Higher-Dimensional Nonlinear Thermodynamic Formalism

Abstract

We introduce a higher-dimensional version of the nonlinear thermodynamic formalism introduced by Buzzi and Leplaideur, in which a potential is replaced by a family of potentials. In particular, we establish a corresponding variational principle and we discuss the existence, characterization, and number of equilibrium measures for this higher-dimensional version.

1 Introduction

Recently, Buzzi and Leplaideur [7] introduced a variation of the thermodynamic formalism, which they called nonlinear thermodynamic formalism. Roughly speaking, this amounts to compute the topological pressure replacing Birkhoff sums by images of them under a given function (that may be nonlinear and thus the name). Our main aim is twofold:

• (1) to introduce a higher-dimensional version of their notion of topological pressure, replacing a potential by a family of potentials, and to establish a corresponding variational principle;

• (2) to discuss the existence, characterization, and number of equilibrium measures, with special attention to the new phenomena that occur in this higher-dimensional version.

We also give a characterization of the nonlinear pressure as a Carathéodory dimension, which allows us to extend the notion to noncompact sets.

The most basic notion of the mathematical thermodynamic formalism is topological pressure. It was introduced by Ruelle [24] for expansive maps and by Walters [28] in the general case. For a continuous map $T :X\rightarrow X$ on a compact metric space, the topological pressure of a continuous function $\varphi :X\rightarrow {\mathbb {R}}$ is defined by

$$\begin{aligned} P(\varphi )=\lim _{\varepsilon \rightarrow 0} \limsup _{n \rightarrow \infty } \frac{1}{n}\log \sup _E\sum _{x\in E}\exp S_n\varphi (x), \end{aligned}$$

(1)

with the supremum taken over all $(n,\varepsilon )$-separated sets E and where $S_n\varphi = \sum _{k=0}^{n-1}\varphi \circ T^k$. An important relation between the topological pressure and the Kolmogorov–Sinai entropy is given by the variational principle

$$\begin{aligned} P(\varphi )=\sup _\mu \left( h_\mu (T)+\int _X \varphi \, d \mu \right) , \end{aligned}$$

(2)

with the supremum taken over all T-invariant probability measures $\mu$ on X and where $h_\mu (T)$ denotes the entropy with respect to $\mu$. This was established by Ruelle [24] for expansive maps and by Walters [28] in the general case. The theory is now a broad and active independent field of study with many connections to other areas of mathematics. We refer the reader to the books [2, 6, 14, 15, 20, 21, 25, 29] for many developments.

Building on work on the Curie–Weiss mean-field theory in [17], the nonlinear topological pressure was introduced in [7] as a generalization of (1) as follows (more precisely, we give an equivalent formulation using separated sets instead of covers). Given a continuous function $F:{\mathbb {R}}\rightarrow {\mathbb {R}}$, the nonlinear topological pressure of a continuous function $\varphi :X \rightarrow {\mathbb {R}}$ is given by

$$\begin{aligned} P_F(\varphi ) = \lim _{\varepsilon \rightarrow 0}\limsup _{n \rightarrow \infty }\frac{1}{n}\log \sup _E\sum _{x \in E}\exp \left[ nF\left( \frac{S_n\varphi (x)}{n}\right) \right] , \end{aligned}$$

(3)

with the supremum taken over all $(n,\varepsilon )$-separated sets E. For $F(x)=x$ we recover the classical topological pressure. Buzzi and Leplaideur also established a version of the variational principle in (2). Namely, assuming that the pair $(T, \Phi )$ has an abundance of ergodic measures (see Sect. 2.1 for the definition), they proved that

$$\begin{aligned} P_F(\varphi ) = \sup _\mu \left( h_\mu (T) + F\biggl (\int _X\varphi \,d\mu \biggr )\right) , \end{aligned}$$

(4)

with the supremum taken over all T-invariant probability measures $\mu$ on X. In addition, they characterized the equilibrium measures of this thermodynamic formalism, that is, the invariant probability measures at which the supremum in (4) is attained, and they showed that a new type of phase transition can occur. Namely, one may have more than one equilibrium measure, although we still have a central limit theorem (see also [16, 26]).

As described above, our main aim in the paper is to understand whether and how the results in [7] extend to the higher-dimensional case. This corresponds to replace the functions F and $\varphi$ in (3), respectively, by a continuous function $F:{\mathbb {R}}^d \rightarrow {\mathbb {R}}$ and by a family $\Phi = \{\varphi _1, \ldots , \varphi _d\}$ of continuous functions $\varphi _i:X \rightarrow {\mathbb {R}}$ for $i=1,\ldots ,d$. The nonlinear topological pressure of $\Phi$ is then defined by

$$\begin{aligned} P_F(\Phi ) = \lim _{\varepsilon \rightarrow 0}\limsup _{n \rightarrow \infty }\frac{1}{n}\log \sup _{E}\sum _{x \in E}\exp \left[ nF\biggl (\frac{S_n\varphi _1(x)}{n},\ldots , \frac{S_n\varphi _d(x)}{n}\biggr )\right] , \end{aligned}$$

with the supremum taken over all $(n,\varepsilon )$-separated sets E. Whenever possible, we follow a similar approach to obtain a variational principle and to discuss the existence, characterization, and number of equilibrium measures.

In particular, assuming that the pair $(T, \Phi )$ has an abundance of ergodic measures, we establish the variational principle

$$\begin{aligned} P_F(\Phi ) = \sup _\mu \left( h_\mu (T) + F\biggl (\int _X\varphi _1 \,d\mu ,\ldots ,\int _X \varphi _d \,d\mu \biggr )\right) , \end{aligned}$$

(5)

with the supremum taken over all T-invariant probability measures $\mu$ on X. As in [7], for a certain class of pairs $(T,\Phi )$ we also characterize the equilibrium measures, that is, the invariant probability measures at which the supremum in (5) is attained. Consider the sets

$$\begin{aligned} L(\Phi ) = \biggl \{\biggl (\int _X\varphi _1 \,d\mu ,\ldots ,\int _X \varphi _d \,d\mu \biggr ) : \mu \text { is }T\text {-invariant}\biggr \} \subset {\mathbb {R}}^d \end{aligned}$$

and

$$\begin{aligned} \mathcal {M}(z)= \biggl \{\mu \text { is }T\text {-invariant}: \biggl (\int _X\varphi _1 \,d\mu ,\ldots ,\int _X \varphi _d \,d\mu \biggr ) = z\biggr \}. \end{aligned}$$

We reduce the problem of finding equilibrium measures to the problem of finding maximizers of the function $E:L(\Phi ) \rightarrow {\mathbb {R}}$ defined by

$$\begin{aligned} E(z) = h(z) + F(z), \end{aligned}$$

where

$$\begin{aligned} h(z)= \sup \bigl \{h_\mu (T): \mu \in \mathcal {M}(z)\bigr \}. \end{aligned}$$

We note that the function E first appeared in [17]. It turns out that h(z) coincides with the topological entropy of the map T on the set

$$\begin{aligned} C_z(\Phi ) = \biggl \{x \in X: \biggl (\lim _{n \rightarrow \infty }\frac{S_n\varphi _1(x)}{n},\ldots ,\lim _{n \rightarrow \infty }\frac{S_n\varphi _d(x)}{n}\biggr ) = z\biggr \} \end{aligned}$$

(see (6)). In general $C_{z}(\Phi )$ need not be compact and so here we need the notion of topological entropy for noncompact sets (see Sect. 2.4 for the definition). In fact, we show that for each $z \in {{\,\mathrm{int}\,}}L(\Phi )$ maximizing E there exists a unique equilibrium measure $\nu _z$. This is actually a classical equilibrium measure for a certain function $\psi _z$ that depends on the family of functions $\Phi$. In addition, we give conditions for the uniqueness of the equilibrium measures, both for $d=1$ and for $d>1$ (see Theorems 10 and 11).

Before proceeding, we highlight the main elements and difficulties of passing to the higher-dimensional case. To the possible extent, our streamlined proof of the variational principle follows arguments in [7] for a single function, considering covers by balls instead of covers by intervals. Our main result (Theorem 7) gives a characterization of equilibrium measures and uses in an essential way the higher-dimensional multifractal analysis developed in [3] (see the following paragraph for further details). It was crucial to make sure that all was prepared so that we could apply this higher-dimensional theory, which allows us to give a description of the equilibrium measures for the nonlinear topological pressure as equilibrium measures of certain functions in ${{\,\mathrm{span}\,}}\{\varphi _1,\ldots ,\varphi _d,1\}$. In addition, in Sect. 5 we describe criteria for the uniqueness of equilibrium measures both for $d=1$ and $d>1$, and we give conditions for the coincidence of equilibrium measures for two systems in terms of the notion of cohomology.

As noted above, to a relevant extent we use in the proofs the higher-dimensional multifractal analysis developed in [3]. This gives once more a connection between the thermodynamic formalism and multifractal analysis, which is a principal characteristic of our work. In particular, that other work includes a conditional variational principle, which shows for example that the topological entropy of the level sets of pointwise dimensions, local entropies, and Lyapunov exponents can be approximated simultaneously by the entropy of ergodic measures. More precisely, for a continuous map $T :X\rightarrow X$ on a compact metric space with upper semicontinuous entropy, it is shown in [3] that if $\Phi = \{\varphi _1, \ldots , \varphi _d\}$ is composed of continuous functions such that each element of ${{\,\mathrm{span}\,}}\{\varphi _1,\ldots ,\varphi _d,1 \}$ has unique equilibrium measure (for the classical topological pressure), then for each $z\in {{\,\mathrm{int}\,}}L(\Phi )$ the set $C_z(\Phi )$ is nonempty and has topological entropy

$$\begin{aligned} h(T|_{C_z(\Phi )})=\sup \bigl \{h_\mu (T): \mu \in \mathcal {M}(z)\bigr \} =\inf _{q\in {\mathbb {R}}^d}P\bigl (\langle q, \Phi -z \rangle \bigr ). \end{aligned}$$

(6)

In addition, there exists an ergodic equilibrium measure $\mu _z\in \mathcal {M}(z)$ with

$$\begin{aligned} \mu _z(C_z(\Phi ))=1\quad \text {and}\quad h_{\mu _z}(T)=h(T|_{C_z(\Phi )}). \end{aligned}$$

We note that some phenomena absent in classical multifractal analysis for a single potential may occur in a higher-dimensional multifractal spectrum. For example, the domain of the spectrum may not be convex and its interior may be empty or have more than one connected component.

Finally, we also detail further the motivation for introducing the nonlinear thermodynamical formalism and for our own work with a higher-dimensional version of this formalism. In statistical mechanics, particularly in the study of magnetic systems, the Curie–Weiss–Potts model is generally seen as an extension of the Curie–Weiss model, which can be considered as a mean-field version of the Ising model (see for example [11,12,13] for detailed discussions). Leplaideur and Watbled traced a parallel between statistical mechanics and ergodic theory for general spin spaces, introducing a generalized Curie–Weiss model in [17] and a generalized Curie–Weiss–Potts model in [18] (the latter model can be seen as a higher-dimensional generalized Curie–Weiss model). When $X = \{-1,1\}^{{\mathbb {N}}}$, T is the shift map, $\varphi$ is a Hölder continuous function and $F:{\mathbb {R}}\rightarrow {\mathbb {R}}$ is given by

$$\begin{aligned} F(z) = \frac{\beta }{2}z^2, \quad \text {where }\beta \ge 0\text { is a physical parameter,} \end{aligned}$$

we recover the generalized Curie–Weiss model. Again for the shift map T on X, when $\Phi = \{\varphi _1,\ldots ,\varphi _d\}$ is a family of Hölder continuous functions and $F:{\mathbb {R}}^{d} \rightarrow {\mathbb {R}}$ is given by

$$\begin{aligned} F(z) = \frac{\beta }{2}\Vert z\Vert ^{2}, \quad \text {where }\beta \ge 0\text { is a physical parameter} \end{aligned}$$

and $\Vert \cdot \Vert$ is a given norm on ${\mathbb {R}}^d$, we recover the generalized Curie–Weiss–Potts model. In this sense, while [7] and more recently [8] extend the study of the generalized Curie–Weiss model for any continuous function $F:{\mathbb {R}}\rightarrow {\mathbb {R}}$ and any map T, analogously our work extends the study of the generalized Curie–Weiss–Potts model to include any continuous function $F:{\mathbb {R}}^{d} \rightarrow {\mathbb {R}}$ and any map T, both under suitable assumptions.

