Here again Jarzynski's identity combined with Markov's inequality (recall (2.12)) yields a strengthening

$$ \mu \left( \left\{ x \in \Omega \left| \frac{1}{\tau} \sum_{t=0}^{\tau-1} \sigma \circ \varphi^t(x) \le -s \right. \right\} \right) \le e^{-s\tau + 2R}. $$

**Remark.** The full phase space $\Omega = T^*M$ of a Hamiltonian system is unbounded and the natural reference measure there is the Liouville one. This forces

$$ \inf_{x \in \Omega} \rho(x) = 0, $$

and incurs the failure of Assumption (3.38). The primary purpose of a thermostat is to avoid this problem by constraining the motion to a compact submanifold $\Sigma \subset \Omega$. Possible choices include surfaces of constant kinetic or total energy. One expects that, in the thermodynamic limit, the statistical properties of the motion do not depend too much on the choice of $\Sigma$, much in the spirit of the equivalence of ensembles in equilibrium statistical mechanics (see [Rue00] and references therein).

The subtler FT and asymptotic FR were first discussed in [GC95a; GC95b], on the basis of the *Chaotic Hypothesis*. From our point of view, the two major outcomes of these works are the formulation of the FT as a large deviation principle and its connection, through the chaotic hypothesis, with the thermodynamic formalism of dynamical systems (see also [Rue99]).

Under the chaotic hypothesis of Gallavotti and Cohen, $(\Omega, \varphi, \mu)$ is a transitive Anosov dynamical system: $\varphi$ is a diffeomorphism of class $C^2$ for which the compact set $\Omega$ is hyperbolic, i.e., the tangent bundle of $\Omega$ has a $\varphi'$-invariant decomposition

$$ T_x \Omega = E_x^u \oplus E_x^s $$

into unstable and stable subbundles such that, for all $x \in \Omega$ and $t \in \mathcal{L}_+$,

$$ \| (\varphi^{-t})'(x) \|_{E_x^u} \le C\lambda^t, \quad \| (\varphi^t)'(x) \|_{E_x^s} \le C\lambda^t $$

for some positive constants $C$ and $0 < \lambda < 1$. Transitivity further means that for any nonempty open sets $U, V \subset \Omega$ and any $\tau \in \mathcal{L}_+$ there is $\mathcal{L}_+ \ni t > \tau$ such that $\varphi^{-t}(U) \cap V \ne \emptyset$.

For $v \in \mathcal{P}_\varphi(\Omega)$, denote by $h_\varphi(v)$ the Kolmogorov-Sinai entropy of $v$ w.r.t. $\varphi$. The topological pressure of $F \in C(\Omega)$ w.r.t. $\varphi$ is defined by the variational expression

$$ \mathfrak{P}_{\varphi}(F) := \sup_{v \in \mathcal{P}_{\varphi}(\Omega)} (\langle F, v \rangle + h_{\varphi}(v)) \quad (3.39) $$

(see, e.g., [Wal82]).

Under the Chaotic Hypothesis, the Rényi entropy $e_t(\alpha) = \text{Ent}_\alpha(\mathbb{P}_{\mu,\tau} | \hat{\mathbb{P}}_{\mu,\tau})$ has a large-time limit

$$ e(\alpha) = \lim_{t \to \infty} \frac{1}{t} e_t(\alpha) $$
which defines a real-analytic function on $\mathbb{R}$. More precisely, setting

$$v(x) := -\log|\det(\varphi'(x)|_{E_x^u})|,$$

one has<fnref n="12" />

$$e(\alpha) = \mathfrak{p}_{\varphi}(v - \alpha\sigma).$$

By the Gärtner-Ellis theorem, the FT holds for the family $(\mathbb{P}_{\mu,\tau}, \hat{\mathbb{P}}_{\mu,\tau})_{\tau \in \mathcal{L}_+}$ with scale $r_t = t$ and rate

$$I(s) = \sup_{\alpha \in \mathbb{R}} (\alpha s - e(-\alpha)). \qquad (3.40)$$

By Assumption (3.38), the additive functional

$$\tilde{\sigma}_{\tau}(\mathbf{x}) := \sum_{t=0}^{\tau-1} \sigma \circ \varphi^t(x(0))$$

is physically equivalent to the entropy production $\sigma_{\tau}$. It therefore also satisfies the FT w.r.t. the family $(\mathbb{P}_{\mu,\tau}, \hat{\mathbb{P}}_{\mu,\tau})_{\tau \in \mathcal{L}_+}$ with the same scale and rate. Moreover, there is a unique SRB measure $\mu_+$, an ergodic element of $\mathcal{P}_{\varphi}(\Omega)$ characterized by the fact that

$$\text{w-lim}_{\tau \to \infty} \frac{1}{\tau} \sum_{t=0}^{\tau-1} \delta_{\varphi^t(x)} = \mu_+$$

for $\mu$-a.e. $x \in \Omega$. Thus, $\mu_+$ is a natural candidate for the nonequilibrium steady state (NESS) of the system. It is also the unique maximizer in (3.39) for the potential $v$, i.e.,

$$\mathfrak{p}_{\varphi}(v) = \langle v, \mu_{+} \rangle + h_{\varphi}(\mu_{+}) = e(0) = 0.$$

In the language of topological dynamics, $\mu_+$ is the unique equilibrium state of $(\Omega, \varphi)$ for the potential $v$. This state is typically quite different from $\mu$. In fact, provided the entropy production rate is non-vanishing,

$$\text{epr} := \lim_{\tau \to \infty} \frac{1}{\tau} \sum_{t=0}^{\tau-1} \langle \sigma, \mu_t \rangle = \langle \sigma, \mu_+ \rangle > 0,$$

$\mu$ and $\mu_+$ are mutually singular. Nevertheless, one can show that the cumulant-generating function

$$e_{+,\tau}(\alpha) = \log\langle e^{-\alpha \tilde{\sigma}_{\tau}}, \mu_{+} \rangle$$

satisfies

$$\lim_{\tau \to \infty} \frac{1}{\tau} e_{+,\tau}(\alpha) = e(\alpha)$$

for all $\alpha \in \mathbb{R}$ [JPR11, Proposition 11.4]. Thus, the additive functional $\tilde{\sigma}_{\tau}$ also satisfies the FT w.r.t. the family $(\mathbb{P}_{\mu,\tau}, \hat{\mathbb{P}}_{\mu,\tau})_{\tau \in \mathcal{L}_+}$ with the rate (3.40).

<footnote n="12">¹² See [MV03] and [JPR11, Proposition 11.2, Theorem 10.10].</footnote>
# Appendix: Level-2 FR for Langevin Dynamics

Under the standing assumptions of Subsection 3.4, the following holds.

**Lemma 3.1** Fix $\ell > 2$ and $\delta \in ]0, 1/2]$. There exists $\varepsilon_0 > 0$ such that, for any $\varepsilon \in ]0, \varepsilon_0[$ there exists $c > 0$ such that, for any function $h \in C^2(\mathbb{R})$ that is bounded below and satisfies $0 \le h' \le 1$, the functions

$$ \Psi(x) := \frac{1}{2} S(x) + \varepsilon p \cdot \Gamma^{-1} \nabla V(q) + h \circ \phi(x) $$

and

$$ \hat{\Psi}(x) := \frac{1}{2} S(x) + \varepsilon p \cdot \Gamma^{-1} \nabla V(q) - \frac{1}{1+\delta} \phi(x) $$

are bounded below and satisfy

$$ G := -(L\Psi + \gamma(\Psi)) \geq -\frac{1}{c} + c|\nabla S|^2 + (h'(1 - \ell h') - h'') \circ \phi \gamma(\phi) $$

and

$$ \hat{G} := -(L\hat{\Psi} + \gamma(\hat{\Psi})) \geq -\frac{1}{c} + c|\nabla S|^2. $$

**Proof.** We first show that $\Psi$ and $\hat{\Psi}$ are bounded below. Minimizing $\Psi$ over $p$, we get

$$ \Psi \geq \frac{1}{2} \left( \tilde{V} - \frac{1}{2} |M \nabla \tilde{V}|^2 \right) + h \circ \phi, \quad (3.41) $$

where

$$ \tilde{V} := \sum_{i} \frac{V_i}{T_i}, \quad M := 2\varepsilon\Gamma^{-1}T^{3/2}. $$

Setting $u := M^2 \nabla \tilde{V}(q)$ in the 2nd-order Taylor formula

$$ \tilde{V}(q-u) = \tilde{V}(q) - u \cdot \nabla \tilde{V}(q) + \int_{0}^{1} u \cdot (D^2 \tilde{V})(q - su)u(1-s)ds $$

we get, with $C := \max_i \sup_q |T_i^{-1} V_i''(q)|$,

$$ \tilde{V}(q) - |M \nabla \tilde{V}(q)|^2 \geq \tilde{V}(q - M^2 \nabla \tilde{V}(q)) - \frac{1}{2} C |M^2 \nabla \tilde{V}(q)|^2, $$

which we rewrite as

$$ \tilde{V}(q) - M \nabla \tilde{V}(q) \cdot \left( I - \frac{1}{2}CM^2 \right) M \nabla \tilde{V}(q) \geq \tilde{V}(q - M^2 \nabla \tilde{V}(q)) \geq 0. $$

Now for small enough $\varepsilon$ one has $CM^2 \le I$ and hence

$$ \tilde{V} - \frac{1}{2} |M \nabla \tilde{V}|^2 \geq 0. $$
From (3.41) we deduce that since $h$ is bounded below, so is $\Psi$. To deal with $\hat{\Psi}$ we note that, by [EPR99a, Proposition 3.5], the function $e^{S/2}\rho_{st} = e^{-S/2+\phi\circ\theta}$ belongs to the Schwartz class. It follows that

$$ \phi \le \hat{C} + \frac{1}{2}S \qquad (3.42) $$

for some constant $\hat{C}$ and hence

$$ \hat{\Psi} \ge \frac{\delta}{1+\delta} \left( \frac{1}{2} S + \varepsilon(1+\delta^{-1})p \cdot \Gamma^{-1} \nabla V(q) \right) - \frac{\hat{C}}{1+\delta}. $$

Repeating the previous argument gives that, for any given $\delta > 0$, $\hat{\Psi}$ is bounded below provided $\varepsilon$ is small enough.

We now turn to the lower bounds for $G$. An explicit calculation gives

$$ G(x) = p \cdot \left( \frac{1}{4} T^{-1} \Gamma T^{-1} - \varepsilon \Gamma^{-1/2} (D^2 V)(q) \Gamma^{-1/2} \right) p + \varepsilon(1-\varepsilon) |\Gamma^{-1/2} (\nabla V)(q)|^2 \\ + (h'(1-h') - h'') \circ \phi(x) \gamma(\phi)(x) \\ - (\Gamma^{1/2} T^{-1} p + 2\varepsilon\Gamma^{-1/2} (\nabla V)(q)) \cdot (\Gamma^{1/2} \nabla_p h \circ \phi)(x) + \mathcal{O}(|(\nabla S)(x)|), $$

where the remainder $\mathcal{O}(|\nabla S|)$ is uniform in $h$ for $0 \le h' \le 1$. Invoking the Cauchy-Schwarz inequality on the first term of the last line in the previous expression further yields, for arbitrary positive $\kappa$ and $\lambda$,

$$ - (\Gamma^{1/2} T^{-1} p + 2\varepsilon\Gamma^{-1/2} (\nabla V)(q)) \cdot (\Gamma^{1/2} \nabla_p h \circ \phi)(x) \\ \geq -\frac{1}{2\kappa} |\Gamma^{1/2} T^{-1} p + 2\varepsilon\Gamma^{-1/2} (\nabla V)(q)|^2 - \frac{\kappa}{2} (h' \circ \phi(x))^2 \gamma(\phi)(x) \\ \geq -\frac{1}{2\kappa} \left(1 + \frac{1}{\lambda}\right) |\Gamma^{1/2} T^{-1} p|^2 - \frac{2\varepsilon^2}{\kappa} (1 + \lambda) |\Gamma^{-1/2} (\nabla V)(q)|^2 \\ - \frac{\kappa}{2} (h' \circ \phi(x))^2 \gamma(\phi)(x), $$

so that

$$ G(x) \ge p \cdot \left( \left( \frac{1}{4} - \frac{1}{2\kappa} \left( 1 + \frac{1}{\lambda} \right) \right) T^{-1} \Gamma T^{-1} - \varepsilon \Gamma^{-1/2} (D^2 V)(q) \Gamma^{-1/2} \right) p \\ + \varepsilon \left( 1 - \varepsilon \left( 1 + \frac{2\varepsilon}{\kappa} (1 + \lambda) \right) \right) |\Gamma^{-1/2} \nabla V|^2 \\ + \left( h' \left( 1 - \left( 1 + \frac{\kappa}{2} \right) h' \right) - h'' \right) \circ \phi(x) \gamma(\phi)(x) + \mathcal{O}(|(\nabla S)(x)|). $$

It follows that for $\kappa > 2$, $\lambda > \kappa/(\kappa-2)$ and small enough $\varepsilon$ there exists $c > 0$ such that

$$ G(x) \ge c |(\nabla S)(x)|^2 + \left( h' \left( 1 - \left( 1 + \frac{\kappa}{2} \right) h' \right) - h'' \right) \circ \phi(x) \gamma(\phi)(x) + \mathcal{O}(|(\nabla S)(x)|), $$
which yields the claimed lower bound of G. The proof of the lower bound of $\hat{G}$ is similar, and we omit it. □

Setting

$$ \mathcal{F}_Q[\varphi] := \langle L\varphi - \gamma(\varphi), Q \rangle, \quad \hat{\mathcal{F}}_Q[\varphi] := \langle \hat{L}\varphi - \gamma(\varphi), Q \rangle, $$

the Donsker-Varadhan level-2 rate functions for the direct and reversed processes are given by<fnref n="13" />

$$ I(Q) = \sup_{\varphi \in C^2(\mathcal{X}), \varphi \le 0} \mathcal{F}_Q[\varphi], \quad \hat{I}(Q) = \sup_{\varphi \in C^2(\mathcal{X}), \varphi \le 0} \hat{\mathcal{F}}_Q[\varphi]. $$

The following result gives the important properties of these rate functions.

**Lemma 3.2** For any $Q \in \mathcal{P}(\mathcal{X})$ the following hold:

(1) If either $I(Q) < \infty$ or $\hat{I}(Q) < \infty$, then $|\nabla S| \in L^2(\mathcal{X}, dQ)$.

(2) $\hat{I}(Q) = I(Q \circ \theta)$.

(3) If $\gamma(\varphi) \in L^1(\mathcal{X}, dQ)$, then

$$ I(Q \circ \theta) = I(Q) + \langle \sigma, Q \rangle. $$

(4) If $I(Q) < \infty$, then $\gamma(\varphi) \in L^1(\mathcal{X}, dQ)$.

**Proof.** (1) is a direct consequence of [Wu01, Corollary 2.2] and the previous lemma.

(2) follows by contraction from the level-3 identity $\hat{I}(Q) = I(\hat{Q})$.

(3) Recall that the generator of the time-reversed stationary process is given by $\hat{L} = L + 2\gamma(\varphi, \cdot)$. It follows that

$$ \begin{aligned} \hat{L}\varphi - \gamma(\varphi) &= L\varphi + 2\gamma(\varphi, \varphi) - \gamma(\varphi) = L\varphi - \gamma(\varphi - \varphi) + \gamma(\varphi) \\ &= L(\varphi - \varphi) - \gamma(\varphi - \varphi) + L\varphi + \gamma(\varphi) = L(\varphi - \varphi) - \gamma(\varphi - \varphi) + \sigma, \end{aligned} $$

and hence

$$ \hat{\mathcal{F}}_Q[\varphi] = \mathcal{F}_Q[\varphi - \varphi] + \langle \sigma, Q \rangle. $$

Thus, by (2), it is enough to show that

$$ \sup_{\varphi \in C_b^2(\mathcal{X})} \mathcal{F}_Q[\varphi - \varphi] = \sup_{\varphi \in C_b^2(\mathcal{X})} \mathcal{F}_Q[\varphi]. $$

Let $\psi \in C^\infty(\mathbb{R})$ be such that

$$ \psi(u) = \begin{cases} -1 & \text{for } u \le -3/2; \\ u & \text{for } |u| \le 1/2; \\ 1 & \text{for } u \ge 3/2, \end{cases} $$

<footnote n="13">¹³ Since the functionals $\mathcal{F}_Q$ and $\hat{\mathcal{F}}_Q$ are invariant under addition of a constant, the condition $\varphi \le 0$ can be replaced by $\varphi$ being bounded above. This will be used without further comment in the sequel.</footnote>
and set $\psi_r(u) = r\psi(u/r)$, $\chi_r(u) = u - \psi_r(u)$ for $r > 0$. Then $\phi_r = \psi_r \circ \phi \in C_b^2(\mathcal{X})$, and it suffices to show that $\tilde{\mathcal{F}}_Q[\varphi - \phi_r] \to \tilde{\mathcal{F}}_Q[\varphi - \phi]$ as $r \to \infty$ for any $\varphi \in C_b^2(\mathcal{X})$. From

$$ \tilde{\mathcal{F}}_Q[\varphi - \phi_r] - \tilde{\mathcal{F}}_Q[\varphi - \phi] = \langle L(\chi_r \circ \phi) + \gamma(\chi_r \circ \phi) + 2\gamma(\phi_r - \varphi, \chi_r \circ \phi), Q \rangle, $$

an explicit calculation, using the chain rule (3.18), yields that

$$ |\tilde{\mathcal{F}}_Q[\varphi - \phi_r] - \tilde{\mathcal{F}}_Q[\varphi - \phi]| \le 4 \int_{|\phi|>r/2} (|\sigma| + \gamma(\varphi) + \gamma(\phi)) \, dQ, $$

and the result follows.

