2.1 Dynamics of skew products

Let F be a continuous skew product on $X\times Y$ ; that is, there are continuous maps $f\colon X\to X$ and $\{g_x\colon Y\to Y|\ x\in X\}$ such that

$$ \begin{align*}F(x,y)=(f(x),g_x(y)). \end{align*} $$

To understand the dynamics of F on $X\times Y$ , define for any $n\geq 0$ and $x\in X$ , the map

$$ \begin{align*}g_x^n:=\ g_{f^{n-1}x}\circ\dots\circ g_x\colon Y_x\to Y_{f^nx}.\end{align*} $$

Then for any $(x,y)\in X\times Y$ , the behavior of the system can be investigated through the sequence $F^n(x,y)=(f^n(x),g_x^n(y)).$

For each $n\geq 0$ , define the nth Bowen metric as

$$ \begin{align*}d_n((x,y),(x',y'))=\max_{0\leq i\leq n}\{d(F^i(x,y),F^i(x',y'))\}.\end{align*} $$

Also denote the nth Bowen ball centered at $(x,y)$ of radius $\delta>0$ by

$$ \begin{align*}B_n((x,y),\delta)=\{(x',y')\colon d_n((x,y),(x',y'))<\delta\}.\end{align*} $$

2.2 Non-uniform expansion along fibers

The following describes our assumptions of non-uniform expansion along the fibers on $X\times Y$ . We shall assume that F is a topologically exact local homeomorphism and that there is a continuous function $(x,y)\mapsto L(x,y)$ such that the following statements hold.

1. (A1) There exists open $U\ni (x,y)$ such that $F|_U$ is invertible and

   $$ \begin{align*}d(F^{-1}(u_1,u_2),F^{-1}(v_1,v_2))\leq L(x,y)d((u_1,u_2),(v_1,v_2))\end{align*} $$
   for all $(u_1,u_2),(v_1,v_2)\in F(U)$ .

2. (A2) There exist constants $\gamma>1$ and $L\geq 1$ , and an open region $\mathcal {A}\subset X\times Y$ such that $L(x,y)\leq L$ for every $(x,y)\in \mathcal {A}$ and $L(x,y)<\gamma ^{-1}$ for all $(x,y)\not \in \mathcal {A}$ , and L is close enough to 1 so that inequality (6) below is satisfied.

Assumption (A1) gives us control of preimages in the image of small balls for which the product map is invertible. Assumption (A2) says that F is uniformly expanding outside of some region $\mathcal {A}$ and not too contracting in $\mathcal {A}$ . Thus, if $\mathcal {A}$ is empty, then everything is reduced to the uniformly expanding case.

We say that an open cover $\mathcal {U}$ of $X\times Y$ separates curves if, given a distance-minimizing geodesic segment c on $X\times Y$ , each element of $\mathcal {U}$ can intersect at most one curve in $F^{-1}(c)$ . Denote by the collection of open covers $\mathcal {U}$ of $X\times Y$ that separates curves and each $U\in \mathcal {U}$ satisfies assumption (A1). We further assume the following.

1. (A3) There exists a finite covering such that $\mathcal {A}$ can without loss of generality be covered by the first $q<\deg (F)$ elements of $\mathcal {U}$ .

2. (A4) For every $x\in X$ , there exists a finite covering $\mathcal {U}_x$ of $Y_x$ which separates curves by sets in and $\mathcal {A}\cap Y_x$ can be covered by the first $q<d$ elements of $\mathcal {U}_x$ .

Assumption (A3) ensures that every point has at least one preimage in the expanding region. Note that (A3) is a strengthened version of assumption H2 from Castro and Varandas [Reference Castro and Varandas2]. Assumption (A4) guarantees that every point in a single fiber has at least preimage in the expanding region. This will be crucial to the arguments in §3 as discussed below.

A map $f\colon X\to X$ is uniformly expanding if there exist $C, \delta _f>0$ and $\gamma>1$ such that

$$ \begin{align*}d(f^n(x),f^n(x'))\geq C\gamma^n d(x,x')\end{align*} $$

whenever $d_n(x,x')\leq \delta _f$ . One can assume without loss of generality that $C=1$ by passing to an adapted metric. This reduces the expanding property to

$$ \begin{align*}d(f(x),f(x'))\geq\gamma d(x,x')\end{align*} $$

whenever $d(x,x')\leq \delta _f$ .

Proof. Fix $(x,y)\in X\times Y$ . Choose $x'\in X$ such that $(f(x'),g_x(y))\in F(U{(x,y)})$ . Since F is a local homeomorphism, there exists $y'\in Y$ such that $(x',y')\in U{(x,y)}$ and $F(x',y')=(f(x'),g_x(y))$ . Thus,

$$ \begin{align*} d(x,x')&\leq d((x,y),(x',y'))\\ &\leq L(x,y)d(F(x,y),F(x',y'))\\ &\leq L(x,y)d((f(x),g_x(y)),(f(x'),g_x(y)))\\ &\leq L(x,y)d(fx,fx'). \end{align*} $$

This finishes the proof since (A4) implies $\inf _{y\in Y}L(x,y)<1$ .

We remark that it would be interesting to see how the proofs in this paper would need to be changed if we removed assumption (A4). In such a setting, the base map could be non-uniformly expanding. It is known that the geometric potential for the Manneville–Pomeau map is Hölder continuous and has two equilibrium states. Uniqueness of the equilibrium state on the base is crucial to the proofs in §4 about the conditional measures of the equilibrium state $\mu $ . It is well known that Hölder potentials associated to expanding maps admit a unique equilibrium states. The uniform expansion in the base given by Lemma 2.1 is essential to proving that the potential $\Phi $ defined in equation (1) is Hölder continuous.

As a consequence of Lemma 2.1, we see that the number of preimages along fibers must be fixed. Indeed, since F is a local homeomorphism on a compact connected manifold, F is a covering map. Thus, we have that $\deg (F)$ is constant. Similarly, since f is expanding on a compact connected manifold onto itself, f is a covering map. Thus, $\hat {d}:={\deg (f)}$ is constant in x. Thus, $d:={\deg (g_x)}$ is constant for all $x\in X$ and $y\in Y_x$ and $\deg (F)=\hat {d}d$ .

The following example shows that there is a robust class of systems that satisfies the given assumptions.

