Proof. The openness is clear from the definition. To show the density, we will show that given any $H \in C^{0}(X, {{\mathcal {H}}})$ and an arbitrary $\epsilon> 0$ , we can construct some $H' \in C^{0}(X, {{\mathcal {H}}})$ such that $H'$ is contractible and $D_{\mathcal {H}}(H, H') < \epsilon $ .

If $({\mathcal {H}}, d_{\mathcal {H}}, \iota _{\mathcal {H}})$ is in Example 1, then it suffices to take $H' = H$ .

Now assume that $({\mathcal {H}}, d_{\mathcal {H}}, \iota _{\mathcal {H}})$ is in Example 3. Denote $A = \{ w \in {\mathbb T} \mid P"(w) = 0 \}$ . Since P is a non-constant analytic function, A is a finite set. For any $\epsilon> 0$ , there exists $\check {H} \in \mathcal B_{\mathcal {H}}(H, \epsilon )$ such that, by denoting $\check {f} = \Phi (\check {H})$ , we have $\check {f}_{x_0}(A) \cap A = \emptyset $ for some $x_0 \in X$ . By continuity, for every x sufficiently close to $x_0$ , we have $\check {f}_{x}(A) \cap A = \emptyset $ . We can then easily deduce that $\check {H}$ is contractible by the minimality of g.

Proof. This lemma is clear if $({\mathcal {H}}, d_{\mathcal {H}}, \iota _{\mathcal {H}})$ is in Example 1, since we can make perturbations using the projective action on the circle by $SL(2, {\mathbb R})$ .

Now we assume that $({\mathcal {H}}, d_{\mathcal {H}}, \iota _{\mathcal {H}})$ is in Example 3. Let us denote $f = \Phi (H)$ , and let k be as in Definition 4.1.

Fix an arbitrary $x \in X$ . We define a function $c: X \times {\mathbb T} \times {\mathbb R} \to {\mathbb R}$ by

$$ \begin{align*} c(x, w, t) = P(w) + H(x) + t + P(w + P(w) + H(x) + t) + H(g(x)). \end{align*} $$

It is clear that c is continuous. By definition, we have

(4.1) $$ \begin{align} (f_{g(x)})_{-c(x, w, t) } \circ (f_x)_t(w) = w. \end{align} $$

By straightforward computation, given any $w_0 \in {\mathbb T}$ , we have

(4.2) $$ \begin{align} \partial_t \partial_w \{ ( f_{g(x)})_{-c(x, w_0, t)} \circ (f_x)_t(w) \}|_{t = 0, w = w_0} = P"( f_x(w_0) )( 1 + P'(w_0)) \neq 0 \end{align} $$

as long as $P"( f_x(w_0) ) \neq 0$ . In this case, the above term has the same sign as $P"( f_x(w_0) )$ .

By compactness and our hypothesis that H is contractible, there exists a finite open covering $\{ S_{\alpha } \times T_{\alpha } \}_{\alpha \in I}$ of $X \times {\mathbb T}$ , such that for each $\alpha \in I$ , there exists an integer ${0 \leq m_{\alpha } \leq k - 2}$ such that $P"( (f^{m_{\alpha } + 1})_{x}(w)) \neq 0$ for every $(x, w) \in S_{\alpha } \times T_{\alpha }$ . Since X is locally compact and Hausdorff, there exists a partition of unity $\{ \rho _{\alpha } \}_{\alpha \in I}$ subordinated to $\{ S_{\alpha } \times T_{\alpha } \}_{\alpha \in I}$ . Moreover, again by compactness and by equation (4.2), there exists some $\delta> 0$ such that

(4.3) $$ \begin{align} &\inf_{t \in (0, \delta]} \inf_{\alpha \in I} \inf_{(x, w_0) \in S_{\alpha} \times T_{\alpha}} t^{-1} | \partial_w \{ (f_{g(x)})_{-c(x, w_0, t)} \circ (f_x)_t(w) \}|_{ w = (f^{m_{\alpha}})_{x}(w_0)} | \nonumber \\ &\quad\geq \inf_{\alpha \in I} \inf_{(x, w_0) \in S_{\alpha} \times T_{\alpha}} | P"((f^{m_{\alpha} + 1})_{x}(w_0))(1 + P'((f^{m_{\alpha}})_{x}(w_0))) + o(1) |> 0. \end{align} $$

For any $0 \leq i \leq k -1$ , we denote

$$ \begin{align*} e_{k, i} = (\delta_{0, i}, \delta_{1, i}, \ldots, \delta_{k-1, i}) \in {\mathbb R}^{k}. \end{align*} $$

We define a map $Q: X \times {\mathbb T} \times [0,1] \to {\mathbb R}^{k}$ by

$$ \begin{align*} Q(x, w, t) = \sum_{\alpha \in I} \rho_{\alpha}(x, w) \mathrm{sgn}( P"((f^{m_{\alpha} + 1})_x(w)) )( \delta t e_{k, m_{\alpha}} - c(x, w, \delta t ) e_{k, m_{\alpha} + 1}). \end{align*} $$

We may define

$$ \begin{align*} E(\sigma, w, x, \check{H}) = (\check{H}(g^{i}(x)) )_{i=0}^{k - 1} - Q(x, w, \sigma). \end{align*} $$

We clearly have item (1). We can deduce item (2) from equation (4.1). We can deduce item (3) for $\check {H} = H$ by a straightforward computation using equations (4.2) and (4.3). Then we can verify item (3) for a general $\check {H} \in \mathcal B_{\mathcal {H}}(H, \epsilon )$ by continuity.

Proof. The statement is clear if $({\mathcal {H}}, d_{\mathcal {H}}, \iota _{\mathcal {H}})$ is in Example 1.

Now we assume that $({\mathcal {H}}, d_{\mathcal {H}}, \iota _{\mathcal {H}})$ is in Example 3. Let t be a constant close to $0$ to be determined. We set

$$ \begin{align*} p_0 = H(x) + t,\quad p_1 = H(g(x)) - c(x, w_0, t). \end{align*} $$

Following the computations in Lemma 4.3, we see that

$$ \begin{align*} \iota_{\mathcal{H}}(p_1) \iota_{\mathcal{H}}(p_0) (w) &= w + P(w) + P( w + P(w) + H(x) + t)\\ &\quad - P(w_0) - P( w_0 + P(w_0) + H(x) + t). \end{align*} $$

Then it is clear that $\iota _{\mathcal {H}}(p_1) \iota _{\mathcal {H}}(p_0) (w_0) = w_0$ . Since $w_0 \neq w_1$ , we have $ w_1 + P(w_1) + H(x) \neq w_0 + P(w_0) + H(x)$ . Then since P has no smaller period, $\iota _{\mathcal {H}}(p_1) \iota _{\mathcal {H}}(p_0) (w_1)$ is a non-constant real analytic function of t. Hence, there exists t arbitrarily close to $0$ such that $\iota _{\mathcal {H}}(p_1) \iota _{\mathcal {H}}(p_0) (w_1) \neq w_1$ . This concludes the proof.

Proof of Theorem 1.10

Let us assume that $C^0(X, {\mathcal {H}}) \neq \overline {\mathcal {ML}({\mathcal {H}})}$ . For any $\epsilon> 0$ , we denote

$$ \begin{align*}\mathcal{U}_\epsilon := \{ H \in C^0(X, {\mathcal{H}}) \setminus \overline{ \mathcal{ML}({\mathcal{H}}) } \mid \vert L_+( H )\vert, \vert L_-(H)\vert<\epsilon\}.\end{align*} $$

Given an arbitrary $\epsilon> 0$ , by the upper-, respectively lower-, semicontinuity of $L_+$ , respectively $L_-$ , we see that $\mathcal {U}_\epsilon $ is open. By Proposition 4.5 and Lemma 4.2, $\mathcal {U}_\epsilon $ is dense. Then the set $\mathcal {U}_0 := \bigcap _{n\geq 1} \mathcal {U}_{{1}/{n}}$ is a residual subset of $C^0(X, {\mathcal {H}}) \setminus \overline {\mathcal {ML}({\mathcal {H}})}$ . By definition, every $H \in \mathcal {U}_0$ satisfies $L_+( H ) = L_- ( H ) = 0$ .

