2.1 Classification of attracting periodic orbits

Let $f\colon X\to X$ be a unimodal map, and let $\{f^k(x)\}_{k=0}^{p-1}$ be a periodic orbit of f with prime period p. We say that $\{f^k(x)\}_{k=0}^{p-1}$ is:

• • hyperbolic attracting if $|(f^p)'(x)|<1$ ;

• • neutral if $|(f^p)'(x)|=1$ ;

• • hyperbolic repelling if $|(f^p)'(x)|>1$ .

The basin of the periodic orbit $\{f^k(x)\}_{k=0}^{p-1}$ is the set of points in X whose omega-limit set is contained in the set $\{f^k(x)\}_{k=0}^{p-1}$ . We say that $\{f^k(x)\}_{k=0}^{p-1}$ is attracting if its basin contains an open set. In this case, the union of the connected components of the basin containing a point from $\{f^k(x)\}_{k=0}^{p-1}$ is called the immediate basin of $\{f^k(x)\}_{k=0}^{p-1}$ . Each connected component of the immediate basin contains exactly one point from $\{f^k(x)\}_{k=0}^{p-1}$ .

If $f\colon X\to X$ is S-unimodal, a neutral periodic point x with prime period p satisfies either

1. (a) $(f^p)"(x)\neq 0$ , or

2. (b) $(f^p)"(x)= 0$ and $(f^p)"'(x)/ (f^p)'(x)<0$ .

In case (a), the periodic point x is locally attracting from only one side and in case (b) it is attracting from both sides. Hence, a periodic point is attracting if and only if it is hyperbolic attracting or neutral. The immediate basin of an attracting periodic orbit contains the critical point [Reference de Melo and van Strien10, Ch. II, Lemma 6.1, Theorem 6.1]. Therefore, there is at most one attracting periodic orbit, denoted by $O(f)$ . Let $B(f)$ denote the immediate basin of $O(f)$ . If f has no attracting periodic orbit, we put $B(f)=\emptyset $ for convenience. Let $\delta _{O(f)}$ denote the element of $\mathcal M(f)$ that is supported on $O(f)$ . The attracting periodic orbit $O(f)$ is called two-sided attracting if it is hyperbolic attracting or neutral in case (b). Otherwise (i.e., if it is neutral in case (a)), an attracting periodic orbit $\{f^k(x)\}_{k=0}^{p-1}$ is called one-sided attracting.

2.2 Lyapunov exponents

Since our definition of the Lyapunov exponent in (1.1) is a little non-standard, we will use the next lemma in §3.

2.3 Renormalization of S-unimodal maps

Let $f\colon X\to X$ be a unimodal map with a non-flat critical point c. A proper closed subinterval J of X is restrictive of period $p\geq 2$ if the following conditions hold (cf. [Reference de Melo and van Strien10, pp. 139]):

• • the interiors of $J,\ldots ,f^{p-1}(J)$ are pairwise disjoint;

• • $f^p(J)\subset J$ and $f^p(\partial J)\subset \partial J$ ;

• • one of the intervals $J,\ldots ,f^{p-1}(J)$ contains c in its interior;

• • J is maximal with respect to these properties—if $J'\supset J$ is a closed interval which is strictly contained in X and satisfies the previous three properties with the same integer p, then $J'=J$ .

We define a strictly decreasing sequence of closed intervals

$$ \begin{align*} X=J_0\supsetneq J_1\supsetneq J_2\supsetneq\cdots \end{align*} $$

which contain the critical point c, and a strictly increasing sequence of integers

$$ \begin{align*} 1=p_0<p_1<p_2<\cdots \end{align*} $$

so that $J_m$ is restrictive of period $p_m$ for $m\ge 1$ , inductively as follows. Given $J_m$ and $p_m$ for some $m\geq 0$ , note that $f^{p_{m}}|_{J_m}\colon J_m\to J_m$ is a unimodal map. If $f^{p_{m}}|_{J_m}$ has a restrictive interval, then define $J_{m+1}$ to be the restrictive interval of $f^{p_{m}}|_{J_m}$ containing c and with the smallest period $r_m$ . We define $p_{m+1}=p_{m}r_m$ . If $f^{p_{m}}|_{J_m}$ has no restrictive interval, then we do not define $J_{m+1}$ and stop the definition setting $\bar m(f)=m$ . If this inductive definition continues for arbitrarily large m, we set $\bar m(f)=\infty $ . We say that f is:

• • non-renormalizable if $\bar m(f)=0$ ;

• • renormalizable if $\bar m(f)\geq 1$ ;

• • only finitely renormalizable if $\bar m(f)<\infty $ ;

• • infinitely renormalizable if $\bar m(f)=\infty $ .

We assume that f has negative Schwarzian derivative, and review the dynamics of the unimodal map $f^{p_m}|_{J_m}$ , $m\geq 0$ . We classify such dynamics into the following cases:

1. (A) $f^{p_m}|_{J_m}$ is non-renormalizable and has no attracting fixed point;

2. (B) $f^{p_m}|_{J_m}$ is renormalizable;

3. (C) $f^{p_m}|_{J_m}$ is non-renormalizable and has an attracting fixed point.

For each $m\geq 0$ , let $L_m$ denote the closed interval in $J_m$ bordered by $f^{p_m}(c)$ and $f^{2p_m}(c)$ . Note that $L_m$ is possibly a singleton, that is, a degenerate closed interval.

In case (A), we have $f^{p_m}(L_m)= L_m$ and $f^{p_m}|_{L_m}$ is topologically exact: for any relatively open subset U of $L_{m}$ there is an integer multiple $n\geq 1$ of $p_m$ such that ${f^{n}(U)=L_{m}}$ . One can check this by combining [Reference de Melo and van Strien10, Theorem V.1.3] and [Reference Ruette30, Theorem 2.19 and Proposition 2.34], for example.

In case (B), we have $c\in \mathrm {int}(L_m)$ and $f^{p_m}(L_m)\subset L_m$ , and the orbit of any point in $\mathrm {int}(J_m)$ eventually falls into $L_m$ . There exists a restrictive interval $J_{m+1}$ that contains the critical point c and the set $\bigcup _{k=0}^{p_{m+1}/p_m}f^{kp_m}(J_{m+1})$ is forward invariant with respect to $f^{p_m}$ . We will consider two subcases:

1. (B1) $p_{m+1}/p_m\neq 2$ ;

2. (B2) $p_{m+1}/p_m=2$ .

Compare these in Figure 1. In case (B2) we have $L_{m}=f^{p_m}(J_{m+1})\cup f^{2p_m}(J_{m+1})$ and $L_m\subset \bigcup _{k=0}^{p_{m+1}-1} f^k(J_{m+1})$ .

Figure 1

The graphs of the renormalized unimodal maps $f^{p_{m}}|_{J_{m}}\colon J_m\to J_m$ .

Figure 1 Long description

A multi-panel figure containing six line graphs labeled A, B 1, B 2, C 1, C 2, and C 3. Each graph is contained within a square frame and features a dashed diagonal line from the bottom-left to the top-right corner.

Panel A: Shows a high-arched unimodal curve. A thick horizontal line segment on the x-axis is labeled L sub m. Dotted lines indicate cobwebbing or iterative paths within the curve.

Panel B 1: Similar to A, but the x-axis features three distinct sub-intervals marked with brackets. Curved arrows point between these intervals, and the central region is labeled J sub m plus 1. The overall interval is labeled L sub m.

Panel B 2: Shows a slightly lower unimodal curve. The x-axis is divided into two main sub-intervals with a double-headed arrow between them, labeled J sub m plus 1 and L sub m.

Panel C 1: Features a much lower unimodal curve that intersects the dashed diagonal line at a point labeled z sub f. The curve peaks well below the top of the frame.

Panel C 2: Shows the curve from C 1 shifted so that the intersection point z sub f is located at the origin (bottom-left corner). The curve is flatter.

Panel C 3: Similar to C 2, showing a further flattened unimodal curve starting from the origin point z sub f, illustrating the progression of the renormalization process.

In case (C), $m=\bar m(f)$ and $J_m$ contains a unique point $z_f$ from $O(f)$ . Indeed, if $J_m$ contained two points from $O(f)$ , then one would find a restrictive interval for $f^{p_m}|_{J_m}$ as an immediate basin of $O(f)$ . In particular, the point $z_f\in O(f)\cap J_m$ is a fixed point of the unimodal map $f^{p_m}|_{J_m}$ . We will consider three subcases:

1. (C1) $z_f\in \mathrm {int}(J_m)$ ;

2. (C2) $z_f\in \partial J_{m}$ and $O(f)$ is neutral;

3. (C3) $z_f\in \partial J_{m}$ and $O(f)$ is hyperbolic attracting.

Compare these in Figure 1. In case (C1), $O(f)$ is hyperbolic attracting and $\mathrm {int}(J_{m})$ is the connected component of $B(f)$ containing $z_f$ . For any closed interval J contained in $\mathrm {int}(J_{m})$ , the following uniform convergence holds:

(2.1) $$ \begin{align}{\lim_{n\to\infty}\sup_{x\in J}}|f^{p_{m}n}(x)-z_f|=0.\end{align} $$

In cases (C2) and (C3), since f has negative Schwarzian derivative, $f^{p_{m}}(c)$ lies in between $z_f$ and c and the following uniform convergence on $J_m$ holds:

(2.2) $$ \begin{align} {\lim_{n\to\infty}\sup_{x\in J_{m}}}|f^{p_{m}n}(x)-z_f|=0. \end{align} $$

If, moreover, $z_f$ is two-sided attracting and $m\geq 1$ , the dynamics of the previous renormalization $f^{p_{m-1}}|_{J_{m-1}}$ is similar to that in case (C1) from Lemma 2.2 below. For this reason, if $1\leq \bar m(f)<\infty $ and $\partial J_{\bar m(f)}$ contains a two-sided attracting fixed point of $f^{p_{\bar m(f)}}|_{J_{\bar m(f)}}$ , then we zoom out to the previous renormalization by setting

$$ \begin{align*} m(f)=\bar m(f)-1. \end{align*} $$

In all other cases, we set

$$ \begin{align*} m(f)=\bar m(f). \end{align*} $$

2.4 Structures associated with renormalization

Let $f\colon X\to X$ be an S-unimodal map with a non-flat critical point c such that $m(f)\geq 1$ . For each integer m with $0\leq m< m(f)$ , we define the mth renormalization cycle $K_m$ by

$$ \begin{align*} K_{m}=\bigcup_{k=0}^{p_{m}-1} f^k(J_{m}). \end{align*} $$

For each integer m with $0\leq m<m(f)-1$ , we denote by $\tilde {\mathscr {P}}_{m}$ the collection of the connected components of $K_m\setminus \bigcup _{k=0}^{p_{m+1}-1}\mathrm {int}(f^k(J_{m+1}))$ .

In the case $m(f)<\infty $ , we set

(2.3) $$ \begin{align}K_{m(f)}=\begin{cases} \displaystyle\bigcup_{k=0}^{p_{m(f)}-1} f^k(J_{m(f)}) &\text{ if } f \text{ has an attracting periodic orbit},\\ \displaystyle\bigcup_{k=0}^{p_{m(f)}-1} f^k(L_{m(f)}) &\text{ if } f \text{ has no attracting periodic orbit.}\end{cases} \end{align} $$

Moreover, we define $\tilde {\mathscr {P}}_{m(f)-1}$ to be the collection of the connected components of the following sets: $K_{m(f)-1}\setminus \bigcup _{k=0}^{p_{m(f)}-1}\mathrm {int}(f^k(J_{m(f)}))$ if f has an attracting periodic orbit; $K_{m(f)-1}\setminus \bigcup _{k=0}^{p_{m(f)}-1}\mathrm {int}(f^k(L_{m(f)}))$ if f has no attracting periodic orbit.

The elements of $\tilde {\mathscr {P}}_m$ with $0\leq m<m(f)$ are closed intervals in $K_m$ , possibly singletons, and $\tilde {\mathscr {P}}_m$ has the Markov property: if $P,Q\in \tilde {\mathscr {P}}_m$ and $f(P)\cap Q\neq \emptyset $ then ${f(P)\supset Q}$ . Let $\mathscr {P}_m$ denote the set of elements of $\tilde {\mathscr {P}}_m$ that are contained in $\bigcup _{k=0}^{p_m-1}f^k(L_m)$ . Note that $\mathscr {P}_m$ is non-empty.

For each integer m with $0\leq m< m(f)$ , we define

$$ \begin{align*}K_{m,m+1}=\mathrm{cl}(K_m\setminus K_{m+1})\end{align*} $$

and

$$ \begin{align*} \Gamma_{m}= \bigcap_{n=0}^\infty f^{-n}\bigg(\!\bigcup_{P\in \mathscr{P}_m}P\bigg).\end{align*} $$

Note that $\Gamma _m$ for $0\leq m< m(f)$ contains an attracting periodic orbit only if ${m=m(f)-1}$ , and this periodic orbit is one-sided attracting. In this case, $K_{m,m+1}$ is disjoint from the interior of the immediate basin of the attracting periodic orbit. For convenience we set $\Gamma _{-1}=\partial X.$ In the case $m(f)<\infty $ we further define

$$ \begin{align*}\Gamma_{m(f)}=\begin{cases} O(f)&\quad\text{if } f \text{ has a two-sided attracting periodic orbit, }\\ \displaystyle\bigcup_{k=0}^{p_{m(f)}-1}f^k(L_{m(f)})&\quad\text{otherwise.} \end{cases}\end{align*} $$

The sets $\Gamma _m$ , $-1\leq m\leq m(f)$ are non-empty closed sets, and $\Gamma _m\cap \Gamma _{m'}=\emptyset $ holds for all distinct integers $m,m'$ in $\{-1,\ldots ,m(f)-1\}$ . Note that $\Gamma _{m(f)-1}$ intersects $\Gamma _{m(f)}$ if and only if $L_{m(f)}=J_{m(f)}$ .