Example 2.2. The Manneville–Pomeau map on is a classic example of a system that displays non-uniform expansion. Define a map $F\colon X\times Y\to X\times Y$ by taking the base map f to be the doubling map on and Manneville–Pomeau maps in the fibers where $p(x)>0$ varies continuously in the base point. Each of these maps has two branches so $d=2$ . Note that $g_x'(y)>g_x'(0)=1$ for all $y\not =0$ . Let $\mathcal {A}$ be any small neighborhood around . Then on $\mathcal {A}^c$ the product map F does not decrease distances; that is, if $(x,y),(x',y')\in \mathcal {A}^c$ , then

$$ \begin{align*}d((f(x),g_x(y)),(f(x'),g_{x'}(y')))\geq \gamma d((x,y),(x',y')).\end{align*} $$

So $q=1$ . Then $F(x,y)=(f(x),g_x(y))$ satisfies assumptions (A1) and (A2) and thus Theorem A holds for this example.

Lemma 2.3. If F satisfies (A1) and (A2), then for any $x,x'\in X$ and $y,y'\in Y$ , we can pair off the preimages of $g_x^{-1}(y)=\{y_1,\ldots ,y_{d}\}$ and $g_{x'}^{-1}(y')=\{y_1',\ldots ,y_{d}'\}$ where for any $k=1,2,\ldots ,q$ ,

$$ \begin{align*}d((x,y_k),(x,y_k'))\leq Ld((fx,y),(fx',y'))\end{align*} $$

while for any $k=q+1,\ldots ,d$ ,

$$ \begin{align*}d((x,y_k),(x',y_k'))\leq \gamma^{-1}d((fx,y),(fx',y')).\end{align*} $$

2.3 Existence and uniqueness of equilibrium states

We say is $\alpha $ -Hölder continuous for some $\alpha>0$ if

$$ \begin{align*}|\varphi|_\alpha:=\sup_{(x,y)\not=(x',y')}\frac{|\varphi(x,y)-\varphi(x',y')|}{d((x,y),(x',y'))^\alpha}<\infty.\end{align*} $$

We denote by $C^\alpha (X\times Y)$ the Banach space of $\alpha $ - $\mathrm {H}\ddot{\mathrm{o}}\mathrm{lder}$ continuous functions on $X\times Y$ . The nth Birkhoff sum is defined as $S_n\varphi (x,y)=\sum _{k=0}^n\varphi \circ F^k(x,y)$ .

We denote by $\mathcal {M}(X\times Y)$ the space of Borel probability measures on $X\times Y$ and $\mathcal {M}(X\times Y,F)$ those that are F-invariant. Given a continuous map $F\colon X\times Y\to X\times Y$ and a potential , the variational principle asserts that

(2) $$ \begin{align} P(\varphi)=\sup\!\bigg\{h_\nu(F)+\int\varphi\, d\nu \colon \nu\in\mathcal{M}(X\times Y,F) \bigg\} \end{align} $$

where $P(\varphi )$ denotes the topological pressure of F with respect to $\varphi $ and $h_\mu (F)$ denotes the metric entropy of F. An equilibrium state for F with respect to $\varphi $ is an invariant measure that achieves the supremum in the right-hand side of equation (2). For uniformly expanding maps, every equilibrium state $\mu $ satisfies the Gibbs property: for any $\varepsilon>0$ , there exists a $C>0$ such that

$$ \begin{align*}C^{-1}\leq\dfrac{\mu(B_n((x,y),\varepsilon))}{e^{-nP(\varphi)+S_n\varphi(x,y)}}\leq C\end{align*} $$

for any $(x,y)\in X\times Y$ and .

For our purposes in this paper, we fix a Holder potential $\varphi \in {C^\alpha (X\times Y)}$ satisfying

(P) $$ \begin{align} \sup\varphi-\inf\varphi<\varepsilon_\varphi\quad \text{and}\quad |e^\varphi|_\alpha<\varepsilon_\varphi e^{\inf\varphi} \end{align} $$

for some $\varepsilon _\varphi>0$ satisfying inequality (3) and equation (4) below (see §3). Almost constant potentials satisfy (P) so Theorem A applies to an open set of potentials. In particular, Theorem A holds for measures of maximal entropy. We assume that L is close enough to $1$ and $0<\varepsilon _\varphi <\log d -\log q$ so that

(3) $$ \begin{align} e^{\varepsilon_\varphi}\cdot\bigg(\dfrac{(d-q)\gamma^{-\alpha}+qL^\alpha}{d}\bigg)<1. \end{align} $$

Under these assumptions, it is known that there is a unique equilibrium state $\mu $ for $\varphi $ on $X\times Y$ .

We will not use the non-lacunary property of $\nu $ or $J_\nu F$ . For more details, see [Reference Varandas and Viana12].

Denote by $\hat {\mu }=\mu \circ \pi _X^{-1}$ the pushforward of the equilibrium state $\mu $ onto the base X. Throughout this paper, we shall refer to this measure as the transverse measure for our skew product.

Our main result is the following theorem.

We remark that the existence in item $(1)$ follows closely proofs from Piraino [Reference Piraino8] on SFTs. However, the proof of the Hölder continuity of $\Phi $ took new ideas on compact, connected manifolds (see §3.4). It is worth noting that Stadlbauer, Varandas and Zhang proved a similar result to item $(3)$ for conformal measure of Ruelle expanding iterated function systems.

2.4 Fiberwise transfer operators for skew products

As is common in the literature, we will utilize Ruelle operators to study the equilibrium state on $(X\times Y,F)$ . Define the transfer operator $\mathcal {L}_\varphi $ acting on $ C(X\times Y)$ by sending $\psi \in C(X\times Y)$ to

Note that under the skew product representation of F, we may write

$$ \begin{align*}\sum_{(\overline{x},\overline{y})\in F^{-1}(x,y)}e^{\varphi(\overline{x},\overline{y})}\psi(\overline{x},\overline{y})=\sum_{\overline{x}\in f^{-1}x}\sum_{\overline{y}\in g_{\overline{x}}^{-1}y}e^{\varphi(\overline{x},\overline{y})}\psi(\overline{x},\overline{y}).\end{align*} $$

This gives rise to a fiberwise transfer operator on the fibers of $X\times Y$ .

