\bigg\}. \end{align*} $$

Then, let $\mathcal F^b_\beta (\mathbb C)$ denote the subspace of bounded potentials and let $\mathcal B(\mathcal F^b_\beta (\mathbb C))$ be the space of bounded operators on $\mathcal F^b_\beta (\mathbb C)$ equipped with the operator norm

$$ \begin{align*} \|\mathcal L\|_{\mathrm{{op}}}=\sup\bigg\{\frac{\|\mathcal L(\phi)\|_\beta}{\|\phi\|_\beta}:\ \phi\in \mathcal F^b_\beta(\mathbb C)\bigg\}. \end{align*} $$

Note that $(\mathcal F^b_\beta (\mathbb C),\|\cdot \|_\beta )$ and $(\mathcal B(\mathcal F^b_\beta (\mathbb C)),\|\cdot \|_{\mathrm {{op}}})$ are Banach spaces.

Let $f\colon \Sigma ^+\to \mathbb R$ be locally Hölder continuous and set $ S(f,a)=\sup \{f(\underline {x})\colon \underline {x}\in \Sigma ^{+}, x_{1}=a\}.$ Then, recall from the introduction that f has a strong entropy gap at infinity if the series

$$ \begin{align*} Z_{1}(f,s)=\sum_{a\in\mathcal{A}}e^{-sS(f,a)} \end{align*} $$

has a finite critical exponent $d(f)>0$ and diverges when $s=d(f)$ .

Let $\mathcal M_\sigma $ denote the space of shift-invariant probability measures on $\Sigma ^+$ and, for any $\mu \in \mathcal M_\sigma $ , let $h_\sigma (\mu )$ denote the measure-theoretic entropy of $\sigma $ with respect to $\mu $ . The (topological) pressure of the locally Hölder continuous potential f is

$$ \begin{align*} P(f)=\sup\bigg\{h_\sigma(\mu)+\int_{\Sigma^+}f\,d\mu\colon \mu\in\mathcal M_\sigma\text{ and }-\int_{\Sigma^+}f\, d\mu<\infty\bigg\}. \end{align*} $$

An equilibrium state for a potential f is a $\sigma $ -invariant Borel probability measure $\mu $ on $\Sigma ^+$ such that

$$ \begin{align*} P(f) = h_\sigma(\mu) + \int_{\Sigma^+} f \, d\mu. \end{align*} $$

A Gibbs state for f is a Borel probability measure $\nu $ on $\Sigma ^+$ for which there exists a constant $Q> 1$ such that

$$ \begin{align*} \frac{1}{Q} \leq \frac{e^{S_n f(\underline{x}) - nP(f)}}{\nu(p)} \leq Q \end{align*} $$

for any n-cylinder p and any $\underline {x}$ in p.

The Gibbs states and equilibrium states of potentials with good regularity (if they exist) are closely related. Specifically, we have the following result.

Two locally Hölder continuous potentials f and g are (Livšic) cohomologous if there exists a locally Hölder continuous potential h such that $f-g=h-h\circ \sigma $ . The real-valued potential f is strictly positive if there exists a constant $B> 0$ such that $f(\underline {x})> B$ for all $\underline {x} \in \Sigma ^+$ .

A potential $f\colon \Sigma ^+\to \mathbb R$ is arithmetic if the additive subgroup of $\mathbb R$ generated by $\{S_nf(\underline {x})\colon \underline {x}\in \mathrm {Fix}^n,\ n\in \mathbb N\}$ is cyclic. Two potentials are independent if $af+bg$ is arithmetic only if $a=b=0$ . Note that if f and g are independent, then each one of them is non-arithmetic.

2.3 Transfer operators

We are interested in certain weighted sums of f and g that have finite pressure. More specifically, we will consider weights in the set

(2.1) $$ \begin{align} D=D(\mathbf{f})&:=(\{(z_{1},z_{2})\in\mathbb{R}^{2}\colon z_1\leq 0,z_2\leq 0,z_1+z_2<0\}\nonumber\\ &\qquad\cup \{(z_{1},z_{2})\in\mathbb{R}^{2}\colon z_1> 0,z_2<0,z_1c(f)+z_2(c(f)-A)<0\}\nonumber\\ &\qquad\cup \{(z_{1},z_{2})\in\mathbb{R}^{2}\colon z_1<0,z_2>0,z_1(c(g)-A)+z_2c(g)<0\})\nonumber\\ &\quad\cap\{(z_{1},z_{2})\in\mathbb{R}^{2}\colon d(-z_{1}f-z_{2}g)<1\}, \end{align} $$

where $A>0$ is such that $\|f-g\|_{\infty }=A<\infty $ , $c(f)=\inf _{\Sigma ^+}f>0$ , and $c(g)=\inf _{\Sigma ^+}g>0$ . Notice that for any $(z_{1},z_{2})\in D$ , the potential $-(z_{1}f+z_{2}g)$ is locally Hölder continuous, strictly positive, and with strong entropy gap at infinity. In particular, [Reference Bray, Canary, Kao and Martone5, Lemma 3.3(1)] implies that $D\subset \{(z_{1},z_{2})\in \mathbb {R}^{2}\colon P(z_{1}f+z_{2}g)<\infty \}$ . We now list some important properties of the weights in D.

Proof. The first assertion follows from the assumption $d(-z_1f-z_2g)<1$ together with [Reference Bray, Canary, Kao and Martone5, Lemma 3.4]. For the second assertion, since $-(z_{1}f+z_{2}g)$ has a strong entropy gap at infinity, the series

$$ \begin{align*} Z_{1}(-z_{1}f-z_{2}g,s)=\sum_{a\in\mathcal{A}}e^{-sS(-z_{1}f-z_{2}g,a)} \end{align*} $$

converges for all $s>d(-(z_{1}f+z_{2}g)).$ Since we assume $(z_1,z_2)\in D$ , we can find $\epsilon>0$ small enough such that $1-\epsilon>d(-(z_{1}f+z_{2}g))$ , and thus,

$$ \begin{align*} \sum_{a\in\mathcal{A}}e^{-(1-\epsilon)S(-z_{1}f-z_{2}g,a)}<\infty. \end{align*} $$

Then,

$$ \begin{align*} \infty>\sum_{a\in\mathcal{A}}e^{-(1-\epsilon)S(-z_{1}f-z_{2}g,a)} & =\sum_{a\in\mathcal{A}}e^{-S(-z_{1}f-z_{2}g,a)}e^{\epsilon S(-z_{1}f-z_{2}g,a)}\\ & =\sum_{a\in\mathcal{A}}e^{-S(-z_{1}f-z_{2}g,a)}\sum_{n\geq1}\frac{\epsilon^{n}}{n!}(S(-z_{1}f-z_{2}g,a))^{n}\\ & =\sum_{n\geq1}\sum_{a\in\mathcal{A}}\frac{\epsilon^{n}}{n!}(S(-z_{1}f-z_{2}g,a))^{n}e^{-S(-z_{1}f-z_{2}g,a)}, \end{align*} $$

where we can swap the series because $(z_1,z_2)\in D$ , $-z_1f-z_2g$ is strictly positive; thus, guaranteeing absolute convergence. It follows that for all $n\in \mathbb {N}$ ,

$$ \begin{align*} \sum_{a\in\mathcal{A}}(S(-z_{1}f-z_{2}g,a))^{n}e^{-S(-z_{1}f-z_{2}g,a)}<\infty. \end{align*} $$

Now, for all $a\in \mathcal A$ , let $[a]$ denote the cylinder of words starting with the letter a and, for any locally Hölder continuous potential h, let $I(h,a)$ denote the infimum of $\{h(\underline {x})\colon \underline {x}\in [a] \}$ . By the Gibbs property of $\mu _{z_1f+z_2g}$ , we know that there exists $Q>1$ such that

$$ \begin{align*} e^{-I(-z_{1}f-z_{2}g,a)}= e^{S(z_{1}f+z_{2}g,a)}\geq\frac{e^{P(z_{1}f+z_{2}g)}}{Q}\mu_{z_{1}f+z_{2}g}([a]). \end{align*} $$

Hence,

$$ \begin{align*} & 0<\int|-z_{1}f-z_{2}g|^{n}\ d\mu_{z_{1}f+z_{2}g}\\ &\leq\frac{Q}{e^{P(z_{1}f+z_{2}g)}}\sum_{a\in\mathcal{A}}(S(-z_{1}f-z_{2}g,a))^{n}e^{-I(-z_{1}f-z_{2}g,a)}\\ &\leq \frac{Q}{e^{P(z_{1}f+z_{2}g)}}\sum_{a\in\mathcal{A}}(S(-z_{1}f-z_{2}g,a))^{n}e^{-S(-z_{1}f-z_{2}g,a)+A}<\infty, \end{align*} $$

where we used that the potential $-z_1f-z_2g$ is locally Hölder continuous to see that

$$ \begin{align*} |I(-z_{1}f-z_{2}g,a)-S(-z_{1}f-z_{2}g,a)|<A \end{align*} $$

for some uniform constant $A>0$ . From here, we deduce that $f,g\in L^{n}(\mu _{z_{1}f+z_{2}g})$ for all $n\in \mathbb {N}$ .

The last assertion is an immediate consequence of the formula

$$ \begin{align*}d(-z_1f-z_2g){}=\frac{d(f)}{-(z_1+z_2)}\end{align*} $$

established in the proof of [Reference Bray, Canary, Kao and Martone5, Theorem C*].

Consider a locally Hölder continuous potential $h\colon \Sigma ^+\to \mathbb K$ . The transfer operator $\mathcal {L}_{h}$ of h (also known as the Ruelle–Perron–Frobenius operator) is defined by

$$ \begin{align*} \mathcal{L}_{h}\phi(\underline{x})=\sum_{\underline{y}\in\sigma^{-1}(\underline{x})}e^{h(\underline{y})}\phi(\underline{y}), \end{align*} $$

where $\phi \colon \Sigma ^+\to \mathbb K$ is a bounded locally Hölder continuous potential. The eigenvalues and eigenvectors of the transfer operator are related to the pressure and equilibrium states as follows.

Theorem 2.6. (Mauldin and Urbański [Reference Mauldin and Urbański30, Corollary 2.7.5], Sarig [Reference Sarig40, Theorem 4.9])

If $u :\Sigma ^+\to \mathbb R$ is locally Hölder continuous, $P(u)<+\infty $ and $\sup u<+\infty $ , then there exist unique probability measures $\mu _u$ and $\nu _u$ on $\Sigma ^+$ and a positive function $h_u:\Sigma ^+\to \mathbb R$ so that

$$ \begin{align*}\mu_u=h_u\nu_u,\quad \mathcal L_uh_u=e^{P(u)}h_u\quad\text{and}\quad \mathcal L_u^*\nu_u=e^{P(u)}\nu_u.\end{align*} $$

Moreover, $h_u$ is bounded away from both $0$ and $+\infty $ , and $\mu _u$ is an equilibrium state for u.

We will be interested in complex analytic perturbations of the transfer operator. To this end, consider the complex domain

$$ \begin{align*} \overline{D}=\overline{D}(\mathbf{f}):=\{(z,w)\subset\mathbb{C}\times\mathbb{C}:\ (\Re{z},\Re{w})\in D\},\end{align*} $$

which contains $D=D(\mathbf {f})$ .

We recall the well-known analytic perturbation theorem (cf. for example, [Reference Kato21], [Reference Parry and Pollicott31, Proposition 4.6], [Reference Sarig40, Theorem 5.6], or [Reference Kloeckner24, Theorem]).

The following corollary is a consequence of the analytic perturbation theorem discussed above, which can be viewed as a complex version of the Ruelle–Perron–Frobenius theorem. See, for example, Sarig [Reference Sarig40, Theorem 5.8] for the standard Ruelle–Perron–Frobenius theorem for the unperturbed transfer operator in the countable setting. This corollary plays a key role in our local asymptotic estimates in §6.