Proof of Proposition 4.5 assuming Proposition 4.7

By Lemma 2.7, ${L_-( H ) \kern1.3pt{\leq}\kern1.3pt 0 \kern1.3pt{\leq}\kern1.3pt L_+( H )}$ . Without loss of generality, we can assume $L_+( H )-L_-( H ) \geq \epsilon $ , for otherwise, we can let $H' = H$ . Without loss of generality, let us assume that

$$ \begin{align*} \mathcal B_{\mathcal{H}}(H, 2\epsilon) \subset C^0(X, {\mathcal{H}}) \setminus \overline{ \mathcal{ML}({\mathcal{H}}) }. \end{align*} $$

Denote

$$ \begin{align*} L = \log (\Vert {\Phi} (H) \Vert_{C^{0,1}} + 1) + 3. \end{align*} $$

Define $H_0 = H$ . Assume that for some integer $n\geq 0$ , we have constructed some contractible $H_n \in C^0(X, {\mathcal {H}}) \setminus \overline { \mathcal {ML}({\mathcal {H}}) }$ so that $D_{\mathcal {H}}(H_n, H)\leq (2^{-1}- 2^{-n-1})\epsilon $ and $L_+( H_n ) - L_-( H_n ) \geq \epsilon $ . Without loss of generality, we may assume that $\epsilon $ is sufficiently small so that we have by Remark 1.6 that

$$ \begin{align*} \| \Phi(H_n) \|_{C^{0,1}} \leq \| \Phi(H) \|_{C^{0,1}} + C D_{\mathcal{H}}(H_n, H) \leq \| \Phi(H) \|_{C^{0,1}} + 1. \end{align*} $$

Then, by Proposition 4.7, we can find a contractible $H_{n+1}\in C^0(X, {\mathcal {H}}) \setminus \overline { \mathcal {ML}({\mathcal {H}}) }$ so that $D_{\mathcal {H}}(H_{n+1}, H_n)\leq 2^{-n-2}\epsilon $ and

$$ \begin{align*} \max(-L_-( H_{n+1}), L_+( H_{n+1})) < \max(-L_-( H_n), L_+( H_n))\bigg(1- 10^{-6}\bigg(\frac{\epsilon}{ L}\bigg)^2 \bigg).\end{align*} $$

Notice that we have $D_{\mathcal {H}}(H_{n+1}, H)\kern1.4pt{\leq}\kern1.4pt (2^{-1}\kern1.4pt{-}\kern1.4pt 2^{-n-1})\epsilon \kern1.4pt{+}\kern1.4pt 2^{-n-2}\epsilon \kern1.4pt{\leq}\kern1.4pt (2^{-1}\kern1.4pt{-}\kern1.4pt 2^{-n-2})\epsilon $ . Then for some integer $m> 0$ , we would have $D_{\mathcal {H}}(H_m, H) < \epsilon $ and ${L_+(H_m) - L_-(H_m) < \epsilon }$ . We let $H' = H_m$ and this concludes the proof.

Proof of Proposition 4.7

Let us fix some contractible $H \in C^0(X, {\mathcal {H}}) \setminus \overline { \mathcal {ML}({\mathcal {H}}) }$ such that $L_+( H )>L_-( H) $ .

By Lemma 2.8, we set

$$ \begin{align*} N^{\prime}_0 = N^{\prime}_0(H, 1), \end{align*} $$

then for any $x \in X$ , for any $n> N^{\prime }_0$ , we have that

$$ \begin{align*} \mathbf{L}_+( (H(g^{i}(x)))_{i=0}^{n-1} ) \leq L_+(H) + 1,\quad \mathbf{L}_-( (H(g^{i}(x)))_{i=0}^{n-1} ) \geq L_-(H) - 1. \end{align*} $$

We set

$$ \begin{align*} N_4 = N_4\big(H, \tfrac{1}{4}\epsilon\big), \end{align*} $$

where the function $N_4$ is given by Lemma 4.8. We fix an arbitrary integer

(4.5) $$ \begin{align} N> N_4 + N^{\prime}_0. \end{align} $$

Denote by $\nu $ the unique g-invariant measure. Under the hypothesis of g, we can choose a subset $B\subset X$ by [Reference Avila and Bochi1, Lemma 6] such that the return time from B to itself via g equals to either N or $N + 1$ , and $\nu (\partial B) = 0$ . We fix such B from now on.

By reducing the size of $\epsilon $ if necessary, we may assume that for any $H' \in \mathcal B_{\mathcal {H}}(H, \epsilon )$ , we have, for $n \in \{ N, N+1 \}$ , that

(4.6) $$ \begin{align} \mathbf{L}_+( (H'(g^{i}(x)))_{i=0}^{n-1} ) \leq L_+(H) + 2,\quad \mathbf{L}_-( (H'(g^{i}(x)))_{i=0}^{n-1} ) \geq L_-(H) - 2. \end{align} $$

Let $\delta> 0$ be a small constant such that

$$ \begin{align*} \sup_{x\in X}\sup_{x'\in B(x,\delta), 0\leq i\leq N} d_{\mathcal{H}}(H(g^i(x')), H(g^i(x)) )<\tfrac{1}{2}\epsilon. \end{align*} $$

Cover the closure of B by open sets $W_1,\ldots ,W_{k_1}$ with diameters less than $\delta /2$ . By [Reference Avila and Bochi1, Lemma 3], we can choose $W_i$ so that $\nu (\partial W_i) = 0$ for all $1\leq i \leq k_1$ . Let ${U_i = W_i \setminus \bigcup _{j<i} W_j}$ . After discarding those $U_k$ which are disjoint from B, and rearranging the indexes, we can assume that for some integer $k_0> 0$ , for each $1\leq k\leq k_0$ , $B\cap U_k\not =\emptyset $ ; and the union of $U_k$ over $1\leq k\leq k_0$ covers B.

By our choice, we have $B=B_N\cup B_{N+1}$ , where $B_l$ denotes the set of points in B whose first return time to B equals to l. Let

$$ \begin{align*} V_0 = \bigcup^{N+1}_{l=N}\bigcup^{l-1}_{i=0}\partial(B_l\cap U_i). \end{align*} $$

Then $\nu (V_0 )=0.$

Denote by $\Gamma $ the set of $(k,l)$ such that $1\leq k\leq k_0, l\in \{N,N+1\}$ and $B_l \cap U_k\not =\emptyset $ . For each $(k,l)\in \Gamma $ , we choose a point $w_{k,l}\in B_l\cap U_k \setminus V_0$ .

Fix an arbitrary $(k, l) \in \Gamma $ . Note that $l \geq N> N_4$ . By Lemma 4.8 for $(\epsilon /2, H, w_{k, l}, l)$ in place of $(\epsilon , H, x, N)$ , we obtain $\mathbf {p} = (p^{k,l}_{0},\ldots , p^{k,l}_{l-1})\in {\mathcal {H}}^l$ such that:

1. (1) we have $d_{\mathcal {H}}(p^{k,l}_{i},H(g^i(w_{k,l})))<\tfrac 12\epsilon $ for any $0\leq i\leq l-1$ ;

2. (2) we have

   (4.7)

   

Now let $\eta> 0$ be a sufficiently small constant to be determined later. By the unique ergodicity of g, there exist an open set $V\supset V_0 $ and an integer $n_0>0$ such that

(4.8) $$ \begin{align} \frac{1}{n}| \{ j\mid 0\leq j\leq n-1, g^j(x)\in V \} | < \frac{\eta}{N+1}\quad \text{for all } x \in X\quad \text{for all } n\geq n_0. \end{align} $$

Moreover, we can assume that $w_{k,l}\not \in V$ for any $(k, l) \in \Gamma $ .