We disintegrate $\varphi $ and get the family of fiberwise potentials $\{\varphi _x(\cdot )=\varphi (x,\cdot )\}_{x\in X}$ . For every $x\in X$ , let $\mathcal {L}_x\colon C(Y_x)\to C(Y_{fx})$ be defined by

$$ \begin{align*} \mathcal{L}_x\psi_x(y)=\sum_{\overline{y}\in g_x^{-1}y}e^{\varphi_x(\overline{y})}\psi_x(\overline{y}) \end{align*} $$

for any $\psi \in C(X\times Y)$ . We shall iterate the transfer operator by letting

$$ \begin{align*}\mathcal{L}_x^n=\mathcal{L}_{f^{n-1}x}\circ\cdots\circ \mathcal{L}_x\colon C(Y_x)\to C(Y_{f^nx}).\end{align*} $$

Along with each of these fiberwise operators, we define its dual $\mathcal {L}_x^*$ by sending a probability measure $\eta \in \mathcal {M}(Y_{fx})$ to the measure $\mathcal {L}_x^*\eta \in \mathcal {M}(Y_x)$ such that for any $\psi \in C(X\times Y)$ ,

$$ \begin{align*}\int \psi\,d(\mathcal{L}_x^*\eta)=\int \mathcal{L}_x\psi\,d\eta.\end{align*} $$

3.1 Birkhoff contraction theorem

It is not hard to check that the Ruelle operator preserves the Banach space of Hölder continuous potentials $ C^\alpha (X\times Y),\ 0<\alpha <1$ . A subset $\Lambda \subset {C^\alpha (X\kern1.3pt{\times}\kern1.3pt Y)}$ is called a cone if $a\Lambda \kern1.3pt{=}\kern1.3pt\Lambda $ for all $a\kern1.3pt{>}\kern1.3pt0$ . A cone $\Lambda $ is convex if $\psi +\zeta \in \Lambda $ for all $\psi ,\ \zeta \in \Lambda $ . We say that $\Lambda $ is a closed cone if $\Lambda \cup \{0\}$ is closed with respect to the Hölder norm. We assume our cones are closed, convex, and $\Lambda \cap (-\Lambda )=\emptyset $ . For any probability measure $\eta $ and Hölder potential $\psi $ , let $\langle \psi ,\eta \rangle \kern1.2pt{=}\kern0.2pt\int \psi\, d\eta $ . Given a closed cone $\Lambda {\kern-1pt}\subset{\kern-1pt} {C^\alpha (X{\kern-1pt}\times{\kern-1pt} Y)}$ , we can define the dual cone $\Lambda ^*{\kern-1pt}={\kern-1pt}\{\eta {\kern-1pt}\in{\kern-1pt} ({C^\alpha (X{\kern-1pt}\times{\kern-1pt} Y)})^*\colon \langle \psi ,\eta \rangle \geq 0 \text {for\ all\ } \psi {\kern-1pt}\in{\kern-1pt} \Lambda \}$ . For more on cones, see §4 in [Reference Naud7] or the appendices in [Reference Piraino8].

Define a partial ordering $\preceq $ on ${C^\alpha (X\times Y)}$ by saying that $\phi \preceq \psi $ if and only if $\psi -\phi \in \Lambda \cup \{0\}$ for any $\phi ,\psi \in {C^\alpha (X\times Y)}$ . Let

$$ \begin{align*}A=A(\phi,\psi)=\sup\{t>0\colon t\phi\preceq\psi\}\quad \text{and}\quad B=B(\phi,\psi)=\inf\{t>0\colon \psi\preceq t\phi\}.\end{align*} $$

The Hilbert projective metric with respect to a closed cone $\Lambda $ is defined as

$$ \begin{align*}\Theta(\phi,\psi)=\log\frac{B}{A}.\end{align*} $$

The following lemma is useful when calculating distances in the Hilbert metric. For a proof, see §4 in [Reference Naud7].

Lemma 3.1. Let $\Lambda $ be a closed cone and $\Lambda ^*$ its dual. For any $\phi ,\psi \in \Lambda $ ,

$$ \begin{align*}\Theta(\phi,\psi)=\log\!\bigg(\!\!\sup\!\bigg\{\dfrac{\langle\phi,\sigma\rangle\langle\psi,\eta\rangle}{\langle\psi,\sigma\rangle\langle\phi,\eta\rangle}\colon \sigma,\eta\in \Lambda^*\ \text{and\ } \langle\psi,\sigma\rangle\langle\phi,\eta\rangle\neq 0\bigg\}\!\bigg).\end{align*} $$

The main idea of the proof of Theorem A is to find a cone on which the fiberwise transfer operator is a contraction. To accomplish this, we will need the Birkhoff contraction theorem.

Theorem 3.2. (Birkhoff [Reference Birkhoff1])

Let $\Lambda _1, \Lambda _2$ be closed cones and $\mathcal {L}\colon \Lambda _1\to \Lambda _2$ a linear map such that $\mathcal {L} \Lambda _1\subset \Lambda _2$ . Then for all $\phi ,\psi \in \Lambda _1$ .

$$ \begin{align*}\Theta_{\Lambda_2}(\mathcal{L}\phi,\mathcal{L}\psi)\leq \tanh\bigg(\frac{\operatorname{\mathrm{diam}}_{\Lambda_2}(\mathcal{L} \Lambda_1)}{4}\bigg)\Theta_{\Lambda_1}(\phi,\psi)\end{align*} $$

where $\operatorname {\mathrm {diam}}_{\Lambda _2}(\mathcal {L} \Lambda _1)=\sup \{\Theta _{\Lambda _2}(\mathcal {L}\phi ,\mathcal {L}\psi )\colon \phi ,\psi \in \Lambda _1\}$ and $\tanh \infty =1$ .

3.2 Existence of $\Phi $

We will use cones of the form

$$ \begin{align*}\Lambda_K={\Lambda_K(\alpha)}=\{\psi\in {C^\alpha(X\times Y)}\colon\psi>0\ \text{and\ } |\psi|_\alpha\leq K\inf\psi\}\cup\{0\}.\end{align*} $$

It can be shown that $\Lambda _K$ is a closed cone in ${C^\alpha (X\times Y)}$ . For these cones, we get an alternate way of calculating distances in the Hilbert metric.