(4) For $R > 0$, define $h_R \in C^2(\mathbb{R})$ by

$$ h_R(u) := \begin{cases} -3R/2 & \text{for } u < -2R; \\ -3R/2 - 4u - 6u^2/R - 3u^3/R^2 - u^4/2R^3 & \text{for } -2R \le u \le -R; \\ u & \text{for } u > -R. \end{cases} $$

One easily shows that

$$ -\frac{3R}{2} \le h_R(u), \quad 0 \le h'_R(u) \le 1, \quad 0 \le h''_R(u) \le \frac{3}{2R}. \quad (3.43) $$

By Lemma 3.1, for sufficiently small $\epsilon$ there is a positive constant $c$ such that the $C^2$-function defined by $\Psi(x) = S(x)/2 + \epsilon p \cdot \Gamma^{-1}\nabla V(q) + (h_R \circ \phi(x))/4$ is bounded below and satisfies

$$ -(L\Psi + \gamma(\Psi)) \ge -\frac{1}{c} + c|\nabla S|^2 + \frac{1}{4} \left( h'_{R} \left( 1 - \frac{3}{4} h'_{R} \right) - h''_{R} \right) \circ \phi \gamma(\phi) $$

for all $R > 0$. It follows that

$$ -\frac{1}{c} + c \int |\nabla S|^2 dQ + \frac{1}{4} \int \left( h'_{R} \left( 1 - \frac{3}{4} h'_{R} \right) - h''_{R} \right) \circ \phi \gamma(\phi) dQ \le \int (-L\Psi - \gamma(\Psi)) dQ \le I(Q). $$

Thus, if $I(Q) < \infty$, then there is $\alpha < \infty$ such that

$$ \int \left( \frac{1}{4} h'_{R} - h''_{R} \right) \circ \phi \gamma(\phi) dQ \le \int \left( h'_{R} \left( 1 - \frac{3}{4} h'_{R} \right) - h''_{R} \right) \circ \phi \gamma(\phi) dQ \le \alpha $$

for all $R > 0$. Using the explicit form of $h_R$ and the inequalities (3.43), one easily shows that

$$ \frac{1}{4} h'_{R}(u) - h''_{R}(u) \ge f_{R}(u) := \begin{cases} 0 & \text{for } u < -2R; \\ -\frac{3}{2R} & \text{for } -2R \le u < -R; \\ \frac{1}{4} & \text{for } u \ge -R, \end{cases} $$

from which we deduce that
$$ \langle f_R \circ \phi \gamma(\phi), Q \rangle \le \alpha $$

for all $R > 0$. Now we set

$$ f(u) := \sum_{n=0}^{\infty} 2^{-n} f_{2^n R}(u) = \sum_{2^n \ge -u/R} 2^{-n-2} - \sum_{-u/2R \le 2^n < -u/R} \frac{3}{R} 2^{-2n-1}, $$

so that

$$ \langle f \circ \phi \gamma(\phi), Q \rangle \le \frac{\alpha}{2}. $$

Observing that $f(u) = \frac{1}{2}$ for $u \ge -R$, while

$$ f(u) \ge \sum_{n \ge \log_2(|u|/R)} 2^{-n-2} - \sum_{n \ge \log_2(|u|/R)-1} \frac{3}{R} 2^{-2n-1} \ge \frac{R}{4|u|} - \frac{8R}{|u|^2} = \frac{R}{4|u|} \left(1 - \frac{32}{|u|}\right) $$

is non-negative for $u < -R \le -32$, we conclude that for $R \ge 32$ one has

$$ \frac{1}{2} \int_{\phi \ge -R} \gamma(\phi) dQ \le \langle f \circ \phi \gamma(\phi), Q \rangle \le \frac{\alpha}{2}. $$

Letting $R \to \infty$ yields the result. $\square$

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**Chapter 4**

**Level-2 LDP and Fluctuation Theorem**
**for Chaotic Maps**

**Abstract** We restrict our attention to chaotic dynamical systems on compact metric spaces and derive FT/FR for their periodic orbit ensembles. The discussion is based on Ruelle–Walters thermodynamic formalism, which is reviewed in some detail. This chapter is motivated by [MV03].

**4.1 A Class of Continuous Dynamical Systems**

Let $(M, d)$ be a compact metric space with Borel $\sigma$-algebra $\mathfrak{M}$. We recall that $C(M)$ denotes the Banach space of real-valued continuous functions on $M$ with the sup-norm $\|\cdot\|$. The set $\mathcal{P}(M)$ of Borel probability measures on $M$ is equipped with the topology of weak convergence (denoted $\rightarrow$) and the corresponding Borel $\sigma$-algebra. This makes $\mathcal{P}(M)$ a convex compact metrizable subspace of the dual of $C(M)$ endowed with its weak-$*$ topology. Given $v \in C(M)$ and $Q \in \mathcal{P}(M)$, we denote by $\langle v, Q \rangle$ the integral of $v$ with respect to $Q$. The Dirac mass at $x \in M$ is denoted by $\delta_x \in \mathcal{P}(M)$.

In the following, we shall always assume:

(C) $\varphi: M \to M$ is a continuous map.

On occasions, we shall strengthen the above standing assumption to:

(H) $\varphi: M \to M$ is a homeomorphism,

and we shall always explicitly mention when Condition (H) is assumed. We set $\mathcal{T} = \mathbb{N}$ in case (C) and $\mathcal{T} = \mathbb{Z}$ in case (H), so that, in both cases, the orbit of $x \in M$ is $(\varphi^t(x))_{t \in \mathcal{T}}$. The set of $\varphi$-invariant elements of $\mathcal{P}(M)$ is denoted by $\mathcal{P}_{\varphi}(M)$. It is a closed convex subset of $\mathcal{P}(M)$ and a Choquet simplex whose extreme points are the elements of $\mathcal{E}_{\varphi}(M)$, the set of $\varphi$-ergodic measures. We denote by $h_{\text{Top}}(\varphi)$ the topological entropy of $\varphi$ and by $h_{\varphi}(\mathbb{P})$ the Kolmogorov–Sinai entropy of $\mathbb{P} \in \mathcal{P}_{\varphi}(M)$ w.r.t. $\varphi$. The following assumptions will play a central role in our discussion.
(USCE) $h_{\text{Top}}(\varphi) < \infty$, and the map $\mathcal{P}_{\varphi}(M) \ni \mathbb{P} \mapsto h_{\varphi}(\mathbb{P})$ is upper semicontinuous.

(ED) The set $\mathcal{E}_{\varphi}(M)$ is entropy-dense in $\mathcal{P}_{\varphi}(M)$, i.e., for any $\mathbb{P} \in \mathcal{P}_{\varphi}(M)$ there exists a sequence $(\mathbb{P}_n)_{n \in \mathbb{N}} \subset \mathcal{E}_{\varphi}(M)$ such that

$$ \mathbb{P}_n \to \mathbb{P} \quad \text{and} \quad h_{\varphi}(\mathbb{P}_n) \to h_{\varphi}(\mathbb{P}) $$

as $n \to \infty$

The topological pressure of $v \in C(M)$ w.r.t. $\varphi$ is

$$ \mathfrak{p}_{\varphi}(v) := \sup_{\mathbb{P} \in \mathcal{P}_{\varphi}(M)} (\langle v, \mathbb{P} \rangle + h_{\varphi}(\mathbb{P})). \quad (4.1) $$

Moreover, $\mathbb{P} \in \mathcal{P}_{\varphi}(M)$ is called equilibrium measure for $v$ whenever it achieves the supremum in this variational formula, i.e.,

$$ \langle v, \mathbb{P} \rangle + h_{\varphi}(\mathbb{P}) = \mathfrak{p}_{\varphi}(v). \quad (4.2) $$

**Proposition 4.1** Under Condition (C) the following hold:

(1) $\mathfrak{p}_{\varphi}(0) = h_{\text{Top}}(\varphi)$.

(2) Either $\mathfrak{p}_{\varphi}(v)$ is infinite for all $v \in C(M)$, or $h_{\text{Top}}(\varphi)$ is finite and so is $\mathfrak{p}_{\varphi}(v)$ for any $v \in C(M)$. Moreover, in the latter case, the map $C(M) \ni v \mapsto \mathfrak{p}_{\varphi}(v) \in \mathbb{R}$ is convex and satisfies

$$ |\mathfrak{p}_{\varphi}(v) - \mathfrak{p}_{\varphi}(u)| \le \|v - u\| $$

for any $u, v \in C(M)$.

If, in addition, Condition (USCE) holds, then one further has:

(3) The dual variational expression

$$ h_{\varphi}(\mathbb{P}) = \inf_{v \in C(M)} (\mathfrak{p}_{\varphi}(v) - \langle v, \mathbb{P} \rangle) $$

holds for all $\mathbb{P} \in \mathcal{P}_{\varphi}(M)$.

(4) For any $v \in C(M)$, the set of equilibrium measures for $v$ is a non-empty compact Choquet simplex. It is also a face of $\mathcal{P}_{\varphi}(M)$ whose extreme points are ergodic measures.

(5) For any $\mathbb{P} \in \mathcal{E}_{\varphi}(M)$ there exists $v \in C(M)$ such that $\mathbb{P}$ is the unique equilibrium measure for $v$.

We refer the reader to [Wal82] for more details and basic properties of the pressure and equilibrium states. Parts (1)–(3) of Proposition 4.1 can be found there. For Parts (4)–(5) see [Rue04, Theorem 2] and [Phe02, Theorem 1].

We finish this section with a brief discussion of our main assumptions (USCE) and (ED).

Given $\varepsilon > 0$ and an integer $t \ge 1$ we denote by
$$B_t(x, \varepsilon) := \{y \in M \mid d(\varphi^s(x), \varphi^s(y)) < \varepsilon \text{ for } 0 \le s < t\}$$

the Bowen ball of radius $\varepsilon$ centered at $x \in M$. A map $\varphi$ satisfying Condition (C) (respectively Condition (H)) is called forward expansive (respectively expansive) whenever there exists $r > 0$ such that,

$$\Gamma(x, r) := \bigcap_{t \in \mathcal{T}} \overline{B_t(x, r)} = \{x\} \quad (4.3)$$

for any $x \in M$. The number $r$ is called the expansiveness constant of $\varphi$. It depends on the metric $d$, but expansiveness itself only depends on the induced topology of $M$. Expansiveness is sufficient to ensure Condition (USCE), see [Wal82, Corollary 7.11.1] and [Bar11, Section 2.4].

For $I = [a, b] \subset \mathcal{T}$, we denote orbit segments by

$$\varphi^I(x) := (\varphi^t(x))_{t \in I},$$

and call specification a finite family of such segments

$$\xi = (\varphi^{I_i}(x_i))_{i \in [1, n]}.$$

The integers $n$ and $L(\xi) := \max\{|t-t'| : t, t' \in I \in [1, n] \}$ are called the rank and the length of $\xi$ respectively. The specification $\xi$ is N-separated whenever

$$d(I_i, I_j) = \min_{t_i \in I_i, t_j \in I_j} |t_i - t_j| \ge N$$

for all distinct $i, j \in [1, n]$. It is $\varepsilon$-shadowed by $x \in M$ whenever

$$\max_{i \in [1, n]} \max_{t \in I_i} d(\varphi^t(x), \varphi^t(x_i)) < \varepsilon.$$

(S) $\varphi$ has the Specification Property<fnref n="1" /> if, for any $\delta > 0$, there is $N(\delta) \ge 1$ such that any $N(\delta)$-separated specification $\xi = (\varphi^{I_i}(x_i))_{i \in [1, n]}$ is $\delta$-shadowed by some point $x \in M$.

Property (S) implies Condition (ED), see [EKW94, Theorem B] whose proof, given for case (H), extends without change to case (C). For a detailed discussion of Property (ED), we refer the reader to [Com17].

## 4.2 A Large Deviation Principle for Empirical Measures

To any $x \in M$ we associate the sequence $(\mu_t^x)_{t \in \mathbb{N}}$ of empirical measures, defined by

<footnote n="1">¹ Many variants of the specification property appear in the literature; see [KLO16] for a review.</footnote>
$$ \mu_t^x := \frac{1}{t} \sum_{0 \le s < t} \delta_{\varphi^s(x)}, \qquad (4.4) $$

so that

$$ \langle v, \mu_t^x \rangle = \frac{1}{t} S_t v(x) $$

where

$$ S_t v := v + v \circ \varphi + \dots + v \circ \varphi^{t-1}. $$

Given $\mathbb{P} \in \mathcal{P}_\varphi(M)$, we shall consider $\mu_t^x$ as a $\mathcal{P}(M)$-valued random variable. The main result in this section is the LDP for the sequence $(\mu_t^x)_{t \in \mathbb{N}}$ w.r.t appropriate sequences of measures.

**Definition 4.2** Given $v \in C(M)$, we say that $(\mathbb{P}_t)_{t \in \mathbb{N}} \subset \mathcal{P}(M)$ is a $(v, \varphi)$-sequence whenever

$$ \lim_{t \to \infty} \frac{1}{t} \log \langle e^{S_t u}, \mathbb{P}_t \rangle = \mathfrak{p}_\varphi(v+u) - \mathfrak{p}_\varphi(v) $$

holds for all $u \in C(M)$.

The following is a direct application of [Com09, Theorem 5.2].

**Theorem 4.3** Assume (C), (USCE), (ED) and let $(\mathbb{P}_t)_{t \in \mathbb{N}}$ be a $(v, \varphi)$-sequence for some $v \in C(M)$. Then, w.r.t. this sequence, the empirical measures $(\mu_t^x)_{t \in \mathbb{N}}$ satisfy the LDP

$$ - \inf_{Q \in \bar{\Gamma}} \| (Q) \| \le \liminf_{t \to \infty} \frac{1}{t} \log \mathbb{P}_t \{ \mu_t^x \in \Gamma \} \le \limsup_{t \to \infty} \frac{1}{t} \log \mathbb{P}_t \{ \mu_t^x \in \Gamma \} \le - \inf_{Q \in \bar{\Gamma}} \| (Q) \| $$

for any Borel set $\Gamma \subset \mathcal{P}(M)$.<fnref n="2" /> The rate $\| \cdot \| : \mathcal{P}(M) \to [0, \infty]$ is given by

$$ \| (Q) \| := \begin{cases} \mathfrak{p}_\varphi(v) - \langle v, Q \rangle - h_\varphi(Q) & \text{if } Q \in \mathcal{P}_\varphi(M); \\ +\infty & \text{otherwise.} \end{cases} $$

We note that since the entropy function $Q \mapsto h_\varphi(Q)$ is affine on $\mathcal{P}_\varphi(M)$,<fnref n="3" /> so is the rate $\| \cdot \|$. The inequality $\| (Q) \| \ge 0$ follows from Proposition 4.1 (3). Moreover, comparing with (4.2) we conclude that $\| (Q) \| = 0$ iff $Q$ is an equilibrium measure for $v$. Finally, we observe that $\| \cdot \|$ is the restriction to $\mathcal{P}(M)$ of the Fenchel-Legendre transform of $u \mapsto \mathfrak{p}_\varphi(v+u) - \mathfrak{p}_\varphi(v)$.

By contraction, one immediately obtains the following

**Corollary 4.4** Under the assumptions of Theorem 4.3, and for any $u \in C(M)$, one has the LDP

$$ - \inf_{r \in \bar{B}} I(r) \le \liminf_{t \to \infty} \frac{1}{t} \log \mathbb{P}_t \{t^{-1} S_t u \in B\} \le \limsup_{t \to \infty} \frac{1}{t} \log \mathbb{P}_t \{t^{-1} S_t u \in B\} \le - \inf_{r \in \bar{B}} I(r) $$

<footnote n="2">² Recall that $\bar{\Gamma}/\bar{\Gamma}$ denotes the interior/closure of $\Gamma$.</footnote>

<footnote n="3">³ See [Wal82, Theorem 8.1].</footnote>
for any Borel set $B \subset \mathbb{R}$. The good convex rate $I: \mathbb{R} \to [0, \infty]$ is given by

$$I(r) = \inf \{ \|Q\| \mid Q \in \mathcal{P}_{\varphi}(M), \langle u, Q \rangle = r \}.$$

We finish this section with some general remarks on $(v, \varphi)$-sequences.

**Remark 1.** If $(\mathbb{P}_t)_{t \in \mathbb{N}}$ is a $(v, \varphi)$-sequence and $v' \in C(M)$, then one easily checks that the sequence $(\mathbb{P}_t^{(v')})_{t \in \mathbb{N}}$ defined by

$$\frac{d\mathbb{P}_t^{(v')}}{d\mathbb{P}_t} = \frac{e^{S_t v'}}{\langle e^{S_t v'}, \mathbb{P}_t \rangle}$$

is a $(v + v', \varphi)$-sequence.

**Remark 2.** If $(\mathbb{P}_t)_{t \in \mathbb{N}}$ is a $(v, \varphi)$-sequence and $(v_t)_{t \in \mathbb{N}} \subset C(M)$ is such that

$$\|v_t - S_t v\| = o(t)$$

as $t \to \infty$, then it is again straightforward to check that the sequence $(\tilde{\mathbb{P}}_t)_{t \in \mathbb{N}}$ defined by

$$\frac{d\tilde{\mathbb{P}}_t}{d\mathbb{P}_t} = \frac{e^{v_t - S_t v}}{\langle e^{v_t - S_t v}, \mathbb{P}_t \rangle}$$

is also a $(v, \varphi)$-sequence. In view of the equivalence of asymptotically additive sequences with additive ones [Cun20], this shows that Theorem 4.3 and Corollary 4.4 also apply to asymptotically additive sequences.

## 4.3 Periodic Orbit Ensemble

In order to exploit the results of the preceding section, we need some concrete scheme to construct $(v, \varphi)$-sequences. In this section we shall invoke the following elementary result:

**Lemma 4.5** If the sequence $(E_t)_{t \in \mathbb{N}}$ of subsets of $M$ is such that, for all $u \in C(M)$,

$$\lim_{t \to \infty} \frac{1}{t} \log \sum_{x \in E_t} e^{S_t u(x)} = \mathfrak{p}_{\varphi}(u), \qquad (4.5)$$

then, for any $v \in C(M)$, setting

$$\mathbb{P}_t := Z_t^{-1} \sum_{x \in E_t} e^{S_t v(x)} \delta_x, \quad Z_t = Z_t(v) := \sum_{x \in E_t} e^{S_t v(x)}, \qquad (4.6)$$

defines a $(v, \varphi)$-sequence.

Among the possible candidates for the subsets $E_t$, we will consider the sets

$$\mathrm{Fix}(\varphi^t) := \{x \in M \mid \varphi^t(x) = x\}$$
of *t*-periodic orbits, and call *periodic orbit ensemble* the corresponding measures (4.6). The significance of periodic orbits in the modern theory of dynamical systems goes back to seminal works of Bowen [Bow70] and Manning [Man71]. In the context of the FT, periodic orbits played an important role in the early numerical works [ECM93]. Ruelle's proof of the Gallavotti-Cohen fluctuation theorem for Anosov diffeomorphisms [Rue99] was centered around periodic orbits. Further insights were obtained in [MV03] where, following the general scheme of [LS99; Mae99], the pairs $(\mathbb{P}_t, \hat{\mathbb{P}}_t)$ and the entropy production observable $\sigma$ were introduced, and the transient fluctuation relation was discussed. The work [MV03] primarily concerned Gibbsian type FT for regular potentials $v$. Finally, let us remark that in the context of lattice spin systems periodic orbits arise naturally from the consideration of periodic boundary conditions. Other boundary conditions lead to asymptotically additive perturbations to which the last remark of Section 4.2 apply. Thus, the periodic orbit ensemble coincides with the canonical ensemble of statistical mechanics.

To formulate the main result of this section, let us introduce the following condition.

