2.1 Transitive subshifts of finite type

For a homeomorphism $f:K\to K$ of a compact metric space $(K,d)$ , we denote by $\mathcal {M}(f,K)$ the space of f-invariant probability measures. Additionally, let $h_{\mathrm {top}}(f,\Lambda )$ denote the topological entropy of an f-invariant compact set $\Lambda $ and $h_{\mu }(f)$ denote the metric entropy of an f-invariant measure. In [Reference Dong, Hou and Tian15], a result on the ‘multi-horseshoe’ dense property of homeomorphisms was obtained.

Let k be a positive integer. Consider the two-side full symbolic space

$$ \begin{align*} \Sigma=\prod_{-\infty}^{\infty} \{0, 1, \ldots, k-1\}, \end{align*} $$

and the shift homeomorphism $\sigma : \Sigma \to \Sigma $ defined by

$$ \begin{align*} (\sigma(w))_{n}=w_{n+1}, \end{align*} $$

where $w =(w_{n})_{-\infty }^{\infty }.$ A metric on $\Sigma $ is defined by $d(x, y)=2^{-m}$ if m is the largest positive integer with $x_{n}=y_{n}$ for any $|n|<m$ , and $d(x, y)=1$ if $x_{0} \neq y_{0}.$ If $\Delta $ is a closed subset of $\Sigma $ with $\sigma (\Delta )=\Delta ,$ then $\sigma |_{\Delta }: \Lambda \to \Delta $ is called a subshift. We usually write this as $\sigma : \Delta \to \Delta .$ A subshift $\sigma : \Delta \rightarrow \Delta $ is said to be of finite type if there exists some positive integer N and a collection of blocks of length $N+1$ with the property that $x=(x_{n})_{-\infty }^{\infty } \in \Delta $ if and only if each block $(x_{i}, \ldots , x_{i+N})$ in x of length $N+1$ is one of the prescribed blocks.

Recall from [Reference Walters, Markley, Martin and Perrizo38] a subshift satisfies the shadowing property if and only if it is a subshift of finite type. As a subsystem of two-side full shift, it is expansive. So we have the following corollary.

2.2 Suspension flows over transitive subshifts of finite type

Let $f\colon K\to K$ be a homeomorphism on a compact metric space $(K,d)$ and consider a continuous roof function $\rho \colon K\to (0,+\infty )$ . We define the suspension space to be

$$ \begin{align*}K_{\rho}=\{(x,s)\in K\times [0,+\infty) \colon 0\leq s\leq \rho(x)\}/\sim,\end{align*} $$

where the equivalence relation $\sim $ identifies $(x,\rho (x))$ with $(f(x),0)$ for all $x\in K$ . Denote $\pi $ the quotient map from $K\times [0,+\infty )$ to $K_{\rho }$ . We define the suspension flow over $f\colon K\to K$ with roof function $\rho $ by

$$ \begin{align*} f_t(x,s)=\pi(x,s+t). \end{align*} $$

For any function $g\colon K_{\rho }\to \mathbb {R}$ , we associate the function $\varphi _g\colon K\to \mathbb {R}$ by

$$ \begin{align*} \displaystyle\varphi_g(x)=\int_0^{\rho(x)}g(x,t)\,{d}t. \end{align*} $$

Since the roof function $\rho $ is continuous, $\varphi _g$ is continuous as long as g is. Moreover, to each invariant probability measure $\mu $ , we associate the measure $\mu _{\rho }$ given by

$$ \begin{align*} \displaystyle\int_{K_{\rho}}g\,{d}\mu_{\rho}=\frac{\int_K\varphi_g\, {d}\mu}{\int_K\rho\,{d}\mu} \quad \text{for all } g\in C(K_{\rho},\mathbb{R}). \end{align*} $$

Observe that not only the measure $\mu _{\rho }$ is $\mathfrak {F}$ -invariant (that is, $\mu _{\rho }(f_t^{-1}A)=\mu _{\rho }(A)$ for all $t\geq 0$ and measurable sets A), but also using that $\rho $ is bounded away from zero, the map

$$ \begin{align*} \mathcal{R}\colon\mathcal{M}(f,K)\to \mathcal{M}(\mathfrak{F},K_{\rho}) \quad\text{given by} \quad \mu\mapsto\mu_{\rho} \end{align*} $$

is a bijection. Abramov’s theorem [Reference Abramov1, Reference Parry and Pollicott31] states that $h_{\mu _{\rho }}(\mathfrak {F})=h_{\mu }(f)/\int \rho \,{d }\mu $ and hence, the topological entropy $h_{\mathrm {top}}(\mathfrak {F})$ of the flow satisfies

$$ \begin{align*} h_{\mathrm{top}}(\mathfrak{F})=\sup\{h_{\mu_{\rho}}(\mathfrak{F})\colon\mu_{\rho}\in \mathcal{M}(\mathfrak{F},K_{\rho})\}=\sup\bigg\{\frac{h_{\mu}(f)}{\int\rho\,{d}\mu}\colon\mu\in \mathcal{M}(f,K)\bigg\}. \end{align*} $$

Throughout, we will use the notation $\Phi =(\phi _t)_{t}$ for a flow on a compact metric space and $\mathfrak {F}=(f_t)_{t}$ for a suspension flow.

Consider a suspension flow $f_{t}: \Delta _{\rho } \rightarrow \Delta _{\rho }$ over a transitive subshift of finite type $(\Delta ,\sigma )$ with a continuous roof function $\rho .$ A metric on $\Delta $ is defined by $d(x, y)=2^{-m}$ if m is the largest positive integer with $x_{n}=y_{n}$ for any $|n|<m$ , and $d(x, y)=1$ if $x_{0} \neq y_{0}.$ There is a natural metric $d_{\Delta _{\rho }}$ , known as the Bowen–Walters metric and we have $d_{\Delta _{\rho }}((x,0),(y,0))=d(x,y)$ for any $x,y\in \Delta $ by [Reference Barreira and Saussol6]. Now we state a result on the ‘multi-horseshoe’ dense property of suspension flows.

Proof. Each $\mathfrak {F}$ -invariant probability measure $\mu _{\rho }^i$ is determined by a $\sigma $ -invariant probability measure $\mu _i$ . Take $\tilde {\zeta },{\kern-1pt}\tilde {\eta }{\kern-1pt}>{\kern-1pt}0$ small enough such that if $\mu ,{\kern-1pt}\nu {\kern-1pt}\in{\kern-1pt} \mathcal {M}(\sigma ,{\kern-1pt}\Delta )$ satisfies $d^*(\mu ,{\kern-1pt}\nu ){\kern-1pt}<{\kern-1pt}\tilde {\zeta },$ then one has

(2.1) $$ \begin{align} d^*(\mathcal{R}(\mu),\mathcal{R}(\nu))<\zeta\quad \text{and}\quad\frac{\int \rho\, d\mu}{\int\rho \,d\nu}\bigg(h_{\mathcal{R}(\mu)}(\mathfrak{F})-\frac{2\tilde{\eta}}{\int\rho\,d \mu}\bigg)> h_{\mathcal{R}(\mu)}(\mathfrak{F})-\eta. \end{align} $$

For the $\tilde {F}=\operatorname {\mathrm {cov}}\{\mu _i\}_{i=1}^m\subseteq \mathcal {M}(\sigma ,\Delta )$ , $x\in \Delta $ and $\tilde {\eta }, \tilde {\zeta },\varepsilon>0,$ by Corollary 2.7, there exist compact invariant subsets $\Delta _i\subseteq \Xi \subsetneq \Delta $ such that for each $1\leq i\leq m$ :

1. (a) $(\Delta _i,\sigma |_{\Delta _i})$ and $(\Xi ,\sigma |_{\Xi })$ are transitive subshifts of finite type;

2. (b) $h_{\mathrm {top}}(\sigma , \Delta _i)>h_{\mu _i}(\sigma )-\tilde {\eta }$ and consequently, $h_{\mathrm {top}}(\sigma , \Xi )>\sup \{h_{\kappa }(\sigma ):\kappa \in F\}-\tilde {\eta };$

3. (c) $d_H(\tilde {F}, \mathcal {M}(\sigma , \Xi ))<\tilde {\zeta }$ , $d_H(\mu _i, \mathcal {M}(\sigma , \Delta _i))<\tilde {\zeta }$ ;

4. (d) there is a positive integer L such that for any $z\in \Xi $ , one has $d(\sigma ^{j+mL}(z),x)<\varepsilon $ for some $0\leq j\leq L-1$ and any $m\in \mathbb {Z}$ .

Then by items (a) and (d), we have items (1) and (4). By equation (2.1) and item (c), we have $d_H(\mu _{\rho }^i, \mathcal {M}(\mathfrak {F}, \Delta ^i_{\rho }))<\zeta $ and thus F is contained in the $\zeta $ -neighborhood of $\mathcal {M}(\mathfrak {F}, \Xi _{\rho }).$ Note that for any $\theta _i\in [0, 1]$ with $\sum _{i=1}^m\theta _i=1$ , one has

$$ \begin{align*} \mathcal{R}\bigg(\sum_{i=1}^{m}\theta_i\mu_i\bigg)=\sum_{i=1}^{m}\frac{\theta_i\int \rho\,d\mu_i}{\sum_{j=1}^{m}\theta_j\int\rho\,d\mu_j}\mathcal{R}(\mu_i). \end{align*} $$

So $\mathcal {M}(\mathfrak {F}, \Xi _{\rho })$ is contained in the $\zeta $ -neighborhood of F and thus we have item (3).

By the variational principle of the topological entropy, there is $\nu _i\in \mathcal {M}(\sigma ,\Delta _i)$ such that $h_{\nu _i}(\sigma )>h_{\mathrm {top}}(\sigma , \Delta _i)-\tilde {\eta }>h_{\mu _i}(\sigma )-2\tilde {\eta }.$ Then

$$ \begin{align*} h_{\mathrm{top}}(\Delta^i_{\rho})\geq h_{\mathcal{R}(\nu_i)}(\mathfrak{F})=\frac{h_{\nu_i}(\sigma)}{\int \rho\,d\nu_i}>\frac{\int\rho\,{d}\mu_i}{\int\rho\,{d}\nu_i} \cdot \Big(h_{\mu^i_{\rho}}(\mathfrak{F}) -\frac{2\tilde{\eta}}{\int\rho\,{d}\mu_i}\Big) >h_{\mu^i_{\rho}}(\mathfrak{F}) -\eta \end{align*} $$

by equation (2.1). We obtain item (2).