Lemma 3.3. For any $\phi ,\psi \in \Lambda _K$ ,

$$ \begin{align*}A(\phi,\psi)=\inf_{z_1,z_2,z_3\in X\times Y}\dfrac{Kd(z_1,z_2)^\alpha\psi(z_3)-(\psi(z_1)-\psi(z_2))}{Kd(z_1,z_2)^\alpha\phi(z_3)-(\phi(z_1)-\phi(z_2))}\end{align*} $$

and

$$ \begin{align*}B(\phi,\psi)=\sup_{z_1,z_2,z_3\in X\times Y}\dfrac{Kd(z_1,z_2)^\alpha\psi(z_3)-(\psi(z_1)-\psi(z_2))}{Kd(z_1,z_2)^\alpha\phi(z_3)-(\phi(z_1)-\phi(z_2))}.\end{align*} $$

Denote by

$$ \begin{align*}\Lambda_K^x=\{\psi\in C^\alpha(X\times Y)\colon\psi_x(\cdot)>0\ \text{and\ } |\psi|_\alpha\leq K\inf\psi\}\cup\{0\}\end{align*} $$

the cross-section of $\Lambda _K$ that lives on $Y_x$ .

Let

$$ \begin{align*}s:=e^{\varepsilon_\varphi}\cdot\bigg(\dfrac{(d-q)\gamma^{-\alpha}+qL^\alpha}{d}\bigg)<1\end{align*} $$

as in inequality (3). We assume that $\varepsilon _\varphi>0$ is small enough that

(4) $$ \begin{align} \zeta:=s+2s\varepsilon_\varphi\operatorname{\mathrm{diam}}(Y)^\alpha <1. \end{align} $$

Then we have the following lemma based on similar arguments from Castro and Varandas [Reference Castro and Varandas2] and Stadlbauer, Suzuki and Varandas [Reference Stadlbauer, Suzuki and Varandas11].

Proof. Fix $x\in X$ and $K>0$ . Denote by $\{y_k\}$ and $\{y_k'\}$ the preimages of y and $y'$ in $Y_x$ , respectively, as given by Lemma 2.3. Now fix $\psi \in \Lambda _K$ . Since $\inf \mathcal {L}_x\psi \geq de^{\inf \varphi }\inf \psi $ and

$$ \begin{align*}&\mathcal{L}_{x}\psi_{fx}(y)-\mathcal{L}_{x}\psi_{fx}(y')\\ &\quad= \sum_{k=1}^d(e^{{\varphi}(x,y_k)}(\psi(x,y_k)-\psi(x,y_k'))+(e^{{\varphi}(x,{y_k})}-e^{{\varphi}(x,y_k')})\psi(x,y_k')),\end{align*} $$

we have

$$ \begin{align*} \dfrac{|\mathcal{L}_x\psi(fx,y)-\mathcal{L}_x\psi(fx,y')|}{\inf\mathcal{L}_x\psi}&\leq d^{-1}\sum_{k=1}^d e^{\varphi(x,y_k)-\inf\varphi}|\psi(x,y_k)-\psi(x,y_k')|(\inf\psi)^{-1}\\&\quad+d^{-1}\sum_{k=1}^d(\sup\psi/\inf\psi)e^{-\inf\varphi}|e^{\varphi(x,y_k)}-e^{\varphi(x,y_k')}|\\&=:I_1+I_2. \end{align*} $$

Note that $|\psi (x,{\kern-1pt}y_k){\kern-1pt}-{\kern-1pt}\psi (x,{\kern-1pt}y_k')|{\kern-1pt}\leq{\kern-1pt} |\psi |_\alpha d(y_k,{\kern-1pt}y^{\prime }_k)^\alpha {\kern-1pt}\leq{\kern-1pt} K\inf \psi\, d(y_k,{\kern-1pt}y^{\prime }_k)^\alpha $ . By Lemma 2.3, $d(y_k,y_k')\leq Ld(y,y')$ for any $1\leq k\leq q $ and $d(y_k,y_k')\leq \gamma ^{-1}d(y,y')$ for $q<k\leq d$ , so

$$ \begin{align*} I_1&\leq d^{-1}\sum_{k=1}^d e^{\varphi(x,y_k)-\inf\varphi}Kd(y_k,y^{\prime}_k)^\alpha\\ &\leq d^{-1}e^{\varepsilon_\varphi}K\sum_{k=1}^d d(y_k,y^{\prime}_k)^\alpha\\ &\leq K e^{\varepsilon_\varphi}d^{-1}(L^\alpha q+(d-q)\gamma^{-\alpha})d(y,y')^\alpha\\ &\leq sKd(y,y')^\alpha \end{align*} $$

where the second inequality holds by (P).

To estimate $I_2$ , note that $|e^{\varphi (x,y_k)}-e^{\varphi (x,y_k')}|\leq |e^{\varphi _x}|_\alpha d(x,y_k),(x,y_k'))^\alpha $ and

$$ \begin{align*}\sup\psi\leq\inf\psi+|\psi|_\alpha\operatorname{\mathrm{diam}}(Y)^\alpha\leq(1+K\operatorname{\mathrm{diam}}(Y)^\alpha)\inf\psi\end{align*} $$

implies that

$$ \begin{align*}\sup\psi/\inf\psi\leq 1+K\operatorname{\mathrm{diam}}(Y)^\alpha\leq 2K\operatorname{\mathrm{diam}}(Y)^\alpha\end{align*} $$

provided that K is sufficiently large. Then (P) implies that

$$ \begin{align*} I_2&\leq 2K\operatorname{\mathrm{diam}}(Y)^\alpha e^{-\inf\varphi}d^{-1}\sum_{k=1}^d|e^{\varphi_x}|_\alpha d(x,y_k),(x,y_k'))^\alpha\\ &\leq 2K\operatorname{\mathrm{diam}}(Y)^\alpha\varepsilon_\varphi d^{-1}\sum_{k=1}^d (L^\alpha q+(d-q)\gamma^{-\alpha})d(y,y')^\alpha\\ &\leq 2K\operatorname{\mathrm{diam}}(Y)^\alpha s\varepsilon_\varphi d(y,y')^\alpha\\ &\leq 2s\varepsilon_\varphi\operatorname{\mathrm{diam}}(Y)^\alpha K d(y,y')^\alpha. \end{align*} $$

Therefore, if we let $\zeta :=s+2s\varepsilon _\varphi \operatorname {\mathrm {diam}}(Y)^\alpha $ , we have that

$$ \begin{align*}|\mathcal{L}_x\psi|_\alpha\leq (s+2s\varepsilon_\varphi\operatorname{\mathrm{diam}}(Y)^\alpha)K\inf\mathcal{L}_x\psi\leq \zeta K\inf\mathcal{L}_x\psi\end{align*} $$

so $\mathcal {L}_x\psi \in \Lambda ^{fx}_{\zeta K}$ .