For each integer m with $-1\leq m\leq m(f)$ we set

$$ \begin{align*}\mathcal M_m(f)=\{\mu\in\mathcal M(f)\colon\mathrm{supp}(\mu)\subset \Gamma_m\},\end{align*} $$

where $\mathrm {supp}(\mu )$ denotes the smallest closed subset of X with full $\mu $ -measure. The sets $\mathcal M_m(f)$ are pairwise disjoint, and if $m(f)<\infty $ then $\mathcal M(f)$ is the convex hull of $\bigcup _{m=-1}^{m(f)}\mathcal M_m(f)$ . If $m(f)<\infty $ and f has a one-sided attracting periodic orbit, then $L_{m(f)}$ is contained in the basin of this attracting periodic orbit, and so $\mathcal M_{m(f)}(f)=\emptyset $ . The set $\mathcal M_{-1}(f)$ is a singleton that is supported on the fixed point in $\partial X$ .

2.5 Symbolic dynamics on each cycle

Let $f\colon X\to X$ be an S-unimodal map such that $m(f)\geq 1$ . For each integer m with $0\leq m< m(f)$ , there is a topological Markov chain over the finite alphabet $\tilde {\mathscr {P}}_m$ determined by the transition matrix

$$ \begin{align*} (M_{PQ})_{P,Q\in\tilde{\mathscr{P}}_m},\quad M_{PQ}=\begin{cases} 1& \text{if } f(P)\supset Q,\\ 0& \text{otherwise}. \end{cases} \end{align*} $$

Let $n\geq 2$ be an integer and let $P_0,P_1,\ldots ,P_{n-1}\in \tilde {\mathscr {P}}_m$ . The word $P_0P_1\cdots P_{n-1}$ of length n is admissible if $M_{P_kP_{k+1}}=1$ holds for $0\leq k\leq n-2$ . Let $E^n_m$ denote the set of admissible words of elements of $\tilde {\mathscr {P}}_m$ of length n. Let $\Sigma _m$ denote the set of one-sided infinite sequences $\{P_k\}_{k=0}^\infty $ of elements of $\tilde {\mathscr {P}}_m$ such that $P_0\cdots P_{n-1}\in E_m^n$ holds for every $n\geq 2$ . We endow $\Sigma _m$ with the restriction of the product topology of the discrete topology of $\tilde {\mathscr {P}}_m$ . Let $\sigma _m\colon \Sigma _m\to \Sigma _m$ denote the left shift: $\sigma _m(\{P_k\}_{k=0}^\infty )=\{P_k\}_{k=1}^\infty $ . For each $P_0P_1\cdots P_{n-1}\in E^n_m$ , we set $I_{P_0P_1\cdots P_{n-1}}=\bigcap _{k=0}^{n-1}f^{-k}(P_k)$ , and define

$$ \begin{align*} \pi_m\colon \Sigma_m\to \bigcap_{n=0}^{\infty} f^{-n}\bigg(\!\bigcup_{P\in \tilde{\mathscr{P}}_m}P\bigg) \end{align*} $$

by

(2.4) $$ \begin{align} \pi_m(\{P_k\}_{k=0}^\infty)\in\bigcap_{n=1}^\infty I_{P_0\cdots P_{n-1} }. \end{align} $$

By [Reference Lyubich21, Main Theorem], f has no wandering interval, and so any homterval is contained in the basin of an attracting periodic orbit. Since $\bigcap _{n=1}^\infty I_{P_0\cdots P_{n-1}}$ is not contained in the basin of an attracting periodic orbit, it is not a homterval, namely, a singleton. Hence, $\pi _m$ is well defined.

We will use the following lemma a few times.

Lemma 2.6. Let $f\colon X\to X$ be an S-unimodal map with a non-flat critical point such that $m(f)\geq 1$ . For each integer m with $0\leq m< m(f)$ , we have

$$ \begin{align*} \lim_{n\to\infty}\sup\{|I_{\omega}|:\omega\in E^n_{m}\}=0.\end{align*} $$

2.6 Distortion estimate on each cycle

Let Y be a non-degenerate compact interval in X and let $g\colon Y\to X$ be a $C^1$ map. Let J be a subinterval of Y such that the restriction of g to J is a diffeomorphism onto its image. By a distortion of g on J we mean the quantity

$$ \begin{align*}\sup_{x,y\in J}\frac{|g'(x)|}{|g'(y)|}. \end{align*} $$

We will frequently use the following distortion estimates on each cycle.

Proof. Since f is $C^2$ , the restriction of $\log |f'|$ to the union of the elements of $\tilde {\mathscr {P}}_m$ not containing a point from an attracting periodic orbit is Lipschitz continuous. Moreover, by Mañé’s theorem [Reference Mañé23, Theorem A], the maximal f-invariant set in this union is a hyperbolic set. Hence, (a) holds.

As for (b), if $\Gamma _m$ contains a neutral periodic orbit, then $m=m(f)-1$ and $\partial J_{m(f)}$ contains a point from the neutral periodic orbit. Let $\varepsilon>0$ , let $n\geq 2$ and let W be a connected component of $\bigcap _{k=0}^{n-1}f^{-k}(K_{m,m+1})$ . There exists $\omega \in E^n_{m}$ such that $W=I_\omega $ . Since $K_{m,m+1}$ does not contain the critical point of f, the infimum of $|f'|$ over this set is positive. For all $x,y\in I_\omega $ we have

$$ \begin{align*} \log\frac{|(f^n)'(x)|}{|(f^n)'(y)|}\leq \bigg(\!\sup_{K_{m,m+1} }\frac{|f"|}{|f'|}\bigg)\sum_{k=0}^{n-1}|f^k(I_\omega)\hspace{-1pt}|\leq\bigg(\!\sup_{K_{m,m+1} }\frac{|f"|}{|f'|}\bigg) \sum_{k=0}^{n-1}\sup_{\omega\in E^{n-k}_m}|I_{\omega}\hspace{-1pt}|. \end{align*} $$

By Lemma 2.6, the last number is bounded by $\varepsilon n$ for a sufficiently large n. The proof of Proposition 2.7 is complete.

2.7 Topological exactness on each cycle

Let Y be a non-degenerate interval in X and $g\colon Y\to Y$ be a continuous map. Let U be a non-degenerate subinterval of Y and let $n\in \mathbb N$ . Each connected component of $g^{-n}(U)$ is called a pullback of U by $g^n$ . If V is a pullback of U by $g^n$ and $g^n|_V$ is a diffeomorphism, then V is called a diffeomorphic pullback of U by $g^n$ .

The next lemma will be used later in §3.2 for the proof of the lower bound in Theorem A, in order to glue different orbits in different cycles together to form one orbit with required properties. It will also be used later in §4.2 for the proof of the upper bound in Theorem A.

Proof. We assume $m=0$ in the proof below. In the case $m>0$ , we can apply the same argument to the renormalized map $f^{p_m}\colon J_m \to J_m$ to obtain the conclusion of the lemma.

Without loss of generality we may assume $X=[0,1]$ and the critical point c is the maxima of f. We claim that the topological entropy $h_{\mathrm {top}}(f)$ of $f\colon X\to X$ is greater than or equal to $\log \sqrt {2}$ . Indeed, the assumption $p_1=p_1/p_0>2$ implies that if we take an interval $[x',x]$ such that $x'<c<x$ and $f(x)=x=f(x')$ , then $f^2(x)=x$ , $f^2(c)<x'$ and $[x',x]\subset f^2([x',x])$ . Since $[x',x]$ contains an $f^2$ -invariant set on which $f^2$ is topologically conjugate to the full shift on two symbols, we obtain $h_{\mathrm {top}}(f)= h_{\mathrm {top}}(f^2)/2\ge \log \sqrt {2}$ .

By the Milnor–Thurston kneading theory (see [Reference de Melo and van Strien10, Ch. 2, Theorem 8.1]), there exists a semiconjugacy from $f\colon X\to X$ to a piecewise linear map with slopes $\pm s$ where $s=e^{h_{\mathrm {top}}(f)}\ge \sqrt {2}$ . More precisely, there exists a continuous and surjective monotone map $\unicode{x3bb} \colon X\to X$ satisfying $T\circ \unicode{x3bb} =\unicode{x3bb} \circ f$ for the tent map

$$ \begin{align*} T\colon X\to X, \quad T(x)=\min\{sx,s(1-x)\}. \end{align*} $$

The map $\unicode{x3bb} $ sends a non-trivial interval in X to a point only if it is eventually mapped by f into the restrictive interval $J_1$ . (For this, we use the non-existence of wandering intervals; see [Reference de Melo and van Strien10, III.4, Proposition 4.3].) Hence, $\unicode{x3bb} $ maps an open interval $J\subset L_0$ that intersects the Cantor set $\Gamma _0$ onto a non-trivial interval $J'$ .

If $s=\sqrt {2}$ , then T is renormalizable with period $2$ and so is f, contradicting the assumption $p_1>2$ . So we may suppose $s> \sqrt {2}$ . Then there is $n\ge 0$ such that both $T^n(J')$ and $T^{n+1}(J')$ contain the turning point $1/2$ of T in their interiors (see [Reference de Melo and van Strien10, III.4, Proposition 4.4 and Exercise 4.1]). Consequently, $f^n(J)$ and $f^{n+1}(J)$ contain c in their interiors and therefore $f^{n+2}(J)\supset L_0=[f^{2}(c),f(c)]$ . Then we have ${f^{n'}(J)\supset f^{n'-n-2}(L_0)=L_0}$ for $n'\ge n+2$ .

Since f is unimodal, for any non-degenerate subinterval I of X we can take a subinterval $I'\subset I$ such that $f\colon I' \to f(I)$ is a homeomorphism. Applying this procedure to the sequence of maps

we find an interval $J'\subset J$ such that $f^{n'}\colon J' \to L_0$ is a homeomorphism. Then $\mathrm {int}(J')$ is a diffeomorphic pullback of $\mathrm {int}(L_0)$ by $f^{n'}$ , as required in the lemma.

Let $f\colon X\to X$ be an S-unimodal map with a non-flat critical point such that $m(f)\geq 1$ . Suppose $p_{m+1}/p_m\neq 2$ for some $0\leq m<m(f)$ . In view of Lemma 2.8, for each ${P\in \mathscr {P}_m}$ that is contained in $L_m$ we fix $q(P)\in \mathbb N$ such that $f^{p_m q(P)}(P)=L_m$ , and for any ${n\ge q(P)}$ there exists a diffeomorphic pullback of $\mathrm {int}(L_m)$ by $f^{p_m n}$ that is contained in P. For later uses and convenience we set

(2.5) $$ \begin{align}M_m=\max\{p_mq(P)\colon P\in\mathscr{P}_m \text{ and } P\subset L_m\} \quad\text{and}\quad M_{-1}=2.\end{align} $$

2.8 Notation

Let $C(X)$ denote the set of real-valued continuous functions on X. For $\phi \in C(X)$ and $n\in \mathbb N$ , write $S_n\phi $ for the sum $\sum _{k=0}^{n-1}\phi \circ f^k$ . For $\ell \in \mathbb N$ define

$$ \begin{align*} C(X)^\ell=\{\vec{\phi}=(\phi_1,\ldots,\phi_\ell)\colon\phi_j\in C(X)\text{ for }1\leq j\leq\ell\}. \end{align*} $$

For $\vec {\phi }=(\phi _1,\ldots ,\phi _\ell )\in C(X)^{\ell }$ , $\vec {\alpha }=(\alpha _1,\ldots ,\alpha _\ell )\in \mathbb R^{\ell }$ and a measure $\mu \in \mathcal M$ , the expression $\int \vec {\phi } \,d\mu>\vec {\alpha }$ indicates that $\int \phi _j \,d\mu>\alpha _j$ holds for $1\leq j\leq \ell $ . We set

$$ \begin{align*} A_n(\vec{\phi},\vec{\alpha})= \bigg\{x\in X: \int\vec\phi \,d\delta_x^n>\vec\alpha\bigg\} \end{align*} $$

and

$$ \begin{align*}S_n\vec{\phi}=(S_n\phi_1,\ldots,S_n\phi_\ell).\end{align*} $$

For $a\in \mathbb R$ and $\vec v=(v_1,\ldots ,v_\ell )\in \mathbb R^{\ell }$ , we write

$$ \begin{align*} \vec{a}=(a,a,\ldots,a)\in\mathbb R^{\ell}\quad \text{and}\quad \|\vec{v}\|=\max_{1\leq j\leq\ell}|v_j\hspace{-1pt}|. \end{align*} $$

For $\vec v,\vec w\in \mathbb R^{\ell }$ , the expression $\vec v\geq \vec w$ indicates that $v_j\geq w_j$ holds for $1\leq j\leq \ell $ .