(PAP) $\varphi$ has the Periodic Approximation Property whenever $E_t = \text{Fix}(\varphi^t)$ is finite for all $t \ge 1$ and satisfies Relation (4.5) for all $u \in C(M)$.

Theorem 4.3 has the immediate

**Corollary 4.6** Suppose that Conditions (C), (ED), (USCE) and (PAP) are satisfied. Then, for any $v \in C(M)$, the conclusions of Theorem 4.3 and Corollary 4.4 hold for the periodic orbit ensemble

$$ \mathbb{P}_t := Z_t^{-1} \sum_{x \in \text{Fix}(\varphi^t)} e^{S_t v(x)} \delta_x, \quad Z_t = Z_t[v] := \sum_{x \in \text{Fix}(\varphi^t)} e^{S_t v(x)}. $$

As seen in Section 4.1, if $\varphi$ is forward expansive or expansive and has the specification property (S), then Conditions (ED) and (USCE) hold. To describe a condition ensuring (PAP) we need to introduce a variant of the specification property (S).

(WPS) $\varphi$ has the Weak Periodic Specification Property if, for any $\delta > 0$, there is a sequence of integers $(m_\delta(t))_{t \ge t_0}$ such that for all $t \ge t_0$,

$$ 0 \le m_{\delta}(t) < t, \quad \lim_{\delta \downarrow 0} \lim_{t \to \infty} \frac{m_{\delta}(t)}{t} = 0, $$

and for any $x \in M$, we have

$$ \text{Fix}(\varphi^t) \cap B_{t-m_\delta(t)}(x, \delta) \neq \emptyset. $$
**Theorem 4.7** If $\varphi$ is forward expansive or expansive and satisfies Condition (WPS), then Condition (PAP) holds.

**Proof.** The argument is based on characterizations of the pressure that we recall now, referring the reader to [Wal82, Section 9.1] for details.

Let $E$ be a finite subset of $M$ and $\varepsilon > 0$. $E$ is called $(\varepsilon, t)$-separated if $y \notin B_t(x, \varepsilon)$ for any distinct $x, y \in E$, and $(\varepsilon, t)$-spanning if the family $(B_t(x, \varepsilon))_{x \in E}$ covers $M$. For $v \in C(M)$ define

$$ \underline{P}_t(v, \varepsilon) := \inf\left\{\sum_{x \in E} e^{S_t v(x)} \mid E \subset M \text{ is } (\varepsilon, t)\text{-spanning}\right\}, \\ \bar{P}_t(v, \varepsilon) := \sup\left\{\sum_{x \in E} e^{S_t v(x)} \mid E \subset M \text{ is } (\varepsilon, t)\text{-separated}\right\}. \qquad (4.7) $$

For any $v \in C(M)$, the four limiting quantities

$$ \lim_{\varepsilon \downarrow 0} \limsup_{t \to \infty} \frac{1}{t} \log \underline{P}_t(v, \varepsilon), \quad \lim_{\varepsilon \downarrow 0} \liminf_{t \to \infty} \frac{1}{t} \log \bar{P}_t(v, \varepsilon), \\ \lim_{\varepsilon \downarrow 0} \limsup_{t \to \infty} \frac{1}{t} \log \bar{P}_t(v, \varepsilon), \quad \lim_{\varepsilon \downarrow 0} \liminf_{t \to \infty} \frac{1}{t} \log \underline{P}_t(v, \varepsilon), \qquad (4.8) $$

coincide with the topological pressure $p_\varphi(v)$.

Let $r$ denote the expansiveness constant of $\varphi$. For fixed $t \ge 1$, uniform continuity and periodicity imply that there is $\delta > 0$ such that if $x, y \in \text{Fix}(\varphi^t)$ and $d(x, y) \le \delta$, then $d(\varphi^t(x), \varphi^t(y)) \le r$ for all $t \in \mathcal{T}$. It follows that $d(x, y) > \delta$ for any distinct $x, y \in \text{Fix}(\varphi^t)$ and compactness implies that $\text{Fix}(\varphi^t)$ may contain only finitely many points. Let $v \in C(M)$, and let $\varepsilon \in ]0, r[$. Then, in view of periodicity, for any $t \ge 1$ the set $\text{Fix}(\varphi^t)$ is $(\varepsilon, t)$-separated. It follows that

$$ \sum_{x \in \text{Fix}(\varphi^t)} e^{S_t v(x)} \le \bar{P}_t(v, \varepsilon) $$

whence, by the above representation of the pressure, we conclude that

$$ \limsup_{t \to \infty} \frac{1}{t} \log \sum_{x \in \text{Fix}(\varphi^t)} e^{S_t v(x)} \le p_\varphi(v). \qquad (4.9) $$

Since $\varphi$ satisfies Condition (WPS), the set $\text{Fix}(\varphi^t)$ is $(\varepsilon, t - m_\varepsilon(t))$-spanning. It follows that

$$ \sum_{x \in \text{Fix}(\varphi^t)} e^{S_t v(x)} \ge \underline{P}_{t-m_\varepsilon(t)}(v, \varepsilon) $$

Combining this with the above representation of the pressure, and using the condition on $m_\varepsilon(t)$, we see that

$$ \liminf_{t \to \infty} \frac{1}{t} \log \sum_{x \in \text{Fix}(\varphi^t)} e^{S_t v(x)} \ge p_\varphi(v), $$
which, together with (4.9), ends the proof.

<figure><img src="image_16.png" /></figure>

By Proposition 4.1 (4), Condition (USCE) ensures that the set of equilibrium states for $v \in C(M)$ is never empty. A non-constructive proof just argues that an upper semicontinuous function achieves its maximum on a compact set. The following result provides an explicit construction.

**Proposition 4.8** If Conditions (USCE) and (PAP) hold, then any weak limit point of the periodic orbit ensemble $(\mathbb{P}_t)_{t \in \mathbb{N}}$ is an equilibrium state for $v$.

**Proof.** Let the sequence $t_n \to \infty$ be such that $\mathbb{P}_{t_n} \to \mathbb{P}$. Since $\mathbb{P}_t \in \mathcal{P}_\varphi(M)$ for all $t$, so does $\mathbb{P}$. In view of Hölder's inequality, the function $v \mapsto \log Z_t[v] \in \mathbb{R}$ is convex. Moreover, for any $u, v \in C(M)$, the function $\alpha \mapsto f(\alpha) := \log Z_t[v + \alpha u]$ is differentiable, and its derivative at zero is given by $f'(0) = \langle S_t u, \mathbb{P}_t \rangle$. By convexity, we have $f(1) - f(0) \ge f'(0)$, which gives $\log Z_t[v+u] - \log Z_t[v] \ge \langle S_t u, \mathbb{P}_t \rangle$. Setting $t = t_n$ in the above inequality, dividing it by $t_n$, and passing to the limit yields $\mathbb{P}_\varphi(v+u) - \mathbb{P}_\varphi(v) \ge \langle u, \mathbb{P} \rangle$ which can be rewritten as

$$ \mathbb{P}_{\varphi}(v+u) - \langle v+u, \mathbb{P} \rangle \ge \mathbb{P}_{\varphi}(v) - \langle v, \mathbb{P} \rangle. $$

Taking the supremum over $u \in C(M)$ and invoking Proposition 4.1 (3) completes the proof.

<figure><img src="image_17.png" /></figure>

## 4.4 Fluctuation Theorems for the Periodic Orbit Ensemble

In this subsection we describe FRs and FTs for the periodic orbit ensemble. We consider first the general setting of Definition 2.2: let $v, \hat{v} \in C(M)$ and denote by $(\mathbb{P}_t)_{t \in \mathbb{N}}$ and $(\hat{\mathbb{P}}_t)_{t \in \mathbb{N}}$ the corresponding periodic orbit ensemble for the dynamical system $(M, \varphi)$. Without loss of generality, we assume that $\mathbb{P}_\varphi(v) = \mathbb{P}_\varphi(\hat{v}) = 0$. Under the assumptions of Corollary 4.6, we deduce from Theorem 4.3 that the empirical measures $(\mu'_t)_{t \in \mathbb{N}}$ satisfy the LDP w.r.t. both $(\mathbb{P}_t)_{t \in \mathbb{N}}$ and $(\hat{\mathbb{P}}_t)_{t \in \mathbb{N}}$, with respective rate functions

$$ \mathbb{I}(\mathbb{Q}) = -\langle v, \mathbb{Q} \rangle - h_{\varphi}(\mathbb{Q}), \quad \hat{\mathbb{I}}(\mathbb{Q}) = -\langle \hat{v}, \mathbb{Q} \rangle - h_{\varphi}(\mathbb{Q}). $$

Thus, we have the level-2 FT with the FR

$$ \hat{\mathbb{I}}(\mathbb{Q}) = \mathbb{I}(\mathbb{Q}) + \text{ep}(\mathbb{Q}), $$

where, obviously, $\text{ep}(\mathbb{Q}) = \langle v - \hat{v}, \mathbb{Q} \rangle$. Recalling that $\mathbb{I}(\mathbb{Q}) = 0$ iff $\mathbb{Q}$ is an equilibrium measure for $v$, we deduce that $\text{ep}(\mathbb{Q}) \ge 0$ for any such measure, while $\text{ep}(\hat{\mathbb{Q}}) \le 0$ whenever $\hat{\mathbb{Q}}$ is an equilibrium measure for $\hat{v}$.

The entropy production associated to the pair $(\mathbb{P}_t, \hat{\mathbb{P}}_t)$ is

$$ \sigma_t = \log \frac{d\mathbb{P}_t}{d\hat{\mathbb{P}}_t} = S_t(v - \hat{v}) - \log \frac{Z_t[v]}{Z_t[\hat{v}]}. $$
Condition (PAP) yields

$$ \lim_{t \to \infty} \frac{1}{t} \log \frac{Z_t[\nu]}{Z_t[\hat{\nu}]} = \mathfrak{p}_{\varphi}(\nu) - \mathfrak{p}_{\varphi}(\hat{\nu}) = 0, $$

which shows that $(\sigma_t)_{t \in \mathbb{N}}$ is physically equivalent to $(\tilde{\sigma}_t)_{t \in \mathbb{N}}$, where

$$ \tilde{\sigma}_t := S_t(\nu - \hat{\nu}). $$

Thus, $(\frac{1}{t}\sigma_t)_{t \in \mathbb{N}}$ and $(\frac{1}{t}\tilde{\sigma}_t)_{t \in \mathbb{N}}$ satisfy the same LDP w.r.t. $(\mathbb{P}_t)_{t \in \mathbb{N}}$ and $(\hat{\mathbb{P}}_t)_{t \in \mathbb{N}}$ with respective rate functions

$$ I(s) = \inf\{\|\mathbb{Q}\| \mid \mathbb{Q} \in \mathcal{P}_{\varphi}(M), \mathrm{ep}(\mathbb{Q}) = s\}, \qquad \hat{I}(s) = \inf\{\|\hat{\mathbb{Q}}\| \mid \hat{\mathbb{Q}} \in \mathcal{P}_{\varphi}(M), \mathrm{ep}(\hat{\mathbb{Q}}) = s\}. $$

Moreover, the FR

$$ \hat{I}(s) = I(s) + s $$

holds for all $s \in \mathbb{R}$. The entropic pressures

$$ e(\alpha) = \lim_{t \to \infty} \frac{1}{t} \log \langle e^{-\alpha \sigma_t}, \mathbb{P}_t \rangle, \qquad \hat{e}(\alpha) = \lim_{t \to \infty} \frac{1}{t} \log \langle e^{\alpha \sigma_t}, \hat{\mathbb{P}}_t \rangle, $$

exist for all $\alpha \in \mathbb{R}$, are given by

$$ e(\alpha) = \mathfrak{p}_{\varphi}((1-\alpha)\nu + \alpha\hat{\nu}), \qquad \hat{e}(\alpha) = \mathfrak{p}_{\varphi}((1-\alpha)\hat{\nu} + \alpha\nu), $$

and satisfy

$$ \hat{e}(\alpha) = e(1-\alpha). $$

They are related to the rate function by Fenchel-Legendre transformation

$$ I(s) = \sup_{\alpha \in \mathbb{R}} (\alpha s - e(-\alpha)), \qquad \hat{I}(s) = \sup_{\alpha \in \mathbb{R}} (\alpha s - \hat{e}(\alpha)). $$

The interpretation of these results rests on the general picture described in Section 2.7: let $\mathbb{P}$ and $\hat{\mathbb{P}}$ denote equilibrium states for $\nu$ and $\hat{\nu}$ such that $\mathbb{P}_{t_n} \to \mathbb{P}$ and $\hat{\mathbb{P}}_{t_n} \to \hat{\mathbb{P}}$ for some subsequence $t_n \to \infty$.<fnref n="4" /> Then $\mathbb{P}_{t_n}$ and $\hat{\mathbb{P}}_{t_n}$ are equivalent measures for all $n$, but their limit points $\mathbb{P}$ and $\hat{\mathbb{P}}$ may be mutually singular, and they definitely are whenever $\mathrm{ep}(\mathbb{P}) > 0$. Then the various exponents described in Section 2.7 quantify the rate at which the supports of $\mathbb{P}_{t_n}$ and $\hat{\mathbb{P}}_{t_n}$ separate.

Next, let us specialize the discussion to pairs of periodic orbit ensemble in involution. To this end, we make the following additional assumption:

(Sym) There is a continuous involution $\theta: M \to M$ such that:

* $\varphi \circ \theta = \theta \circ \varphi$ in case (C);

* $\varphi \circ \theta = \theta \circ \varphi^{-1}$ in case (H).

<footnote n="4">⁴ In absence of phase transition, i.e., when both $\nu$ and $\hat{\nu}$ have a unique equilibrium measure, one has $\mathbb{P}_t \to \mathbb{P}$ and $\hat{\mathbb{P}}_t \to \hat{\mathbb{P}}$ as $t \to \infty$.</footnote>
For $v \in C(M)$ and $Q \in \mathcal{P}_{\varphi}(M)$, we then set $\hat{v} := v \circ \theta$ and $\hat{Q} := Q \circ \theta^{-1}$, so that $\langle \hat{v}, Q \rangle = \langle v, \hat{Q} \rangle$. Let us remark that $\hat{Q} \in \mathcal{P}_{\varphi}(M)$. Indeed, for any $v \in C(M)$ we have

$$ \langle v \circ \varphi, \hat{Q} \rangle = \langle v \circ \varphi \circ \theta, Q \rangle = \langle v \circ \theta \circ \varphi^{\pm 1}, Q \rangle = \langle v \circ \theta, Q \rangle = \langle v, \hat{Q} \rangle. $$

In case (H), Assumption (Sym) is the standard time-reversal invariance condition that appears in most of the works on the FR and FT. To the best of our knowledge, it was not previously observed that Assumption (Sym) also leads to FT in case (C). Since in the latter case $\varphi$ is not required to be invertible, this allows us to extend FR/FT to a large class of irreversible dynamical systems.

Observe that Condition (Sym) implies

$$ S_t(v \circ \theta) = \sum_{0 \le s < t} v \circ \theta \circ \varphi^s = \sum_{0 \le s < t} v \circ \varphi^{\pm s} \circ \theta = (S_t v) \circ \theta_t \quad (4.10) $$

where

$$ \theta_t := \begin{cases} \theta & \text{in case (C)}; \\ \theta \circ \varphi^{t-1} & \text{in case (H)}, \end{cases} $$

is involutive. In the case (C) this is immediate, and in the case (H) it follows from

$$ \theta_t^{-1} = \varphi^{1-t} \circ \theta^{-1} = \varphi^{1-t} \circ \theta = \theta \circ \varphi^{t-1} = \theta_t. $$

In addition, we have

**Lemma 4.9** Under Condition (Sym), the following holds for any $Q \in \mathcal{P}_{\varphi}(M)$ and any $v \in C(M)$:

$$ h_{\varphi}(\hat{Q}) = h_{\varphi}(Q), \quad (4.11) $$

$$ \mathfrak{p}_{\varphi}(\hat{v}) = \mathfrak{p}_{\varphi}(v). \quad (4.12) $$

Moreover, $\theta$ and $\theta_t$ act bijectively on $\text{Fix}(\varphi^t)$.

**Proof.** Since the Kolmogorov–Sinai entropy is a conjugacy invariant (see [Wal82, Theorem 4.11]), we deduce

$$ h_{\varphi^{\pm 1}}(Q) = h_{\varphi}(Q \circ \theta^{-1}) = h_{\varphi}(\hat{Q}) $$

from the fact that $\theta^{-1} \circ \varphi \circ \theta = \varphi^{\pm 1}$, which proves (4.11) in case (C). To deal with case (H), it suffices to take into account [Wal82, Theorem 4.13] which states that $h_{\varphi^{-1}}(Q) = h_{\varphi}(Q)$.

In order to prove (4.12), we observe that, by (4.1) and (4.11),

$$ \begin{aligned} \mathfrak{p}_{\varphi}(\hat{v}) &= \sup_{Q \in \mathcal{P}_{\varphi}(M)} (\langle v \circ \theta, Q \rangle + h_{\varphi}(Q)) = \sup_{Q \in \mathcal{P}_{\varphi}(M)} (\langle v, \hat{Q} \rangle + h_{\varphi}(\hat{Q})) \\ &= \sup_{Q \in \mathcal{P}_{\varphi}(M)} (\langle v, Q \rangle + h_{\varphi}(Q)) = \mathfrak{p}_{\varphi}(v). \end{aligned} $$
To prove the last statement, it suffices to show that Fix($\varphi^t$) is invariant under $\theta$, a property that follows immediately from Condition (Sym). $\square$

In the following we shall again assume, without loss of generality, that $v \in C(M)$ is such that $p_{\varphi}(v) = 0$. Relation (4.10) and the last statement of Lemma 4.9 immediately imply that $Z_t[\hat{v}] = Z_t[v] = Z_t$ and<fnref n="5" />

$$ \hat{\mathbb{P}}_t = \mathbb{P}_t \circ \theta_t^{-1} = Z_t^{-1} \sum_{x \in \text{Fix}(\varphi^t)} e^{S_t \hat{v}(x)} \delta_x $$

for all $t$. In particular, the measures $\mathbb{P}_t$ and $\hat{\mathbb{P}}_t$ are equivalent, and the entropy production

$$ \sigma_t = \log \frac{d\mathbb{P}_t}{d\hat{\mathbb{P}}_t} = S_t(v - \hat{v}) \quad (4.13) $$

satisfies $\sigma_t \circ \theta_t = -\sigma_t$. For $\alpha \in \mathbb{R}$, the entropic pressure is

$$ e(\alpha) = \lim_{t \to \infty} \frac{1}{t} \log \langle e^{-\alpha \sigma_t}, \mathbb{P}_t \rangle = p_{\varphi}((1-\alpha)v + \alpha \hat{v}). $$

For $\mathbb{Q} \in \mathcal{P}_{\varphi}(M)$ one has $\text{ep}(\mathbb{Q}) = \langle v - \hat{v}, \mathbb{Q} \rangle = \langle v, \mathbb{Q} - \hat{\mathbb{Q}} \rangle$, so that

$$ \text{ep}(\hat{\mathbb{Q}}) = -\text{ep}(\mathbb{Q}). $$

**Theorem 4.10** In addition to the hypotheses of Corollary 4.6, suppose that Condition (Sym) holds. Then the rate function $\|\cdot\|$ of the LDP for the empirical measures (4.4) under the periodic orbit ensemble $(\mathbb{P}_t)_{t \in \mathbb{N}}$ satisfies the FR

$$ \|\hat{\mathbb{Q}}\| = \|\mathbb{Q}\| + \text{ep}(\mathbb{Q}) $$

for any $\mathbb{Q} \in \mathcal{P}_{\varphi}(M)$. Furthermore, the sequence $(\frac{1}{t}\sigma_t)_{t \in \mathbb{N}}$ under the laws $(\mathbb{P}_t)_{t \in \mathbb{N}}$ satisfies the LDP with the good convex rate function given by

$$ I(s) = \inf\{\|\mathbb{Q}\| : \mathbb{Q} \in \mathcal{P}_{\varphi}(M), \text{ep}(\mathbb{Q}) = s\}, $$

which is also the Legendre transform of the function $\alpha \mapsto e(-\alpha)$. Moreover, the FR

$$ I(-s) = I(s) + s \quad (4.14) $$

holds for all $s \in \mathbb{R}$.