2.3 Basic sets and proof of Theorem 2.5

Following the classical arguments of Bowen [Reference Bowen7, Reference Bowen8] on Axiom A vector fields, every basic set is semi-conjugate to a suspension flow over a transitive subshift of finite type with a continuous roof function. Now we recall some basic results from [Reference Bowen7, Reference Bowen8]. Readers can also see [Reference Barreira and Saussol6].

Given $X\in \mathscr {X}^1(M)$ and a basic set $\Lambda ,$ by [Reference Bowen8, Theorem 2.5], there is a family of closed sets $R_{1}, \ldots , R_{k}$ and a positive number $\alpha $ so that the following properties hold:

1. (1) $\Lambda =\bigcup _{t \in [- \alpha ,0]} \phi _{t}(\bigcup _{i=1}^{k} R_{i});$

2. (2) there exists a transitive subshift of finite type $(\Delta ,\sigma )$ and a continuous and onto map $\pi :\Delta \to \bigcup _{i=1}^{k} R_{i}$ such that $\pi \circ \sigma =T\circ \pi $ (T is the transfer map whose definition will be recalled later);

3. (3) $\pi $ is one-to-one off $\bigcup _{n\in \mathbb {Z}} \sigma ^n(\pi ^{-1}(\bigcup _{i=1}^{k} \partial R_{i}))$ .

By item (1), we define the transfer function $\tau : \Lambda \rightarrow [0, \infty )$ by

$$ \begin{align*} \tau(x)=\min \bigg\{t>0: \phi_{t}(x) \in \bigcup_{i=1}^{k} R_{i}\bigg\}. \end{align*} $$

There is a positive number $\beta $ such that $\tau (x)\geq \beta $ for any $x\in \Lambda .$ Additionally, there is a metric on $\Delta $ such that $\tau \circ \pi $ is Lipschitz. Let $T: \Lambda \rightarrow \bigcup _{i=1}^{k} R_{i}$ be the transfer map given by $T(x)=\phi _{\tau (x)}(x)$ . The restriction of T to $\bigcup _{i=1}^{k} R_{i}$ is invertible and $\bigcup _{i=1}^{k} R_{i}$ is a T-invariant set. We can obtain a suspension flow $f_t:\Delta _{\rho }\to \Delta _{\rho }$ over the transitive subshift of finite type $(\Delta ,\sigma )$ with Lipschitz roof function $\rho =\tau \circ \pi .$ Then we extend $\pi $ to a finite-to-one surjection $\pi : \Delta _{\rho } \rightarrow \Lambda $ by $\pi (x, s)=(\phi _{s} \circ \pi )(x)$ for every $(x, s) \in \Delta _{\rho }.$ We have

$$ \begin{align*} \pi \circ f_{t}=\phi_{t} \circ \pi \end{align*} $$

and $\pi $ is one-to-one off $\bigcup _{t\in \mathbb {R}} f_t(\pi ^{-1}(\bigcup _{i=1}^{k} \partial R_{i}))$ .

The boundary of every $R_i$ consists of two parts $\partial R_i=\partial ^s R_i\cup \partial ^u R_i.$ Denote

$$ \begin{align*} \Delta^{s} \Lambda=\phi_{[0, \alpha]} \bigg(\bigcup_{i=1}^{k}\partial^s R_i\bigg)\quad\text{and}\quad \Delta^{u} \Lambda=\phi_{[-\alpha,0]} \bigg(\bigcup_{i=1}^{k}\partial^u R_i\bigg). \end{align*} $$

By [Reference Bowen8, Proposition 2.6], one has $\phi _t(\Delta ^{s} \Lambda )\subset \Delta ^{s} \Lambda $ and $\phi _{-t}(\Delta ^{u} \Lambda )\subset \Delta ^{u} \Lambda $ for any $t\geq 0.$ In fact, in the proof of [Reference Bowen8, Proposition 2.6], it is also shown that

(2.2) $$ \begin{align} T\bigg(\bigcup_{i=1}^{k}\partial^s R_i\bigg)\subset \bigcup_{i=1}^{k}\partial^s R_i\quad\text{and}\quad T^{-1}\bigg(\bigcup_{i=1}^{k}\partial^u R_i\bigg)\subset \bigcup_{i=1}^{k}\partial^u R_i. \end{align} $$

Now we give the proof of Theorem 2.5.

Proof of Theorem 2.5

Let $f_t:\Delta _{\rho }\to \Delta _{\rho }$ be the associated suspension flow of the basic set $\Lambda $ . Since $\pi ^{-1}(\bigcup _{i=1}^{k}\partial ^s R_i)$ is a proper closed subset of $\Delta ,$ then there is $\tilde {x}\in \Delta $ and $\tilde {\varepsilon }>0$ such that

$$ \begin{align*} B(\tilde{x},\tilde{\varepsilon})\cap\pi^{-1}\bigg( \bigcup_{i=1}^{k}\partial^s R_i\bigg)=\emptyset, \end{align*} $$

where $B(\tilde {x},\tilde {\varepsilon })=\{y\in \Delta : d(\tilde {x},y)<\tilde {\varepsilon }\}.$

Fix $F=\operatorname {\mathrm {cov}}\{\mu _i\}_{i=1}^m\subseteq \mathcal {M}(\Phi ,\Lambda )$ and $\eta , \zeta>0.$ Then there is $\tilde {\mu }_i\in \mathcal {M}(\mathfrak {F},\Delta _{\rho })$ such that $\mu _i=\tilde {\mu }_i\circ \pi ^{-1}.$ Since $\pi $ is continuous, there is $\tilde {\zeta }>0$ such that $d^{*}(\tilde {\mu }\circ \pi ^{-1},\tilde {\nu }\circ \pi ^{-1})<\zeta $ for any $\tilde {\mu },\tilde {\nu }\in \mathcal {M}(\mathfrak {F},\Delta _{\rho })$ with $d^*(\tilde {\mu },\tilde {\nu })<\tilde {\zeta }.$ For $\tilde {F}=\operatorname {\mathrm {cov}}\{\tilde {\mu }_i\}_{i=1}^m\subseteq \mathcal {M}(\mathfrak {F},\Delta _{\rho })$ , $\tilde {x}$ and $\eta ,\tilde {\zeta },\tilde {\varepsilon }>0,$ by Proposition 2.8, there exist compact $\mathfrak {F}$ -invariant subsets $\Delta ^i_{\rho }\subseteq \Xi _{\rho }\subsetneq \Delta _{\rho }$ such that for each $1\leq i\leq m$ :

1. (a) $f_{t}|_{\Delta ^i_{\rho }}: \Delta ^i_{\rho } \rightarrow \Delta ^i_{\rho }$ and $f_{t}|_{\Xi _{\rho }}: \Xi _{\rho }\rightarrow \Xi _{\rho }$ are suspension flows over transitive subshifts of finite type $(\Delta _i,\sigma |_{\Delta _i})$ and $(\Xi ,\sigma |_{\Xi })$ with the roof function $\rho ;$

2. (b) $h_{\mathrm {top}}(\Delta ^i_{\rho })>h_{\tilde {\mu }_i}(\mathfrak {F})-\eta $ and consequently, $h_{\mathrm {top}}(\Xi _{\rho })>\sup \{h_{\kappa }(\mathfrak {F}):\kappa \in F\}-\eta ;$

3. (c) $d_H(\tilde {F}, \mathcal {M}(\mathfrak {F}, \Xi _{\rho }))<\tilde {\zeta }$ , $d_H(\tilde {\mu }_i, \mathcal {M}(\mathfrak {F}, \Delta ^i_{\rho }))<\tilde {\zeta }$ ;

4. (d) for any $z\in \Xi $ , there is an increasing sequence $\{t_m\}_{m=-\infty }^{+\infty }$ such that $f_{t_m}((z,0))\in \Delta \times \{0\}$ , $d_{\Delta _{\rho }}(f_{t_m}((z,0)),(\tilde {x},0))<\tilde {\varepsilon }$ for any $m\in \mathbb {Z},$ and $\lim \nolimits _{m\to +\infty }t_m=\lim \nolimits _{m\to -\infty }-t_m=+\infty $ .

By Claim 2.9 and item (d), we have

$$ \begin{align*}\Delta_i\cap \bigcup_{n\in\mathbb{Z}} \sigma^n\bigg(\pi^{-1}\bigg(\bigcup_{i=1}^{k} \partial R_{i}\bigg)\bigg)=\emptyset\quad\text{and}\quad\Xi\cap \bigcup_{n\in\mathbb{Z}} \sigma^n\bigg(\pi^{-1}\bigg(\bigcup_{i=1}^{k} \partial R_{i}\bigg)\bigg)=\emptyset. \end{align*} $$

Then

$$ \begin{align*}\Delta^i_{\rho}\cap \bigcup_{t\in\mathbb{R}} f_t\bigg(\pi^{-1}\bigg(\bigcup_{i=1}^{k} \partial R_{i}\bigg)\bigg)=\emptyset\quad\text{and}\quad \Xi_{\rho}\cap \bigcup_{t\in\mathbb{R}} f_t\bigg(\pi^{-1}\bigg(\bigcup_{i=1}^{k} \partial R_{i}\bigg)\bigg)=\emptyset. \end{align*} $$

This implies $\pi $ is one-to-one on $\Delta ^i_{\rho }$ and $\Xi _{\rho }.$ So $\Lambda _i=\pi (\Delta ^i_{\rho })$ and $\Theta =\pi (\Xi _{\rho })$ are the horseshoes we want. $\Box $

3.1 Non-additive thermodynamic formalism for flows

In this subsection, we recall a few basic notions and results on the non-additive thermodynamic formalism for flows. Consider a continuous flow $\Phi =(\phi _t)_{t\in \mathbb {R}}$ on a compact metric space $(M,d).$ Let $\Lambda \subset M$ be a compact $\phi _t$ -invariant set. Given $x\in \Lambda $ and $t,\varepsilon>0$ , we consider the set

$$ \begin{align*} B_{t}(x, \varepsilon)=\{y \in \Lambda: d(\phi_{s}(y), \phi_{s}(x))<\varepsilon \text { for any } s \in[0, t]\}. \end{align*} $$

Moreover, let $a=(a_t)_{t\geq 0}$ be a family of continuous functions $a_t:\Lambda \to \mathbb {R}$ with tempered variation, that is, such that

$$ \begin{align*} \lim _{\varepsilon \rightarrow 0} \limsup_{t \rightarrow \infty} \frac{\gamma_{t}(a, \varepsilon)}{t}=0, \end{align*} $$

where

$$ \begin{align*} \gamma_{t}(a, \varepsilon)=\sup \{|a_{t}(y)-a_{t}(x)|: y \in B_{t}(x, \varepsilon),\ x\in \Lambda \}. \end{align*} $$