The (asymptotic) covariance of f and g with respect to the shift-invariant measure $\mu $ is given by

$$ \begin{align*} \mathrm{Cov}(f,g,\mu):=\lim_{n\to\infty}\frac{1}{n}\int S_{n}\bigg(f-\int f\, d\mu\bigg)S_{n}\bigg(g-\int g\, d\mu\bigg)\, d\mu \end{align*} $$

and the (asymptotic) variance of f with respect to $\mu $ is $\text {Var}(f,\mu ):=\text {Cov}(f,f,\mu ).$

2.4 The Legendre transform

Recall (see Assumptions 2.3) that f and g are strictly positive locally Hölder continuous potentials with strong entropy gaps at infinity and $\|f-g\|_\infty $ is finite. We assume further that f and g are independent. For ${\mathbf {f}}=(f,g)$ , we consider the domain D defined in (2.1).

Part (4) in Corollary 2.10 implies that the map ${\mathbb {P}: D \to \mathbb {R}}$ is convex. The Legendre transform of ${\mathbb {P}}$ is defined for all ${\mathbf {x}\in \nabla \mathbb {P}(}D)$ as

$$ \begin{align*} \mathbb{P^{*}}({\textbf{x}})=\sup_{{\textbf{z}\in}D}(\langle{\textbf{x}},{\textbf{z}}\rangle-\mathbb{P}({\textbf{z}})), \end{align*} $$

where $\langle \cdot ,\cdot \rangle $ denotes the standard inner product on $\mathbb R^2$ .

Proof. Since $\mathbb {P}$ is analytic and strictly convex in D, all but the last assertion are classical properties of the Legendre transform (cf. [Reference Rockafellar and Wets37, Example 11.9] for a proof).

The last assertion follows from the derivative formula for the pressure, namely Corollary 2.10(2). Recall that $\mu =\mu _{z_{1}f+z_{2}g}=\mu _{\langle {\mathbf {z}},{\mathbf {f}}\rangle }$ is the equilibrium state of $z_{1}f+z_{2}g$ ; thus,

$$ \begin{align*}P(z_{1}f+z_{2}g)=h_{\mu}(\sigma)+\int {(z_{1}f+ z_{2}g)}\, d\mu.\end{align*} $$

Then, since by assumption

$$ \begin{align*} {\textbf{x}}=\nabla\mathbb{P}({\textbf{z}})=(\partial_{z_{1}}P(z_{1}f+z_{2}g), \partial_{z_{2}}P(z_{1}f+z_{2}g))=\bigg(\!\int f\, d\mu,\int g\, d\mu\bigg), \end{align*} $$

we apply part (5) and compute

$$ \begin{align*} \hspace{-12pt}\mathbb{P}^{*}({\textbf{x}})= \langle{\textbf{x}},{\textbf{z}}\rangle-\mathbb{P}({\textbf{z}}) = \int z_{1}f\, d\mu+\int z_{2}g\, d\mu-P(z_{1}f+z_{2}g) = -h_{\mu}(\sigma).\\[-42pt] \end{align*} $$

Proof. For part (1), we compute

$$ \begin{align*} \frac{d}{dt}(-t\mathbb{P}^{*}({\textbf{x}}/t)) & =-\mathbb{P}^{*}({\textbf{x}}/t)+(-t)\langle\nabla\mathbb{P}^{*}({\textbf{x}}/t),-{\textbf{x}}/t^{2}\rangle\\ & =-\mathbb{P}^{*}({\textbf{x}}/t)+\langle\nabla\mathbb{P}^{*}({\textbf{x}}/t),{\textbf{x}}/t\rangle\\ & =\mathbb{P}(\nabla\mathbb{P}^{*}({\textbf{x}}/t)), \end{align*} $$

where the last equality follows from Proposition 2.11(5).

For part (2), we take another derivative of the above equation. Then,

$$ \begin{align*} \frac{d^{2}}{dt^{2}}(-t\mathbb{P}^{*}({\textbf{x}}/t)) & =\frac{d}{dt}\mathbb{P}(\nabla\mathbb{P}^{*}({\textbf{x}}/t))\\ & =\nabla\mathbb{P}(\nabla\mathbb{P}^{*}({\textbf{x}}/t))\cdot\nabla^{2}\mathbb{P}^{*}({\textbf{x}}/t)\cdot (-{\textbf{x}}/t^2) =-\frac{1}{t}(\nabla^{2}\mathbb{P}^{*}({\textbf{x}}/t))\langle{\textbf{x}}/t, {\textbf{x}}/t\rangle. \end{align*} $$

The second derivative is smaller than zero because $\nabla ^{2}\mathbb {P}^{*}$ is positive definite (as by Corollary 2.10, $\nabla ^{2}\mathbb {P}$ is positive definite).

2.5 The Manhattan curve

The Manhattan curve of $\mathbf {f}$ is given by

$$ \begin{align*} \mathcal C(\mathbf{f})=\mathcal{C}(f,g)=\{(a,b)\in\mathbb{R}_{\geq 0}^{2}:\ P(-af-bg)=0,\ a+b>0\}, \end{align*} $$

and the extended Manhattan curve of $\mathbf {f}$ is

$$ \begin{align*} \mathcal C^*(\mathbf{f})=\{(a,b)\in \mathcal D(\mathbf{f}) :\ P(-af-bg)=0\}, \end{align*} $$

where $\mathcal D(\mathbf {f})=-D(\mathbf {f})$ and $D(\mathbf {f})$ is defined in (2.1).

Before we list important properties of the Manhattan curve, we recall that under our assumptions on the potentials f and g (see Assumptions 2.3), there exist $\delta _f,\delta _g>d(f)=d(g)>0$ , which are the unique numbers satisfying Bowen’s formula $P(-\delta _f f)=P(-\delta _g g)=0$ (cf. [Reference Bray, Canary, Kao and Martone5, Corollary 1.2]). Recall that we are assuming that f and g are independent.

By Theorem 2.14, we can parameterize $\mathcal {C}^*({\mathbf {f}})$ by $(s,q(s))$ for $q=q(s)$ an analytic function and s in an open interval containing $[0,\delta _f]$ . We define the set of admissible slopes as

$$ \begin{align*} \mathcal{S}(\mathbf{f}):=\{m\colon m \text{ is the slope of the normal to } \mathcal{C}^*({\mathbf{f}}) \text{ for some point } (a,b)\in\mathcal{C}^*({\mathbf{f}})\}. \end{align*} $$

Notation 2.15. When the potentials are independent, by the strict convexity of $\mathcal {C}^*({\mathbf {f})}$ , for any $m\in \mathcal {S}({\mathbf {f})}$ , there exists a unique point $(a_m,b_m)\in \mathcal {C}^*({\mathbf {f})}$ such that the slope of the normal at $(a_m,b_m)$ is m. Then, define

$$ \begin{align*} t_{m}:=\bigg(\!\int f\, d\mu_{-a_{m}f-b_{m}g}\bigg)^{-1},\quad {\textbf{m}}:=(1,m),\quad {\textbf{x}}_{m}:={\textbf{m}}/t_{m}, \quad\textbf{z}_m:=(-a_m,-b_m). \end{align*} $$

Proof. Theorem 2.14(2) implies that $\mathcal S(\mathbf {f})$ is an open interval containing the interval with extremes given by the slopes of the normal at the points $(\delta _f,0)$ and $(0,\delta _g)$ . Item (1) follows from the definition of $\mathcal {C}^*(\mathbf {f})$ and the observation (cf. [Reference Bray, Canary, Kao and Martone5, Lemma 3.3(2)]) that $P(\langle \mathbf {z}_m, \mathbf {f} \rangle ) = 0$ implies $d(-\langle \mathbf {z}_m, \mathbf {f} \rangle ) < 1$ . For item (2), we notice that ${\textbf {x}}_{m}=(1/t_{m},m/t_{m})=(\int f\, d\mu ,\int g\, d\mu )$ , where $\mu =\mu _{-a_{m}f-b_{m}g}.$ Thus, by part (4) in Proposition 2.11, it is sufficient to show

$$ \begin{align*} \nabla\mathbb{P}(-a_{m},-b_{m})=\bigg(\!\int f\, d\mu,\int g\, d\mu\bigg). \end{align*} $$

However, this follows from the formula for the derivative of the pressure given in Corollary 2.10(2). The third assertion follows from Proposition 2.11 and the second assertion. The last assertion is an immediate consequence of assertions (1) and (2) and part (2) in Lemma 2.12.

3.1 Properties of the correlation number

In this subsection, we establish some of the properties of $H_{\mathbf {f}}(m)$ including Corollary 1.4 and Theorem 1.5 from the introduction.

Corollary 1.4. (Correlation number rigidity)

For $m_*=\delta _f/\delta _g$ , we have

(3.1) $$ \begin{align} H_{\mathbf{f}}(m_*)\leq\delta_f. \end{align} $$

Moreover, the equality holds if and only if $m_*S_{n}f(\underline {x})=S_{n}g(\underline {x})$ for all $\underline {x}\in \mathrm {Fix}^{n}.$

Proof. Since the Manhattan curve $\mathcal {C}(\mathbf {f})$ is convex and analytic, the point $(a,b)\in \mathcal {C}(\mathbf {f})$ is below or on the secant passing through the points $(\delta _f,0)$ and $(0,\delta _g)$ . Hence,

$$ \begin{align*} a\delta_g+b\delta_f\leq\delta_f\delta_g, \end{align*} $$

and the equality holds if and only if $\mathcal {C}(\mathbf {f})$ is a straight line. See Figure 1.

Figure 1

The Manhattan curve and the secant between its intercepts with the axes (colour online).

We obtain (3.1) by specializing at the point $(a_{m_*},b_{m_*})\in \mathcal {C}(\mathbf {f})$ since

$$ \begin{align*} H_{\mathbf{f}}(m_*) & =a_{m_*}+m_*b_{m_*}\leq\frac{\delta_f\delta_g}{\delta_g}=\delta_f. \end{align*} $$

The rigidity statement follows from Theorem 2.14(3) as $\mathcal {C}(\mathbf {f})$ is a straight line segment if and only if

$$ \begin{align*} S_{n}f(\underline{x})=\frac{\delta_g}{\delta_f}S_{n}g(\underline{x}) \end{align*} $$

for all $\underline {x}\in \mathrm {Fix}^{\textit {n}}.$

Recall that for any $\alpha , \beta> 0$ , the $(\alpha , \beta )$ -Bishop–Steger entropy of ${\mathbf {f}}$ is defined as

$$ \begin{align*} h_{(\alpha, \beta)}^{\mathrm{BS}}({\textbf{f}}) := \lim_{T \to \infty} \frac{1}{T} \log \sum_{n} \frac{1}{n} \#\{ \underline{x} \in \mathrm{ Fix}^{n}\colon \alpha S_{n} f(\underline{x}) + \beta S_{n} g(\underline{x}) \leq T \}. \end{align*} $$

Theorem 1.5. Let $m \in \mathcal {S}(\mathbf {f})$ be an admissible slope. Then,

$$ \begin{align*} \frac{H_{{\textbf{f}}}(m)}{\alpha + m\beta} \leq h_{(\alpha, \beta)}^{\mathrm{BS}}({\textbf{f}}), \end{align*} $$

and the equality holds if and only if m is the unique slope satisfying $a_{m}/b_{m} = \alpha /\beta $ .