<footnote n="5">⁵ The notation $\hat{\mathbb{P}}_t = \mathbb{P}_t \circ \theta_t^{-1}$ is not conflicting with our previous convention $\hat{\mathbb{Q}} = \mathbb{Q} \circ \theta^{-1}$ which we have restricted to $\mathbb{Q} \in \mathcal{P}_{\varphi}(M)$. Indeed, for such $\mathbb{Q}$ one has $\mathbb{Q} \circ \theta_t^{-1} = \mathbb{Q} \circ \theta^{-1}$.</footnote>
## 4.5 Weak Gibbs Measures

We continue our study of FT/FR in chaotic dynamical systems, now focusing on Gibbs and weak Gibbs measures.

**Definition 4.11** We say that $\mathbb{P} \in \mathcal{P}(M)$ is a weak Gibbs measure for $v \in C(M)$ if for any $t \ge 1$ and any $\varepsilon > 0$ there is $K_t(\varepsilon) \ge 1$ such that

$$K_t(\varepsilon)^{-1} e^{S_t v(x)-t\mathfrak{p}_\varphi(v)} \le \mathbb{P}(B_t(x, \varepsilon)) \le K_t(\varepsilon) e^{S_t v(x)-t\mathfrak{p}_\varphi(v)}, \quad (4.15)$$

for every $x \in M$, and

$$\lim_{\varepsilon \downarrow 0} \limsup_{t \to \infty} \frac{1}{t} \log K_t(\varepsilon) = 0. \quad (4.16)$$

If $K_t(\varepsilon)$ does not depend on $t$ and $\varepsilon$, then $\mathbb{P}$ is a Gibbs measure for $v$.

We emphasize that the definition of (weak) Gibbs measure does not require $\mathbb{P} \in \mathcal{P}_\varphi(M)$. Notice also that it follows from (4.15) that the support of $\mathbb{P}$ coincides with $M$. The following lemma shows that if the latter property is satisfied, then it suffices to require the validity of (4.15) for $\mathbb{P}$-a.e. $x \in M$. This observation is technically useful when transfer operators are used to construct weak Gibbs measures; see [Kes01, Section 2], [CLI01, Appendix B].

**Lemma 4.12** Let $v \in C(M)$ and $\mathbb{P} \in \mathcal{P}(M)$ be such that $\text{supp}(\mathbb{P}) = M$. Assume that for all $t \ge 1$ and all $\varepsilon > 0$, (4.15) holds for $\mathbb{P}$-a.e. $x \in M$, with $K_t(\varepsilon)$ satisfying (4.16). Then $\mathbb{P}$ is weak Gibbs for $v$.

**Proof.** Given $t \ge 1$ and $\varepsilon > 0$, let $A \subset M$ be a set of full $\mathbb{P}$-measure on which (4.15) holds and let $x \in M$ be an arbitrary point. Since $\bar{A} = M$, there is $x' \in A \cap B_t(x, \varepsilon/2)$, such that

$$\mathbb{P}(B_t(x, \varepsilon)) \ge \mathbb{P}(B_t(x', \varepsilon/2)) \ge K_t(\varepsilon/2)^{-1} e^{S_t v(x') - t\mathfrak{p}_\varphi(v)} \ge K'_t(\varepsilon)^{-1} e^{S_t v(x) - t\mathfrak{p}_\varphi(v) - t\gamma_\varepsilon(v)},$$

and

$$\mathbb{P}(B_t(x, \varepsilon)) \le \mathbb{P}(B_t(x', 2\varepsilon)) \le K_t(2\varepsilon) e^{S_t v(x') - t\mathfrak{p}_\varphi(v)} \le K'_t(\varepsilon) e^{S_t v(x) - t\mathfrak{p}_\varphi(v) + t\gamma_\varepsilon(v)},$$

where

$$K'_t(\varepsilon) := \max(K_t(2\varepsilon), K_t(\varepsilon/2))$$

and

$$\gamma_\varepsilon(v) := \max\{|v(x) - v(y)| \mid x, y \in M, d(x, y) \le \varepsilon\}. \quad (4.17)$$

The condition (4.16) and the uniform continuity of $v$ give

$$\lim_{\varepsilon \downarrow 0} \limsup_{t \to \infty} \frac{1}{t} \log K'_t(\varepsilon) = 0, \quad \lim_{\varepsilon \downarrow 0} \gamma_\varepsilon(v) = 0,$$

from which the statement follows. $\square$
We next show that invariant weak Gibbs measures are equilibrium states (the converse is not true in general).

**Lemma 4.13** Let $v \in C(M)$ and assume that $\mathbb{P} \in \mathcal{P}_\varphi(M)$ is a weak Gibbs measure for $v$. Then $\mathbb{P}$ is an equilibrium measure for $v$.

**Proof.** By (4.15) we have, for all $x \in M$,

$$ \lim_{\varepsilon \downarrow 0} \limsup_{t \to \infty} \frac{1}{t} (S_t v(x) - \log \mathbb{P}(B_t(x, \varepsilon))) = \mathfrak{p}_\varphi(v). \quad (4.18) $$

On the other hand, the Brin–Katok local entropy formula [BK83] implies that for $\mathbb{P}$-a.e. $x \in M$,

$$ \lim_{\varepsilon \downarrow 0} \limsup_{t \to \infty} \frac{1}{t} (S_t v(x) - \log \mathbb{P}(B_t(x, \varepsilon))) = h_\varphi(\mathbb{P}, x) + \bar{v}(x), \quad (4.19) $$

where $h_\varphi(\mathbb{P}, \cdot)$ and $\bar{v}$ are $\varphi$-invariant functions such that $\langle h_\varphi(\mathbb{P}, \cdot), \mathbb{P} \rangle = h_\varphi(\mathbb{P})$ and $\langle \bar{v}, \mathbb{P} \rangle = \langle v, \mathbb{P} \rangle$. Equating the right-hand sides of (4.18) and (4.19) and integrating with respect to $\mathbb{P}$, we arrive at (4.2). $\square$

**Lemma 4.14** Suppose that Condition (USCE) holds and that $\mathbb{P} \in \mathcal{P}(M)$ is a weak Gibbs measure for $v \in C(M)$. Then, setting $\mathbb{P}_t = \mathbb{P}$ for all $t \in \mathbb{N}$ yields a $(v, \varphi)$-sequence.

**Proof.** The proof follows that of [Kif90, Proposition 3.2] (see also [JPR11, Proposition 10.3]).

For any $u \in C(M)$, $\varepsilon > 0$, $t \ge 1$, and any $(\varepsilon, t)$-spanning set $E$, using (4.15) we derive

$$ \begin{aligned} \langle e^{S_t u}, \mathbb{P} \rangle &\le \sum_{x \in E} \langle \mathbf{1}_{B_t(x, \varepsilon)} e^{S_t u}, \mathbb{P} \rangle \le e^{t\gamma_\varepsilon(u)} \sum_{x \in E} e^{S_t u(x)} \mathbb{P}(B_t(x, \varepsilon)) \\ &\le K_t(\varepsilon) e^{-t(\mathfrak{p}_\varphi(v) - \gamma_\varepsilon(u))} \sum_{x \in E} e^{S_t(v+u)(x)}, \end{aligned} $$

where the modulus of continuity $\gamma_\varepsilon$ is as in (4.17). Taking the infimum over all $(\varepsilon, t)$-spanning set $E$ and recalling the definition (4.7) yields

$$ \frac{1}{t} \log \langle e^{S_t u}, \mathbb{P} \rangle \le \gamma_\varepsilon(u) - \mathfrak{p}_\varphi(v) + \frac{1}{t} \log K_t(\varepsilon) + \frac{1}{t} \log \underline{P}_t(v+u, \varepsilon). $$

By uniform continuity, $\gamma_\varepsilon(u) \to 0$ as $\varepsilon \downarrow 0$. Together with (4.16) and (4.8), this gives

$$ \limsup_{t \to \infty} \frac{1}{t} \log \langle e^{S_t u}, \mathbb{P} \rangle \le \mathfrak{p}_\varphi(v+u) - \mathfrak{p}_\varphi(v). $$

To prove the reverse inequality, we proceed similarly, observing that for any $(\varepsilon, t)$-separated set $E$ we have
$$ \langle e^{S_t u}, \mathbb{P} \rangle \geq K_t^{-1}(\epsilon) e^{-t(\mathfrak{p}_{\varphi}(v) + \gamma_{\epsilon}(u))} \sum_{x \in E} e^{S_t(v+u)(x)}. $$

Taking the supremum over all $(\epsilon, t)$-separated sets and repeating the above argument we obtain the desired result. □

The following result is again a direct consequence of Lemma 4.14, Theorem 4.3 and Corollary 4.4.

**Proposition 4.15** Assume that Conditions (C), (USCE) and (ED) hold and that $\mathbb{P}$ is a weak Gibbs measure for $v \in C(M)$. Then the conclusions of Theorem 4.3 and Corollary 4.4 hold with $\mathbb{P}_t$ replaced by $\mathbb{P}$.

Recall that $\sigma_t$ is defined by (4.13). By Lemma 4.14, the limit

$$ e(\alpha) := \lim_{t \to \infty} \frac{1}{t} \log \int_M e^{-\alpha \sigma_t} d\mathbb{P} $$

exists for all $\alpha \in \mathbb{R}$ and is given by

$$ e(\alpha) = \mathfrak{p}_{\varphi}((1 - \alpha)v + \alpha v \circ \theta) - \mathfrak{p}_{\varphi}(v). $$

The proof of the following result is exactly the same as that of Theorem 4.10.

**Theorem 4.16** If in addition to the hypotheses of Theorem 4.15, Condition (Sym) holds, then all the conclusions of Theorem 4.10 hold with $\mathbb{P}_t$ replaced by $\mathbb{P}$.

**Remark 1.** Our assumptions do not exclude the situation where the sequence $(\frac{1}{t}\sigma_t)_{t \in \mathbb{N}}$ is exponentially equivalent to 0, i.e., the situation where $I(0) = 0$ and $I(s) = +\infty$ for all $s \neq 0$. In this case the symmetry (4.14) is trivial, and $ep(\mathbb{P}) = 0$ (note that (4.14) always implies that $ep(\mathbb{P}) \geq 0$). It is a non-trivial question to determine whether a given system is truly out of equilibrium, i.e., if $ep(\mathbb{P}) > 0$. For $\mathbb{P} \in \mathcal{P}_{\varphi}(M)$, we have $ep(\mathbb{P}) = \langle \sigma, \mathbb{P} \rangle$, where $\sigma = v - v \circ \theta$. In the context of the chaotic hypothesis of Gallavotti-Cohen, Section 3.5.3, when $\mathbb{P}$ is the SRB measure, the inequality $ep(\mathbb{P}) > 0$ is called dissipativity and is often assumed (see [BGM00] for a model where dissipativity can be proved and [JPR11, Proposition 11.5] for generic results in this direction). See also [Rue96; Bon+06] for discussions and [MV03, Theorem 5.2], [JPR11, Corollary 10.16 (7)].

**Remark 2.** Weak Gibbs measures have been extensively studied in the recent literature on multifractal formalism; see [Cun+19] for references and additional information.

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## Chapter 5
# The Tent Map

**Abstract** To illustrate the variety and depth of mathematical tools needed to study FR/FT in the realm of chaotic dynamics, in this chapter we present a detailed study of the tent map. This example is rich enough to allow for the discussion of interesting phenomena such as phase transitions and their consequences, and provides a blueprint for related study of other interval maps.

### 5.1 Basic Properties

The tent map is the piecewise affine unimodal map of the unit interval $M := [0, 1]$ defined by

$$ \varphi(x) := 1 - |1 - 2x|. $$

It has 0 and 2/3 as unstable fixed points and the orbit of the critical point $c = 1/2$ is $c \to 1 \to 0 \to 0 \cdots$. Since 2/7 has period 3, $\varphi$ has periodic orbits of any period by Šarkovskii's theorem [CE80, Theorem II.3.10]. Moreover, $|\varphi'(x)| = 2 > 1$ for $x \neq c$ implies that all these periodic orbits are unstable. One easily shows that, for $t \in \mathbb{N}$ and $j \in [0, 2^{t-1}]$,

$$ |2^{t-1}x - j| \le \frac{1}{2} \Rightarrow \varphi^t(x) = 2|2^{t-1}x - j|, $$

from which one deduces

$$ \mathrm{Fix}(\varphi^t) = \left\{ \frac{2(j-1)}{2^t-1} \mid j \in [1, 2^{t-1}] \right\} \cup \left\{ \frac{2j}{2^t+1} \mid j \in [1, 2^{t-1}] \right\}, $$

and in particular $|\mathrm{Fix}(\varphi^t)| = 2^t$. The set of $\varphi$-periodic points is dense in $M$, more precisely $\mathrm{dist}(x, \mathrm{Fix}(\varphi^t)) \le 2^{-t}$ holds for any $x \in M$. For later reference, let us introduce

$$ \mathcal{C} := \bigcup_{n \in \mathbb{Z}_+} \varphi^{-n}(\{c\}) = \{j2^{-k} \mid k \in \mathbb{N}, j \in [0, 2^k], j \text{ odd}\}, $$

the dense set of preimages of the critical point $c$, and $M_0 := M \setminus \mathcal{C}$.
<figure><img src="image_18.png" /><figcaption>Fig. 5.1 The tent map $\varphi$, its first iterates, and the induced map $\psi$ which will be central in Section 5.3.</figcaption></figure>

Since $\varphi^t$ has $2^t$ intervals of monotonicity, the Misiurewicz formula [MS80, Theorem 1] gives the topological entropy of the tent map

$$h_{\text{Top}}(\varphi) = \log 2. \qquad (5.1)$$

The elementary calculation

$$\int_{0}^{1} f \circ \varphi(x) dx = \int_{0}^{1/2} f(2x) dx + \int_{1/2}^{1} f(2(1-x)) dx = \int_{0}^{1} f(x) dx$$

shows that Lebesgue measure $\lambda$ is $\varphi$-invariant. By Rokhlin's formula

$$h_{\varphi}(\lambda) = \int_{0}^{1/2} \log \frac{d\lambda \circ \varphi|_{[0,1/2]}}{d\lambda} d\lambda + \int_{1/2}^{1} \log \frac{d\lambda \circ \varphi|_{[1/2,1]}}{d\lambda} d\lambda = \log 2, \qquad (5.2)$$

so $\lambda$ has maximal entropy. One easily infers from the above mentioned structure of periodic orbits that $\varphi$ is topologically transitive, i.e., for any open subsets $U, V \subset M$ there exists $t \ge 0$ such that $U \cap \varphi^t(V) \ne \emptyset$. By [Hof81, Theorem 4], $\varphi$ is intrinsically ergodic: $\lambda$ is the unique $\varphi$-invariant measure of maximal entropy. In particular, $\lambda$ is the only $\varphi$-ergodic measure which is absolutely continuous w.r.t. the Lebesgue measure on $M$.

Since $\varphi$ has periodic points of period 3, it follows from [Rue17, Theorem 2.20] that it is topologically mixing: for any open subsets $U, V \subset M$ there exists $t \ge 0$ such that $U \cap \varphi^s(V) \ne \emptyset$ holds for all $s \ge t$. By [Rue17, Theorem 3.4], it follows that $\varphi$ has the specification property (S) and hence satisfies Condition (ED).

The symmetry $\varphi(1-x) = \varphi(x)$ implies that, for any $r > 0$, $1-x \in \Gamma(x, r)$ whenever $|x-c| \le r/2$ (recall the definition (4.3)). Hence, the tent map fails to be forward expansive. It is also easy to see that $\text{Fix}(\varphi^t)$ is not $(\varepsilon, t)$-separated for $\varepsilon > 2/(4^t - 1)$. Thus, we cannot apply the results of Chapter 4 directly.

We shall see how to recover Properties (USCE) and (PAP) in the next section. In Sections 5.3–5.4 we will focus on the basic properties of the tent map's pressure $p_{\varphi}(v)$. This will lead us, in Section 5.5, to formulate a simple criterion for the occurrence of phase transitions, i.e., singularities of the pressure $p_{\varphi}(\kappa v)$, as a function of the coupling strength $\kappa$. We will encounter an obstruction to the
fluctuation theorem for the tent map in Section 5.6, when discussing the last necessary ingredient of Theorem 4.10, namely Assumption (Sym). We will overcome this problem by “squaring” the system. In the last two sections we will discuss a specific example.