Given $\varepsilon>0$ , we say that a set $\Gamma \subset \Lambda \times \mathbb {R}_{0}^{+}$ covers $\Lambda $ if

$$ \begin{align*} \bigcup_{(x, t) \in \Gamma} B_{t}(x, \varepsilon) \supset \Lambda, \end{align*} $$

and we write

$$ \begin{align*} a(x, t, \varepsilon)=\sup \{a_{t}(y): y \in B_{t}(x, \varepsilon)\} \quad \text { for }(x, t) \in \Gamma. \end{align*} $$

For each $\alpha \in \mathbb {R}$ , let

(3.1) $$ \begin{align} M(\Lambda, a, \alpha, \varepsilon)=\lim _{T \rightarrow+\infty} \inf _{\Gamma} \sum_{(x, t) \in \Gamma} \exp (a(x, t, \varepsilon)-\alpha t) \end{align} $$

with the infimum taken over all countable sets $\Gamma \subset \Lambda \times [T,+\infty )$ covering $\Lambda $ . When $\alpha $ goes from $-\infty $ to $+\infty $ , the quantity in equation (3.1) jumps from $+\infty $ to 0 at a unique value and so one can define

$$ \begin{align*} P(a, \Lambda, \varepsilon)=\inf \{\alpha \in \mathbb{R}: M(\Lambda, a, \alpha, \varepsilon)=0\}. \end{align*} $$

Moreover, the limit

$$ \begin{align*} P(a, \Lambda)=\lim _{\varepsilon \rightarrow 0} P(a, \Lambda, \varepsilon) \end{align*} $$

exists and is called the non-additive topological pressure of the family a on the set $\Lambda $ .

We recall that a family of functions $a=(a_t)_{t\geq 0}$ on $\Lambda $ is said to be almost additive with respect to the flow $(\phi _t)_{t\in \mathbb {R}}$ if there is a constant $C>0$ such that

$$ \begin{align*} -C+a_{t}+a_{s} \circ \phi_{t} \leqslant a_{t+s} \leqslant a_{t}+a_{s} \circ \phi_{t}+C \end{align*} $$

for every $t, s\geq 0$ . Let $A(\Phi ,\Lambda )$ be the set of all almost additive families of continuous functions $a=(a_{t})_{t \geqslant 0}$ on $\Lambda $ with tempered variation such that

$$ \begin{align*} \sup _{t \in[0, s]}\|a_{t}\|_{\infty}<\infty \quad \text {for some } s>0. \end{align*} $$

From [Reference Barreira and Holanda5, Proposition 4], the limit $\lim \nolimits _{t \rightarrow \infty } {1}/{t} \int _{\Lambda } a_{t}\,{d} \mu $ exists for any $a=(a_{t})_{t \geq 0}\in A(\Phi ,\Lambda )$ and any $\mu \in \mathcal {M}(\Phi ,\Lambda ),$ and the function

(3.2) $$ \begin{align} \mathcal{M}(\Phi,\Lambda) \ni \mu \mapsto \lim _{t \rightarrow \infty} \frac{1}{t} \int_{\Lambda} a_{t}\,{d} \mu \end{align} $$

is continuous with the weak $^*$ topology. Let $a=(a_{t})_{t \geq 0}\in A(\Phi ,\Lambda ).$ Then by [Reference Barreira and Holanda4, Theorem 2.1 and Lemma 2.4], we have

(3.3) $$ \begin{align} \begin{aligned} P(a,\Lambda)&=\sup_{\mu\in\mathcal{M}(\Phi,\Lambda)}\bigg(h_{\mu}(\Phi)+\lim _{t \rightarrow \infty} \frac{1}{t} \int_{M} a_{t}\,d \mu\bigg)\\ &=\sup_{\mu\in\mathcal{M}_{\mathrm{erg}}(\Phi,\Lambda)}\bigg(h_{\mu}(\Phi)+\lim _{t \rightarrow \infty} \frac{1}{t} \int_{M} a_{t}\,d \mu\bigg). \end{aligned} \end{align} $$

A measure $\mu \in \mathcal {M}(\Phi ,\Lambda )$ is said to be an equilibrium measure for the almost additive family a if

$$ \begin{align*}P(a,\Lambda)=h_{\mu}(\Phi)+\lim\limits_{t \rightarrow \infty} \frac{1}{t} \int_{M} a_{t}\,d \mu.\end{align*} $$

We say that a has bounded variation if for every $\kappa>0$ , there exists $\varepsilon>0$ such that

$$ \begin{align*} |a_{t}(x)-a_{t}(y)|<\kappa \quad \text { whenever } y \in B_{t}(x, \varepsilon). \end{align*} $$

Following the arguments of [Reference Barreira and Holanda4, §3.2], we have the following result on the uniqueness of equilibrium measure for suspension flows.

Proof. Define a sequence of functions $c_n:\Delta \to \mathbb {R}$ by

$$ \begin{align*}c_n(x)=a_{\rho_n(x)}(x),\end{align*} $$

where $\rho _n(x)=\sum _{i=0}^{n-1}\rho (\sigma ^i(x)).$ Following the arguments of [Reference Barreira and Holanda4, §3.2], we have:

1. (1) $c=(c_n)_{n\in \mathbb {N}}$ is an almost additive sequence of continuous functions, that is, there is a constant $\tilde {C}>0$ such that for every $n, m \in \mathbb {N}$ , we have

   $$ \begin{align*} -\tilde{C}+c_{n}+c_{m} \circ \sigma^{n} \leqslant c_{n+m} \leqslant \tilde{C}+c_{n}+c_{m} \circ \sigma^{n}; \end{align*} $$

2. (2) $c=(c_n)_{n\in \mathbb {N}}$ has bounded variation, that is, there exists $\varepsilon>0$ for which $ \sup _{n \in \mathbb {N}} \gamma _{n}(c, \varepsilon )<\infty $ with

   $$ \begin{align*}\gamma_{n}(c, \varepsilon)=\sup \{|c_{n}(x)-c_{n}(y)|: x,y\in\Delta,\ d(\sigma^{k} (x), \sigma^{k} (y))<\varepsilon \text { for } k=0, \ldots, n\};\end{align*} $$

3. (3) for any ergodic measure $\nu \in \mathcal {M}(\sigma ,\Delta )$ and $\mu =\mathcal {R}(\nu )$ (recall §2.2),

   $$ \begin{align*} h_{\mu}(\mathfrak{F})+\lim _{t \rightarrow \infty} \frac{1}{t} \int_{\Delta_{\rho}} a_{t}\,d \mu=(h_{\nu}(\sigma)+\lim _{n \rightarrow \infty} \frac{1}{n} \int_{\Delta} c_{n}\,d \nu) \bigg/ \int_{\Delta} \rho\,d \nu. \end{align*} $$

By item (3), we have $h_{\mu }(\mathfrak {F})+\lim \nolimits _{t \rightarrow \infty } ({1}/{t}) \int _{\Delta _{\rho }} a_{t}\,d \mu =0$ if and only if $h_{\nu }(\sigma )+\lim \nolimits _{n \rightarrow \infty } ({1}/{n})\int _{\Delta } c_{n}\,d \nu =0.$ Since $P(a,\Delta _{\rho })=0$ , then $\mu $ is an equilibrium measure for a if and only if $\nu $ is an equilibrium measure for c by equation (3.3).

It is proved in [Reference Dong, Hou and Tian15, Theorem 5.3] that for every homeomorphism which is transitive, expansive, and has the shadowing property, and for every almost additive sequence of continuous functions with bounded variation, there is a unique equilibrium measure. Recall from [Reference Walters, Markley, Martin and Perrizo38] a subshift satisfies the shadowing property if and only if it is a subshift of finite type. As a subsystem of a two-side full shift, it is expansive. Then there is a unique equilibrium measure $\nu _{c}$ for c, and $\mathcal {R}(\mu _c)$ is the unique equilibrium measure for a.

Proof. For any $t \geq 0,$ define

$$ \begin{align*} b_{t}=a_{t}-P(a) t. \end{align*} $$

Then b is an almost additive family with bounded variation and satisfies $\sup _{t \in [0, s]}\|b_{t}\|_{\infty }<\infty $ and $P(b)=0$ . For each $\mathfrak {F}$ -invariant probability measure $\mu $ on $\Delta _{\rho }$ , we have

$$ \begin{align*} \frac{1}{t} \int_{X} a_{t}\,d \mu=\frac{1}{t} \int_{X} b_{t}\,d \mu-P(a). \end{align*} $$

This implies that a and b have the same equilibrium measures. Then by Lemma 3.1, we complete the proof.

Proof. Define $\tilde {a}_t=a_t\circ \pi $ for any $t\geq 0.$ Since $\pi $ is continuous, $\tilde {a}=(\tilde {a}_{t})_{t \geq 0}$ is an almost additive family of continuous functions on $\Delta _{\rho }$ with bounded variation and $\sup _{t \in [0, s]}\|\tilde {a}_{t}\|_{\infty }<\infty .$ By Corollary 3.2, there is a unique equilibrium measure $\mu _{\tilde {a}}$ for $\tilde {a}$ . Since $\pi $ is finite-to-one, we have $h_{\mu _{\tilde {a}}\circ \pi ^{-1}}(\Phi )=h_{\mu _{\tilde {a}}}(\mathfrak {F})$ and thus

$$ \begin{align*}h_{\mu_{\tilde{a}}\circ\pi^{-1}}(\Phi)+\lim\limits_{t\to\infty}\frac{1}{t}\int a_t\, d\mu_{\tilde{a}}\circ\pi^{-1}=h_{\mu_{\tilde{a}}}(\mathfrak{F})+\lim\limits_{t\to\infty}\frac{1}{t}\int \tilde{a}_t\, d\mu_{\tilde{a}}=P(\tilde{a})\geq P(a).\end{align*} $$

Then by equation (3.3), $\mu _{\tilde {a}}\circ \pi ^{-1}$ is an equilibrium measure for $a.$ To show that the measure is unique, suppose $\nu $ is an equilibrium measure for $a,$ and choose $\mu \in \mathcal {M}(\mathfrak {F},\Delta _{\rho })$ with $\mu \circ \pi ^{-1}=\nu .$ Then

$$ \begin{align*}h_{\mu}(\mathfrak{F})+\lim\limits_{t\to\infty}\frac{1}{t}\int \tilde{a}_t\,d\mu=h_{\nu}(\Phi)+\lim\limits_{t\to\infty}\frac{1}{t}\int a_t\,d\nu= P(a)=P(\tilde{a}).\end{align*} $$

Thus, $\mu =\mu _{\tilde {a}}$ by the uniqueness of $\mu _{\tilde {a}}.$ So $\nu =\mu \circ \pi ^{-1}=\mu _{\tilde {a}}\circ \pi ^{-1}$ is the unique equilibrium measure for a.