We note that our results and the higher-dimensional results in [8] are mostly overlapping, but there are some differences. Namely, in [8] the authors use tools from convex analysis, for example to characterize the nonlinear equilibrium measures. They also discuss phase transitions in the nonlinear context, study equidistribution of Gibbs ensembles, and obtain a generic result on the uniqueness of nonlinear equilibrium measures. On the other hand, we characterize the nonlinear equilibrium measures using a conditional variational principle coming from higher-dimensional multifractal analysis. For instance, in our characterization we are able to specify the support of the nonlinear equilibrium measure. Moreover, we study in more detail the cohomology relations in the nonlinear context, and we obtain a simple criteria for uniqueness of nonlinear equilibrium measures.

2 Nonlinear Topological Pressure

2.1 Basic Notions

We first recall the notion of nonlinear topological pressure introduced by Buzzi and Leplaideur in [7] as an extension of the classical topological pressure. Let $T:X \rightarrow X$ be a continuous map on a compact metric space $X=(X,d)$. For each $n\in {\mathbb {N}}$ we consider the distance

$$\begin{aligned} d_n(x,y)=\max \bigl \{d(T^k(x),T^k(y)):k=0,\ldots ,n-1\bigr \}. \end{aligned}$$

Take $n\in {\mathbb {N}}$ and $\varepsilon >0$. A set $C\subset X$ is said to be an $(n,\varepsilon )$-cover of X if $\bigcup _{x \in C}B_n(x,\varepsilon ) = X$, where

$$\begin{aligned} B_n(x,\varepsilon ) = \bigl \{y \in X: d_n(y,x) < \varepsilon \bigr \} \end{aligned}$$

(usually the set $B_n(x,\varepsilon )$ is called a Bowen ball). Given a continuous function $F:{\mathbb {R}}\rightarrow {\mathbb {R}}$, the nonlinear topological pressure of a continuous function $\varphi :X \rightarrow {\mathbb {R}}$ is defined by

$$\begin{aligned} P_F(\varphi ) = \lim _{\varepsilon \rightarrow 0}\limsup _{n \rightarrow \infty }\frac{1}{n}\log \inf _{C}\sum _{x \in C}\exp \left[ nF\biggl (\frac{S_n\varphi (x)}{n}\biggr )\right] , \end{aligned}$$

where $S_n\varphi = \sum _{k=0}^{n-1}\varphi \circ T^k$, with the infimum taken over all $(n,\varepsilon )$-covers C.

Let $\mathcal {M}$ be the set of T-invariant probability measures on X. Following [7], we say that the pair $(T,\varphi )$ has an abundance of ergodic measures if for each $\mu \in \mathcal {M}$, $h<h_\mu (T)$ and $\varepsilon >0$ there exists an ergodic measure $\nu \in \mathcal {M}$ such that

$$\begin{aligned} h_{\nu }(T) > h\quad \text {and}\quad \biggl |\int _X\varphi \,d\nu - \int _X\varphi \,d\mu \biggr |< \varepsilon . \end{aligned}$$

Assuming that $(T,\varphi )$ has an abundance of ergodic measures, they obtained the variational principle

$$\begin{aligned} P_F(\varphi ) = \sup _{\mu \in \mathcal {M}}\biggl \{h_\mu (T) + F\biggl (\int _X\varphi \,d\mu \biggr )\biggr \}. \end{aligned}$$

(7)

They also established (7) when F is a convex function (without assuming that the pair $(T,\varphi )$ has an abundance of ergodic measures). We say that $\nu \in \mathcal {M}$ is an equilibrium measure for $(F,\varphi )$ with respect to T if

$$\begin{aligned} P_F(\varphi ) = h_{\nu }(T) + F\biggl (\int _X\varphi \,d\nu \biggr ). \end{aligned}$$

2.2 Higher-Dimensional Version

In this paper we consider a higher-dimensional generalization of the nonlinear topological pressure.

Given $n\in {\mathbb {N}}$ and $\varepsilon > 0$, a set $E \subset X$ is said to be $(n,\varepsilon )$-separated if $d_{n}(x,y) > \varepsilon$ for every $x, y \in E$ with $x \ne y$. Since X is compact, any $(n,\varepsilon )$-separated set has finite cardinality. Let $F:{\mathbb {R}}^d \rightarrow {\mathbb {R}}$ be a continuous function and let $\Phi = \{\varphi _1, \ldots , \varphi _d\}$ be a family of continuous functions $\varphi _i:X \rightarrow {\mathbb {R}}$ for $i=1,\ldots , d$. The nonlinear topological pressure of the family $\Phi$ is defined by

$$\begin{aligned} P_F(\Phi ) = \lim _{\varepsilon \rightarrow 0}\limsup _{n \rightarrow \infty }\frac{1}{n}\log \sup _{E}\sum _{x \in E}\exp \left[ nF\biggl (\frac{S_n\varphi _1(x)}{n},\ldots , \frac{S_n\varphi _d(x)}{n}\biggr )\right] \end{aligned}$$

(8)

with the supremum taken over all $(n,\varepsilon )$-separated sets E. One can easily verify that the function

$$\begin{aligned} \varepsilon \mapsto \limsup _{n \rightarrow \infty }\frac{1}{n}\log \sup _{E}\sum _{x \in E}\exp \left[ nF\biggl (\frac{S_n\varphi _1(x)}{n},\ldots , \frac{S_n\varphi _d(x)}{n}\biggr )\right] \end{aligned}$$

is nondecreasing and so $P_F(\Phi )$ is well defined. Notice that we only need to consider F on the compact set

$$\begin{aligned} \bigl [-\Vert \varphi _1\Vert _{\infty },\Vert \varphi _1\Vert _{\infty }\bigr ]\times \cdots \times \bigl [-\Vert \varphi _d\Vert _{\infty },\Vert \varphi _d\Vert _{\infty }\bigr ] \subset {\mathbb {R}}^{d}. \end{aligned}$$

We also describe briefly a characterization of the nonlinear topological pressure using $(n,\varepsilon )$-covers. Let

$$\begin{aligned} \mathcal {W}_n(C)= \sum _{x \in C}\exp \left[ nF\biggl (\frac{S_n\varphi _1(x)}{n}, \ldots ,\frac{S_n\varphi _d(x)}{n}\biggr )\right] . \end{aligned}$$

Following closely arguments in [2], one can show that

$$\begin{aligned} P_F(\Phi ) = \lim _{\varepsilon \rightarrow 0}\limsup _{n \rightarrow \infty }\frac{1}{n}\log \inf _{C}\mathcal {W}_n(C) = \lim _{\varepsilon \rightarrow 0}\liminf _{n \rightarrow \infty }\frac{1}{n}\log \inf _{C}\mathcal {W}_n(C), \end{aligned}$$

(9)

with the infimum taken over all $(n,\varepsilon )$-covers C of X.

2.3 Dependence on the Potentials

In this section we discuss briefly how the nonlinear topological pressure depends on the potentials. Given a family of continuous functions $\Phi = \{\varphi _1,\ldots ,\varphi _d\}$, we define the norm

$$\begin{aligned} \Vert \Phi \Vert = \max _{j \in \{1,\ldots ,d\}}\Vert \varphi _j\Vert _{\infty }. \end{aligned}$$

Recall that F is said to be Hölder continuous with constants $C, \alpha > 0$ if

$$\begin{aligned} |F(x)-F(y)| \le C\Vert x - y\Vert _{\infty }^{\alpha }\quad \text {for every} \ x, y \in {\mathbb {R}}^{d}. \end{aligned}$$

Proposition 1

Let $\Phi$ and $\Psi$ be families of continuous functions and let $F:{\mathbb {R}}^{d} \rightarrow {\mathbb {R}}$ be a continuous function. Then the following properties hold:

• (1) the map $\Phi \mapsto P_F(\Phi )$ is continuous;

• (2) if F is Hölder continuous with constants $C,\alpha > 0$, then

  $$\begin{aligned} |P_F(\Phi ) - P_{F}(\Phi )| \le C\Vert \Phi - \Psi \Vert ^{\alpha }; \end{aligned}$$

  (10)

  in particular, if F is Lipschitz, then $\Phi \mapsto P_{F}(\Phi )$ is also Lipschitz with the same Lipschitz constant.

Proof

We first prove property (1). By the uniform continuity of F, given $\varepsilon > 0$, there exists $\delta > 0$ such that $|F(x)-F(y)| < \varepsilon$ whenever $\Vert x-y\Vert _{\infty } < \delta$. Consider a family of continuous functions $\Psi$ such that $\Vert \Psi - \Phi \Vert < \delta$. Then

$$\begin{aligned} \biggl \Vert \biggl (\frac{S_n\psi _1(x)}{n},\ldots ,\frac{S_n\psi _d(x)}{n}\biggr ) - \biggl (\frac{S_n\varphi _1(x)}{n},\ldots , \frac{S_n\varphi _d(x)}{n}\biggr )\biggr \Vert _\infty < \delta \end{aligned}$$

for any $n \in {\mathbb {N}}$ and $x \in X$. By the uniform continuity of F, we obtain

$$\begin{aligned} nF\biggl (\frac{S_n\psi _1(x)}{n},\ldots ,\frac{S_n\psi _d(x)}{n}\biggr ) < nF\biggl (\frac{S_n\varphi _1(x)}{n},\ldots ,\frac{S_n\varphi _d(x)}{n}\biggr ) + n\varepsilon \end{aligned}$$

for any $n \in {\mathbb {N}}$ and $x \in X$. It follows from the definition of the topological pressure in (8) that $P_F(\Psi ) - P_{F}(\Phi )< \varepsilon$. One can show in the same manner that $P_F(\Phi ) - P_{F}(\Psi )< \varepsilon$. Therefore,

$$\begin{aligned} |P_F(\Psi ) - P_{F}(\Phi )| < \varepsilon , \end{aligned}$$

which establishes the first property in the proposition.

Now assume that F is Hölder continuous with constants $C, \alpha > 0$. Then for any families of continuous functions $\Phi$ and $\Psi$ we have

$$\begin{aligned} \biggl |F\biggl (\frac{S_n\varphi _1(x)}{n},\ldots ,\frac{S_n\varphi _d(x)}{n}\biggr ) - F\biggl (\frac{S_n\psi _1(x)}{n},\ldots ,\frac{S_n\psi _d(x)}{n}\biggr )\biggr | \le C\Vert \Phi -\Psi \Vert ^{\alpha } \end{aligned}$$

for any $n \in {\mathbb {N}}$ and $x \in X$. Proceeding as in the proof of property (1), we readily obtain inequality (10). $\square$

2.4 Extension to Noncompact Sets

Based on work of Pesin and Pitskel’ in [22], we give a characterization of the nonlinear topological pressure as a Carathéodory dimension. In particular, this allows us to extend the notion to noncompact sets. We expect that this extension plays an important role in an appropriate version of multifractal analysis associated with the nonlinear topological pressure.

We continue to consider a continuous map $T:X \rightarrow X$ on a compact metric space. Given a finite open cover $\mathcal {U}$ of X, for each $n \in {\mathbb {N}}$ let $\mathcal {X}_n$ be the set of strings $U = (U_1,\ldots , U_n)$ with $U_i \in \mathcal {U}$ for $i=1,\ldots ,n$. We write $l(U) = n$ and we define

$$\begin{aligned} X(U)= \bigl \{x \in X: T^{k-1} \in U_k \text { for } k = 1,\ldots ,l(U) \bigr \}. \end{aligned}$$

We say that $\Gamma \subset \bigcup _{n \in {\mathbb {N}}}\mathcal {X}_n$ covers a set $Z \subset X$ if $Z \subset \bigcup _{U \in \Gamma }X(U)$.