## 5.2 Symbolic Dynamics

The tent map admits a simple and well-known coding associated to the Markov partition of M given by $J_0 := [0, 1/2]$ and $J_1 := [1/2, 1]$. Denote by

$$x = 0.b_0 b_1 b_2 \cdots = \sum_{k \in \mathbb{Z}_+} \frac{b_k}{2^{k+1}} \qquad (5.3)$$

the binary expansion of $x \in M$. We will call the sequence $b = (b_k)_{k \in \mathbb{Z}_+}$ the **b-code** of $x$. Note that

$$1 - x = \sum_{k \in \mathbb{Z}_+} \frac{1 - b_k}{2^{k+1}} = 0.\hat{b}_0 \hat{b}_1 \hat{b}_2 \cdots,$$

where $\hat{b}_k := 1 - b_k$. Since $x \in J_{b_0}$, one has

$$\varphi(x) = \begin{cases} 0.b_1 b_2 b_3 \cdots & \text{if } b_0 = 0; \\ 0.\hat{b}_1 \hat{b}_2 \hat{b}_3 \cdots & \text{if } b_0 = 1. \end{cases}$$

Thus, denoting by $j$ the surjection $\Sigma = \{0, 1\}^{\mathbb{Z}_+} \ni \mathbf{b} = (b_k)_{k \in \mathbb{Z}_+} \mapsto x \in M$ defined in (5.3), we can write

$$\varphi \circ j = j \circ \zeta,$$

where $\zeta : \Sigma \to \Sigma$ is given by $\zeta(\mathbf{b}) := (b_0 \oplus b_{k+1})_{k \in \mathbb{Z}_+}$, the symbol $\oplus$ denoting the “exclusive or” (xor) boolean operator.<fnref n="1" />

The so-called Gray coding $\varkappa : \Sigma \to \Sigma$ defined by $\mathbf{s} = \varkappa(\mathbf{b})$, with

$$s_0 := b_0, \quad s_k := b_{k-1} \oplus b_k,$$

maps **b-code** into **s-code** in such a way that $\varkappa \circ \zeta(\mathbf{b}) = \varkappa(\mathbf{b}') = \mathbf{s}'$, where

$$s_0' := b_0' = b_0 \oplus b_1 = s_1, \quad s_k' := b_{k-1}' \oplus b_k' = (b_0 \oplus b_k) \oplus (b_0 \oplus b_{k+1}) = b_k \oplus b_{k+1} = s_{k+1}.$$

Clearly,

$$\varkappa \circ \zeta = \sigma \circ \varkappa,$$

where $\sigma$ denotes the left shift on $\Sigma$ and one easily shows that $\varkappa$ is a bijection whose inverse is given by $\mathbf{b} = \varkappa^{-1}(\mathbf{s})$, with

<footnote n="1">¹ Note that $((0, 1), \oplus)$, as an Abelian group, is isomorphic to $(\mathbb{Z}_2, +)$. In particular, $b_1 \oplus b_2 \oplus \cdots \oplus b_n$ is 1 whenever the word $b_1 \cdots b_n$ contains an even number of 1's, and 0 otherwise.</footnote>
$$b_0 := s_0, \quad b_k := s_k \oplus b_{k-1}.$$

Setting $\ell = j \circ \kappa^{-1}$, we finally get

$$\varphi \circ \ell = \ell \circ \sigma. \qquad (5.4)$$

The product topology makes $\Sigma$ a compact metrizable space, and we shall equip this space with the compatible metric

$$\Sigma \times \Sigma \ni (a, b) \mapsto d(a, b) := 2^{-\inf\{k \in \mathbb{Z}_+ | a_k \neq b_k\}}.$$

We further denote by $\Sigma^* := \{w = b_1 \cdots b_k \in \{0, 1\}^k \mid k \in \mathbb{N}\}$ the set of finite-length words, and by $|w|$ the length of such a word. Given $w_1 \in \Sigma^*$ and $w_2 \in \Sigma^* \cup \Sigma$, we write their concatenation as $w_1 w_2$ and the periodic repetition of $w_1$ as $\overline{w_1} \in \Sigma$.

The following lemma summarizes the basic properties of the map $\ell$.

**Lemma 5.1** (1) $\ell: \Sigma \to M$ is onto. For any $x \in M$, $x = \ell(s)$ iff $\varphi^k(x) \in J_{s_k}$ holds for all $k \in \mathbb{Z}_+$.

(2) For any $s, s' \in \Sigma$, $|\ell(s') - \ell(s)| \le d(s', s)$.

(3) $\Sigma_0 := \ell^{-1}(M_0) = \Sigma \setminus \{w_1 \overline{0} \in \Sigma \mid w \in \Sigma^*, |w| \ge 1\}$.

(4) The map $\ell: \Sigma_0 \to M_0$ is a continuous bijection with a measurable inverse.

(5) Fix($\sigma^t$) = $\overline{w}$ | $w \in \Sigma^*$, $|w| = t \subset \Sigma_0$ and Fix($\varphi^t$) = $\ell$(Fix($\sigma^t$)).

(6) For any $\mathbb{P} \in \mathcal{P}_\varphi(M)$ one has $\mathbb{P}(\mathcal{C}) = 0$ and $h_\varphi(\mathbb{P}) = h_\sigma(\mathbb{P} \circ \ell)$.

**Proof.** (1) Is a direct consequence of the above construction.

(2) The claim follows from

$$|j(s') - j(s)| \le \sum_{k \ge -\log_2 d(s', s)} 2^{-k-1} = d(s', s),$$

after noticing that $\kappa$ and its inverse are isometries.

(3) One has $\ell^{-1}(M \setminus \mathcal{C}) = \Sigma \setminus \ell^{-1}(\mathcal{C})$ and $s \in \ell^{-1}(\mathcal{C})$ iff

$$\kappa^{-1}(s) \in j^{-1}(\mathcal{C}) = \{w_0 \overline{1}, w_1 \overline{0} \mid w \in \Sigma^*\}, \qquad (5.5)$$

and hence $s \in \{\kappa(w_0 \overline{1}), \kappa(w_1 \overline{0}) \mid w \in \Sigma^*\}$ from which the result immediately follows.

(4) Given (2), the isometry of $\kappa$ and its inverse mentioned in its proof, and the identity in (5.5), it suffices to show that the map $j: \Sigma \setminus \{w_0 \overline{1}, w_1 \overline{0} \mid w \in \Sigma^*\} \to M_0$ is injective with a measurable inverse. To deal with the first property, suppose that $s, s' \in \Sigma$ are such that $j(s') = j(s)$. W.l.o.g. we can assume that for some $n \in \mathbb{Z}_+$ one has $d(s', s) = 2^{-n}, s'_n = 0$ and $s_n = 1$. It follows that

$$\frac{s_n - s'_n}{2^{n+1}} = \frac{1}{2^{n+1}} = \sum_{k>n} \frac{s'_k - s_k}{2^{k+1}},$$
from which one easily concludes that, for any $k > n$, $s'_k - s_k = 1$ and hence $s'_k = 1$ and $s_k = 0$. This means that $\{s', s\} \subset j^{-1}(\mathscr{C})$. The measurability of the inverse map follows at once from the fact that $\ell(\mathbf{s}) = x \in M_0$ iff $s_k = 1_{[1/2, 1]} \circ \varphi^k(x)$.

(5) The first identity is obvious. The inclusion following it is a direct consequence of (3). Since $\mathrm{Fix}(\varphi^t) \cap \mathscr{C} = \emptyset$, the last assertion follows from (4).

(6) Let $\mathbb{P} \in \mathcal{P}_{\varphi}(M)$ and set $m := \mathbb{P}(\{1/2\})$. For $n \in \mathbb{Z}_+$, further set $\mathscr{C}_n := \varphi^{-n}(\{1/2\})$. It follows that $\mathbb{P}(\mathscr{C}_n) = m$ for all $n \in \mathbb{Z}_+$, and since $\mathscr{C}$ is the disjoint union of all $\mathscr{C}_n$

$$1 \geq \mathbb{P}(\mathscr{C}) = \sum_{n \in \mathbb{Z}_+} \mathbb{P}(\mathscr{C}_n) = \sum_{n \in \mathbb{Z}_+} m,$$

which gives that $m = 0$ and hence $\mathbb{P}(\mathscr{C}) = 0$. Hence, for any $\mathbb{P} \in \mathcal{P}_{\varphi}(M)$, the map $\ell^{-1}: M_0 \to \Sigma_0$ induces an isomorphism between the measurable dynamical systems $(M, \varphi, \mathbb{P})$ and $(\Sigma, \sigma, \mathbb{P} \circ \ell)$, and the result follows. $\square$

It follows from (5.4) and the previous lemma that the tent map $(M, \varphi)$ is a (topological) factor of the full shift $(\Sigma, \sigma)$. Consequently, the inequalities [Wal82, Section 4.4 and Theorem 9.8 (4)]

$$h_{\varphi}(\mathbb{P}) \leq h_{\sigma}(\mathbb{P} \circ \ell^{-1}), \quad \mathfrak{p}_{\varphi}(\nu) \leq \mathfrak{p}_{\sigma}(\nu \circ \ell),$$

hold for all $\mathbb{P} \in \mathcal{P}_{\varphi}(M)$ and all $\nu \in C(M)$. Since the shift $\sigma$ is forward expansive and has the specification property (S), it satisfies condition (PAP) and

$$\mathfrak{p}_{\sigma}(\nu \circ \ell) = \lim_{t \to \infty} \frac{1}{t} \log \sum_{x \in \mathrm{Fix}(\sigma^t)} e^{\nu \circ \ell(x) + \nu \circ \ell \circ \sigma(x) + \dots + \nu \circ \ell \circ \sigma^{t-1}(x)}.$$

Invoking Lemma 5.1 (5), we can write

$$\mathfrak{p}_{\varphi}(\nu) \leq \mathfrak{p}_{\sigma}(\nu \circ \ell) = \lim_{t \to \infty} \frac{1}{t} \log \sum_{x \in \mathrm{Fix}(\varphi^t)} e^{S^t \nu(x)}. \quad (5.6)$$

**Theorem 5.2** *The tent map $(M, \varphi)$ satisfies both the (USCE) and the (PAP) properties.*

**Proof.** Since the full shift on $\Sigma$ is forward expansive with expansiveness constant $r = 1/2$, the map $\mathbb{Q} \to h_{\sigma}(\mathbb{Q})$ is upper semicontinuous [Wal82, Theorem 8.2]. We claim that $\mathbb{P}_k \circ \ell - \mathbb{P} \circ \ell$ in $\mathcal{P}_{\sigma}(\Sigma)$ whenever $\mathbb{P}_k \to \mathbb{P}$ in $\mathcal{P}_{\varphi}(M)$. The upper semi-continuity of the map $\mathbb{P} \to h_{\varphi}(\mathbb{P})$ then follows from Lemma 5.1 (6). To prove our claim, we have to show that $\langle f \circ \ell^{-1}, \mathbb{P}_k \rangle \to \langle f \circ \ell^{-1}, \mathbb{P} \rangle$ for all $f \in C(\Sigma)$. By a density argument, it suffices to consider functions of the form $f(s) = F(s_0, s_1, \dots, s_{n-1})$ for $n \ge 1$. For $x \in M_0$, we then have $f \circ \ell^{-1}(x) = F \circ S_n(x)$ where

$$S_n(x) := (1_{[1/2,1]}(x), 1_{[1/2,1]} \circ \varphi(x), \dots, 1_{[1/2,1]} \circ \varphi^{n-1}(x)).$$
Observing that the function $S_n$ is constant on each $I_m = ]m2^{-n}, (m+1)2^{-n}[$, we deduce that the set of discontinuities of $F \circ S_n$ is $\{m2^{-n} | m \in ]0, 2^n[\}$ $\subset \mathscr{C}$. Since $\mathbb{P}(\mathscr{C}) = 0$, applying [Bil68, Theorem 5.2]$^2$ leads to the desired relation

$$ \langle f \circ \ell^{-1}, \mathbb{P}_k \rangle = \int_M F(S_n(x))\mathbb{P}_k(dx) \rightarrow \int_M F(S_n(x))\mathbb{P}(dx) = \langle f \circ \ell^{-1}, \mathbb{P} \rangle. $$

By Lemma 5.1 (4) $\mathscr{P}_\varphi(M)$ is the image of $\mathscr{P}_\sigma(\Sigma)$ by the map $\mathbb{Q} \rightarrow \mathbb{P} = \mathbb{Q} \circ \ell^{-1}$. Therefore,

$$ \mathbb{P}_{\sigma}(v \circ \ell) = \sup_{\mathbb{Q} \in \mathcal{P}_{\sigma}(\Sigma)} (\langle v, \mathbb{Q} \circ \ell^{-1} \rangle + h_{\sigma}(\mathbb{Q})) = \sup_{\mathbb{P} \in \mathcal{P}_{\varphi}(M)} (\langle v, \mathbb{P} \rangle + h_{\sigma}(\mathbb{P} \circ \ell)), $$

and Lemma 5.1 (6) yields that $\mathbb{P}_\varphi(v) = \mathbb{P}_\sigma(v \circ \ell)$. The result follows from the identity (5.6). $\square$

The tent map thus satisfies all the requirements of Corollary 4.6.<fnref n="3" /> Consequently, the LDPs of Theorem 4.3 and Corollary 4.4 hold for the corresponding periodic orbit ensemble. Before discussing a related fluctuation theorem, we shall discuss some interesting features of the pressure map.

## 5.3 Inducing

This section introduces the inducing scheme which will allow us, through the representation of the tent map dynamics as a shift over a countable alphabet, to achieve some control over the pressure $\mathbb{P}_\varphi$. This will give us the opportunity to discuss some interesting features of the LDP associated to the periodic orbit ensemble of the tent map.

Setting $I_t := ]2^{-t-1}, 2^{-t}]$ for $t \in \mathbb{Z}_+$ and $I_\infty := \{0\}$ we have

$$ I_{t+1} = [0, 1/2] \cap \varphi^{-1}(I_t), $$

and so, whenever $x \in I_t$ for some $t \ge 1$, one has $\varphi^s(x) \in I_{t-s}$ for $s \in [0, t]$, i.e., $t$ is the first return time of $x$ to $I_0$.

For $x \in M$, let

$$ \tau(x) := \sum_{n \in \mathbb{Z}_+} 1_{I_n}(x)(n+1), $$

and define $\psi: [0, 1] \rightarrow [0, 1]$ by setting<fnref n="4" /> $\psi(x) := \varphi^{\tau(x)}(x)$. With $M := \mathbb{Z}_+ \cup \{\infty\}$, denote by $s$ the left shift on the subshift

<footnote n="2">² Theorem 5.2 of the 1st edition of Billingsley's book disappeared from the 2nd edition. It is, however, a simple consequence of [Bil99, Theorem 2.7].</footnote>

<footnote n="3">³ Recall that (S) implies (ED).</footnote>

<footnote n="4">⁴ Note that $\tau(0) = 0$, so that $\psi(0) = 0$. Note also that $\varphi^{\tau(x)}(x) < 1$ for all $x \in M$. See Figure 5.1.</footnote>
$$ \mathcal{S} := \{ \mathbf{t} = (t_k)_{k \in \mathbb{Z}_+} \in \mathbb{M}^{\mathbb{Z}_+} \mid A_{t_k t_{k+1}} = 1 \text{ for all } k \in \mathbb{Z}_+ \} $$

associated to the transition matrix

$$ A_{st} := \begin{cases} 1 & \text{if } s = 0 \text{ and } t \in \mathbb{Z}_+; \\ 1 & \text{if } s \in \mathbb{N} \text{ and } t \in \mathbb{M}; \\ 1 & \text{if } s = t = \infty; \\ 0 & \text{otherwise.} \end{cases} $$

Define the **t-code** through the map $T: [0,1] \to \mathcal{S}$,

$$ T(x) = (t_k(x))_{k \in \mathbb{Z}_+}, \quad t_k(x) := \begin{cases} s \in \mathbb{Z}_+ & \text{if } \psi^k(x) \in I_s; \\ \infty & \text{if } \psi^k(x) = 0, \end{cases} $$

and the map $X: \mathcal{S} \to [0,1[$ by

$$ X(\mathbf{t}) := \sum_{k \in \mathbb{Z}_+} 2^{-(t_0+\cdots+t_k)} (-2)^{-k}, $$

where it is understood that $2^{-\infty} = 0$.

The following result states that $T = X^{-1}$ provides a bijective coding which conjugates the induced map $\psi$ to the left shift $s$.

**Proposition 5.3** (1) $T \circ \psi = s \circ T$.

(2) $\psi \circ X = X \circ s$.

(3) For any $\mathbf{t} \in \mathcal{S}$, one has

$$ \{X(\mathbf{t})\} = \bigcap_{k \in \mathbb{Z}_+} \psi^{-k}(I_{t_k}). $$

(4) $X \circ T = \text{Id}$, in particular, $X$ is surjective and $T$ is injective.

(5) $T \circ X = \text{Id}$, in particular, $T$ is surjective and $X$ is injective.

(6) For $t_0, t_1, \dots, t_{s-1} \in \mathbb{Z}_+$ one has<fnref n="5" />

$$ X(\overline{t_0 t_1 \cdots t_{s-1}}) = \frac{\sum_{r=0}^{s-1} 2^{-(t_0+\cdots+t_r)} (-2)^{-r}}{1 - 2^{-(t_0+\cdots+t_{s-1})} (-2)^{-s}}, $$

and for $u \in \mathbb{Z}_+$

$$ X(u \overline{t_0 t_1 \cdots t_{s-1}}) = 2^{-u} \left( 1 - \frac{1}{2} X(\overline{t_0 t_1 \cdots t_{s-1}}) \right). $$

**Proof. (1)** Is an immediate consequence of the definition of the map $T$.

<footnote n="5">⁵ Here, as above, $\overline{w}$ denotes the periodic repetition of the word $w$.</footnote>
(2) Using the crude estimate

$$|X(\boldsymbol{t})| \leq \sum_{k \in \mathbb{Z}_+} 2^{-k} = 2$$

which holds for $\boldsymbol{t} \in \mathfrak{S}$, and observing that the inequality is strict whenever $\boldsymbol{t} \neq \bar{0}$ while $X(\bar{0}) = 2/3$, we conclude that $|X(\boldsymbol{t})| < 2$. Invoking twice the relation

$$X(\boldsymbol{t}) = 2^{-t_0} \left(1 - \frac{1}{2} X \circ \mathfrak{s}(\boldsymbol{t})\right), \qquad (5.7)$$

one shows that

$$X(\boldsymbol{t}) \in 2^{-t_0} (1 - 2^{-t_1}) [0, 1], \qquad (5.8)$$

from which one easily deduces that $X(\boldsymbol{t}) \in I_{t_0}$, provided $t_1 \neq 0$. In the case $t_1 = 0$, Relation (5.8) gives $X(\boldsymbol{t}) \in ]0, 2^{-t_0}[$. Since we must have $t_2 < \infty$, repeated applications of (5.7) yield $X \circ \mathfrak{s}^2(\boldsymbol{t}) > 0$, $X \circ \mathfrak{s}(\boldsymbol{t}) < 1$, and $X(\boldsymbol{t}) > 2^{-t_0-1}$, so that finally, $X(\boldsymbol{t}) \in I_{t_0}$ and hence $\tau \circ X(\boldsymbol{t}) = t_0 + 1$. In particular, using the fact that $\varphi^{t_0+1}|_{I_{t_0}} : x \mapsto 2^{t_0+1}(2^{-t_0} - x)$, it follows from (5.7) that, for any $\boldsymbol{t} \in \mathfrak{S}$,

$$\psi \circ X(\boldsymbol{t}) = \varphi^{t_0+1}(X(\boldsymbol{t})) = 2^{t_0+1}(2^{-t_0} - X(\boldsymbol{t})) = X \circ \mathfrak{s}(\boldsymbol{t}).$$

(3) The last relation gives $\psi^k \circ X(\boldsymbol{t}) = X \circ \mathfrak{s}^k(\boldsymbol{t}) \in I_{t_k}$, and hence

$$X(\boldsymbol{t}) = \sum_{k \in \mathbb{Z}_+} 2^{-(t_0+\cdots+t_k)} (-2)^{-k} \in \bigcap_{k \in \mathbb{Z}_+} \psi^{-k}(I_{t_k}).$$

To prove the stated equality, we have to show that whenever

$$\{x, y\} \subset \bigcap_{k \in \mathbb{Z}_+} \psi^{-k}(I_{t_k}), \qquad (5.9)$$

we must have $x = y$. If $t_0 = \infty$, then $x = y = 0$. In the case $t_r = \infty$ for some $r > 0$ with finite $t_1, \dots, t_{r-1}$, it is easy to conclude that in $\psi^k(x) = \psi^k(y) = 2^{-t_k}$ for $k < r$, and in particular $x = y$. Let us consider now the case where all $t_k$ are finite. From the fact that $\{x, y\} \subset I_{t_0}$ we deduce

$$|x - y| < 2^{-t_0 - 1}, \qquad |\psi(x) - \psi(y)| = 2^{t_0 + 1} |x - y|. \qquad (5.10)$$

In the same way, since $\{\psi(x), \psi(y)\} \subset I_{t_1}$, one has

$$|\psi(x) - \psi(y)| < 2^{-t_1 - 1}, \qquad |\psi^2(x) - \psi^2(y)| = 2^{t_1 + 1} |\psi(x) - \psi(y)|. \qquad (5.11)$$

Combining the last identity of (5.10) with the first inequality of (5.11) we conclude that $|x - y| < 2^{-t_0 - t_1 - 2}$. Iterating this argument leads to

$$t_k(x) = t_k(y) \text{ for } 0 \le k < n \implies |x - y| < 2^{-t_0 - t_1 - \dots - t_{n-1} - n} \le 2^{-n}, \qquad (5.12)$$

and letting $n \to \infty$ yields again that $x = y$.
(4) Let $x \in [0, 1]$. By definition $\mathbf{t} = (t_k)_{k \in \mathbb{Z}_+} = T(x)$ is such that

$$x \in \bigcap_{k \in \mathbb{Z}_+} \psi^{-k}(I_{t_k}). \qquad (5.13)$$

By Part (3), $y = X \circ T(x) = X(\mathbf{t})$ satisfies

$$y \in \bigcap_{k \in \mathbb{Z}_+} \psi^{-k}(I_{t_k}).$$

Thus, (5.9) holds and by the proof of Part (3) one has $y = x$.

