2. Thermodynamic formalism and the CLT

In this section, we collect some basic notions and results in thermodynamic formalism for later use. We start from the pressure. Let $\Lambda $ be some symbolic set with finite symbols, and $\Lambda ^{\mathbb {N}}$ be the shift space equipped with the metric

$$ \begin{align*} d(x,y)=\frac{1}{2^{l(x,y)}} \end{align*} $$

for distinct $x=x_0x_1x_2\ldots , y=y_0y_1y_2\ldots \in \Lambda ^{\mathbb {N}}$ , where

$$ \begin{align*} l(x,y)=\min\{i\in\mathbb{N}: x_i\neq y_i\}. \end{align*} $$

For a continuous potential $\phi : \Lambda ^{\mathbb {N}}\rightarrow \mathbb {R}$ on the compact metric space $\Lambda ^{\mathbb {N}}$ , let

$$ \begin{align*} S_{m,\phi}(x)=\sum_{i=0}^{m-1}\phi\circ\sigma^i(x) \end{align*} $$

for $m\in \mathbb {N}$ , where $\sigma $ is the shift map.

Definition 2.1. The pressure $P(\phi )$ of a continuous potential $\phi $ on $\Lambda ^{\mathbb {N}}$ is defined to be

$$ \begin{align*} P(\phi)=\lim_{m\rightarrow\infty}\frac{1}{m}\log \sum_{\sigma^m(x)=x}e^{S_{m,\phi}(x)}. \end{align*} $$

One can refer to [Reference WaltersWal1, p. 208] for a definition for continuous potentials on general compact metric spaces. It satisfies the well-known variational formula

$$ \begin{align*} P(\phi)=\sup\bigg\{h(\mu)+\int\phi\,d\mu: \mu \text{ is a } \sigma\text{-invariant measure on }\Lambda^{\mathbb{N}}\bigg\}. \end{align*} $$

Let $C^0(\Lambda ^{\mathbb {N}})$ be the collection of all the continuous potentials on $\Lambda ^{\mathbb {N}}$ . Two potentials $\psi , \phi \in C^0(\Lambda ^{\mathbb {N}})$ are said to be cohomologous [Reference WaltersWal2] in the case where there exists a continuous map $\zeta : \Lambda ^{\mathbb {N}}\rightarrow \mathbb {R}$ such that

$$ \begin{align*} \psi(x)-\phi(x)=\zeta(x)-\zeta\circ\sigma(x). \end{align*} $$

We write $\psi \sim \phi $ to denote the equivalence relationship between two potentials cohomologous to each other. The maps in

$$ \begin{align*} \{\zeta(x)-\zeta\circ\sigma(x): \zeta\in C^0(\Lambda^{\mathbb{N}})\} \end{align*} $$

are called coboundaries. The importance of the cohomologous relationship is revealed in the following result.

Another important tool in thermodynamic formalism is the Ruelle operator.

Definition 2.3. For a continuous potential $\psi : \Lambda ^{\mathbb {N}}\rightarrow \mathbb {R}$ , define the Ruelle operator $\mathcal {L}_\psi $ acting on $C^0(\Lambda ^{\mathbb {N}})$ as

$$ \begin{align*} (\mathcal{L}_\psi f)(x)=\sum_{y:\sigma(y)=x}e^{\psi(y)}f(y) \end{align*} $$

for $f\in C^0(\Lambda ^{\mathbb {N}})$ .

One can see easily that its compositions satisfy

(2.1) $$ \begin{align} (\mathcal{L}^m_\psi f)(x)=\sum_{y:\sigma^m(y)=x}e^{S_{m,\psi}(y)}f(y) \end{align} $$

for any $m\in \mathbb {N}$ . For $\psi \in C^{h}(\Lambda ^{\mathbb {N}})$ , it admits a simple maximum isolated eigenvalue ${\unicode{x3bb} =e^{P(\psi )}}$ such that

(2.2) $$ \begin{align} (\mathcal{L}_\psi w_\psi)(x)=e^{P(\psi)}w_\psi(x) \end{align} $$

for some eigenfunction $w_\psi (x)\in C^{h}(\Lambda ^{\mathbb {N}})$ , refer to [Reference RuelleRue1]. It then follows that

(2.3) $$ \begin{align} (\mathcal{L}^m_\psi w_\psi)(x)=e^{mP(\psi)}w_\psi(x) \end{align} $$

for $w_\psi (x)\in C^{h}(\Lambda ^{\mathbb {N}})$ . A potential $\psi $ is said to be normalised if

$$ \begin{align*} P(\psi)=0\quad \text{and}\quad w_\psi=1_{\Lambda^{\mathbb{N}}}, \end{align*} $$

where $1_{\Lambda ^{\mathbb {N}}}$ is the identity map on $\Lambda ^{\mathbb {N}}$ . In the case of $\psi $ being not normalised, we call

$$ \begin{align*} \bar{\psi}=\psi+\log w_\psi-\log w_\psi\circ\sigma-P(\psi) \end{align*} $$

the normalisation of $\psi $ . It is easy to check that $\bar {\psi }$ is a normalised potential. Moreover, $\bar {\psi }$ and $\psi $ share the same equilibrium state.

The unique equilibrium measure for a Hölder potential $\psi $ is denoted by $\mu _\psi $ in the following. Now we turn to the CLT for the random process $\{\phi \circ \sigma ^j(x)\}_{j=0}^\infty $ with the equilibrium measure $\mu _\psi $ defined by some Hölder potential $\psi $ , while $\phi $ is also assumed to be Hölder. It deals with the asymptotic behaviour of the distribution of ${S_{m,\phi }}/{\sqrt {m}}$ with respect to $\mu _\psi $ as $m\rightarrow \infty $ . The Ruelle operator comes in here, see [Reference Coelho and ParryCP, Reference LalleyLal, Reference Rousseau-EgèleRou]. Let

$$ \begin{align*} G_m(y)=\mu_\psi \bigg\{x\in \Lambda^{\mathbb{N}}: \frac{S_{m,\phi}(x)}{\sqrt{m}}<y\bigg\} \end{align*} $$

for $y\in \mathbb {R}$ . For $a,b\in \mathbb {R}$ and $b>0$ , let $N_{a,b}(y)$ be the normal distribution with expectation a and standard deviation $\sqrt {b}$ on $\mathbb {R}$ , that is,

$$ \begin{align*} \frac{dN_{a,b}(y)}{dy}=\frac{1}{\sqrt{2\pi b}}e^{-(y-a)^2/2b} \end{align*} $$

for $y\in \mathbb {R}$ . For Hölder potentials $\psi , \phi $ on a shift space, since the pressure $P(\psi +t\phi )$ is analytic in a small neighbourhood around $0$ , denote by

$$ \begin{align*} \Delta_m=P^{(m)}(\psi+t\phi)|_{t=0} \end{align*} $$

for $m\in \mathbb {N}$ for convenience, while the readers can understand its dependence on $\psi , \phi $ easily from the contexts in the following. Let

$$ \begin{align*} P(\psi+t\phi)=\sum_{m=0}^\infty\frac{\Delta_m}{m!}t^m=\sum_{m=0}^3\frac{\Delta_m}{m!}t^m+t^4\kappa(t), \end{align*} $$

where $\kappa (t)=\sum _{m=0}^\infty ({\Delta _{m+4}}/{(m+4)!})t^m$ .

We now come to one of the key ingredients in the proofs of Theorems 1.1 and 1.2.

Central Limit Theorem. Let $\psi , \phi $ be Hölder potentials on a shift space with $\phi $ being not cohomologous to a constant. If $\int \phi \,d\mu _\psi =0$ , we have

$$ \begin{align*} G_m(y)=N_{0,\Delta_2}(y)+O(1/\sqrt{m}), \end{align*} $$

where

(2.4) $$ \begin{align} O(1/\sqrt{m})\leq\frac{9|\Delta_3|+2|\Delta_4|}{\sqrt{2\pi^3 m}(\Delta_2)^{3/2}}. \end{align} $$

The convergence is uniform with respect to y. In the case of $\phi $ being generic, we have

(2.5) $$ \begin{align} G_m(y)=N_{0,\Delta_2}(y)+H_m(y)+o(1/\sqrt{m}), \end{align} $$

where $H_m(y)= ({\Delta _3}/{6\sqrt {m}}) (1-({y^2}/{\Delta _2}))e^{-({y^2}/{2\Delta _2})}$ .

The bounds of error terms in equations (2.4) and (2.5) will be used in the proofs of Theorems 1.1 and 1.2, respectively. In the following, we will justify equation (2.4), while equation (2.5) follows from existing results.

This fits into special cases of the Berry–Esseen theorem [Reference FellerFel]. A significant point in the version here comparing with [Reference Coelho and ParryCP, Theorems 2, 3] or [Reference Parry and PollicottPP, Theorem 4.13] is the explicit bound on the tail term $O(1/\sqrt {m})$ in equation (2.4). In the following, we justify this explicit bound. To do this, let

$$ \begin{align*} \chi_m(z)=\displaystyle\int e^{iz({S_{m,\phi}}/{\sqrt{m}})}\,d\mu_\psi \end{align*} $$

be the Fourier transformation of $G_m(y)$ . Note that the Fourier transformation of $N_{0,\Delta _2}(y)$ is $e^{-({z^2\Delta _2}/{2})}$ .

Lemma 2.4. Let $\psi , \phi $ be Hölder potentials on a shift space with $\phi $ being not cohomologous to a constant. For $\epsilon>0$ small enough, we have

(2.6) $$ \begin{align} \frac{1}{2\pi}\int_{0}^{\epsilon\sqrt{m}}\frac{1}{z}|\chi_m(z)-e^{-({z^2\Delta_2}/{2})}|\,dz\leq \frac{\sqrt{2} |\Delta_3|}{12\sqrt{\pi m}(\Delta_2)^{3/2}} \end{align} $$

for any $m\in \mathbb {N}$ large enough.

Proof. According to [Reference Parry and PollicottPP, equation (4.6)], we have

$$ \begin{align*} \displaystyle\int_{0}^{\epsilon\sqrt{m}}\frac{1}{z} \bigg|\chi_m(z)-e^{-({z^2\Delta_2}/{2})}+\frac{iz^3\Delta_3}{6\sqrt{m}}e^{- ({z^2\Delta_2}/{2})}\bigg|\,dz=O(1/m) \end{align*} $$

for $\epsilon>0$ small enough. So

(2.7) $$ \begin{align} \frac{1}{2\pi}\displaystyle\int_{0}^{\epsilon\sqrt{m}}\frac{1}{z} \bigg|\chi_m(z)-e^{- ({z^2\Delta_2}/{2})}\bigg|\,dz\leq O(1/m)+\frac{|\Delta_3|}{12\pi\sqrt{m}}\displaystyle\int_{0}^{\epsilon\sqrt{m}}z^2e^{-({z^2\Delta_2}/{2})}\,dz. \end{align} $$

By

$$ \begin{align*} \displaystyle\int_{-\infty}^{\infty}z^2e^{-({z^2\Delta_2}/{2})}\,dz=\frac{\sqrt{2\pi}}{(\Delta_2)^{3/2}}, \end{align*} $$

we obtain equation (2.6) from equation (2.7).

Equipped with Lemma 2.4, we can justify the explicit bound on the tail term in the CLT in equation (2.4).

Proof of the tail term in CLT.