Then we have the following theorem, combining §2.3 and Definition 2.2.

3.2 Conditional variational principles

3.2.1 Conditional variational principle of almost additive families

Consider a continuous flow $\Phi =(\phi _t)_{t\in \mathbb {R}}$ on a compact metric space $(M,d).$ Let $\Lambda \subset M$ be a compact $\phi _t$ -invariant set. Let $d \in \mathbb {N}$ and $(A, B) \in A(\Phi ,\Lambda )^{d} \times A(\Phi ,\Lambda )^{d}$ , where

$$ \begin{align*} A=(a^{1}, \ldots, a^{d}) \quad\text {and}\quad B=(b^{1}, \ldots, b^{d}), \end{align*} $$

and $a^{i}=(a_{t}^{i})_{t\in \mathbb {R}}$ and $b^{i}=(b_{t}^{i})_{t\in \mathbb {R}}$ . Assume that

(3.4) $$ \begin{align} \liminf _{t \rightarrow \infty} \frac{b_{t}^{i}(x)}{t}>0 \quad \text {and} \quad b_{t}^{i}(x)>0 \end{align} $$

for every $i=1, \ldots , d, x \in \Lambda $ , and $t \in \mathbb {R}$ . Given $\alpha =(\alpha _1, \ldots , \alpha _{d}) \in \mathbb {R}^{d}$ , we define

$$ \begin{align*} R_{A,B}(\alpha)=\bigcap_{i=1}^{d}\bigg\{x \in \Lambda: \lim _{t \rightarrow \infty} \frac{a_{t}^{i}(x)}{b_{t}^{i}(x)}=\alpha_{i}\bigg\}. \end{align*} $$

We also define the map $\mathcal {P}_{A,B}: \mathcal {M}(\Phi ,\Lambda ) \rightarrow \mathbb {R}$ by

(3.5) $$ \begin{align} \mathcal{P}_{A,B}(\mu) =\lim _{t \rightarrow \infty}\bigg(\frac{\int a_{t}^{1}\,d \mu}{\int b_{t}^{1}\,d \mu}, \ldots, \frac{\int a_{t}^{d}\,d \mu}{\int b_{t}^{d}\,d \mu}\bigg). \end{align} $$

Equation (3.2) ensures that the function $\mathcal {P}_{A,B}$ is continuous. Denote

$$ \begin{align*}L_{A,B}=\{\mathcal{P}_{A,B}(\mu):\mu\in \mathcal{M}(\Phi,\Lambda)\}.\end{align*} $$

Let $E(\Phi ,\Lambda ) \subset A(\Phi ,\Lambda )$ be the set of families with a unique equilibrium measure. Define the sequence of constant functions $u=(u_t)_{t\geq 0}$ with $u_t\equiv t$ for any $t\geq 0.$ In [Reference Barreira and Holanda5], L. Barreira and C. Holanda give the conditional variational principle as the following theorem.

Theorem 3.5. [Reference Barreira and Holanda5, Theorem 9]

Suppose that $\Phi =(\phi _t)_{t\in \mathbb {R}}$ is a continuous flow on a compact metric space $(M,d)$ and $\Lambda \subset M$ is a compact invariant set such that the entropy function $\mathcal {M}(\Phi ,\Lambda ) \ni \mu \mapsto h_{\mu }(\Phi )$ is upper semi-continuous. Let $d \in \mathbb {N}$ and $(A, B) \in A(\Phi ,\Lambda )^{d} \times A(\Phi ,\Lambda )^{d}$ such that B satisfies equation (3.4) and $ \text {span}\{a^1, b^1,\ldots ,a^d,b^d,u\} \subseteq E(\Phi ,\Lambda ). $ If $\alpha \in \mathrm {Int}(L_{A,B})$ , then $R_{A,B}(\alpha )\neq \emptyset $ , and the following properties hold.

1. (1) $h_{\mathrm {top}}(R_{A,B}(\alpha ))$ satisfies the variational principle:

   $$ \begin{align*} h_{\mathrm{top}}(R_{A,B}(\alpha))=\max \{h_{\mu}(\Phi): \mu \in \mathcal{M}(\Phi,\Lambda) \text { and } \mathcal{P}_{A,B}(\mu)=\alpha\}. \end{align*} $$

2. (2) There is an ergodic measure $\mu _{\alpha } {\kern-1pt}\in{\kern-1pt} \mathcal {M}(\Phi ,\Lambda ) $ with $\mathcal {P}_{A,B}(\mu _{\alpha })=\alpha , \mu _{\alpha }(R_{A,B}(\alpha ))=1$ , and

   $$ \begin{align*} h_{\mathrm{top}}(R_{A,B}(\alpha))=h_{\mu_{\alpha}}(\Phi). \end{align*} $$

Remark 3.6. In fact, Barreira and Holanda in [Reference Barreira and Holanda5, Theorem 9] give the proof of Theorem 3.5 under the assumption $d=1.$ However, Barreira and Doutor in [Reference Barreira and Doutor3, Theorem 3] proved the case of discrete time for any $d\geq 1.$ Combining the two proofs, one can obtain Theorem 3.5 for any $d\geq 1.$

3.2.2 Conditional variational principle of asymptotically additive families

Let $\Phi =(\phi _t)_{t \in \mathbb {R}}$ be a continuous flow on a compact metric space $(M,d)$ and $\Lambda \subset M$ be a compact $\phi _t$ -invariant set. A family of functions $a=(a_t)_{t \geq 0}$ is said to be asymptotically additive with respect to $\Phi $ on $\Lambda $ if for each $\varepsilon>0$ , there exists a function $b_{\varepsilon }: \Lambda \rightarrow \mathbb {R}$ such that

$$ \begin{align*} \limsup _{t \rightarrow \infty} \frac{1}{t}\sup_{x\in\Lambda}\bigg|a_t(x)-\int_0^t(b_{\varepsilon} \circ \phi_s(x))\,d s\bigg| \leq \varepsilon. \end{align*} $$

Let $AA(\Phi ,\Lambda )$ be the set of all asymptotically additive families of continuous functions $a=(a_{t})_{t \geqslant 0}$ on $\Lambda $ with tempered variation such that

$$ \begin{align*} \sup _{t \in[0, s]}\|a_{t}\|_{\infty}<\infty \quad \text {for some } s>0. \end{align*} $$

Proceeding as in [Reference Feng and Huang16], one can see that every almost additive family of functions is asymptotically additive.

Now assume that $\Phi $ is expansive on $\Lambda $ . Holanda in [Reference Holanda20, Corollary 6] proved that for any $a=(a_t)_{t \geq 0}\in AA(\Phi ,\Lambda ),$ there exists a continuous function $b: \Lambda \rightarrow \mathbb {R}$ such that

$$ \begin{align*} \lim \limits_{t \rightarrow \infty} \frac{1}{t} \int_{\Lambda} a_{t}\,{d} \mu=\int_{\Lambda} b\,d\mu \end{align*} $$

for any $\mu \in \mathcal {M}(\Phi ,\Lambda ).$ This implies that the function

(3.6) $$ \begin{align} \mathcal{M}(\Phi,\Lambda) \ni \mu \mapsto \lim _{t \rightarrow \infty} \frac{1}{t} \int_{\Lambda} a_{t}\,{d} \mu \end{align} $$

is continuous with the weak $^*$ topology. Let $d \in \mathbb {N}$ and $(A, B) \in AA(\Phi ,\Lambda )^{d} \times AA(\Phi ,\Lambda )^{d}$ , where

$$ \begin{align*} A=(a^{1}, \ldots, a^{d}) \quad\text {and}\quad B=(b^{1}, \ldots, b^{d}), \end{align*} $$

and $a^{i}=(a_{t}^{i})_{t\in \mathbb {R}}$ and $b^{i}=(b_{t}^{i})_{t\in \mathbb {R}}$ . Assume that

(3.7) $$ \begin{align} \begin{aligned} &\lim _{t \rightarrow \infty} \frac{1}{t} \int_{\Lambda} b_t^i\,d \mu \geq 0 \quad \text {for all } \mu \in \mathcal{M}(\Phi,\Lambda) \end{aligned} \end{align} $$

with equality only permitted when

(3.8) $$ \begin{align} \lim _{t \rightarrow \infty} \frac{1}{t} \int_{\Lambda} a_t^i\,d \mu \neq 0 \end{align} $$

for every $i=1, \ldots , d.$ Given $\alpha =(\alpha _1, \ldots , \alpha _{d}) \in \mathbb {R}^{d}$ , we define

$$ \begin{align*} R_{A,B}(\alpha)=\bigcap_{i=1}^{d}\bigg\{x \in \Lambda: \lim _{t \rightarrow \infty} \frac{a_{t}^{i}(x)}{b_{t}^{i}(x)}=\alpha_{i}\bigg\}. \end{align*} $$

We also define the map $\mathcal {P}_{A,B}: \mathcal {M}(\Phi ,\Lambda ) \rightarrow \mathbb {R}$ by

(3.9) $$ \begin{align} \mathcal{P}_{A,B}(\mu) =\lim _{t \rightarrow \infty}\bigg(\frac{\int a_{t}^{1}\,d \mu}{\int b_{t}^{1}\,d \mu}, \ldots, \frac{\int a_{t}^{d}\,d \mu}{\int b_{t}^{d}\,d \mu}\bigg). \end{align} $$

Equation (3.6) ensures that the function $\mathcal {P}_{A,B}$ is continuous. Denote

$$ \begin{align*}L_{A,B}=\{\mathcal{P}_{A,B}(\mu):\mu\in \mathcal{M}(\Phi,\Lambda)\}.\end{align*} $$

Without using the uniqueness of equilibrium assumption, C. Holanda obtains a conditional variational principle for asymptotically additive families of continuous functions.

Theorem 3.7. [Reference Holanda20, Theorem 11]

Given $X\in \mathscr {X}^1(M)$ and an invariant compact set $\Lambda $ , assume that $\Lambda $ is a basic set or a horseshoe of $\phi _{t}.$ Let $d \in \mathbb {N}$ and $(A, B) \in AA(\Phi ,\Lambda )^{d} \times AA(\Phi ,\Lambda )^{d}$ satisfying equation (3.7). If $\alpha \in \mathrm {Int}(L_{A,B})$ , then $R_{A,B}(\alpha )\neq \emptyset $ , and the following properties hold.