We define a map $H' \in C^0(X, {{\mathcal {H}}})$ by using Proposition 2.6 so that $D_{\mathcal {H}}(H, H') < \epsilon $ , and for any $(k,l)\in \Gamma $ , any $0\leq i\leq l-1$ , we have $H'(g^i(x)) = p^{k,l}_{i}$ for all $x \in B_l\cap U_k \setminus V$ .

By letting $\eta> 0$ be sufficiently small depending only on $H, \epsilon $ (this can be realized by choosing V of sufficiently small measure, and by letting $n_0$ be sufficiently big), we can ensure by equations (4.6), (4.7), and (4.8) that

By a similar argument, we obtain an analogous bound for $ - L_-(H')$ . Consequently, we have

We see that $H'$ satisfies the conclusion of Proposition 4.7.

Proof. Let us denote for simplicity $f = {\Phi } (H)$ and then F is a lift of f. Without loss of generality, we assume $\epsilon \in (0,1)$ is sufficiently small to apply Lemma 4.3.

We fix a small constant $\sigma _0 = \sigma _0(f, \epsilon )> 0$ such that for every $\check {H} \in \mathcal B_{\mathcal {H}}(H, \epsilon )$ , $x \in X$ , and $w \in {\mathbb T}$ , we have

(5.1) $$ \begin{align} E(\sigma_0, w, x, \check{H}) \in \prod_{i=0}^{k-1} \mathcal B_{\mathcal{H}}( \check{H}(g^{i}(x)), \epsilon), \end{align} $$

where the map E is given by Lemma 4.3. We let $r_0, \epsilon _2$ be given by Lemma 4.3(3) for $\sigma _0$ .

Since $L_+(f) = L_-(f) = 0$ , there exists $N' =N'(H,\epsilon )>0 $ such that

$$ \begin{align*} \sup_{x\in X, w \in \mathbb{T}} |\!\log D(f^n)_x(w)|< \frac{n\epsilon_2}{4k}\quad \text{for all } n>N'. \end{align*} $$

Then we choose $\kappa ' = \kappa '(H,\epsilon )> 0$ to be sufficiently small so that for any $H' \in \mathcal B_{\mathcal {H}}(H, \kappa ')$ , by denoting $f' = {\Phi } (H')$ , we have

(5.2) $$ \begin{align} \sup_{x\in X, w \in \mathbb{T}} |\!\log D(f')^n_x(w)|< \frac{n\epsilon_2}{2k}\quad \text{for all } n>N'. \end{align} $$

We let $\epsilon _0 = \epsilon _0(H, \epsilon ) \in (0, \epsilon )$ be given by Lemma 2.5. We let $\kappa " = \kappa _1(H,\epsilon _0)$ be given by Corollary 2.4. By Lemma 2.3, there exists some $N" = N"(H,\epsilon )> 0$ such that for any $n> N"$ , we have

(5.3) $$ \begin{align} \sup_{x\in X, y\in\mathbb{R}}|(F^n)_x(y)-y-n\rho(F)|< \frac{({1}/{60})\epsilon_2}{k( \log(\Vert f\Vert_{C^{0,1}}+1)+1) }n\kappa". \end{align} $$

We choose $\kappa "'> 0$ to be sufficiently small, depending only on $H, \kappa "$ and $\epsilon _2$ , so that for any $H' \in \mathcal B_{\mathcal {H}}(H, \kappa "')$ , for any $n> N"$ , we have

(5.4) $$ \begin{align} \sup_{x\in X, y\in\mathbb{R}}|(F')^n_x(y)- (F^n)_x(y) |< \frac{({1}/{20})\epsilon_2}{k(\log(\Vert f\Vert_{C^{0,1}}+1)+1)}n\kappa" , \end{align} $$

where $F'$ denotes the lift of $f' = {\Phi } (H')$ that is close to F.

We define

(5.5) $$ \begin{align} \kappa_3& = \frac{1}{2}\min\bigg(\epsilon_0, \kappa',\kappa"', \frac{({1}/{100})\epsilon_2}{k ( \log(\Vert f\Vert_{C^{0,1}}+1)+1 )}\kappa"\bigg), \end{align} $$

(5.6) $$ \begin{align} N_5 &= 2N' + 2N" + 100 \frac{k}{\epsilon_2}(\log(\Vert f\Vert_{C^{0,1}}+1) + 1)N_1(H,\epsilon_0) + 100 \frac{k}{\epsilon_2}. \end{align} $$

Let $N> N_5$ and $(x, y, z)\in \Omega _N(F, \kappa _3)$ . We define

(5.7) $$ \begin{align} \bar{N} &= \bigg\lceil \frac{1}{k}\bigg( 1 - \frac{\epsilon_2}{10 k (\log(\Vert f\Vert_{C^{0,1}} + 1))}\bigg)N \bigg\rceil, \end{align} $$

(5.8) $$ \begin{align} r_3 &= (\Vert f\Vert_{C^{0,1}}+1)^{-N} r_0. \end{align} $$

Let $\check {H}$ be as in the proposition and let $\check {F}$ be the unique lift of $\check {f} = {\Phi } (\check {H})$ that is close to F. For any $\sigma \in [0, \sigma _0]$ , we define

$$ \begin{align*} ( v^\sigma_{ik + j})_{j=0}^{k-1} = E( \sigma, ( \check{f}^{ik})_x(y\,\mod \!1), g^{ik}(x), \check{H} )\quad \text{for all } 0 \leq i \leq \bar{N}-1. \end{align*} $$

By Lemma 4.3(2), (3), and equation (5.1), for every $\sigma \in [0, \sigma _0]$ , we have that

(5.9) $$ \begin{align} d_{\mathcal{H}}(v^{\sigma}_l, \check{H}(g^{l}(x))) < \epsilon\quad \text{for all } 0 \leq l \leq \bar{N}k-1 \end{align} $$

and

(5.10) $$ \begin{align} V^\sigma_{ik-1}\cdots V^\sigma_0(y)= (\check{F}^{ik})_x(y)\quad \text{for all } 1\leq i\leq \bar{N}, \end{align} $$

where $V^\sigma _j$ is the unique lift of $\iota _{\mathcal {H}}(v^\sigma _j)$ that is close to $\check {F}_{g^j(x)}$ .

We have the following.