(5) Let $\mathbf{t} = (t_k)_{k \in \mathbb{Z}_+} \in \mathcal{S}$. By Part (3), $x = X(\mathbf{t})$ satisfies (5.13) and hence $T(x) = \mathbf{t}$.

(6) Write

$$X(\overline{t_0 t_1 \cdots t_{s-1}}) = \sum_{n \in \mathbb{Z}_+} \sum_{k=ns}^{(n+1)s-1} 2^{-(t_0+\cdots+t_k)} (-2)^{-k},$$

with $t_j = t_r$ for $j \in \mathbb{Z}_+$, $r \in [0, s-1]$ and $j \equiv r \pmod{s}$. Then, setting $k = ns + r$,

$$\begin{align*}
X(t_0 t_1 \cdots t_{s-1}) &= \sum_{n \in \mathbb{Z}_+} \sum_{r=0}^{s-1} 2^{-(t_0+\cdots+t_{s-1})n-(t_0+\cdots+t_r)} (-2)^{-ns-r} \\
&= \sum_{n \in \mathbb{Z}_+} [2^{-(t_0+\cdots+t_{s-1})} (-2)^{-s}]^n \sum_{r=0}^{s-1} 2^{-(t_0+\cdots+t_r)} (-2)^{-r} \\
&= \frac{\sum_{k=0}^{s-1} 2^{-(t_0+\cdots+t_k)} (-2)^{-k}}{1 - 2^{-(t_0+\cdots+t_{s-1})} (-2)^{-s}}
\end{align*}$$

yields the first formula. The second one now follows from (5.7). □

**Remark 1.** The following construction is left as an exercise. Given the Gray-code $\mathbf{s} = \ell^{-1}(x)$ of $x \in M_0$, one gets its $\mathbf{t}$-code $\mathbf{t} = T(x)$ by counting the number of 0's before the first 1 and the number of 0's separating the following 1's. For example the $\mathbf{s}$-code

$$\ell^{-1}(2^{-1/2}) = 11101111100001101000101010\cdots$$

yields

<figure><img src="image_19.png" /></figure>

and hence the corresponding $\mathbf{t}$-code is $T(2^{-1/2}) = 00010000401311\cdots$.

**Remark 2.** From the properties of the intervals $I_t (t \in \mathbb{Z}_+)$ described at the beginning of this section, one easily sees that the tent map on the non-compact set $M_0$ can also be coded by the *renewal shift*: the left shift on the space

$$\mathcal{R} := \{ \mathbf{u} = (u_k)_{k \in \mathbb{Z}_+} \in \mathbb{M}^{\mathbb{Z}_+} \mid u_{k+1} = u_k - 1 \text{ if } u_k > 0 \}.$$
The **u**-code is defined by

$$
\mathbf{u} = U(x) = (u_k(x))_{k \in \mathbb{Z}_+}, \quad u_k(x) := \begin{cases} s \in \mathbb{Z}_+ & \text{if } \varphi^k(x) \in I_s; \\ \infty & \text{if } \varphi^k(x) = 0, \end{cases}
$$

and is obtained from the **t**-code by substituting each $t_k$ by the word

$t_k(t_k - 1) \cdots 10,$

for example

$$
U(2^{-1/2}) = 0001000004321001032101010\dots
$$

5.4 Grand Canonical Picture

Let $v \in C(M)$. Considering the Lebesgue measure $\lambda$ as well as the Dirac masses
$\delta_0$ and $\delta_{2/3}$, Definition (4.1) and Formulas (5.1), (5.2) yield the bounds

$$
\max(\log 2 + \langle v, \lambda \rangle, v(0), v(2/3)) \le \mathbb{P}_{\phi}(v) \le \max_{x \in M} (\log 2 + v(x)).
$$

By Theorem 5.2,

$$
\mathbb{P}_{\varphi}(v) = \limsup_{t \to \infty} \frac{1}{t} \log \sum_{x \in \text{Fix}(\varphi^t)} e^{S_t v(x)},
$$

and since

$$
\sum_{x \in \text{Fix}(\varphi^t)} e^{S_t v(x)} = e^{t v(0)} + \sum_{x \in \text{Fix}(\varphi^t) \setminus \{0\}} e^{S_t v(x)},
$$

we can write

$$
\mathbb{P}_{\varphi}(v) = \max \left( v(0), \limsup_{t \to \infty} \frac{1}{t} \log \sum_{x \in \text{Fix}(\varphi^t) \setminus \{0\}} e^{S_t v(x)} \right). \quad (5.14)
$$

Focusing on the second argument in the above max, let us denote by $\text{Orb}_t(\varphi)$ the
set of $t$-periodic orbits of $\varphi$. We have

$$
\sum_{x \in \text{Fix}(\varphi^t)} e^{S_t v(x)} = \sum_{o \in \text{Orb}_t(\varphi)} |o| e^{\sum_{x \in o} v(x)},
$$

and

$$
\sum_{x \in \text{Fix}(\varphi^t) \cap I_0} e^{S_t v(x)} = \sum_{o \in \text{Orb}_t(\varphi)} |o \cap I_0| e^{\sum_{x \in o} v(x)}.
$$

Noticing that $1 \le |o \cap I_0| \le |o| \le t$ for any $o \in \text{Orb}_t(\varphi)$ distinct from the orbit of 0,
one easily shows that
$$ \limsup_{t \to \infty} \frac{1}{t} \log \sum_{x \in \text{Fix}(\varphi^t) \setminus \{0\}} e^{S_t \nu(x)} = \limsup_{t \to \infty} \frac{1}{t} \log \sum_{x \in \text{Fix}(\varphi^t) \cap I_0} e^{S_t \nu(x)}. $$

To proceed, we further partition the set $\text{Fix}(\varphi^t) \cap I_0$ according to the total number of visits of the segment $(\varphi^r(x))_{r \in [0, t-1]}$ to $I_0$, i.e.,

$$ \text{Fix}(\varphi^t) \cap I_0 = \bigcup_{s=1}^{t} M_{s,t}, $$

where

$$ M_{s,t} := \left\{ \varphi^{\tau(x)-1}(x) \mid x \in \text{Fix}(\psi^s), \sum_{k=0}^{s-1} \tau \circ \psi^k(x) = t \right\}. $$

In terms of **t** and **u**-codes, one has

$$ T(M_{s,t}) = \left\{ \mathbf{t} = 0 \overline{t_0 \cdots t_{s-1}} \mid \sum_{k=0}^{s-1} t_k = t-s \right\}, \\ U(M_{s,t}) = \left\{ \mathbf{u} = 0 \overline{t_0(t_0-1) \cdots 10 \cdots t_{s-1}(t_{s-1}-1) \cdots 10} \mid \sum_{k=0}^{s-1} t_k = t-s \right\}. $$

Associating to $v$ the function

$$ \mathcal{S} \ni \mathbf{t} = t_0 t_1 \cdots \mapsto \nu^{\#}(\mathbf{t}) := \sum_{r=0}^{t_0} \nu \circ \varphi^r \circ X(\mathbf{t}), \qquad (5.15) $$

and setting

$$ S^s \nu^{\#} := \sum_{r=0}^{s-1} \nu^{\#} \circ s^r, \qquad (5.16) $$

one easily checks that, for $\mathbf{t} = 0 \overline{t_0 \cdots t_{s-1}} \in T(M_{s,t})$,

$$ (S_t \nu) \circ X(\mathbf{t}) = S^s \nu^{\#}(\overline{t_0 \cdots t_{s-1}}), $$

each side of this identity being the sum of the values taken by $v$ on a segment of length $t$ of the orbit of $X(\mathbf{t})$. Thus, we can write

$$ \sum_{x \in \text{Fix}(\varphi^t) \cap I_0} e^{S_t \nu(x)} = \sum_{s=1}^{t} \sum_{\mathbf{t} \in T(M_{s,t})} e^{(S_t \nu) \circ X(\mathbf{t})} = \sum_{s=1}^{t} \sum_{\substack{t_0, \dots, t_{s-1} \in \mathbb{Z}_+ \\ t_0 + \dots + t_{s-1} = t-s}} e^{S^s \nu^{\#}(\overline{t_0 \cdots t_{s-1}})}. $$

Invoking the Cauchy-Hadamard formula allows us to express the pressure $p_{\varphi}(v)$ in terms of the radius of convergence $\rho_{\text{gc}}(v)$ of the (grand canonical) partition function

$$ \mathfrak{P}_{\varphi}(v, z) := \sum_{t=1}^{\infty} z^t \sum_{x \in \text{Fix}(\varphi^t) \cap I_0} e^{S_t \nu(x)}, $$

namely
$$p_{\varphi}(v) = \max(v(0), -\log \rho_{\text{gc}}(v)). \qquad (5.17)$$

To control the function $\mathfrak{P}_{\varphi}$ we shall express it in terms of the countable Markov shift $(\mathcal{S}, \mathfrak{s})$,

$$ \begin{aligned} \mathfrak{P}_{\varphi}(v, z) &= \sum_{t=1}^{\infty} z^t \sum_{x \in \text{Fix}(\varphi^t) \cap I_0} e^{S_t v(x)} \\ &= \sum_{t=1}^{\infty} z^t \sum_{s=1}^{t} \sum_{\mathbf{t} \in T(M_{t,s})} e^{S^s v^{\#}(\mathbf{t})} \\ &= \sum_{s=1}^{\infty} z^s \sum_{t=s}^{\infty} z^{t-s} \sum_{\mathbf{t} \in T(M_{t,s})} e^{S^s v^{\#}(\mathbf{t})} \\ &= \sum_{s=1}^{\infty} z^s \Xi_s(v, z), \end{aligned} \qquad (5.18) $$

with

$$ \Xi_s(v, z) := \sum_{\mathbf{t} \in \text{Fix}(\mathfrak{s}^s)} z^{t_0 + \dots + t_{s-1}} e^{S^s v^{\#}(\mathbf{t})} = \sum_{t_0, \dots, t_{s-1} \in \mathbb{Z}_+} z^{t_0 + \dots + t_{s-1}} e^{S^s v^{\#}(\overline{t_0 \dots t_{s-1}})}. \quad (5.19) $$

Besides this standard combinatorial device, we need to impose some more stringent regularity conditions on the potential $v$. The $k$-variation of a function $f: \mathcal{S} \to \mathbb{R}$ is defined as

$$ \mathrm{Var}_k(f) := \sup \{|f(\mathbf{t}) - f(\mathbf{s})| \mid \mathbf{t}, \mathbf{s} \in \mathcal{S}, t_j = s_j \text{ for } 0 \le j < k\}. \quad (5.20) $$

This function has summable variations whenever

$$ \mathrm{Var}(f) := \sum_{k \ge 1} \mathrm{Var}_k(f) < \infty, $$

and is $\theta$-Hölder, for $\theta \in ]0, 1[$, whenever

$$ \sup_{k \ge 1} \theta^{-k} \mathrm{Var}_k(f) < \infty. $$

Clearly, the latter condition implies the former.

**Proposition 5.4** Suppose that the potential $v \in C(M)$ is such that $v^{\#}$ has summable variations.

(1) For $s \in \mathbb{N}$, the radius of convergence of the series (5.19) is given by $e^{-v(0)}$. The limit

$$ p(v, \mu) := \lim_{s \to \infty} \frac{1}{s} \log \Xi_s(v, e^{-\mu}) \qquad (5.21) $$

exists for $\mu > v(0)$ and defines a non-increasing function of $\mu$.

(2) The function $\mu \mapsto p(v, \mu) - \mu$ is strictly decreasing, moreover

$$ \lim_{\mu \to +\infty} p(v, \mu) - \mu = \inf_{\mu > v(0)} p(v, \mu) - \mu = -\infty, \qquad (5.22) $$
and

$$ \Delta(v) := \lim_{\mu \uparrow v(0)} p(v, \mu) - \mu = \sup_{\mu > v(0)} p(v, \mu) - \mu \in [-\infty, +\infty]. \quad (5.23) $$

(3) The following alternative holds:

* either $\Delta(v) \le 0$, $\rho_{gc}(v) \ge e^{-v(0)}$, and $\mathfrak{p}_{\varphi}(v) = v(0)$;

* or $\Delta(v) > 0$,

$$ \rho_{gc}(v) = e^{-\mu^*} := \inf\{e^{-\mu} \mid p(v, \mu) - \mu > 0\} < e^{-v(0)}, $$

$$ \text{and } \mathfrak{p}_{\varphi}(v) = \mu^* > v(0). $$

(4) The function $\kappa \mapsto \Delta(\kappa v)$ is convex and takes the value $+\infty$ at $\kappa = 0$. If $v^\#$ is $\theta$-Hölder for some $\theta \in ]0, 1[$, then

$$ \kappa_c(v) := \sup\{\kappa \ge 0 \mid \Delta(\kappa v) > 0\} \in ]0, \infty], $$

and the function $\kappa \mapsto \mathfrak{p}_{\varphi}(\kappa v)$ is real-analytic on the interval $]0, \kappa_c(v)[$.

**Remark.** In the dynamical system literature on countable Markov shifts, $p(v, 0)$ is called *Gurevich pressure* of the potential $v^\#$ for the induced system

$$ p(v, 0) := P_G(v^\#). $$

Thus, the function $\mu \mapsto p(v, \mu) - \mu$, which plays a central role in the preceding result, is related to the Gurevich pressure by

$$ P_G((v - \mu)^\#) = p(v, \mu) - \mu. $$

In particular

$$ \Delta(v) = P_G((v - v(0))^\#) $$

is the so-called *discriminant* of the potential $v$ (see [Sar01; PZ06]). In the cases where $\Delta(v) > 0$, the potential $v$ is said to be *positive recurrent*.

**Proof.** (1) We first consider the case $s = 1$. The radius of convergence of the series

$$ \Xi_1(v, z) = \sum_{t=0}^{\infty} z^t e^{v^\#(\bar{t})}, $$

is given by the Cauchy-Hadamard formula

$$ \varrho = \liminf_{t \to \infty} e^{-v^\#(\bar{t})/t}. $$

By (5.15) and Proposition 5.3 (6) we have

$$ \frac{1}{t} v^{\#}(\bar{t}) = \frac{1}{t} \sum_{s=0}^{t} v \circ \varphi^{s} \circ X(\bar{t}) = \frac{1}{t} \sum_{s=0}^{t} v \left( \frac{2^{-s}}{1 + 2^{-t-1}} \right). \quad (5.24) $$
Noticing that

$$ \max_{s \in [0, t]} \left| \frac{2^{-s}}{1 + 2^{-t-1}} - 2^{-s} \right| \le 2^{-t}, $$

and invoking the uniform continuity of $v$ and Cesàro's limit theorem we deduce from (5.24) that

$$ \lim_{t \to \infty} \frac{1}{t} v^{\#}(\bar{t}) = v(0). \qquad (5.25) $$

This yields $\rho = e^{-v(0)}$, as claimed.