Proof. Without loss of generality, suppose $\psi $ is normalised and $\int \phi \,d\mu _\psi =0$ . It suffices for us to justify equation (2.4) by [Reference Coelho and ParryCP, Theorems 2, 3]. Similar to the proof of [Reference Coelho and ParryCP, Theorem 2], apply [Reference FellerFel, Lemma 2] with the cumulative functions $G_m(y)$ and $N_{0,\Delta _2}(y)$ , in our case, one gets (cf. [Reference Coelho and ParryCP, (20)])

(2.8) $$ \begin{align} |G_m(y)-N_{0,\Delta_2}(y)|\leq\frac{1}{2\pi}\int_{0}^{\epsilon\sqrt{m}} \frac{1}{z}\bigg|\chi_m(z)-e^{-({z^2\Delta_2}/{2})}\bigg|\,dz + \frac{24}{\epsilon\sqrt{2m\pi^3\Delta_2}}. \end{align} $$

Now let us take

$$ \begin{align*} \frac{1}{\epsilon}=\frac{2}{\Delta_2}\bigg(\frac{|\Delta_3|}{6}+\frac{|\Delta_4|}{24}+\delta\bigg) \end{align*} $$

for some small $\delta>0$ , such that it satisfies (cf. [Reference Coelho and ParryCP, (10)])

$$ \begin{align*} \frac{1}{\epsilon}>\max\bigg\{\frac{2}{\Delta_2}\bigg(\frac{|\Delta_3|}{6}+t\kappa(t)\bigg), \frac{2}{\Delta_2}\kappa(t)\bigg\} \end{align*} $$

for any $|t|<\epsilon $ in equation (2.8). By equation (2.6), we have

(2.9) $$ \begin{align} & |G_m(y)-N_{0,\Delta_2}(y)|\nonumber\\[1mm]&\quad \leq \frac{\sqrt{2} |\Delta_3|}{12\sqrt{\pi m}(\Delta_2)^{3/2}}+\frac{24}{\sqrt{2m\pi^3\Delta_2}} \frac{2}{\Delta_2}\bigg(\frac{|\Delta_3|}{6}+\frac{|\Delta_4|}{24}+\delta\bigg)\nonumber\\&\quad = \frac{\sqrt{2} |\Delta_3|}{12\sqrt{\pi m}(\Delta_2)^{3/2}}+\frac{8|\Delta_3|}{\sqrt{2\pi^3 m}(\Delta_2)^{3/2}}+\frac{2|\Delta_4|}{\sqrt{2\pi^3 m}(\Delta_2)^{3/2}}+\frac{48\delta}{\sqrt{2\pi^3 m}(\Delta_2)^{3/2}}\\[1mm]&\quad \leq \frac{9|\Delta_3|}{\sqrt{2\pi^3 m}(\Delta_2)^{3/2}}+\frac{2|\Delta_4|}{\sqrt{2\pi^3 m}(\Delta_2)^{3/2}}+\frac{48\delta}{\sqrt{2\pi^3 m}(\Delta_2)^{3/2}}.\nonumber \end{align} $$

Finally, letting $\delta \rightarrow 0$ in equation (2.9), we get equation (2.4).

We will deal with the pressure function $P(\psi +t\phi )$ for $t\geq 0$ and $\psi , \phi \in C^{h}(\Lambda ^{\mathbb {N}})$ for some $0<h\leq 1$ in the following sections. By [Reference RuelleRue2], $P(\psi +t\phi )$ depends analytically on t in the case that $\psi , \phi $ are Hölder. We will often assume that

$$ \begin{align*} \int\phi\,d\mu_\psi=0 \end{align*} $$

in the following when dealing with the higher derivatives of $P(\psi +t\phi )$ because if $\int \phi \,d\mu _\psi =c\neq 0$ , we have

$$ \begin{align*} P(\psi + t(\phi-c)) = P(\psi+t\phi) - ct, \end{align*} $$

then

(2.10) $$ \begin{align} \frac{d^n P(\psi+t(\phi-c))}{dt^n}=\frac{d^n P(\psi+t\phi)}{dt^n} \end{align} $$

for any $n\geq 2$ while $\int (\phi -c)\,d\mu _\psi =0$ . We can also assume that $\psi $ is normalised when dealing with the derivatives of $P(\psi +t\phi )$ . If this is not the case, we can simply change $\psi $ to its normalisation $\bar {\psi }$ while

(2.11) $$ \begin{align} \frac{d^n P(\psi+t\phi)}{dt^n}=\frac{d^n P(\bar{\psi}+t\phi)}{dt^n} \end{align} $$

for $n\geq 1$ because

$$ \begin{align*} P(\bar{\psi}+t\phi)=P(\psi+t\phi)-P(\psi) \end{align*} $$

for any $t\in \mathbb {R}$ .

3. Derivatives of the pressures of Hölder potentials

In this section, we formulate some explicit expressions for the derivatives of the pressure $P(t\phi )=P(t)$ in terms of the derivatives of the eigenfunction of $\mathcal {L}_{t\phi }$ for $\phi \in C^{h}(\Lambda ^{\mathbb {N}})$ with respect to t. We give basically two expressions of the derivatives, one of which allows the introduction of the random stochastic process $\{\phi \circ \sigma ^j(x)\}_{j=0}^m$ for $m\in \mathbb {N}$ . The CLT for the random process $\{\phi \circ \sigma ^j(x)\}_{j=0}^\infty $ takes core role in our proofs of Theorems 1.1 and 1.2.

First we define some basics to deal with the higher derivatives of compositional functions by the Faà di Bruno’s formula. For an integer $j\in \mathbb {N}$ , we say

$$ \begin{align*} \tau=\tau_1\tau_2\cdots\tau_q \end{align*} $$

with $q\in \mathbb {N}$ is a partition of j if the non-increasing sequence of positive integers $j\geq \tau _1\geq \tau _2\geq \cdots \geq \tau _q\geq 1$ satisfies $\sum _{i=1}^q\tau _i=j$ . Denote the collection of all the possible partitions of j by $\mathfrak {P}(j)$ . For example, Table 1 lists all the partitions in $\mathfrak {P}(5)$ .

Table 1

Partitions of $5$ .

We sometimes simply write $\tau $ to denote the set $\{\tau _1,\tau _2,\ldots ,\tau _q\}$ for convenience in the following, so $\#\tau =q$ . Now for $\tau $ being a partition of $j\geq 1$ , let $\{B_j^\tau \}$ be the number of different choices of dividing a set of j different elements into $\#\tau =q$ sets of sizes $\{\tau _i\}_{i=1}^q$ (with no order on the sets of partitions). Set $B_0^0=1$ for convenience. For example, consider the cases $j=5$ and $\tau =3,1,1$ , the number of different choices of dividing a set of $5$ different elements into $q=3$ sets of sizes $3,1,1$ respectively is

$$ \begin{align*} C_5^3=10=B_5^{3,1,1}. \end{align*} $$

Table 2 lists all the numbers $\{B_5^\tau \}_{\tau \in \mathfrak {P}(5)}$ .

Table 2

The coefficients $B_5^\tau $ .

For a smooth map $f: \mathbb {R}\rightarrow \mathbb {R}$ on the real line (which suffices for our purposes in this work) and some partition $\tau =\tau _1,\tau _2,\ldots ,\tau _q\in \mathfrak {P}(j)$ with $j\geq 1$ , let

$$ \begin{align*} f^{(\tau)}(x)=f^{(\tau_1)}(x)f^{(\tau_2)}(x)\cdots f^{(\tau_q)}(x) \end{align*} $$

be the product of the derivatives. For $j=0$ and $\tau =0\in \mathfrak {P}(0)$ , set $f^{(0)}(x)=1$ . Then for two smooth functions $f: \mathbb {R}\rightarrow \mathbb {R}$ and $g:\mathbb {R}\rightarrow \mathbb {R}$ , we have

(3.1) $$ \begin{align} \frac{d^j(g\circ f(x))}{dx^j}=\sum_{\tau\in\mathfrak{P}(j)}B_j^\tau g^{(\#\tau)}(f(x))f^{(\tau)}(x) \end{align} $$

by virtue of Faà di Bruno’s formula.

Now we turn to the higher derivatives of the pressure function. We start from some standard case, then extend the result to the general case.

Theorem 3.1. Let $\psi , \phi \in C^{h}(\Lambda ^{\mathbb {N}})$ with $\psi $ being normalised for some finite symbolic set $\Lambda $ . Assume $\int \phi \,d\mu _{\psi }=0$ , where $\mu _{\psi }$ is the equilibrium state of $\psi $ . Let $w(t,x)$ be the eigenfunction of the maximum isolated eigenvalue $e^{P(\psi +t\phi )}$ of $\mathcal {L}_{\psi +t\phi }$ , which depends analytically on t in a small neighbourhood of $0$ . Considering the derivatives of the pressure function $P(\psi +t\phi )$ at $t=0$ , we have

(3.2) $$ \begin{align} \nonumber P^{(n)}(\psi+t\phi)|_{t=0}&=\sum_{j=1}^n C_n^j\int_{\Lambda^{\mathbb{N}}}(\phi(x))^j w^{(n-j)}(0,x)\,d\mu_{\psi}(x) \\\nonumber &\quad -\sum_{j=2}^{n-2}C_n^j\sum_{\tau\in\mathfrak{P}(j),1\notin\tau} B_j^\tau P^{(\tau)}(\psi+t\phi)|_{t=0}\int_{\Lambda^{\mathbb{N}}}w^{(n-j)}(0,x)\,d\mu_{\psi}(x)\\&\quad -\sum_{\tau\in\mathfrak{P}(n),\{1,n\}\cap\tau=\emptyset} B_n^\tau P^{(\tau)}(\psi+t\phi)|_{t=0} \end{align} $$

for any $n\geq 2$ .

Proof. According to the above notation, note that

(3.3) $$ \begin{align} (\mathcal{L}_{\psi+t\phi} w(t,\cdot))(x)=e^{P(\psi+t\phi)}w(t,x). \end{align} $$

The nth derivative of $(\mathcal {L}_{\psi +t\phi } w(t,\cdot ))(x)=\sum _{y:\sigma (y)=x}e^{\psi (y)+t\phi (y)}w(t,y)$ gives

(3.4) $$ \begin{align} \nonumber &\frac{d^n \mathcal{L}_{\psi+t\phi} w(t,\cdot)(x)}{dt^n}\\ \nonumber &\quad =\sum_{y:\sigma(y)=x}\sum_{j=0}^{n}C_n^j \frac{d^j e^{(\psi+t\phi)(y)}}{dt^j}w^{(n-j)}(t,y)\\ &\quad =\sum_{y:\sigma(y)=x}\sum_{j=0}^{n}C_n^j e^{(\psi+t\phi)(y)}(\phi(y))^j w^{(n-j)}(t,y)\\ \nonumber &\quad =\sum_{j=0}^{n}C_n^j \mathcal{L}_{\psi+t\phi} ((\phi(\cdot))^j w^{(n-j)}(t,\cdot)). \end{align} $$

All derivatives are with respect to t. In the case of $t=0$ , this means

(3.5) $$ \begin{align} \frac{d^n \mathcal{L}_{\psi+t\phi} w(t,\cdot)(x)}{dt^n}\bigg|_{t=0}=\sum_{j=0}^{n}C_n^j \mathcal{L}_{\psi} ((\phi(\cdot))^j w^{(n-j)}(0,\cdot)). \end{align} $$

Note that the dual operator $\mathcal {L}_{\psi }^*$ fixes $\mu _{\psi }$ , so integration of both sides of equation (3.5) gives

(3.6) $$ \begin{align} \int\frac{d^n \mathcal{L}_{\psi+t\phi} w(t,\cdot)(x)}{dt^n}\bigg|_{t=0}\,d\mu_{\psi}(x)=\sum_{j=0}^{n}C_n^j \int(\phi(x))^j w^{(n-j)}(0,x)\mu_{\psi}(x). \end{align} $$

To get the nth derivative of $P(\psi +t\phi )$ , differentiating $e^{P(\psi +t\phi )}w(t,x)$ for n times by equation (3.1), we get