Note that $\sup \mathcal {L}_x\psi \leq (1+\zeta K(\operatorname {\mathrm {diam}} Y)^\alpha )\inf \mathcal {L}_x\psi $ . Let $y_1,y_2,y_3\in Y$ . Then since $|\mathcal {L}_x\psi |_\alpha \leq \zeta K\inf \mathcal {L}_x\psi $ , we have

$$ \begin{align*} &\dfrac{Kd(y_1,y_2)^\alpha\mathcal{L}_x\psi(y_3)-(\mathcal{L}_x\psi(y_1)-\mathcal{L}_x\psi(y_2))}{Kd(y_1,y_2)^\alpha\mathcal{L}_x\phi(y_3)-(\mathcal{L}_x\phi(y_1)-\mathcal{L}_x\phi(y_2))}\\&\quad\leq\dfrac{(K\sup\mathcal{L}_x\psi+\zeta K\inf\mathcal{L}_x\psi)d(y_1,y_2)^\alpha}{(K\inf\mathcal{L}_x\phi-\zeta K\inf\mathcal{L}_x\phi)d(y_1,y_2)^\alpha}. \end{align*} $$

Thus, $B(\mathcal {L}_x\psi ,\mathcal {L}_x\phi )\leq ({K\sup \mathcal {L}_x\psi +\zeta K\inf \mathcal {L}_x\psi })/({K\inf \mathcal {L}_x\phi -\zeta K\inf \mathcal {L}_x\phi })$ . A similar calculation gives a lower bound on $A(\mathcal {L}_x\psi ,\mathcal {L}_x\phi )$ . So by Lemma 3.3, we have

$$ \begin{align*} \Theta(\mathcal{L}_x\psi,\mathcal{L}_x\phi) &\leq \log\!\bigg(\dfrac{K\sup\mathcal{L}_x\phi+\zeta K\inf\mathcal{L}_x\phi}{K\inf\mathcal{L}_x\phi-\zeta K\inf\mathcal{L}_x\phi}\cdot\dfrac{K\sup\mathcal{L}_x\psi+\zeta K\inf\mathcal{L}_x\psi}{K\inf\mathcal{L}_x\psi-\zeta K\inf\mathcal{L}_x\psi}\bigg)\\ &\leq \log\!\bigg(\dfrac{K(1+\zeta K\operatorname{\mathrm{diam}}(Y)^\alpha)(1+\zeta)\inf\mathcal{L}_x\phi}{K(1-\zeta)\inf\mathcal{L}_x\phi}\bigg)\\&\quad+\log\!\bigg(\dfrac{K(1+\zeta K\operatorname{\mathrm{diam}}(Y)^\alpha)(1+\zeta)\inf\mathcal{L}_x\psi}{K(1-\zeta)\inf\mathcal{L}_x\psi}\bigg)\\ &\leq 2\log\!\bigg(\dfrac{1+\zeta}{1-\zeta}\bigg)+2\log(1+\zeta K\operatorname{\mathrm{diam}}(Y)^\alpha)<\infty. \end{align*} $$

This proves the existence of M.

Theorem 3.5. Let There exist $0<\tau <1$ and $C_1>0$ such that for all , and any probability measures $\sigma _n$ on $Y_{f^nx}$ and $\sigma _m$ on $Y_{f^mx}$ , we have

$$ \begin{align*}|\Phi_n^{\sigma_n}(x)-\Phi_m^{\sigma_m}(x)|\leq C_1\ \tau^k.\end{align*} $$

Thus, $\Phi (x)=\lim _{n\to \infty }\Phi _n^{\sigma _n}(x)$ exists and $|\Phi _n^{\sigma _n}(x)-\Phi (x)|\leq C_1\tau ^n$ .

Proof. Fix $x\in X$ . Suppose $n,m\geq k \geq 1$ . Then

where $\sigma _{fx,n}=(\mathcal {L}_{f^{k+1}x})^*\cdots (\mathcal {L}_{f^nx})^*\sigma _n$ . By Lemma 3.1, we see that

Clearly,

for any $K>0$ . Then

by Lemma 3.4. Fix K large and M as in Lemma 3.4. Set $\tau =\tanh {(M/4)}$ . By Theorem 3.2, we have

Let $C_1=M/\tau $ . Hence, the sequence $\{\Phi _n\}_{n\geq 0}$ is Cauchy and the limit exists at every $x\in X$ .

This proves the existence of $\Phi $ .

3.3 Fiber measures

To completely understand the equilibrium state $\mu $ on $(X\times Y,F)$ , we need to understand how it gives weight to the fibers $\{Y_x\}_{x\in X}$ . The first step is the following non-stationary RPF theorem adapted from [Reference Climenhaga and Hemenway3], whose proof we include here for completeness (see Hafouta [Reference Hafouta5] for a similar result when the base is invertible).

Theorem 3.6 is a consequence of the following two propositions.

Proposition 3.7. Let $C_1>0$ and $0<\tau <1$ be as in Theorem 3.5. Given any $x\in X$ , , and $\sigma _k\in \mathcal {M}(Y_{f^kx})$ , define $\nu _{x,k}\in \mathcal {M}(Y_{x})$ by . If $m,n\geq k$ and $\psi \in \Lambda _K$ , we have

$$ \begin{align*}\bigg|\kern-2pt\int\psi\, d\nu_{x,n}-\int\psi\, d\nu_{x,m}\bigg|\leq C_1\|\psi\|\tau^k.\end{align*} $$

In particular, $\langle \psi ,\nu _x\rangle :=\lim _{n\to \infty }\langle \psi ,\nu _{x,n}\rangle $ exists and defines a probability measure $\nu _x$ on $Y_x$ with

$$ \begin{align*}\bigg|\kern-2pt\int\psi\, d\nu_{x,n}-\int\psi\, d\nu_x\bigg|\leq C_1\|\psi\|\tau^n.\end{align*} $$

Proof. Let

and

. Note that

. So

A similar computation shows that $b_k\leq \langle \psi ,\nu _{x,n}\rangle $ . Then $b_k\leq \langle \psi ,\nu _{x,n}\rangle \leq c_k$ for all $n\geq k$ . Therefore, $|\langle \psi ,\nu _{x,n}\rangle -\langle \psi ,\nu _{x,m}\rangle |\leq c_k-b_k$ for all $n,m\geq k$ . Lemma 3.4 implies that