Given a family of continuous functions $\Phi = \{\varphi _1,\ldots , \varphi _d\}$, for each $n \in {\mathbb {N}}$ we define $S_n\Phi = (S_n\varphi _1,\ldots , S_n\varphi _d)$. Moreover, given a function $F:{\mathbb {R}}^d \rightarrow {\mathbb {R}}$, for each $U \in \mathcal {X}_n$ let

$$\begin{aligned} F_{\Phi }(U) = {\left\{ \begin{array}{ll} \sup _{X(U)} n F\left( \frac{1}{n}S_n\Phi \right) &{} \text {if } X(U) \ne \emptyset , \\ -\infty &{} \text {if } X(U) = \emptyset . \end{array}\right. } \end{aligned}$$

Finally, given a set $Z \subset X$ and a number $\alpha \in {\mathbb {R}}$, we define

$$\begin{aligned} M_Z(\alpha , \Phi , \mathcal {U})= \lim _{n \rightarrow \infty } \inf _{\Gamma }\sum _{U \in \Gamma }\exp (-\alpha l(U) + F_{\Phi }(U)), \end{aligned}$$

with the infimum taken over all $\Gamma \subset \bigcup _{k\ge n}\mathcal {X}_k$ covering Z and with the convention that $\exp (-\infty ) = 0$. One can easily verify that the map $\alpha \mapsto M_Z(\alpha , \Phi , \mathcal {U})$ goes from $+\infty$ to zero at a unique $\alpha \in {\mathbb {R}}$ and so one can define

$$\begin{aligned} P_F(Z,\Phi , \mathcal {U}) = \inf \bigl \{\alpha \in {\mathbb {R}}: M_Z(\alpha , \Phi , \mathcal {U}) = 0\bigr \}. \end{aligned}$$

One can proceed as in the proof of Theorem 2.2.1 in [2] to show that the limit

$$\begin{aligned} P_F(Z,\Phi )= \lim _{{{\,\mathrm{diam}\,}}\mathcal {U}\rightarrow 0} P_F(Z,\Phi , \mathcal {U}) \end{aligned}$$

exists. One could also introduce the number $P_F(Z,\Phi )$ using $(n,\varepsilon )$-separated sets or $(n,\varepsilon )$-covers (covers by Bowen balls), in a similar manner to that, for example, in Appendix D in [1].

When $Z=X$ we recover the notion of nonlinear topological pressure for any convex function F.

Theorem 2

If the function F is convex, then $P_F(\Phi ) = P_F(X,\Phi )$.

Proof

The proof is obtained modifying arguments in Sect. 4.2.3 of [2] and so we only give a brief sketch. Given a finite open cover $\mathcal {U}$ of X, we define

$$\begin{aligned} Z_n(\Phi ,\mathcal {U})= \inf _{\Gamma }\sum _{U \in \Gamma }\exp F_{\Phi }(U), \end{aligned}$$

with the infimum taken over all $\Gamma \subset \mathcal {X}_n$ covering X. Given $\Gamma _1 \subset \mathcal {X}_{n_1}$ and $\Gamma _2 \subset \mathcal {X}_{n_2}$, let

$$\begin{aligned} \Gamma ' = \bigl \{UV: U \in \Gamma _1 \text { and } V \in \Gamma _2\bigr \}. \end{aligned}$$

Note that if $\Gamma _1$ and $\Gamma _2$ cover X, then $\Gamma '$ also covers X. Moreover, since F is convex, it follows readily from the identity

$$\begin{aligned} \frac{S_{m+n}\varphi (x)}{m+n}=\frac{m}{m+n}\cdot \frac{S_{m}\varphi (x)}{m} +\frac{n}{m+n}\cdot \frac{S_{n}\varphi (T^m(x))}{n} \end{aligned}$$

that

$$\begin{aligned} F_{\Phi }(UV) \le F_{\Phi }(U) + F_{\Phi }(V) \end{aligned}$$

for each $UV \in \Gamma '$. We have

$$\begin{aligned} \begin{aligned} Z_{n_1 + n_2}(\Phi , \mathcal {U})&\le \sum _{UV \in \Gamma '}\exp F_{\Phi }(UV)\\&\le \sum _{U \in \Gamma _1}\exp F_{\Phi }(U) \sum _{V \in \Gamma _2}\exp F_{\Phi }(V) \end{aligned} \end{aligned}$$

and so

$$\begin{aligned} Z_{n_1+n_2}(\Phi ,\mathcal {U}) \le Z_{n_1}(\Phi ,\mathcal {U})Z_{n_2}(\Phi ,\mathcal {U}). \end{aligned}$$

Therefore, one can define

$$\begin{aligned} Z(\Phi ,\mathcal {U})= \lim _{n \rightarrow \infty }\frac{1}{n}\log Z_n(\Phi ,\mathcal {U}). \end{aligned}$$

Finally, it follows as in Lemmas 2.2.5 and 2.2.6 in [2] that

$$\begin{aligned} \lim _{{{\,\mathrm{diam}\,}}\mathcal {U}\rightarrow 0}Z(\Phi ,\mathcal {U}) = P_F(X,\Phi ) \end{aligned}$$

and

$$\begin{aligned} P_F(\Phi ) = \lim _{{{\,\mathrm{diam}\,}}\mathcal {U}\rightarrow 0}Z(\Phi ,\mathcal {U}). \end{aligned}$$

This yields the desired result. $\square$

Taking $F=\mathrm{id}$ and $\Phi =0$ we recover the notion of topological entropy

$$\begin{aligned} h(T|_Z)=P_\mathrm{id}(Z,0) \end{aligned}$$

of T on the set Z introduced by Pesin and Pitskel’ in [22]. It coincides with the notion of topological entropy for noncompact sets introduced earlier by Bowen in [4]. We emphasize that Z need not be compact nor T-invariant. When $Z=X$ we recover the usual notion of topological entropy.

3 Variational Principle

In this section we establish a variational principle for the nonlinear topological pressure.

Let $T:X \rightarrow X$ be a continuous map on a compact metric space and let $\Phi = \{\varphi _1,\ldots ,\varphi _d\}$ be a family of continuous functions. We say that the pair $(T,\Phi )$ has an abundance of ergodic measures if for each $\mu \in \mathcal {M}$, $h<h_\mu (T)$ and $\varepsilon >0$ there exists an ergodic measure $\nu \in \mathcal {M}$ such that $h_{\nu }(T) > h$ and

$$\begin{aligned} \biggl |\int _X\varphi _i \,d\nu - \int _X\varphi _i \,d\mu \biggr |< \varepsilon \quad \text {for} \ i =1,\ldots ,d. \end{aligned}$$

Moreover, we say that T has entropy density of ergodic measures if for every $\mu \in \mathcal {M}$ there exist ergodic measures $\nu _n\in \mathcal {M}$ for $n\in {\mathbb {N}}$ such that $\nu _n \rightarrow \mu$ in the $\hbox {weak}^*$ topology and $h_{\nu _n}(T) \rightarrow h_\mu (T)$ when $n\rightarrow \infty$. Note that if T has entropy density of ergodic measures, then the pair $(T,\Phi )$ has an abundance of ergodic measures for any family of continuous functions $\Phi$.

In order to give examples of pairs with an abundance of ergodic measures we first recall a few notions. Given $\delta >0$, we say that T has weak specification at scale $\delta$ if there exists $\tau \in {\mathbb {N}}$ such that for every $(x_1,n_1),\ldots ,(x_k,n_k) \in X \times {\mathbb {N}}$ there are $y \in X$ and times $\tau _1,\ldots , \tau _{k-1} \in {\mathbb {N}}$ such that $\tau _i \le \tau$ and

$$\begin{aligned} d_{n_i}(T^{s_{i-1} + \tau _{i-1}}(y),x_i) < \delta \quad \text {for} \ i=1,\ldots , k, \end{aligned}$$

where $s_i = \sum _{i=1}^{i}n_i + \sum _{i=1}^{i-1}\tau _i$ with $n_0 = \tau _0 = 0$. When one can take $\tau _i = \tau$ for $i =1,\ldots ,k-1$, we say that T has specification at scale $\delta$. Finally, we say that T has weak specification if it has weak specification at every scale $\delta$ and, analogously, we say that T has specification if it has specification at every scale $\delta$.

It was shown earlier in [10, Theorem B] and [23, Theorem 2.1] that mixing subshifts of finite type and mixing locally maximal hyperbolic sets have entropy density of ergodic measures. More recently, it was shown in [9] that a continuous map $T:X \rightarrow X$ on a compact metric space with the weak specification property such that the entropy map $\mu \mapsto h_\mu (T)$ is upper semicontinuous, has entropy density of ergodic measures. In particular, this implies that the pair $(T,\Phi )$ has an abundance of ergodic measures for any family of continuous functions $\Phi$. Some examples of maps with an abundance of ergodic measures include expansive maps with specification or with weak specification, topologically transitive locally maximal hyperbolic sets for diffeomorphisms, and transitive topological Markov chains.

The following theorem establishes a variational principle for the nonlinear topological pressure.

Theorem 3

Let $T:X \rightarrow X$ be a continuous map on a compact metric space and let $\Phi =\{\varphi _1,\ldots , \varphi _d\}$ be a family of continuous functions. Given a continuous function $F:{\mathbb {R}}^d \rightarrow {\mathbb {R}}$, if the pair $(T,\Phi )$ has an abundance of ergodic measures, then

$$\begin{aligned} P_F(\Phi ) = \sup _{\mu \in \mathcal {M}}\biggl \{h_\mu (T) + F\biggl (\int _X\Phi \,d\mu \biggr ) \biggr \}, \end{aligned}$$

(11)

where $\int _X\Phi \,d\mu = \big (\int _X\varphi _1 \,d\mu ,\ldots , \int _X\varphi _{d} \,d\mu \big )$.

Proof

To the possible extent we follow arguments in [7] for a single function. We divide the proof into two lemmas.

Lemma 1

Let $T:X \rightarrow X$ be a continuous map on a compact metric space and let $\Phi$ be a family of continuous functions. Then:

• (1)
  $$\begin{aligned} P_F(\Phi ) \ge \sup _{\mu \in \mathcal {M}_{\mathrm{erg}}}\biggl \{h_\mu (T) + F\biggl (\int _X\Phi \,d\mu \biggr )\biggr \}; \end{aligned}$$

• (2) if, in addition, the pair $(T,\Phi )$ has an abundance of ergodic measures, then

  $$\begin{aligned} P_F(\Phi ) \ge \sup _{\mu \in \mathcal {M}}\biggl \{h_\mu (T) + F\biggl (\int _X\Phi \,d\mu \biggr )\biggr \}. \end{aligned}$$

Proof of the lemma

Given $r>0$, since X is compact there exist $\delta ,\varepsilon > 0$ such that

$$\begin{aligned} |\varphi _{i}(x) - \varphi _{i}(y)|<\delta /2 \quad \text {whenever} \ d(x,y) < \varepsilon \end{aligned}$$

for $i=1,\ldots ,d$ and

$$\begin{aligned} |F(v) - F(w)|< r \quad \text {whenever} \ \Vert v-w\Vert < \delta . \end{aligned}$$

For definiteness we shall take the $\ell ^\infty$ norm on ${\mathbb {R}}^d$. Now let $\mu \in \mathcal {M}$ be an ergodic measure. By Birkhoff’s ergodic theorem and the Brin–Katok local entropy formula, together with Egorov’s theorem, there exist a set $A\subset X$ of measure $\mu (A) > 1- r$ and $N \in {\mathbb {N}}$ such that

$$\begin{aligned} \biggl |\frac{S_n\varphi _i(x)}{n} - \int _X\varphi _i \,d\mu \biggr |<\delta /2 \end{aligned}$$

(12)

for all $i=1,\ldots ,d$ and

$$\begin{aligned} \biggl |\frac{1}{n}\log \mu (B_n(x,2\varepsilon )) + h_\mu (T) \biggr |< r, \end{aligned}$$

(13)

for $x \in A$ and $n > N$.