Claim 5.2. For any $0\leq i\leq \bar {N} - 1$ , we have

$$ \begin{align*} D(V^{\sigma_0}_{(i+1)k-1}\cdots V^{\sigma_0}_{ik})(y') &< e^{- \epsilon_2} D((\check{F}^k)_x)( (\check{F}^{ik})_x(y))\\ &\qquad\quad \text{for all } y' \in ( (\check{F}^{ik})_x(y) - r_0, (\check{F}^{ik})_x(y) + r_0), \end{align*} $$

and for any $r"\in (0, r_0 )$ and any $0\leq \sigma _1 <\sigma _2 \leq \sigma _0$ , we have

$$ \begin{align*} V^{\sigma_2}_{(i+1)k-1}\cdots V^{\sigma_2}_{ik}( (\check{F}^{ik})_x(y)+[-r",r"])\Subset V^{\sigma_1}_{(i+1)k-1}\cdots V^{\sigma_1}_{ik}( (\check{F}^{ik})_x(y)+(-r",r")). \end{align*} $$

In particular, for any $\bar {r}\in (0, r_3)$ , we have

$$ \begin{align*} &&V^{\sigma_2}_{\bar{N} k -1}\cdots V_0^{\sigma_2}([y-\bar{r},y+ \bar{r}]) \Subset V^{\sigma_1}_{\bar{N}k - 1}\cdots V_0^{\sigma_1}([y-\bar{r}, y+\bar{r}]). \nonumber \end{align*} $$

By equation (5.4), $(x, y, z)\in \Omega _N(F, \kappa _3)$ , and equation (5.5), we have

$$ \begin{align*} \begin{aligned} | (\check{F}^N)_x(y)-z| &\leq | ( F^N)_x(y)-z| + | (\check{F}^N)_x(y) - (F^N)_x(y)|\\ &\leq N\bigg( \kappa_3 + \frac{({1}/{20})\epsilon_2}{k (\log(\Vert f\Vert_{C^{0,1}}+1)+1)}\kappa"\bigg)<(N-\bar{N} k)\kappa". \end{aligned} \end{align*} $$

Hence, we have $(g^{\bar {N} k}(x), (\check {F}^{\bar {N} k})_x(y),z) \in \Omega _{N-\bar {N} k}(F, \kappa ")$ . Moreover, by equation (5.6) and $N>N_5$ , we have $N-\bar {N}k>N_1(f,\epsilon _0)$ . Then we can apply Lemma 2.5 to define

$$ \begin{align*} (u_{\bar{N} k },\ldots, u_{N-1}) = \Phi^{\check{H}}_{N-\bar{N} k}(g^{\bar{N} k}(x), (\check{F}^{\bar{N} k})_x(y),z).\end{align*} $$

We have

(5.11) $$ \begin{align} d_{\mathcal{H}}(u_i, \check{H}(g^i(x))) < 2\epsilon\quad \text{for all } \bar{N} k \leq i\leq N-1. \end{align} $$

For every $\bar {N} k \leq i\leq N-1$ , let us denote by $U_i$ the unique lift of $\iota _{\mathcal {H}}(u_i)$ that is close to $\check {F}_{g^{i}(x)}$ .

Claim 5.3. For any $\bar {r}\in (0, r_3]$ , we have

$$ \begin{align*} U_{N-1}\cdots U_{\bar{N} k}V^{\sigma_0}_{\bar{N} k - 1}\cdots V_0^{\sigma_0}([y-\bar{r},y+ \bar{r}])\subset [z-\tfrac{1}{10}\bar{r}, z+\tfrac{1}{10}\bar{r}]. \end{align*} $$

Proof. By equation (5.2) and $\kappa _3 < \kappa '$ , we see that

(5.12) $$ \begin{align} \sup_{x \in X, y \in {\mathbb R}} |\!\log D (\check{F}^n)_x(y)| < \frac{n\epsilon_2}{2k}\quad \text{for all } n> N'. \end{align} $$

By equation (5.8), $\bar r \leq r_3$ , equation (5.12), and by repeatedly applying Claim 5.2, we obtain

$$ \begin{align*} D(V^{\sigma_0}_{\bar{N} k -1}\cdots V_0^{\sigma_0})(y')< e^{-\bar{N}\epsilon_2/2}\quad \text{for all } y'\in (y-\bar{r}, y+\bar{r}). \end{align*} $$

Thus,

(5.13) $$ \begin{align} V^{\sigma_0}_{\bar{N}k -1}\cdots V_0^{\sigma_0}((y-\bar{r}, y+\bar{r}))\subset V^{\sigma_0}_{\bar{N}k -1}\cdots V_0^{\sigma_0}(y)+e^{-\bar{N} \epsilon_2/2}(-\bar{r},\bar{r}). \end{align} $$

By equations (5.6), (5.7), (5.11), and (5.13), we have

(5.14) $$ \begin{align} &U_{N-1}\cdots U_{\bar{N} k} V^{\sigma_0}_{\bar{N} k - 1}\cdots V_0^{\sigma_0}([y-\bar{r},y+\bar{r}]) \\\subset\ &z+(\Vert f\Vert_{C^{0,1}}+1)^{N-\bar{N}k}e^{-\bar{N}\epsilon_2/2}[-\bar{r},\bar{r}]\subset [z-\tfrac{1}{10}\bar{r}, z+\tfrac{1}{10}\bar{r}], \nonumber \end{align} $$

since we have

$$ \begin{align*} (N-\bar{N}k) \log (\Vert f\Vert_{C^{0,1}}+1) - \bar{N}\epsilon_2/2 \leq - \bar{N}\epsilon_2/4 \leq - \log 10.\\[-34pt] \end{align*} $$

Let $\bar {r}\in (0, r_3)$ be given by the proposition. We define

$$ \begin{align*}\sigma_1 \!=\! \inf\{\sigma\!\in\! [0,\sigma_0]\mid U_{N-1}\cdots U_{\bar{N} k } V^\sigma_{\bar{N} k -1} \cdots V^\sigma_0([y-\bar{r}, y+\bar{r}])\! \subset\! [z-\tfrac{1}{10}\bar{r}, z+\tfrac{1}{10}\bar{r}]\}.\end{align*} $$

By Claim 5.3, we see that $\sigma _1$ is well defined. By the last inclusion in Claim 5.2, $\sigma _1$ depends continuously on $(x, y, z)$ . Let us define

$$ \begin{align*} p_i = \begin{cases} v^{\sigma_1}_{i}, & 0 \leq i \leq \bar{N}k - 1, \\ u_i, & \bar{N} k \leq i \leq N-1. \end{cases} \end{align*} $$

Conclusions (1)–(3) then follow from the construction.

Proof of Theorem 1.8

Given an arbitrary $H \in C^0(X, {\mathcal {H}})$ , we will show that H can be approximated by elements in $\mathcal {ML}({\mathcal {H}})$ .

Without loss of generality, we may assume that every element in a neighborhood of H in $C^0(X, {\mathcal {H}})$ is contractible, since by Lemma 4.2, every element of $C^0(X, {\mathcal {H}})$ is a limit of elements with this property.

By Theorem 1.10, either H is already in $\overline {\mathcal {ML}({\mathcal {H}})}$ , in which case there is nothing left to prove, or, up to replacing H by an arbitrarily close element, we may assume without loss of generality that H is contractible and $L_+( H )= L_-( H )=0$ . It remains to show that for any $\epsilon> 0$ , there exists $H' \in \mathcal {ML}({\mathcal {H}})$ such that $D_{\mathcal {H}}(H, H')<\epsilon $ . To simplify the notation, let us denote $f = {\Phi } (H)$ and let F denote a lift of f.