. So $1\leq {c_k}/{b_k}\leq e^{M\tau ^{k-1}}$ . Thus, $b_k\leq c_k\leq b_ke^{M\tau ^{k-1}}$ which implies that $c_k-b_k\leq b_k(e^{M\tau ^{k-1}}-1)$ . Moreover, for all $y\in Y$ , we have

So $b_k\leq \|\psi \|$ . Hence,

$$ \begin{align*}|\langle\psi,\nu_{x,n}\rangle-\langle\psi,\nu_{x,m}\rangle|\leq c_k-b_k\leq\|\psi\|(e^{M\tau^{k-1}}-1).\end{align*} $$

Thus, $\{\nu _{x,n}\}$ is a Cauchy sequence. Then there is a constant $C_1>0$ such that

$$ \begin{align*}|\langle\psi,\nu_{x,n}\rangle-\langle\psi,\nu_{x}\rangle|\leq C_1\|\psi\|\tau^n\end{align*} $$

for all $n\geq 0$ .

Proof. For all $\psi \in C(Y_x)$ , we have

Observe that

This completes the proof of Theorem 3.7.

As in Denker and Gordin [Reference Denker and Gordin4], we call a system of $\{\nu _x\colon x\in X\}$ of conditional probabilities for $(X\times Y,F)$ a family of Gibbs measures for a continuous function if there exists a positive measurable function with the following property: for all $x\in X$ , the Jacobian of $\mu _x$ with respect to the map F is given by

$$ \begin{align*}\dfrac{d\nu_{f(x)}\circ g_x}{d\nu_x}=A(x)e^{-\varphi_x}.\end{align*} $$

Proof. Choose $A\in Y_x$ such that $g_x|_A$ is invertible. This implies

Therefore,

. Since this holds for $x\in X$ , $\{\nu _x\colon x\in X\}$ forms a family of Gibbs measures.

3.4 Regularity of $\Phi $

Now we will show that $\Phi $ is Hölder continuous. A direct consequence of Lemma 3.4 is the following lemma, which we will need to prove the Hölder continuity of $\Phi $ . For convenience, we write

Proof. Let $\overline {\varphi }_x=\varphi _x-\log \unicode{x3bb} _x$ and write

. Theorem 3.7 gives $(\overline {\mathcal {L}}_{x})^*\nu _{fx}=\nu _x$ for all $x\in X$ . Inductively, we get that $(\overline {\mathcal {L}}^{n}_x)^*\nu _{f^{n}x}=\nu _{x}$ . Then for any $k,\ell $ ,

Let $\Lambda ^+$ be the cone of strictly positive continuous functions on $X\times Y$ . Since $\Lambda _K\subset \Lambda ^+$ , the projective metrics of the two cones satisfy $\Theta ^+(\phi ,\psi )\leq \Theta (\phi ,\psi )$ . Write

. Then

, so $1\leq {\sup \psi _k}/{\inf \psi _k}\leq e^M$ . We know that

. This implies that $e^{-M}\leq \psi _k\leq e^M$ for all

. Thus,

Let and $x,x'\in X$ , and $y\in Y$ . Let $\mathcal {W}_n=\{1,\ldots ,d\}^n$ . By Lemma 2.3, we can write

$$ \begin{align*}g^{-1}_{f^{n-1}x}(y)=\{y_1,\ldots,y_d\}\quad \text{and}\quad g^{-1}_{f^{n-1}x'}(y)=\{y^{\prime}_1,\ldots,y^{\prime}_d\}\end{align*} $$

such that

$$ \begin{align*}d((f^{n-1}x,y_k),(f^{n-1}x',y^{\prime}_k))\leq L_kd_X(f^nx,f^nx')\end{align*} $$

where $L_k=L$ if $1\leq k\leq q$ and $L_k=\gamma ^{-1}$ if $q< k\leq d$ . Continuing in this way, we get that

$$ \begin{align*}g^{-n}_{x}(y)=\{y_w\in Y_x\colon w\in\mathcal{W}_n\}\quad \text{and}\quad g^{-n}_{x'}(y)=\{y^{\prime}_w\in Y_{x^{\prime}_w}\colon w\in \mathcal{W}_n\}\end{align*} $$

such that for all $0\leq k\leq n$ ,

$$ \begin{align*}d(F^{k}(x,y_w),F^{k}(x',y^{\prime}_w))\leq L_{w_{k+1}}\cdots L_{w_{n}}d_X(f^nx,f^nx').\end{align*} $$

Let and $0<\iota <1$ . A pair of inverse branches for F of length n starting from $(f^nx,y)$ and $(f^nx',y)$ and labeled by $w\in \mathcal {W}_n$ is good if for all such that $jm\leq n$ , we have

$$ \begin{align*}\#\{n-jm< i\leq n\colon w_i\leq q\}\leq\iota jm.\end{align*} $$

This means that the last $jm$ iterates of an orbit segment of length n will be in the contraction region at most $\iota jm$ times. We will denote the collection of words corresponding to good trajectories by

$$ \begin{align*}\mathcal{W}^{\mathcal{G}}_n=\mathcal{W}^{\mathcal{G}}_n(m)=\{w\in\mathcal{W}_n\colon\ \text{for all}\ j\leq n/m, \#\{n-jm< i\leq n\colon w_i\leq q\}\leq\iota jm\}\end{align*} $$

and the collection of words for bad trajectories by

$$ \begin{align*}\mathcal{W}^{\mathcal{B}}_n&=\mathcal{W}^{\mathcal{B}}_n(m)\\&=\{w\in\mathcal{W}_n\colon \text{there exists}\ j\leq n/m \ni \#\{n-jm< i\leq n\colon w_i\leq q\}\geq\iota jm\}.\end{align*} $$

The following lemma due to Varandas and Viana gives us a way to count the number of words that code bad trajectories of a given length. Let $I(\iota ,n)=\{w\in \mathcal {W}_n\colon \#\{1\leq k\leq n\colon w_i\leq q\}\geq \iota n\}$ .

Lemma 3.11. Given $\varepsilon>0$ , there exists a $\iota _0\in (0,1)$ such that

$$ \begin{align*} \limsup_{n\to\infty}\frac{1}{n}\log\#I(\iota,n) <\log q+\varepsilon \end{align*} $$

for all $\iota \in (\iota _0,1)$ . Therefore, there exists a $C>0$ such that $\#I(\iota ,n) \leq Cq^ne^{\varepsilon n}$ for all n.