Now let C be an arbitrary $(n,\varepsilon )$-cover and let $D \subset C$ be a minimal $(n,\varepsilon )$-cover of A. For each $x \in D$, the ball $B_n(x,\varepsilon )$ intersects A at some point y (otherwise one could discard the point x in D). Note that

$$\begin{aligned} d(T^k(x),T^k(y))<\varepsilon \quad \text {for} \ k=0,\ldots ,n-1. \end{aligned}$$

Hence, it follows from (12) that

$$\begin{aligned} \begin{aligned} \biggl |\frac{S_n\varphi _{i}(x)}{n} - \int _X\varphi _{i} \,d\mu \biggr |&\le \frac{1}{n}|S_n\varphi _{i}(x) - S_n\varphi _{i}(y)|\\&\quad + \biggl |\frac{S_n\varphi _{i}(y)}{n} - \int _X\varphi _{i} \,d\mu \biggr |< \delta /2 + \delta /2 = \delta \end{aligned} \end{aligned}$$

for $i =1,\ldots ,d$ and so

$$\begin{aligned} \biggl |F\biggl (\frac{S_n\varphi _1(x)}{n},\ldots ,\frac{S_n\varphi _d(x)}{n}\biggr )- F\biggl (\int _X\Phi \,d\mu \biggr )\biggr |< r. \end{aligned}$$

Moreover, $B_n(x,\varepsilon ) \subset B_n(y,2\varepsilon )$ and so it follows from (13) that

$$\begin{aligned} 1-r< \mu (A) \le |D|\max _{x \in D}\mu (B_n(x,\varepsilon )) \le |D|e^{-n(h_\mu (T) - r)}, \end{aligned}$$

where |D| denotes the cardinality of D. Therefore,

$$\begin{aligned} \begin{aligned} \mathcal {W}_n(C)&\ge |D|\exp \bigg [nF\biggl (\int _X\Phi \,d\mu \biggr ) - r\bigg ]\\&\ge (1-r)\exp [n(h_\mu (T) - r)] \exp \bigg [nF\biggl (\int _X\Phi \,d\mu \biggr ) - r\bigg ] \end{aligned} \end{aligned}$$

for any sufficiently large $n \in {\mathbb {N}}$. It follows from (9) that

$$\begin{aligned} P_F(\Phi ) \ge h_\mu (T) + F\biggl (\int _X\Phi \,d\mu \biggr ) - 2r. \end{aligned}$$

Finally, by the arbitrariness of $r > 0$ we obtain

$$\begin{aligned} P_F(\Phi ) \ge h_\mu (T) + F\biggl (\int _X\Phi \,d\mu \biggr ). \end{aligned}$$

(14)

This yields the first property in the lemma.

Now we consider an arbitrary measure $\nu \in \mathcal {M}$. If $(T,\Phi )$ has an abundance of ergodic measures, then for each $h<h_{\nu }(T)$ and $\varepsilon > 0$ there exists an ergodic measure $\mu \in \mathcal {M}$ such that

$$\begin{aligned} \biggl |F\biggl (\int _X\Phi \,d\nu \biggr ) - F\biggl (\int _X\Phi \,d\mu \biggr )\biggr |< \varepsilon \quad \text {and} \quad h_\mu (T) > h \end{aligned}$$

(since F is continuous). By (14) we obtain

$$\begin{aligned} P_F(\Phi ) \ge h_\mu (T) + F\biggl (\int _X\Phi \,d\mu \biggr ) > h + F\biggl (\int _X\Phi \,d\nu \biggr ) - \varepsilon \end{aligned}$$

(15)

and it follows from the arbitrariness of h and $\varepsilon$ that

$$\begin{aligned} P_F(\Phi ) \ge h_\nu (T) + F\biggl (\int _X\Phi \,d\nu \biggr ). \end{aligned}$$

This yields the second property in the lemma. $\square$

Now we obtain the reverse inequality, without requiring that there are an abundance of ergodic measures.

Lemma 2

Let $T:X \rightarrow X$ be a continuous map on a compact metric space and let $\Phi$ be a family of continuous functions. Then

$$\begin{aligned} P_F(\Phi ) \le \sup _{\mu \in \mathcal {M}}\biggl \{h_\mu (T) + F\biggl (\int _X\Phi \,d\mu \biggr )\biggr \}. \end{aligned}$$

Proof of the lemma

Given $p < P_{F}(\Phi )$, take $\varepsilon > 0$ such that

$$\begin{aligned} \limsup _{n \rightarrow \infty }\frac{1}{n}\log \inf _{C}\mathcal {W}_n(C) > p \end{aligned}$$

with the infimum taken over all $(n,\varepsilon )$-covers C. Since each maximal $(n,\varepsilon )$-separated set $E_n$ is an $(n,\varepsilon )$-cover, we have

$$\begin{aligned} \limsup _{n \rightarrow \infty }\frac{1}{n}\log \mathcal {W}_n(E_n) > p \end{aligned}$$

and given $r > 0$, there exists a diverging subsequence $(n_k)_{k \in {\mathbb {N}}}$ such that

$$\begin{aligned} \mathcal {W}_{n_k}(E_{n_{k}}) \ge \exp [n_k (p - r)]\quad \text {for} \ k\in {\mathbb {N}}. \end{aligned}$$

(16)

We cover the compact set $\Phi (X)$ by balls $B(z_i, r_i)$ for $i=1,\ldots , L$ such that $|F(z)-F(z_i)|<r$ for all $z\in B(z_i, r_i)$ and $i=1,\ldots , L$. Now let

$$\begin{aligned} \Lambda ^{i}_{k} = \biggl \{x \in E_{n_k}: \biggl (\frac{S_{n_k}\varphi _1(x)}{n_k},\ldots , \frac{S_{n_k}\varphi _{d}(x)}{n_k}\biggr ) \in B(z_i,r_i)\biggr \}. \end{aligned}$$

Note that

$$\begin{aligned} \mathcal {W}_{n_k}(E_{n_k}) \le \sum _{i=1}^{L}\mathcal {W}_{n_k}(\Lambda ^{i}_{k}) \le L\mathcal {W}_{n_k}(\Lambda ^{i}_{k}) \quad \hbox { for some}\ i \in \{1,\ldots ,L\} \end{aligned}$$

and so it follows from (16) that

$$\begin{aligned} \begin{aligned} \exp [n_k(p - r)]&\le \mathcal {W}_{n_k}(E_{n_k})\\&\le L\mathcal {W}_{n_k}(\Lambda ^i_k) \le L|\Lambda ^i_k|\exp [n_k (F(z_i) + r)]. \end{aligned} \end{aligned}$$

This implies that

$$\begin{aligned} |\Lambda ^i_k| \ge \exp [n_k(p - F(z_i) -3r)] \end{aligned}$$

(17)

for any sufficiently large k. Proceeding as in the proof of the variational principle in [19], we also consider the measures

$$\begin{aligned} \mu ^i_k= \frac{1}{|\Lambda ^i_k|}\sum _{x \in \Lambda ^i_k}\delta _{x} \quad \text {and} \quad \nu ^i_k= \frac{1}{n_k}\sum _{j=0}^{n_k - 1}\mu ^i_k T^{-j}. \end{aligned}$$

Without loss of generality, one can assume that $\nu ^i_k$ converges to a T-invariant measure $\mu ^i$ in the $\hbox {weak}^*$ topology satisfying

$$\begin{aligned} h_{\mu ^i}(T) \ge \limsup _{n_k \rightarrow \infty }\frac{1}{n_k}\log |\Lambda ^i_k|. \end{aligned}$$

(18)

By the definition of $\nu ^i_k$ we have

$$\begin{aligned} \begin{aligned} \int _X\Phi \,d\mu ^i&= \lim _{k \rightarrow \infty }\int _X\Phi \,d\nu ^i_k \\&=\lim _{k \rightarrow \infty }\biggl (\int _X\frac{S_{n_k}\varphi _1}{n_k} \,d\mu ^i_k,\ldots ,\int _X\frac{S_{n_k} \varphi _{d}}{n_k} \,d\mu ^i_k\biggr ) \in \overline{B(z_i,r_i)}. \end{aligned} \end{aligned}$$

Hence, by (17) and (18) we obtain

$$\begin{aligned} \begin{aligned} h_{\mu ^i}(T) + F\biggl (\int _X\Phi \,d\mu ^i \biggr )&\ge p - F(z_i) -3r + F(z_i) - r\\&= p- 4r. \end{aligned} \end{aligned}$$

The desired result follows from the arbitrariness of r and p. $\square$

Lemmas 1 and 2 establish the statement in the theorem.$\square$

For a general continuous map T, we obtain a variational principle for an arbitrary convex function F.

Theorem 4

Let $T:X \rightarrow X$ be a continuous map on a compact metric space and let $\Phi =\{\varphi _1,\ldots , \varphi _d\}$ be a family of continuous functions. If $F:{\mathbb {R}}^d \rightarrow {\mathbb {R}}$ is a convex continuous function, then identity (11) holds.

Proof

It follows from the first property in Lemma 1 that

$$\begin{aligned} P_F(\Phi ) \ge h_\mu (T) + F\biggl (\int _X\Phi \,d\mu \biggr ) \end{aligned}$$

for every ergodic measure $\mu \in \mathcal {M}$. Now let $\nu \in \mathcal {M}$ be an arbitrary measure and consider its ergodic decomposition with respect to T. It is described by a probability measure $\tau$ on $\mathcal {M}$ that is concentrated on the subset of ergodic measures $\mathcal {M}_{\mathrm{erg}}$. We recall that for every bounded measurable function $\psi :X \rightarrow {\mathbb {R}}$ we have

$$\begin{aligned} \int _X \psi \,d\nu = \int _\mathcal {M}\biggl (\int _X \psi \,d\mu \biggr ) \,d\tau (\mu ). \end{aligned}$$

For a convex function F one can use Jensen’s inequality to obtain

$$\begin{aligned} \begin{aligned} F\biggl (\int _X\Phi \,d\nu \biggr )&= F\biggl (\int _\mathcal {M}\biggl (\int _X\varphi _1 \,d\mu \biggr ) d\tau (\mu ),\ldots ,\int _\mathcal {M}\biggl ( \int _X\varphi _d \,d\mu \biggr )d\tau (\mu )\biggr )\\&\le \int _\mathcal {M}F\biggl (\int _X\Phi \,d\mu \biggr )d\tau (\mu ). \end{aligned} \end{aligned}$$

Moreover, we also have

$$\begin{aligned} h_{\nu }(T) = \int _\mathcal {M}h_{\mu }(T)\, d\tau (\mu ) \end{aligned}$$

(see for example Theorem 9.6.2 in [27]). Hence,

$$\begin{aligned} h_{\nu }(T) + F\biggl (\int _X\Phi \,d\nu \biggr ) \le \int _\mathcal {M}\biggl [h_\mu (T) + F\biggl (\int _X\Phi \,d\mu \biggr )\biggr ]d\tau (\mu ) \le P_F(\Phi ). \end{aligned}$$

(19)

The desired result follows now readily from Lemma 2. $\square$

We also obtain a variational principle over the ergodic measures.

Corollary 5

Let $T:X \rightarrow X$ be a continuous map on a compact metric space, let $\Phi =\{\varphi _1,\ldots , \varphi _d\}$ be a family of continuous functions, and let $F:{\mathbb {R}}^d \rightarrow {\mathbb {R}}$ be a continuous function. If the pair $(T,\Phi )$ has an abundance of ergodic measures or F is convex, then

$$\begin{aligned} P_F(\Phi ) = \sup _{\mu \in \mathcal {M}_{\mathrm{erg}}}\biggl \{h_\mu (T) + F\biggl (\int _X\Phi \,d\mu \biggr ) \biggr \}. \end{aligned}$$

(20)

Proof

Since $\mathcal {M}_{\mathrm{erg}}\subset \mathcal {M}$, we have

$$\begin{aligned} \sup _{\mu \in \mathcal {M}} \biggl \{h_\mu (T) + F\biggl (\int _X\Phi \,d\mu \biggr )\biggr \} \ge \sup _{\mu \in \mathcal {M}_{\mathrm{erg}}}\biggl \{h_\mu (T) + F\biggl (\int _X\Phi \,d\mu \biggr )\biggr \}. \end{aligned}$$

Now we assume that the pair $(T,\Phi )$ has an abundance of ergodic measures and we establish the reverse inequality. It follows from (15) that for each $\nu \in \mathcal {M}$, $h<h_\nu (T)$ and $\varepsilon >0$, there exists an ergodic measure $\mu \in \mathcal {M}$ such that

$$\begin{aligned} h_\mu (T) + F\biggl (\int _X\Phi \, d\mu \biggr ) > h + F\biggl (\int _X\Phi \, d\nu \biggr )-\varepsilon . \end{aligned}$$

Since h and $\varepsilon$ are arbitrary, this readily implies that

$$\begin{aligned} \sup _{\mu \in \mathcal {M}_{\mathrm{erg}}}\biggl \{h_{\mu }(T) + F\biggl (\int _X\Phi \,d\mu \biggr )\biggr \}\ge \sup _{\nu \in \mathcal {M}}\biggl \{h_\nu (T) + F\biggl (\int _X\Phi \,d\nu \biggr )\biggr \}. \end{aligned}$$

Finally, it follows from Theorem 3 that identity (20) holds.