Proof. We ease notation by writing $h_{\mathrm {BS}}$ for $h_{(\alpha , \beta )}^{\mathrm {BS}}({\mathbf {f}})$ . From [Reference Bray, Canary, Kao and Martone5, Theorem A], we know that $P(-h_{\mathrm {BS}}(\alpha f + \beta g)) = 0$ , which implies $(a_{\mathrm {BS}}, b_{\mathrm {BS}}) := (\alpha h_{\mathrm {BS}}, \beta h_{\mathrm {BS}}) \in \mathcal {C}({\mathbf {f}})$ . Observe that $a_{\mathrm {BS}}/b_{\mathrm {BS}} = \alpha /\beta $ and denote by $m_{\mathrm {BS}}$ the slope of the normal to $\mathcal {C}({\mathbf {f}})$ at $(a_{\mathrm {BS}}, b_{\mathrm {BS}})$ . We claim that for any $m\in \mathcal {S}(\mathbf {f})$ ,

$$ \begin{align*} \frac{a_{m} + m b_{m}}{\alpha + m\beta} \leq \frac{a_{m_{\mathrm{BS}}} + m_{\mathrm{BS}} b_{m_{\mathrm{BS}}}}{\alpha + m_{\mathrm{BS}} \beta} = h_{\mathrm{BS}}. \end{align*} $$

The last equality follows directly from the definition of $h_{\mathrm {BS}}$ . To prove the inequality, note that $\mathcal {C}({\mathbf {f}})$ can be parameterized as $(s, q(s))$ , where $q(s)$ is an analytic function, and we denote the slope of the normal to $\mathcal {C}({\mathbf {f}})$ at $(s, q(s))$ by $m(s)$ so that $q'(s) = -1/m(s)$ . Now, consider the function

$$ \begin{align*} F(s) = \frac{s + m(s) q(s)}{\alpha + m(s) \beta} = \frac{s - q(s)/q'(s)}{\alpha -\beta / q'(s)} = \frac{s q'(s) - q(s)}{\alpha q'(s) - \beta}. \end{align*} $$

Differentiating, we get

$$ \begin{align*} F'(s) = \frac{q"(s)(\alpha q(s) - \beta s)}{(\alpha q'(s) - \beta)^{2}}. \end{align*} $$

Since $\mathcal {C}({\mathbf {f}})$ is strictly convex and real analytic, we know that $q"(s)> 0$ . Moreover, $F'(0)>0$ , while $F'(\delta _f)<0$ , so $F(s)$ achieves its maximum when $\alpha q(s) = \beta s$ , which implies $s / q(s) = \alpha / \beta $ . By the strict convexity of $\mathcal {C}({\mathbf {f}})$ , this solution is unique, occurring at $(a_{\mathrm {BS}}, b_{\mathrm {BS}}) = (\alpha h_{\mathrm {BS}}, \beta h_{\mathrm {BS}})$ . This completes the proof of the claim and the theorem.

We conclude this section by establishing technical properties of $H_{\mathbf {f}}(m)$ , which will be useful for the estimates carried out in the next two sections.

Proof. For the first assertion, given $m\in \mathcal {S}(\mathbf {f})$ , consider $(a_{m},b_{m})\in \mathcal {C}(\mathbf {f})$ , so that $P(-a_{m}f-b_mg)=0$ , $H_{\mathbf {f}}(m)=a_{m}+b_{m}m$ , and

$$ \begin{align*} m=\frac{\int g\, d\mu_{-a_{m}f-b_{m}g}}{\int f\, d\mu_{-a_{m}f-b_{m}g}}. \end{align*} $$

Similarly, for $m'\in \mathcal {S}(g,f)$ , there is $(b_{m'},a_{m'})\in \mathcal {C}(g,f)$ , that is, $P(-b_{m'}g-a_{m'}f)=0$ , and

$$ \begin{align*} m'=\frac{\int f\, d\mu_{-a_{m'}f-b_{m'}g}}{\int g\, d\mu_{-a_{m'}f-b_{m'}g}}. \end{align*} $$

Moreover, $ H_{(g,f)}(m')=b_{m'}+a_{m'}m'.$ By Theorem 2.14(4), we see that $m\in \mathcal {S}(\mathbf {f})$ if and only if $1/m\in \mathcal {S}(g,f)$ , and that when $m'=1/m$ , we have $(a_{m'},b_{m'})=(b_m,a_m)$ . Thus,

$$ \begin{align*} H_{(g,f)}\bigg(\dfrac{1}{m}\bigg)=b_m+a_{m}\frac{1}{m}=\frac{1}{m}(a_m+mb_m)=\frac{1}{m}H_{\mathbf{f}}(m). \end{align*} $$

For item (2), by Corollary 2.16(2), we know $\nabla \mathbb {P}(-a_{m},-b_{m})={\textbf {x}}_m$ and

$$ \begin{align*} \mathbb{P}^{*}({\textbf{x}}_m)=-h_{-a_{m}f-b_{m}g}(\sigma). \end{align*} $$

Hence, using the fact that $\mu _{-a_{m}f-b_{m}g}$ is the equilibrium state of $-a_{m}f-b_{m}g$ , we compute

$$\begin{align*} -t_{m}\mathbb{P}^{*}({\textbf{x}}_m) & =\dfrac{h_{-a_{m}f-b_{m}g}(\sigma)}{\int f\, d\mu_{-a_{m}f-b_{m}g}}\\ & =\dfrac{P(-a_{m}f-b_{m}g)-\int(-a_{m}f-b_{m}g)\, d\mu_{-a_{m}f-b_{m}g}}{\int f\, d\mu_{-a_{m}f-b_{m}g}}\\ & =a_{m}+b_{m}\dfrac{\int g\, d\mu_{-a_{m}f-b_{m}g}}{\int f\, d\mu_{-a_{m}f-b_{m}g}}=a_{m}+b_{m}m\\ & =H_{{\textbf{f}}}(m). \end{align*}$$

The last assertion follows from Proposition 2.11(4) as $\nabla \mathbb {P}^{*}({\textbf {x}}_{m})=(-a_{m},-b_{m})$ and

$$ \begin{align*} \nabla^{2}\mathbb{P}^{*}({\textbf{x}}_{m})=(\nabla^{2}\mathbb{P}(-a_{m},-b_{m}))^{-1}= \frac{1}{\mathrm{det\nabla^{2}\mathbb{P}}}\begin{pmatrix} \mathrm{Var}(g,\mu) & \mathrm{Cov}(f,g,\mu)\\ \mathrm{Cov}(f,g,\mu) & \mathrm{Var}(f,\mu) \end{pmatrix} ^{-1} \end{align*} $$

is positive definite (as by Corollary 2.10(4), $\nabla ^{2}\mathbb {P}$ is positive definite).

Remark 3.4. Using the same suspension flow interpretation as in Remark 3.2, one can relate the term in Lemma 3.3(3) with the second derivative of the pressure over the suspension flow. Explicitly,

$$ \begin{align*} \overline{\mathbb{P}}_{m}":=t_{m}^{3}\cdot({\textbf{x}}_{m})^\top\cdot\nabla^{2}\mathbb{P}^{*}({\textbf{x}}_{m})\cdot {\textbf{x}}_{m}=\bigg(\frac{d^{2}}{dz^{2}}P_{\phi^f}|_{z=-b_{m}}(z\Psi^g)\bigg). \end{align*} $$

The convenient notation $\overline {\mathbb {P}}_{m}"$ will be used in §6.2.

4.1 Preimages of sample points and a priori estimates

Fix an admissible slope $m\in {\mathcal {S}(\mathbf {f})}$ , $k\in \mathbb {Z}_{>0}$ , and a non-periodic word $\underline {z}_{p}\in \Sigma ^+$ in each k-cylinder $p\in \Lambda _{k}$ . For every $n\in \mathbb Z_{>0}$ and $\xi>0$ , define

$$ \begin{align*} &{\mathcal{W}}(n,p,t;\mathbf{f},m,\xi)=p\cap\sigma^{-n}(\underline{z}_{p})\cap\{\underline{x}\in\Sigma^+\colon S_{n}\mathbf{f}(\underline{x})\in I^2_{\xi,m}(t)\}\quad\text{and} \\ &\quad W(n,p,t)=W(n,p,t;\mathbf{f},m,\xi):=\#\mathcal{W}(n,p,t;\mathbf{f},m,\xi). \end{align*} $$

Propositions 4.1 and 4.6 relate $M(n,t)$ to $W(n,p,t)$ , which is advantageous because $W(n,p,t)$ is itself related to the transfer operator via the Laplace–Fourier transform (defined below).

First, we introduce the following convenient notation. For any $\mathbf {y}\in \mathbb R^2$ , define

$$ \begin{align*} M(n,\mathbf{y}):=M(n,{\textbf{y}};\mathbf{f})=\sum_{\underline{x}\in\mathrm{Fix}^{n}}{\mathbf{1}_{\{\underline{x}\colon S_{n}\mathbf{f}(\underline{x})=\mathbf{y}\}}(\underline{x})} \end{align*} $$

and for any open bounded rectangle $U\subset \mathbb R^2$ , let

$$ \begin{align*} M(n,U)=\#\{\underline{x}\in\mathrm{Fix}^{n}\colon S_{n}\mathbf{f}(\underline{x})\in U\} =\int_{U}M(n,\mathbf{y})\, d\mathbf{y}. \end{align*} $$

Note that with this notation, $M(n,t)=M(n,I^2_{\xi ,m}(t))$ . Define $W(n,p,\mathbf {y})$ and $W(n,p,U)$ analogously and let

$$ \begin{align*} W(n,U)=\sum_{p\in\Lambda_{k}}W(n,p,U). \end{align*} $$

Finally, for any open rectangle $U=(a,b)\times (c,d)$ and sufficiently small $\epsilon>0$ , consider the open rectangles

$$ \begin{align*} U_{\pm\epsilon}:=(a\mp\epsilon,b\pm\epsilon)\times(c\mp\epsilon,d\pm\epsilon). \end{align*} $$

We will need to complement the results in Proposition 4.1 with an exponential control of the quantities $\sum _{p\in \Lambda _{k}}W(n,p,t)$ and $M(n,t)$ for $k\in \mathbb {Z}_{>0}$ and $n<k$ , similar to [Reference Bray, Canary, Kao and Martone5, Lemma 5.1(iv)]. However, in the current setting, we will need a different approach which relies on Lemmas 4.2 and 4.4.

Lemma 4.2. Let $f,g\colon \Sigma ^{+}\to \mathbb {R}$ be strictly positive locally Hölder continuous potentials with strong entropy gaps at infinity such that $\|f-g\|_\infty $ is finite. If $m\geq 1$ is an admissible slope, then

$$ \begin{align*} d(f)<H_{\mathbf{f}}(m). \end{align*} $$

Proof. By [Reference Bray, Canary, Kao and Martone5, Lemma 3.3], it suffices to show that $P(-H_{\mathbf {f}}(m)f)<\infty $ . Recall that, by definition, there exists a point $(a_{m},b_{m})$ on the Manhattan curve $\mathcal {C}(\mathbf {f})$ such that $H_{\mathbf {f}}(m)=a_{m}+m\cdot b_{m}$ . Since $m\geq 1$ and the pressure function is increasing by Corollary 2.10(2), we see that

$$ \begin{align*} P(-H_{\mathbf{f}}(m)f)=P(-a_{m}f-m\cdot b_{m}f)\leq P(-a_{m}f-b_{m}f). \end{align*} $$

Then, since $\|f-g\|_\infty =C<\infty $ , and P is increasing and linear on constants, we see that

$$ \begin{align*} P(-a_{m}f-b_{m}f)\leq P(-a_{m}f-b_{m}g+b_{m}C){\kern-1pt}={\kern-1pt}P(-a_{m}f-b_{m}g)+b_{m}C=b_{m}C{\kern-1pt}<{\kern-1pt}\infty, \end{align*} $$

where we used the fact that $P(-a_{m}f-b_{m}g)=0$ in the last equality.

Lemma 4.4. Let $f,g\colon \Sigma ^{+}\to \mathbb {R}$ be strictly positive locally Hölder continuous potentials with strong entropy gaps at infinity such that $\|f-g\|_\infty $ is finite. Suppose $m\geq 1$ . Then, for all $k\in \mathbb {N}$ and $b\in (d(f),H_{\mathbf {f}}(m))$ , there exists $C=C(k,b,\xi )>0$ such that for all $n<k$ and $t>0$ , we have

$$ \begin{align*} \sum_{p\in\Lambda_{k}}W(n,p,t)\leq Ce^{bt} \quad\text{and}\quad M(k,t)\leq Ce^{bt}. \end{align*} $$

Proof. We notice that, by [Reference Bray, Canary, Kao and Martone5, Lemma 5.1(iv)], for all $b>d(f)$ , there exists $C=C(k,b,\xi )$ such that

$$\begin{align*} \sum_{p\in\Lambda_{k}}W(n,p,t) & \leq\sum_{p\in\Lambda_{k}}\sum_{\underline{y}\in\sigma^{-n}(\underline{z}_{p})}\mathbf{1}_{p}(\underline{y})\mathbf{1}_{\{S_{n}f(\underline{y})\leq t+\xi\}}(\underline{y})\leq Ce^{bt}. \end{align*}$$

As $m\geq 1$ , we have that $H_{\mathbf {f}}(m)>d(f)$ by Lemma 4.2, and hence, we may choose any $b\in (d(f),H_{\mathbf {f}}(m)).$ The same argument shows that for all $b\in (d(f),H_{\mathbf {f}}(m))$ , there exists $C=C(k,b,\xi )$ such that

$$ \begin{align*} \text{ }M(k,t)\leq Ce^{bt}.\\[-38pt] \end{align*} $$

We are now ready to bound $\sum _{p\in \Lambda _{k}}W(n,p,t)$ and $M(k,t)$ for any $k\in \mathbb {Z}_{>0}$ and $n<k$ .