2.9 Infinitely renormalizable maps

Let $f\colon X\to X$ be an infinitely renormalizable S-unimodal map with a non-flat critical point. The omega-limit set of Lebesgue almost every initial point $x\in X$ coincides with the closed invariant set

$$ \begin{align*} \Lambda=\bigcap_{m=0}^\infty K_m, \end{align*} $$

and the empirical measure $\delta _x^n$ converges to the post-critical measure $\mu _\infty $ supported on $\Lambda $ . The restriction of f to $\Lambda $ is uniquely ergodic and the post-critical measure $\mu _\infty $ is the unique invariant measure on $\Lambda $ . Moreover, $h(\mu _\infty )=\chi ^+(\mu _\infty )=0$ holds (see [Reference Bruin and Keller4, Theorem 3.4(b)]).

In order to ‘approximate’ $\mu _\infty $ , we will use the following lemma, which in particular asserts that the post-critical measure is approximated by empirical measures along sufficiently long orbit segments in sufficiently deep cycles.

For infinitely renormalizable maps, the Lyapunov exponent depends continuously on invariant measures.

3.1 Key lower bound

The next proposition allows us to approximate a given invariant measure with finitely many intervals in a particular sense.

Proposition 3.1. Let $f\colon X\to X$ be an S-unimodal map with a non-flat critical point. Let $\ell \in \mathbb N$ , $\vec {\phi }\in C(X)^{ \ell }$ , $\vec {\alpha }\in \mathbb R^{\ell }$ and let $\mu \in \mathcal M(f)$ satisfy $\int \vec \phi \,d\mu>\vec {\alpha }$ . For any $\varepsilon>0$ there exists $N\geq 1$ such that for each integer $n\geq N$ , there exists a finite collection $\mathscr {A}_{n}$ of closed intervals in $A_{n}(\vec {\phi },\vec {\alpha })$ with pairwise disjoint interiors such that

(a) $$ \begin{align} \#\mathscr{A}_{n}&>\exp((h(\mu)-\varepsilon)n)\quad and \qquad\qquad\ \ \quad\qquad\end{align} $$

(b) $$ \begin{align} |A|&>\exp(-(\chi^+(\mu)+\varepsilon)n)\quad\text{for every }A\in\mathscr{A}_{n}. \end{align} $$

In order to prove Proposition 3.1, we approximate the given measure $\mu $ with a finite convex combination of measures each supported on some $\Gamma _m$ or $\Lambda $ , and then for each of these measures construct a collection of finitely many intervals. We then glue orbits from these intervals together to construct a collection intervals with the desired properties.

The proof of Proposition 3.1 is given in §3.4. Below we deduce a corollary to Proposition 3.1 and complete the proof of the lower bound in Theorem A.

Corollary 3.2. Let $f\colon X\to X$ be an S-unimodal map with a non-flat critical point. Let $\ell \in \mathbb N$ , $\vec {\phi }\in C(X)^{\ell }$ , $\vec {\alpha }\in \mathbb R^{\ell }$ and let $\mu \in \mathcal M(f)$ satisfy $\int \vec {\phi } \,d\mu>\vec {\alpha }$ . Then

$$ \begin{align*} \liminf_{n\to\infty}\dfrac{1}{n}\log |A_n(\vec{\phi},\vec{\alpha})|\geq F(\mu). \end{align*} $$

Proof of the lower bound in Theorem A

Note that subsets of $\mathcal M$ of the form ${\{\mu \in \mathcal M\colon \int \vec \phi \,d\mu>\vec \alpha \}}$ with $\ell \in \mathbb N$ , $\vec {\phi }\in C(X)^{ \ell }$ , $\vec {\alpha }\in \mathbb R^{\ell }$ constitute a base of the weak* topology of $\mathcal M$ . So, any non-empty open subset $\mathcal G$ of $\mathcal M$ is written as the union $\mathcal G=\bigcup _{\unicode{x3bb} }\mathcal G_\unicode{x3bb} $ of sets $\mathcal G_\unicode{x3bb} $ of this form. For each $\mathcal G_\unicode{x3bb} $ , Corollary 3.2 gives

$$ \begin{align*} \liminf_{n\to\infty}\dfrac{1}{n}\log|\{x\in X:\delta_x^n\in\mathcal G_\unicode{x3bb}\}| \geq \sup_{\mathcal G_\unicode{x3bb}}F. \end{align*} $$

Hence, we obtain

$$ \begin{align*}\liminf_{n\to\infty}\dfrac{1}{n}\log| \{x\in X:\delta_x^n\in\mathcal G\}| \geq\sup_{\unicode{x3bb}}\sup_{\mathcal G_\unicode{x3bb}} F= \sup_{\mathcal{G}} F= -\inf_{\mathcal G} I, \end{align*} $$

as required in Theorem A.

3.2 Approximation of a measure on each cycle

The rest of this section is entirely dedicated to a proof of Proposition 3.1. We fix $\ell \in \mathbb N$ , $\vec {\phi }\in C(X)^{\ell }$ , $\vec {\alpha }\in \mathbb R^{\ell }$ in the statement of Proposition 3.1 until the end of §3.4, and assume $f\colon X\to X$ is an S-unimodal map with a non-flat critical point c.

The next lemma allows us to approximate an invariant measure on each single cycle with a finite collection of intervals.

Lemma 3.3. Let $f\colon X\to X$ satisfy $m(f)\geq 1$ . Let $0\leq m< m(f)$ and suppose $p_{m+1}/p_m\neq 2$ . Let $\mu \in \mathcal M_m(f)$ , $P\in \mathscr {P}_m$ satisfy $P\subset L_m$ and $\mu (P)>0$ , and let $P'$ be a non-degenerate closed interval in $L_{m}$ . For any $\varepsilon>0$ , there exists $N\geq 1$ such that for each integer $q\geq N$ , there exists a finite collection $\mathscr {B}_{p_mq}(P,P')$ of pullbacks of $P'$ by $f^{p_mq}$ that are contained in P such that

(a) $$ \begin{align} \qquad\qquad \bigg|\frac{1}{p_mq}\log\# \mathscr{B}_{p_mq}(P,P')-h(\mu)\bigg| & <\varepsilon, \end{align} $$

(b) $$ \begin{align} \sup_{B\in\mathscr{B}_{p_mq}(P,P')}\sup_{B}\bigg\|\frac{1}{p_mq}S_{p_mq}\vec\phi-\int\vec\phi \,d\mu\bigg\| & <\varepsilon. \end{align} $$

If, moreover, $P'\cap \{f(c),\ldots ,f^{M_m-1}(c)\}=\emptyset $ for the integer $M_m\geq 1$ in (2.5), then every pullback $B\in \mathscr {B}_{p_mq}(P,P')$ is diffeomorphic and satisfies

(c) $$ \begin{align} &\sup_{B}\bigg|\frac{1}{p_mq}\log|(f^{p_mq})'|-\chi^+(\mu)\bigg|<\varepsilon. \end{align} $$

Proof. From the definition of $\pi _m$ in (2.4) and Proposition 2.5, $f|_{\Gamma _m}$ is topologically conjugate to a topological Markov chain via the finite Markov partition $\mathscr {P}_m$ . For $q\in \mathbb N$ let $\tilde {\mathscr {B}}_{p_m q}(P)$ denote the collection of the pullbacks of elements of $\mathscr {P}_m$ by $f^{p_m q}$ that are contained in P. Lemma 2.6 gives $\lim _{q\to \infty }\sup _{B\in \tilde {\mathscr {B}}_{p_mq}(P)}|B|=0$ , and so for any continuous function $\varphi \colon K_{m,m+1}\to \mathbb R$ we have

(3.1) $$ \begin{align} \lim_{q\to\infty}\sup_{B\in\tilde{\mathscr{B}}_{p_m q}(P)}\sup_{x,y\in B}\frac{1}{p_m q}|S_{p_m q}\varphi(x)-S_{p_m q}\varphi(y)|=0. \end{align} $$

The proof of Lemma 3.3 divides into two cases. We first treat the case where $\mu $ is ergodic. Since $m<m(f)$ , $\mu $ is not supported on the orbit of a hyperbolic attracting periodic orbit (see §2.4). Let $\varepsilon>0$ . Since $\mu (L_m)=1/p_m$ , the normalized restriction of $\mu $ to $L_m$ is an $f^{p_m}|_{L_m}$ -invariant Borel probability measure with entropy $p_mh(\mu )$ . For $q\in \mathbb N$ , we set $r=q-M_m /p_m$ . Since $P\in \mathscr {P}_m$ , $\tilde {\mathscr {B}}_{p_m r}(P)$ coincides with the collection of the pullbacks of elements of $\mathscr {P}_m$ by $f^{p_m r}$ that are contained in P. We claim that for all sufficiently large q there exists a subset $\mathscr {B}_{p_m r}(P)$ of $\tilde {\mathscr {B}}_{p_m r}(P)$ for which the following inequalities hold:

$$ \begin{align*} \bigg|\frac{1}{p_m r}\log\#\mathscr{B}_{p_m r}(P)-h(\mu)\bigg| & <\frac{\varepsilon}{3};\\ \sup_{B\in\mathscr{B}_{p_m r}(P)}\sup_{B}\bigg\|\frac{1}{p_m r}S_{p_m r}\vec\phi-\int \vec\phi \,d\mu\bigg\| & <\frac{\varepsilon}{3};\\ \sup_{B\in\mathscr{B}_{p_m r}(P)}\sup_{B}\bigg|\frac{1}{p_m r}\log|(f^{p_m r})'|-\chi^+(\mu)\bigg| & <\frac{\varepsilon}{3}. \end{align*} $$

Indeed, the first inequality follows from Shannon, McMillan and Breiman’s theorem [Reference Cornfeld, Fomin and Sinaĭ9] for the normalized restriction. The second one follows from (3.1) and Birkhoff’s ergodic theorem. The third one follows from Proposition 2.7, Birkhoff’s ergodic theorem and $\chi ^+(\mu )=\int \log |f'|\,d\mu $ by Lemma 2.1.

By the definition of $M_m$ in (2.5), if $Q\in \mathscr {P}_m$ and $Q\subset L_m$ then $f^{M_m}(Q)=L_m$ . For each $ B\in \mathscr {B}_{p_m r}(P)$ , the interval $f^{p_m r}(B)$ is an element of $\mathscr {P}_m$ that is contained in $L_m$ . We fix a connected component $W_B$ of $f^{p_m r}(B)\cap f^{-M_m}(P')$ . Let $B'$ denote the diffeomorphic pullback of $W_B$ by $f^{p_m r}$ that is contained in B. We define

$$ \begin{align*}\mathscr{B}_{p_mq}(P,P')=\{B'\colon B\in\mathscr{B}_{p_mr}(P)\}.\end{align*} $$

Clearly we have $\#\mathscr {B}_{p_mq}(P,P')=\#\mathscr {B}_{p_m r}(P)$ , and the elements of $\mathscr {B}_{p_mq}(P,P')$ are pullbacks of $P'$ by $f^{p_m q}$ . If $P'\cap \{f(c),\ldots ,f^{M_m-1}(c)\}=\emptyset $ then for every $ B\in \mathscr {B}_{p_m r}(P)$ , $W_B$ is a diffeomorphic pullback of $P'$ by $f^{M_m}$ . So, every element of $\mathscr {B}_{p_mq}(P,P')$ is a diffeomorphic pullback of $P'$ by $f^{p_m q}$ . Therefore, for all sufficiently large $q\geq 1$ we obtain inequalities (a), (b), (c) of Lemma 3.3 from the corresponding three estimates for $\mathscr {B}_{p_m r}(P)$ as above. To deduce (c) we have noted that the orbits of points in elements of $\mathscr {B}_{p_mq}(P,P')$ up to time $p_mq$ are uniformly bounded away from the critical point c by a distance independent of q.

It remains to treat the case where $\mu $ is non-ergodic. By Lemma 2.1, the map ${\nu \in \mathcal {M}_m(f)\mapsto \chi ^+(\nu )\in \mathbb {R}}$ is affine. By virtue of the ergodic decomposition theorem and Jacobs’s theorem on the decomposition of entropy [Reference Jacobs16], for any $\varepsilon>0$ there exist a finite number of ergodic measures $\mu _1,\ldots ,\mu _s$ in $\mathcal M_m(f)$ and constants $\rho _1,\ldots ,\rho _s$ in $(0,1)$ adding up to $1$ such that the measure $\mu '=\sum _{i=1}^s\rho _i\cdot \mu _i$ in $\mathcal M_m(f)$ satisfies

$$ \begin{align*} |h(\mu)-h(\mu')|<\varepsilon, \quad \bigg\|\!\int\vec{\phi}\,d\mu-\int\vec{\phi}\,d\mu'\bigg\|<\varepsilon,\quad |\chi^+(\mu)-\chi^+(\mu')|<\varepsilon, \end{align*} $$

and $\mu '(P)$ is sufficiently close to $\mu (P)$ so that $\mu '(P)>0$ . With no loss of generality we may assume that $\mu $ is written as a convex combination of ergodic measures $\mu _i\in \mathcal M_m(f)$ :

$$ \begin{align*} \mu=\rho_1\cdot\mu_1+\rho_2\cdot\mu_2+\cdots+\rho_s\cdot\mu_s, \end{align*} $$

with $\mu _1(P)>0$ and $\rho _1,\ldots ,\rho _s$ adding up to $1$ . We fix $P_1,\ldots ,P_s\in \mathscr {P}_m$ so that $P_1=P$ , $P_i\subset L_m$ and $\mu _i(P_i)>0$ for $1\leq i\leq s$ . We set $P_{s+1}=P'$ .