Now assume that F is convex. It follows from (19) that

$$\begin{aligned} h_{\nu }(T) + F\biggl (\int _X\Phi \,d\nu \biggr ) \le \sup _{\mu \in \mathcal {M}_{\mathrm{erg}}}\biggl \{h_\mu (T) + F\biggl (\int _X\Phi \,d\mu \biggr )\biggr \} \end{aligned}$$

for each $\nu \in \mathcal {M}$. Therefore,

$$\begin{aligned} \sup _{\nu \in \mathcal {M}}\biggl \{h_{\nu }(T) + F\biggl (\int _X\Phi \,d\nu \biggr )\biggr \} \le \sup _{\mu \in \mathcal {M}_{\mathrm{erg}}}\biggl \{h_\mu (T) + F\biggl (\int _X\Phi \,d\mu \biggr )\biggr \} \end{aligned}$$

and applying Theorem 4 we also obtain identity (20). $\square$

Remark

Without the assumptions of abundance of ergodic measures or convexity of the function F we are only able to show that

$$\begin{aligned} \sup _{\mu \in \mathcal {M}}\biggl \{h_{\mu }(T) + F\biggl (\int _X\Phi d\mu \biggr )\biggr \} \ge P_{F}(\Phi ) \ge \sup _{\mu \in \mathcal {M}_{erg}}\biggl \{h_{\mu }(T) + F\biggl (\int _X\Phi d\mu \biggr )\biggr \}. \end{aligned}$$

In fact, if we drop both assumptions, then the variational principle may fail (see Example 2.5 in [7]).

4 Equilibrium Measures: Existence and Characterization

In this section we consider the problem of characterizing the equilibrium measures of the nonlinear topological pressure.

4.1 Existence of Equilibrium Measures

In view of Theorem 3, we say that $\mu \in \mathcal {M}$ is an equilibrium measure for $(F,\Phi )$ with respect to T if

$$\begin{aligned} P_F(\varphi ) = h_\mu (T) + F\biggl (\int _X\Phi \,d\mu \biggr ). \end{aligned}$$

We first formulate a result on the existence of equilibrium measures.

Theorem 6

Let $T:X \rightarrow X$ be a continuous map on a compact metric space such that the map $\mu \mapsto h_\mu (T)$ is upper semicontinuous, let $\Phi =\{\varphi _1,\ldots , \varphi _d\}$ be a family of continuous functions, and let $F:{\mathbb {R}}^d \rightarrow {\mathbb {R}}$ be a continuous function. If the pair $(T,\Phi )$ has an abundance of ergodic measures or F is convex, then there exists at least one equilibrium measure for $(F,\Phi )$.

Proof

Since the map $\mu \mapsto h_\mu (T)$ is upper semicontinuous, F is continuous and the map $\mu \mapsto \int _X\psi \,d\mu$ is continuous for each continuous function $\psi :X \rightarrow {\mathbb {R}}$, we conclude that $\mu \mapsto h_\mu (T) + F(\int _X\Phi \,d\mu )$ is upper semicontinuous. Together with the compactness of $\mathcal {M}$, this guarantees that there exists a measure $\mu _{\Phi } \in \mathcal {M}$ such that

$$\begin{aligned} \sup _{\mu \in \mathcal {M}} \biggl \{h_\mu (T) + F\biggl (\int _X\Phi \,d\mu \biggr )\biggr \} = h_{\mu _{\Phi }}(T) + F\biggl (\int _X\Phi \,d\mu _{\Phi }\biggr ). \end{aligned}$$

Hence, it follows from the variational principles in Theorems 3 and 4 that $\mu _{\Phi }$ is an equilibrium measure for $(F,\Phi )$. $\square$

In some cases one can pass to the one-dimensional setting of the nonlinear thermodynamic formalism.

Example 1

Consider the function $F:{\mathbb {R}}^d \rightarrow {\mathbb {R}}$ defined by

$$\begin{aligned} F(z_1,\ldots ,z_d) = f(\alpha _1z_1 + \cdots + \alpha _dz_d), \end{aligned}$$

where $f:{\mathbb {R}}\rightarrow {\mathbb {R}}$ is a continuous function and $\alpha _j \in {\mathbb {R}}$ for $j =1,\ldots ,d$. Then

$$\begin{aligned} F\biggl (\int _X\Phi \,d\mu \biggr ) = F\biggl (\int _X\varphi _1 \,d\mu ,\ldots ,\int _X\varphi _d \,d\mu \biggr ) = f\biggl (\int _X\varphi \,d\mu \biggr ) \end{aligned}$$

for every $\mu \in \mathcal {M}$, where

$$\begin{aligned} \varphi =\alpha _1\varphi _1 + \cdots + \alpha _d\varphi _d. \end{aligned}$$

Moreover, $P_F(\Phi ) = P_f(\varphi )$ and this implies that $(F,\Phi )$ and $(f,\varphi )$ have the same equilibrium measures. In other words, for a function F as above the study of equilibrium measures can be reduced to the case when $d=1$.

Of course, in general the continuous function F can be much more complicated. For instance, as mentioned in the introduction, the Curie–Weiss–Potts model involves the study of the topological pressure for the function

$$\begin{aligned} F(z_1,\ldots ,z_d) = \frac{\beta }{2}(z_1^2 + \cdots + z_d^2)^{1/2}, \end{aligned}$$

where $\beta \ge 0$ is a physical parameter.

For the following example, we recall the notion of cohomology. We say that a function $\varphi :X \rightarrow {\mathbb {R}}$ is cohomologous to a function $\psi :X \rightarrow {\mathbb {R}}$ (with respect to T) if there exists a measurable bounded function $q:X \rightarrow {\mathbb {R}}$ such that

$$\begin{aligned} \varphi (x) = \psi (x) + q(T(x)) - q(x)\quad \text {for} \ x \in X. \end{aligned}$$

Example 2

(Reduction of dimension via cohomology) Let T be a continuous map on a compact metric space and let $\Phi =\{\varphi _1,\ldots , \varphi _d\}$ be a family of continuous functions such that the pair $(T,\Phi )$ has an abundance of ergodic measures. Let $F:{\mathbb {R}}^d \rightarrow {\mathbb {R}}$ be a continuous function and assume that $\varphi _1$ is cohomologous to $\varphi _d$. This implies that $\int _X\varphi _1 \,d\mu = \int _X\varphi _{d} \,d\mu$ for every $\mu \in \mathcal {M}$. Therefore,

$$\begin{aligned} \begin{aligned} F\biggl (\int _X\Phi \,d\mu \biggr )&= F\biggl (\int _X\varphi _1 \,d\mu , \int _X\varphi _2 \,d\mu ,\ldots ,\int _X\varphi _1 \,d\mu \biggr ) \\&= G\biggl (\int _X\varphi _1 \,d\mu ,\ldots ,\int _X\varphi _{d-1} \,d\mu \biggr ) \end{aligned} \end{aligned}$$

for every $\mu \in \mathcal {M}$, where

$$\begin{aligned} G(z_1,\ldots ,z_{d-1}) = F(z_1, z_2,\ldots ,z_{d-1},z_1) \end{aligned}$$

for each $(z_1,\ldots ,z_d) \in {\mathbb {R}}^d$. The cohomology assumption also implies that

$$\begin{aligned} \Vert S_n\varphi _1-S_n\varphi _{d}\Vert _{\infty }/n \rightarrow 0\quad \text {when} \ n \rightarrow \infty . \end{aligned}$$

Together with the continuity of F, this implies that $P_{F}(\Phi ) = P_{G}(\Psi )$, where $\Psi = \{\varphi _1,\ldots ,\varphi _{d-1}\}$. Hence, the pairs $(F,\Phi )$ and $(G,\Psi )$ have the same equilibrium measures. More generally, in order to further reduce the dimension of the problem, one could consider additional cohomology relations between any two functions in $\Phi$. For instance, if $\varphi _1$ is cohomologous to all functions $\varphi _j \in \Phi$, then the problem reduces to the one-dimensional case.

Example 3

(Reduction to the classical case via cohomology) Let T be a continuous map on a compact metric space and let $\Phi = \{\varphi _1, \varphi _2\}$ be a pair of continuous functions such that $(T,\Phi )$ has an abundance of ergodic measures. Moreover, assume that $\varphi _1$ is cohomologous to $\varphi _2$ and consider the function $F:{\mathbb {R}}^2 \rightarrow {\mathbb {R}}$ given by $F(z_1,z_2) = (z_1^3+ z_2^3)^{1/3}$. This implies that $\int _X\varphi _1 \,d\mu = \int _X\varphi _2 \,d\mu$ for every $\mu \in \mathcal {M}$ and so

$$\begin{aligned} \begin{aligned} h_\mu (T) + F\biggl (\int _X\Phi \,d\mu \biggr )&= h_\mu (T) + F\biggl (\int _X\varphi _1 \,d\mu ,\int _X\varphi _1 \,d\mu \biggr ) \\&= h_\mu (T) + \int _X2^{1/3}\varphi _1\, d\mu \end{aligned} \end{aligned}$$

for every $\mu \in \mathcal {M}$. Letting $\psi = 2^{1/3}\varphi _1$, it follows from the definitions that $P_F(\Phi ) = P(\psi )$, where P denotes the classical topological pressure. Hence, $\nu$ is an equilibrium measure for $(F,\Phi )$ if and only if $\nu$ is an equilibrium measure for $\psi$.

Recall that a continuous function $\varphi :X \rightarrow {\mathbb {R}}$ is said to have the Bowen property if there exist $K > 0$ and $\varepsilon > 0$ such that whenever

$$\begin{aligned} d(T^k(x),T^k(y)) < \varepsilon \quad \text {for} \ k = 0,1,\ldots ,n-1 \end{aligned}$$

we have

$$\begin{aligned} |S_n\varphi (x) - S_n\varphi (y)| \le K. \end{aligned}$$

If $T:X \rightarrow X$ is an expansive map with specification and $\varphi _1$ (or $\varphi _2$) is a continuous function with the Bowen property, then there exists a unique equilibrium measure $\mu _\psi$ for $\psi$ (see [5]). Therefore, $\mu _\psi$ is also the unique equilibrium measure for $(F,\Phi )$.

We observe that this example can be easily generalized to the case when

$$\begin{aligned} F(z_1,\ldots ,z_d)^n = H_{n}(z_1,\ldots ,z_d), \end{aligned}$$

where $H_n$ is a homogeneous polynomial of degree n, assuming additional cohomology relations between some pairs of functions in $\Phi$.

4.2 Characterization of Equilibrium Measures

Now we consider the problem of characterizing the equilibrium measures. Given a pair $(T,\Phi )$, we consider the set

$$\begin{aligned} L(\Phi )=\biggl \{\int _X\Phi \,d\mu :\mu \in \mathcal {M}\biggr \}. \end{aligned}$$

Since the map $\mu \mapsto \int _X\psi \,d\mu$ is continuous for each continuous function $\psi :X\rightarrow {\mathbb {R}}$ and $\mathcal {M}$ is compact and connected, the set $L(\Phi )$ is a compact and connected subset of ${\mathbb {R}}^d$. For each $z \in {\mathbb {R}}^d$, we also consider the level sets

$$\begin{aligned} \mathcal {M}(z)= \biggl \{\mu \in \mathcal {M}: \int _X\Phi \,d\mu = z\biggr \} \end{aligned}$$

and

$$\begin{aligned} C_z(\Phi )= \biggl \{x \in X: \lim _{n \rightarrow \infty }\frac{S_n\Phi (x)}{n} = z \biggr \}. \end{aligned}$$

(21)

Following closely [7], we say that the pair $(T,\Phi )$ is $C^r$ regular (for some $2 \le r \le \omega$, where $\omega$ refers to the analytic case) if the following holds:

• (1) each function in ${{\,\mathrm{span}\,}}\{\varphi _1,\ldots ,\varphi _d,1 \}$ has a unique equilibrium measure for the classical topological pressure and ${{\,\mathrm{int}\,}}L(\Phi ) \ne \emptyset$;

• (2) for each $z \in {{\,\mathrm{int}\,}}L(\Phi )$ the map $q \mapsto P(\langle q, \Phi - z\rangle )$, where P is the classical topological pressure and $\langle \cdot , \cdot \rangle$ is the usual inner product, takes only finite values, is of class $C^r$, is strictly convex, and its second derivative is a positive definite bilinear form for each $q \in {\mathbb {R}}^d$;

• (3) the entropy map $\mu \mapsto h_\mu (T)$ is upper semicontinuous and bounded.