1. (1) $h_{\mathrm {top}}(R_{A,B}(\alpha ))$ satisfies the variational principle:

   $$ \begin{align*} h_{\mathrm{top}}(R_{A,B}(\alpha))=\sup \{h_{\mu}(\Phi): \mu \in \mathcal{M}(\Phi,\Lambda) \text { and } \mathcal{P}_{A,B}(\mu)=\alpha\}. \end{align*} $$

2. (2) For any $\varepsilon>0$ , there exists an ergodic measure $\mu _{\alpha } \in \mathcal {M}(\Phi ,\Lambda ) $ such that $\mathcal {P}_{A,B}(\mu _{\alpha })=\alpha , \mu _{\alpha }(R_{A,B}(\alpha ))=1$ and

   $$ \begin{align*} |h_{\mathrm{top}}(R_{A,B}(\alpha))-h_{\mu_{\alpha}}(\Phi)|<\varepsilon. \end{align*} $$

Remark 3.8. Using the work of Climenhaga [Reference Climenhaga11, Theorem 3.3] and Cuneo [Reference Cuneo13, Theorem 1.2], and the conclusion that every Hölder continuous function has a unique equilibrium measure with respect to the map T (recall that T is the transfer map, see §2.3), Holanda in [Reference Holanda20, Theorem 11] gives the proof of Theorem 3.7 under the assumption $d=1$ and $\Lambda $ is a mixing basic set. However, [Reference Climenhaga11, Theorem 3.3] is stated for any $d\geq 1$ and by [Reference Bowen9, Example 2], every Hölder continuous function has a unique equilibrium measure with respect to the map T when $\Lambda $ is just a transitive basic set. So one can obtain Theorem 3.7 for any $d\geq 1$ and any basic set $\Lambda .$ When $\Lambda $ is a horseshoe of $\phi _{t},$ we can verify the results also hold following the argument of [Reference Holanda20, Theorem 11].

3.3 Abstract conditions on which Theorems A and B hold

Let $\Phi =(\phi _t)_{t \in \mathbb {R}}$ be a continuous flow on a compact metric space $(M,d),$ and $\Lambda \subset M$ be a compact $\phi _t$ -invariant set. Assume that $\Phi $ is expansive on $\Lambda .$ Let $d \in \mathbb {N}$ and $(A, B) \in AA(\Phi ,\Lambda )^{d} \times AA(\Phi ,\Lambda )^{d}.$ For any $\alpha \in L_{A,B},$ denote

$$ \begin{align*}M_{A,B}(\alpha)&=\{\mu\in \mathcal{M}(\Phi,\Lambda):\mathcal{P}_{A,B}(\mu)=\alpha\}, \\M_{A,B}^{\mathrm{erg}}(\alpha) &=\{\mu\in \mathcal{M}_{\mathrm{erg}} (\Phi,\Lambda):\mathcal{P}_{A,B}(\mu)=\alpha\}. \end{align*} $$

Then $M_{A,B}(\alpha )$ is closed in $\mathcal {M}(\Phi ,\Lambda )$ since the map $\mathcal {P}_{A,B}$ is continuous.

Let $\chi :\mathcal {M}(\Phi ,\Lambda )\to \mathbb {R}$ be a continuous function. We define the pressure of $\chi $ with respect to $\mu $ by $P(\Phi ,\chi ,\mu )=h_{\mu }(\Phi )+\chi (\mu ).$ Now we give a result in the context of asymptotically additive families.

For any $\alpha \in L_{A,B},$ denote

$$ \begin{align*}H_{A,B}(\Phi,\chi,\alpha)=\sup\{P(\Phi,\chi,\mu):\mu\in M_{A,B}(\alpha)\}.\end{align*} $$

In particular, when $\chi \equiv 0,$ we write

$$ \begin{align*}H_{A,B}(\Phi,\alpha)=H_{A,B}(\Phi,0,\alpha)=\sup\{h_{\mu}(\Phi):\mu\in M_{A,B}(\alpha)\}.\end{align*} $$

We list two conditions for $\chi :$

1. (A.1) for any $\mu _1, \mu _2 \in \mathcal {M}(\Phi ,\Lambda )$ with $P(\Phi ,\chi ,\mu _1) \neq P(\Phi ,\chi ,\mu _2)$ ,

   (3.10) $$ \begin{align} \beta(\theta):=P(\Phi,\chi,\theta \mu_1+(1-\theta) \mu_2)\text{ is strictly monotonic on }[0,1]; \end{align} $$

2. (A.2) for any $\mu _1, \mu _2 \in \mathcal {M}(\Phi ,\Lambda )$ with $P(\Phi ,\chi ,\mu _1) = P(\Phi ,\chi ,\mu _2)$ ,

   (3.11) $$ \begin{align} \beta(\theta):=P(\Phi,\chi,\theta \mu_1+(1-\theta) \mu_2)\text{ is constant on }[0,1]. \end{align} $$

Now we give abstract conditions on which Theorems A and B hold in the context of asymptotically additive families.

3.4 Proof of Theorem 3.9

3.4.1 Some lemmas

We establish several auxiliary results. For any $r\in \mathbb {R},$ denote $r^+=\{s\in \mathbb {R}:s>r\}$ and $r^-=\{s\in \mathbb {R}:s<r\}.$ For any $d \in \mathbb {N}$ , $r=(r_1, \ldots , r_{d})\in \mathbb {R}^d$ , and $\xi =(\xi _1, \ldots , \xi _{d})\in \{+,-\}^d$ , we define

$$ \begin{align*}r^{\xi}=\{s=(s_1, \ldots, s_{d})\in\mathbb{R}^d:s_i\in r_i^{\xi_i} \text{ for }i=1,2,\ldots,d\}.\end{align*} $$

We denote $F^d=\{({p_1}/{q_1}, \ldots , {p_d}/{q_d}):p_i,q_i\in \mathbb {R}\text { and }q_i>0 \text { for any }1\leq i\leq d\}.$ It is easy to check the following lemma.

Lemma 3.13. Let $b_i={p^i}/{q^i}\in F^1$ for $i=1,2.$

1. (1) If $b_1=b_2,$ then $({\theta p^1+(1-\theta )p^2})/({\theta q^1+(1-\theta )q^2})=b_1=b_2$ for any $\theta \in [0,1].$

2. (2) If $b_1\neq b_2,$ then $({\theta p^1+(1-\theta )p^2})/({\theta q^1+(1-\theta )q^2})$ is strictly monotonic on $\theta \in [0,1].$

We can obtain the following result using mathematical induction.

Lemma 3.14. Let $d \in \mathbb {N}$ and $a=({p_1}/{q_1}, \ldots , {p_d}/{q_d}) \in F^{d}.$ If $\{b_{\xi }=({p_1^{\xi }}/{q_1^{\xi }}, \ldots , {p_d^{\xi }}/ {q_d^{\xi }})\}_{\xi \in \{+,-\}^d}\subseteq F^d$ are $2^d$ numbers satisfying $b_{\xi }\in a^{\xi }$ for any $\xi \in \{+,-\}^d,$ then there are $2^d$ numbers $\{\theta _{\xi }\}_{\xi \in \{+,-\}^d}\subseteq [0,1]$ such that $\sum _{\xi \in \{+,-\}^d}\theta _{\xi }=1$ and

$$ \begin{align*} \frac{\sum_{\xi\in\{+,-\}^d}\theta_{\xi} p^{\xi}_i}{\sum_{\xi\in\{+,-\}^d}\theta_{\xi} q^{\xi}_i}=\frac{p_i}{q_i}\quad\text{for any }1\leq i\leq d.\end{align*} $$

From Lemma 3.14, we have the following corollary.

Corollary 3.15. Suppose that $\Phi =(\phi _t)_{t\in \mathbb {R}}$ is a continuous flow on a compact metric space $(M,d),$ and $\Lambda \subset M$ is a compact invariant set such that $\Phi $ is expansive on $\Lambda .$ Let $d \in \mathbb {N}$ and $(A, B) \in AA(\Phi ,\Lambda )^{d} \times AA(\Phi ,\Lambda )^{d}$ , satisfying equation (3.7). Then for any $\alpha \in \mathrm {Int}(L_{A,B}),$ and $2^d$ invariant measures $\{\mu _{\xi }\}_{\xi \in \{+,-\}^d}$ with

$$ \begin{align*}\mathcal{P}_{A,B}(\mu_{\xi})\in \alpha^{\xi} \quad\text{for any }\xi\in \{+,-\}^d,\end{align*} $$

there are $2^d$ numbers $\{\theta _{\xi }\}_{\xi \in \{+,-\}^d}\subseteq [0,1]$ such that

$$ \begin{align*} \sum _{\xi \in \{+,-\}^d}\theta _{\xi }=1 \quad \text{and}\quad \mathcal {P}_{A,B} \left(\sum _{\xi \in \{+,-\}^d}\theta _{\xi }\mu _{\xi }\right) = \alpha . \end{align*} $$

Lemma 3.16. Suppose that $\Phi =(\phi _t)_{t\in \mathbb {R}}$ is a continuous flow on a compact metric space $(M,d),$ and $\Lambda \subset M$ is a compact invariant set such that $\Phi $ is expansive on $\Lambda .$ Let $d \in \mathbb {N}$ and $(A, B) \in AA(\Phi ,\Lambda )^{d} \times AA(\Phi ,\Lambda )^{d}$ satisfying equation (3.7). Then for any $\alpha \in \mathrm {Int}(L_{A,B}),$ any $\mu \in M_{A,B}(\alpha )$ , and any $\eta ,\zeta>0$ , there are $2^d$ invariant measures $\{\mu _{\xi }\}_{\xi \in \{+,-\}^d}$ such that for any $\xi \in \{+,-\}^d$ ,

$$ \begin{align*}\mathcal{P}_{A,B}(\mu_{\xi})\in \alpha^{\xi}, h_{\mu_{\xi}}(\Phi)>h_{\mu}(f)-\eta \quad\text{and}\quad d^*(\mu_{\xi},\mu)<\zeta.\end{align*} $$

Proof. By $\alpha \in \mathrm {Int}(L_{A,B})$ , there is $\nu _{\xi }\in \mathcal {M}(\Phi ,\Lambda )$ such that $\mathcal {P}_{A,B}(\nu _{\xi })\in a^{\xi }$ for any $\xi \in \{+,-\}^d.$ We choose $\tau _{\xi }\in (0,1)$ close to $1$ such that $\mu _{\xi }=\tau _{\xi }\mu +(1-\tau _{\xi })\nu _{\xi }$ satisfies

$$ \begin{align*}h_{\mu_{\xi}}(\Phi)>h_{\mu}(\Phi)-\eta \quad\text{and}\quad d^*(\mu_{\xi},\mu)<\zeta \quad\text{for any }\xi\in \{+,-\}^d.\end{align*} $$

Then we have $\mathcal {P}_{A,B}(\mu _{\xi })\in \alpha ^{\xi }$ by $\tau ^{\xi }>0$ and Lemma 3.13(2).