Since it is assumed that $\varepsilon _\varphi <\log d-\log q$ , it follows that ${qe^{\varepsilon _\varphi }}/{d}<1$ . Choose $\varepsilon>0$ such that

(5) $$ \begin{align}\theta:=\frac{qe^\varepsilon e^{\varepsilon_\varphi}}{d}<1.\end{align} $$

Let $\iota =\iota (\varepsilon ,d,q)\in (0,1)$ be given by Lemma 3.11. We now further our assumption on the constant L by assuming that it is close enough to $1$ so that there is a $c>0$ satisfying

(6) $$ \begin{align} 0<\gamma^{-(1-\iota)}L^\iota<e^{-2c}<1. \end{align} $$

Lemma 3.12. There is a $Q>0$ such that for all , if $(x,\overline {y})$ and $(x',\overline {y}')$ are preimages coded by a word in $\mathcal {W}_n^{\mathcal {G}}{(m)}$ , then

$$ \begin{align*}d(F^k(x,\overline{y}),F^k(x',\overline{y}'))\leq Q^me^{-2c(n-k)}d(f^nx,f^nx')\end{align*} $$

for all $0\leq k<n$ .

Proof. Fix . Write $n-k=jm+i$ for $0\leq i<m$ . Since our preimage branches are assumed to be good, we get

$$ \begin{align*} d(F^k(x,\overline{y}),F^k(x',\overline{y}'))&\leq L_{w_{k+1}}\cdots L_{w_{n}}d(F^{k+i}(x,\overline{y}),F^{k+i}(x',\overline{y}'))\\ &\leq L_{w_{k+1}}\cdots L_{w_{k+i}}(L^{\iota jm}\gamma^{-(1-\iota ){jm}})d(f^nx,f^nx').\end{align*} $$

Recall from (6) that we can choose $c>0$ so that $0<\gamma ^{-(1-\iota )}L^\iota <e^{-2c}<1$ . Thus,

$$ \begin{align*} d(F^k(x,\overline{y}),F^k(x',\overline{y}'))&\leq L^m e^{-2cjm}d(f^nx,f^nx')\\ &\leq (Le^{2c})^{m}e^{-2c(n-k)}d(f^nx,f^nx').\\[-3.1pc] \end{align*} $$

In what follows it will be convenient to write $a\colon \mathcal {W}_n\to Y$ so that $a(w)=y_w$ and $a'\colon \mathcal {W}_n\to Y$ so that $a'(w)=y^{\prime }_w$ . Lemma 2.3 gives us bijections $b_j\colon \mathcal {W}_{jm}\to g^{-jm}_{f^{n-jm}x}(y)$ such that $b_j(v)=g_{x}^{n-jm}(a(uv))$ and $c_v\colon \mathcal {W}_{n-jm}\to g^{-(n-jm)}_{x}(b_j(v))$ such that $c_v(u)=a(uv)$ as well as their associated maps $b_j'$ and $c_v'$ . See Figure 1 for reference.

Figure 1

The maps a, $b_j$ , and $c_v$ on words of corresponding lengths.

Lemma 3.13. Let $\theta $ as in (5) above. There exists $C_2>0$ such that

$$ \begin{align*}\sum_{w\in{\mathcal{W}_n^{\mathcal{B}}(m)}}e^{S_n\varphi_x(a(w))}\leq C_2\theta^m\sum_{w\in{\mathcal{W}_n^{\mathcal{G}}(m)}}e^{S_n\varphi_x(a(w))}\end{align*} $$

for all , $x,x'\in X$ , and $y\in Y$ .

Proof. Let $x\in X$ and $y\in Y$ . For any $w\in \mathcal {W}_n^{\mathcal {B}}$ , there is $1\leq j\leq n/m$ such that $w=uv$ for some $u\in \mathcal {W}_{n-jm}$ and $v\in I(\iota ,jm)$ . Thus,

by Lemma 3.10. Note that for any $j\leq {n}/{m}$ ,

Lemma 3.11 implies that $\#I(\iota ,n)\leq Cq^{n}e^{\varepsilon n}$ for all $n\geq 0$ . Then since $\#\mathcal {W}_n$ is finite,

where the last inequality holds by Lemma 3.11 and (P). Let $\theta ={qe^\varepsilon e^{\varepsilon _\varphi }}/{d}<1$ . Then

$$ \begin{align*}\dfrac{\sum_{w\in{\mathcal{W}_n^{\mathcal{B}}}}e^{S_n\varphi_x(a(w))}}{\sum_{w\in\mathcal{W}_n}e^{S_n\varphi_x(a(w))}}\leq e^{2M}\sum_{j=1}^{\infty}\bigg(\dfrac{qe^{\varepsilon}e^{\varepsilon_\varphi}}{d}\bigg)^{jm}= e^{2M}\bigg(\dfrac{\theta^m}{1-\theta^m}\bigg).\end{align*} $$

But

$$ \begin{align*}\sum_{w\in\mathcal{W}_n}e^{S_n\varphi_x(a(w))}=\sum_{w\in{\mathcal{W}_n^{\mathcal{B}}}}e^{S_n\varphi_x(a(w))}+\sum_{w\in{\mathcal{W}_n^{\mathcal{G}}}}e^{S_n\varphi_x(a(w))}.\end{align*} $$

Choose m such that $1-\theta ^m\geq \tfrac 12$ . Then

$$ \begin{align*}\sum_{w\in{\mathcal{W}_n^{\mathcal{B}}}}e^{S_n\varphi_x(a(w))}\leq 2e^{2M}C\theta^m\bigg(\sum_{w\in{\mathcal{W}_n^{\mathcal{B}}}}e^{S_n\varphi_x(a(w))}+\sum_{w\in{\mathcal{W}_n^{\mathcal{G}}}}e^{S_n\varphi_x(a(w))}\bigg).\end{align*} $$

So if we increase m so that $2e^{2M}C\theta ^m<\tfrac 12$ , then

$$ \begin{align*}\sum_{w\in\mathcal{W}_n^{\mathcal{B}}}e^{S_n\varphi_x(a(w))}\leq 4e^{2M}C\theta^m\sum_{w\in\mathcal{W}_n^{\mathcal{G}}}e^{S_n\varphi_x(a(w))}.\end{align*} $$

This achieves the desired result.