Let $\varepsilon>0$ . Let $q\ge s$ be a large integer, and write it as a sum of positive integers

$$ \begin{align*} q= q_1+q_2+\cdots+q_s, \end{align*} $$

with $|p_mq_i-p_m\rho _i q|\leq p_m$ for $1\leq i\leq s.$ From the previous argument in the ergodic case, if q is sufficiently large then for each i there is a finite collection $\mathscr {B}_{p_mq_i}(P_i,P_{i+1})$ of diffeomorphic pullbacks of $P_{i+1}$ by $f^{p_m q_i}$ that is contained in $P_i$ for which the following three inequalities hold:

(3.2) $$ \begin{align} \qquad\qquad\qquad \bigg|\frac{1}{p_m q_i}\log\# \mathscr{B}_{p_mq_i}(P_i,P_{i+1})-h(\mu_i)\bigg| & <\frac{\varepsilon}{2}; \end{align} $$

(3.3) $$ \begin{align} \qquad\quad \sup_{B\in\mathscr{B}_{p_mq_i}(P_i,P_{i+1}) }\sup_{B}\bigg\|\frac{1}{p_m q_i}S_{p_m q_i}\vec\phi-\int\vec\phi \,d\mu_i\bigg\| & < \frac{\varepsilon}{2}; \end{align} $$

(3.4) $$ \begin{align} \sup_{B\in\mathscr{B}_{p_mq_i}(P_i,P_{i+1}) }\sup_{B} \bigg|\frac{1}{p_m q_i}\log|(f^{p_m q_i})'|-\chi^+(\mu_i)\bigg| & < \frac{\varepsilon}{2}. \end{align} $$

We now define $\mathscr {B}_{p_mq}(P,P')$ to be the collection of the diffeomorphic pullbacks of $P'$ by the composition $g_s\circ g_{s-1}\circ \cdots \circ g_1$ of diffeomorphisms $g_i$ of the form $f^{p_m q_i}|_{B}$ with $B\in \mathscr {B}_{p_mq_i}(P_i,P_{i+1})$ . If q is sufficiently large, then by (3.2) we have

$$ \begin{align*} \bigg|\frac{1}{p_m q}\log\#\mathscr{B}_{p_mq}(P,P')-h(\mu) \bigg|&\leq\frac{1}{p_m q}\bigg|\! \sum_{i=1}^s\log\#\mathscr{B}_{p_mq_i}(P_i,P_{i+1}) -p_m q \!\sum_{i=1}^s\rho_i h(\mu_i)\bigg|\\&\leq \frac{1}{p_m q}\sum_{i=1}^s |\!\log\#\mathscr{B}_{p_mq_i}(P_i,P_{i+1}) -p_m q_i h(\mu_i) |\\& \quad + \sum_{i=1}^s\bigg|\frac{q_i}{q}-\rho_i\bigg| h(\mu_i) <\frac{\varepsilon}{2}+\frac{s}{q}h_{\mathrm{top}}(f) <\varepsilon, \end{align*} $$

where $h_{\mathrm {top}}(f)$ denotes the topological entropy of f. Moreover, for each $B\in \mathscr {B}_{p_mq}(P,P')$ , using (3.3) for $1\leq i\leq s$ , we have

$$ \begin{align*} \sup_B\bigg\|\frac{1}{p_m q}S_{p_m q}\vec\phi- \int\vec\phi \,d\mu\bigg\|&\leq\frac{1}{p_m q}\sum_{i=1}^s \sup_{B'\in\mathscr{B}_{p_mq_i}(P_i,P_{i+1}) }\sup_{B'}\bigg\|S_{p_m q_i}\vec\phi-p_m q_i\int\vec\phi \,d\mu_i\bigg\|\\&\quad +\sum_{i=1}^s \bigg|\frac{q_i}{q}-\rho_i\bigg| \|\vec\phi \| < \frac{\varepsilon}{2}+\frac{s}{q}\|\vec \phi\| < \varepsilon. \end{align*} $$

Since $\varepsilon>0$ is arbitrary, the above two estimates respectively imply (a) and (b) in Lemma 3.3. We can prove (c) in the same way as (b), using (3.4) and noting that the orbits of points in elements of $\mathscr {B}_{p_mq}(P,P')$ up to time $p_mq$ are uniformly bounded away from the critical point c by a distance independent of q. This completes the proof of Lemma 3.3.

3.3 The initial transition

Let $\mu _{-1}$ denote the element of the singleton $\mathcal M_{-1}(f)$ . Recall that $\mu _{-1}$ is the unit point mass at the fixed point in $\partial X$ (see §2.4). Put $p_{-1}=1$ . Recall that $L_0$ is the closed interval in X bordered by $f(c)$ and $f^{2}(c)$ (see §2.3). Let $x_f$ denote the fixed point of f in $\partial X$ . If $L_0$ is strictly contained in X, let $P_{-1}$ denote the connected component of $X\setminus L_0$ that contains $x_f$ .

Lemma 3.4. Let $f\colon X\to X$ be S-unimodal such that $L_0$ is strictly contained in X and is not a singleton. Let P be a non-degenerate closed interval in $\mathrm {int}(X)$ . Suppose that either

1. (i) f is renormalizable and $P\subset L_0$ , or

2. (ii) f is non-renormalizable and P is contained in the open interval bordered by $x_f$ and the unique fixed point $z_f$ of f in $\mathrm {int}(X)$ (see (C1) in Figure 1).

For any $\varepsilon>0$ , there exists $N\geq 1$ such that for each integer $n\geq N$ , there exists a pullback B of P by $f^{n}$ that is contained in $P_{-1}$ such that

(a) $$ \begin{align} &\sup_{B}\bigg\|\dfrac{1}{n}S_{n}\vec\phi-\int\vec\phi \,d\mu_{-1}\bigg\|<\varepsilon. \end{align} $$

If, moreover, $f(c)\notin P$ , then the pullback B is diffeomorphic and satisfies

(b) $$ \begin{align} &\sup_{B}\bigg|\dfrac{1}{n}\log|(f^{n})'|-\chi^+(\mu_{-1})\bigg|<\varepsilon. \end{align} $$

3.4 Proof of Proposition 3.1

If $L_0=X$ then f is non-renormalizable, and the conclusion of Proposition 3.1 was already shown in the proof of [Reference Chung, Rivera-Letelier and Takahasi7, Proposition 2.1]. If $L_0$ is a singleton, then $f(c)=c$ and any element of $\mathcal M(f)$ is a convex combination of $\mu _{-1}$ and the unit point mass at c. Hence, it is not difficult to show the conclusion of Proposition 3.1 by slightly modifying the argument in Case 2 below.

We assume $L_0$ is strictly contained in X and not a singleton. Let $\mu \in \mathcal M(f)$ satisfy $\int \vec {\phi } \,d\mu>\vec \alpha $ . Let $\varepsilon>0$ be small enough so that

(3.5) $$ \begin{align}\int\vec{\phi}\,d\mu-\vec\varepsilon>\vec{\alpha}.\end{align} $$

In what follows we treat four cases separately. We start with the case where f is infinitely renormalizable. Arguments for the remaining cases will proceed pretty much in parallel to the infinitely renormalizable case.

Case 1: $m(f)=\infty $ . Let $m_*$ , $N_*$ be positive integers for which the conclusion of Lemma 2.9 holds with $\varepsilon $ replaced by $\varepsilon /3$ . By the continuity of the Lyapunov exponent in Lemma 2.10, we may assume with no loss of generality that there exist constants ${\rho _{-1},\ldots ,\rho _{m_*}\in (0,1)}$ that add up to $1$ , and for each $m\in \{-1,\ldots ,m_*-1\}$ a measure $\mu _m\in \mathcal M_{m}(f)$ such that

$$ \begin{align*} \mu=\rho_{-1}\cdot\mu_{-1}+\rho_{0}\cdot\mu_{0}+\cdots+\rho_{m_*-1}\cdot\mu_{m_*-1}+\rho_{m_*} \cdot\mu_\infty. \end{align*} $$

For each $m\in \{0,\ldots ,m_*-1\}$ we take a non-degenerate closed interval $P_m$ so that the following statements hold: if $p_{m+1}/p_{m}\neq 2$ then $P_m\in \mathscr {P}_{m}$ , $P_m\subset L_{m}$ and $\mu _m(P_m)>0$ ; if $p_{m+1}/p_{m}= 2$ then $P_m\subset L_{m}$ , $P_m$ contains the periodic point in $\partial J_{m+1}$ in its interior and satisfies $P_{m}\cap \{f(c),\ldots ,f^{M_{m-1}-1}(c)\}=\emptyset $ (recall (2.5)). We set $P_{m_*}=J_{m_*}$ .

For each $m\in \{-1,\ldots , m_*-1\}$ and for sufficiently large $q\in \mathbb N$ , we define a collection $\mathscr {B}_{p_{m}q}(P_m,P_{m+1})$ of non-degenerate closed subintervals of $P_m$ as follows.

• • $m=-1$ : $\mathscr {B}_{p_{-1}q}(P_{-1},P_{0})$ is the singleton consisting of the pullback of $P_0$ by $f^{p_{-1}q}$ that is obtained by applying Lemma 3.4 with $P=P_0$ . Since $f(c)\notin P_0$ by the choice of $P_0$ , this pullback is diffeomorphic by Lemma 3.4.

• • $m\geq 0$ : if $p_{m+1}/p_{m}\neq 2$ then $\mathscr {B}_{p_{m}q}(P_m,P_{m+1})$ is the collection of the diffeomorphic pullbacks of $P_{m+1}$ by $f^{p_{m}q}$ contained in $P_m$ that are obtained by applying Lemma 3.3 with $\mu =\mu _m$ , $P=P_m$ , $P'=P_{m+1}$ and $\varepsilon $ replaced by $\varepsilon /2$ ; if $p_{m+1}/p_{m}=2$ then $\mathscr {B}_{p_{m}q}(P_m,P_{m+1})$ is the singleton consisting of the diffeomorphic pullback of $P_{m+1}$ by $f^{p_{m}q}$ that is contained in $P_m$ and closest to the periodic point in $\partial J_{m+1}$ among all such diffeomorphic pullbacks.

We claim that for every $-1\leq m\leq m_*-1$ the following inequalities hold provided q is sufficiently large:

(3.6) $$ \begin{align}\qquad\ \ \quad\qquad \bigg|\frac{1}{p_{m}q}\log\# \mathscr{B}_{p_{m}q}(P_m,P_{m+1})-h(\mu_m)\bigg| & <\frac{\varepsilon}{2}; \end{align} $$

(3.7) $$ \begin{align}\qquad \sup_{B\in\mathscr{B}_{p_{m}q}(P_m,P_{m+1})}\sup_{B}\bigg\|\frac{1}{p_{m}q}S_{p_{m}q}\vec\phi-\int\vec\phi \,d\mu_m\bigg\| & <\frac{\varepsilon}{2}; \end{align} $$

(3.8) $$ \begin{align} \sup_{B\in\mathscr{B}_{p_{m}q}(P_m,P_{m+1})}\sup_{B}\bigg|\frac{1}{p_{m}q}\log|(f^{p_{m}q})'|-\chi^+(\mu_m)\bigg| & <\frac{\varepsilon}{2}. \end{align} $$

Indeed, for $m=-1$ , (3.6) is trivial since $\mu _{-1}$ is the unit point mass at the fixed point of f in $\partial X$ , and (3.7) and (3.8) are consequences of Lemma 3.4. Let $m\geq 0$ . If $p_{m+1}/p_{m}=2$ then $\mu _m$ is the empirical measure on the orbit of the hyperbolic repelling periodic point in $\partial J_{m+1}$ . Therefore, (3.6), (3.7) and (3.8) hold in these two cases. If $p_{m+1}/p_{m}\neq 2$ then (3.6), (3.7) and (3.8) are consequences of Lemma 3.3.

We write $n\in \mathbb N$ as a linear combination of non-negative integers $q_{-1},\ldots ,q_{m_*-1},r$ in the form

$$ \begin{align*} n= p_{-1}q_{-1}+p_0q_0+\cdots+p_{m_*-1}q_{m_*-1}+r, \end{align*} $$

with $|p_{m}q_m- \rho _m n|\le 1$ for $-1\leq m\leq m_*-1$ and $|r- \rho _{m_*} n|\le 1.$ In what follows we assume n is sufficiently large so that $r\geq N_*$ , and for each $-1\leq m\leq m_*-1$ , $\mathscr {B}_{p_{m}q_m}(P_m,P_{m+1})$ satisfies (3.6), (3.7) and (3.8). Lemma 2.9(b) gives

(3.9) $$ \begin{align} \sup_{K_{m_*}}\bigg\|\dfrac{1}{r}S_r\vec\phi-\int\vec\phi \,d\mu_\infty\bigg\|<\frac{\varepsilon}{2}. \end{align} $$

Let $\mathscr {A}_n$ denote the collection of the pullbacks of $P_{m_*}$ by maps $h_{m_*-1}\circ \cdots \circ h_{-1}$ , where each $h_m$ is a diffeomorphism of the form $f^{p_{m}q_m}|_{B}$ with $B\in \mathscr {B}_{p_{m}q_m}(P_m,P_{m+1})$ . By $\#\mathscr {A}_n=\prod _{m=-1}^{m_*-1}\#\mathscr {B}_{p_{m}q_m}(P_m,P_{m+1})$ , (3.6) and $h(\mu _{\infty })=0$ , we have

(3.10) $$ \begin{align} \bigg| \dfrac{1}{n}\log\#\mathscr{A}_n-h(\mu) \bigg|<\varepsilon. \end{align} $$

A similar argument to the proofs of (b) and (c) in Lemma 3.3 based on (3.7) and (3.9) shows that

(3.11) $$ \begin{align} \sup_{A\in\mathscr{A}_n}\sup_A\bigg\| \dfrac{1}{n}S_{n}\vec\phi-\int\vec\phi \,d\mu \bigg\|<\varepsilon. \end{align} $$

From $\chi ^+(\mu _\infty )=0$ , $\chi ^+(\mu )=\sum _{m=-1}^{m_*-1}\rho _m \chi ^+(\mu _m)$ by Lemma 2.1 and (3.8), we obtain

(3.12) $$ \begin{align} \sup_{A\in\mathscr{A}_n}\sup_A\bigg| \dfrac{1}{n-r}\log|(f^{n-r})'|-\chi^+(\mu) \bigg|<\frac{\varepsilon}{2}. \end{align} $$

The inequality in Proposition 3.1(a) follows from (3.10). From (3.5) and (3.11), the elements of $\mathscr {A}_n$ are contained in $A_{n}(\vec {\phi },\vec {\alpha })$ . The inequality in Proposition 3.1(b) follows from (3.12).