(3.7) $$ \begin{align} \nonumber &\frac{d^n (e^{P(\psi+t\phi)}w(t,x))}{dt^n}\\\nonumber &\quad =\sum_{j=0}^{n}C_n^j \frac{d^j e^{P(\psi+t\phi)}}{dt^j}w^{(n-j)}(t,x)\\\nonumber &\quad =\sum_{j=0}^{n-1}C_n^j \frac{d^j e^{P(\psi+t\phi)}}{dt^j}w^{(n-j)}(t,x)+\frac{d^n e^{P(\psi+t\phi)}}{dt^n}w(t,x)\\\nonumber &\quad =\sum_{j=0}^{n-1}C_n^j \sum_{\tau\in\mathfrak{P}(j)} B_j^\tau P^{(\tau)}(\psi+t\phi)e^{P(\psi+t\phi)}w^{(n-j)}(t,x)\\\nonumber &\qquad +\sum_{\tau\in\mathfrak{P}(n)} B_n^\tau P^{(\tau)}(\psi+t\phi)e^{P(\psi+t\phi)}w(t,x)\\\nonumber &\quad =\!\sum_{j=0}^{n-1}C_n^j \bigg(\!\sum_{\tau\in\mathfrak{P}(j),1\notin\tau}\! B_j^\tau P^{(\tau)}(\psi+t\phi)+ \!\sum_{\tau\in\mathfrak{P}(j),1\in\tau}\! B_j^\tau P^{(\tau)}(\psi+t\phi)\bigg)\\\nonumber & \qquad \times e^{P(\psi+t\phi)}w^{(n-j)}(t,x)\\&\qquad +\sum_{\tau\in\mathfrak{P}(n),n\notin\tau} B_n^\tau P^{(\tau)}(\psi+t\phi)e^{P(\psi+t\phi)}w(t,x)+P^{(n)}(\psi+t\phi)e^{P(\psi+t\phi)}w(t,x). \end{align} $$

Remember $P(\psi )=0$ and $w(0,x)=1$ as $\psi $ is normalised [Reference Parry and PollicottPP, p. 66]. Taking $t=0$ in equation (3.7), we get

(3.8) $$ \begin{align} \nonumber &\frac{d^n (e^{P(\psi+t\phi)}w(t,x))}{dt^n}\bigg|_{t=0}\\\nonumber &\quad =\sum_{j=0}^{n-1}\!C_n^j \bigg(\sum_{\tau\in\mathfrak{P}(j),1\notin\tau}\! B_j^\tau P^{(\tau)}(\psi+t\phi)|_{t=0}+\sum_{\tau\in\mathfrak{P}(j),1\in\tau}\! B_j^\tau P^{(\tau)}(\psi+t\phi)|_{t=0}\bigg)\\\nonumber & \qquad \times w^{(n-j)}(0,x)\\&\qquad +\sum_{\tau\in\mathfrak{P}(n),n\notin\tau} B_n^\tau P^{(\tau)}(\psi+t\phi)|_{t=0}+P^{(n)}(\psi+t\phi)|_{t=0}. \end{align} $$

Since $\int \phi \,d\mu _{\psi }=P'(\psi +t\phi )|_{t=0}=0$ and $\int w'(0,x)\,d\mu _{\psi }=0$ [Reference Parry and PollicottPP, p. 66], integrating both sides of equation (3.8) with respect to $\mu _{\psi }$ , we get

(3.9) $$ \begin{align} \nonumber &\displaystyle\int\frac{d^n (e^{P(\psi+t\phi)}w(t,x))}{dt^n}\bigg|_{t=0}\,d\mu_{\psi}\\ \nonumber &\quad =\sum_{j=0}^{n-1}C_n^j \sum_{\tau\in\mathfrak{P}(j),1\notin\tau} B_j^\tau P^{(\tau)}(\psi+t\phi)|_{t=0}\int w^{(n-j)}(0,x)\,d\mu_{\psi}\\ &\qquad +\sum_{\tau\in\mathfrak{P}(n),\{1,n\}\cap\tau=\emptyset} B_n^\tau P^{(\tau)}(\psi+t\phi)|_{t=0}+P^{(n)}(\psi+t\phi)|_{t=0}. \end{align} $$

Finally, combining equations (3.6) and (3.9) together, we get equation (3.2).

Remark 3.2. The terms

$$ \begin{align*} -\sum_{j=2}^{n-2}C_n^j\sum_{\tau\in\mathfrak{P}(j),1\notin\tau} B_j^\tau P^{(\tau)}(\psi+t\phi)|_{t=0}\int_{\Lambda^{\mathbb{N}}}w^{(n-j)}(0,x)\,d\mu_{\psi}(x) \end{align*} $$

and

$$ \begin{align*} -\sum_{\tau\in\mathfrak{P}(n),\{1,n\}\cap\tau=\emptyset} B_n^\tau P^{(\tau)}(\psi+t\phi)|_{t=0} \end{align*} $$

in equation (3.2) are null in the case of $n\leq 3$ . This also applies to the corresponding terms later.

One can find some description of derivatives of the pressure function by covariance of the sequence of functions $\{\phi \circ \sigma ^j\}_{j\in \mathbb {N}}$ in [Reference Kucherenko and QuasKS1, Corollary 1] for smooth $\phi $ . Without the assumptions of $\psi $ being normalised and $\int \phi \,d\mu _{\psi }=0$ , Theorem 3.1 evolves into the following form.

Corollary 3.4. Let $\psi , \phi \in C^{h}(\Lambda ^{\mathbb {N}})$ with some finite symbolic set $\Lambda $ . Here, $\mathcal {L}_{\psi +t\phi }$ admits a maximum isolated eigenvalue $e^{P(\psi +t\phi )}$ close to $e^{P(\psi )}$ with eigenfunction $w(t,x)$ whose projection depends analytically on t in a small neighbourhood of $0$ . Considering the derivatives of the pressure $P(\psi +t\phi )$ at $t=0$ , we have

(3.10) $$ \begin{align} \nonumber P^{(n)}(\psi+t\phi)|_{t=0}&=\sum_{j=1}^n C_n^j\displaystyle\int_{\Lambda^{\mathbb{N}}}(\phi(x)-\int \phi\,d\mu_{\psi})^j w^{(n-j)}(0,x)\,d\mu_{\psi}(x)\\ \nonumber &\quad -\sum_{j=2}^{n-2}C_n^j\sum_{\tau\in\mathfrak{P}(j),1\notin\tau} B_j^\tau P^{(\tau)}(\psi+t\phi)|_{t=0}\displaystyle\int_{\Lambda^{\mathbb{N}}}w^{(n-j)}(0,x)\,d\mu_{\psi}(x)\\ &\quad -\sum_{\tau\in\mathfrak{P}(n),\{1,n\}\cap\tau=\emptyset} B_n^\tau P^{(\tau)}(\psi+t\phi)|_{t=0} \end{align} $$

for any $n\geq 2$ .

Proof. Let

$$ \begin{align*} \bar{\psi}=\psi+\log w_\psi(x)-\log w_\psi\circ\sigma-P(\psi), \end{align*} $$

where $w_\psi (x)$ is the eigenfunction of $\mathcal {L}_{\psi }$ corresponding to the eigenvalue $e^{P(\psi )}$ . Taking pressure in the following equation:

$$ \begin{align*} \bar{\psi}+t\phi=\psi+t\phi+\log w_\psi(x)-\log w_\psi\circ\sigma-P(\psi), \end{align*} $$

then applying Proposition 2.2, we see that

$$ \begin{align*} P(\bar{\psi}+t\phi)=P(\psi+t\phi)-P(\psi). \end{align*} $$

This implies

(3.11) $$ \begin{align} \frac{d^n P(\bar{\psi}+t\phi)}{dt^n}=\frac{d^nP(\psi+t\phi)}{dt^n} \end{align} $$

for any $n\geq 1$ . Now applying Theorem 3.1 to the normalised potential $\bar {\psi }$ and ${\phi -\int \phi \,d\mu _{\psi }}$ , (note that $\int (\phi -\int \phi \,d\mu _{\psi })\,d\mu _{\psi }=0$ and $\mu _{\psi }=\mu _{\bar {\psi }}$ ), we justify the corollary by equation (3.11).

In the following, we present some concrete formulae of some special order n by virtue of Theorem 3.1 for later use.

Corollary 3.5. Let $\psi , \phi \in C^{h}(\Lambda ^{\mathbb {N}})$ with $\psi $ being normalised. Let $\mu _{\psi }$ be the equilibrium state of $\psi $ and $\int \phi \,d\mu _{\psi }=0$ . Let $e^{P(\psi +t\phi )}$ be the maximum eigenvalue of $\mathcal {L}_{\psi +t\phi }$ with eigenfunction $w(t,x)$ for small t. Then we have

(3.12) $$ \begin{align} P{"'}(\psi+t\phi)|_{t=0}=3\int\phi w{"}(0,x)\,d\mu_{\psi}+3\int \phi^2 w{'}(0,x)\,d\mu_{\psi}+\int \phi^3\,d\mu_{\psi}. \end{align} $$

Corollary 3.6. Let $\psi , \phi \in C^{h}(\Lambda ^{\mathbb {N}})$ with $\psi $ being normalised. Let $\mu _{\psi }$ be the equilibrium state of $\psi $ and $\int \phi \,d\mu _{\psi }=0$ . Let $e^{P(\psi +t\phi )}$ be the maximum eigenvalue of $\mathcal {L}_{\psi +t\phi }$ with eigenfunction $w(t,x)$ for small t. Then we have

(3.13) $$ \begin{align} \nonumber & P{""}(\psi+t\phi)|_{t=0}\\ \nonumber &\quad = 4\int \phi w{"'}(0,x)\,d\mu_{\psi}+6\int \phi^2 w{"}(0,x)\,d\mu_{\psi}+4\int \phi^3 w{'}(0,x)\,d\mu_{\psi}+\int \phi^4\,d\mu_{\psi}\\ \nonumber &\qquad -6P{"}(\psi+t\phi)|_{t=0}\int w{"}(0,x)\,d\mu_{\psi}-3(P{"}(\psi+t\phi)|_{t=0})^2\\ \nonumber &\quad = 4\int \phi w{"'}(0,x)\,d\mu_{\psi}+6\int \phi^2 w{"}(0,x)\,d\mu_{\psi}+4\int \phi^3 w{'}(0,x)\,d\mu_{\psi}+\int \phi^4\,d\mu_{\psi}\\ \nonumber &\qquad -6\bigg(\int \phi^2\,d\mu_{\psi}+2\int \phi w'(0,x)\,d\mu_{\psi}\bigg)\int w{"}(0,x)\,d\mu_{\psi}\\ &\qquad -3\bigg(\int \phi^2\,d\mu_{\psi}+2\int \phi w'(0,x)\,d\mu_{\psi}\bigg)^2. \end{align} $$

Proof. The first equality follows instantly from Theorem 3.1 with $n=4$ along with some direct computations on the Faà di Bruno’s coefficients $\{B_4^\tau \}_{\tau \in \mathfrak {P}(4)}$ . The second one is true as

$$ \begin{align*} P{"}(\psi+t\phi)|_{t=0}=\int \phi^2\,d\mu_{\psi}+2\int \phi w'(0,x)\,d\mu_{\psi}. \end{align*} $$

The latter description depends only on $\phi (x), \{w^{(j)}(0,x)\}_{j=1}^3$ and $\mu _{\psi }(x)$ .

One can also get some precise formulae for some particular n in Corollary 3.4, and some non-inductive ones as we indicate in Remark 3.3. While equations (3.2), (3.10), (3.12), (3.13) all give interesting descriptions of the derivatives of the pressure function $P(\psi +t\phi )$ , it seems to us difficult to discover any essential rigid restriction on them, or relationships between them. In the following, we turn to the description of them by the random stochastic process $\{\phi \circ \sigma ^j(x)\}_{j=0}^\infty $ . This is not a new idea on exploring the regularity of the pressure function $P(\psi +t\phi )$ , as one can recall it from many others’ work in thermodynamic formalism. Again, we first consider some standard case, then extend to the general case.