Examples of $C^r$ regular pairs $(T,\Phi )$ include topologically mixing subshifts of finite type, $C^{1+\varepsilon }$ expanding maps, and $C^{1+\varepsilon }$ diffeomorphisms with a locally maximal hyperbolic set, with $\Phi$ composed of Hölder continuous functions. Finally, we say that the family of functions $\Phi = \{\varphi _1,\ldots ,\varphi _d\}$ is cohomologous to a constant $c = (c_1,\ldots ,c_d)$ if $\varphi _{i}$ is cohomologous to $c_i$ for $i=1,\ldots ,d$. Then $L(\Phi ) = \{c\}$ and so ${{\,\mathrm{int}\,}}L(\Phi ) = \emptyset$.

The following theorem is our main result. Given a function $F:{\mathbb {R}}^d \rightarrow {\mathbb {R}}$, we consider the set

$$\begin{aligned} K(F,\Phi )= \biggl \{\int _X\Phi \,d\mu : \mu \text { is an equilibrium measure for } (F,\Phi )\biggr \} \subset L(\Phi ). \end{aligned}$$

We also consider the function $h:L(\Phi ) \rightarrow {\mathbb {R}}$ defined by

$$\begin{aligned} h(z)= \sup \bigl \{h_\mu (T): \mu \in \mathcal {M}(z)\bigr \}. \end{aligned}$$

(22)

Theorem 7

Let $T:X \rightarrow X$ be a continuous map on a compact metric space and let $\Phi =\{\varphi _1,\ldots , \varphi _d\}$ be a family of continuous functions such that the pair $(T,\Phi )$ is $C^1$ regular. For each continuous function $F:{\mathbb {R}}^d \rightarrow {\mathbb {R}}$, the following properties hold:

• (1) $K(F,\Phi )$ is a nonempty compact set;

• (2) $K(F,\Phi )$ is the set of maximizers of the function $z \mapsto h(z) + F(z)$;

• (3) if $K(F,\Phi ) \subset {{\,\mathrm{int}\,}}L(\Phi )$, then the equilibrium measures for $(F,\Phi )$ are the elements of $\{\nu _{z}: z \in K(F,\Phi )\}$, where each $\nu _z\in \mathcal {M}$ satisfies:

  • $\nu _z$ is ergodic;

  • $\nu _z$ is the unique invariant measure in $\mathcal {M}(z)$ supported on the level set $C_z(\Phi )$ such that $h_{\nu _z}(T) = h(z)$;

  • $\nu _z$ is the unique equilibrium measure for a function

    $$\begin{aligned} \psi _z= \langle q(z), \Phi - z\rangle \end{aligned}$$

    (23)

    in ${{\,\mathrm{span}\,}}\{\varphi _1,\ldots ,\varphi _d,1\}$, for some $q(z) \in {\mathbb {R}}^{d}$.

Proof

We divide the proof into steps.

Lemma 3

$K(F,\Phi )$ is a nonempty compact subset of $L(\Phi )$.

Proof of the lemma

Let $(z_n)_{n\in {\mathbb {N}}}$ be a sequence in $K(F,\Phi )$ converging to a point $z \in L(\Phi )$. For each $n \in {\mathbb {N}}$ there exists an equilibrium measure $\mu _n \in \mathcal {M}$ for $(F,\Phi )$ such that $z_n= \int _X\Phi \,d\mu _n$. Passing eventually to a subsequence, we may assume that there exists $\mu \in \mathcal {M}$ such that $\mu _n \rightarrow \mu$ when $n\rightarrow \infty$ in the $\hbox {weak}^*$ topology. Since the map $\mu \mapsto h_\mu (T)$ is upper semicontinuous, we obtain

$$\begin{aligned} P_F(\Phi ) = \limsup _{n \rightarrow \infty }\bigg [h_{\mu _n}(T) + F\biggl (\int _X\Phi \,d\mu _n\biggr )\bigg ] \le h_\mu (T) + F\biggl (\int _X\Phi \,d\mu \biggr ), \end{aligned}$$

which implies that $\mu$ is an equilibrium measure for $(F,\Phi )$. Since $z = \int _X\Phi \,d\mu$, we conclude that $z \in K(F,\Phi )$. Hence, $K(F,\Phi )$ is closed. Moreover, since

$$\begin{aligned} K(F,\Phi ) \subset [-\Vert \varphi _1\Vert _{\infty }, \Vert \varphi _1\Vert _{\infty }] \times \cdots \times [-\Vert \varphi _d\Vert _{\infty }, \Vert \varphi _d\Vert _{\infty }], \end{aligned}$$

the set $K(F,\Phi )$ is also bounded. By Theorem 6 it is nonempty. $\square$

Lemma 4

For each $z \in {{\,\mathrm{int}\,}}L(\Phi )$ there exists an ergodic measure $\nu _{z} \in \mathcal {M}$ such that $\int _X\Phi \,d\nu _z = z$. In fact, $\nu _z$ is the unique equilibrium measure for the function $\psi _z$ given by (23).

Proof of the lemma

For each $z \in L(\Phi )$ we consider the function

$$\begin{aligned} \Delta _z(q)= P\bigl (\langle q, \Phi -z \rangle - h(T|_{C_z(\Phi )})\bigr ), \end{aligned}$$

where P is the classical topological pressure and $h(T|_{C_z(\Phi )})$ is the topological entropy of T on the set $C_z(\Phi )$ (see Sect. 2.4 for the definition). By Lemmas 1 and 2 in [3] we have

$$\begin{aligned}&\inf _{q \in {\mathbb {R}}^d}\Delta _z(q) \ge 0 \quad \hbox { for}\ z \in L(\Phi ),\\&\inf _{q \in {\mathbb {R}}^d}\Delta _z(q) = 0 \quad \hbox { for}\ z \in {{\,\mathrm{int}\,}}L(\Phi ), \end{aligned}$$

and there exists at least one point $q(z) \in {\mathbb {R}}^{d}$ such that $\Delta _z(q(z)) = 0$. Since the map $q \mapsto \Delta _z(q)$ is of class $C^{1}$ and $\Delta _z$ has a minimum at q(z), we conclude that $\partial _q \Delta _z(q(z)) = 0$. Now let $\nu _z$ be the equilibrium measure of the function $\psi _z$ in (23). One can proceed as in the proof of Theorem 8 in [3] to verify that $\nu _z$ is ergodic with

$$\begin{aligned} \nu _z(C_z(\Phi )) = 1\quad \text {and}\quad \int _X\Phi \,d\nu _z = z. \end{aligned}$$

Moreover, since $\psi _z \in {{\,\mathrm{span}\,}}\{\varphi _1,\ldots ,\varphi _{d},1\}$, it follows from the notion of $C^1$ regular pair that $\nu _z$ is the unique equilibrium measure for $\psi _z$. $\square$

Lemma 5

For each $z \in L(\Phi )$ there exists $\mu \in \mathcal {M}(z)$ with $h(z) = h_\mu (T)$. Moreover, when $z \in {{\,\mathrm{int}\,}}L(\Phi )$ this measure is unique and coincides with $\nu _z$.

Proof of the lemma

Take $z \in L(\Phi )$. By the definition of $L(\Phi )$, there exists $\mu \in \mathcal {M}$ such that $\int _X\Phi \, d\mu = z$, that is, $\mathcal {M}(z) \ne \emptyset$. By the compactness of $\mathcal {M}(z)$ and the upper semicontinuity of the map $\mu \mapsto h_\mu (T)$, there exists $\mu \in \mathcal {M}(z)$ maximizing the metric entropy.

Now take $z \in {{\,\mathrm{int}\,}}L(\Phi )$. By Lemma 4, there exists a measure $\nu _z \in \mathcal {M}$ such that $\int _X\Phi \,d\nu _z = z$, where $\nu _z$ is the unique equilibrium measure for the function $\psi _z$ in (23). Let $\mu \in \mathcal {M}(z)$ be a measure maximizing the metric entropy. Since $\int _X\Phi \,d\mu = \int _X\Phi \,d\nu _z$, it follows readily from (23) that

$$\begin{aligned} \int _X\psi _z \,d\mu = \int _X\psi _z \,d\nu _z. \end{aligned}$$

Therefore,

$$\begin{aligned} h_\mu (T) + \int _X\psi _z \,d\mu \ge h_{\nu _z}(T) + \int _X\psi _z \,d\nu _z = P(\psi _z), \end{aligned}$$

which implies that $\mu$ is also an equilibrium measure for $\psi _z$ (for the classical topological pressure). Since $\psi _z$ has a unique equilibrium measure, we conclude that $\mu = \nu _z$. $\square$

Now consider the function $E:L(\Phi ) \rightarrow {\mathbb {R}}$ defined by $E(z) = h(z) + F(z)$.

Lemma 6

$z \in K(F,\Phi )$ if and only if z maximizes the function E.

Proof of the lemma

First assume that $z \in L(\Phi )$ maximizes the function E. By Lemma 5, there exists $\mu \in \mathcal {M}(z)$ such that $h(z) = h_\mu (T)$ and so

$$\begin{aligned} h_\mu (T) + F\biggl (\int _X\Phi \,d\mu \biggr ) = h(z) + F(z) = \sup _{\mu \in \mathcal {M}}\biggl \{h_\mu (T) + F\biggl (\int _X\Phi \,d\mu \biggr )\biggr \}. \end{aligned}$$

This implies that $\mu$ is an equilibrium measure for $(F,\Phi )$ and so $z \in K(F,\Phi )$.

Now assume that $z \in K(F,\Phi )$. Then there exists an equilibrium measure $\mu$ for $(F,\Phi )$ such that $z = \int _X\Phi \,d\mu$ and so

$$\begin{aligned} \begin{aligned} E(z)&= h(z) + F(z) \\&\ge h_\mu (T) + F\biggl (\int _X\Phi \,d\mu \biggr )\\&= \sup _{\nu \in \mathcal {M}}\biggl \{h_\nu (T) + F\biggl (\int _X\Phi \,d\nu \biggr )\biggr \}\\&= \sup _{w \in L(\Phi )} \sup _{\nu \in \mathcal {M}(w)}\biggl \{h_\nu (T) + F\biggl (\int _X\Phi \,d\nu \biggr )\biggr \}\\&= \sup _{w \in L(\Phi )} \sup _{\nu \in \mathcal {M}(w)}\{h_{\nu }(T) + F(w)\} \\&= \sup _{w \in L(\Phi )}\{h(w) + F(w)\} = \sup _{w \in L(\Phi )}E(w). \end{aligned} \end{aligned}$$

This shows that z maximizes E. $\square$

Lemmas 3 and 6 give items (1) and (2) in the theorem. Now we establish item (3). For each $z \in K(F,\Phi )$ there exists an equilibrium measure $\mu$ for $(F,\Phi )$ such that $\int _X\Phi \,d\mu = z$. When $K(F,\Phi ) \subset {{\,\mathrm{int}\,}}L(\Phi )$, it follows from Lemmas 4 and 5 that $\mu$ is the unique measure with $\int _X\Phi \,d\mu = z$ and that $\mu = \nu _z$, where $\nu _z$ is ergodic and is the unique equilibrium measure for some function $\psi _z$. $\square$

It is shown in [8] that the condition $K(F,\Phi ) \subset {{\,\mathrm{int}\,}}L(\Phi )$ in the last property of Theorem 7 holds for a certain class of pairs $(T,\Phi )$ that they call $C^r$ Legendre (we refer to that paper for the definition).

Remark

In the proof of Lemma 4, for each $z \in {{\,\mathrm{int}\,}}L(\Phi )$ the point q(z) minimizing $\Delta _z(q)$ might not be unique. Therefore, one may have more than one function $\psi _z$ as in (23). On the other hand, Lemma 5 guarantees that all possible functions $\psi _z$ have the same equilibrium measure $\nu _z$.

5 Number of Equilibrium Measures

In this section we consider the problem of how many equilibrium measures a $C^r$ regular system has.

5.1 Preliminary Results

We start with some auxiliary results about the function h in (22). Note that

$$\begin{aligned} h(z)=\sup \left\{ h_\mu (T):\int _X\Phi \,d\mu = z\hbox { with}\ \mu \in \mathcal {M}\right\} . \end{aligned}$$

Proposition 8

For a $C^1$ regular pair $(T, \Phi )$ the function $h:L(\Phi ) \rightarrow {\mathbb {R}}$ is upper semicontinuous, concave and finite.