Case 2: $m(f)<\infty $ and $\mathrm {int}(J_{m(f)})$ contains a two-sided attracting periodic point (Remark 2.3(C-I)). Recall that $\delta _{O(f)}$ denotes the element of $\mathcal M(f)$ supported on $O(f)$ . With no loss of generality we may assume there exist constants $\rho _{-1},\ldots ,\rho _{m(f)}\in (0,1)$ adding up to $1$ and for each $m\in \{-1,\ldots ,m(f)-1\}$ a measure $\mu _m\in \mathcal M_{m}(f)$ such that

$$ \begin{align*}\mu=\rho_{-1}\cdot\mu_{-1}+\rho_0\cdot\mu_0+ \cdots+\rho_{m(f)}\cdot\delta_{O(f)}. \end{align*} $$

For each $m\in \{0,\ldots ,m(f)\}$ we take a non-degenerate closed interval $P_m$ so that the following assertions hold: if $m<m(f)$ and $p_{m+1}/p_{m}\neq 2$ then $P_m\in \mathscr {P}_{m}$ , $P_m\subset L_{m}$ and $\mu _m(P_m)>0$ ; if $m<m(f)$ and $p_{m+1}/p_{m}= 2$ then $P_m\subset L_m$ , $P_m$ contains the periodic point in $\partial J_{m+1}$ in its interior and satisfies $P_{m}\cap \{f(c),\ldots ,f^{M_{m-1}-1}(c)\}=\emptyset $ ; $P_{m(f)}$ is contained in the connected component of the immediate basin of the attracting periodic orbit that contains c. In the case of $m(f)=0$ , we take $P_{m(f)}$ to be a closed interval contained in the open interval bordered by $x_f$ and the unique fixed point in $z_f$ .

For each $m\in \{-1,\ldots , m(f)-1\}$ and for sufficiently large $q\in \mathbb N$ , we define a collection $\mathscr {B}_{p_{m}q}(P_m,P_{m+1})$ of non-degenerate closed subintervals of $P_m$ as follows.

• • $m=-1$ : $\mathscr {B}_{p_{-1}q}(P_{-1},P_{0})$ is the singleton consisting of the pullback of $P_0$ by $f^{p_{-1}q}$ that is obtained by applying Lemma 3.4 with $P=P_0$ . Since $f(c)\notin P_0$ by the choice of $P_0$ , this pullback is diffeomorphic by Lemma 3.4.

• • $m\geq 0$ : if $p_{m+1}/p_{m}\neq 2$ then $\mathscr {B}_{p_{m}q}(P_m,P_{m+1})$ is the collection of the diffeomorphic pullbacks of $P_{m+1}$ by $f^{p_{m}q}$ contained in $P_m$ that are obtained by applying Lemma 3.3 with $\mu =\mu _m$ , $P=P_m$ , $P'=P_{m+1}$ and $\varepsilon $ replaced by $\varepsilon /2$ ; if $p_{m+1}/p_{m}=2$ then $\mathscr {B}_{p_{m}q}(P_m,P_{m+1})$ is the singleton consisting of the diffeomorphic pullback of $P_{m+1}$ by $f^{p_{m}q}$ that is contained in $P_m$ and closest to the periodic point in $\partial J_{m+1}$ among all such diffeomorphic pullbacks.

Note that the second case $m\geq 0$ occurs only if $m(f)\geq 1$ .

Let $n\geq 1$ be an integer, and write it as a linear combination of positive integers $q_1,\ldots ,q_{m(f)-1}$ in the form

$$ \begin{align*} n= p_{-1}q_{-1}+\cdots+p_{m(f)-1}q_{m(f)-1}+r, \end{align*} $$

with $|p_{m}q_m- \rho _m n|\le 1$ for $-1\leq m\leq m(f)-1$ and $|r- \rho _{m(f)} n|\le 1.$ In what follows we assume n is sufficiently large so that for each $m\in \{-1,\ldots ,m(f)-1\}$ , $\mathscr {B}_{p_{m}q_m}(P_m,P_{m+1})$ satisfies (3.6) $^{\prime }$ , (3.7) $^{\prime }$ and (3.8) $^{\prime }$ which are exactly the same as (3.6), (3.7) and (3.8) respectively under the different assumption on f, and

(3.13) $$ \begin{align} \sup_{P_{m(f)}}\bigg\|\frac{1}{r }S_{r }\vec\phi-\int\vec\phi \,d\delta_{O(f)}\bigg\|<\frac{\varepsilon}{2}. \end{align} $$

Let $\mathscr {A}_n$ denote the collection of the pullbacks of $P_{m(f)}$ by maps $h_{m(f)-1}\circ \cdots \circ h_0\circ h_{-1}$ where $h_m$ for $-1\leq m\leq m(f)-1$ is a diffeomorphism of the form $f^{p_{m}q}|_{B}$ with $B\in \mathscr {B}_{p_{m}q}(P_m,P_{m+1})$ . From $\#\mathscr {A}_n=\prod _{m=-1}^{m(f)-1}\#\mathscr {B}_{p_{m}q_m}(P_{m},P_{m+1})$ and (3.6) $^{\prime }$ we obtain Proposition 3.1(a). From (3.5), (3.7) $^{\prime }$ and (3.13), the elements of $\mathscr {A}_n$ are contained in $A_{n}(\vec {\phi },\vec {\alpha })$ . Since the measure $\delta _{O(f)}$ is supported on $O(f)$ , we have $\chi ^+(\delta _{O(f)})=0$ from the definition (1.1) of the Lyapunov exponent and $\chi ^+(\mu )=\sum _{m=-1}^{m(f)-1}\rho _m\chi ^+(\mu _m)$ by Lemma 2.1. Hence, (3.8) $^{\prime }$ implies Proposition 3.1(b).

Case 3: $m(f)<\infty $ and $\partial J_{m(f)}$ contains a one-sided attracting periodic point (Remark 2.3(C-II)). This case is treated by a slight modification of the argument for Case 2.

Case 4: $m(f)<\infty $ and f has no attracting periodic orbit. We may assume there exist constants $\rho _{-1},\ldots ,\rho _{m(f)}\in (0,1)$ adding up to $1$ , and for each $m\in \{-1,\ldots ,m(f)\}$ a measure $\mu _m\in \mathcal M_{m}(f)$ such that

$$ \begin{align*} \mu=\rho_{-1}\cdot\mu_{-1}+\rho_0\cdot\mu_0+\cdots+\rho_{m(f)}\cdot\mu_{m(f)}. \end{align*} $$

Set $g=f^{p_{m(f)}}|_{L_{m(f)}}$ , and let $\nu $ denote the normalized restriction of $\mu _{m(f)}$ to $L_{m(f)}$ . Let $h(g,\nu )$ , $\chi ^+(g,\nu )$ denote the entropy and Lyapunov exponent of the measure $\nu $ with respect to g. By Lemma 2.1 we have $\chi ^+(g,\nu )=\int \log |Dg|d\nu $ . As in §2.3, g is topologically exact. From the proof of [Reference Chung, Rivera-Letelier and Takahasi7, Proposition 2.1], for all sufficiently large $q\geq 1$ there exist a closed interval $P_{m(f)}$ in $L_{m(f)}$ and a finite collection $\mathscr {B}_{p_{m(f)}q}(P_{m(f)},P_{m(f)})$ of diffeomorphic pullbacks of $P_{m(f)}$ by $g^q$ that are contained in $P_{m(f)}$ such that

(3.14) $$ \begin{align} \#\mathscr{B}_{p_{m(f)}q}(P_{m(f)},P_{m(f)})&>\exp\bigg(\bigg( h(g,\nu)-\frac{\varepsilon}{2}\bigg)q\bigg) \nonumber \\&=\exp\bigg(\bigg(p_{m(f)}h(\mu_{m(f)})-\frac{\varepsilon}{2}\bigg)q\bigg), \end{align} $$

and for every $B\in \mathscr {B}_{p_{m(f)}q}(P_{m(f)},P_{m(f)})$ ,

(3.15) $$ \begin{align} \sup_{ B}&\bigg\|\dfrac{1}{q}\sum_{k=0}^{q-1}\vec \phi\circ g^k-\int\vec\phi \,d\nu\bigg\|=\sup_{ B}\bigg\|\dfrac{1}{q}S_{p_{m(f)}q}\vec\phi-p_{m(f)}\int\vec\phi \,d\mu_{m(f)}\bigg\|<\frac{\varepsilon}{2} \end{align} $$

and

(3.16) $$ \begin{align} |B|>\exp\bigg(\!-\bigg(\chi^+(g,\nu)+\frac{\varepsilon}{2}\bigg)q\bigg)=\exp\bigg(\!-\bigg(p_{m(f)}\chi^+(\mu_{m(f)})+\frac{\varepsilon}{2}\bigg)q\bigg). \end{align} $$

For each $m\in \{0,\ldots , m(f)-1\}$ , we take a non-degenerate closed interval $P_m$ such that the following assertions hold: if $p_{m+1}/p_{m}\neq 2$ then $P_m\in \mathscr {P}_{m}$ , $P_m\subset L_{m}$ and $\mu _m(P_m)>0$ : if $p_{m+1}/p_{m}= 2$ then $P_m\subset L_m$ , $P_m$ contains the periodic point in $\partial J_{m+1}$ in its interior and satisfies $P_{m}\cap \{f(c),\ldots ,f^{M_{m-1}-1}(c)\}=\emptyset $ .

For each $m\in \{-1,\ldots , m(f)-1\}$ and for sufficiently large $q\in \mathbb N$ , we define a collection $\mathscr {B}_{p_{m}q}(P_m,P_{m+1})$ of non-degenerate closed subintervals of $P_m$ as follows.

• • $m=-1$ : $\mathscr {B}_{p_{-1}q}(P_{-1},P_{0})$ is the singleton consisting of the pullback of $P_0$ by $f^{p_{-1}q}$ that is obtained by applying Lemma 3.4 with $P=P_0$ . Since $f(c)\notin P_0$ by the choice of $P_0$ , this pullback is diffeomorphic by Lemma 3.4.

• • $m\geq 0$ : if $p_{m+1}/p_{m}\neq 2$ then $\mathscr {B}_{p_{m}q}(P_m,P_{m+1})$ is the collection of the diffeomorphic pullbacks of $P_{m+1}$ by $f^{p_{m}q}$ contained in $P_m$ obtained by applying Lemma 3.3 with $\mu =\mu _m$ , $P=P_m$ , $P'=P_{m+1}$ and $\varepsilon $ replaced by $\varepsilon /2$ ; if $p_{m+1}/p_{m}=2$ then $\mathscr {B}_{p_{m}q}(P_m,P_{m+1})$ is the singleton consisting of the diffeomorphic pullback of $P_{m+1}$ by $f^{p_{m}q}$ that is contained in $P_m$ and closest to the periodic point in $\partial J_{m+1}$ among all such diffeomorphic pullbacks.

Note that the second case $m\geq 0$ occurs only if $m(f)\geq 1$ .