Theorem 3.7. Let $\psi , \phi \in C^{h}(\Lambda ^{\mathbb {N}})$ with $\psi $ being normalised. Let $\mu _{\psi }$ be the equilibrium state of $\psi $ and $\int \phi \,d\mu _{\psi }=0$ . Let $e^{P(\psi +t\phi )}$ be the maximum isolated eigenvalue of $\mathcal {L}_{\psi +t\phi }$ with eigenfunction $w(t,x)$ whose projection depends analytically on t. Considering the derivatives of the pressure $P(\psi +t\phi )$ at $t=0$ , we have

(3.14) $$ \begin{align} \nonumber &P^{(n)}(\psi+t\phi)|_{t=0}\\ \nonumber &\quad =\lim_{m\rightarrow\infty}\frac{1}{m}\bigg(\sum_{j=2}^n C_n^j\displaystyle\int_{\Lambda^{\mathbb{N}}}(S_{m,\phi}(x))^j w^{(n-j)}(0,x)\,d\mu_{\psi}(x)\\ \nonumber &\qquad -\sum_{j=2}^{n-2}C_n^j\sum_{\tau\in\mathfrak{P}(j),1\notin\tau} m^{\#\tau}B_j^\tau P^{(\tau)}(\psi+t\phi)|_{t=0}\displaystyle\int_{\Lambda^{\mathbb{N}}}w^{(n-j)}(0,x)\,d\mu_{\psi}(x)\\ &\qquad -\sum_{\tau\in\mathfrak{P}(n),\{1,n\}\cap\tau=\emptyset} m^{\#\tau} B_n^\tau P^{(\tau)}(\psi+t\phi)|_{t=0}\bigg) \end{align} $$

for any $n\geq 2$ .

Proof. The proof follows the routine of Proof of Theorem 3.1. Considering equation (2.1), we take n-derivatives on both sides of equation (2.3), take $t=0$ , then integrate both sides with respect to $\mu _{\psi }(x)$ , divided by m, and we get

(3.15) $$ \begin{align} \nonumber &P^{(n)}(\psi+t\phi)|_{t=0}\\\nonumber &\quad =\frac{1}{m}\bigg(\sum_{j=1}^n C_n^j\displaystyle\int_{\Lambda^{\mathbb{N}}}(S_{m,\phi}(x))^j w^{(n-j)}(0,x)\,d\mu_{\psi}(x)\\\nonumber &\qquad\qquad -\sum_{j=2}^{n-2}C_n^j\sum_{\tau\in\mathfrak{P}(j),1\notin\tau} m^{\#\tau}B_j^\tau P^{(\tau)}(\psi+t\phi)|_{t=0}\displaystyle\int_{\Lambda^{\mathbb{N}}}w^{(n-j)}(0,x)\,d\mu_{\psi}(x)\\&\qquad\qquad -\sum_{\tau\in\mathfrak{P}(n),\{1,n\}\cap\tau=\emptyset} m^{\#\tau} B_n^\tau P^{(\tau)}(\psi+t\phi)|_{t=0}\bigg) \end{align} $$

as equation (3.2). Now since $w^{(n-1)}(0,x)$ is bounded on X, the ergodic theorem guarantees

(3.16) $$ \begin{align} \lim_{m\rightarrow\infty}\frac{1}{m}\int_{\Lambda^{\mathbb{N}}}S_{m,\phi}(x) w^{(n-1)}(0,x)\,d\mu_{\psi}(x)=0. \end{align} $$

Then equation (3.14) follows from equation (3.15) as $m\rightarrow \infty $ by equation (3.16).

Theorem 3.7 establishes some link between the derivatives of the pressure function and the process $\{\phi \circ \sigma ^j(x)\}_{j=0}^\infty $ through $S_{m,\phi }$ with respect to the equilibrium state $\mu _{\psi }$ . We also formulate a general version of the result.

Corollary 3.8. Let $\psi , \phi \in C^{h}(\Lambda ^{\mathbb {N}})$ with $\mu _{\psi }$ be the equilibrium state of $\psi $ . Here, $\mathcal {L}_{\psi +t\phi }$ admits a maximum isolated eigenvalue $e^{P(\psi +t\phi )}$ close to $e^{P(\psi )}$ with eigenfunction $w(t,x)$ whose projection depends analytically on t in a small neighbourhood of $0$ . Considering the derivatives of the pressure function $P(\psi +t\phi )$ at $t=0$ , we have

(3.17) $$ \begin{align} \nonumber &P^{(n)}(\psi+t\phi)|_{t=0}\\ \nonumber &\quad =\lim_{m\rightarrow\infty}\frac{1}{m}\bigg(\sum_{j=2}^n C_n^j\displaystyle\int_{\Lambda^{\mathbb{N}}}\bigg(S_{m,\phi}-m\int\phi\,d\mu_{\psi}\bigg)^j w^{(n-j)}(0,x)\,d\mu_{\psi}(x)\\ \nonumber &\qquad -\sum_{j=2}^{n-2}C_n^j\sum_{\tau\in\mathfrak{P}(j),1\notin\tau} m^{\#\tau}B_j^\tau P^{(\tau)}(\psi+t\phi)|_{t=0}\displaystyle\int_{\Lambda^{\mathbb{N}}}w^{(n-j)}(0,x)\,d\mu_{\psi}(x)\\ &\qquad -\sum_{\tau\in\mathfrak{P}(n),\{1,n\}\cap\tau=\emptyset} m^{\#\tau} B_n^\tau P^{(\tau)}(\psi+t\phi)|_{t=0}\bigg) \end{align} $$

for any $n\geq 2$ .

The following is a classical result on the second derivative of the pressure [Reference Parry and PollicottPP, Ch. 4].

Corollary 3.9. Let $\psi , \phi \in C^{h}(\Lambda ^{\mathbb {N}})$ with $\psi $ being normalised. Let $\mu _{\psi }$ be the equilibrium state of $\psi $ and $\int \phi \,d\mu _{\psi }=0$ . Let $e^{P(\psi +t\phi )}$ be the maximum eigenvalue of $\mathcal {L}_{\psi +t\phi }$ with eigenfunction $w(t,x)$ for small t. Then we have

(3.18) $$ \begin{align} P{"}(\psi+t\phi)|_{t=0}=\lim_{m\rightarrow\infty}\frac{1}{m}\int S_{m,\phi}^2\,d\mu_{\psi}. \end{align} $$

Now we give some precise descriptions of the third and fourth derivatives of $P(\psi +t\phi )$ by virtue of Theorem 3.7.

Corollary 3.11. Let $\psi , \phi \in C^{h}(\Lambda ^{\mathbb {N}})$ with $\psi $ being normalised. Let $\mu _{\psi }$ be the equilibrium state of $\psi $ and $\int \phi \,d\mu _{\psi }=0$ . Let $e^{P(\psi +t\phi )}$ be the maximum eigenvalue of $\mathcal {L}_{\psi +t\phi }$ with eigenfunction $w(t,x)$ for small t. Then we have

(3.19) $$ \begin{align} P{"'}(\psi+t\phi)|_{t=0}=\lim_{m\rightarrow\infty}\frac{3}{m}\int S_{m,\phi}^2 w{'}(0,x)\,d\mu_{\psi}+\lim_{m\rightarrow\infty}\frac{1}{m}\int S_{m,\phi}^3\,d\mu_{\psi}. \end{align} $$

Corollary 3.12. Let $\psi , \phi \in C^{h}(\Lambda ^{\mathbb {N}})$ with $\psi $ being normalised. Let $\mu _{\psi }$ be the equilibrium state of $\psi $ and $\int \phi \,d\mu _{\psi }=0$ . Let $e^{P(\psi +t\phi )}$ be the maximum eigenvalue of $\mathcal {L}_{\psi +t\phi }$ with eigenfunction $w(t,x)$ for small t. Then we have

(3.20) $$ \begin{align} \nonumber & P^{(4)}(\psi+t\phi)|_{t=0}\\\nonumber &\quad = \lim_{m\rightarrow\infty}\bigg(\frac{6}{m}\displaystyle\int S_{m,\phi}^2 w{"}(0,x)\,d\mu_{\psi}+\frac{4}{m}\int S_{m,\phi}^3 w{'}(0,x)\,d\mu_{\psi}+\frac{1}{m}\int S_{m,\phi}^4\,d\mu_{\psi}\\\nonumber &\qquad -6P{"}(\psi+t\phi)|_{t=0}\displaystyle\int w{"}(0,x)\,d\mu_{\psi}-3m(P{"}(\psi+t\phi)|_{t=0})^2\bigg)\\\nonumber &\quad = \lim_{m\rightarrow\infty}\bigg(\frac{6}{m}\displaystyle\int S_{m,\phi}^2 w{"}(0,x)\,d\mu_{\psi}+\frac{4}{m}\int S_{m,\phi}^3 w{'}(0,x)\,d\mu_{\psi}+\frac{1}{m}\int S_{m,\phi}^4\,d\mu_{\psi}\\&\qquad -\frac{6}{m}\displaystyle\int S_{m,\phi}^2\,d\mu_{\psi}\displaystyle\int w{"}(0,x)\,d\mu_{\psi}-\frac{3}{m}\bigg(\int S_{m,\phi}^2\,d\mu_{\psi}\bigg)^2\bigg). \end{align} $$

Through the above formulae, we see the importance of the asymptotic distribution of the random variable $S_{m,\phi }$ with respect to $\mu _{\psi }$ , which is described by the CLT for the process $\{\phi \circ \sigma ^j(x)\}_{j=0}^\infty $ . Equipped with all the above results, now we are in a position to prove the rigidity results on the third derivatives of $P(\psi +t\phi )$ using Corollary 3.11. We first show Theorem 1.2.

Proof of Theorem 1.2

From now on, we fix $t_{*}\in (0,\infty )$ . Let $\psi =t_{*}\phi $ . Simply by making a change of variable, we can see that

$$ \begin{align*} P^{(n)}(t_{*})=P^{(n)}(t\phi)|_{t=t_{*}}=P^{(n)}(\psi+t\phi)|_{t=0} \end{align*} $$

for any $n\geq 0$ . So equation (1.2) is equivalent to

(3.21) $$ \begin{align} |P"'(\psi+t\phi)|_{t=0} (1-\sqrt{2\pi}(P"(\psi+t\phi)|_{t=0})^{3/2})|\leq 3 M_\phi P"(\psi+t\phi)|_{t=0}. \end{align} $$

We can assume $\psi $ is normalised as otherwise we can change it to its normalisation by equation (2.11). Moreover, it suffices for us to prove it under the assumption $\int \phi \,d\mu _\psi =0$ by virtue of equation (2.10). If $P"(\psi +t\phi )|_{t=0}=0$ , then $\phi $ is cohomologous to a constant according to [Reference Parry and PollicottPP, Proposition 4.12]. This forces $P"'(\psi +t\phi )|_{t=0}=0$ , so equation (3.21) is satisfied in this case. In the following, we assume $P"(\psi +t\phi )|_{t=0}>0$ . We resort to Corollary 3.11 to justify equation (3.21) under the above assumptions. We first estimate the term ${1}/{m}\int S_{m,\phi }^3\,d\mu _{\psi }$ in equation (3.19). Since we are assuming the potential is generic, we can apply the CLT with equation (2.5):

$$ \begin{align*} &\frac{1}{m}\displaystyle\int S_{m,\phi}^3\,d\mu_{\psi}\\ &\quad = \sqrt{m}\displaystyle\int \bigg(\frac{S_{m,\phi}}{\sqrt{m}}\bigg)^3\,d\mu_{\psi}\\ &\quad = \sqrt{m}\displaystyle\int y^3\,d G_m(y)\\ &\quad = \sqrt{m}\displaystyle\int y^3\,d N_{0,P"(\psi+t\phi)|_{t=0}}(y)+\sqrt{m}\int y^3\,dH_m(y)+\sqrt{m}\cdot o(1/\sqrt{m})\\ &\quad = \sqrt{m}\cdot 0+\displaystyle\int y^3\,d\bigg(\frac{P"'(\psi+t\phi)|_{t=0}}{6}\bigg(1-\frac{y^2}{P"(\psi+t\phi)|_{t=0}}\bigg)e^{-y^2/2P" (\psi+t\phi)|_{t=0}}\bigg)\\ &\qquad +\sqrt{m}\cdot o(1/\sqrt{m})\\ &\quad = P"'(\psi+t\phi)|_{t=0}\sqrt{2\pi}(P"(\psi+t\phi)|_{t=0})^{3/2}+\sqrt{m}\cdot o(1/\sqrt{m}). \end{align*} $$