Proposition 4.6. (Negligible parts)

Let $f,g\colon \Sigma ^{+}\to \mathbb {R}$ be strictly positive locally Hölder continuous potentials with strong entropy gaps at infinity such that $\|f-g\|_\infty $ is finite. Then, for any $k>0$ , there exist constants $\zeta =\zeta (m), C=C(k,m,\xi )>0$ such that for all $n<k$ ,

$$ \begin{align*} \sum_{p\in\Lambda_{k}}W(n,p,t)\leq Ce^{(H_{\mathbf{f}}(m)-\zeta)t} \quad\text{and}\quad M(k,t)\leq Ce^{(H_{\mathbf{f}}(m)-\zeta)t}. \end{align*} $$

Proof. For $m\geq 1$ , a stronger version is given by Lemma 4.4. Thus, we assume $m<1$ and consider $H_{(g,f)}(m')$ with $m'=1/m>1$ . Applying Lemma 4.4 to the pair $(g,f)$ , if we pick $\zeta $ small enough such that $H_{(g,f)}(m')-m'\zeta \in (d(g),H_{(g,f)}(m'))$ , then we know that there exists $C=C(k,m,\xi )$ such that

$$ \begin{align*} & \sum_{p\in\Lambda_{k}}W(n,p,\widetilde{t};(g,f),m',\xi)\\ &\quad =\#(p\cap\sigma^{-n}(\underline{z}_{p})\cap\{\underline{x}\colon\ S_{n}(g,f)(\underline{x})\in I_{\xi,m'}^{2}(\widetilde{t})\})\leq Ce^{(H_{(g,f)}(m')-m'\zeta)\widetilde{t}} \end{align*} $$

for all $n<k$ and $\widetilde {t}\geq 0$ . Therefore, setting $\widetilde {t}=mt$ , we have

$$ \begin{align*} \sum_{p\in\Lambda_{k}}W(n,p,t;\mathbf{f},m,\xi) & =\#(p\cap\sigma^{-n}(\underline{z}_{p})\cap\{\underline{x}\colon S_{n}\mathbf{f}(\underline{x})\in I^2_{\xi,m}(t)\})\\ & =\sum_{p\in\Lambda_{k}}W\bigg(n,p,mt;(g,f),\frac{1}{m},\xi\bigg)\\ & \leq Ce^{(H_{(g,f)}({1}/{m})-({1}/{m})\zeta)mt}\\ & =Ce^{(mH_{(g,f)}({1}/{m})-\zeta)t}\\ & =Ce^{(H_{\mathbf{f}}(m)-\zeta)t}, \end{align*} $$

where the last equality follows from Lemma 3.3(1). Applying the same argument to $M(k,t)$ , we obtain

$$ \begin{align*} M(k,t)\leq Ce^{(H_{\mathbf{f}}(m)-\zeta)t}, \end{align*} $$

which ends the proof.

The Laplace–Fourier transform of a continuous function $F\colon \mathbb {R}^{2}\to \mathbb {R}$ is

(4.2) $$ \begin{align} \widehat{F}(\mathbf{z})=\int_{\mathbb{R}^{2}}e^{\langle{\mathbf{z},\mathbf{x}}\rangle}F(\mathbf{x})\, d{\textbf{x}}. \end{align} $$

Write $\widehat {W}(n,p,\mathbf {z})$ , $\widehat {W}(n,\mathbf {z})$ , and $\widehat {M}(n,\mathbf {z})$ for the Laplace–Fourier transforms of the functions $\mathbf {x}\mapsto W(n,p,\mathbf {x})$ , $\mathbf {x}\mapsto W(n,\mathbf {x})$ , $\mathbf {x}\mapsto M(n,\mathbf {x})$ , respectively. The key observation is that $\widehat {W}(n,p,\mathbf {z})$ is related to the transfer operator as follows:

(4.3) $$ \begin{align} \widehat{W}(n,p,\mathbf{z}) & =\int_{\mathbb{R}^{2}}e^{\langle\mathbf{z},\mathbf{x}\rangle}\ W(n,p,\mathbf{x})\, d{\textbf{x}}\nonumber \\ & =\int_{\mathbb{R}^{2}}e^{\langle\mathbf{z},\mathbf{x}\rangle}\sum_{\underline{y}\in\sigma^{-n}(\underline{z}_{p})}{\mathbf{1}}_{p}(\underline{y}){\mathbf{1}}_{\{\underline{y}\colon S_{n}\mathbf{f}(\underline{y})=\mathbf{x}\}}(\underline{y})\, d\mathbf{x}\nonumber \\ & =\sum_{\underline{y}\in\sigma^{-n}(\underline{z}_{p})}e^{\langle\mathbf{z},S_{n}\mathbf{f}(\underline{y})\rangle}{\mathbf{1}}_{p}(\underline{y})\nonumber\\ & =(\mathcal{L}_{\langle\mathbf{z},\mathbf{f}\rangle}^{n} {\mathbf{1}}_{p})(\underline{z}_{p}). \end{align} $$

Assume $\mathbf {z}\in D(\mathbf {f})$ so that $\langle \mathbf {z},\mathbf {f}\rangle $ is a locally Hölder continuous potential with finite pressure. By Proposition 2.5(1), $\mu _{\langle \mathbf {z},\mathbf {f}\rangle }$ is the unique invariant Gibbs measure and equilibrium state for $\langle \mathbf {z},\mathbf {f}\rangle $ , and we denote by $Q>1$ its Gibbs constant.

Lemma 4.7. Consider $k\in \mathbb N$ and $\mathbf {z}\in D(\mathbf {f})$ . For all $n\geq k,$ and $p\in \Lambda _{k}$ , we have that

$$ \begin{align*} \widehat{W}(n,p,\mathbf{z})\leq Q\mu_{\langle\mathbf{z},\mathbf{f}\rangle}(p)e^{n\mathbb{P}(\mathbf{z})}, \end{align*} $$

where $\mu _{\langle \mathbf {z},\mathbf {f}\rangle }$ is the equilibrium state for $\langle \mathbf {z},\mathbf {f}\rangle $ with Gibbs constant Q.

Proof. Notice that

$$ \begin{align*} e^{-n\mathbb{P}(\mathbf{z})}\cdot\widehat{W}(n,p,\mathbf{z})=e^{-n\mathbb{P}(\mathbf{z})}(\mathcal{L}_{\langle\mathbf{z},\mathbf{f}\rangle}^{n}{\mathbf{1}}_{p})(\underline{z}_{p}) & =\sum_{\underline{y}\in\sigma^{-n}(\underline{z}_{p})}e^{\langle\mathbf{z},S_{n}\mathbf{f}(\underline{y})\rangle-n\mathbb{P}(\mathbf{z})}{\mathbf{1}}_{p}(\underline{y})\\ & =\sum_{\underline{y}\in\sigma^{-n}(\underline{z}_{p})\cap p}e^{\langle\mathbf{z},S_{n}\mathbf{f}(\underline{y})\rangle-n\mathbb{P}(\mathbf{z})}. \end{align*} $$

Since $n\geq k$ , we have $\sigma ^{-n}(\underline {z}_{p})\cap p\neq \emptyset $ . Then, for any $\underline {y}\in \sigma ^{-n}(\underline {z}_{p})\cap p$ , there exists an n-cylinder $p_{\underline {y}}$ such that $\underline {y}\in p_{\underline {y}}\subset p$ . Moreover, $p_{\underline {y}_1}=p_{\underline {y}_2}$ implies $\underline {y}_1=\underline {y}_2$ . Thus, we can repeatedly use the Gibbs property to see that

$$ \begin{align*} \sum_{\underline{y}\in\sigma^{-n}(\underline{z}_{p})\cap p}e^{\langle\mathbf{z},S_{n}\mathbf{f}(\underline{y})\rangle-n\mathbb{P}(\mathbf{z})}\leq Q\sum_{\underline{y}\in\sigma^{-n}(\underline{z}_{p})\cap p}\mu_{\langle\mathbf{z},\mathbf{f}\rangle}(p_{\underline{y}})\leq Q\mu_{\langle\mathbf{z},\mathbf{f}\rangle}(p), \end{align*} $$

which concludes the proof.

Finally, we apply Lemma 4.7 to estimate the quantity

$$ \begin{align*} W(n,U):=\sum_{p\in\Lambda_{k}}W(n,p,U). \end{align*} $$

Lemma 4.8. Assume $\mathbf {z}\in D(\mathbf {f})$ , U is an open bounded rectangle in $\mathbb R^2$ , and $n\geq k$ . Let $\mathbf {x}\in \mathbb R^2$ be such that $\langle \mathbf {z},\mathbf {y}\rangle \geq \langle \mathbf {z},\mathbf {x}\rangle $ for all $\mathbf {y}\in U$ . Then,

$$ \begin{align*} W(n,U)\leq Qe^{n\mathbb{P}(\mathbf{z})-\langle\mathbf{z},\mathbf{x}\rangle}. \end{align*} $$

Proof. The statement follows from the proof of [Reference Lalley25, Proposition 1]. Indeed,

$$ \begin{align*} W(n,U) &\leq\sum_{p\in\Lambda_{k}}\int_{U}e^{\langle\mathbf{z},\mathbf{y}\rangle-\langle\mathbf{z},\mathbf{x}\rangle}W(n,p,\mathbf{y}) \,d\textbf{y} \\&\leq\sum_{p\in\Lambda_{k}}e^{-\langle\mathbf{z},\mathbf{x}\rangle}\int_{\mathbb{R}^{2}}e^{\langle\mathbf{z},\mathbf{y}\rangle}W(n,p,\mathbf{y}) \,d\textbf{y} \\ & \leq\sum_{p\in\Lambda_{k}}\widehat{W}(n,p,\mathbf{z})e^{-\langle\mathbf{z},\mathbf{x}\rangle}\leq Qe^{n\mathbb{P}(\mathbf{z})-\langle\mathbf{z},\mathbf{x}\rangle}, \end{align*} $$

where we apply Lemma 4.7 to obtain the last inequality.

6.1 The proofs of Theorems 1.1, 1.2, and 1.3

Theorem 6.1 is the last ingredient needed to establish Theorems 1.1, 1.2, and 1.3 from the introduction. In this subsection, we record the proofs of these results while assuming Theorem 6.1.