Let integer $d> 0$ be given by Lemma 3.1. Let $\kappa _3$ be given by Proposition 5.1. Without loss of generality, we may assume that $\kappa _3(f, \delta )$ is monotonically increasing in $\delta $ . We inductively define positive constants $0 < \epsilon _{-1} < \epsilon _{0} < \cdots < \epsilon _{d}$ by the following formula:

$$ \begin{align*} \epsilon_d{\kern-1pt}={\kern-1pt}\frac{\epsilon}{4(d+1)}, \ \epsilon_{d-k}=\min \bigg(\frac{1}{2}\epsilon_{d-k+1},\frac{1}{2(d+1)}\kappa_3(f,\epsilon_{d-k+1})\bigg)\quad\! \text{for all } 1{\kern-1pt}\leq{\kern-1pt} k{\kern-1pt}\leq{\kern-1pt} d{\kern-1pt}+{\kern-1pt}1. \end{align*} $$

Then we have

(6.1) $$ \begin{align} 2(\epsilon_0+\cdots+\epsilon_d)<\epsilon, \ 2(\epsilon_0+\cdots+\epsilon_k)< \kappa_3(f,\epsilon_{k+1})\quad \text{for all } 0\leq k\leq d-1. \end{align} $$

We set $\kappa ' = \inf _{1\leq i\leq d} \kappa _3(f, \epsilon _i)$ . Recall that by our hypothesis, g is uniquely ergodic. Let us denote by $\nu $ the unique g-invariant measure on X. By our hypothesis that $G(g)$ (see Definition 1.4) is dense in $\mathbb {R}$ , we can choose a constant

(6.2) $$ \begin{align} \rho'\in(\rho(F)-\tfrac{1}{4}\kappa',\quad \rho(F)+\tfrac{1}{4}\kappa')\cap G(g). \end{align} $$

By Definition 1.4, there exist continuous maps $\phi : X\rightarrow \mathbb {R}$ and $\psi : X \rightarrow \mathbb {R}/\mathbb {Z}$ so that $\rho ' = \int \phi \,d\nu $ and $\psi (g(x))-\psi (x)=\phi (x)\,\mod \!1.$

Since g is uniquely ergodic, we can choose $n_0 = n_0(f,\epsilon )> 0$ to be a large integer so that

(6.3) $$ \begin{align} n_0>N_5(H,\epsilon_i)\quad \text{for all } -1\leq i\leq d \quad (\mbox{see Proposition}\ {5.1}\ \mbox{for}\ N_5) \end{align} $$

and for any integer $n> n_0$ , we have

(6.4) $$ \begin{align} \sup\limits_{x\in X}\vert \Sigma^{n-1}_{i=0} \phi(g^i(x))-n\rho' \vert < \tfrac{1}{4}n\kappa', \ \sup\limits_{x\in X, y \in {\mathbb R}} \vert (F^{n})_{x}( y ) - y - n \rho(F) \vert < \tfrac{1}{4}n\kappa'. \end{align} $$

We choose an open set $U \subset X$ such that $\phi |_U$ admits a continuous lift $\hat {\psi }: U \rightarrow {\mathbb R}$ . Namely, $\hat {\psi }$ is continuous and $\psi (x) = \hat {\psi }(x)\,\mod \!1$ for any $x \in U$ .

By Lemma 3.1 and by enlarging $n_0$ if necessary, we can choose a compact set $K \subset U$ that is d-mild, $n_0$ -good, and M-spanning for some $M> 0$ .

We choose an arbitrary $\bar {r}\in (0, \tfrac 14)$ such that

(6.5) $$ \begin{align} \bar{r}<r_3(H,l, \epsilon)\quad \text{for all } n_0< l\leq M, \end{align} $$

where $r_3$ is given by Proposition 5.1.

Let $\{ K^i \}^d_{i=-1}, \{Z^i \}^d_{i=0}$ and $l: X \to {\mathbb Z}_+$ be defined as in §3, associated to K. We will define a sequence of $H^{(i)} \in C^0(X, {\mathcal {H}}), 1 \leq i\leq d $ by induction.

We define $H^{(-1)} = H$ , $f^{(-1)} = f = {\Phi } (H^{(-1)})$ , and $F^{(-1)} = F.$ Assume that we have defined $H^{(k)}$ for some $-1 \leq k \leq d-1$ such that, let $F^{(k)}$ be the lift of $f^{(k)} = {\Phi } (H^{(k)})$ that is close to $F^{(k-1)}$ if $k \geq 0$ , then:

1. (f1) $D_{\mathcal {H}}(H^{(k)}, H)\leq 2(\epsilon _0+\cdots +\epsilon _k);$

2. (f2) for any $x\in K^k$ , we have $((F^{(k)})^{l(x)})_x(\hat {\psi }(x))=\hat {\psi }(g^{l(x)}(x))+\Sigma _{j=0}^{l(x)-1}\phi (g^j(x));$

3. (f3) for any $x\ {\in}\ K^k$ , we have $((F^{(k)})^{l(x)})_x(\hat {\psi }(x)\ {+}\ [-\bar {r},\bar {r}])\ {\subset}\ \hat {\psi }(g^{l(x)}(x))\ {+} \Sigma _{j=0}^{l(x)-1} \phi (g^j(x))+[-({1}/{10})\bar {r}, ({1}/{10})\bar {r}].$

Note that the above properties are true for $k = -1$ simply because $K^{-1} = \emptyset $ by Lemma 3.2(4).

For each $-1\leq j\leq d$ , we let

$$ \begin{align*}W^j = \bigcup_{x\in K^j} \bigcup_{0\leq i < l(x)} \{g^i(x)\}.\end{align*} $$

Since $K^d = K$ is M-spanning, we have $W^d = X$ . Recall that we have the following lemma.

By equation (6.1) and item $(f1)$ , we have $D_{\mathcal {H}}(H^{(k)}, H) < \kappa _3(f,\epsilon _{k+1})$ . By equations (6.3) and (6.5), we can apply Proposition 5.1 to $(\epsilon _{k+1}, l, H, H^{(k)},\bar {r})$ in place of $(\epsilon , N, H, \check {H}, \bar {r})$ to define $\Psi _l$ for all $n_0 <l\leq M$ .

We define a continuous map $\tilde {H} : W^{k+1} \to {\mathcal {H}}$ such that $\tilde {H}(x) = H^{(k)}(x)$ for every ${x \in W^k}$ in the following way. Let

(6.6) $$ \begin{align} \tilde{H}(x) = H^{(k)}(x)\quad \text{for all } x\in W^k. \end{align} $$

For any $x\in Z^{k+1}$ , we have $n_0 <l(x)\leq M$ . By equations (6.2) and (6.4), we have

$$ \begin{align*} & \vert \hat{\psi}(x) + \sum\limits_{i=0}^{l(x)-1} \phi(g^i(x)) - (F^{l(x)})_{x}( \hat{\psi}(x) ) \vert \\&\quad\leq \bigg\vert\! \sum\limits_{i=0}^{l(x)-1} \phi(g^i(x))-l(x)\rho(F)\bigg\vert + \vert (F^{l(x)})_{x}( \hat{\psi}(x) ) - \hat{\psi}(x) - l(x)\rho(F)\vert \\&\quad< 3 \times \frac{1}{4}l(x)\kappa' < l(x)\kappa_3(f,\epsilon_{k+1}). \end{align*} $$

Then we can define

(6.7) $$ \begin{align} (\tilde{H}(x), \ldots, \tilde{H}(g^{l(x)-1}(x))) =\Psi_{l(x)}(x,\hat\psi(x),\hat\psi(x)+\sum^{l(x)-1}_{i=0}\phi(g^i(x)) ). \end{align} $$

By Proposition 5.1(1), we have

(6.8) $$ \begin{align} d_{\mathcal{H}}(\tilde{H}(g^i(x)), H^{(k)}(g^i(x))) < 2\epsilon_{k+1}\quad \text{for all } x\in Z^{k+1}, 0\leq i< l(x). \end{align} $$

For $x \in K^{k+1}$ , for each $0 \leq i < l(x)$ , we let $\tilde {F}_{i}$ be the lift of $\iota _{\mathcal {H}}(\tilde {H}(g^{i}(x)))$ that is close to $F^{(k)}_{g^{i}(x)}$ . By items $(f2), (f3)$ for k, and by Proposition 5.1(2), for any $x\in K^{k+1} = K^{k} \cup Z^{k+1}$ , we have

$$ \begin{align*} \tilde{F}_{l(x)-1} \cdots \tilde{F}_{0}(\hat\psi(x)) &= \hat\psi(x) + \sum^{l(x)-1}_{i=0} \phi(g^i(x)),\\\tilde{F}_{l(x)-1} \cdots \tilde{F}_{0}([\hat\psi(x)-\bar{r}, \hat\psi(x)+\bar{r}]) &\subset \hat\psi(x) + \sum^{l(x)-1}_{i=0} \phi(g^i(x))+\bigg[-\frac{1}{10}\bar{r},\dfrac{1}{10}\bar{r}\bigg]. \end{align*} $$

We have the following.