By taking $m\rightarrow \infty $ , we get

(3.22) $$ \begin{align} \lim_{m\rightarrow\infty} \frac{1}{m}\displaystyle\int S_{m,\phi}^3\,d\mu_{\psi}=P"'(\psi+t\phi)|_{t=0}\sqrt{2\pi}(P"(\psi+t\phi)|_{t=0})^{3/2}. \end{align} $$

By equation (3.19), we have

(3.23) $$ \begin{align} P"'(\psi+t\phi)|_{t=0} (1-\sqrt{2\pi}(P"(\psi+t\phi)|_{t=0})^{3/2}) = \lim_{m\rightarrow\infty}\frac{3}{m}\int S_{m,\phi}^2 w{'}(0,x)\,d\mu_{\psi}. \end{align} $$

Since $w{'}(0,x)$ depends continuously on $x\in X$ , there exists some $M_\phi $ depending on $\phi $ , such that

(3.24) $$ \begin{align} |w{'}(0,x)|\leq M_\phi. \end{align} $$

Now taking absolute values on both sides of equation (3.23), we justify equation (3.21) by equations (3.24) and (3.18).

The proof of Theorem 1.1 on the pressure functions of non-generic Hölder potentials follows a similar way.

Proof of Theorem 1.1

Fixing $t_{*}\in (0,\infty )$ , we can simply assume $\psi =t_{*}\phi $ is normalised and $\int \phi \,d\mu _\psi =0$ . In the case where $P"(\psi +t\phi )|_{t=0}=0$ , so $\phi $ is cohomologous to a constant, equation (1.1) holds obviously. In the following, we assume $\phi $ is not cohomologous to a constant, so $P"(\psi +t\phi )|_{t=0}>0$ . We again resort to Corollary 3.11 to justify equation (1.1) under these assumptions. Now for the term ${1}/{m}\int S_{m,\phi }^3\,d\mu _{\psi }$ in equation (3.19), by virtue of the CLT with equation (2.4),

(3.25) $$ \begin{align} \nonumber &\frac{1}{m}\displaystyle\int S_{m,\phi}^3\,d\mu_{\psi}\\ \nonumber &\quad = \sqrt{m}\displaystyle\int \bigg(\frac{S_{m,\phi}}{\sqrt{m}}\bigg)^3\,d\mu_{\psi}\\ \nonumber &\quad = \sqrt{m}\displaystyle\int y^3\,d G_m(y)\\ \nonumber &\quad \leq \sqrt{m}\displaystyle\int y^3\,d N_{0,P"(\psi+t\phi)|_{t=0}}(y)+\sqrt{m}\frac{9|P"'(\psi+t\phi)|_{t=0}| +2|P^{(4)}(\psi+t\phi)|_{t=0}|}{\sqrt{2\pi^3 m}(P"(\psi+t\phi)|_{t=0})^{3/2}}\\ \nonumber &\quad = \sqrt{m}\cdot 0+\frac{9|P"'(\psi+t\phi)|_{t=0}|+2|P^{(4)}(\psi+t\phi)|_{t=0}|}{\sqrt{2\pi^3} (P"(\psi+t\phi)|_{t=0})^{3/2}}\\ &\quad = \frac{9|P"'(\psi+t\phi)|_{t=0}|+2|P^{(4)} (\psi+t\phi)|_{t=0}|}{\sqrt{2\pi^3}(P"(\psi+t\phi)|_{t=0})^{3/2}} \end{align} $$

for m large enough. By taking $m\rightarrow \infty $ in equation (3.25), we get

(3.26) $$ \begin{align} \lim_{m\rightarrow\infty} \frac{1}{m}\displaystyle\int S_{m,\phi}^3\,d\mu_{\psi}\leq \frac{9|P"'(\psi+t\phi)|_{t=0}|+2|P^{(4)}(\psi+t\phi)|_{t=0}|}{\sqrt{2\pi^3}(P"(\psi+t\phi)|_{t=0})^{3/2}}. \end{align} $$

Taking modulus on both sides of equation (3.26), we get

(3.27) $$ \begin{align} \nonumber & |P"'(\psi+t\phi)|_{t=0}|\\ \nonumber &\quad \leq \bigg|\lim_{m\rightarrow\infty}\frac{1}{m}\displaystyle\int S_{m,\phi}^3\,d\mu_{\psi}\bigg| + \bigg|\lim_{m\rightarrow\infty}\frac{3}{m}\int S_{m,\phi}^2 w{'}(0,x)\,d\mu_{\psi}\bigg|\\ \nonumber &\quad \leq \frac{9|P"'(\psi+t\phi)|_{t=0}|+2|P^{(4)}(\psi+t\phi)|_{t=0}|}{\sqrt{2\pi^3} (P"(\psi+t\phi)|_{t=0})^{3/2}}+3M_\phi \bigg|\lim_{m\rightarrow\infty}\frac{3}{m}\displaystyle\int S_{m,\phi}^2\,d\mu_{\psi}\bigg|\\ &\quad = \frac{9|P"'(\psi+t\phi)|_{t=0}|+2|P^{(4)}(\psi+t\phi)|_{t=0}|}{\sqrt{2\pi^3} (P"(\psi+t\phi)|_{t=0})^{3/2}}+3M_\phi P"(\psi+t\phi)|_{t=0} \end{align} $$

for some $|w{'}(0,x)|\leq M_\phi $ , which results in equation (1.1).

One can predict from Corollary 3.12, Theorem 3.7, and the proof of Theorems 1.1, 1.2 that some more rigid relationships between higher derivatives of the pressure function $\{P^{(n)}(t\phi )\}_{n\in \mathbb {N}}$ are possible. These rigidity relationships impose restrictions on fitting convex analytic functions whose supporting lines intersect the vertical axis in some bounded set in $[0,\infty )$ by pressures of Hölder potentials.

4. Global fitting of convex analytic functions via pressures of Hölder potentials

This section is dedicated to the proof of Theorem 1.7. We start from the following result on some global behaviour of the pressure functions of generic Hölder potentials.

Theorem 4.1. Let $\alpha>0$ . If there exists a strictly convex analytic function $F(t)$ on $(\alpha ,\infty )$ , with its supporting lines intersecting the vertical axis in $[\underline {\gamma },\overline {\gamma }]\subset [0,\infty )$ , such that

(4.1) $$ \begin{align} \sup_{t\in(\alpha,\infty)}\bigg\{\bigg|\frac{F{"'}(t)}{F{"}(t)}-\sqrt{2\pi F{"}(t)}\bigg|\bigg\}=\infty, \end{align} $$

then there does not exist a shift space with finite symbols with a generic Hölder potential $\phi $ satisfying

$$ \begin{align*} P(t\phi)= F(t) \end{align*} $$

on $(\alpha ,\infty )$ .

Proof. This follows directly from Theorem 1.2 in fact. Suppose on the contrary that there exist some shift space X with finite symbols and some generic Hölder potential $\phi \in C^{h}(X)$ satisfying $P(t\phi )= F(t)$ on $(\alpha ,\infty )$ , then according to Theorem 1.2, we have

$$ \begin{align*} \sup_{t\in(\alpha,\infty)}\bigg\{\bigg|\frac{F{"'}(t)}{F{"}(t)}-\sqrt{2\pi F{"}(t)} \bigg|\bigg\}\leq 3M_\phi \end{align*} $$

for some finite $M_\phi>0$ . This contradicts equation (4.1).

Be careful that we cannot exclude the possibility that one can locally fit some convex analytic function through the pressure of some generic Hölder potential on some shift space with finite symbols by Theorem 1.2. This is because for any strictly convex analytic function $F(t)$ on $(\alpha ,\infty )$ and $\alpha \leq \underline {\alpha }\leq \overline {\alpha }$ , we always have

$$ \begin{align*} \sup_{\underline{\alpha}\leq t\leq \overline{\alpha}}\bigg\{\bigg|\frac{F{"'}(t)}{F{"}(t)}-\sqrt{2\pi F{"}(t)}\bigg|\bigg\}<\infty. \end{align*} $$

So one cannot exclude the possibility that there exists some generic Hölder potential $\phi $ on some shift space satisfying

$$ \begin{align*} P(t\phi)= F(t) \end{align*} $$

on $[\underline {\alpha },\overline {\alpha }]$ through Theorem 1.2. See §5 for more results on the problem of local fitting of some convex analytic functions through the pressures of Hölder potentials.

Now for $\alpha>0$ , let

$$ \begin{align*} \mathcal{F}_\alpha&=\{\kern-1pt F(t)\kern1.2pt{:}\kern1.2pt F(t) \text{ is a strictly convex analytic function on } (\alpha,\infty) \text{ satisfying equation } (4.1), \\[-1pt] &\qquad\ \text{its supporting lines intersect the vertical axis in a bounded interval in}\ [0,\infty)\}. \end{align*} $$

We will show that $\mathcal {F}_\alpha \neq \emptyset $ for any $\alpha>0$ in the following.

Proposition 4.2. For any $\alpha>0$ , we have

$$ \begin{align*} \tilde{\mathcal{F}}_\alpha =\bigg\{F_{a,b,c}(t)=\frac{at^2+bt+te^{-ct^2}+e^{-ct^2}}{t}\bigg|_{(\alpha,\infty)}\bigg\}_{a,b>0, c>1/2\sqrt{2}}\subset {\mathcal{F}}_\alpha. \end{align*} $$

Proof. The restricted functions on $(\alpha ,\infty )$ are of course analytic. By considering the second derivative of a function $F_{a,b,c}(t)\in \tilde {\mathcal {F}}_\alpha $ , we have

$$ \begin{align*} F_{a,b,c}^{\prime\prime}(t)=4c^2t^2e^{-ct^2}+4c^2te^{-ct^2}-2ce^{-ct^2}+2ct^{-1}e^{-ct^2}+2t^{-3}e^{-ct^2} \end{align*} $$

for $t\in (0,\infty )$ . Now since

$$ \begin{align*} 4c^2t+2ct^{-1}\geq 2\sqrt{8c^3}>2c, \end{align*} $$

considering $c>1/2\sqrt {2}$ , we can see that $F_{a,b,c}^{\prime \prime }(t)>0$ on $(0,\infty )$ . This shows that for any $\alpha>0$ , $F_{a,b,c}(t)\in \tilde {\mathcal {F}}_\alpha $ is a convex function. Considering the third derivative of a function $F_{a,b,c}(t)\in \tilde {\mathcal {F}}_\alpha $ , we have

$$ \begin{align*} F_{a,b,c}^{\prime\prime\prime}(t)=-8c^3t^3e^{-ct^2}-8c^3t^2e^{-ct^2} +12c^2te^{-ct^2}-6ct^{-2}e^{-ct^2}-6t^{-4}e^{-ct^2} \end{align*} $$

for $t\in (0,\infty )$ . Then we have

$$ \begin{align*} \lim_{t\rightarrow\infty} \bigg(\frac{F^{\prime\prime\prime}_{a,b,c}(t)}{F^{\prime\prime}_{a,b,c}(t)}-\sqrt{2\pi F^{\prime\prime}_{a,b,c}(t)}\bigg)=\lim_{t\rightarrow\infty} \frac{-8c^3t^3e^{-ct^2}}{4c^2t^2e^{-ct^2}}=-\infty. \end{align*} $$

This means that $F_{a,b,c}(t)\in \tilde {\mathcal {F}}_\alpha $ satisfies equation (4.1). To see that the supporting lines of a function $F_{a,b,c}(t)\in \tilde {\mathcal {F}}_\alpha $ intersect the vertical axis in a bounded domain in $[0,\infty )$ , write the function as

$$ \begin{align*} F_{a,b,c}(t)=at+b+e^{-ct^2}+t^{-1}e^{-ct^2}. \end{align*} $$

Its graph on $(0,\infty )$ is a strictly convex smooth curve with asymptotes $y=at+b$ and $t=0$ .