Theorem 6.3. (Theorem 1.1 and Theorem 1.3)

For any $m\in \mathcal {S}({\mathbf {f}})$ , we have

$$ \begin{align*} \lim_{t\to\infty}\frac{1}{t}\log M(t;{\mathbf{f},m,\xi)}=H_{\mathbf{f}}(m). \end{align*} $$

Proof. Fix $k\in \mathbb N$ . For $n\geq k$ , we apply Proposition 4.1(3) with $U=I^2_{\xi ,m}(t)$ to deduce that for any $p\in \Lambda _k$ ,

(6.1) $$ \begin{align} W(n,p,t;\mathbf{f},m,\xi-\epsilon_k)\leq \#\mathcal{M}_p(n,t;{\mathbf{f},m,\xi)}\leq W(n,p,t;\mathbf{f},m,\xi+\epsilon_k), \end{align} $$

where $(\epsilon _k)_k$ is the sequence from Proposition 4.1(2). Then, we apply Proposition 4.6 to see that there exists $\zeta =\zeta (m)>0$ and $C_{1}=C_1(k,m,\xi )>0$ such that for any $n<k$ ,

(6.2) $$ \begin{align} \#\mathcal{M}(n,t;{\textbf{f}},m,\xi)\leq C_{1}e^{t(H_{\mathbf{f}}(m)-\zeta)}. \end{align} $$

Moreover, Lemma 5.4 and Theorem 6.1 show that there exists $C_{2}=C_2(m,p,\xi )$ and $C_{3}=C_3(k,m,\xi )$ such that

$$ \begin{align*} \frac{e^{tH_{\mathbf{f}}(m)}}{t^{{3}/{2}}}(C_2+o(1))= \sum_{n\geq 1} \frac{1}{n} W(n,p,t)\leq \sum_{p\in\Lambda_k}\sum_{n\geq 1} \frac{1}{n} W(n,p,t)\leq C_{3}\frac{e^{tH_{\mathbf{f}}(m)}}{t}. \end{align*} $$

Thus, recalling that $M(t;\mathbf {f},m,\xi )=\sum _{p\in \Lambda _k}\sum _{n\geq 1}({1}/{n})\#\mathcal M_p(n,t;\mathbf {f},m,\xi )$ , we obtain

$$ \begin{align*} \lim_{t\to\infty}\frac{1}{t}\log M(t;{\textbf{f},m,\xi)}=H_{\mathbf{f}}(m).\\[-42pt] \end{align*} $$

Proof of Theorem 1.2

By Theorem 6.1, we know

$$ \begin{align*} \sum_{n\geq1}\frac{1}{n}W(n,p,t;{\textbf{f},m,\xi)=\frac{e^{tH_{\mathbf{f}}(m)}}{t^{{3}/{2}}}C(m,p,\xi)(1+o(1))}, \end{align*} $$

where $C(m,p,\xi )=(2\pi \overline {\mathbb {P}}^{\prime \prime }_{m})^{-1/2}\cdot C_{p}(m)\cdot \int _{0}^{\xi }e^{a_{m}t}\, dt\cdot \int _{0}^{\xi }e^{b_{m}t}\, dt.$ Moreover, by (6.1) and (6.2), we get

$$ \begin{align*} & \frac{e^{tH_{\mathbf{f}}(m)}}{t^{{3}/{2}}}C(m,p,\xi-\epsilon_k)(1+o(1))\\ &\quad \leq\sum_{n\geq1}\frac{1}{n}\#\mathcal{M}_p(n,t;{\textbf{f},m,\xi)\leq\frac{e^{tH_{\mathbf{f}}(m)}}{t^{{3}/{2}}}C(m,p,\xi+\epsilon_k)(1+o(1))}. \end{align*} $$

To conclude the proof, recall that $\epsilon _k\to 0$ as $k\to \infty $ by Proposition 4.1(2).

6.2 Proof of Theorem 6.1

This proof consists of three steps. First, for any $\epsilon>0$ , we break the sum into two parts:

$$ \begin{align*} \sum_{n\geq1}\frac{1}{n}W(n,p,t)=\sum_{n:\ |({n}/{t})-t_{m}|\ge\epsilon}\frac{1}{n}W(n,p,t)+\sum_{n:\ |({n}/{t})-t_{m}|<\epsilon}\frac{1}{n}W(n,p,t). \end{align*} $$

By Lemma 5.3, for any $\epsilon>0$ ,

$$ \begin{align*} \limsup_{t\to\infty}e^{-tH_{{\textbf{f}}}(m)}\cdot t^{{3}/{2}}\bigg(\sum_{n:\ |({n}/{t})-t_{m}|\geq\epsilon}\frac{1}{n}W(n,p,t)\bigg)\leq\limsup_{t\to\infty}C_{2}t^{{3}/{2}}e^{-t\eta}=0. \end{align*} $$

Therefore, for any $\epsilon>0$ , it suffices to study the asymptotic behavior for $t\to \infty $ of

$$ \begin{align*} \sum_{n:\ |({n}/{t})-t_{m}|<\epsilon}\frac{1}{n}W(n,p,t). \end{align*} $$

Since the characteristic function of the rectangle $(0,\xi )\times (0,\xi )$ can be approximated by a non-negative smooth function u with compact support, it is enough to show that

(6.3) $$ \begin{align} &\lim_{\epsilon\to0}\lim_{t\to\infty}e^{-tH_{{\textbf{f}}}(m)}\, t^{{3}/{2}}\bigg(\sum_{n:\,|({n}/{t})-t_{m}|<\epsilon}\frac{1}{n} \int_{\mathbb{R}^{2}}u({\textbf{y}}-t{\textbf{m}}) W(n,p,{\textbf{y}})\, d{\textbf{y}}\bigg)\nonumber\\ &\quad =(2\pi\overline{\mathbb{P}}^{\prime\prime}_{m})^{-({1}/{2})}C_{p}(m)\int_{\mathbb{R}^{2}}u({\textbf{y}})\, e^{-\langle \mathbf{z}_{m}, {\textbf{y}} \rangle}\, d{\textbf{y}}. \end{align} $$

We will need Lemma 6.5 below, whose proof is postponed to §6.3. For $N\in \mathbb N$ , denote by $C_c^N(\mathbb R^2)$ the space of N-differentiable functions on $\mathbb R^2$ with compact support.

Lemma 6.5. Let K be a compact neighborhood of $\mathbf {x}_m$ in $\nabla \mathbb P(D)$ . Suppose ${\mathbf {x}=\nabla }\mathbb {P}({\textbf {z})}\in K$ and $u\in C_{c}^{N}(\mathbb {R}^{2})$ is non-negative, then as $n\to \infty $ ,

$$ \begin{align*} \int u({\textbf{y}-n{\textbf{x}})W(n,p,\mathbf{y})\, d \mathbf{y}\sim\frac{e^{-n\mathbb{P}^{*}({\textbf{x}})}}{2\pi n}(\det\nabla\mathbb{P}^{*}({\textbf{x}}))^{1/2}\cdot C_{p}(\mathbf{z})\cdot\int_{\mathbb{R}^{2}}u({\textbf{y})}e^{-\langle{\textbf{z}},{\textbf{y}}\rangle}\, d{\textbf{y}},} \end{align*} $$

where $C_{p}(\mathbf {z})=h_{\langle {\mathbf {z},\mathbf {f}}\rangle }(\underline {z}_{p})\nu _{\langle {\mathbf {z},\mathbf {f}}\rangle }(p)$ . Furthermore, the above convergence is uniform in K.

We now begin with the first step. We plan to apply Lemma 6.5 to points ${\mathbf {x}}$ near ${\mathbf {x}}_{m} = {\textbf {m}}/t_{m}$ (cf. Notation 2.15), in particular, to those of the form ${\mathbf {x}} = {\textbf {m}}/(n/t)$ whenever $|n/t - t_{m}| < \epsilon $ for small $\epsilon $ . To do this, we need to ensure that these points lie in $\nabla \mathbb {P}(D)$ . To see this, note first that D is an open set (by Proposition 2.5(3)), and that $\nabla \mathbb {P}$ is a homeomorphism on D. The latter follows from the inverse function theorem and because $\nabla ^{2}\mathbb {P}$ is positive definite (Corollary 2.10(4)). Consequently, $\nabla \mathbb {P}(D) \subset \mathbb {R}^{2}$ is open.

Since ${\mathbf {x}}_{m}$ lies in the open set $\nabla \mathbb {P}(D)$ , there exists $\epsilon> 0$ sufficiently small such that whenever $|n/t - t_{m}| < \epsilon $ , the point ${\textbf {m}}/(n/t)$ lies in a neighborhood of ${\mathbf {x}}_{m}$ contained in $\nabla \mathbb {P}(D)$ . Thus, there exists $ {\mathbf {z}}^{*} := \nabla \mathbb {P}^{*}\!({{\textbf {m}}}/{n/t}) \in D. $

Then, summing the estimate from Lemma 6.5 over the set $\{n:|{n}/{t}-t_{m}|<\epsilon \}$ , for $t\to \infty $ , we have

(6.4) $$ \begin{align} & \sum_{|{n}/{t}-t_{m}|<\epsilon}\frac{1}{n}\int u({\textbf{y}}-t{\textbf{m}})W(n,p,{\textbf{y}})\, d{\textbf{y}}\nonumber\\ & \quad \sim\underset{:=~(*)}{\underbrace{\sum_{|({n}/{t})-t_{m}|<\epsilon}\frac{e^{-n\mathbb{P}^{*}(t{\textbf{m}}/n)}}{2\pi n^{2}}\cdot(\det\nabla^{2}\mathbb{P}^{*}(t{\textbf{m}}/n))^{1/2}\cdot\mathbf{Q}_{p}(t{\textbf{m}/n})}}, \end{align} $$

where $\mathbf {Q}_{p}(t{\textbf {m}}/n)\colon =C_p(\mathbf {z}^*)\int _{\mathbb R^2} u(\mathbf {y})e^{-\langle {\mathbf {z}}^*,\mathbf {y}\rangle }$ . Now, the last step in the proof of Theorem 6.1 consists of using a standard probability trick, Lemma 6.7 below, to simplify the right-hand side of the above equation.

Lemma 6.7. (Lalley [Reference Lalley25, Lemma 6])

Suppose F is continuous on a closed interval $[t_{1},t_{2}]$ and $t_{1}<t_{m}<t_{2}$ . Let $\sigma>0$ . Then, for any $\epsilon>0$ such that $(t_m-\epsilon ,t_m+\epsilon )\subset [t_{1},t_{2}]$ , we have

(6.5) $$ \begin{align} \lim_{t\to\infty}\sum_{n\colon|({n}/{t})-t_m|<\epsilon}\frac{1}{t^{1/2}}F\bigg(\frac{n}{t}\bigg)e^{-({t(({n}/{t})-t_m)^{2}}/{2\sigma^{2}})}=F(t_m)(2\pi)^{1/2}\sigma. \end{align} $$

Moreover, (6.5) holds uniformly for $t_m$ and $\sigma $ in a compact subset of $(0,\infty )$ , and F in a compact subset of $C([t_{1},t_{2}])$ with $t_{1}<t_m<t_{2}$ .

Recall that by Lemma 2.12(2) and Corollary 2.16(4), $s\mapsto -s\mathbb {P^{*}}({\textbf {m}}/s)$ is concave down for $s>0$ and achieves its maximum uniquely at $s=t_{m}$ . Expanding $-s\mathbb {P^{*}}({\textbf {m}}/s)$ near $t_{m}$ , we have

$$\begin{align*} -\frac{n}{t}\mathbb{P}^{*}\bigg({\textbf{m}}\frac{t}{n}\bigg)&= -t_{m}\mathbb{P}^{*}({\textbf{m}}/t_{m})-t_{m}^{-1}({\textbf{m}}/t_{m})^\top\cdot\nabla^{2}\mathbb{P}^{*}({\textbf{m}}/t_{m})\cdot({\textbf{m}}/t_{m})\cdot \frac{|{n}/{t}-t_{m}|^{2}}{2}\\& \qquad+R(\zeta,m)\bigg|\dfrac{n}{t}-t_{m}\bigg|^{3}\\& = \underset{H_{{\textbf{f}}}(m)}{\underbrace{-t_{m}\mathbb{P}^{*}({\mathbf{x}}_{m})}}-\underset{\text{main contribution}}{\underbrace{t_{m}^{-1}({\textbf{x}}_{m})^\top\cdot\nabla^{2}\mathbb{P}^{*}({\textbf{x}}_{m})\cdot{\textbf{x}}_{m}\cdot \frac{|{n}/{t}-t_{m}|^{2}}{2}}}+\underset{\text{negligible term}}{\underbrace{R(\zeta,m)\bigg|\dfrac{n}{t}-t_{m}\bigg|^{3}}}, \end{align*}$$

where $R(\zeta , m)$ is the remainder term from the Taylor expansion for some $\zeta \in (({n}/{t}),\, t_{m})$ . Since the function $s\mapsto -s\mathbb {P^{*}}({\textbf {m}}/s)$ is analytic around $s=t_{m}$ , there exists a constant $C_{m,\epsilon }>0$ such that $|R(\zeta ,x)|\leq C_{m,\epsilon }$ . Recalling that

$$ \begin{align*} t_{m}^{-1}({\textbf{x}}_{m})^\top\cdot\nabla^{2}\mathbb{P}^{*}({\textbf{x}}_{m})\cdot{\textbf{x}}_{m}=t_{m}^{-4}\overline{\mathbb{P}}^{\prime\prime}_{m}, \end{align*} $$