Let $n\geq 1$ be an integer that is written as a linear combination of non-negative integers $q_{-1},\ldots ,q_{m(f)},r$ in the form

$$ \begin{align*} n= p_{-1}q_{-1}+\cdots+p_{m(f)-1}q_{m(f)-1}+p_{m(f)}q_{m(f)}+r, \end{align*} $$

with $|p_{m}q_m- \rho _m n|\le 1$ for $-1\leq m\leq m(f)$ and $0\leq r<p_{m(f)}$ . We assume n is sufficiently large so that for each m, $\mathscr {B}_{p_{m}q_m}(P_{m},P_{m+1})$ satisfies (3.6) $^{\prime \prime }$ , (3.7) $^{\prime \prime }$ and (3.8) $^{\prime \prime }$ which are exactly the same as (3.6), (3.7) and (3.8) respectively under the different assumption on f, and $\mathscr {B}_{p_{m(f)}q_{m(f)}}(P_{m(f)},P_{m(f)})$ satisfying (3.14), (3.15) and (3.16) exists. We set $P_{m(f)+1}=P_{m(f)}$ for convenience. Define $\mathscr {A}_n$ to be the collection of the pullbacks of $P_{m(f)}$ by maps $ h_{m(f)}\circ \cdots \circ h_0\circ h_{-1}$ , where each $h_m$ , $-1\leq m\leq m(f)$ is a diffeomorphism of the form $f^{p_{m} q_m}|_{B}$ with $B\in \mathscr {B}_{p_{m}q_m}(P_{m},P_{m+1})$ . From $\#\mathscr {A}_n=\prod _{m=-1}^{m(f)}\#\mathscr {B}_{p_{m}q_m}(P_{m},P_{m+1})$ , (3.6) $^{\prime \prime }$ and (3.14) we obtain the inequality in Proposition 3.1(a). From (3.5), (3.7) $^{\prime \prime }$ and (3.15), the elements of $\mathscr {A}_n$ are contained in $A_{n}(\vec {\phi },\vec {\alpha })$ . By Lemma 2.1 we have $\chi ^+(\mu )=\sum _{m=-1}^{m(f)}\rho _m\chi ^+(\mu _m)$ . From (3.8) $^{\prime \prime }$ and (3.16) we obtain the inequality in Proposition 3.1(b). The proof of Proposition 3.1 is complete.

4.1 Key upper bound

Let $\ell \in \mathbb N$ , $\vec {\phi }\in C(X)^{ \ell }$ , $\vec {\alpha }\in \mathbb R^{\ell }$ . Define

$$ \begin{align*} \bar {A}_n(\vec{\phi},\vec{\alpha})= \bigg\{x\in X: \int\vec\phi \,d\delta_x^n\geq\vec\alpha\bigg\}. \end{align*} $$

Proposition 4.1. Let $f\colon X\to X$ be an S-unimodal map with a non-flat critical point. Let $\ell \in \mathbb N$ , $\vec {\phi }\in C(X)^{ \ell }$ , $\vec {\alpha }\in \mathbb R^{\ell }$ . For any $\varepsilon>0$ there exists $N\in \mathbb N$ such that if $n\geq N$ and $\bar {A}_n(\vec {\phi },\vec {\alpha })\neq \emptyset $ , then there exists a measure $\mu \in \mathcal M(f)$ such that

(a) $$ \begin{align} |\bar {A}_n(\vec{\phi},\vec{\alpha})|&\leq \exp ((F(\mu)+\varepsilon)n)\quad and \end{align} $$

(b) $$ \begin{align} \int\vec\phi \,d\mu&>\vec\alpha-\vec\varepsilon. \qquad\qquad\ \ \qquad \end{align} $$

We finish the proof of the upper bound in Theorem A assuming Proposition 4.1.

Proof of the upper bound in Theorem A

Let $\mathcal C$ be a non-empty closed subset of $\mathcal M$ . Let $\mathcal G$ be an arbitrary open set containing $\mathcal C$ . Since $\mathcal M$ is metrizable and $\mathcal C$ is compact, we can choose $\varepsilon>0$ and finitely many closed sets $\mathcal {C}_1,\ldots ,\mathcal C_s$ of the form $\mathcal C_i=\{\mu \in \mathcal M\colon \int \vec \phi _i \,d\mu \geq \vec \alpha _i\}$ with $\ell _i\in \mathbb N$ , $\vec \phi _i\in C(X)^{ \ell _i}$ , $\vec \alpha _i\in \mathbb R^{\ell _i}$ for $1\leq i\leq s$ so that

(4.1) $$ \begin{align} \mathcal C \subset \bigcup_{i=1}^s \mathcal C_i \subset \bigcup_{i=1}^s \mathcal C_i(\varepsilon)\subset \mathcal G, \end{align} $$

where $\mathcal C_i(\varepsilon )=\{\mu \in \mathcal M\colon \int \vec \phi _i \,d\mu>\vec \alpha _i-\vec {\varepsilon }\}$ . Since $F(\mu )\leq -I(\mu )$ for $\mu \in \mathcal M$ , Proposition 4.1 implies

$$ \begin{align*} \limsup_{n\to\infty}\dfrac{1}{n}\log |\{x\in X:\delta_x^n\in\mathcal C_i\}|\leq -\inf_{\mathcal C_i(\varepsilon)}I +\varepsilon\quad \text{for }1\leq i\leq s. \end{align*} $$

These inequalities and (4.1) give

$$ \begin{align*} \limsup_{n\to\infty}\dfrac{1}{n}\log |\{x\in X:\delta_x^n\in \mathcal C\}| \le \max_{1\leq i\leq s} \Big(\!-\inf_{\mathcal C_i(\varepsilon)}I\Big)+\varepsilon \le -\inf_{\mathcal G}I+\varepsilon. \end{align*} $$

Since $\varepsilon>0$ is arbitrary and $\mathcal G$ is an arbitrary open set containing $\mathcal C$ , it follows that

$$ \begin{align*} \limsup_{n\to\infty}\dfrac{1}{n}\log |\{x\in X:\delta_x^n\in \mathcal C\}|\leq \inf_{\mathcal G \supset \mathcal C} \Big(-\inf_{\mathcal G} I\Big)= -\inf_{\mathcal C}I,\end{align*} $$

as required. The last equality is due to the lower semicontinuity of I.

Standing hypotheses from §4.2 to §4.4

The rest of this section is entirely dedicated to the proof of Proposition 4.1. We assume $f\colon X\to X$ is an S-unimodal map with a non-flat critical point c, and $\ell \in \mathbb N$ , $\vec {\phi }\in C(X)^{\ell }$ , $\vec {\alpha }\in \mathbb R^{\ell }$ in the statement of Proposition 4.1 are fixed till the end of §4.4.

4.2 Escape estimate in a single renormalization cycle

Suppose $m(f)\geq 1$ , and let m be an integer with $0\leq m\leq m(f)-1$ . For $\vec \beta \in \mathbb R^{\ell }$ , $P\subset K_{m,m+1}$ and $n\in \mathbb N$ , define

$$ \begin{align*} A_{n}(\vec{\phi},\vec{\beta},P)= \bar {A}_{n}(\vec{\phi},\vec{\beta})\cap P\cap\bigcap_{k=0}^{n-1}f^{-k}(K_{m,m+1}). \end{align*} $$

This subsection is dedicated to proving of the following lemma.

Lemma 4.2. Suppose $m(f)\geq 1$ and let $0\leq m\leq m(f)-1$ be an integer. For any $\varepsilon>0$ there exists $N\geq 1$ such that if $\vec \beta \in \mathbb R^{\ell }$ , $P\in \tilde {\mathscr {P}}_m$ , $n\geq N$ satisfy $P\subset K_{m,m+1}$ and $A_{n}(\vec {\phi },\vec {\beta },P)\neq \emptyset $ , then there exists a measure $\mu \in \mathcal M(f)$ such that

(a) $$ \begin{align} |A_{n}(\vec{\phi},\vec{\beta},P)|&\leq \exp((F(\mu)+\varepsilon)n)|P|\quad and \end{align} $$

(b) $$ \begin{align} \int\vec\phi \,d\mu&>\vec\beta-\vec\varepsilon. \qquad\qquad\ \ \qquad\end{align} $$

Proof. Let $\varepsilon>0$ and let $P\in \tilde {\mathscr {P}}_m$ . Since $P\subset K_{m,m+1}$ , if $p_{m+1}/p_m= 2$ then $f(P)$ contains a point from the orbit of the periodic point in $\partial J_m$ . Then we can verify (a) and (b) easily, taking the measure $\mu $ to be the equidistribution on this periodic orbit. For the rest of this proof, we suppose $p_{m+1}/p_m\neq 2$ , and split the proof into two cases, in which either $P\in \mathscr {P}_m$ or $P\in \tilde {\mathscr {P}}_m\setminus \mathscr {P}_m$ .

Case 1: $P\in \mathscr {P}_m$ . We perform an escape estimate relative to P. Let $n\in \mathbb N$ and suppose $A_{n}(\vec {\phi },\vec {\beta },P)\neq \emptyset $ . Let $\mathscr {P}_{m,n}(P)$ denote the collection of the diffeomorphic pullbacks of elements of $\mathscr {P}_m$ by $f^{n}$ that are contained in P and intersect $\bar {A}_{n}(\vec {\phi },\vec {\beta })$ . Note that every element of $\mathscr {P}_{m,n}(P)$ is contained in $\bigcap _{k=0}^{n-1} f^{-k}(K_{m,m+1})$ , and the union of elements of $\mathscr {P}_{m,n}(P)$ contains $A_{n}(\vec {\phi },\vec {\beta },P)$ .

Sublemma 4.3. For any $\varepsilon>0$ there exists $N\geq 1$ such that for every ${Q\in \mathscr {P}_m}$ , every ${n\geq N}$ and every diffeomorphic pullback W of Q by $f^{n}$ contained in P, there exist $k\in \mathbb Z$ with $|k|\leq p_m-1$ and a diffeomorphic pullback Y of P by $f^{n+k+M_m }$ that is contained in W and satisfies

$$ \begin{align*} |Y|\geq \exp(-\varepsilon n)|W|. \end{align*} $$

Proof. Since P, $Q\in \mathscr {P}_m$ , there exist $k_P$ , $k_Q\in \{0,\ldots ,p_m-1\}$ such that $f^{k_P}(P)$ and $f^{k_Q}(Q)$ are elements of $\mathscr {P}_m$ contained in $L_m$ . By the definition of $M_m$ in (2.5), there exists a diffeomorphic pullback $W_P$ of $f^{k_P}(P)$ by $f^{M_m }$ that is contained in $f^{k_Q}(Q)$ . Let $W^{\prime }_P$ be the diffeomorphic pullback of $W_P$ by $f^{k_Q}$ contained in Q. This is a diffeomorphic pullback of P by $f^{M_m+k_Q-k_P}$ . Let Y be the pullback of $W^{\prime }_P$ by $f^n|_{W}$ . Then Y is a diffeomorphic pullback of P by $f^{n+M_m+k_Q-k_P}$ .

By Proposition 2.7 with $\varepsilon $ replaced by $\varepsilon /2$ , if n is sufficiently large we have

$$ \begin{align*} \frac{|Y|}{|W|} & \geq \exp\bigg(\!-\frac{\varepsilon n}{2}\bigg)\frac{|f^{n}(Y)|}{|f^{n}(W)|}=\exp\bigg(\!-\frac{\varepsilon n}{2}\bigg)\frac{|W_P'|}{|Q|}\\ &\geq \exp\bigg(\!-\frac{\varepsilon n}{2}\bigg)C\frac{|W_{P}|}{|f^{k_Q}(Q)|}\geq \exp(-\varepsilon n) \end{align*} $$

as required, where $C>0$ is a constant independent of P and Q.

In view of Sublemma 4.3, for each $W\in \mathscr {P}_{m,n}(P)$ we fix an integer $k(W)$ with $|k(W)|\leq p_m-1$ and a diffeomorphic pullback $Y_W$ of P by $f^{n+k(W)+M_m}$ that is contained in W and satisfies $|Y_W\hspace{-1pt}|\geq \exp (-\varepsilon n/3)|W|.$ Set

$$ \begin{align*} \mathscr{P}_{m,n,k}(P)=\{W\in \mathscr{P}_{m,n}(P)\colon k(W)=k\}.\end{align*} $$

If n is sufficiently large, then for all $-p_m+1\leq k\leq p_m-1$ we have

(4.2) $$ \begin{align} \bigcup_{W\in \mathscr{P}_{m,n,k}(P)}Y_W\subset A_{n+k+M_m}\bigg(\vec{\phi},\vec{\beta}-\dfrac{1}{2}\vec{\varepsilon}\bigg) \end{align} $$

and

(4.3) $$ \begin{align} |A_{n}(\vec{\phi},\vec{\alpha},P)| \leq\sum_{W\in \mathscr{P}_{m,n}(P)}|W|\leq \exp\bigg(\frac{\varepsilon n}{3}\bigg)\sum_{W\in \mathscr{P}_{m,n}(P)}|Y_W\hspace{-1pt}|. \end{align} $$

Choose $\underline {k}\in \{-p_m+1,\ldots ,p_m-1\}$ such that

(4.4) $$ \begin{align} \sum_{W\in \mathscr{P}_{m,n}(P)}|Y_W\hspace{-1pt}|\leq(2p_m-1)\sum_{W\in \mathscr{P}_{m,n,\underline{k}}(P)}|Y_W\hspace{-1pt}|,\end{align} $$

and set $\tilde n=n+\underline {k}+M_m$ . The restriction of $f^{\tilde n}$ to $\bigcup _{W\in \mathscr {P}_{m,n,\underline {k}}(P)} Y_W$ induces a fully branched expanding Markov map onto P with finitely many branches. Let $\Delta \subset \Gamma _m$ denote its maximal invariant set, namely

$$ \begin{align*} \Delta=\bigcap_{j=0}^\infty (f^{\tilde n})^{-j}\bigg(\!\bigcup_{W\in \mathscr{P}_{m,n,\underline{k}}(P)}Y_W\bigg). \end{align*} $$

The map $f^{\tilde n}|_{\Delta }\colon \Delta \to \Delta $ is topologically conjugate to the full shift $\sigma \colon \Sigma \to \Sigma $ over the finite alphabet $\mathscr {P}_{m,n,\underline {k}}(P)$ . We write $\pi \colon \Sigma \to \Delta $ for the conjugacy map, and define the induced potential $\Phi \colon \Sigma \to \mathbb R$ by $\Phi (\omega )=-\log |(f^{\tilde n})'(\pi (\omega ))|$ . From Lemma 2.6, $\Phi $ is continuous with respect to the shift metric. The variational principle [Reference Bowen2, pp. 40, 2.17] gives