In Figure 1, we provide the readers with the graph of the function

$$ \begin{align*} F_{2,3,1}(t)=\frac{2t^2+3t+te^{-t^2}+e^{-t^2}}{t} \end{align*} $$

on $(0,\infty )$ .

Figure 1

Graph of $F_{2,3,1}(t)$ .

This means that any function in the family $\tilde {\mathcal {F}}_\alpha $ cannot be fitted by any generic Hölder potential on any shift space globally, by Theorem 4.1. In the following, we exclude the possibility that they can be fitted by non-generic Hölder potentials on shift spaces with finite symbols. One can show that [Reference Parry and PollicottPP, Ch. 4] if $\phi $ is non-generic, then there exists a continuous function $u: X \to \mathbb R$ , $c_\phi \in \mathbb R$ , and a locally constant potential $\tilde {\phi }: X \to \mathbb R$ , such that

(4.2) $$ \begin{align} \phi(x) = u\circ\sigma(x)-u(x) + c_\phi + \tilde{\phi}(x). \end{align} $$

Proposition 4.3. For any $\alpha>0$ and any $F_{a,b,c}(t)\in \tilde {\mathcal {F}}_\alpha $ with $a,b>0, c > ({1}/{2\sqrt {2}})$ , there does not exist a shift space with finite symbols with a non-generic Hölder potential $\phi $ satisfying

$$ \begin{align*} P(t\phi)= F(t) \end{align*} $$

on $(\alpha ,\infty )$ .

Proof. Note that for a non-generic Hölder potential $\phi $ on a shift space with finite symbols, according to equation (4.2), we have

$$ \begin{align*} P(t\phi)=tc_\phi + P(t\tilde{\phi}), \end{align*} $$

where $\tilde {\phi }$ is some locally constant potential. By the explicit formula (see for example [Reference WaltersWal1, p. 214]) for the pressure functions of locally constant potentials on shift spaces with finite symbols, we see that any $F_{a,b,c}(t)$ cannot be fitted by pressure of any non-generic Hölder potential $\phi $ globally.

Equipped with all the above results, Theorem 1.7 follows instantly from Propositions 4.2 and 4.3.

5. Local fitting of prescribed germs via pressures of locally constant potentials

In this section, we deal with the local fitting of analytic functions by the pressures of Hölder potentials, especially the pressures of piecewise constant ones. First, we borrow some notion originating from analytic continuation.

Definition 5.1. A germ of level $\iota (\iota \in \mathbb {N}\cup \{\infty \})$ at $t_{*}$ is the formal power series

$$ \begin{align*} g(t)=a_0+a_1(t-t_{*})+ \frac{a_2}{2!}(t-t_{*})^2+\frac{a_3}{3!}(t-t_{*})^3+\cdots+\frac{a_\iota}{\iota!}(t-t_{*})^\iota \end{align*} $$

for some $(a_0,a_1,\ldots ,a_\iota )\in \mathbb {R}^{\iota +1}$ .

The convergent radius (the superior of values $\delta \geq 0$ on $[t_{*}-\delta ,t_{*}+\delta ]$ such that the germ converges) of the power series is called the radius of the germ. Any finite-level germ admits infinite radius while an infinite-level germ may admit some finite radius. We are only interested in germs of radius $\delta>0$ . The following problem will be our concern in this section.

Problem 5.2. For a germ

$$ \begin{align*} g(t)=a_0+a_1(t-t_{*})+ \frac{a_2}{2!}(t-t_{*})^2+\cdots+\frac{a_\iota}{\iota!}(t-t_{*})^\iota \end{align*} $$

of level $\iota \ (\iota \in \mathbb {N}\cup \{\infty \})$ at $t_{*}$ with some strictly positive radius, does there exist some Hölder potential $\phi $ on some shift space with finite symbols and some $\delta>0$ , such that

$$ \begin{align*} P(t\phi)=g(t)+O((t-t_{*})^{\iota+1}) \end{align*} $$

on $[t_{*}-\delta ,t_{*}+\delta ]$ ?

We assume $O((t-t_{*})^\infty )=0$ in Problem 5.2. The question can still be understood in Katok’s flexibility program in the class of symbolic dynamical systems, or even in some smooth systems. Obvious conditions to guarantee a positive answer to the problem are equation (1.5) and

(5.1) $$ \begin{align} a_2>0 \end{align} $$

in the case $\iota \geq 2$ . The condition in equation (5.1) guarantees convexity of the germ (in some neighbourhood of $t_{*}$ ), while equation (1.5) guarantees the supporting lines of the germ intersect the vertical axis in a bounded set in $[0,\infty )$ (also in some neighbourhood of $t_{*}$ ). We are especially interested in its answer when the Hölder potential in Problem 5.2 is required to be a locally constant one. We have seen the importance of the family of locally constant potentials in approximating convex analytic functions in Corollary 1.4. In fact, Corollary 1.4 has some interesting interpretation in approximation theory [Reference TimanTim, Ch. I] when we consider the explicit expressions of the pressures of locally constant potentials on the shift space. For $n\in \mathbb {N}$ , recall that

$$ \begin{align*} \Lambda_n=\{1,2,\ldots,n\}. \end{align*} $$

Lemma 5.3. For an integer $k\geq 0$ , consider some locally constant potential

$$ \begin{align*} \phi(x)=c_{x_{-k}x_{-k+1}\cdots x_0\cdots x_{k-1}x_{k}} \end{align*} $$

for $x=\cdots x_{-1}x_{0}x_{1}\cdots \in [x_{-k}\cdots x_{k}]$ on the shift space $\Lambda _n^{\mathbb {Z}}$ , we have

$$ \begin{align*} P(t\phi)=\log\sum_{(x_{-k},\ldots,x_{k})\in\Lambda_n^{2k+1}}e^{tc_{x_{-k}\cdots x_{k}}} \end{align*} $$

for any $t\in (-\infty ,\infty )$ .

Now combining Corollary 1.4 and Lemma 5.3, we have the following result.

Corollary 5.4. Let $F(t)$ be a convex Lipschitz function on $(\alpha ,\infty )$ for some $\alpha>0$ , such that its supporting lines intersect the vertical axis in $[\underline {\gamma },\overline {\gamma }]$ with $0\leq \underline {\gamma }\leq \overline {\gamma }<\infty $ . Then there exists some $K\in \mathbb {N}$ and some sequences of constants

$$ \begin{align*} \{c_{n,j}\}_{j=1}^{K^n}, \end{align*} $$

such that

(5.2) $$ \begin{align} \lim_{n\rightarrow\infty} \log \sum_{j=1}^{K^n} e^{tc_{n,j}}=F(t) \end{align} $$

for any $t\in (\alpha ,\infty )$ .

Proof. Take $K=\#\Lambda $ for the symbolic set in the proof of Corollary 1.4, then the locally constant potential $\phi _{n}(x)=\phi _{n,-}(x)$ admits $K^n$ constant values on corresponding level-n cylinder sets. Denote these values by $\{c_{n,j}\}_{j=1}^{K^n}$ for $n\in \mathbb {N}$ . According to Lemma 5.3,

$$ \begin{align*} P(t\phi_{n,-})=\log \sum_{j=1}^{K^n} e^{tc_{n,j}} \end{align*} $$

for any $n\geq 1$ . This gives equation (5.2) by virtue of equation (1.3).

Corollary 5.4 indicates that logarithm of the finite sums of the exponential maps in the family $\{e^{tc}\}_{c\in \mathbb {R}}$ are dense in the space of certain convex Lipschitz maps on $(\alpha ,\infty )$ . The above approximation is uniform with respect to t in a bounded set. This makes the family $\{e^{tc}\}_{c\in \mathbb {R}}$ (family of locally constant potentials) important in detecting the properties of certain convex Lipschitz maps (among continuous or Hölder potentials).

From now on, we turn our attention to Problem 5.2, but with restriction to locally constant potentials. We focus on locally constant potentials defined on the level- $0$ cylinder sets, whose theory is equivalent to those defined on the deeper cylinder sets, where the symbols are replaced by words (recoding). On the shift space $\Lambda _n^{\mathbb {Z}}$ with $n\geq 2$ , consider the locally constant potential

$$ \begin{align*} \phi(x)=z_{x_0} \end{align*} $$

for $x=\cdots x_{-1}x_{0}x_{1}\cdots \in [x_0]$ , where $\{z_i\}_{1\leq i\leq n}$ are all constants. Let

$$ \begin{align*} Q_0(t,z_1,z_2,\ldots,z_n)=\sum_{i=1}^n e^{tz_i}, \end{align*} $$

so

$$ \begin{align*} P(t\phi)=\log Q_0(t,z_1,\ldots,z_n) \end{align*} $$

by Lemma 5.3. Let

$$ \begin{align*} Q_1(t,z_1,z_2,\ldots,z_n)=\sum_{i=1}^n z_ie^{tz_i} \end{align*} $$

and

$$ \begin{align*} Q_2(t,z_1,z_2,\ldots,z_n)=\sum_{1\leq i<j\leq n}(z_i-z_j)^2e^{t(z_i+z_j)}. \end{align*} $$

Through some elementary calculations, one can check that

$$ \begin{align*} P'(t\phi)=\frac{dP(t\phi)}{dt}=\frac{Q_1(t,z_1,\ldots,z_n)}{Q_0(t,z_1,\ldots,z_n)}, \end{align*} $$

while

(5.3) $$ \begin{align} P"(t\phi)=\frac{d^2P(t\phi)}{dt^2}=\frac{Q_2(t,z_1,\ldots,z_n)}{Q_0^2(t,z_1,\ldots,z_n)}. \end{align} $$

Let

$$ \begin{align*} R_2(t,z_1,z_2,\ldots,z_n)=\sum_{i=1}^n z_i^2e^{tz_i}, \end{align*} $$

one can check that

$$ \begin{align*} Q_2(t,z_1,\ldots,z_n)=Q_0(t,z_1,\ldots,z_n)R_2(t,z_1,\ldots,z_n)-Q_1^2(t,z_1,\ldots,z_n). \end{align*} $$

In the following, we will often fix $t=t_{*}>0$ , so we will frequently write

$$ \begin{align*} Q_0(t_{*},z_1,z_2,\ldots,z_n)=Q_0(z_1,z_2,\ldots,z_n) \end{align*} $$

with $t_{*}$ omitted for convenience. Similar notation apply to other terms above. Let

(5.4) $$ \begin{align} \kern-15pt Q_0(z_1,\ldots,z_n)&=\sum_{i=1}^n e^{t_{*}z_i}=e^{a_0}, \end{align} $$

(5.5) $$ \begin{align} Q_1(z_1,\ldots,z_n)&=\sum_{i=1}^n z_ie^{t_{*}z_i}=a_1e^{a_0} \end{align} $$

be two equations with unknowns $\{z_1, z_2, \ldots , z_n\}$ for fixed $t_{*}>0, (a_0, a_1)\in \mathbb {R}^2$ and some $n\geq 2$ . Let

$$ \begin{align*} \Gamma_{5.4}^n=\{(z_1, z_2, \ldots, z_n)\in\mathbb{R}^n: z_1, z_2, \ldots, z_n \text{ satisfy equation }(5.4)\} \end{align*} $$

and

$$ \begin{align*} \Gamma_{5.5}^n=\{(z_1, z_2, \ldots, z_n)\in\mathbb{R}^n: z_1, z_2, \ldots, z_n \text{ satisfy equation }(5.5)\}. \end{align*} $$

They are both $n-1$ dimensional smooth hypersurfaces. We first present readers with the following result on fitting an analytic function

$$ \begin{align*} a_0+a_1(t-t_{*})+O((t-t_{*})^2) \end{align*} $$

with $t_{*}, a_0, a_1$ subject to equation (1.5) around some fixed $t_{*}>0$ by pressures of locally constant potentials on general shift spaces.