for n such that $|({n}/{t})-t_{m}|<\epsilon $ , we estimate

(6.6) $$ \begin{align} (*)\leq\sum_{|({n}/{t})-t_{m}|<\epsilon}&\underset{=e^{tH_{{\textbf{f}}}(m)}\cdot e^{-({t|({n}/{t})-t_{m}|^{2}})/{2\sigma_{+}^{2}}}} {\underbrace{e^{tH_{{\textbf{f}}}(m)-tt_{m}^{-4}\overline{\mathbb{P}}^{\prime\prime}_{m}\cdot({|({n}/{t})-t_{m}|^{2}})/{2}+ tC_{m,\epsilon}\epsilon|({n}/{t})-t_{m}|^{2}}}}\nonumber\\ & \quad\cdot\underset{=({1}/{t^{2}})F(n/t)}{\underbrace{\dfrac{1}{2\pi n^{2}}\cdot(\det\nabla^{2}\mathbb{P}^{*}(t{\textbf{m}}/n))^{1/2}\cdot\mathbf{Q}_{p}(t{\textbf{m}}/n)}} \end{align} $$

$$ \begin{align*} (*)\geq\sum_{|({n}/{t})-t_{m}|<\epsilon}&\underset{=e^{tH_{{\textbf{f}}}(m)}\cdot e^{-({t|({n}/{t})-t_{m}|^{2}})/({2\sigma_{-}^{2}})}} {\underbrace{e^{tH_{{\textbf{f}}}(m)-tt_{m}^{-4}\overline{\mathbb{P}}^{\prime\prime}_{m}\cdot({|({n}/{t})-t_{m}|^{2}})/{2}-tC_{m,\epsilon}\epsilon|({n}/{t})-t_{m}|^{2}}}}\\ & \quad \cdot\underset{=({1}/{t^{2}})F(n/t)}{\underbrace{{1}/{2\pi n^{2}}\cdot(\det\nabla^{2}\mathbb{P}^{*}(t{{\textbf{m}}}/n))^{{1}/{2}}\cdot\mathbf{Q}_{p}(t{\textbf{m}}/n)}}, \end{align*} $$

where $F(s)=({1}/{2\pi s^{2}})(\det \nabla ^{2}\mathbb {P}^{*}({\textbf {m}}/s))^{1/2}\cdot \mathbf {Q}_{p}({\textbf {m}}/s)$ and $\sigma _{\pm }=(t_{m}^{-4}\overline {\mathbb {P}}^{\prime \prime }_{m}\mp 2\epsilon C_{m,\epsilon })^{-1/2}$ .

By Lemma 6.7, we get

(6.7) $$ \begin{align} & \lim_{t\to\infty}\frac{1}{F(t_{m})}\sum_{|({n}/{t})-t_{m}|<\epsilon}t^{-1/2}e^{-({t|({n}/{t})-t_{m}|^{2}})/({2\sigma_{+}^{2}})}\cdot F(n/t)\nonumber\\ & \quad =\sqrt{2\pi}\sigma_{+} =\sqrt{2\pi}(t_{m}^{-4}\overline{\mathbb{P}}^{\prime\prime}_{m}-2\epsilon C_{m,\epsilon})^{-({1}/{2})} \end{align} $$

and similarly,

$$ \begin{align*} & \lim_{t\to\infty}\frac{1}{F(t_{m})}\sum_{|({n}/{t})-t_{m}|<\epsilon}t^{-1/2}e^{-({t|({n}/{t})-t_{m}|^{2}})/({2\sigma_{-}^{2}})}\cdot F(n/t)\\ & \quad =\sqrt{2\pi}\sigma_{-}=(t_{m}^{-4}\overline{\mathbb{P}}^{\prime\prime}_{m}+2\epsilon C_{m,\epsilon})^{-{1}/{2}}. \end{align*} $$

Finally, we put all the estimates above together and obtain, for any $\epsilon $ small enough (depending on m),

$$ \begin{align*} \begin{aligned} & \limsup_{t\to\infty}e^{-tH_{{\textbf{f}}}(m)}\cdot t^{{3}/{2}}\bigg(\sum_{|({n}/{t})-t_{m}|\leq\epsilon}\frac{1}{n}\int u({\textbf{y}}-t{\textbf{m}})\cdot W(n,p,{\textbf{y})\, d{\textbf{y}}}\bigg)\\& \quad =\limsup_{t\to\infty}e^{-tH_{{\textbf{f}}}(m)}\cdot t^{{3}/{2}}(*)\cdot (1+o(1)) & (\text{by~}(6.4))\\&\quad \leq\lim_{t\to\infty}e^{-tH_{{\textbf{f}}}(m)}\cdot t^{{3}/{2}}\cdot\bigg((1+o(1)) \cdot t^{-{3}/{2}}e^{tH_{{\textbf{f}}}(m)} \\& \qquad \qquad\qquad\times \bigg({\sum_{|({n}/{t})-t_{m}|<\epsilon}t^{-{1}/{2}}e^{-({t|({n}/{t})-t_{m}|^{2}})/({2\sigma_{+}^{2}})}\cdot F(n/t)}\bigg)\bigg) & (\text{by~}(6.6))\\&\quad= F(t_{m})\sqrt{2\pi}\sigma_{+}. & (\text{by~}(6.7)). \end{aligned} \end{align*} $$

Similarly, we have

$$ \begin{align*} \liminf_{T\to\infty}e^{-tH_{{\textbf{f}}}(m)}\cdot t^{{3}/{2}}\bigg(\sum_{|({n}/{t})-t_{m}|\leq\epsilon}\frac{1}{n}\int u({\textbf{y}}-t{\textbf{m}})\cdot W(n,p,\textbf{y})\, d{\textbf{y}}\bigg) \geq F(t_{m})\sqrt{2\pi}\sigma_{-}. \end{align*} $$

Then, (6.3) follows by taking the limit

$$ \begin{align*} \lim_{\epsilon\to0}\sigma_{\pm}=\lim_{\epsilon\to0}(t_{m}^{-4}\overline{\mathbb{P}}^{\prime\prime}_{m}\mp2\epsilon C_{m,\epsilon})^{-{1}/{2}}=t_{m}^{-2}\overline{\mathbb{P}}^{\prime\prime}_{m}, \end{align*} $$

which concludes the proof of Theorem 6.1.

6.3 The proof of Lemma 6.5

In this subsection, we aim to prove the following key asymptotic estimate.

Lemma 6.5

Let K be a compact neighborhood of $\mathbf {x}_m$ in $\nabla \mathbb P(D)$ . Suppose ${\mathbf {x}=\nabla }\mathbb {P}({\mathbf {z})}\in K$ and $u\in C_{c}^{N}(\mathbb {R}^{2})$ is non-negative. Then, as $n\to \infty $ ,

$$ \begin{align*} \int u({\textbf{y}-n{\textbf{x}})W(n,p,\mathbf{y})\, d \mathbf{y}\sim\frac{e^{-n\mathbb{P}^{*}({\textbf{x}})}}{2\pi n}(\det\nabla\mathbb{P}^{*}({\textbf{x}}))^{1/2}\cdot C_{p}(\mathbf{z})\cdot\int_{\mathbb{R}^{2}}u({\textbf{y})}e^{-\langle{\textbf{z}},{\textbf{y}}\rangle}\, d{\textbf{y}},} \end{align*} $$

where $C_{p}(\mathbf {z})=h_{\langle {\mathbf {z},\mathbf {f}}\rangle }(\underline {z}_{p})\nu _{\langle {\mathbf {z},\mathbf {f}}\rangle }(p)$ . Furthermore, the above convergence is uniform in K.

In general, verifying Lemma 6.5 across the entire space $C_{c}^{N}(\mathbb {R}^{2})$ is challenging. However, as noted by Babillot and Ledrappier [Reference Babillot and Ledrappier2, Lemma 2.4], it suffices to check Lemma 6.5 on a smaller, well-behaved function class $\mathcal {H}$ . Specifically, the function class $\mathcal {H}$ is the linear span of functions of the form $\{ e^{i\langle {\mathbf {z}}, \cdot \rangle } h(\cdot ) : h \in \mathcal {H}^{+}, {\mathbf {z}} \in \mathbb {R}^{2} \}$ , where a function $h: \mathbb {R}^{2} \to \mathbb {C}$ belongs to the class $\mathcal {H}^{+}$ if:

1. (1) h is real-valued and non-negative; and

2. (2) the Laplace–Fourier transform (see (4.2)) ${\boldsymbol {\Theta }} \mapsto \widehat {h}(-i{\boldsymbol {\Theta }})$ belongs to $C_{c}^{N}(\mathbb {R}^{2})$ .

Remark 6.8. The statement of Babillot and Ledrappier [Reference Babillot and Ledrappier2, Lemma 2.4] is formulated in terms of convergence of measures on $\mathbb {R}^{2}$ . As explained immediately before [Reference Babillot and Ledrappier2, Lemma 2.4], by considering the Borel measures $\nu _{n}^{\mathbf {x}}$ on $\mathbb {R}^{2}$ , defined (via the Riesz representation theorem) by

$$ \begin{align*} \nu_{n}^{\mathbf{x}}(G) := \frac{2\pi n\, e^{n\mathbb{P}^{*}(\mathbf{x})}} {C_{p}(\mathbf{z}) (\det\nabla\mathbb{P}^{*}(\mathbf{x}))^{1/2}} \int_{\mathbb{R}^{2}} G(\mathbf{y}-n\mathbf{x})\, W(n,p,\mathbf{y}) \, d\mathbf{y}, \end{align*} $$

for any continuous compactly supported function G on $\mathbb {R}^{2}$ , and by noting that the limiting measure $\unicode{x3bb} $ (in the notation of [Reference Babillot and Ledrappier2, Lemma 2.4]) is given by

$$ \begin{align*} \unicode{x3bb}(G) = \int_{\mathbb{R}^{2}} G(\mathbf{y})\, e^{-\langle \mathbf{z}, \mathbf{y} \rangle} \, d\mathbf{y}, \end{align*} $$

the proof of [Reference Babillot and Ledrappier2, Lemma 2.4] applies verbatim in our setting.

The remainder of this section is devoted to deriving the following lemma.

Lemma 6.9. Let K be a compact neighborhood of $\mathbf {x}_m$ in $\nabla \mathbb P(D)$ . Suppose ${\mathbf {x}=\nabla }\mathbb {P}({\mathbf {z})}\in K$ and $u\in \mathcal {H}$ . Then, as $n\to \infty $ ,

$$ \begin{align*} \int u({\textbf{y}-n{\textbf{x}})W(n,p,\mathbf{y})\, d \mathbf{y}\sim\frac{e^{-n\mathbb{P}^{*}({\textbf{x}})}}{2\pi n}(\det\nabla\mathbb{P}^{*}({\textbf{x}}))^{1/2}\cdot C_{p}(\mathbf{z})\cdot\int_{\mathbb{R}^{2}}u({\textbf{y})}e^{-\langle{\textbf{z}},{\textbf{y}}\rangle}\, d{\textbf{y}},} \end{align*} $$

where $C_{p}(\mathbf {z})=h_{\langle {\mathbf {z},\mathbf {f}}\rangle }(\underline {z}_{p})\nu _{\langle {\mathbf {z},\mathbf {f}}\rangle }(p)$ . Furthermore, the above convergence is uniform in K.

6.3.1 The saddle-point method and the proof of Lemma 6.9

This subsection aims to prove Lemma 6.9. Before beginning the proof, we first introduce one of our main tools: the saddle-point method.

Let $F:\Omega \colon =\{\|\boldsymbol {\Theta }\|<\epsilon \}\subset {\mathbb {R}}^{2}\to {\mathbb {R}}$ be an analytic function such that $\nabla F(\mathbf {0})=0$ and $\nabla ^{2}F(\mathbf {0})$ is positive definite. As F is analytic $\Omega \subset \mathbb {R}^{2}$ , we can extend F to a holomorphic function on $\Omega _{\mathbb {C}}=\{{\boldsymbol {\Theta }\in \mathbb {C}^{2}:\ }\|\boldsymbol {\Theta }\|<\epsilon \}$ . Here, we will abuse notation and continue to call the extended holomorphic function by $F:\Omega _{\mathbb {C}}\to \mathbb {C}$ .