Proof. It is enough to show that for any $\{x_n\}, \{l_n\}, x', l'$ in Lemma 6.1 with $j = k + 1$ , we have

(6.9) $$ \begin{align} \tilde{H}(g^{l_n}(x_n)) \rightarrow \tilde{H}(g^{l'}(x')),\quad n\rightarrow \infty. \end{align} $$

We first assume that conclusion (1) in Lemma 6.1 is true, namely, $x'\in Z^{k+1}$ . Then equation (6.9) follows immediately from Lemma 3.2(3) and the continuity of $\Psi _{l(x')}$ .

Now assume that conclusion (2) in Lemma 6.1 is true, namely, $x'\in K^k$ . It is enough to prove equation (6.9) in the following two cases: (1) $x_n \in K^k$ for all n; (2) $x_n \in Z^{k+1}$ for all n. In the first case, we have $g^{l_n}(x_n) \in W^k$ for all $ n$ . By Lemma 6.1, we have ${g^{l'}(x')\in W^k}$ . Then equation (6.9) follows from equation (6.6) and the fact that $F^{(k)}$ is continuous.

Assume that the second case is true, namely, $x_n\in Z^{k+1}$ for all n. Moreover, after passing to a subsequence, we can assume that there exists $l_0$ such that $l(x_n) = l_0$ for all n. By equation (6.7), we have

$$ \begin{align*} \tilde{H}(g^{l'}(x_n)) = \mbox{the}\ l'\mbox{th coordinate of } \Psi_{l_0}(x_n). \end{align*} $$

Then by the continuity of $\Psi _{l_0}$ and the fact that $x_n\rightarrow x', l_n\rightarrow l'$ as n tends to infinity, we have that

$$ \begin{align*} \tilde{H}(g^{l_n}(x_n)) \rightarrow \mbox{the}\ l'\mbox{th coordinate of } \Psi_{l_0}(x'), n\rightarrow \infty. \end{align*} $$

It is then enough to show that the $l'$ th coordinate of $\Psi _{l_0}(x')$ equals $F^{(k)}_{g^{l'}(x')}$ . By Proposition 5.1(3), it is enough to verify that

(6.10) $$ \begin{align} (({F}^{(k)})^{l_0})_{x'}(\hat\psi(x')) &= \hat\psi(x') + \sum^{l_0-1}_{i=0} \phi(g^i(x')), \end{align} $$

(6.11) $$ \begin{align} ((\tilde{F}^{(k)})^{l_0})_{x'}(\hat\psi(x')+[-\bar{r},\bar{r}]) &\subset \hat\psi(x') + \sum^{l_0-1}_{i=0} \phi(g^i(x'))+\bigg[-\dfrac{1}{10}\bar{r},\dfrac{1}{10}\bar{r}\bigg]. \end{align} $$

These follow from items $(f2), (f3)$ , and Lemma 3.2(5).

By equation (6.8) and Proposition 2.6, we can choose $H^{(k+1)}\in C^0(X, {\mathcal {H}})$ so that $D_{\mathcal {H}} (H^{(k+1)}, H^{(k)}) < 2\epsilon _{k+1}$ and satisfies that $H^{(k+1)}(x) = \tilde {H}(x)$ for all $x\in W^{k+1}$ . It is straightforward to verify items $(f1)$ – $(f3)$ for $k + 1$ . This completes the induction.

We let $H' = H^{(d)}$ , $f' = \Phi (H')$ , and let $\mathcal {R}= \bigcup _{x\in K}\{x\}\times (\psi (x)-\bar {r}, \psi (x)+\bar {r})$ . By item $(f3)$ , we can see that $f^{\prime }_K(\overline {{\mathcal R}})\subset \mathcal {R}$ . By Lemma 3.3, $f'$ is mode-locked, and hence $H' \in \mathcal {ML}({\mathcal {H}})$ . This concludes the proof.

Proof of Theorem 1.7

By Theorem 1.8, it remains to consider the case where $({\mathcal {H}}, d_{\mathcal {H}}, \iota _{\mathcal {H}})$ is given by Example 2. In this case, ${\mathcal {H}} = {\mathbb R}$ and $d_{\mathcal {H}} = d_{{\mathbb R}}$ . Moreover, we may assume that the function P for defining $({\mathcal {H}}, d_{\mathcal {H}}, \iota _{\mathcal {H}})$ (see Example 3) has a smaller period, for otherwise, $({\mathcal {H}}, d_{\mathcal {H}}, \iota _{\mathcal {H}})$ is given by Example 3, and we could already conclude by Theorem 1.8.

Now let $R^{-1} \in (0, 1)$ be the smallest positive period of P, where $R \in {\mathbb Z}_{> 0}$ (such R exists since P is non-constant). Then there exists a non-constant real analytic function $P \in C^{\omega }({\mathbb T})$ with no smaller period such that $\tilde {P}(R w) = RP(w)$ for every $w \in {\mathbb T}$ . Notice that we have $\| \tilde P' \| = \| P' \| < 1$ . For each $h \in {\mathbb R}$ , we define

$$ \begin{align*} \tilde{\iota}_{\mathcal{H}}(h)(w) = w + \tilde P(w) + h. \end{align*} $$

Then we have

$$ \begin{align*} R \iota_{\mathcal{H}}(h)(w) = \tilde\iota_{\mathcal{H}}(Rh)(Rw). \end{align*} $$

We define a continuous map $\tilde \Phi : C^0(X, {\mathcal {H}}) \to \mathrm {{Diff}}^{\,0,1}_g(X \times {\mathbb T})$ by

$$ \begin{align*} \tilde\Phi(H)(x, w) = (g(x), \tilde\iota_{\mathcal{H}}( H(x))(w)). \end{align*} $$

By definition, it is straightforward to deduce the equation

$$ \begin{align*} \rho(\tilde\Phi(R \cdot H)) = R \rho (\Phi(H)) \in {\mathbb R}/{\mathbb Z}, \end{align*} $$

where $R \cdot H$ denotes the function $x \mapsto R H(x)$ in $C^0(X, {\mathcal {H}})$ . By Lemma 2.1, we see that $\tilde \Phi (R \cdot H) \in \mathcal {ML}$ if and only if $\Phi (H) \in \mathcal {ML}$ . By definition, $({\mathcal {H}}, d_{\mathcal {H}}, \tilde {\iota }_{\mathcal {H}})$ is given by Example 3. By Theorem 1.8, the set $\mathcal {ML}(H, \tilde \iota _{\mathcal {H}})$ is dense. Hence, $\mathcal {ML}(H, \iota _{\mathcal {H}})$ is also dense. This concludes the proof.

Proof. Take a constant $\kappa> 0$ such that $\mathcal B_{\mathcal {H}}(H, 2 \kappa ) \subset C^0(X, {\mathcal {H}}) \setminus \overline { \mathcal {ML}({\mathcal {H}}) }$ .