3.4.2 Proof of Theorem 3.9

Fix $\alpha \in \mathrm {Int}(L_{A,B})$ , $\mu _0\in M_{A,B}(\alpha )$ , and $\eta , \zeta>0.$ Since $\Phi $ is expansive on $\Lambda ,$ the metric entropy $\mathcal {M}(\Phi ,\Lambda ) \ni \mu \mapsto h_{\mu }(\Phi )$ is upper semi-continuous. Hence, there is $0<\zeta '<\zeta $ such that for any $\omega \in \mathcal {M}(\Phi ,\Lambda )$ with $d^*(\mu _0,\omega )<\zeta '$ , we have

(3.12) $$ \begin{align} h_{\omega}(\Phi)<h_{\mu_0}(\Phi)+\frac{3\eta}{8} \quad\text{and}\quad|\chi(\omega)-\chi(\mu_0)|<\frac{\eta}{2}. \end{align} $$

By Lemma 3.16, there are $2^d$ invariant measures $\{\mu _{\xi }\}_{\xi \in \{+,-\}^d}$ such that for any $\xi \in \{+,-\}^d$ ,

(3.13) $$ \begin{align} \mathcal{P}_{A,B}(\mu_{\xi})\in \alpha^{\xi},\quad h_{\mu_{\xi}}(\Phi)>h_{\mu_0}(\Phi)-\frac{\eta}{8} \quad\text{and}\quad d^*(\mu_{\xi},\mu_0)<\frac{\zeta'}{2}. \end{align} $$

Since the map $\mathcal {P}_{A,B}$ is continuous, there is $0<\zeta "<\zeta '$ such that for any $\omega _{\xi } \in \mathcal {M}(\Phi ,\Lambda )$ with $d^*(\omega _{\xi },\mu _{\xi })<\zeta "$ , one has

(3.14) $$ \begin{align} \mathcal{P}_{A,B}(\omega_{\xi})\in \alpha^{\xi}. \end{align} $$

For the $2^d$ invariant measures $\{\mu _{\xi }\}_{\xi \in \{+,-\}^d},$ there are compact invariant subsets $\Lambda _{\xi }\subseteq \Theta \subsetneq \Lambda $ such that for each $\xi \in \{+,-\}^d$ :

1. (1) for any $a\in \mathrm {Int}(\mathcal {P}_{A,B}(\mathcal {M}(\Phi ,\Theta )))$ and any $\varepsilon>0,$ there exists an ergodic measure $\mu _a$ supported on $\Theta $ with $\mathcal {P}_{A,B}(\mu _a)=a$ such that $|h_{\mu _a}(\Phi )-H(\Phi ,a,\Theta )|<\varepsilon ;$

2. (2) $h_{\mathrm {top}}( \Lambda _{\xi })>h_{\mu _{\xi }}(\Phi )-{\eta }/{8};$

3. (3) $d_H(\operatorname {\mathrm {cov}}\{\mu _{\xi }\}_{\xi \in \{+,-\}^d}, \mathcal {M}(\Phi ,\Theta ))<{\zeta "}/{2}$ , $d_H(\mu _{\xi }, \mathcal {M}(\Phi ,\Lambda _{\xi }))<{\zeta "}/{2}.$

By item (2) and the variational principle, there is $\nu _{\xi }\in \mathcal {M}(\Phi ,\Lambda _{\xi })$ such that

$$ \begin{align*}h_{\nu_{\xi}}(\Phi)>h_{\mathrm{top}}( \Lambda_{\xi})-\frac{\eta}{8}>h_{\mu_{\xi}}(\Phi)-\frac{2\eta}{8}>h_{\mu_0}(\Phi)-\frac{3\eta}{8}.\end{align*} $$

Then by item (3) and equation (3.14), we have $\mathcal {P}_{A,B}(\nu _{\xi })\in \alpha ^{\xi }.$ By Corollary 3.15, there are $2^d$ numbers $\{\theta _{\xi }\}_{\xi \in \{+,-\}^d}\subseteq [0,1]$ such that $\sum _{\xi \in \{+,-\}^d}\theta _{\xi }=1$ and $\mathcal {P}_{A,B}(\nu ')= \alpha $ , where $\nu '=\sum _{\xi \in \{+,-\}^d}\theta _{\xi }\nu _{\xi }.$ Then on one hand, we have

(3.15) $$ \begin{align} H(\Phi,\alpha,\Theta) \geq h_{\nu'}(\Phi)\geq \min\{h_{\nu_{\xi}}(\Phi):\xi\in \{+,-\}^d\}> h_{\mu_0}(\Phi)-\frac{3\eta}{8}. \end{align} $$

On the other hand, by item (3), and equations (3.13) and (3.12), we have

(3.16) $$ \begin{align} H(\Phi,\alpha,\Theta)<h_{\mu_0}(\Phi)+\frac{3\eta}{8}. \end{align} $$

Now by item (1), there exists an ergodic measure $\nu $ supported on $\Theta $ with $\mathcal {P}_{A,B}(\nu )=\alpha $ such that $|h_{\nu }(\Phi )-H(\Phi ,\alpha ,\Theta )|<{\eta }/{8}.$ Then $\nu \in M_{A,B}^{\mathrm {erg}}(\alpha )$ , and by equations (3.15) and (3.16), we have $|h_{\nu }(\Phi )-h_{\mu _0}(\Phi )|<{\eta }/{2}.$ By item (3) and equation (3.13), we have $d^*(\nu ,\mu _0)<\zeta '<\zeta .$ Finally, by equation (3.12), we have $|\chi (\nu )-\chi (\mu _0)|<{\eta }/{2}$ and thus $|P(\Phi ,\chi ,\nu )-P(\Phi ,\chi ,\mu _0)|<\eta .$

3.5 Proof of Theorem 3.10

3.5.1 Some lemmas

Lemma 3.17. Suppose that $\Phi =(\phi _t)_{t\in \mathbb {R}}$ is a continuous flow on a compact metric space $(M,d),$ and $\Lambda \subset M$ is a compact invariant set. Let V be a convex subset of $\mathcal {M}(\Phi ,\Lambda ).$ If there is an invariant measure $\mu _V\in V$ with $S_{\mu _V}=\Lambda ,$ then $\{\mu \in V:S_{\mu }=\Lambda \}$ is residual in $V.$

Proof. Since $\{\mu \in \mathcal {M}(\Phi ,\Lambda ):S_{\mu }=\Lambda \}$ is either empty or a dense $G_{\delta }$ subset of $\mathcal {M}(\Phi ,\Lambda )$ from [Reference Denker, Grillenberger and Sigmund14, Proposition 21.11], if there is an invariant measure $\mu _V\in V$ with $S_{\mu _V}=\Lambda ,$ then $\{\mu \in \mathcal {M}(\Phi ,\Lambda ):S_{\mu }=\Lambda \}$ is a dense $G_{\delta }$ subset of $\mathcal {M}(\Phi ,M).$ Thus, $\{\mu \in V:S_{\mu }=\Lambda \}$ is a $G_{\delta }$ subset of $V.$ In addition, for any $\nu \in V$ and $\theta \in (0,1),$ we have $\nu _{\theta }=\theta \nu + (1-\theta )\mu _V\in V$ and $S_{\nu _{\theta }}=\Lambda .$ So $\{\mu \in V:S_{\mu }=\Lambda \}$ is dense in $V,$ and thus is residual in $V.$

Lemma 3.18. Suppose that $\Phi =(\phi _t)_{t\in \mathbb {R}}$ is a continuous flow on a compact metric space $(M,d),$ and $\Lambda \subset M$ is a compact invariant set such that $\Phi $ is expansive on $\Lambda .$ Let $d \in \mathbb {N}$ and $(A, B) \in AA(\Phi ,\Lambda )^{d} \times AA(\Phi ,\Lambda )^{d}$ , satisfying equation (3.7). If $\chi :\mathcal {M}(\Phi ,\Lambda )\to \mathbb {R}$ is a continuous function satisfying equations (3.10) and (3.11), then for any $\alpha \in \mathrm {Int}(L_{A,B})$ and any $\max _{\mu \in M_{A,B}(\alpha )}\chi (\mu )\leq c< H_{A,B}(\Phi ,\chi ,\alpha ),$ the following properties hold:

1. (1) if $\{\mu \in M_{A,B}^{\mathrm {erg}}(\alpha ):P(\Phi ,\chi ,\mu )\geq c\}$ is dense in $\{\mu \in M_{A,B}(\alpha ):P(\Phi ,\chi ,\mu )\geq c\},$ then $\{\mu {\kern-1pt}\in{\kern-1pt} M_{A,B}^{\mathrm {erg}}(\alpha ):P(\Phi ,{\kern-1pt}\chi ,{\kern-1pt}\mu ){\kern-1pt}\geq{\kern-1pt} c\}$ is residual in $\{\mu {\kern-1pt}\in{\kern-1pt} M_{A,B}(\alpha ):P(\Phi ,{\kern-1pt}\chi ,{\kern-1pt} \mu ){\kern-1pt}\geq{\kern-1pt} c\};$

2. (2) if there is an invariant measure $\tilde {\mu }$ with $S_{\tilde {\mu }}{\kern-1pt}={\kern-1pt}\Lambda $ , then $\{\mu {\kern-1pt}\in{\kern-1pt} M_{A,B}(\alpha ):P(\Phi ,{\kern-1pt}\chi ,{\kern-1pt}\mu ){\kern-1pt}\geq{\kern-1pt} c, S_{\mu }=\Lambda \}$ is residual in $\{\mu \in M_{A,B}(\alpha ):P(\Phi ,\chi ,\mu )\geq c\};$

3. (3) if $\{\mu \in \mathcal {M}(\Phi ,\Lambda ):h_{\mu }(\Phi )=0\}$ is dense in $\mathcal {M}(\Phi ,\Lambda ),$ then $\{\mu \in M_{A,B}(\alpha ):P(\Phi ,\chi ,\mu )= c\}$ is residual in $\{\mu \in M_{A,B}(\alpha ):P(\Phi ,\chi ,\mu )\geq c\}.$