To fix our notation, we write $M_{u}=\sup \{|u(x)|\}$ , $v(t)=O_{F}(t)$ if and only if there exist $t_{0}>0$ and two constants $C_{1}$ and $C_{2}$ only depending on F such that $C_{1}t\leq v(t)\leq C_{2}t$ for all $t>t_{0}$ , and $L_{v}$ is the Lipschitz constant of a Lipschitz function $v,$ that is, $|v(z)-v(w)|\leq L{}_{v}|z-w|.$

Proposition 6.11. (Saddle-point method)

If $G\colon \mathbb {C}^{2}\to \mathbb {C}$ is a Lipschitz function with compact support, then

$$ \begin{align*} \int_\Omega G(i\boldsymbol{\Theta})e^{nF(i\boldsymbol{\Theta})}\, d\boldsymbol{\Theta} & = \frac{e^{nF(\mathbf{0})}G(\mathbf{0})}{n}\cdot\frac{2\pi}{\sqrt{\det \nabla^{2}F(\mathbf{0})}}\\ &\quad +\frac{e^{nF(\mathbf{0})}}{n}\bigg(L_{G}\cdot O_{F}\bigg(\frac{1}{\sqrt{n}}\bigg)+M_{G}\cdot O_{F}\bigg(\frac{1}{\sqrt{n}}\bigg)\bigg). \end{align*} $$

Proof. We include a proof of this statement in Appendix B because we were not able to find an appropriate reference. Our proof is adapted from [Reference Anantharaman1, §5.2].

We establish Lemma 6.9 in two steps. First, Lemma 6.12 allows us to use the saddle-point method to derive one side of the statement of Lemma 6.9. The second step is Lemma 6.14, which relies on the complex Ruelle–Perron–Frobenius theorem (that is, Corollary 2.9).

Lemma 6.12. Let $\mathbf {z}\in D$ and $\mathbf {x}=\nabla \mathbb P (\mathbf {z})$ . Suppose $u \in \mathcal {H}$ and define

$$ \begin{align*} F({\textbf{y}}):=\langle{\textbf{x}},{\textbf{y}}-{\textbf{z}}\rangle+\mathbb{P}({{\textbf{z}}-{\textbf{y}}}) \end{align*} $$

and

$$ \begin{align*} G_{p}({\textbf{y}}):=\widehat{u}({\textbf{y}}-{\textbf{z}})C_{p}({\textbf{z}}-{\textbf{y}}), \end{align*} $$

where $C_{p}({\mathbf {z}}-{\mathbf {y}})=h_{\langle {\mathbf {z}}-{\mathbf {y}},{\mathbf {f}}\rangle }(\underline {z}_{p})\nu _{\langle {{\mathbf {z}}-{\mathbf {y}},\mathbf {f}}\rangle }(p)$ . Then, there exists $\epsilon>0$ such that F is analytic in $\Omega =\{\boldsymbol {\Theta }\colon \|{\boldsymbol {\Theta }}\|<\epsilon \}$ , $F(\mathbf {0})=-\mathbb {P}^{*}({\textbf {x}})$ , $\nabla F(\mathbf {0})=0$ , and $\nabla ^{2}F(\mathbf {0})=\nabla ^{2}\mathbb {P}(\mathbf {z})=(\nabla ^{2}\mathbb {P}^{*}(\mathbf {x}))^{-1}$ is positive definite. Moreover, $G_{p}$ is a bounded Lipschitz continuous function in $\Omega _{\mathbb {C}}$ .

Proof. The analyticity and the formulas for the derivatives of F follow from Corollary 2.10. As $u\in \mathcal {H}$ , we know $\widehat {u}\in C_{c}^{N}(\mathbb R^2)$ . Corollary 2.9 gives us the analyticity of ${{\textbf {y}}\mapsto C_{p}({\mathbf {z}}-{\textbf {y}})}$ provided ${\textbf {y}}\in \Omega _{\mathbb {C}}$ . Hence, $G_{p}$ is bounded Lipschitz in $\Omega _{\mathbb {C}}$ .

An immediate consequence of Proposition 6.11 and Lemma 6.12 is the following corollary.

Corollary 6.13. Let $\Omega =\{ \|\boldsymbol {\Theta }\|<\epsilon \} \subset {\mathbb {R}}^{2}$ , and let F and $G_p$ be as in Lemma 6.12. Then, as $n\to \infty $ ,

(6.8) $$ \begin{align} \int_{\Omega}G_{p}(i\boldsymbol{\Theta})e^{nF(i\boldsymbol{\Theta})}\, d\boldsymbol{\Theta}\sim e^{-n\mathbb{P}^{*}({\textbf{x}})}\bigg(\frac{2\pi}{n}\bigg)(\det\nabla^{2}\mathbb{P}^{*}({\textbf{z}}))^{{1}/{2}}\widehat{u}(-{\textbf{z}})\cdot C_{p}({\textbf{z}}). \end{align} $$

The following lemma shows that the asymptotic growth of $\int _{{\mathbb {R}}^{2}}u({\textbf {y}}-n{\textbf {x}})W(n,p,{\textbf {y}})\, d{\textbf {y}}$ is captured by integrating the transfer operator around zero after applying the Laplace–Fourier transform. The proof is based on estimates of the transfer operator.

Lemma 6.14. Let $\textbf {z} \in D$ and $\textbf {x}=\nabla \mathbb {P}(\mathbf {z})$ . Suppose $u \in \mathcal {H}$ , then as $n\to \infty $ ,

$$ \begin{align*} \int_{\Omega}\widehat{u}(i\boldsymbol{\Theta}-{\textbf{z}})C_{p}({\textbf{z}}-i\boldsymbol{\Theta})e^{n\langle{\textbf{x}},i\boldsymbol{\Theta}-{\textbf{z}}\rangle+n\mathbb{P}({\textbf{z}}-i\boldsymbol{\Theta})}\, d\boldsymbol{\Theta}\sim(2\pi)^{2}\int_{\mathbb R^{2}}u({\textbf{y}}-n{\textbf{x}})W(n,p,{\textbf{y}})\, d{\textbf{y}} \end{align*} $$

where $C_{p}({\mathbf {z}}-i\boldsymbol {\Theta })=h_{\langle {\mathbf {z}}-i\boldsymbol {\Theta },{\mathbf {f}}\rangle }(\underline {z}_{p})\cdot \nu _{\langle {\mathbf {z}}-i\boldsymbol {\Theta },{\mathbf {f}}\rangle }(p)$ . Moreover, the convergence above is uniform for $\mathbf {x}\in \nabla \mathbb P(D)$ in any compact neighborhood of ${\mathbf {x}}$ .

Proof. First, recall the Pancherel–Parseval identity

(6.9) $$ \begin{align} (2\pi)^2\int_{\mathbb{R}^2}U(\mathbf{y})V(\mathbf{y})\,d\mathbf{y}=\int_{\mathbb{R}^2}\widehat U(i\boldsymbol{\Theta})\widehat V(-i\boldsymbol{\Theta})\,d\boldsymbol{\Theta}, \end{align} $$

and that the Laplace–Fourier transform of $W(n,p,\textbf {y})$ is related to the transfer operator by (4.3). Now, we apply 6.9 to the functions

$$ \begin{align*} U(\mathbf{y})=u(\mathbf{y}-n\mathbf{x})e^{-\langle\mathbf{y},\mathbf{z}\rangle}\quad\text{and}\quad V(\mathbf{y})=W(n,p,\mathbf{y})e^{\langle\mathbf{y},\mathbf{z}\rangle} \end{align*} $$

to get

$$ \begin{align*} (2\pi)^{2} & \int_{\mathbb R^{2}}u({\textbf{y}}-n{\textbf{x}})W(n,p,\textbf{y})\,d{\textbf{y}} = \int_{\mathbb R^{2}}\widehat{u}(i\boldsymbol{\Theta}-{\textbf{z}})e^{n\langle{\textbf{x}},i{\boldsymbol{\Theta}}-{\textbf{z}}\rangle}\widehat{W}(n,p,\mathbf{z}-i\boldsymbol{\Theta})\, d\boldsymbol{\Theta}\\ = & \int_{\mathbb R^{2}}\widehat{u}(i\boldsymbol{\Theta}-{\textbf{z}})e^{n\langle{\textbf{x}}, i\boldsymbol{\Theta}-{\textbf{z}}\rangle}(\mathcal{L}_{\langle{\textbf{z}}-i\boldsymbol{\Theta},\mathbf{f}\rangle}^{n}{\mathbf{1}}_{p})(\underline{z}_{p})\, d\boldsymbol{\Theta}. \end{align*} $$

By Corollary 2.9(5), there exists $\delta>0$ such that for any $\boldsymbol {\Theta }\neq \mathbf {0}$ ,

(6.10) $$ \begin{align} \lim_{n\to\infty}(1+\delta)^{n}e^{-n\mathbb{P}({\textbf{z}})}\|\mathcal{L}_{\langle{\textbf{z}}-i\boldsymbol{\Theta},\mathbf{f}\rangle}^{n}{\mathbf{1}}_{p}\|_{\infty}=0, \end{align} $$

so we can find n large enough such that

$$ \begin{align*} \Vert e^{-n\mathbb{P}({\textbf{z}})}(1+\delta)^{n}(\mathcal{L}_{\langle{\textbf{z}}-i\boldsymbol{\Theta},\mathbf{f}\rangle}^{n}{\mathbf{1}}_{p})(\underline{z}_{p})\Vert \leq1. \end{align*} $$

By Corollary 2.9(4), we know that if $\boldsymbol {\Theta }$ is such that $\|\boldsymbol {\Theta }\|<\epsilon $ , there exist $R>0$ and ${\eta \in (0,1)}$ independent of $\boldsymbol {\Theta }$ such that

(6.11) $$ \begin{align} \Vert e^{-n\mathbb{P}({\textbf{z}}-i\boldsymbol{\Theta})}\mathcal{L}_{\langle{\textbf{z}}-i\boldsymbol{\Theta},\mathbf{f}\rangle}^{n}{\mathbf{1}}_{p}-C_{p}({\textbf{z}}-i\boldsymbol{\Theta})\Vert _{\infty}\leq R\eta^{n}. \end{align} $$

Now, we split the integral as

$$ \begin{align*} (2\pi)^{2}\int_{\mathbb R^{2}}u({\textbf{y}}-n{\textbf{x}})W(n,p,{\textbf{y}})\, d{\textbf{y}} &= \int_{\mathbb R^{2}}\underset{U(\boldsymbol{\Theta})}{\underbrace{\widehat{u}(i\boldsymbol{\Theta}-{\textbf{z}})e^{n\langle{\textbf{x}}, i{\boldsymbol{\Theta}}-{\textbf{z}}\rangle}(\mathcal{L}_{\langle{\textbf{z}}-i\boldsymbol{\Theta},\mathbf{f}\rangle}^{n}{\mathbf{1}}_{p})(\underline{z}_{p})}}\, d\boldsymbol{\Theta}\\ &= \int_{\Omega}U(\boldsymbol{\Theta})\, d\boldsymbol{\Theta}+\int_{\Omega^{c}}U(\boldsymbol{\Theta})\, d\boldsymbol{\Theta}. \end{align*} $$

Recall that $\mathbb {P}^{*}({\textbf {x}})=\langle {\textbf {x}},{\textbf {z}}\rangle -\mathbb {P}({\textbf {z}})$ by Proposition 2.11(4), so we have