Assume in contrast that the lemma is false. Then there exist some $\epsilon> 0$ , an increasing sequence $\{ n_k \}_{k \geq 1}$ , a sequence $\{ H_k \in \mathcal B_{\mathcal {H}}(H, \kappa ) \}_{k \geq 1}$ , a sequence $\{ x_k \in X \}_{k \geq 1}$ , and a sequence $\{ y_k \in [0,1) \}_{k \geq 1}$ such that

(7.1) $$ \begin{align} ((F^{(k)}_{\epsilon})^{n_{k}})_{x_k} (y_k) - ((F^{(k)})^{n_k})_{x_k}(y_k) \leq n_k/k, \end{align} $$

where $F^{(k)}$ is a lift of $\Phi (H_k)$ . We may assume without loss of generality that the set $\{ F^{(k)} \}_{k \geq 1}$ is bounded in $C^0({\mathbb R}, {\mathbb R})$ ; and $H_k$ converges to some $\check {H}$ . Then, after passing to a subsequence, we may assume that $F^{(k)}$ converges to a lift $\check {F}$ of $\Phi (\check {H})$ .

Fix an arbitrary integer $m> 0$ . By equation (7.1), for all sufficiently large integer $k> 0$ , there exists some $0 \leq l_k \leq n_k - m$ such that, for $u_k = g^{l_k}(x_k)$ and some $z_k \in [0,1)$ , we have

$$ \begin{align*} ((F^{(k)}_{\epsilon})^{m})_{u_k}(z_k) - ((F^{(k)})^{m})_{u_k}(z_k) \leq 2m/k. \end{align*} $$

By extracting a subsequence, we may assume that $u_k$ converges to some $u \in X$ , and $z_k$ converges to some $z \in [0,1]$ . Then we have

(7.2) $$ \begin{align} (\check{F}_{\epsilon}^m)_{u}(z) - (\check{F}^m)_{u}(z) = 0. \end{align} $$

By our choice of $\kappa $ , it is clear that $\check {H} \in C^0(X, {\mathcal {H}}) \setminus \overline { \mathcal {ML}({\mathcal {H}}) }$ . Thus, equation (7.2) contradicts Corollary 2.4 if m is sufficiently large. Consequently, the lemma must be true.

Proof. Given $x \in X$ and $n> K(\epsilon )$ , we have by Proposition 2.6 that there exists some $\tilde H \in \mathcal B_{\mathcal {H}}(H, \kappa )$ and $\tilde H(g^{i}(x)) = H_i$ for every $0 \leq i \leq n-1$ . Then we can immediately deduce the corollary by Lemma 7.1.

Proof. This lemma is obvious if $({\mathcal {H}}, d_{\mathcal {H}}, \iota _{\mathcal {H}})$ is in Example 1, as we can use the projective action given by the parabolic elements in $SL(2, {\mathbb R})$ to make perturbations.

Now we assume that $({\mathcal {H}}, d_{\mathcal {H}}, \iota _{\mathcal {H}})$ is in Example 3.

Fix an arbitrary $x \in X$ . We define a real analytic function $c_x : {\mathbb T} \times {\mathbb R}^2 \to {\mathbb R}$ by

$$ \begin{align*} c_x(y\,\mod \!1, s, t) = F_{g^2(x)} \circ(F_{s+t})_{g(x)} \circ (F_t)_x(y) - F_{g^2(x)} \circ F_{g(x)} \circ F_x(y). \end{align*} $$

By straightforward computation, we see that

(7.3) $$ \begin{align} (f_{- c_x(w, s, t)})_{g^2(x)} \circ (f_{s + t})_{g(x)} \circ (f_t)_x(w) = f_{g^2(x)} \circ f_{g(x)} \circ f_x(w). \end{align} $$

By continuity, we have

(7.4) $$ \begin{align} \lim_{s \to 0} \sup_{x \in X} \sup_{w \in {\mathbb T}} |c_x(w, s, 0)| = 0. \end{align} $$

Moreover, given any $w_0 \in {\mathbb R}$ , we have

$$ \begin{align*} & \partial_t \{ (f_{- c_x(w_0, s, t)})_{g^2(x)} \circ (f_{s + t})_{g(x)} \circ (f_t)_x(w) \}|_{t = 0} \\ &\quad= - \partial_t c_x(w_0, s, t)|_{t = 0} + \partial_t \{ f_{g^2(x)} \circ (f_{s + t})_{g(x)} \circ (f_t)_x(w) \}|_{t = 0} \\ &\quad= - \partial_t c_x(w_0, s, t)|_{t = 0} + \partial_t \{ c_x(w, s, t) \}|_{t = 0}. \end{align*} $$

Fix some $s \in (0, \epsilon / 2)$ sufficiently close to $0$ such that $\sup _{x \in X} \sup _{w \in {\mathbb T}} |c_x(w, s, 0)| < \epsilon / 2$ , and the function $ \partial _t \{ c_x(w, s, t) \}|_{t = 0}$ , as a non-constant real analytic function of w, reaches its minimum on a finite set $C_x \subset {\mathbb T}$ . Then

$$ \begin{align*} \partial_t \{ (f_{- c_x(w, s, t)})_{g^2(x)} \circ (f_{s + t})_{g(x)} \circ (f_t)_x(w) \}|_{t = 0} \geq 0, \end{align*} $$

and the equality holds if and only if $w \in C_x$ . By compactness, there exists an integer $L> 0$ such that $|C_x| \leq L$ for every $x \in X$ .

Define $M {\kern-1pt}={\kern-1pt} 5L$ . We will inductively construct $p^{(j)}_i$ for $j {\kern-1pt}\in{\kern-1pt} \{0,{\kern-1pt}1 \}$ and $i {\kern-1pt}\in{\kern-1pt} \{ 0, \ldots , M{\kern-1pt}-{\kern-1pt}1 \}$ so that the conclusion of the lemma is satisfied.

Denote $C_x = \{ w_0, \ldots , w_{\ell - 1} \}$ , where $\ell \leq L$ . We set $w_{0, i} = w_i$ for $0 \leq i \leq \ell -1$ .

Given an integer $0 \leq m \leq \ell - 1$ , assume that we have already constructed $p_i$ for $i \in \{ 0, \ldots , 5m - 1 \}$ . We define

(7.5) $$ \begin{align} & (p_{5m}, p_{5m + 1}, p_{5m + 2}) \nonumber\\&\quad = ( H( g^{5m}(x)), H( g^{5m + 1}(x) ), H( g^{5m+2}(x) ) ) + (0, s, -c_{g^{5m}(x)}( (f^{5m})_x (w_0), s, 0)). \nonumber\\ \end{align} $$

By our choice of s, we have $(p_{5m}, p_{5m + 1}, p_{5m + 2}) \in \mathcal B_{\mathcal {H}}( H(g^{5m}(x)), \epsilon ) \times \mathcal B_{\mathcal {H}}( H(g^{5m + 1}(x)), \epsilon ) \times \mathcal B_{\mathcal {H}}( H(g^{5m + 2}(x)), \epsilon )$ . We define

$$ \begin{align*} z_{m, i} = \iota_{\mathcal{H}}( ( p_{5m + k} )_{k = 0}^{2} )(w_{m, i})\quad \text{for all } 0 \leq i \leq \ell-1. \end{align*} $$

By Lemma 4.4, there exists $(p_{5m + 3},{\kern-1pt} p_{5m + 4}) {\kern-1pt}\in{\kern-1pt} \mathcal B_{\mathcal {H}}( H(g^{5m + 3}(x)),{\kern-1pt} \epsilon )\! \times \! \mathcal B_{\mathcal {H}}( H(g^{5m + 4}(x)),{\kern-1pt} \epsilon )$ such that

(7.6) $$ \begin{align} \iota_{\mathcal{H}}((p_{5m + 3}, p_{5m + 4}))(z_{m, 0}) &= \iota_{\mathcal{H}}( (H(g^{5m + 3}(x)), H(g^{5m + 4}(x)) ) )(z_{m, 0}), \end{align} $$