Proof

Take $z \in L(\Phi )$ and consider a sequence $(z_n)_{n \in {\mathbb {N}}}$ in $L(\Phi )$ such that $z_n \rightarrow z$ when $n\rightarrow \infty$. By Lemma 5, eventually passing to a subsequence one can assume that for each $n \in {\mathbb {N}}$ there exists $\mu _n \in \mathcal {M}(z_n)$ such that $h(z_n) = h_{\mu _n}(T)$ and $\mu _n \rightarrow \mu$ when $n\rightarrow \infty$ for some $\mu \in \mathcal {M}$ in the $\hbox {weak}^*$ topology. We also have

$$\begin{aligned} \int _X\Phi \,d\mu = \lim _{n \rightarrow \infty }\int _X\Phi \,d\mu _{n} = \lim _{n \rightarrow \infty }z_n = z \end{aligned}$$

and so $\mu \in \mathcal {M}(z)$. Moreover, since $\mu \mapsto h_\mu (T)$ is upper semicontinuous, we obtain

$$\begin{aligned} \limsup _{n \rightarrow \infty }h(z_n) = \limsup _{n \rightarrow \infty }h_{\mu _n}(T) \le h_\mu (T) \le h(z) \end{aligned}$$

and so h is upper semicontinuous on $L(\Phi )$.

Now we prove the concavity property. Take $z_1, z_2 \in L(\Phi )$ and $\mu _1 \in \mathcal {M}(z_1)$, $\mu _2 \in \mathcal {M}(z_2)$ such that $h(z_1) = h_{\mu _1}(T)$ and $h(z_2) = h_{\mu _2}(T)$. Since the entropy map is affine, for each $t \in [0,1]$ we have

$$\begin{aligned} \begin{aligned} h(tz_1 + (1-t)z_2) \ge h_{t\mu _1 + (1-t)\mu _2}(T)&= th_{\mu _1}(T) + (1-t)h_{\mu _2}(T)\\&= th(z_1) + (1-t)h(z_2). \end{aligned} \end{aligned}$$

The upper semicontinuity of h on $L(\Phi )$ together with the compactness of $L(\Phi )$ and the fact that $\mathcal {M}(z) \ne \emptyset$ for each $z \in L(\Phi )$, guarantee that h is finite on $L(\Phi )$. $\square$

As pointed out in the recent work [30], in strong contrast to what happens for $d=1$, the function $z \mapsto h(z)$ need not be continuous on $L(\Phi )$.

Proposition 9

If the pair $(T,\Phi )$ is $C^r$ regular, then the function $h|_{{{\,\mathrm{int}\,}}L(\Phi )}$ is $C^{r-1}$. Moreover, if $(T,\Phi )$ is $C^{\omega }$ regular, then $h|_{{{\,\mathrm{int}\,}}L(\Phi )}$ is analytic.

Proof

It follows from Theorem 12 in [3] that if $(T,\Phi )$ is $C^r$ regular, then the map ${{\,\mathrm{int}\,}}L(\Phi ) \ni z \mapsto h(T|_{C_z(\Phi )})$ (the topological entropy of T on $C_z(\Phi )$) is of class $C^{r-1}$, and that if the pair is $C^{\omega }$ regular, then this map is analytic. Since $h(z) = h(T|_{C_z(\Phi )})$ for $z \in {{\,\mathrm{int}\,}}L(\Phi )$, we obtain the desired statement. $\square$

For $d=1$, Corollary 1.11 in [7] says that if the pair $(T,\Phi )$ is $C^\omega$ and F is analytic on ${{\,\mathrm{int}\,}}L(\Phi )$, then the set $K(F,\Phi )$ is finite. In particular, there exist finitely many equilibrium measures.

5.2 Equilibrium Measures I

For $d=1$, it was shown in [7] that no point on $\partial L(\Phi )$ maximizes the function $E = h + F$. By Lemma 6, this implies that $K(F,\Phi ) \subset {{\,\mathrm{int}\,}}L(\Phi )$. It is also shown that $h''(z) < 0$ for every $z \in {{\,\mathrm{int}\,}}L(\Phi )$ and so $h:L(\Phi ) \rightarrow {\mathbb {R}}$ is a strictly concave function. Note that for $d = 1$ we have $L(\varphi ) = [A, B]$, where $A = \inf _{\mu \in \mathcal {M}}\int \varphi \,d\mu$ and $B = \sup _{\mu \in \mathcal {M}}\int \varphi \,d\mu$.

The next result is a criterion for uniqueness of equilibrium measures.

Theorem 10

Let $(T,\varphi )$ be a $C^r$ regular pair and let $F:{\mathbb {R}}\rightarrow {\mathbb {R}}$ be a $C^r$ function that is concave on [A, B]. Then there exists a unique equilibrium measure for $(F,\varphi )$. Moreover, the equilibrium measure is ergodic.

Proof

Since F is concave and h is strictly concave, the function $E = h + F$ is strictly concave. This implies that E has at most one maximizer in (A, B). Since there is no maximizer of E on $\partial L(\varphi ) = \{A,B\}$ and $K(F,\varphi ) \ne \emptyset$, we conclude that there exists a unique point $z^* \in (A,B)$ maximizing E. Hence, it follows from Lemma 6 that $K(F,\varphi ) = \{z^*\}$. By Theorem 4.3 in [7] together with Lemma 4, we conclude that there exists a unique equilibrium measure for $(F,\varphi )$ and that this measure is ergodic. $\square$

The following example illustrates various possibilities.

Example 4

Let $\Sigma = \{-1,1\}$ and let $T:\Sigma ^{{\mathbb {Z}}} \rightarrow \Sigma ^{{\mathbb {Z}}}$ be the two-sided shift. We consider the function $\varphi :\Sigma \rightarrow {\mathbb {R}}$ defined by $\varphi (\cdots \omega _{-1} \omega _0 \omega _1 \cdots ) = \omega _0$. Then $L(\varphi ) = [-1,1]$ and the entropy function $h:L(\varphi ) \rightarrow {\mathbb {R}}$ is given by

$$\begin{aligned} h(z) = -\frac{1-z}{2}\log \left( \frac{1-z}{2}\right) -\frac{1+z}{2}\log \left( \frac{1+z}{2}\right) . \end{aligned}$$

(24)

For the function $F:L(\Phi ) \rightarrow {\mathbb {R}}$ defined by $F(z) = \alpha /(z^2 - 2)$, where $\alpha \in {\mathbb {R}}$, we have

$$\begin{aligned} F''(z) = 2\alpha (3z^2 + 2)/(z^2 - 2)^3. \end{aligned}$$

Notice that for $\alpha > 0$ we have $F''< 0$ on ${{\,\mathrm{int}\,}}L(\varphi )$. Since $F \equiv 0$ for $\alpha = 0$, the function F is concave on ${{\,\mathrm{int}\,}}L(\varphi )$ whenever $\alpha \ge 0$. Hence, by Theorem 10 there exists a unique equilibrium measure $\nu _{z^*}$ for $(F, \varphi )$, where $z^* = 0$ (see Fig. 1). For $\alpha < 0$, the number of equilibrium measures may vary and is the number of absolute maximizers of E on $(-1,1)$. For instance, for $\alpha = -1$ there is one equilibrium measure, while for $\alpha = -2.3$ there are two equilibrium measures (see also Fig. 1).

Theorem 10 also shows that in order to have finitely many equilibrium measures it is not necessary that the pair $(T, \varphi )$ is $C^\omega$ and that the function F is analytic. We give an example in the nonanalytic $C^{\infty }$ case.

Fig. 1

The number of equilibrium measures depends on the parameter $\alpha$

Fig. 2

Nonanalytic $C^{\infty }$ case

Example 5

Consider the pair $(T,\varphi )$ in Example 4 and let $F:{\mathbb {R}}\rightarrow {\mathbb {R}}$ be the function given by

$$\begin{aligned} F(z) = {\left\{ \begin{array}{ll} 3\exp (-1/z) &{} \text {if } z > 0, \\ 0 &{} \text {if } z \le 0. \end{array}\right. } \end{aligned}$$

One can show that F is $C^\infty$ but not analytic. For $-1 \le z \le 0$, we have

$$\begin{aligned} E = h + F = h + 0 = h. \end{aligned}$$

It follows from (24) that E has a local maximum $y_1 = 1$ at $z_1^{*} = 0$. For $0 < z \le 1$, one can verify that E has a local maximum $y_2 \approx 1.33$ at $z_2^{*} \approx 0.75$. Since $y_1 < y_2$, the function E has a unique global maximum at $z_2^{*} \in (0, 1) \subset {{\,\mathrm{int}\,}}L(\varphi )$ (see Fig. 2). By Theorem 7, we conclude that $\nu _{z_2^*}$ is the unique equilibrium measure for $(F,\varphi )$.

5.3 Equilibrium Measures II

As in the one-dimensional case, for $d > 1$ no point in $\partial {L(\Phi )}$ can maximize the function E, that is, $K(F,\Phi ) \subset {{\,\mathrm{int}\,}}L(\Phi )$ (see the Claim in the proof of Theorem 4.15 in [8]). This is possible because $C^{r}$ regular pairs are $C^{r}$ Legendre (see Proposition 4.10 in [8]).

The following statement is a version of the uniqueness result in Theorem 10 for $d>1$.

Theorem 11

Let $(T,\Phi )$ be a $C^r$ regular pair and let $F:{\mathbb {R}}^d \rightarrow {\mathbb {R}}$ be a $C^r$ function that is strictly concave on $L(\Phi )$. Then there exists a unique equilibrium measure for $(F,\Phi )$. Moreover, the equilibrium measure is ergodic.

Proof

By Proposition 8, the map $z \mapsto h(z)$ is upper semicontinuous on $L(\Phi )$. Since F is $C^r$ on $L(\Phi )$, we conclude that $z \mapsto E(z)$ is upper semicontinuous on $L(\Phi )$. Together with the compactness of $L(\Phi )$, this guarantees the existence of at least one point in $L(\Phi )$ maximizing the function E. On the other hand, by Propositions 8 and 9 and the strict concavity of F, the function E is strictly concave on $L(\Phi )$ and $C^{r-1}$ on ${{\,\mathrm{int}\,}}L(\Phi )$. The concavity property of E implies that there exists at most one maximizer in $L(\Phi )$. Since there are no maximizers of E in $\partial L(\Phi )$, the unique point $z^*$ maximizing E must be in ${{\,\mathrm{int}\,}}L(\Phi )$. It follows now from Theorem 7 that $K(F,\Phi ) = \{z^*\}$, that is, $\nu _{z^*}$ is the unique equilibrium measure for $(F,\Phi )$. Moreover, by Lemma 4, $\nu _{z^*}$ is an ergodic measure. $\square$

In Example 4, we have $h|_{\partial L(\varphi )} \equiv 0$. It turns out that this behavior at the boundary of $L(\varphi )$ is typical for some $C^r$ regular systems, even for $d>1$. Let $\mathcal {H}_{\theta }$ be the space of Hölder continuous functions with Hölder exponent $\theta > 0$. The following result is a particular case of Theorem 14 in [3].

Theorem 12

Let T be a subshift of finite type, a $C^{1+\varepsilon }$ diffeomorphism with a hyperbolic set, or a $C^{1+\varepsilon }$ map with a repeller, that is assumed to be topologically mixing. Then there exists a residual set ${\mathcal {O}} \subset (\mathcal {H}_{\theta })^{d}$ such that for each $\Phi \in {\mathcal {O}}$ we have

$$\begin{aligned} h|_{\partial L(\Phi )} \equiv 0\quad \text {and} \quad L(\Phi ) = \overline{{{\,\mathrm{int}\,}}L(\Phi )}. \end{aligned}$$

(25)

We also note that in Example 4 with $\alpha > 0$, the function F satisfies

$$\begin{aligned} F|_{{{\,\mathrm{int}\,}}L(\varphi )} > \max _{z \in \partial L(\varphi )}F(z), \end{aligned}$$

(26)

where $F_{{{\,\mathrm{int}\,}}L(\varphi )}$ is the restriction to ${{\,\mathrm{int}\,}}L(\varphi )$. In fact, condition (26) together with the continuity of F implies that F must actually be constant on $\partial L(\Phi )$, as it happens in Example 4. This scenario is a more general situation in which $E = h + F$ attains its maximum on ${{\,\mathrm{int}\,}}L(\Phi )$:

$$\begin{aligned} \max _{z \in {{\,\mathrm{int}\,}}L(\Phi )}E(z) > \max _{z \in \partial L(\Phi )}E(z). \end{aligned}$$

(27)

Note that this condition may depend not only on F, but also on the family of functions $\Phi$.