To deal with the general case $s \in \mathbb{N}$, observe that

$$ \Xi_{s+s'}(v, e^{-\mu}) = \sum_{\mathbf{t} \in \mathbb{Z}_+^s} e^{-\mu(t_0 + \cdots + t_{s-1})} \sum_{\mathbf{r} \in \mathbb{Z}_+^{s'}} e^{-\mu(r_0 + \cdots + r_{s'-1})} e^{S^s v^\#(\bar{\mathbf{r}})} e^{S^{s'} v^\#(\bar{\mathbf{r}}')} $$

Since

$$ S^s v^\#(\bar{\mathbf{t}}) - \sum_{j=1}^s \mathrm{Var}_j(v^\#) \le S^s v^\#(\overline{\mathbf{t}\mathbf{r}}) \le S^s v^\#(\bar{\mathbf{t}}) + \sum_{j=1}^s \mathrm{Var}_j(v^\#), $$

setting $a_s := \log \Xi_s(v, e^{-\mu})$ and $c := 2\mathrm{Var}(v^\#)$, we derive

$$ a_s + a_{s'} - c \le a_{s+s'} \le a_s + a_{s'} + c, \qquad (5.26) $$

from which one easily infers that

$$ \Xi_1(v, e^{-\mu})^s e^{-c(s-1)} \le \Xi_s(v, e^{-\mu}) \le \Xi_1(v, e^{-\mu})^s e^{c(s-1)}, $$

and hence that the radius of convergence of the series (5.19) is equal to $e^{-v(0)}$ for all $s \in \mathbb{N}$. The existence of the limit (5.21) follows from the sub/super-additivity properties expressed in (5.26). Fekete's lemma [PS78, Part I, Problem 99] provides the explicit formulas

$$ p(v, \mu) = \inf_{s \in \mathbb{N}^s} \frac{1}{s} \left( \log \Xi_s(v, e^{-\mu}) + c \right) = \sup_{s \in \mathbb{N}^s} \frac{1}{s} \left( \log \Xi_s(v, e^{-\mu}) - c \right). $$

Thus

$$ \left| \log \Xi_s(v, e^{-\mu}) - s p(v, \mu) \right| \le c, \qquad (5.27) $$

and it follows from (5.18) that, for $|z| = e^{-\mu} < \rho_{gc}(v)$,

$$ \frac{e^{-c}}{1 - e^{p(v, \mu) - \mu}} \le |\mathfrak{P}_{\phi}(v, z)| \le \frac{e^{c}}{1 - e^{p(v, \mu) - \mu}}. \qquad (5.28) $$

As a direct consequence of (5.19), the (positive) function $\mu \mapsto \Xi_s(v, e^{-\mu})^{1/s}$ is non-increasing. By Definition (5.21), the same is true of $\mu \mapsto p(v, \mu)$,
(2) Given the latter property, the claimed strict monotony is obvious. Invoking (5.27), the simple fact that $\Xi_s(v, 0)^{1/s} = e^{v(2/3)}$ yields

$$ \lim_{\mu \to \infty} p(v, \mu) \le \lim_{\mu \to \infty} \frac{1}{s} \log \Xi_s(v, e^{-\mu}) + \frac{c}{s} = \frac{1}{s} \log \Xi_s(v, 0) + \frac{c}{s} = v(2/3) + \frac{c}{s}, $$

which proves (5.22). Starting again with (5.27), and keeping only the first term in the series $\Xi_1$, we derive

$$ p(v, \mu) - \mu \ge \log \Xi_1(v, e^{-\mu}) - \mu - c \ge v(2/3) - \mu - c, $$

which, in the limit $\mu \downarrow v(0)$, proves (5.23).

(3) Suppose that $\Delta(v) \le 0$, fix $\mu > v(0)$ so that $p(v, \mu) - \mu = -\delta < 0$, and notice that it follows from (5.28) that

$$ \sup_{|z|<e^{-\mu}} |\mathfrak{P}_{\varphi}(v, z)| \le \frac{e^c}{1 - e^{-\delta}} < \infty, $$

from which we infer that $\rho_{gc}(v) \ge e^{-\mu}$. Letting $\mu \downarrow v(0)$ gives $\rho_{gc}(v) \ge e^{-v(0)}$ and (5.17) allows us conclude that $p_{\varphi}(v) = v(0)$.

In the case $\Delta(v) > 0$, $\mu^* = \sup\{\mu | v(0) < \mu < p(v, \mu)\} > v(0)$ and for any $v(0) < \mu < \mu^*$ one has $p(v, \mu) - \mu > 0$ and therefore

$$ \limsup_{s \to \infty} (e^{-\mu s} \Xi_s(v, e^{-\mu}))^{1/s} = e^{p(v, \mu) - \mu} > 1. $$

By Cauchy's root test, the grand canonical partition function (5.18) diverges for $|z| \ge e^{-\mu}$. Letting $\mu \uparrow \mu^*$, we derive $\rho_{gc}(v) \le e^{-\mu^*}$. Reciprocally, if $\mu > \mu^*$, then

$$ p(v, \mu) - \mu < 0, $$

and hence

$$ \limsup_{s \to \infty} (e^{-\mu s} \Xi_s(v, e^{-\mu}))^{1/s} < 1. $$

Invoking again the root test, we conclude that (5.18) converges for $|z| \le e^{-\mu}$. Letting $\mu \downarrow \mu^*$, we derive $\rho_{gc}(v) \ge e^{-\mu^*}$.

(4) For reasons of space, we only sketch the proof. The reader should consult [Sar01, Theorem 5] and references therein for more details.

For $\kappa = 0$, an elementary calculation gives $p(0, \mu) - \mu = -\log(e^{\mu} - 1)$, from which we conclude

$$ \Delta(0) = \sup_{\mu > 0} \log \frac{1}{e^{\mu} - 1} = +\infty. $$

By (5.21) and (5.23), we have

$$ \Delta(\kappa v) = \sup \lim_{v \to 0} \left( \frac{1}{s} \log \Xi_s(\kappa v, e^{-v - \kappa v(0)}) - v - \kappa v(0) \right). $$
Observing that $\Xi_s(\kappa v, e^{-v-\kappa v(0)}) = e^{s\kappa v(0)}\Xi_s(\kappa(v-v(0)), e^{-v})$ yields

$$ \Delta(\kappa v) = \sup_{v>0} \lim_{s \in \mathbb{N}} \left( \frac{1}{s} \log \Xi_s(\kappa(v-v(0)), e^{-v}) - v \right). $$

From (5.19) we infer that the function $\kappa \mapsto \Xi_s(\kappa(v-v(0)), e^{-v})$ is log-convex, so that the previous identity immediately implies the convexity of $\kappa \mapsto \Delta(\kappa v)$. We also note, for later reference, that this function is non-increasing provided $\max_{x \in M} v(x) = v(0)$.

By (5.25), given $\varepsilon > 0$ there is $N$ such that, for any $n \ge N$ and $t \in \mathcal{S}$ one has

$$ \begin{aligned} \mu n - \kappa v^{\#}(nt) &\ge \mu n - \kappa v^{\#}(\bar{n}) - \kappa \mathrm{Var}_1(v^{\#}) \\ &\ge (\mu - \kappa v(0)) n - \left| \frac{1}{n} v^{\#}(\bar{n}) - v(0) \right| n - \kappa \mathrm{Var}_1(v^{\#}) \\ &\ge (\mu - \kappa v(0) - \varepsilon) n - \kappa \mathrm{Var}_1(v^{\#}). \end{aligned} $$

It follows that the Ruelle-Perron-Frobenius transfer operator $L_{\kappa,\mu}$, defined on $C_b(\mathcal{S})$ by

$$ (L_{\kappa,\mu} f)(t) := \sum_{n \in \mathbb{Z}_+} e^{-\mu n + \kappa v^{\#}(nt)} f(nt), $$

satisfies

$$ \|L_{\kappa,\mu} 1\|_{\infty} < \infty $$

provided $\mu > \kappa v(0)$. It $v^{\#}$ is $\theta$-Hölder and $\mu > \kappa v(0)$, we deduce that $L_{\kappa,\mu}$ acts as a bounded operator on the Banach space

$$ \mathcal{L}_{\theta} := \{f \in C_b(\mathcal{S}) \mid \|f\|_{\theta} < \infty\}, $$

where

$$ \|f\|_{\theta} = \|f\|_{\infty} + \sup_{t \neq s} \theta^{-\inf\{k \in \mathbb{Z}_+ | t_k \neq s_k\}} |f(t) - f(s)|. $$

Since, for any $t \in \mathcal{S}$ and $s \in \mathbb{N}$,

$$ \begin{aligned} (L_{\kappa, \mu}^s 1)(t) &= \sum_{t_0, \dots, t_{s-1} \in \mathbb{Z}_+} e^{-\mu(t_0 + \dots + t_{s-1}) + \kappa S^s v^{\#}(t_0 \dots t_{s-1} t)} \\ &= \Xi_s(\kappa v, e^{-\mu})(1 + o(s)), \end{aligned} $$

the Perron-Frobenius theorem yields that the logarithm of the spectral radius of the operator $L_{\kappa,\mu}$ is

$$ p(\kappa v, \mu) = \limsup_{n \to \infty} \frac{1}{n} \log(L_{\kappa, \mu}^n 1)(t). $$

The crucial point is that one can show that $e^{p(\kappa,\mu)}$ is a simple discrete eigenvalue of $L_{\kappa,\mu}$, so that the claimed analyticity follows from regular perturbation theory. $\square$

<figure><img src="image_20.png" /></figure>
## 5.5 Phase Transitions

Whenever the critical coupling $\kappa_c(v)$ is finite, Proposition 5.4 implies that the function $\kappa \mapsto p_\phi(\kappa v)$ is real-analytic on $]0, \kappa_c(v)[ \cup ]\kappa_c(v), \infty[$, but fails to be analytic at $\kappa = \kappa_c(v)$. In thermodynamics, such a phenomenon is related to phase transition and is accompanied by a multiplicity of the equilibrium states. In this section we elaborate on the occurrence of such a phase transition for the tent map.

To this end, we shall consider lower and upper bounds on the partition function. For $v \in C(M)$ and $t \in \mathbb{Z}_+$, let

$$v_{-}(t) := \inf_{x \in I_{t}} v(x), \quad v_{+}(t) := \sup_{x \in I_{t}} v(x). \qquad (5.29)$$

From the definition (5.15) and Proposition 5.3 we deduce, for $t \in \mathbb{S}$,

$$\underline{v}(t_0) := \sum_{s=0}^{t_0} v_{-}(s) \le v^{\#}(\mathbf{t}) \le \sum_{s=0}^{t_0} v_{+}(s) := \bar{v}(t_0),$$

and hence, by (5.16),

$$\sum_{j=0}^{s-1} \underline{v}(t_j) \le S^s v^{\#}(\mathbf{t}) \le \sum_{j=0}^{s-1} \bar{v}(t_j).$$

By Definition (5.19),

$$\left( \sum_{t \in \mathbb{Z}_+} e^{-\mu(t+1)+\kappa \underline{v}(t)} \right)^s \le e^{-\mu s} \Xi_s(\kappa v, e^{-\mu}) \le \left( \sum_{t \in \mathbb{Z}_+} e^{-\mu(t+1)+\kappa \bar{v}(t)} \right)^s,$$

and we derive from (5.21–5.23) that

$$\log \left( \sum_{t \in \mathbb{Z}_+} e^{-\mu(t+1)+\kappa \underline{v}(t)} \right) \le p(\kappa, \mu) - \mu \le \log \left( \sum_{t \in \mathbb{Z}_+} e^{-\mu(t+1)+\kappa \bar{v}(t)} \right).$$

In particular, setting $w = v - v(0)$, we obtain

$$\log \left( \sum_{t \in \mathbb{Z}_+} e^{\kappa \underline{w}(t)} \right) \le \Delta(\kappa) \le \log \left( \sum_{t \in \mathbb{Z}_+} e^{\kappa \bar{w}(t)} \right). \qquad (5.30)$$

The following are elementary consequences of these inequalities.

**Lemma 5.5** If $\kappa_c(v) < \infty$, then

$$\lim_{t \to \infty} \underline{v}(t) = -\infty.$$

Moreover, if $\bar{v}(t) < 0$ for any $t \in \mathbb{Z}_+$ and

$$\sum_{t \in \mathbb{Z}_+} e^{\kappa \bar{v}(t)} < \infty$$
for some $\kappa > 0$, then $\kappa_c(v) < \infty$.

## 5.6 A Map of the Square

One easily checks that the map of the unit interval defined by

$$x = 0.b_0 b_1 b_2 b_3 \cdots \mapsto \theta(x) := 0.\hat{b}_0 \hat{b}_1 \hat{b}_2 \hat{b}_3 \cdots,$$

is involutive and satisfies $\varphi \circ \theta = \theta \circ \varphi$. However, this map is discontinuous on the dense set $\mathcal{C}$. In fact, there is no non-trivial continuous involution $\theta$ of the unit interval satisfying Assumption (Sym) for the tent map. Indeed, injectivity forces such a $\theta$ to be either strictly increasing or strictly decreasing. The only increasing involution is the identity, and hence is trivial.<fnref n="6" /> If $\theta$ is a decreasing involution, then surjectivity demands $\theta(0) = 1$ and since

$$\varphi \circ \theta(0) = \varphi(1) = 0 \neq 1 = \theta(0) = \theta \circ \varphi(0),$$

Assumption (Sym) fails.<fnref n="7" />

Thus, even though the tent map satisfies all the requirements for the results of Section 4.4 to apply,<fnref n="8" /> the only possible application turns out to be trivial. To reach a non-trivial conclusion, we shall consider the map of the unit square $M \times M$ defined by $\varphi(x, y) = (\varphi(x), \varphi(y))$, for which all the requirements of Corollary 4.6 also hold. In this case, Assumption (Sym) is verified by the involution

$$\theta(x, y) := (y, x).$$

Given $v_1, v_2 \in C(M)$, the potential

$$v(x, y) := v_1(x) + v_2(y),$$

and given the associated periodic orbit ensemble, the empirical measure

$$\mu_t^{x,y} := \frac{1}{t} \sum_{0 \le s < t} \delta_{\phi^s(x,y)},$$

satisfies the LDP with rate

$$\mathbb{I}(\mathbb{Q}) = \mathfrak{p}_{\phi}(v) - \langle v, \mathbb{Q} \rangle - h_{\phi}(\mathbb{Q}).$$

The conclusions of Theorem 4.10, and in particular the FR

$$\mathbb{I}(\hat{\mathbb{Q}}) = \mathbb{I}(\mathbb{Q}) + \text{ep}(\mathbb{Q}),$$

<footnote n="6">⁶ A continuous $\theta: [0, 1] \to [0, 1]$ is involutive iff its graph is symmetric w.r.t. the diagonal.</footnote>

<footnote n="7">⁷ One faces a similar issue in the study of Anosov systems, see [MV03, Section 6]</footnote>

<footnote n="8">⁸ Recall the discussion at the end of Section 5.2.</footnote>
hold, with the entropy production

$$ ep(Q) = \int (v_1 - v_2)(x)Q(dx, dy) - \int (v_1 - v_2)(y)Q(dx, dy). $$

Since $Fix(\varphi^t) = Fix(\varphi^t) \times Fix(\varphi^t)$, one has

$$ \mathfrak{p}_{\varphi}(v) = \lim_{t \to \infty} \frac{1}{t} \log \sum_{x, y \in \text{Fix}(\varphi^t)} e^{S_t v_1(x) + S_t v_2(y)} = \mathfrak{p}_{\varphi}(v_1) + \mathfrak{p}_{\varphi}(v_2), $$

from which we deduce that the entropic pressure is given by

$$ e(\alpha) = \mathfrak{p}_{\varphi}((1 - \alpha)v_1 + \alpha v_2) + \mathfrak{p}_{\varphi}((1 - \alpha)v_2 + \alpha v_1) - \mathfrak{p}_{\varphi}(v_1) - \mathfrak{p}_{\varphi}(v_2). $$

## 5.7 Bracketing the Critical Point and the Pressure

We finish this chapter with an analysis of a specific potential exhibiting phase transitions, i.e., singularities of the pressure function. Numerical calculation of the critical point $\kappa_c$ and of the pressure is a delicate matter. Rigorous estimates on these quantities are more tractable. It is fairly easy to get lower and upper bounds on the partition function and to derive from these estimates bounds on the critical point and the pressure.

The *Hurwitz zeta function* is defined by the convergent expansion

$$ \zeta(p, z) := \sum_{n \in \mathbb{Z}_+} (n + z)^{-p}, \quad \text{Re } p > 1, \text{Re } z > 0, \qquad (5.31) $$

and extends to an analytic function on $(p, z) \in \mathbb{C}^2 \setminus \{(1) \times \mathbb{C} \setminus \{0\}\}$. The special value $\zeta(p, 1)$ coincides with *Riemann zeta function* $\zeta(p)$ on $\mathbb{C} \setminus \{1\}$. The functions

$$ D_{-}(\kappa, \mu) := \sum_{n \in \mathbb{Z}_+} e^{-\mu(n+1) - \kappa(\zeta(p) - \zeta(p, n+3) - 1)} - 1, \qquad (5.32) $$

$$ D_{+}(\kappa, \mu) := \sum_{n \in \mathbb{Z}_+} e^{-\mu(n+1) - \kappa(\zeta(p) - \zeta(p, n+2))} - 1, \qquad (5.33) $$

play the central role in

**Lemma 5.6** Let the potential $v \in C(M)$ be given by

$$ v(x) := -(1 - \log_2 x)^{-p}, \qquad (5.34) $$

with constant $p \in ]0, 1[$.

(1) The induced function $v^#$ defined in (5.15) is $\theta$-Hölder for $\theta \ge 1/2$, and hence has summable variations, more precisely
$$ \mathrm{Var}_k(v^\#) \le \frac{2p}{\log 2} \zeta(p+1,2)2^{-k}, \quad \mathrm{Var}(v^\#) \le \frac{3p}{2\log 2} \zeta(p+1,2). $$

(2) The function $(\kappa, \mu) \mapsto D_\pm(\kappa, \mu)$ is analytic in $\mathcal{O} := \{(\kappa, \mu) \in \mathbb{C}^2 \mid \mathrm{Re}(\mu) > 0\}$ and continuous on $\mathcal{O} \cap \{(\kappa, \mu) \in \mathbb{C}^2 \mid \mathrm{Re}(\kappa) > 0\}$. The function $\kappa \mapsto D_\pm(\kappa, 0)$ is strictly decreasing on $]0, \infty[$, diverges to $+\infty$ as $\kappa \downarrow 0$ and tends to $-1$ as $\kappa \to +\infty$. In particular, it has a unique zero $\kappa_\pm(p) \in ]0, \infty[$. For fixed $\kappa < \kappa_\pm(p)$, the function $\mu \mapsto D_\pm(\kappa, \mu)$ is strictly decreasing on $]0, \infty[$, tends to $D_\pm(\kappa, 0) > 0$ as $\mu \downarrow 0$ and to $-1$ as $\mu \to +\infty$. In particular, it has a unique zero $\mu_\pm(p, \kappa) \in ]0, \infty[$.