Theorem 5.5. Let $t_{*}>0, (a_0, a_1)\in \mathbb {R}^2, n\geq 2$ satisfying equation (1.5) and

(5.6) $$ \begin{align} \frac{a_0-\log n}{t_{*}}<a_1. \end{align} $$

Then there exists some $\delta _n>0$ and some sequence $\{r_{i,n}\}_{i=1}^n\subset \mathbb {R}$ , such that the locally constant potential

$$ \begin{align*} \phi(x)=r_{x_0} \end{align*} $$

for $x=\cdots x_{-1}x_{0}x_{1}\cdots \in [x_0]$ on the full shift space $\Lambda _n^{\mathbb {Z}}$ satisfies

$$ \begin{align*} P(t\phi)=a_0+a_1(t-t_{*})+O((t-t_{*})^2) \end{align*} $$

on $[t_{*}-\delta _n,t_{*}+\delta _n]$ .

Proof. In fact, it suffices for us to show that the system of equations

$$ \begin{align*} \left\{\!\!\begin{array}{ll} \mathrm{equation}\ (5.4),\\ \mathrm{equation}\ (5.5), \end{array} \right. \end{align*} $$

with unknowns $\{z_1, z_2, \ldots , z_n\}$ admits a solution under conditions of the theorem. Without loss of generality, we assume

(5.7) $$ \begin{align} z_1\leq z_2\leq \cdots\leq z_n. \end{align} $$

Under this assumption, it is easy to see that

$$ \begin{align*} \frac{a_0-\log n}{t_{*}}\leq z_n < \frac{a_0}{t_{*}}. \end{align*} $$

Now we estimate the values of $Q_1(z_1,\ldots ,z_n)$ with $z_n$ approaching the terminals. When $z_n$ approaches the right terminal from below, we have

$$ \begin{align*} \lim_{(z_1, z_2, \ldots, z_n)\in\Gamma_{5.4}^n,\ z_n\nearrow {a_0}/{t_{*}}} Q_1(z_1,\ldots,z_n)= \frac{a_0}{t_{*}}e^{a_0}>a_1e^{a_0} \end{align*} $$

by virtue of equation (1.5). When $z_n$ approaches the left terminal from above, we have

$$ \begin{align*} \lim_{(z_1, z_2, \ldots, z_n)\in\Gamma_{5.4}^n,\ z_n\searrow {a_0}/{t_{*}}} Q_1(z_1,\ldots,z_n)=\frac{a_0-\log n}{t_{*}}e^{a_0}<a_1e^{a_0} \end{align*} $$

by virtue of equation (5.6). Since $\Gamma _{5.4}^n$ is a smooth hypersurface, by the mean value theorem, there exists some $(r_{1,n}, r_{2,n}, \ldots , r_{n,n})\in \Gamma _{5.4}^n$ satisfying equations (5.4) and (5.5) simultaneously. At last, for $x=\cdots x_{-1}x_{0}x_{1}\cdots \in [x_0]$ on the full shift space $\Lambda _n^{\mathbb {Z}}$ , let

$$ \begin{align*} \phi(x)=r_{x_0,n} \end{align*} $$

be the locally constant potential. As $P(t\phi )$ is analytic, there exists some $\delta _n>0$ such that

$$ \begin{align*} P(t\phi)=a_0+a_1(t-t_{*})+O((t-t_{*})^2) \end{align*} $$

for $t\in [t_{*}-\delta _n,t_{*}+\delta _n]$ .

Care must be taken that the $\{r_{i,n}\}_{i=1}^n$ all depend on n. Theorem 5.5 induces the following interesting flexibility result on fitting certain analytic functions locally by pressures of locally constant potentials on general shift spaces.

Corollary 5.7. Let $t_{*}>0$ and $(a_0, a_1)\in \mathbb {R}^2$ satisfy equation (1.5). Then there exists some $N\in \mathbb {N}$ , such that for any $n\geq N$ , there exist some $\delta _n>0$ and some sequence $\{r_{i,n}\}_{i=1}^n\subset \mathbb {R}$ , such that the locally constant potential

$$ \begin{align*} \phi(x)=r_{x_0,n} \end{align*} $$

for $x=\cdots x_{-1}x_{0}x_{1}\cdots \in [x_0]$ on the full shift space $\Lambda _n^{\mathbb {Z}}$ satisfies

$$ \begin{align*} P(t\phi)=a_0+a_1(t-t_{*})+O((t-t_{*})^2) \end{align*} $$

on $[t_{*}-\delta _n,t_{*}+\delta _n]$ .

Proof. Under conditions of the corollary, for the given values $t_{*}, a_0, a_1$ satisfying equation (1.5), choose $N\in \mathbb {N}$ large enough such that

$$ \begin{align*} \frac{a_0-\log N}{t_{*}}<a_1. \end{align*} $$

This means that for any $n>N$ , the condition in equation (5.6) is satisfied for $t_{*}, a_0, a_1, n$ . Then the conclusion follows from Theorem 5.5.

Note that on some particular symbolic spaces, Theorems 5.5 and 5.7 may be trivial. For example, for given $(t_{*}, a_0, a_1)\in \mathbb {R}^3$ without any constraints, by choosing $\beta =e^{a_0-t_{*}a_1}$ , consider the constant potential

$$ \begin{align*} \phi(x)= a_1 \end{align*} $$

on the $\beta $ -shift space with symbols $\{0,1,\ldots , \lfloor \beta \rfloor \}$ . It is easy to see that

$$ \begin{align*} P(t\phi)=a_0-t_{*}a_1+a_1 t=a_0+a_1(t-t_{*}) \end{align*} $$

on $(-\infty ,\infty )$ . However, our results guarantee conclusions on general shift spaces beyond these specific ones.

From now on, we go towards the proof of Theorem 1.9. For fixed $t_{*}>0, (a_0, a_1)\in \mathbb {R}^2$ and $n\geq 3$ , let

$$ \begin{align*} &\Gamma_{5.4,5.5}^n=\Gamma_{5.4}^n\cap\Gamma_{5.5}^n\\& \quad =\{(z_1, z_2, \ldots, z_n)\in\mathbb{R}^n: z_1, z_2, \ldots, z_n \text{ satisfy equations } (5.4) \text{ and } (5.5)\}. \end{align*} $$

We describe some topological properties of the set $\Gamma _{5.4,5.5}^n$ in the following result.

Proof. The Jacobian of the functions $Q_0(z_1,\ldots ,z_n)-e^{a_0}$ and $Q_1(z_1,\ldots ,z_n)-a_1e^{a_0}$ with respect to $z_1, z_2, \ldots , z_n$ is

$$ \begin{align*} J=\begin{pmatrix} t_{*}e^{t_{*}z_1} & t_{*}e^{t_{*}z_2} & \cdots & t_{*}e^{t_{*}z_n}\\ e^{t_{*}z_1}+t_{*}z_1e^{t_{*}z_1} & e^{t_{*}z_2}+t_{*}z_2e^{t_{*}z_2} & \cdots & e^{t_{*}z_n}+t_{*}z_ne^{t_{*}z_n} \end{pmatrix}. \end{align*} $$

Its rank is strictly less than $2$ if and only if

$$ \begin{align*} z_1=z_2=\cdots=z_n. \end{align*} $$

Since $a_1\neq ({a_0-\log n}/{t_{*}})$ , this is excluded from points in $\Gamma _{5.4,5.5}$ . By the implicit function theorem [Reference LangLan, Theorem 5.9], if $\Gamma _{5.4,5.5}^n$ is not empty, it is an $(n-2)$ -dimension smooth manifold locally. The gradient of the function $Q_0(z_1,\ldots ,z_n)-e^{a_0}$ is

$$ \begin{align*} \bigtriangledown(Q_0(z_1,\ldots,z_n)-e^{a_0})=(t_{*}e^{t_{*}z_1}, t_{*}e^{t_{*}z_2}, \ldots, t_{*}e^{t_{*}z_n}), \end{align*} $$

whose individual components will always be strictly positive. The gradient of the function $Q_1(z_1,\ldots ,z_n)-a_1e^{a_0}$ is

$$ \begin{align*} \bigtriangledown(Q_1(z_1,\ldots,z_n)-a_1e^{a_0})=(e^{t_{*}z_1}+t_{*}z_1e^{t_{*}z_1}, e^{t_{*}z_2}+t_{*}z_2e^{t_{*}z_2}, \ldots, e^{t_{*}z_n}+t_{*}z_ne^{t_{*}z_n}), \end{align*} $$

with the ith individual component vanishing if and only if $z_i=- ({1}/{t_{*}})$ for $1\leq i\leq n$ . So $\Gamma _{5.4}^n$ and $\Gamma _{5.5}^n$ cannot be tangent to each other. Moreover, note that

$$ \begin{align*} e^{t_{*}z_i}+t_{*}z_ie^{t_{*}z_i}>0 \end{align*} $$

if $z_i>- ({1}/{t_{*}})$ , while

$$ \begin{align*} e^{t_{*}z_i}+t_{*}z_ie^{t_{*}z_i}<0 \end{align*} $$

if $z_i<- ({1}/{t_{*}})$ for any $1\leq i\leq n$ . These force the intersection of zeros of the two functions $Q_0(z_1,\ldots ,z_n)-e^{a_0}$ and $Q_1(z_1,\ldots ,z_n)-a_1e^{a_0}$ to be connected if the intersection is not empty. This implies $\Gamma _{5.4,5.5}$ is a manifold globally in the case of being non-empty. Here, $\Gamma _{5.4,5.5}^n$ is compact since it is a bounded set.

Let

$$ \begin{align*} \Gamma_{5.4,1,2,1}^3=\{(z_1, z_2, z_3)\in\mathbb{R}^3: z_1, z_2, z_3 \text{ satisfy } e^{z_1}+e^{z_2}+e^{z_3}=e^2\} \end{align*} $$

and

$$ \begin{align*} \Gamma_{5.5,1,2,1}^3=\{(z_1, z_2, z_3)\in\mathbb{R}^3: z_1, z_2, z_3 \text{ satisfy } z_1e^{z_1}+z_2e^{z_2}+z_3e^{z_3}=e^2\} \end{align*} $$

be the corresponding surfaces with $t_{*}=1, a_0=2, a_1=1$ . Figure 2 depicts parts of the two $2$ -dimension surfaces, whose intersection will be a $1$ -dimension smooth curve.

Figure 2

$\Gamma _{5.4,1,2,1}^3$ (lighter) and $\Gamma _{5.5,1,2,1}^3$ (darker).

Equipped with all the above results, now we are ready to prove Theorem 1.9.