Proof. (1) From [Reference Denker, Grillenberger and Sigmund14, Proposition 5.7], $\mathcal {M}_{\mathrm {erg}}(\Phi ,\Lambda )$ is a $G_{\delta }$ subset of $\mathcal {M}(\Phi ,\Lambda ).$ Then $\{\mu \in M_{A,B}^{\mathrm {erg}}(\alpha ):P(\Phi ,\chi ,\mu )\geq c\}$ is a $G_{\delta }$ subset of $\{\mu \in M_{A,B}(\alpha ):P(\Phi ,\chi ,\mu )\geq c\}.$ If $\{\mu \in M_{A,B}^{\mathrm {erg}}(\alpha ):P(\Phi ,\chi ,\mu )\geq c\}$ is dense in $\{\mu \in M_{A,B}(\alpha ):P(\Phi ,\chi ,\mu )\geq c\},$ then $\{\mu \in M_{A,B}^{\mathrm {erg}}y(\alpha ):P(\Phi ,\chi ,\mu )\geq c\}$ is residual in $\{\mu \in M_{A,B}(\alpha ):P(\Phi ,\chi ,\mu )\geq c\}.$

(2) Since $\{\mu \in \mathcal {M}(\Phi ,\Lambda ):S_{\mu }=\Lambda \}$ is either empty or a dense $G_{\delta }$ subset of $\mathcal {M}(\Phi ,\Lambda )$ from [Reference Denker, Grillenberger and Sigmund14, Proposition 21.11], if there is an invariant measure $\tilde {\mu }$ with $S_{\tilde {\mu }}=\Lambda $ , then $\{\mu \in \mathcal {M}(\Phi ,\Lambda ):S_{\mu }=\Lambda \}$ is a dense $G_{\delta }$ subset of $\mathcal {M}(\Phi ,\Lambda ).$

Now we show that $\{\mu \in M_{A,B}(\alpha ):P(\Phi ,\chi ,\mu )\geq c,\ S_{\mu }=\Lambda \}$ is non-empty. By Lemma 3.16, there exist $2^d$ invariant measures $\{\mu _{\xi }\}_{\xi \in \{+,-\}^d}$ such that

$$ \begin{align*}\mathcal{P}_{A,B}(\mu_{\xi})\in \alpha^{\xi} \quad\text{for any }\xi\in \{+,-\}^d.\end{align*} $$

Since $\{\mu \in \mathcal {M}(\Phi ,\Lambda ):S_{\mu }=\Lambda \}$ is dense in $ \mathcal {M}(\Phi ,\Lambda )$ and the map $\mathcal {P}_{A,B}$ is continuous, then there exists $\omega _{\xi }\in \mathcal {M}(\Phi ,\Lambda )$ close to $\mu _{\xi }$ such that

$$ \begin{align*} \mathcal{P}_{A,B}(\omega_{\xi})\in \alpha^{\xi},\quad S_{\omega_{\xi}}=\Lambda\quad\text{for any }\xi\in \{+,-\}^d. \end{align*} $$

By Corollary 3.15, there are $2^d$ numbers $\{\theta _{\xi }\}_{\xi \in \{+,-\}^d}\subseteq [0,1]$ such that $\sum _{\xi \in \{+,-\}^d}\theta _{\xi }=1$ and $\mathcal {P}_{A,B}(\omega )= \alpha $ , where $\omega =\sum _{\xi \in \{+,-\}^d}\theta _{\xi }\mu _{\omega }$ . Then $S_{\omega }=\Lambda .$ Since $c< H_{A,B}(\Phi ,\chi ,\alpha ),$ there is $\nu \in M_{A,B}(\alpha )$ such that $P(\Phi ,\chi ,\nu )>c.$ By equation (3.10), we can choose $\theta \in (0,1)$ close to $1$ such that $\mu '=\theta \nu +(1-\theta )\omega $ satisfies $P(\Phi ,\chi ,\mu ')>c.$ Then $\mu '\in \{\mu \in M_{A,B}(\alpha ):P(\Phi ,\chi ,\mu )\geq c,\ S_{\mu }=\Lambda \}.$ Note that $\{\mu \in M_{A,B}(\alpha ):P(\Phi ,\chi ,\mu )\geq c\}$ is a convex set by equations (3.10), (3.11), and Lemma 3.13. So by Lemma 3.17, we complete the proof of item (2).

(3) Fix $\mu _0\in M_{A,B}(\alpha )$ with $P(\Phi ,\chi ,\mu _0)\geq c$ and $\zeta>0.$ By Lemma 3.16, there exist $2^d$ invariant measures $\{\mu _{\xi }\}_{\xi \in \{+,-\}^d}$ such that

(3.17) $$ \begin{align} \mathcal{P}_{A,B}(\mu_{\xi})\in \alpha^{\xi} \quad\text{and}\quad d^*(\mu_{\xi},\mu_0)<\frac{\zeta}{2} \quad\text{for any }\xi\in \{+,-\}^d. \end{align} $$

Since $\{\mu \in \mathcal {M}(\Phi ,\Lambda ):h_{\mu }(\Phi )=0\}$ is dense in $\mathcal {M}(\Phi ,\Lambda )$ and the function $\mathcal {P}_{A,B}$ is continuous, then there exists $\nu _{\xi }\in \mathcal {M}(\Phi ,\Lambda )$ close to $\mu _{\xi }$ such that

(3.18) $$ \begin{align} \mathcal{P}_{A,B}(\nu_{\xi})\in \alpha^{\xi},\quad h_{\nu_{\xi}}(\Phi)=0 \quad\text{and}\quad d^*(\nu_{\xi},\mu_{\xi})<\frac{\zeta}{2} \quad\text{for each } \xi\in \{+,-\}^d. \end{align} $$

By Corollary 3.15, there are $2^d$ numbers $\{\theta _{\xi }\}_{\xi \in \{+,-\}^d}\subseteq [0,1]$ such that $\sum _{\xi \in \{+,-\}^d} \theta _{\xi } =1$ and $\mathcal {P}_{A,B}(\nu ')= \alpha $ , where $\nu '=\sum _{\xi \in \{+,-\}^d}\theta _{\xi }\nu _{\xi }.$ Then by equation (3.18), $h_{\nu '}(\Phi ){\kern-1pt}={\kern-1pt}0.$ By equations (3.17) and (3.18), we have $d^*(\nu ',\mu _0)<\zeta .$ Now by equation (3.10), we choose $\theta \in [0,1]$ such that $\nu =\theta \mu _0+(1-\theta )\nu '$ satisfies $P(\Phi ,\chi ,\nu )=c.$ Then by Lemma 3.13(1),

(3.19) $$ \begin{align} \mathcal{P}_{A,B}(\nu)=\alpha \quad\text{and}\quad d^*(\nu,\mu_0)<\zeta. \end{align} $$

So $\{\mu \in M_{A,B}(\alpha ):P(\Phi ,\chi ,\mu )= c\}$ is dense in $\{\mu \in M_{A,B}(\alpha ):P(\Phi ,\chi ,\mu )\geq c\}.$ Since $\Phi $ is expansive on $\Lambda ,$ the metric entropy $\mathcal {M}(\Phi ,\Lambda ) \ni \mu \mapsto h_{\mu }(\Phi )$ is upper semi-continuous. Hence, $\{\mu \in \mathcal {M}(\Phi ,\Lambda ):P(\Phi ,\chi ,\mu )\in [c,c+{1}/{n})\}$ is open in $\{\mu \in \mathcal {M}(\Phi ,\Lambda ):P(\Phi ,\chi ,\mu )\geq c\}$ for any $n\in \mathbb {N^{+}}.$ Then

$$ \begin{align*} &\bigg\{\mu\in M_{A,B}(\alpha):P(\Phi,\chi,\mu)\in\bigg[c,c+\frac{1}{n}\bigg)\bigg\}\\&\quad \text{ is open and dense in }\{\mu\in M_{A,B}(\alpha):P(\Phi,\chi,\mu)\geq c\}, \end{align*} $$

and thus $\{\mu \in M_{A,B}(\alpha ):P(\Phi ,\chi ,\mu )=c\}$ is residual in $\{\mu \in M_{A,B}(\alpha ):P(\Phi , \chi ,\mu ) \geq c\}.$

3.5.2 Proof of Theorem 3.10

(1) Fix $\alpha {\kern-1pt}\in{\kern-1pt} \mathrm {Int}(L_{A,B})$ , $\mu _0{\kern-1pt}\in{\kern-1pt} M_{A,B}(\alpha )$ , $\max _{\mu \in M_{A,B}(\alpha )} \alpha (\mu ) \leq c\leq P(\Phi ,\chi ,\mu _0)$ , and $\eta , \zeta>0.$ By Lemma 3.18(3), there exists $\nu '\in M_{A,B}(\alpha )$ such that $P(\Phi ,\chi ,\nu ')=c$ and $d^*(\nu ',\mu _0)< {\zeta }/{2}.$ For $\alpha \in \mathrm {Int}(L_{A,B})$ , $\nu '\in M_{A,B}(\alpha )$ , and $\eta , {\zeta }/{2}>0,$ there is $\nu \in M_{A,B}^{\mathrm {erg}}(\alpha )$ such that $d^*(\nu ,\nu ') < {\zeta }/{2}$ and $|P(\Phi ,\chi ,\nu )-P(\Phi ,\chi ,\nu ')|<\eta .$ Then we complete the proof of item (1).