(3) The following estimates hold for the critical coupling strength $\kappa_c(p)$ and the pressure $p_\varphi(\kappa V)$,

$$ \kappa_-(p) \le \kappa_c(p) \le \kappa_+(p), $$

$$ \min(\mu_-(p, \kappa), \mu_+(p, \kappa)) \le p_\varphi(\kappa v) \le \max(\mu_-(p, \kappa), \mu_+(p, \kappa)) $$

$$ \text{for } \kappa \le \kappa_c(p). $$

**Proof. (1)** By (5.15) and (5.20) one has

$$ \mathrm{Var}_k(v^\#) \le \sup \left\{ \sum_{q=0}^{t_0(x)} |v \circ \varphi^q(x) - v \circ \varphi^q(y)| \mid x, y \in M, t_j(x) = t_j(y) \text{ for } 0 \le j < k \right\}. $$

Taking into account the convexity of $v$, we can write

$$ \sup_{x,y \in I_t} \frac{|v(x) - v(y)|}{|x - y|} \le |v'(2^{-t-1})| = \frac{p}{\log 2} \frac{2^{t+1}}{(t+2)^{p+1}}, $$

so that, for $t_0(x) = t_0(y)$ and $0 \le q \le t_0(x)$,

$$ |v \circ \varphi^q(x) - v \circ \varphi^q(y)| \le \frac{p}{\log 2} \frac{2^{t_0(x)-q+1}}{(t_0(x)-q+2)^{p+1}} |\varphi^q(x) - \varphi^q(y)|. $$

For $x, y \in M$ such that $t_j(x) = t_j(y)$ for $0 \le j < k$, taking (5.12) into account, we derive

$$ \begin{aligned} |v \circ \varphi^q(x) - v \circ \varphi^q(y)| &\le \frac{p}{\log 2} \frac{2^{t_0(x)-q+1}}{(t_0(x)-q+2)^{p+1}} 2^{-(t_0(x)-q+t_1(x)+\cdots+t_{k-1}(x)+k)} \\ &\le \frac{2p}{\log 2} \frac{1}{(t_0(x)-q+2)^p} 2^{-k}, \end{aligned} $$

and hence

$$ \sum_{q=0}^{t_0(x)} |v \circ \varphi^q(x) - v \circ \varphi^q(y)| \le \frac{2p}{\log 2} 2^{-k} \sum_{q=2}^{t_0(x)+2} \frac{1}{q^{p+1}} \le \frac{2p}{\log 2} \zeta(p+1, 2) 2^{-k}, $$

i.e.,

$$ \mathrm{Var}_k(v^\#) \le \frac{2p}{\log 2} \zeta(p+1,2)2^{-k}. $$
Summing over $k \ge 1$ yields

$$ \mathrm{Var}(v^#) \le \frac{2p}{\log 2} \zeta(p+1, 2). $$

(2) We consider $D_+$, the analysis of $D_-$ is similar. The relation

$$ \zeta(p) - \zeta(p, m+1) = \sum_{k=1}^{\infty} k^{-p} - \sum_{k=0}^{\infty} (k+m+1)^{-p} = \sum_{k=1}^{m} k^{-p}, \quad (5.35) $$

which, in view of (5.31), holds for $\mathrm{Re}\,p > 1$ and $m \ge 1$, extends by analyticity to any $p \in \mathbb{C} \setminus \{1\}$ and $m \ge 0$. Thus

$$ \phi_n := \zeta(p) - \zeta(p, n+2) = \sum_{k=1}^{n+1} k^{-p} \ge \int_1^{n+2} x^{-p} dx = \frac{(n+2)^{1-p}-1}{1-p}, $$

and the series in Definition (5.33) admits the majorant

$$ \sum_{n=0}^{\infty} e^{-(n+1) \mathrm{Re}\,\mu - \phi_n \mathrm{Re}\,\kappa}, $$

which is convergent whenever $\mathrm{Re}\,\mu > 0$ or $\mathrm{Re}\,\mu = 0$ and $\mathrm{Re}\,\kappa > 0$. This shows that $D_+$ is analytic in $\mathcal{O}$ and continuous in $\bar{\mathcal{O}} \cap \{(\kappa, \mu) \in \mathbb{C}^2 | \mathrm{Re}(\kappa) > 0\}$. The remaining statements are simple consequences of the fact that the above majorant coincides with the series in (5.33) for real values of $\kappa$ and $\mu$.

(3) For the potential (5.34), Definition (5.29) gives

$$ v_{-}(t) = -(t+1)^{-p}, \quad v_{+}(t) = -(t+2)^{-p}, $$

so that

$$ \underline{v}(t) = - \sum_{n=1}^{t+1} n^{-p} $$

and

$$ \bar{v}(t) = 1 - \sum_{n=1}^{t+2} n^{-p}. $$

Invoking Relation (5.35), we can write

$$ \underline{v}(t) = -(\zeta(p) - \zeta(p, t+2)), $$

$$ \bar{v}(t) = -(\zeta(p) - 1 - \zeta(p, t+3)), $$

and, in view of (5.30), (5.32) and (5.33),

$$ \log(1 + D_{-}(\kappa, 0)) \le \Delta(\kappa v) \le \log(1 + D_{+}(\kappa, 0)), $$

and more generally
$$ \log(1 + D_{-}(\kappa, \mu)) \le p(\kappa \nu, \mu) - \mu \le \log(1 + D_{+}(\kappa, \mu)). $$

According to the discussion in the previous section, the critical point $\kappa_c(p)$ is determined by the condition $\Delta(\kappa) = 0$. Since the function $\kappa \mapsto \Xi_k(\kappa \nu, r)$ is obviously non-increasing, recalling that

$$ \Delta(\kappa \nu) = \lim_{r \uparrow 1} \liminf_{k \to \infty} \Xi_k(\kappa, r)^{-k}, $$

we deduce that the function $\kappa \mapsto \Delta(\kappa \nu)$ is non-decreasing and it follows that

$$ \kappa_{-}(p) \le \kappa_{c}(p) \le \kappa_{+}(p). $$

The same analysis applies to the pressure $p_{\phi}(\kappa \nu)$, which is the zero of the function $\mu \mapsto \rho(\kappa \nu, \mu)e^{\mu} - 1$ and is bracketed by the zeros of $D_{\pm}(\kappa, \mu)$. $\square$

## 5.8 Numerical Calculation of the Pressure

Direct application of (5.14) for numerical evaluation converges slowly and, due to the exponential growth $|M_t| = 2^t$, becomes impractical. Selecting the subsequence of prime $n$'s largely improves the situation. For prime $t$, the set $\text{Fix}(\phi^t)$ decomposes into the two fixed points 0, 2/3 and periodic points of primitive period $t$. Thus

$$ \sum_{x \in \text{Fix}(\phi^t)} e^{\kappa S_t \nu(x)} = e^{t \kappa \nu(0)} + e^{t \kappa \nu(2/3)} + \sum_{x \in \text{Per}_t} e^{\kappa S_t \nu(x)}, $$

where $\text{Per}_t$ denotes the set of points with primitive period $t$. Denoting by $\text{Orb}'_t$ the set of periodic orbits of primitive period $t$, we get

$$ \sum_{x \in \text{Per}_t} e^{\kappa S_t \nu(x)} = \sum_{o \in \text{Orb}'_t} e^{\kappa \nu(o)}, $$

where

$$ \nu(o) := \sum_{x \in o} \nu(x). $$

Thus

$$ \mathfrak{p}_{\phi}(\kappa \nu) = \lim_{t \to \infty, t \text{ prime}} \frac{1}{t} \log \left( e^{t \kappa \nu(0)} + e^{t \kappa \nu(2/3)} + t \sum_{o \in \text{Orb}'_t} e^{\kappa \nu(o)} \right), $$

and hence

$$ \mathfrak{p}_{\phi}(\kappa \nu) = \max \left( \kappa \nu(0), \kappa \nu(2/3), \lim_{t \to \infty, t \text{ prime}} \frac{1}{t} \log \left( t \sum_{o \in \text{Orb}'_t} e^{\kappa \nu(o)} \right) \right). $$
It turns out that the sequence

$$ \text{prime } t \mapsto p_t(\kappa) := \frac{1}{t} \log \left( t \sum_{o \in \text{Orb}_t'} e^{\kappa v(o)} \right) $$

is rapidly converging; The plots of $p_{11}(\kappa)$ and $p_{13}(\kappa)$ are indistinguishable for $\kappa \le 1/2$, and slight departures near $\kappa_c(p)$ appear for $\kappa > 1/2$, see Figure 5.2. Consistency with our rigorous lower/upper bounds is clearly displayed in Figure 5.3; observe that the curvature is small, but increases with $p$. The critical point $\kappa_c(p)$ is easily computed from this approximation and Figure 5.4 shows the result of this calculation.

Going back to the square map of Section 5.6, consider the potential

$$ v(x, y) = \kappa_c(p_1)v_1(x) + \kappa_c(p_2)v_2(y), $$

where $v_1$ and $v_2$ are of the form (5.34) with respective exponents $p_1, p_2 \in ]0, 1[$.

Figure 5.5 shows the entropic pressure $e(\alpha)$ and the associate rate $I(s)$ for $(p_1, p_2) = (0.9, 0.1)$, illustrating the occurrence of phase transitions. The singularities of the entropic pressure materialize as jumps in its first derivative, located on the vertical lines. To each of them corresponds a segment of unexposed points of the rate.

## References

[Bil68] P. BILLINGSLEY: *Convergence of probability measures*. First edition. John Wiley & Sons, Inc., New York, 1968.

[Bil99] ———, *Convergence of probability measures*. Second edition. Wiley Series in Probability and Statistics: Probability and Statistics. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1999, x+277. DOI: 10.1002/9780470316962.

[CE80] P. COLLET AND J.-P. ECKMANN: *Iterated Maps on the Interval as Dynamical Systems*. Birkhäuser, 1980. DOI: 10.1007/978-0-8176-4927-2.

[Hof81] F. HOFBAUER: On intrinsic ergodicity of piecewise monotonic transformations with positive entropy II. Israel J. Math. 38 (1981), 107-115. DOI: 10.1007/bf02761854.

[MS80] M. MISIUREWICZ AND W. SZLENK: Entropy of piecewise monotone mappings. Studia Math. 67 (1980), 45-63. DOI: 10.4064/sm-67-1-45-63.

[MV03] C. MAES AND E. VERBITSKIY: Large deviations and a fluctuation symmetry for chaotic homeomorphisms. Commun. Math. Phys. 233 (2003), 137-151. DOI: 10.1007/s00220-002-0764-6.

[PS78] G. PÓLYA AND G. SZEGŐ: *Problems and Theorems in Analysis. Vol. I*. Springer Study Edition. Series. Integral Calculus. Theory of Functions. Springer, New York-Heidelberg, 1978. DOI: 10.1007/978-3-642-61983-0.

[PZ06] Y. PESIN AND K. ZHANG: Phase transitions for uniformly expanding maps. J. Stat. Phys. 122 (2006), 1095-1110. DOI: 10.1007/s10955-005-9005-7.

[Rue17] S. RUETTE: *Chaos on the Interval*. Vol. 67. University Lecture Series. American Mathematical Society, Providence, RI, 2017. DOI: 10.1090/ulect/067.
[Sar01] O. M. SARIG: Phase transitions for countable Markov shifts. Commun. Math. Phys. **217** (2001), 555–577. DOI: 10.1007/s002200100367.

[Wal82] P. WALTERS: *An Introduction to Ergodic Theory*. Springer, New York, 1982.
<figure><img src="image_21.png" /><figcaption>Fig. 5.2 The sequence {$p_n(\kappa) | n = 3,5,7,11,13,17$} for some values of the exponent $p$.</figcaption></figure>
<figure><img src="image_22.png" /><figcaption>Fig. 5.3 The approximation of the pressure $p_q(\kappa\nu)$ from $p_{23}(\kappa)$ (solid) and the lower/upper bounds $\mu_\pm(\kappa)$ (dashed) for some values of the exponent $p$.</figcaption></figure>
<figure><img src="image_23.png" /><figcaption>Fig. 5.4 The critical point $κ_c(p)$ computed from $p_{23}(κ)$ (solid) and the lower/upper bounds $κ_±(p)$ (dashed).</figcaption></figure>

<figure><img src="image_24.png" /><figcaption>Fig. 5.5 The entropic pressure and the rate for $(p_1, p_2) = (0.9, 0.1)$ (see main text for details).</figcaption></figure>
<figure><img src="image_25.png" /></figure>
Epilogue

The panorama of FRs and FTs presented in this short monograph arose through
the attempts of mathematical physicists to unravel the general structure behind
the respective numerical and theoretical physics discoveries of the 1990s, and to
test the proposed structure on various models of physical interest. The emerging
mathematical problems are in the field of Large Deviation Theory. They are typ-
ically difficult and poorly understood. This should not come as a surprise, since
the rigorous proofs of FRs and FTs amount to the justification of the Second Law
in a given setting.

Boltzmann's ergodic hypothesis provides a historical parallel to our topic. Roughly speaking, it states that the time and ensemble averages are the same, and emerged in Boltzmann's attempts to understand the foundations of statistical mechanics. It has led to the birth of ergodic theory as a mathematical discipline. FRs and FTs deal with fluctuations that accompany time-reversal symmetry breaking in non-equilibrium statistical mechanics. Unlike in the case of Boltzmann's ergodic hypothesis, the mathematical discipline to which they relate, the theory of large deviations, was well-developed in early 1990s at the time of the first discoveries, and has provided the natural structural setting for their study. It remains to be seen whether the novel mathematical problems brought by FRs and FTs will lead to major new mathematical developments.

The follow-up monograph [Cun+] is primarily addressed to a pure mathemat-
ics audience. Just like ergodicity, FRs and FTs can be formulated mathemati-
cally in great generality, and studied without any reference to underlying physics.
However, the existing Large Deviation Theory technology is far from sufficient to
address FRs and FTs in most models of physical interest. The monograph [Cun+]
is focused on the state-of-the-art mathematical techniques related to this study.

The two monographs are complementary to each other and can be read in-
dependently. Neither is encyclopedic nor even a comprehensive review of the
subject, and perhaps at this point such a treatise cannot be written anymore.<fnref n="9" />

<footnote n="9">⁹ Among important missing topics, we mention the derivation of the linear response theory (Green–Kubo formulas, Onsager reciprocity relation, Fluctuation–Dissipation theorem) from</footnote>
We do hope, though, that they may provide a blueprint for future studies of FRs and FTs by both mathematical physicists and mathematicians.

References

[Ben+24] T. BENOIST, L. BRUNEAU, V. JAKŠIĆ, A. PANATI, AND C.-A. PILLET: A note on two-times measurement entropy production and modular theory. Lett. Math. Phys. **114:32** (2024). DOI: 10.1007/s11005-024-01777-0.

[Ben+25a] ————, Entropic fluctuations in statistical mechanics II. Quantum dynamical systems. Commun. Math. Phys. **406** (2025). DOI: 10.1007/s00220-025-05360-z.

[Ben+25b] ————, On the thermodynamic limit of two-time measurement entropy production. Rev. Math. Phys. (2025). DOI: 10.1142/S0129055X24610063.

[Ben+26] ————, Entropic fluctuation theorems for the spin-fermion Model. J. Math. Phys. (2026).

[Cun+] N. CUNEO, V. JAKŠIĆ, V. NERSESYAN, C.-A. PILLET, AND A. SHIRIKYAN: *Mathematical Theory of the Fluctuation Theorem*. In preparation.

[Jak+12] V. JAKŠIĆ, Y. OGATA, Y. PAUTRAT, AND C.-A. PILLET: *Entropic fluctuations in quantum statistical mechanics—An introduction*. Quantum Theory from Small to Large Scales. Ed. by J. Fröhlich, M. Salmhofer, W. de Roeck, V. Mastropietro, and L. Cugliandolo. Vol. 95. Lecture Notes of the Les Houches Summer School. Oxford University Press, Oxford, 2012, 213–410. DOI: 10.1093/acprof:oso/9780199652495.003.0004.

[JPR11] V. JAKŠIĆ, C.-A. PILLET, AND L. REY-BELLET: Entropic fluctuations in statistical mechanics: I. Classical dynamical systems. Nonlinearity **24** (2011), 699–763. DOI: 10.1088/0951-7715/24/3/003.

<footnote>FRs and FTs, see [JPR11] and references therein, and a discussion of extensions of FRs and FTs to the quantum domain, see [Ben+24; Ben+25b; Ben+25a; Ben+26; Jak+12] and references therein.</footnote>
# Index of Notations

## Abbreviations

- **FR** fluctuation relation

- **FT** fluctuation theorem

- **LDP** large deviation principle

- **NESS** nonequilibrium steady state

- **SDE** stochastic differential equation

## Assumptions

- (C) $\varphi$ is continuous, page 83

- (H) $\varphi$ is a homeomorphism, page 83

- (USCE) upper semicontinuity of entropy, page 84

- (S) specification property, page 85

- (WPS) weak periodic specification property, page 88

- (ED) the set $\mathcal{E}_{\varphi}(M)$ is entropy-dense in $\mathcal{P}_{\varphi}(M)$, page 84

- (PAP) periodic approximation of pressure, page 88

- (Sym) reversal hypothesis, page 91
## Sets

N the set of natural numbers

Z the set of integers

[a, b] the set of integers in [a, b]

Z+ the set of non-negative integers

R the set of real numbers

R+ the set of non-negative real numbers

C the set of complex numbers

C(X) space of continuous functions f: X → R

C_b(X) subspace of bounded f ∈ C(X), with the supremum norm ||·||

P(X) set of Borel probability measures on X

P_φ(X) subset of φ-invariant elements of P(X) (for φ: X → X)

E_φ(X) subset of ergodic elements of P_φ(X)

(Ω_λ, F_λ)_{λ∈L} a directed family of measurable spaces

Fix(φ) set of fixed points of the map φ

## Maps

P, \hat{P}, Q, \hat{Q} probability measures

(P_λ, \hat{P}_λ) a pair of probability measures on (Ω_λ, F_λ),

S(P) Shannon entropy of P, Equ. (3.8)

Ent(P|Q) relative entropy, Equ. (2.6)

Ent_α(P|Q) Rényi relative entropy, Equ. (2.9)

σ_λ entropy production observable, Equ. (2.5)

ep_λ entropy production, Equ. (2.7)

e_λ(α) cumulant generating function of -σ_λ, Equ. (2.10)

e(α) entropic pressure, Equ. (2.24)

σ phase space contraction rate, Equ. (3.37)

μ_t^x empirical measures, Equ. (4.4)

h_{Top}(φ) topological entropy of a map φ
$h_{\varphi}(Q)$ Kolmogorov–Sinai entropy of $Q$ w.r.t. $\varphi$

$\mathfrak{p}_{\varphi}(v)$ topological pressure of continuous potential $v$ w.r.t. $\varphi$, Equ. (4.1)

## Misc Symbols

$\partial^{\pm}$ left/right derivative, page 31

$\tilde{X}/\bar{X}$ the interior/closure of a set $X$

$\mathbb{Q} \ll \mathbb{P}$ absolute continuity of $\mathbb{Q}$ w.r.t. $\mathbb{P}$

$\frac{d\mathbb{Q}}{d\mathbb{P}}$ Radon–Nikodym derivative

$(\cdot, \cdot)$ Euclidean inner product on $\mathbb{R}^d$

$\text{Dom}(f)$ essential domain of a convex function $f$

$d(\cdot, \cdot)$ metric on a Polish space

$\mathbb{P} \otimes \mathbb{Q}$ the product measure

$\langle f, \mathbb{P} \rangle$ the integral of $f$ w.r.t. $\mathbb{P}$

$|X|$ cardinality of the set $X$

$1_X$ indicator function of the set $X$

$\{\cdot, \cdot\}$ Poisson bracket