Now we apply the above with various values of m to prove the Hölder continuity of $\Phi $ . First, a bound on $\Phi _n$ .

Lemma 3.14. There exist $C_3>0$ and $\beta>0$ such that

$$ \begin{align*}|\Phi_n(x)-\Phi_n(x')|\leq C_3d(f^nx,f^nx')^{\alpha\beta}\end{align*} $$

for all $x,x'\in X$ and .

Proof. First note that along good orbit pairs, we have by Lemma 3.12 that

$$ \begin{align*} |S_n\varphi(x,\overline{y})-S_n\varphi(x',\overline{y}')|&\leq \sum_{k=0}^{n-1}|\varphi|_\alpha d(F^k(x,\overline{y}),F^k(x',\overline{y}'))^\alpha\nonumber\\ &\leq \sum_{k=0}^{n-1}|\varphi|_\alpha Q^{\alpha m}e^{-2c\alpha (n-k)}d(f^nx,f^nx')^\alpha\nonumber\\ &\leq Q^{\alpha m}d(f^nx,f^nx')^\alpha\cdot\sum_{k=0}^{\infty}|\varphi|_\alpha e^{-2c\alpha (n-k)}.\nonumber \end{align*} $$

Let $V=\sum _{k=0}^{\infty }|\varphi |_\alpha e^{-2c\alpha (n-k)}$ . Then

(7) $$ \begin{align} |S_n\varphi(x,\overline{y})-S_n\varphi(x',\overline{y}')|\leq VQ^{\alpha m}d(f^nx,f^nx')^\alpha. \end{align} $$

For convenience, we write $\Sigma _{\mathcal {G}}=\sum _{w\in \mathcal {W}_n^{\mathcal {G}}}e^{S_n\varphi _x(a(w))}$ and $\Sigma _{\mathcal {B}}=\sum _{w\in \mathcal {W}_n^{\mathcal {B}}}e^{S_n\varphi _x(a(w))}$ as well as $\Sigma _{\mathcal {G}}'$ and $\Sigma _{\mathcal {B}}'$ for the sums of the preimages associated to $a'(w)$ . By Lemma 3.13, we get that

(8)

Note that by (7),

$$ \begin{align*} \dfrac{\Sigma_{\mathcal{G}}}{\Sigma_{\mathcal{G}}'}&=\dfrac{\sum_{w\in\mathcal{W}_n^{\mathcal{G}}}e^{S_n\varphi_x(a(w))}}{\sum_{w'\in\mathcal{W}_n^{\mathcal{G}}}e^{S_n\varphi_{x'}(a'(w))}}\\ &\leq \dfrac{\sum_{w'\in\mathcal{W}_n^{\mathcal{G}}}e^{VQ^{\alpha m}d(f^nx,f^nx')^\alpha}e^{S_n\varphi_{x'}(a'(w))}}{\sum_{w'\in\mathcal{W}_n^{\mathcal{G}}}e^{S_n\varphi_{x'}(a'(w))}}\\ &\leq e^{VQ^{\alpha m}d(f^nx,f^nx')^\alpha}. \end{align*} $$

Let $\Phi _n$ be as in Theorem 3.5 for the delta measure on Y, $\delta _y\ (y\in Y)$ . Then

where the second inequality holds due to (8).

Let $\rho _1={\theta }/{Q^\alpha }$ and note that $\rho _1<1$ . Then there is a such that

(9) $$ \begin{align} \rho_1^{k+1}\leq d(f^nx,f^nx')^\alpha\leq\rho_1^k. \end{align} $$

Now set $m=k$ . Let $\beta =({\log \theta }/{\log \rho _1})$ and note that

$$ \begin{align*}\theta^m=e^{m\log\theta}=e^{\beta m\log\rho_1}=\rho_1^{\beta m}\leq\rho_1^{-\beta}d(f^nx,f^nx')^{\alpha\beta}.\end{align*} $$

Thus, $Q^{\alpha m}d(f^nx,{\kern-1pt} f^nx')^\alpha {\kern-1pt}\leq{\kern-1pt} \theta ^m{\kern-1pt}\leq{\kern-1pt} \rho _1^{-\beta }d(f^nx,{\kern-1pt}f^nx')^{\alpha \beta }$ . Hence, letting $C_3{\kern-1pt}={\kern-1pt}2(V{\kern-1pt}+{\kern-1pt}C)\rho _1^{-\beta }$ yields

$$ \begin{align*}|\Phi_n(x)-\Phi_n(x')|\leq C_3d(f^nx,f^nx')^{\alpha\beta}.\\[-34pt]\end{align*} $$

Proof. Let $C=\max \{C_1,C_3\}$ . For any $n\geq 0$ ,

$$ \begin{align*} |\Phi(x)-\Phi(x')|&\leq |\Phi(x)-\Phi_n(x)|+|\Phi_n(x)-\Phi_n(x')|+|\Phi_n(x')-\Phi(x')|\\ &\leq 2C\tau^n+Cd(f^nx,f^nx')^{\alpha\beta}\\ &\leq 2C\tau^n+C\Gamma^{\alpha\beta n}d(x,x')^{\alpha\beta}. \end{align*} $$

where the second inequality follows from Theorem 3.5 and Lemma 3.14 and $\Gamma $ is the inherited Lipschitz constant for f.

Similarly to the argument in the proof of Lemma 3.14, we need to adjust the Hölder exponent to establish our bound. Let $\rho _2={\tau }/{\Gamma ^{\alpha \beta }}$ . Then there is a k such that $\rho _2^{k+1}\leq d(x,x')^{\alpha \beta }\leq \rho _2^k$ . Let $n=k$ and $\eta ={\log \tau }/{\log \rho _2}$ . Then

$$ \begin{align*}\tau^n=e^{n\log\tau}=e^{\eta n\log\rho_2}=\rho_2^{\eta n}\leq\rho_2^{-\eta}d(x,x')^{\alpha\beta\eta}.\end{align*} $$

So $\Gamma ^{\alpha \beta }d(x,x')^{\alpha \beta }\leq \tau ^n\leq \rho _2^{-\eta }d(x,x')^{\alpha \beta \eta }$ . Therefore,

$$ \begin{align*}|\Phi(x)-\Phi(x')|\leq 2C\rho_2^{-\eta}d(x,x')^{\alpha\beta\eta}.\\[-34pt]\end{align*} $$