(6.12) $$ \begin{align} & \bigg|\!\int_{\Omega^c}U(\boldsymbol{\Theta})\, d\boldsymbol{\Theta}\bigg| \nonumber\\ &\quad \leq\int_{K\cap\Omega^{c}}\widehat{u}(i{\boldsymbol{\Theta}}-\mathbf{z})e^{n(-\langle{\textbf{x}},{\textbf{z}}\rangle+\mathbb{P}({\textbf{z}}))}(1+\delta)^{-n}|e^{-n\mathbb{P}({\textbf{z}})}(1+\delta)^{n}(\mathcal{L}_{\langle{\textbf{z}}-i\boldsymbol{\Theta},\mathbf{f}\rangle}^{n}{\mathbf{1}}_{p})(\underline{z}_{p})|\, d\boldsymbol{\Theta}\nonumber \\ & \quad \leq M\int_{K\cap\Omega^{c}}e^{-n\mathbb{P}^{*}({\textbf{x}})}(1+\delta)^{-n}\, d{\boldsymbol{\Theta}}\nonumber \\ & \quad =Me^{-n\mathbb{P}^{*}({\textbf{x}})}(1+\delta)^{-n}\text{Leb}(K), \end{align} $$

where $\Vert \widehat {u}\Vert _{\infty }\leq M$ and $\widehat u$ vanishes for ${\boldsymbol {\Theta }}$ outside of K. However, we can compute

$$ \begin{align*} \int_{\Omega}U(\boldsymbol{\Theta})\, d\boldsymbol{\Theta}= & \int_{\Omega}\widehat{u}(i\boldsymbol{\Theta}-{\textbf{z}})e^{n\langle{\textbf{x}},i{\boldsymbol{\Theta}}-{\textbf{z}}\rangle}(\mathcal{L}_{\langle{\textbf{z}}-i\boldsymbol{\Theta},\mathbf{f}\rangle}^{n}{\mathbf{1}}_{p})(\underline{z}_{p})\, d\boldsymbol{\Theta}\\ = & \underset{(*)}{\underbrace{\int_{\Omega}\widehat{u}(i\boldsymbol{\Theta}-{\textbf{z}})e^{n\langle{\textbf{x}},i{\boldsymbol{\Theta}}-{\textbf{z}}\rangle+n\mathbb{P}({\textbf{z}}-i\boldsymbol{\Theta})}C_{p}({\textbf{z}}-i\boldsymbol{\Theta})\, d\boldsymbol{\Theta}}}\\ & +\int_{\Omega}\widehat{u}(i\boldsymbol{\Theta}-{\textbf{z}})e^{n\langle{\textbf{x}},i{\boldsymbol{\Theta}}-{\textbf{z}}\rangle}R({\textbf{z}},{\textbf{f}},\boldsymbol{\Theta})\, d\boldsymbol{\Theta}, \end{align*} $$

where $R({\textbf {z}},{\textbf {f}},\boldsymbol {\Theta })=\mathcal {L}_{\langle {\textbf {z}}-i\boldsymbol {\Theta },\mathbf {f}\rangle }^{n}{\mathbf {1}}_{p}(\underline {z}_p)-e^{n\mathbb {P}({\textbf {z}}-i\boldsymbol {\Theta })}C_{p}({\textbf {z}}-i\boldsymbol {\Theta })$ . Observe that by (6.11), we have

(6.13) $$ \begin{align} \bigg|\!\int_{\Omega}\!\widehat{u}(i\boldsymbol{\Theta}-{\textbf{z}})e^{n\langle{\textbf{x}},i{\boldsymbol{\Theta}}-{\textbf{z}}\rangle}R({\textbf{z}},{\textbf{f}},\boldsymbol{\Theta})\, d\boldsymbol{\Theta}\bigg|\! & \leq M\bigg|\!\int_{\Omega}\!e^{n\langle{\textbf{x}},i{\boldsymbol{\Theta}}-{\textbf{z}}\rangle}e^{n\mathbb{P}({\textbf{z}}-i{\boldsymbol{\Theta}})}e^{-n\mathbb{P}({\textbf{z}}-i{\boldsymbol{\Theta}})}R({\textbf{z}},{\textbf{f}},\boldsymbol{\Theta})\, d\boldsymbol{\Theta}\bigg|\nonumber\\ & \leq MR\eta^{n}\bigg|\!\int_{\Omega}e^{n\langle{\textbf{x}},i{\boldsymbol{\Theta}}-{\textbf{z}}\rangle}e^{n\mathbb{P}({\textbf{z}}-i{\boldsymbol{\Theta}})}\, d\boldsymbol{\Theta}\bigg|\nonumber \\ & \leq M'R\eta^{n}\int_{\Omega}e^{n\langle{\textbf{x}},-{\textbf{z}}\rangle}e^{n\mathbb{P}({\textbf{z}})}\, d\boldsymbol{\Theta}\nonumber \\ & \leq\pi\epsilon^{2}M'R\eta^{n}e^{-n\mathbb{P}^{*}({\textbf{x}})}, \end{align} $$

where we used Corollary 2.9(5) to obtain the third inequality. We notice that by Corollary 6.13, we know the exponential growth rate of

$$ \begin{align*} (*):=\int_{\Omega}\widehat{u}(i\boldsymbol{\Theta}-{\textbf{z}})e^{n\langle{\textbf{x}},i{\boldsymbol{\Theta}}-{\textbf{z}}\rangle+n\mathbb{P}({\textbf{z}}-i\boldsymbol{\Theta})}C_{p}({\textbf{z}}-i\boldsymbol{\Theta})\, d\boldsymbol{\Theta} \end{align*} $$

is $\mathbb {P}^{*}({\mathbf {x}})$ . Meanwhile, (6.12) and (6.13) show that the growth rate of these two terms are strictly less then $\mathbb {P}^{*}({\mathbf {x}})$ . Hence,

$$ \begin{align*} \lim_{n\to\infty}\frac{(2\pi)^{2}\int_{\mathbb R^{2}}u({\textbf{y}}-n{\textbf{x}})W(n,p,{\textbf{y}})\, d{\textbf{y}}}{\int_{\Omega}\widehat{u}(i\boldsymbol{\Theta}-{\textbf{z}})e^{n\langle{\textbf{x}},i{\boldsymbol{\Theta}}-{\textbf{z}}\rangle+n\mathbb{P}({\textbf{z}}-i\boldsymbol{\Theta})}C_{p}({\textbf{z}}-i\boldsymbol{\Theta})\, d\boldsymbol{\Theta}}=1.\\[-42pt] \end{align*} $$

We are now ready to present the proof of Lemma 6.9, which follows from Corollary 6.13 and Lemma 6.14.

Proof of Lemma 6.9

By Lemma 6.14, we know

$$ \begin{align*} \lim_{n\to\infty}\frac{(2\pi)^{2}\int_{\mathbb R^{2}}u({\textbf{y}}-n{\textbf{x}})W(n,p,{\textbf{y}})\, d{\textbf{y}}}{\int_{\Omega}\hat{u}(i\boldsymbol{\Theta}-{\textbf{z}})e^{n\langle{\textbf{x}},i{\boldsymbol{\Theta}}-{\textbf{z}}\rangle+n\mathbb{P}({\textbf{z}}-i\boldsymbol{\Theta})}C_{p}({\textbf{z}}-i\boldsymbol{\Theta})\, d\boldsymbol{\Theta}}=1. \end{align*} $$

Moreover, Corollary 6.13 shows that

$$ \begin{align*} \lim_{n\to\infty}\frac{({e^{-n\mathbb{P}^{*}({\textbf{x}})}}/{2\pi n})(\det\nabla^{2}\mathbb{P}^{*}(\mathbf{x}))^{1/2}\widehat{u}(-\mathbf{z})C_{p}(\mathbf{z})}{\int_{\Omega}\widehat{u}(i\boldsymbol{\Theta}-{\textbf{z}})e^{n\langle{\textbf{x}},i{\boldsymbol{\Theta}}-{\textbf{z}}\rangle+n\mathbb{P}({\textbf{z}}-i\boldsymbol{\Theta})}C_{p}({\textbf{z}}-i\boldsymbol{\Theta})\, d\boldsymbol{\Theta}}=1. \end{align*} $$

Combining these two equations, we obtain

$$ \begin{align*} \lim_{n\to\infty}\frac{({e^{-n\mathbb{P}^{*}({\textbf{x}})}}/{2\pi n})(\det\nabla^{2}\mathbb{P}^{*}(\mathbf{x}))^{1/2}\widehat{u}(-\mathbf{z})C_{p}(\mathbf{z})}{(2\pi)^{2}\int_{\mathbb R^{2}}u({\textbf{y}}-n{\textbf{x}})W(n,p,{\textbf{y}})\, d{\textbf{y}}}=1.\\[-42pt] \end{align*} $$

7.1 Cusped Hitchin representations

To define cusped Hitchin representations, we will first need to discuss a notion of positivity on the space of complete flags.

A matrix $U\in \textsf {SL}(d,\mathbb {R})$ is unipotent and totally positive with respect to a basis ${u=(u_1,\ldots ,u_d)}$ of $\mathbb R^d$ if in the basis u, the matrix U is unipotent, upper triangular and the minors of U are strictly positive, unless they are forced to be zero due to the shape of the matrix. Recall that a (complete) flag F in $\mathbb R^d$ is a maximal nested sequence of subspaces of $\mathbb R^d$ . That is, $F=(F^0,F^1,\ldots , F^{d-1},F^d)$ with $\dim F^i=i$ and $F^{i-1}\subset F^{i}$ for all $i=1,\ldots ,d$ . We denote by $\mathcal F_d$ the space of flags in $\mathbb R^d$ . Two flags $F,G$ are transverse if $F^i\cap G^{d-i}=\{0\}$ for all $i=0,\ldots , d$ . A basis $u=(u_1,\ldots , u_d)$ is consistent with a pair of transverse flags $F,G$ if $u_i\in F^i\cap G^{d-i+1}$ for all $i=1,\ldots , d$ . Then, a k-tuple of flags $(F_1,\ldots , F_k)$ is positive if there exist a basis $u=(u_1,\ldots , u_d)$ consistent with $F_1$ and $F_k$ , and unipotent and totally positive matrices $U_2,\ldots , U_{k-1}$ such that ${F_j=U_j\cdot \cdots \cdot U_2\cdot F_2}$ .

Given a torsion-free geometrically finite Fuchsian group, a representation $\rho \colon \Gamma \to \textsf {SL}(d,\mathbb R)$ is cusped Hitchin if there exists a continuous, $\rho $ -equivariant map $\xi \colon \Lambda (\Gamma )\to \mathcal F_d$ which sends any positive tuple flags in $\Lambda (\Gamma )\subset \partial \mathbb H^2\cong \mathcal F_2$ to a positive tuple of flags in  $\mathcal F_d$ .

7.2 Length functions and potentials

Choose

$$ \begin{align*} \mathfrak a=\{x\in\mathbb{R}^d\colon x_1+\cdots+x_d=0\}\quad\text{and}\quad\mathfrak a^+=\{x\in\mathfrak a\colon x_1\geq \cdots\geq x_d\} \end{align*} $$

as a Cartan subspace and a closed positive Weyl chamber for the Lie algebra of $\textsf {SL}(d,\mathbb R)$ , respectively. The Jordan projection is the map

$$ \begin{align*} \unicode{x3bb}\colon\textsf{SL}(d,\mathbb R)\to\mathfrak a^+ \end{align*} $$

which records the logarithms of the moduli of eigenvalues of elements of $\textsf {SL}(d,\mathbb R)$ in decreasing order. If $\gamma \in \Gamma $ is hyperbolic and $\rho $ is a cusped Hitchin representation, then $(\unicode{x3bb} \circ \rho )(\gamma )$ lies in the interior of $\mathfrak a^+$ (see, for example, [Reference Fock and Goncharov16, Corollary 9.2] and [Reference Canary, Zhang and Zimmer9, Theorem 1.4]). Thus, if $\Delta $ is the set of non-zero positive linear combinations of the simple roots for $\mathfrak a^+$ , for any $\phi \in \Delta $ , we define the $\phi $ -length function of $\rho $ as

$$ \begin{align*} \ell^\phi_\rho(\gamma)=(\phi\circ\unicode{x3bb}\circ\rho)(\gamma)\geq 0 \end{align*} $$

and observe that if $\gamma $ is hyperbolic, then $\ell ^\phi _\rho (\gamma )>0$ .

Bray et al [Reference Bray, Canary, Kao and Martone5, Theorem D] construct a potential on the shift space from Theorem 7.1 associated to a cusped Hitchin representation $\rho $ and a choice of length function.

Theorem 7.2. (Bray et al [Reference Bray, Canary, Kao and Martone5])

Let $\Gamma $ be a torsion-free, geometrically finite Fuchsian group which is not convex cocompact, and $\rho :\Gamma \to \textsf {SL}(d,\mathbb R)$ is a cusped Hitchin representation and $\phi \in \Delta $ . Then, there exists a strictly positive locally Hölder continuous potential $\tau _\rho ^\phi \colon \Sigma ^+\to \mathbb R$ with strong entropy gap at infinity such that for every $\underline {x}\in \mathrm {Fix}^n$ ,

$$ \begin{align*} S_n\tau^\phi_\rho(\underline{x})=\ell^\phi_\rho(G(\underline{x})). \end{align*} $$

Moreover, if $\eta \colon \Gamma \to \textsf {SL}(d,\mathbb R)$ is another cusped Hitchin representation, then ${\|\tau _\rho ^\phi -\tau ^\phi _\eta \|_\infty }$ is finite.

In this setting, we can characterize algebraically when a pair of potentials is independent.

We are now ready to prove Thoerem 1.6.