(4.5) $$ \begin{align} \sup_{\tilde\nu\in\mathcal M(\sigma)}\bigg(h(\sigma,\tilde\nu)+ \int\Phi \,d\tilde\nu\bigg)= \lim_{j\to\infty}\dfrac{1}{j}\log\bigg(\!\sum_{\omega\in\sigma^{-j}(\omega')} \exp{\sum_{i=0}^{j-1}\Phi(\sigma^i\omega)}\bigg) \end{align} $$

for any fixed $\omega '\in \Sigma $ , where $\mathcal M(\sigma )$ denotes the space of $\sigma $ -invariant Borel probability measures endowed with the weak* topology, and $h(\sigma ,\tilde \nu )$ denotes the measure-theoretic entropy of $\tilde \nu $ with respect to $\sigma $ . By the distortion estimates in Proposition 2.7, if $n\geq 1$ is sufficiently large then for any $W\in \mathscr {P}_{m,n,\underline {k}}(P)$ and any $\omega \in \Sigma $ such that $\pi (\omega )\in Y_W$ we have

$$ \begin{align*} \exp(\Phi(\omega))\geq \exp\bigg(\!-\frac{\varepsilon n}{2} \bigg)\frac{|Y_W\hspace{-1pt}|}{|P|}. \end{align*} $$

Hence, the series inside the logarithm in (4.5) is bounded from below as follows:

$$ \begin{align*} \sum_{\omega\in \sigma^{-j}(\omega')}\exp\bigg(\!\sum_{i=0}^{j-1}\Phi(\sigma^i\omega)\bigg) &\geq\bigg(\!\inf_{\omega'\in \Sigma}\sum_{\omega\in \sigma^{-1}(\omega')}\exp(\Phi(\omega))\bigg)^j\\ &\geq\bigg(\!\exp\bigg(\!-\frac{\varepsilon n}{2} \bigg)\sum_{W\in \mathscr{P}_{m,n,\underline{k}}(P)}\frac{|Y_W\hspace{-1pt}|}{|P|}\bigg)^j. \end{align*} $$

Taking logarithms of both sides, dividing by j and letting $j\to \infty $ , we have

$$ \begin{align*} \lim_{j\to\infty}\frac{1}{j}\log\bigg(\!\sum_{\omega\in \sigma^{-j}(\omega')} \exp\bigg(\!\sum_{i=0}^{j-1}\Phi(\sigma^i\omega)\bigg)\bigg) \geq\log\sum_{W\in \mathscr{P}_{m,n,\underline{k}}(P) }\frac{|Y_W\hspace{-1pt}|}{|P|}-\frac{\varepsilon n}{2}. \end{align*} $$

Plugging this inequality into (4.5) yields

(4.6) $$ \begin{align}\sup_{\tilde\nu\in\mathcal M(\sigma)}\bigg(h(\sigma,\tilde\nu)+ \int\Phi \,d\tilde\nu\bigg)\geq \log\sum_{W\in\mathscr{P}_{m,n,\underline{k}}(P)}\frac{|Y_W\hspace{-1pt}|}{|P|}-\frac{\varepsilon n}{2}. \end{align} $$

Since $\mathcal M(\sigma )$ is compact in the weak* topology and the entropy function on it is upper semicontinuous, there exists a measure $\tilde \mu \in \mathcal M(\sigma )$ that attains the supremum in (4.6). The measure $\tilde \mu \circ \pi ^{-1}$ on $\Delta $ is $f^{\tilde n}$ -invariant and its spread

$$ \begin{align*} \mu = \frac{1}{\tilde n} \sum_{W\in \mathscr{P}_{m,n,\underline{k}}(P) }\sum_{j=0}^{\tilde n-1} (\tilde\mu|_{Y_W})\circ (f^j\circ\pi)^{-1} \end{align*} $$

belongs to $\mathcal M_m(f)$ . Although $f^{\tilde n}$ may not be the first return map to $\Delta $ , Abramov’s formula, connecting the entropies of $\mu $ and $\tilde \mu \circ \pi ^{-1}$ , and Kac’s formula, connecting the integrals of $-\log |f|$ and $\Phi $ , still hold [Reference Pesin and Senti25, Theorem 2.3] and we have

(4.7) $$ \begin{align} h(\sigma,\tilde\mu)+\int\Phi \,d\tilde\mu=F(\mu)\tilde n. \end{align} $$

From (4.6) and (4.7) we have

$$ \begin{align*} \sum_{W\in \mathscr{P}_{m,n,\underline{k}}(P) } |Y_W\hspace{-1pt}|\leq \exp\bigg(F(\mu)\tilde n+\frac{\varepsilon n}{2 }\bigg)|P|. \end{align*} $$

From this inequality, (4.3) and (4.4), for all sufficiently large $n\geq 1$ we obtain

$$ \begin{align*} |A_{n}(\vec{\phi},\vec{\beta},P)| &\leq (2p_m-1)\exp\bigg(\frac{\varepsilon n}{3} \bigg)\sum_{W\in \mathscr{P}_{m,n,\underline{k}}(P) }|Y_W\hspace{-1pt}|\\ &\leq \exp((F(\mu)+\varepsilon)n)|P|, \end{align*} $$

as required in Lemma 4.2(a). The inequality in Lemma 4.2(b) follows from (4.2).

Case 2: $P\in \tilde {\mathscr {P}}_m\setminus \mathscr {P}_m$ . Let z denote the periodic point of period $p_m$ in $\partial J_m$ . Then z is hyperbolic repelling, and $f^{p_m}$ maps the point in $\partial J_m\setminus \{z\}$ to z. For simplicity we assume $z\in P$ . Otherwise, $z\in f^{p_m}(P)$ holds and so the argument is analogous.

Let $n\in \mathbb N$ satisfy $A_n(\vec {\phi },\vec {\beta },P)\neq \emptyset $ . Our strategy is to use the simplest version of the coarse graining to be formally introduced in §4.3: we begin by splitting $A_{n}(\vec {\phi },\vec {\beta },P)$ into two subsets, one consisting of points which remain in $\bigcup _{k=0}^{p_m-1}f^k(P)$ almost until time n, and the complement of this set. For the first set, the influence of the dynamics near the set $\Gamma _m$ is negligible, and a local analysis near the orbit of z suffices for the estimate of its Lebesgue measure. The second set is influenced by the dynamics near the orbit of z and the dynamics near $\Gamma _m$ . We estimate the Lebesgue measure of each set separately, and unify the estimates at the end to obtain the desired one in Lemma 4.2.

For each $t\in \mathbb N$ , we put

$$ \begin{align*} V_{t}=P\cap\bigg(f^{-t}\bigg(\!\bigcup_{Q\in\mathscr{P}_m} Q\bigg)\setminus\bigcup_{k=0}^{t-1} f^{-k}\bigg(\!\bigcup_{Q\in\mathscr{P}_m} Q\bigg)\bigg), \end{align*} $$

and split $|A_{n}(\vec {\phi },\vec {\beta },P) |=I+II$ where

$$ \begin{align*} I=\sum_{t=1}^{n }|A_{n}(\vec{\phi},\vec{\beta},V_t)|\quad \text{and}\quad II=\sum_{ t= n+1 }^\infty|A_{n}(\vec{\phi},\vec{\beta},V_t)|. \end{align*} $$

The set $V_t$ is non-empty if and only if t is an integer multiple of $p_m$ . If $V_t\neq \emptyset $ then $f^t(V_t)$ contains some element of $\mathscr {P}_m$ , and thus

(4.8) $$ \begin{align} |f^{t}(V_t)|\geq\min\{|Q|\colon Q\in\mathscr{P}_{m}\}>0. \end{align} $$

Since the restriction of $f^t$ to the smallest closed interval that contains $V_t$ and z extends to a diffeomorphism on an open interval, the distortion of $f^{t}$ on the former interval is bounded by a constant $C>0$ that depends only on f. Then we have

(4.9) $$ \begin{align} |V_t\hspace{-1pt}| \leq C\exp(F(\delta_z^{p_m})t)|P|. \end{align} $$

By (4.9) we obtain

(4.10) $$ \begin{align} II\leq\sum_{t=n+1}^\infty|V_t\hspace{-1pt}|\leq \frac{C\exp(F(\delta_z^{p_m})(n+1))}{1-\exp(F(\delta_z^{p_m})p_m)}|P|. \end{align} $$

In order to estimate the sum I, for each $t\in \{1,\ldots , n\}$ let $M_{2,\ell }(t,\mathbb Z_\varepsilon )$ denote the set of $2\times \ell $ matrices $(\beta _{ij})_{i=0,1,1\leq j\leq \ell }$ with entries in $\{(\varepsilon /3) a\colon a\in \mathbb Z\}$ for which the following inequalities hold:

(4.11) $$ \begin{align} \beta_{0j}t+\beta_{1j}(n-t)&>\bigg(\beta_j-\frac{\varepsilon}{3}\bigg)n\quad \text{for }1\leq j\leq\ell, \quad\qquad\end{align} $$

(4.12) $$ \begin{align} \inf\phi_j-\frac{\varepsilon}{3}&<\beta_{ij}\leq \sup\phi_j\quad \text{for }i=0,1\text{ and } 1\leq j\leq\ell. \end{align} $$

For $\mathbf B=(\beta _{ij})\in M_{2,\ell }(t,\mathbb Z_\varepsilon )$ and $i=0,1$ , put $\vec {\beta }_i=(\beta _{i1},\ldots ,\beta _{i\ell })$ and define

$$ \begin{align*} A_{n}(t,\mathbf B)=\bigg\{x\in V_t\colon \dfrac{1}{t}S_{t}\vec\phi(x)\geq\vec\beta_0 \text{ and } \frac{1}{n-t}S_{n-t}\vec\phi(f^{t}(x))\geq\vec\beta_1\bigg\}. \end{align*} $$

We claim that

(4.13) $$ \begin{align} A_{n}(\vec{\phi},\vec\beta,V_t)\subset \bigcup_{\mathbf B\in M_{2,\ell}(t,\mathbb Z_\varepsilon)}A_n(t,\mathbf B). \end{align} $$

Indeed, for each $x\in A_n(\vec {\phi },\vec {\beta },V_t)$ there exists $\mathbf B=(\beta _{ij})\in M_{2,\ell }(t,\mathbb Z_\varepsilon )$ such that for $1\leq j\leq \ell $ we have

$$ \begin{align*} \beta_{0j}\leq \frac{1}{t }S_{t}\phi_j(x)<\beta_{0j}+\frac{\varepsilon}{3}\quad \text{and}\quad \beta_{1j}\leq \frac{1}{n-t }S_{n-t}\phi_j(f^{t}(x))<\beta_{1j}+\frac{\varepsilon}{3}. \end{align*} $$

This implies (4.11) and (4.12), and hence $x\in A_n(t,\mathbf B)$ , which verifies (4.13).

Let $\delta>0$ be such that

(4.14) $$ \begin{align} \|\vec\phi(x)-\vec\phi(y)\|<\frac{\varepsilon}{7}\quad \text{for } x,y\in X \text{ with } |x-y|<\delta. \end{align} $$

In view of what was proved in Case 1 and Lemma 2.6, for the rest of the proof in Case 2 we assume $N\geq 1$ is a sufficiently large integer for which the following assertions hold.

1. (D1) If $n\geq N$ then the inequalities in (a) and (b) of Lemma 4.2 hold for every element of $\mathscr {P}_m$ with $\varepsilon $ replaced by $\varepsilon /4$ .

2. (D2) For every connected component W of $\bigcap _{k=0}^{N-1}f^{-k}(K_{m,m+1})$ we have $|W|<\delta $ .

To finish the proof in Case 2, for all sufficiently large n we establish the inequalities in (a) and (b) of Lemma 4.2 for every element of $\tilde {\mathscr {P}}_m\setminus \mathscr {P}_m$ .