Proof of Theorem 1.9

First, for the given $t_{*}>0$ and $(a_0, a_1)\in \mathbb {R}^2$ satisfying equation (1.5), if n is large enough, $\Gamma _{5.4,5.5}^n$ is not empty according to Corollary 5.7. So $\Gamma _{5.4,5.5}^n$ is a compact $(n-2)$ -dimension smooth manifold for n large enough. We recall here the $R_2$ and $Q_2$ defined after Corollary 5.4. In the following, we always assume n is large enough. Now let

$$ \begin{align*} m_{t_{*},a_0,a_1,n}=\min\bigg\{\frac{R_2(t_{*},z_1,z_2,\ldots,z_n)}{e^{a_0}}-a_1^2:(z_1, z_2, \ldots, z_n)\in\Gamma_{5.4,5.5}^n\bigg\}, \end{align*} $$

while

(5.8) $$ \begin{align} M_{t_{*},a_0,a_1,n}=\max\bigg\{\frac{R_2(t_{*},z_1,z_2,\ldots,z_n)}{e^{a_0}}-a_1^2:(z_1, z_2, \ldots, z_n)\in\Gamma_{5.4,5.5}^n\bigg\}. \end{align} $$

For any $m_{t_{*},a_0,a_1,n}\leq a_2\leq M_{t_{*},a_0,a_1,n}$ , since $\Gamma _{5.4,5.5}^n$ is a smooth manifold, there exist $\{c_{i,n}\}_{i=1}^n\subset \mathbb {R}$ , such that $(c_{1,n},c_{2,n},\ldots ,c_{n,n})$ satisfies equations (5.4), (5.5) and

(5.9) $$ \begin{align} a_2=\frac{Q_2(t_{*},c_{1,n},\ldots,c_{n,n})}{Q_0^2(t_{*},c_{1,n},\ldots,c_{n,n})} =\frac{R_2(t_{*},c_{1,n},\ldots,c_{n,n})}{e^{a_0}}-a_1^2 \end{align} $$

simultaneously. Now let

$$ \begin{align*} \phi(x)=c_{x_0,n} \end{align*} $$

for $x=\cdots x_{-1}x_{0}x_{1}\cdots \in [x_0]$ on the full shift space $\Lambda _n^{\mathbb {Z}}$ . It is a locally constant potential. According to equations (5.3) and (5.9), we have

(5.10) $$ \begin{align} P"(t_{*}\phi)=\frac{Q_2(t_{*},c_{1,n},\ldots,c_{n,n})}{Q_0^2(t_{*},c_{1,n},\ldots,c_{n,n})}=a_2. \end{align} $$

Since $(c_{1,n},c_{2,n},\ldots ,c_{n,n})$ satisfies equations (5.4) and (5.5), we have

(5.11) $$ \begin{align} P(t_{*}\phi)=\frac{Q_2(t_{*},c_{1,n},\ldots,c_{n,n})}{Q_0^2(t_{*},c_{1,n},\ldots,c_{n,n})}=a_0, \end{align} $$

while

(5.12) $$ \begin{align} P'(t_{*}\phi)=\frac{Q_2(t_{*},c_{1,n},\ldots,c_{n,n})}{Q_0^2(t_{*},c_{1,n},\ldots,c_{n,n})}=a_1. \end{align} $$

Note that $P(t\phi )$ is analytic with respect to t on $(\alpha ,\infty )$ for any $\alpha>0$ , so there exists some $\delta _n>0$ , such that equation (1.6) holds on $[t_{*}-\delta _n,t_{*}+\delta _n]$ , by equations (5.10), (5.11) and (5.12).

In the following, we illustrate some dependent relationship between

$$ \begin{align*} \{m_{t_{*},a_0,a_1,n}, M_{t_{*},a_0,a_1,n}\}_{n\in\mathbb{N}} \end{align*} $$

and some particular $t_{*},a_0,a_1,n$ satisfying equation (1.5). There should be some universal relationship between them, while we hope the following observations will provide some hints. The first one is that it is possible for $m_{t_{*},a_0,a_1,n}=0$ for some $t_{*},a_0,a_1,n$ .

Proposition 5.9. Let $t_{*}>0$ and $(a_0,a_1)\in \mathbb {R}^2$ satisfy equation (1.5). Then $m_{t_{*},a_0,a_1,n}=0$ for $n\geq 2$ if and only if

(5.13) $$ \begin{align} a_1=\frac{a_0-\log n}{t_{*}}. \end{align} $$

Proof. Note that $m_{t_{*},a_0,a_1,n}=0$ is equivalent to say that there exists some locally constant potential $\phi $ on $\Lambda _n^{\mathbb {Z}}$ such that $P"(t_{*}\phi )=0$ according to Theorem 1.9. By [Reference Parry and PollicottPP, Proposition 4.12], this happens if and only if $\phi $ is a constant potential on $\Lambda _n^{\mathbb {Z}}$ . In this case, we have

$$ \begin{align*} \phi(x)=\frac{a_0-\log n}{t_{*}} \end{align*} $$

for any $x\in \Lambda _n^{\mathbb {Z}}$ , which implies equation (5.13).

This result does not tell things about the sequence

$$ \begin{align*} \{m_{t_{*},a_0,a_1,n}\}_{\ n\in\mathbb{N} \mbox{ large enough}} \end{align*} $$

for given $t_{*},a_0,a_1$ , since equation (5.13) will never be true for any n large enough for fixed $t_{*},a_0,a_1$ . The following result describes a limiting behaviour of the sequence

$$ \begin{align*} \{M_{t_{*},a_0,a_1,n}\}_{\ n\in\mathbb{N} \mbox{ large enough }} \end{align*} $$

for $t_{*}=1,a_0=2,a_1=1$ .

Proposition 5.10. Let $t_{*}=1,a_0=2,a_1=1$ , in symbols of Theorem 1.9, we have

(5.14) $$ \begin{align} \lim_{n\rightarrow\infty} M_{1,2,1,n}=\infty. \end{align} $$

To justify Proposition 5.10, we first illustrate some basic properties about the function $ze^{t_{*}z}$ for $t_{*}>0$ .

Lemma 5.11. For $t_{*}>0$ , $ze^{t_{*}z}$ is strictly decreasing on $(-\infty ,-({1}/{t_{*}}))$ , strictly increasing on $(-({1}/{t_{*}}),\infty )$ , while it attains its minimum $-({1}/{t_{*}})e^{-1}$ at $z=- ({1}/{t_{*}})$ . It admits one and only one inflection in $(-\infty ,- ({1}/{t_{*}}))$ .

Figure 3

Graph of $\varsigma (z)=ze^{z}$ .

In Figure 3, we depict the graph of $\varsigma (z)=ze^{z}$ .

Proof of Proposition 5.10

Since we are considering the limit behaviour of $M_{1,2,1,n}$ , we always assume n is large enough throughout the proof. Now consider the following two equations:

(5.15) $$ \begin{align} (n-1)e^{z_a}+e^{z_b}=e^2 \end{align} $$

and

(5.16) $$ \begin{align} (n-1)z_ae^{z_a}+z_be^{z_b}=e^2 \end{align} $$

with unknowns $z_a, z_b$ . Let

$$ \begin{align*} \Gamma_{5.15}=\{(z_a, z_b)\in\mathbb{R}^2: z_a, z_b \text{ satisfy equation } (5.15)\} \end{align*} $$

and

$$ \begin{align*} \Gamma_{5.16}=\{(z_a, z_b)\in\mathbb{R}^2: z_a, z_b \mbox{ satisfy equation } (5.16)\}. \end{align*} $$

We describe the graphs of $\Gamma _{5.15}$ and $\Gamma _{5.16}$ separately in the following. Here, $\Gamma _{5.15}$ is a one-dimensional smooth curve with two asymptotes $z_a=2-\log (n-1)$ and $z_b=2$ . It is strictly decreasing when we consider the curve as the graph of the function

$$ \begin{align*} z_b=\log (e^2-(n-1)e^{z_a}) \end{align*} $$

for $z_a\in (-\infty , 2-\log (n-1))$ . Here, $\Gamma _{5.16}$ is also a one-dimensional smooth curve with two asymptotes $z_a=\varsigma ^{-1}({e^2}/({n-1}))$ and $z_b=\varsigma ^{-1}(e^2)$ . When we consider the $\Gamma _{5.16}$ as the graph of the function

$$ \begin{align*} z_b=\eta(z_a) \end{align*} $$

as the implicit function induced by equation (5.16), it is strictly increasing for $z_a\in (-\infty ,-1)$ , strictly decreasing for $z_a\in (-1,\varsigma ^{-1}({e^2}/{n-1}))$ , with its maximum $\varsigma ^{-1}(e^2+(n-1)e^{-1})$ attained at $z_a=-1$ . Let $\varsigma _l^{-1}(-({e^2}/({n-1})))$ be the smaller one of the two intersections of $z_b=2$ and $\Gamma _{5.16}$ , then $\Gamma _{5.15}$ and $\Gamma _{5.16}$ must intersect at some unique point $c_{a,n}\in (-\infty , \varsigma _l^{-1}(-({e^2}/({n-1}))))$ . Obviously,

$$ \begin{align*} \lim_{n\rightarrow\infty} c_{a,n}=-\infty \end{align*} $$

since $\lim _{n\rightarrow \infty } \varsigma _l^{-1}(-({e^2}/({n-1})))=-\infty $ . Now we analyse the order of $c_{a,n}$ with respect to n as $n\rightarrow \infty $ . Let

$$ \begin{align*} z_{a,n}=-\log n-\log\log n+\log 1-1. \end{align*} $$

One can check that

$$ \begin{align*} \lim_{n\rightarrow\infty, z_b\rightarrow 2} ((n-1)e^{z_{a,n}}+e^{z_b})=e^2, \end{align*} $$

while

$$ \begin{align*} \lim_{n\rightarrow\infty, z_b\rightarrow 2} ((n-1)z_{a,n}e^{z_{a,n}}+z_be^{z_b})=e^2. \end{align*} $$

These imply that

$$ \begin{align*} c_{a,n}=-\log n-\log\log n+o(\log\log n). \end{align*} $$

Figure 4

$\Gamma _{5.15}$ and $\Gamma _{5.16}$ .

Table 3

$\{c_{a,n}\}_{n\in \mathbb {N}}$ and $\{\eta (c_{a,n})\}_{n\in \mathbb {N}}$ .

Note that $(c_{a,n},c_{a,n},\ldots , c_{a,n}, \eta (c_{a,n}))\in \Gamma _{5.4,5.5}^n$ for $t_{*}=1, a_0=2, a_1=1$ . Now

$$ \begin{align*} & R_2(c_{a,n},c_{a,n},\ldots, c_{a,n}, \eta(c_{a,n}))\\ &\quad = (n-1)c_{a,n}^2e^{c_{a,n}}+(\eta(c_{a,n}))^2e^{\eta(c_{a,n})}\\ &\quad = (n-1)(-\log n-\log\log n+o(\log\log n))^2e^{-\log n-\log\log n+o(\log\log n)}+4e^2+o(1)\\ &\quad = \log n+o(\log n), \end{align*} $$

from which it is easy to see that

$$ \begin{align*} \lim_{n\rightarrow\infty} R_2(c_{a,n},c_{a,n},\ldots, c_{a,n}, \eta(c_{a,n}))=\infty. \end{align*} $$

This forces

$$ \begin{align*} \lim_{n\rightarrow\infty} M_{1,2,1,n}=\infty, \end{align*} $$

by equation (5.8).

We provide the readers with the curves $\Gamma _{5.15}$ and $\Gamma _{5.16}$ in Figure 4. Obviously, some more general conclusions are available if one considers variations of the parameters $t_{*},a_0,a_1$ in Proposition 5.10. In the proof of Proposition 5.10, we analysed the asymptotic behaviour of $c_{a,n}$ as $n\rightarrow \infty $ . To illustrate this, we provide the readers with some solutions $\{c_{a,n}\}_{n\in \mathbb {N}}$ and $\{\eta (c_{a,n})\}_{n\in \mathbb {N}}$ in Table 3, from which one can see the order of decay and increase of the sequences with respect to n clearly.