A similar idea works for typical $C^r$ regular systems in the sense that they belong to the residual set ${\mathcal {O}}$ in Theorem 12. Let $(T, \Phi )$ be a $C^r$ regular pair satisfying (25). In particular, ${{\,\mathrm{int}\,}}L(\Phi ) \ne \emptyset$. Now let $F:{\mathbb {R}}^d \rightarrow {\mathbb {R}}$ be a function satisfying (26) with $\varphi$ replaced by $\Phi$. Since $h \ge 0$, we have

$$\begin{aligned} \begin{aligned} \max _{z \in \partial L(\Phi )}E(z)&\le \max _{z \in \partial L(\Phi )}h(z) + \max _{z \in \partial L(\Phi )}F(z)\\&\le h|_{{{\,\mathrm{int}\,}}L(\Phi )} + \max _{z \in \partial L(\Phi )}F(z) \\&< h|_{{{\,\mathrm{int}\,}}L(\Phi )} + F|_{{{\,\mathrm{int}\,}}L(\Phi )} = E_{{{\,\mathrm{int}\,}}L(\Phi )}, \end{aligned} \end{aligned}$$

which implies that property (27) holds. Therefore, $K(F, \Phi ) \subset {{\,\mathrm{int}\,}}L(\Phi )$ and so one can apply item (3) of Theorem 7.

It was shown recently in [8] that condition (27) is satisfied for $C^r$ Legendre pairs. This implies that we always have $K(F, \Phi ) \subset {{\,\mathrm{int}\,}}L(\Phi )$ in our setup.

For $d=1$ and $C^r$ regular systems, the function h in (22) is strictly concave. The next example (which should be compared with the Curie–Weiss–Potts model for 3 colors) illustrates that this may still happen for $d>1$, but unfortunately we are not able to describe for which $C^r$ regular pairs the function h is strictly concave.

Example 6

Let $T:X \rightarrow X$ be the two-sided shift with $X = \{1,2,3\}^{\mathbb {Z}}$ and let $\varphi _1 = \chi _{C_1}$ and $\varphi _2 = \chi _{C_3}$, where $C_i$ is the set of all sequences

$$\begin{aligned} (\cdots \omega _{-1}\omega _0\omega _1 \cdots ) \in X \end{aligned}$$

with $\omega _0 = i$. Since $\int _X \varphi _1 d\mu = \mu (C_1)$ and $\int _X \varphi _2 d\mu = \mu (C_3)$ for each $\mu \in \mathcal {M}$, we have

$$\begin{aligned} L(\Phi ) = \bigl \{(\mu (C_1),\mu (C_3)): \mu \in \mathcal {M}\bigr \}. \end{aligned}$$

By Theorem 8 in [3], we obtain

$$\begin{aligned} \begin{aligned} h(z_1, z_2)&= \max _{\mu \in \mathcal {M}}\bigl \{h_\mu (T): (\mu (C_1), \mu (C_2)) = (z_1,z_2)\bigr \}\\&= -z_1\log z_1 - z_2 \log z_2 - z_3\log z_3. \end{aligned} \end{aligned}$$

On the other hand, since $\mu (C_1) + \mu (C_2) + \mu (C_3) = 1$ for each $\mu \in \mathcal {M}$, we have

$$\begin{aligned} L(\Phi )=\bigl \{(z_1,z_2) \in [0,1] \times [0,1]: z_1+z_2 \le 1\bigr \} \end{aligned}$$

and

$$\begin{aligned} h(z_1,z_2) = -z_1\log z_1 - z_2 \log z_2 - (1-z_1-z_2)\log (1-z_1-z_2). \end{aligned}$$

Note that ${{\,\mathrm{int}\,}}L(\Phi ) \ne \emptyset$ and that $\partial L(\Phi )$ is the set

$$\begin{aligned} \bigl (({\mathbb {R}}\times \{0\}) \cup (\{0\}\times {\mathbb {R}}) \cup \{(z_1,z_2): z_1+ z_2 = 1\}\bigr ) \cap ([0,1] \times [0,1]). \end{aligned}$$

For $(z_1,z_2) = (1/2,0)$, (0, 1/2), $(1/2,1/2) \in \partial L(\Phi )$ we have $h(z_1,z_2) = 1 > 0$ and so the system is not typical. On the other hand, one can easily verify that the map $(z_1,z_2) \mapsto h(z_1,z_2)$ is still strictly concave on $L(\Phi )$.

Now consider the function $F(z_1, z_2) = \beta (z_1^2+z_2^2)/2$ with $\beta \in {\mathbb {R}}$. One can verify that the determinant of the Hessian matrix of $E = h + F$ is given by

$$\begin{aligned} \det H_{E}(z_1,z_2) = \beta ^2 + \beta \frac{z_1(1-z_1) + z_2(1-z_2)}{z_1z_2(1-z_1-z_2)} + \frac{1}{z_1z_2(1-z_1-z_2)}. \end{aligned}$$

Since $\det H_{E}(z_1,z_2) > 0$ for $(z_1,z_2) \in {{\,\mathrm{int}\,}}L(\Phi )$ and $\beta \ge 0$, every critical point of E is nondegenerate for all $\beta \ge 0$. Hence, for each $\beta \ge 0$, the function E has at most finitely many critical points. In addition, it was shown in [8] that condition (27) always holds. So E attains its maximal value only at critical points. It follows from Theorem 7 that the pair $(F,\Phi )$ has finitely many equilibrium measures.

On the other hand, for $\beta < 0$ the function F is strictly concave and one can use Theorem 11 to conclude that $(F,\Phi )$ has a unique equilibrium measure.

Remark

In Example 6 the parameter $\beta$ is related to the absolute temperature and the model has physical meaning only when $\beta \ge 0$. However, the general concave case (with $\beta < 0$) might be useful for possible applications in other contexts.

5.4 Coincidence of Equilibrium Measures

The following result gives a sufficient condition so that two systems share equilibrium measures. We say that $\Phi _1 = \{\varphi _{1,1},\ldots ,\varphi _{1,d}\}$ is cohomologous to $\Phi _2 = \{\varphi _{2,1},\ldots ,\varphi _{2,d}\}$ if $\varphi _{1,i}$ is cohomologous to $\varphi _{2,i}$ for $i =1,\ldots ,d$. Then

$$\begin{aligned} \int _X\Phi _1 \,d\mu = \int _X\Phi _2 \,d\mu \quad \text {for each }\mu \in \mathcal {M}, \end{aligned}$$

which readily implies that $L(\Phi _1) = L(\Phi _2)$.

Proposition 13

Let $(T, \Phi _1)$ and $(T,\Phi _2)$ be $C^r$ regular pairs such that $\Phi _1$ is cohomologous to $\Phi _2$ and let $F_1:L(\Phi _1) \rightarrow {\mathbb {R}}$ and $F_2:L(\Phi _2) \rightarrow {\mathbb {R}}$ be continuous functions. If a point $z \in {{\,\mathrm{int}\,}}L(\Phi _1) \cap {{\,\mathrm{int}\,}}L(\Phi _2)$ is simultaneously a maximizer for the functions $E_1= h_1 + F_1$ and $E_2= h_2 + F_2$, then $\nu _{z}$ is an equilibrium measure for $(F_1, \Phi _1)$ and $(F_2,\Phi _2)$.

Proof

Since $\Phi _1$ is cohomologous to $\Phi _2$, we have $L:= L(\Phi _1) = L(\Phi _2)$. Now take $z \in {{\,\mathrm{int}\,}}L$ and consider the functions

$$\begin{aligned} \Delta _1(q)= P(\langle q, \Phi _1 - z\rangle ) - h_1(z)\quad \text {and}\quad \Delta _2(q)= P(\langle q, \Phi _2-z\rangle ) - h_2(z), \end{aligned}$$

where P denotes the classical topological pressure and where each $h_i$ is the corresponding entropy function (see (22)). By the cohomology assumption, we have

$$\begin{aligned} \lim _{n \rightarrow \infty }\frac{\Vert S_n\varphi _{1,i} - S_n\varphi _{2,i}\Vert _{\infty }}{n} = 0 \quad \hbox { for}\ i =1,\ldots ,d \end{aligned}$$

and so $C_{z}(\Phi _1) = C_{z}(\Phi _2)$ for all $z \in {\mathbb {R}}^d$ (see (21)). In particular, this implies that $h := h_1 = h_2$. Therefore,

$$\begin{aligned}{}[\langle q, \Phi _1-z\rangle - h_1(z)]-[\langle q, \Phi _2-z\rangle - h_2(z)] = \langle q, \Phi _1 - \Phi _2\rangle \end{aligned}$$

for $q \in {\mathbb {R}}^d$. Again since $\Phi _1$ is cohomologous to $\Phi _2$, we conclude that

$$\begin{aligned} \Delta _1(q) = \Delta _2(q)\quad \text {for }q \in {\mathbb {R}}^d. \end{aligned}$$

(28)

On the other hand, by the proof of Theorem 8 in [3] the function $q\mapsto \Delta _1(q)$ attains its minimum at a point $q_1(z)$ and $\nu _{1,z}$ is the unique equilibrium measure for the function $\langle q_1(z), \Phi _1 - z\rangle - h(z)$. Similarly, $q \mapsto \Delta _2(q)$ attains its minimum at a point $q_2(z)$ and $\nu _{2,z}$ is the unique equilibrium measure for the function $\langle q_2(z), \Phi _2 - z\rangle - h(z)$. By (28), one can take $q_1(z) = q_2(z)$ and so $\nu _z:= \nu _{1,z} = \nu _{2,z}$. The desired result follows now from Theorem 7. $\square$

A direct consequence of Proposition 13 is that if $\Phi _1$ is cohomologous to $\Phi _2$ and the functions $E_1$ and $E_2$ attain maximal values at the same points, then $(F_1, \Phi _1)$ and $(F_2, \Phi _2)$ have the same equilibrium measures (in particular, this happens when $F_1 = F_2$). For the converse to hold we need stronger conditions so that the coincidence of two equilibrium measures yields a cohomology relation.

Theorem 14

Let X be a topologically mixing locally maximal hyperbolic set for a diffeomorphism T and let $\Phi _1$ and $\Phi _2$ be families of Hölder continuous functions. Moreover, let $F_1$ and $F_2$ be continuous functions. If $(F_1,\Phi _1)$ and $(F_2,\Phi _2)$ have the same equilibrium measures, then for each $z_1$ and $z_2$ maximizing $E_1$ and $E_2$, respectively, there exist $q_1, q_2\in {\mathbb {R}}^d$ such that $\langle q_1, \Phi _1 - z_1\rangle$ is cohomologous to $\langle q_2, \Phi _2-z_2\rangle$.

Proof

By Theorem 7, each equilibrium measure for $(F_i,\Phi _i)$ is a measure $\nu _{z_i}$ with $z_i \in K(F_i,\Phi _i)$ that is the unique equilibrium measure for

$$\begin{aligned} \psi _i = \langle q_i(z_i), \Phi _i - z_i\rangle - h_i(z_i), \end{aligned}$$

where $q_i(z_i)$ is a minimizer of the function

$$\begin{aligned} \Delta _i(q) = P(\langle q, \Phi _i - z_i\rangle ) - h_i(z_i). \end{aligned}$$

Since by hypotheses $\nu _{z_1} = \nu _{z_2}$, the function $\psi _1-\psi _2$ is cohomologous to $P(\psi _1) - P(\psi _2) \in {\mathbb {R}}$. But since

$$\begin{aligned} \Delta _1(q_1(z_1)) = \Delta _2(q_2(z_2)) = 0 \end{aligned}$$

(see Lemma 2 in [3]), we have $P(\psi _1) =P(\psi _2)$. So there exists a continuous function $S= S(z_1,z_2): X \rightarrow {\mathbb {R}}$ such that $\psi _1- \psi _2 = S\circ T - S$, that is,

$$\begin{aligned} \begin{aligned} S \circ T - S&= \langle q_1(z_1), \Phi _1 - z_1\rangle - \langle q_2(z_2), \Phi _2 - z_2\rangle - h_1(z_1) + h_2(z_2). \end{aligned} \end{aligned}$$

Again since $\nu _{z_1} = \nu _{z_2}$, by Lemma 5 we have

$$\begin{aligned} h_1(z_1) = h_{\nu _{z_1}}(T) = h_{\nu _{z_2}}(T) = h_2(z_2). \end{aligned}$$

Hence, for each $z_1$ and $z_2$ maximizing $E_1$ and $E_2$, respectively, there exist points $q_1=q_1(z_1), q_2=q_2(z_2) \in {\mathbb {R}}^d$ as in the statement of the theorem. $\square$