Let $n\geq N$ and let $t\in \{1,\ldots ,n-N+1\}$ . For each matrix $\mathbf B=(\beta _{ij})\in M_{2,\ell }(t,\mathbb Z_\varepsilon )$ with $A_n(t,\mathbf B)\neq \emptyset $ , we estimate the Lebesgue measure $|A_n(t,\mathbf B)|$ from above. Let $\mathscr {P}_m(t,\mathbf B)$ denote the collection of the diffeomorphic pullbacks of elements of $\mathscr {P}_m$ by $f^{n-t}$ that intersect $f^{t}(A_n(t,\mathbf B))$ . The elements of $\mathscr {P}_m(t,\mathbf B)$ intersect $\bar {A}_{n-t}(\vec {\phi },\vec \beta _1)$ and their union contains $f^{t}(A_n(t,\mathbf B))$ , because

$$ \begin{align*}f^{t}(A_{n}(t,\mathbf B))\subset \bar {A}_{n-t}( \vec{\phi},\vec\beta_1)\cap\bigcap_{k=0}^{n-t-1}f^{-k}(K_{m,m+1}). \end{align*} $$

By (D1), for each $Q\in \mathscr {P}_m(t,\mathbf B)$ there exists $\mu _{t,Q}\in \mathcal M_m(f)$ such that

(4.15) $$ \begin{align} |\bar {A}_{n-t}(\vec{\phi},\vec\beta_1)\cap Q |&\leq\exp\bigg(\bigg(F(\mu_{t,Q})+\frac{\varepsilon}{4}\bigg)(n-t)\bigg)|Q|\quad\text{and} \end{align} $$

(4.16) $$ \begin{align} \int\vec\phi \,d\mu_{t,Q}&>\vec\beta_1-\dfrac{1}{4}\vec\varepsilon. \qquad\qquad\qquad\ \ \quad\qquad \end{align} $$

Pick a measure $\mu _{t,\mathbf B}$ in the finite set $\{\mu _{t,Q}\colon Q\in \mathscr {P}_m(t,\mathbf B)\}$ that maximizes the free energy within this finite set. By (4.15) and $\sum _{Q\in \mathscr {P}_m(t,\mathbf B)}|Q|\leq |X|$ we have

$$ \begin{align*} | f^{t}(A_n(t,\mathbf B ))|&\leq \sum_{Q\in\mathscr{P}_m(t,\mathbf B) } |\bar {A}_{n-t}(\vec{\phi},\vec\beta_1)\cap Q|\\&\leq\exp\bigg(\bigg(F(\mu_{t,\mathbf B})+\frac{\varepsilon}{4}\bigg)(n-t)\bigg)|X|. \end{align*} $$

Combining this inequality with (4.8), we obtain

(4.17) $$ \begin{align} \frac{|A_n(t,\mathbf B ) |}{|V_t|}&\leq C \frac{| f^{t}(A_n(t,\mathbf B)) |}{|f^{t}(V_t)|}\leq \frac{C|X|}{|f^{t}(V_t)| }\exp\bigg(\bigg(F(\mu_{t,\mathbf B})+\frac{\varepsilon}{4}\bigg)(n-t)\bigg). \end{align} $$

For the first inequality we have used the bounded distortion.

Sublemma 4.4. If n is sufficiently large, then we have

$$ \begin{align*} \int\vec\phi \,d\delta_z^{p_m}>\vec\beta-\vec\varepsilon. \end{align*} $$

Proof. Take $x\in A_{n}(\vec {\phi },\vec {\beta },P)$ . Since x, $z\in P$ , the same argument as the deduction of (3.1) yields

(4.18) $$ \begin{align}\dfrac{1}{n}S_n\vec\phi(z)\geq \dfrac{1}{n}S_n\vec\phi(x)-\frac{\vec\varepsilon}{2}>\vec\beta-\frac{\vec\varepsilon}{2}\end{align} $$

provided n is sufficiently large.

Write $n=p_mq+r$ , with non-negative integers q, r satisfying $0\leq r\leq q-1$ . Using (4.18), we have

$$ \begin{align*} \int\vec\phi \,d\delta_z^{p_m}&=\frac{1}{p_mq}S_{p_mq}\vec\phi(z)=\frac{n}{p_mq}\dfrac{1}{n}S_n\vec\phi(z)-\frac{1}{p_mq}S_{n-p_mq}\vec\phi(f^{p_mq}(z))\\ &=\dfrac{1}{n}S_n\vec\phi(z)+\frac{n-p_mq}{p_mq}\dfrac{1}{n}S_n\vec\phi(z)-\frac{1}{p_mq}S_{n-p_mq}\vec\phi(f^{p_mq}(z))\\ &>\vec\beta-\frac{\vec\varepsilon}{2}-\frac{2r}{p_mq}\|\vec\phi(z)\|>\vec\beta-\vec\varepsilon \end{align*} $$

provided n is sufficiently large.

Define a measure $\nu _{t,\mathbf B}\in \mathcal M(f)$ by

$$ \begin{align*}\nu_{t,\mathbf B}= \dfrac{t}{n}\cdot\delta_z^{p_m}+\bigg(1-\dfrac{t}{n}\bigg)\cdot\mu_{t,\mathbf B}.\end{align*} $$

Sublemma 4.5. If n is sufficiently large, then we have

$$ \begin{align*}\int\vec\phi \,d\nu_{t,\mathbf B}>\vec\beta-\vec\varepsilon.\end{align*} $$

Proof. From (4.16) we have

(4.19) $$ \begin{align}\int\vec\phi \,d\mu_{t,\mathbf B}>\vec\beta_1-\dfrac{1}{4}\vec\varepsilon.\end{align} $$

We first treat the case $t\sup \Vert \vec \phi \Vert <\varepsilon n/30$ . Clearly we have

(4.20) $$ \begin{align}\bigg\Vert\dfrac{t}{n}\int\vec\phi \,d\delta_z^{p_m}\bigg\Vert<\frac{\varepsilon}{30}. \end{align} $$

From (4.12) we have $\|\vec \beta _{0}\|\leq \sup \|\vec \phi \|+\varepsilon /3$ , which gives

(4.21) $$ \begin{align} \bigg\Vert\dfrac{t}{n}\vec\beta_0\bigg\Vert<\frac{\varepsilon}{3}+\frac{\varepsilon}{30}. \end{align} $$

Combining (4.19), (4.20), (4.21) and then using (4.11), we obtain

$$ \begin{align*} \int\vec\phi \,d\nu_{t,\mathbf B}&=\dfrac{t}{n}\int\vec\phi \,d\delta_z^{p_m}+\bigg(1-\dfrac{t}{n}\bigg)\int\vec\phi \,d\mu_{t,\mathbf B}\\&>-\dfrac{1}{30}\vec\varepsilon+\dfrac{t}{n}\vec\beta_0+\dfrac{n-t}{n}\bigg(\vec\beta_1-\dfrac{1}{4}\vec\varepsilon\bigg)-\dfrac{t}{n}\vec\beta_0\\&\geq-\dfrac{1}{30}\vec\varepsilon+\bigg(\dfrac{t}{n}\vec\beta_0+\dfrac{n-t}{n}\vec\beta_1\bigg)-\dfrac{1}{4}\vec\varepsilon-\dfrac{t}{n}\vec\beta_0\\&>-\dfrac{1}{30}\vec\varepsilon+\vec\beta-\dfrac{1}{3}\vec\varepsilon-\dfrac{1}{4}\vec\varepsilon-\dfrac{1}{3}\vec\varepsilon-\dfrac{1}{30}\vec\varepsilon>\vec\beta-\vec\varepsilon, \end{align*} $$

as required.

It remains to treat the case $t\sup \Vert \vec \phi \Vert \geq \varepsilon n/30$ . We assume n is large enough so that $t\geq N$ . Let $x\in A_{n}(t,\mathbf B)$ . There exists $k\in \{0,\ldots ,p_m-1\}$ such that for every $0\leq s\leq t-1$ , $f^s(x)$ and $f^s(f^k(z))$ belong to the same connected component of $K_{m,m+1}$ . Hence, if $s\leq t-N$ then $f^s(x)$ and $f^s(f^k(z))$ belong to the same connected component of $\bigcap _{\ell =0}^{t-s-1}f^{-\ell }(K_{m,m+1})$ , and so $|f^s(x)-f^s(f^k(z))|<\delta $ by (D2). From (4.14) we have

$$ \begin{align*} \Vert S_{t}\vec\phi(x)-S_{t}\vec\phi(f^k(z))\Vert & \leq \Vert S_{t-N+1}\vec\phi(x)-S_{t-N+1}\vec\phi(f^k(z))\Vert\\ &\quad +\Vert S_{N-1}\vec\phi(f^{t-N+1}(x))-S_{N-1}\vec\phi(f^{t-N+1}(f^k(z)))\Vert\\ & \leq \frac{\varepsilon }{7}(t-N+1)+2(N-1)\sup\|\vec\phi\|<\frac{t\varepsilon}{6} \end{align*} $$

provided n is sufficiently large, and thus

(4.22) $$ \begin{align} \dfrac{1}{t}S_{t}\vec\phi(f^k(z))>\dfrac{1}{t} S_{t}\vec\phi(x)-\dfrac{1}{6}\vec\varepsilon\geq \vec\beta_0-\dfrac{1}{6}\vec\varepsilon. \end{align} $$

Since z is of period $p_m$ , we have

$$ \begin{align*}\bigg\Vert\dfrac{1}{p_m}S_{p_m}\vec\phi(z)- \dfrac{1}{t}S_t\vec\phi(f^k(z))\bigg\Vert\leq \dfrac{p_m}{t}\sup\|\vec\phi\|<\frac{\varepsilon}{6}\end{align*} $$

provided n is sufficiently large, and thus

(4.23) $$ \begin{align}\frac{1}{p_m}S_{p_m}\vec\phi(z)> \dfrac{1}{t}S_t\vec\phi(f^k(z))-\dfrac{1}{6}\vec\varepsilon.\end{align} $$

Combining (4.22) and (4.23), we have

(4.24) $$ \begin{align}\int\vec\phi \,d\delta_z^{p_m}=\frac{1}{p_m}S_{p_m}\vec\phi(z)>\vec\beta_0-\dfrac{1}{3}\vec\varepsilon.\end{align} $$

Combining (4.19), (4.24) and then using (4.11), we obtain

$$ \begin{align*} \int\vec\phi \,d\nu_{t,\mathbf B}&=\dfrac{t}{n}\int\vec\phi \,d\delta_z^{p_m}+ \dfrac{n-t}{n}\int\vec\phi \,d\mu_{t,\mathbf B}\\ &>\dfrac{t}{n}\bigg(\vec\beta_0-\dfrac{1}{3}\vec\varepsilon\bigg)+\frac{n-t}{n}\bigg(\vec\beta_1-\dfrac{1}{4}\vec\varepsilon\bigg)\\ &=\dfrac{t}{n}\vec\beta_0+\frac{n-t}{n}\vec\beta_1-\dfrac{1}{3}\vec\varepsilon-\dfrac{1}{4}\vec\varepsilon>\vec\beta-\vec\varepsilon, \end{align*} $$

as required.

From (4.8) (4.9) and (4.17) we have

(4.25) $$ \begin{align} |A_n(t,\mathbf B)|\leq \frac{C^2|X|}{|f^{t}(V_t)|}\exp\bigg(\bigg( F(\nu_{t,\mathbf B})+\frac{\varepsilon}{4}\bigg)n\bigg)|P|. \end{align} $$

Let $\mu $ be a measure in the finite set $\{\nu _{t,\mathbf B}\colon \mathbf B\in M_{2,\ell }(t,\mathbb Z_\varepsilon )\}\cup \{\delta _z^{p_m}\}$ that maximizes the free energy within this finite set. By (4.12), clearly we have

(4.26) $$ \begin{align} \#M_{2,\ell}(t,\mathbb Z_\varepsilon)\leq\prod_{j=1}^\ell\bigg(\frac{3}{\varepsilon}\Big(\sup\phi_j-\inf\phi_j\Big)+2\bigg)^2. \end{align} $$

By (4.13), if n is sufficiently large then (4.25) and (4.26) together imply

(4.27) $$ \begin{align}|A_{n}(\vec{\phi},\vec\beta,V_t)|\leq\sum_{\mathbf B\in M_{2,\ell}(t,\mathbb Z_\varepsilon)}|A_n(t,\mathbf B)|\leq \exp\bigg(\bigg( F(\mu)+\frac{\varepsilon}{3}\bigg)n\bigg)|P|.\end{align} $$

To finish, from (4.9) and (4.27) we have

$$ \begin{align*} I&=\sum_{t=1}^{n-N}|A_{n}(\vec{\phi},\vec{\beta},V_t)|+ \sum_{t= n-N+1 }^{n}|A_{n}(\vec{\phi},\vec{\beta},V_t)|\\ &\leq\sum_{t=1}^{n-N}|A_{n}(\vec{\phi},\vec{\beta},V_t)|+ \sum_{t= n-N+1}^n |V_t\hspace{-1pt}|\\ &\leq \frac{n}{p_m}\exp\bigg(\bigg( F(\mu)+\frac{\varepsilon}{3}\bigg)n\bigg)|P|+\frac{Cn}{p_m} \exp(F(\mu)(n-N+1))|P|. \end{align*} $$

Combining this estimate with that of the sum $II$ in (4.10), we obtain

$$ \begin{align*} |A_{n}(\vec{\phi},\vec{\beta},P)|=I+II\leq \exp((F(\mu)+\varepsilon)n)|P| \end{align*} $$

for all sufficiently large n, as required in Lemma 4.2(a). The inequality in Lemma 4.2(b) is a consequence of Sublemmas 4.4 and 4.5. The proof of Lemma 4.2 is complete.

4.3 Coarse graining decomposition into clusters

In this subsection, for all sufficiently large $n\geq 1$ we decompose the set $\bar {A}_n(\vec {\phi },\vec {\alpha })$ into a finite number of clusters consisting of points that share the same ‘itinerary’ up to time n, and share almost the same average values of $\vec {\phi }$ along the segments of orbits each contained in a single cycle.

Suppose $m(f)\geq 1$ . Let $\varepsilon>0$ and write $\mathbb Z_\varepsilon =\{(\varepsilon /3) a\colon a\in \mathbb Z\}$ . If $m(f)=\infty $ , then let $m_*$ , $N_*\in \mathbb N$ satisfy the conclusion of Lemma 2.9 with $\varepsilon $ replaced by $\varepsilon /2$ , for any ${\vec {\beta }=(\beta _{1},\ldots ,\beta _{\ell })\in \mathbb Z_\varepsilon ^\ell }$ satisfying $\inf \phi _j-\varepsilon /3<\beta _{j}\leq \sup \phi _j$ for $1\leq j\leq \ell .$ We set