(2) Fix $\alpha \in \mathrm {Int}(L_{A,B})$ and $\max _{\mu \in M_{A,B}(\alpha )}\chi (\mu )\leq c<H_{A,B}(\Phi ,\chi ,\alpha ).$ First we show that

(3.20) $$ \begin{align} \{\mu\in M_{A,B}^{\mathrm{erg}}(\alpha):P(f,\alpha,\mu)\geq c\} \text{ is dense in }\{\mu\in M_{A,B}(\alpha):P(f,\alpha,\mu)\geq c\}. \end{align} $$

Let $\mu _0\in M_{A,B}(\alpha )$ be an invariant measure with $P(\Phi ,\chi ,\mu _0) \geq c$ and $\zeta>0$ . If $P(\Phi ,\chi ,\mu _0)>c$ , then there is $\eta>0$ such that $c<c+\eta <P(\Phi ,\chi ,\mu _0).$ For $\alpha \in \mathrm {Int}(L_{A,B})$ , $\mu _0\in M_{A,B}(\alpha )$ , and $\eta , \zeta>0,$ there exists an ergodic measure $\nu \in M_{A,B}^{\mathrm {erg}}(\alpha )$ such that $d^*(\nu ,\mu _0)<\zeta $ and $|P(\Phi ,\chi ,\nu )-P(\Phi ,\chi ,\mu _0)|<\eta .$ If $P(\Phi ,\chi ,\mu _0)=c$ , then we can pick an invariant measure $\mu '\in M_{A,B}(\alpha )$ such that $c<P(\Phi ,\chi ,\mu ') \leq H_{A,B}(\Phi ,\chi ,\alpha )$ , and next pick a sufficiently small number $\theta \in (0,1)$ such that $d^*(\mu _0, \mu ")<\zeta / 2$ , where $\mu "=(1-\theta ) \mu _0+\theta \mu '.$ By equation (3.10), we have $P(\Phi ,\chi ,\mu ")>c.$ By the same argument, there exists an ergodic measure $\nu \in M_{A,B}^{\mathrm {erg}}y(\alpha )$ such that $d^*(\nu ,\mu ")<\zeta /2$ and $P(\Phi ,\chi ,\nu )> c.$ So $d^*(\nu ,\mu _0)<\zeta .$

By equation (3.20) and Lemma 3.18(1),

(3.21) $$ \begin{align} \{\mu\in M_{A,B}^{\mathrm{erg}}(\alpha):P(\Phi,\chi,\mu)\geq c\}\text{ is residual in }\{\mu\in M_{A,B}(\alpha):P(\Phi,\chi,\mu)\geq c\}. \end{align} $$

By Lemma 3.18(3),

(3.22) $$ \begin{align} \{\mu\in M_{A,B}(\alpha):P(\Phi,\chi,\mu)=c\} \text{ is residual in }\{\mu\in M_{A,B}(\alpha):P(\Phi,\chi,\mu)\geq c\}. \end{align} $$

If there is an invariant measure $\tilde {\mu }$ with $S_{\tilde {\mu }}=\Lambda $ , then by Lemma 3.18(2), we have

(3.23) $$ \begin{align} \{\mu\in M_{A,B}(\alpha):P(\Phi,\chi,\mu)\geq c,\ S_{\mu}=\Lambda\}\text{ is residual in }\{\mu\in M_{A,B}(\alpha):P(\Phi,\chi,\mu)\geq c\}. \end{align} $$

So by equations (3.21), (3.22), and (3.23), we complete the proof of item (2).

(3) Fix $\alpha \in \mathrm {Int}(L_{A,B})$ and $\mu _0\in M_{A,B}(\alpha )$ with $\max \nolimits _{\mu \in M_{A,B}(\alpha )}\chi (\mu )\leq P(\Phi ,\chi ,\mu _0)< H_{A,B}(\Phi ,\chi ,\alpha ).$ Then by item (2), the set $\{\mu \in M_{A,B}^{\mathrm {erg}}y(\alpha ):P(\Phi ,\chi ,\mu )=P(\Phi ,\chi ,\mu _0)\}$ is residual in $\{\mu \in M_{A,B}(\alpha ):P(\Phi ,\chi ,\mu )\geq P(\Phi ,\chi ,\mu _0)\}.$ In particular, there is $\mu _{\alpha }\in M_{A,B}^{\mathrm {erg}}(\alpha )$ so that $P(\Phi ,\chi ,\mu _{\alpha })=P(\Phi ,\chi ,\mu _0).$

5.1 Singular hyperbolicity and geometric Lorenz attractors

First, we recall the definition of singular hyperbolicity which was introduced by Morales, Pacifico, and Pujals [Reference Morales, Pacifico and Pujals29] to describe the geometric structure of Lorenz attractors and these ideas were extended to higher dimensional cases in [Reference Li, Gan and Wen25, Reference Metzger and Morales28].

We recall the concept of a homoclinic class of a hyperbolic periodic orbit.

Definition 5.3. Given a vector field $X\in \mathscr {X}^1(M)$ , an invariant compact subset $\Lambda \subset M$ is a homoclinic class if there exists a hyperbolic periodic point $p\in \Lambda \cap \text {Per}(X)$ so that

$$ \begin{align*} \Lambda\colon=\overline{W^{s}(\operatorname{Orb}(p)) \pitchfork W^{u}(\operatorname{Orb}(p))}, \end{align*} $$

that is, it is the closure of the points of transversal intersection between stable and unstable manifolds of the periodic orbit $\operatorname {Orb}(p)$ of p. We say a homoclinic class is non-trivial if it is not reduced to a single hyperbolic periodic orbit.

From [Reference Araújo and Pacifico2, Theorem 2.17], any non-empty homoclinic class $\Lambda $ contains a dense set of periodic orbits. Then there is an invariant measure $\tilde {\mu }$ with $S_{\tilde {\mu }}=\Lambda $ by [Reference Denker, Grillenberger and Sigmund14, Proposition 21.12].

Now we give the definition of geometric Lorenz attractors following Guckenheimer and Williams [Reference Guckenheimer, Marsden and McCracken18, Reference Guckenheimer and Williams19, Reference Williams39] for vector fields on a closed 3-manifold $M^3$ .

Definition 5.5. We say $X\in \mathscr {X}^r(M^3)$ ( $r\geq 1$ ) exhibits a geometric Lorenz attractor $\Lambda $ if X has an attracting region $U\subset M^3$ such that $\Lambda =\bigcap _{t>0}\phi ^X_t(U)$ is a singular hyperbolic attractor and satisfies the following (see Figure 2).

• • There exists a unique singularity $p\in \Lambda $ with three exponents $\unicode{x3bb} _1<\unicode{x3bb} _2<0<\unicode{x3bb} _3$ , which satisfy $\unicode{x3bb} _1+\unicode{x3bb} _3<0$ and $\unicode{x3bb} _2+\unicode{x3bb} _3>0$ .

• • $\Lambda $ admits a $C^r$ -smooth cross-section S which is $C^1$ -diffeomorphic to $[-1,1]\times [-1,1]$ such that $\Gamma =\{0\}\times [-1,1]=W^s_{\mathrm {loc}}(\sigma )\cap S$ , and for every $z\in U\setminus W^s_{\mathrm {loc}}(\sigma )$ , there exists $t>0$ such that $\phi _t^X(z)\in S$ .

• • Up to the previous identification, the Poincaré map $P:S\setminus \Gamma \rightarrow S$ is a skew-product map

  $$ \begin{align*} P(x,y)=\big( f(x)~,~g(x,y) \big)\quad\text{for all }(x,y)\in[-1,1]^2\setminus \Gamma. \end{align*} $$
  Moreover, it satisfies:

  1. – $g(x,y)<0$ for $x>0$ , and $g(x,y)>0$ for $x<0$ ;

  2. – $\sup _{(x,y)\in S\setminus \Gamma }\big |\partial g(x,y)/\partial y\big |<1$ and $\sup _{(x,y)\in S\setminus \Gamma }\big |\partial g(x,y)/\partial x\big |<1$ ;

  3. – the one-dimensional quotient map $f:[-1,1]\setminus \{0\}\rightarrow [-1,1]$ is $C^1$ -smooth and satisfies $\lim _{x\rightarrow 0^-}f(x)=1$ , $\lim _{x\rightarrow 0^+}f(x)=-1$ , $-1<f(x)<1$ , and $f'(x)>\sqrt {2}$ for every $x\in [-1,1]\setminus \{0\}$ .

Figure 2 Geometric Lorenz attractor and return map.

It has been proved that the geometric Lorenz attractor is a homoclinic class [Reference Araújo and Pacifico2, Theorem 6.8] and $C^2$ -robust [Reference Shi, Tian and Wang34, Proposition 4.7].

For singular hyperbolic attractors, we have the following result.

5.2 Proof of Theorem C

Given a vector field $X\in \mathscr {X}^1(M)$ and an invariant compact set $\Lambda $ , let $\mathcal {N}\subset \mathcal {M}(\Phi ,\Lambda )$ be a convex set. We denote $\mathcal {N}_{\mathrm {erg}}y=\mathcal {M}_{\mathrm {\mathrm {erg}}}(\Phi ,\Lambda )\cap \mathcal {N}.$ Let $d \in \mathbb {N}$ and $(A, B) \in A(\Phi ,\Lambda )^{d} \times A(\Phi ,\Lambda )^{d}$ such that B satisfies equation (3.4). Denote

$$ \begin{align*}L_{A,B}^{\mathcal{N}}=\{\mathcal{P}_{A,B}(\mu):\mu\in \mathcal{N}\}.\end{align*} $$

For any $\alpha \in L_{A,B}^{\mathcal {N}},$ denote

$$ \begin{align*}M_{A,B}^{\mathcal{N}}(\alpha)=\{\mu\in {\mathcal{N}}:\mathcal{P}_{A,B}(\mu)=\alpha\},\quad M_{A,B}^{\mathrm{erg},\mathcal{N}}(\alpha)=\{\mu\in \mathcal{N}_{\mathrm{erg}}y:\mathcal{P}_{A,B}(\mu)=\alpha\}.\end{align*} $$

Let $\chi :\mathcal {M}(\Phi ,\Lambda )\to \mathbb {R}$ be a continuous function. We define the pressure of $\chi $ with respect to $\mu $ by $P(\Phi ,\chi ,\mu )=h_{\mu }(\Phi )+\chi (\mu ).$ For any $\alpha \in L_{A,B}^{\mathcal {N}},$ denote

$$ \begin{align*}H_{A,B}^{\mathcal{N}}(\Phi,\chi,\alpha)=\sup\{P(\phi,\chi,\mu):\mu\in M_{A,B}^{\mathcal{N}}(\alpha)\}.\end{align*} $$

In particular, when $\chi \equiv 0,$ we write

$$ \begin{align*}H_{A,B}^{\mathcal{N}}(\Phi,\alpha)=H_{A,B}^{\mathcal{N}}(\Phi,0,\alpha)=\sup\{h_{\mu}(\Phi):\mu\in M_{A,B}^{\mathcal{N}}(\alpha)\}.\end{align*} $$

If we replace $\mathcal {M}(\Phi ,\Lambda )$ by $\mathcal {N}$ in §§3.3–3.5, the results of Theorems 3.9, 3.10, and Remark 3.12 also hold. In fact, the convexity of $\mathcal {M}(\Phi ,\Lambda )$ is one core property in the proof of Theorems 3.9, 3.10, and Remark 3.12. Since $\mathcal {N}$ is convex, then the arguments of §3.3 are also true. Here, we omit the proof.