(7.7) $$ \begin{align} \iota_{\mathcal{H}}((p_{5m + 3}, p_{5m + 4}))(z_{m, m + 1}) &\not\in C_{g^{5m + 5}(x)} \quad \mbox{if}\ m < \ell - 1. \end{align} $$

We define

$$ \begin{align*} w_{m+1, i} = \iota_{\mathcal{H}}( (p_{5m + 3}, p_{5 m + 4}) )(z_{m, i}) \quad \text{for all } 0 \leq i \leq \ell - 1. \end{align*} $$

We set $p^{(0)}_{k} = p_{k}$ for every $0 \leq k \leq 5\ell -1$ . Let $t> 0$ be a small constant to be determined. For each $0 \leq m \leq \ell -1$ , we define

$$ \begin{align*} (p^{(1)}_{5m}, p^{(1)}_{5m + 1}, p^{(1)}_{5m + 2}) &= ( H( g^{5m}(x)), H( g^{5m + 1}(x) ), H( g^{5m+2}(x) ) ) \\&\quad + (t, s + t, -c_{g^{5m}(x)}( (f^{5m})_x (w_0), s, t)), (p^{(1)}_{5m + 3}, p^{(1)}_{5m + 4} )\\& = (p_{5m + 3}, p_{5m + 4}). \end{align*} $$

We set $p^{(0)}_{k} = p^{(1)}_{k} = H(g^{k}(x))$ for every $5 \ell \leq k \leq M - 1$ . By letting t be sufficiently close to $0$ , we have $d_{\mathcal {H}}(p^{(j)}_k, H(g^{k}(x))) < \epsilon $ for $j \in \{0,1\}$ and $0 \leq k \leq M - 1$ .

Take an arbitrary $y \in {\mathbb R}$ such that $y\,\mod \!1 = w_0$ . It is then straightforward to verify item (1) by equations (7.5) and (7.3). To verify item (2), we take an arbitrary $u \in {\mathbb T}$ and denote $u_k = \iota _{\mathcal {H}}( ( p_i )_{i = 0}^{k - 1} )(u)$ for each $0 \leq k \leq M-1$ . Now we view $p^{(1)}_k$ for each $0 \leq k \leq M-1$ as a function of t. Then we have

$$ \begin{align*} & \partial_t \{ \iota_{\mathcal{H}}( ( p^{(1)}_k )_{k = 0}^{M - 1} )(u) \}|_{t = 0} \\ &\quad = \sum_{ m = 0 }^{ \ell - 1} D\iota_{\mathcal{H}}( ( p_k )_{k = 5m + 3 }^{M-1} )( u_{5m + 3} ) \\ &\qquad \cdot \partial_t \{ (f_{- c_{ g^{5m}(x) }( (f^{5m})_x (w_0) , s, t)})_{g^{5m + 2}(x)} \circ (f_{s + t})_{g^{5m + 1}(x)} \circ (f_t)_{ g^{5m}(x) }( u_{5m} ) \}|_{t = 0}. \end{align*} $$

By construction, we see that for any $u \neq w_0$ , there exists some $0 \leq m \leq \ell - 1$ such that $u_{5m} \notin C_{g^{5m}(x)}$ . Consequently, we have

$$ \begin{align*} \partial_t \{ \iota_{\mathcal{H}}( ( p^{(1)}_k )_{k = 0}^{M - 1} )(u) \}|_{t = 0} \geq 0 \end{align*} $$

with equality if and only if $u = w_0$ . Then it is straightforward to deduce item (2) by compactness.

Proof. Without loss of generality, we assume that $\epsilon \in (0, \min (\kappa (H), 1)/2)$ , where $\kappa (H)$ is given by Lemma 7.1.

We let $\epsilon _0 = \epsilon _0(H, \epsilon /2)$ be given by Lemma 2.5. Let

(7.8) $$ \begin{align} m_0 &= \lceil 3 \kappa_1(f, \epsilon_0)^{-1}N_1(f, \epsilon_0) \rceil +1, \end{align} $$

(7.9) $$ \begin{align} \epsilon_2 &= (\Vert f \Vert_{C^{0,1}} + 1)^{-m_0}\epsilon, \end{align} $$

where $\kappa _1$ and $N_1$ are given by Corollary 2.4.

Let $M = M(H, \epsilon )$ be given by Lemma 7.3. Given $x \in X$ and $y \in {\mathbb R}$ , we apply Lemma 7.3 to obtain $(p^{(j)}_i)_{i=0}^{M-1} \in {{\mathcal {H}}}^{M}$ for $j \in \{ 0,1 \}$ and $0 \leq i \leq M-1$ . We denote $(p^{(j)}_i)_{i=0}^{M-1}$ as $(p^{(j)}_{x, y, i})_{i=0}^{M-1}$ to indicate the dependence on x and y.

By Lemma 7.3, we define

(7.10) $$ \begin{align} \ \,\epsilon_1 &= \delta_{\epsilon, M}( \epsilon_2)> 0,\quad\qquad\qquad \end{align} $$

(7.11) $$ \begin{align} \epsilon^{\prime}_1 &= M^{-1}( \Vert f\Vert_{C^{0,1}} + 1)^{-M} \epsilon_1. \end{align} $$

We have for every $x \in X$ , $y \in {\mathbb R}$ , and $y' \not \in \mathbb {Z} + y + (-\epsilon _2,\epsilon _2)$ that

(7.12) $$ \begin{align} P^{(1)}_{x, y, M - 1} \circ \cdots \circ P^{(1)}_{x, y, 0}(y') - P^{(0)}_{x, y, M - 1} \circ \cdots \circ P^{(0)}_{x, y, 0}(y')> \epsilon_1. \end{align} $$

We let

(7.13) $$ \begin{align} m_1 = 3 \lceil \Delta(\epsilon^{\prime}_1)^{-1} \rceil K(\epsilon^{\prime}_1), \end{align} $$

where functions $\Delta , K$ are given by Lemma 7.1.

We fix some $x \in X$ and $y \in {\mathbb R}$ from now on. We define

(7.14) $$ \begin{align} \hat{p}_{kM + i} &= p^{(1)}_{g^{kM}(x), (F^{kM})_x(y), i}, \notag\\\check{p}_{kM + i} &= p^{(0)}_{g^{kM}(x), (F^{kM})_x(y), i}\quad \text{for all } k \in {\mathbb Z},\ 0 \leq i \leq M-1. \end{align} $$

By Lemma 7.3, we have $d_{\mathcal {H}}( \hat {p}_i, H(g^i(x)) ), d_{\mathcal {H}}( \check {p}_i, H(g^i(x)) ) <\epsilon $ for all $i\in \mathbb {Z}$ . We denote by $\hat {P}_i$ , respectively $\check {P}_i$ , the unique lift of $\hat {p}_i$ , respectively $\check {p}_i$ , close to $F_{g^{i}(x)}$ . By equation (7.14) and Lemma 7.3(1), it is direct to verify that for any $i \geq 0, \hat {P}_{i M -1} \cdots \hat {P}_0(y) = \check {P}_{i M -1} \cdots \check {P}_0(y) = (F^{i M})_{x}(y)$ ; and for any $i < 0, \hat {P}^{-1}_{i M} \cdots \hat {P}^{-1}_{-1}(y) = \check {P}^{-1}_{i M}, \ldots \check {P}^{-1}_{-1}(y) = (F^{i M})_x(y)$ .

Define

$$ \begin{align*}N_2 = m_1 M + m_0.\end{align*} $$

We have the following claim.

Claim 7.5. For any $z\in (y,y+1)$ , there exists $0\leq i\leq m_1$ such that

$$ \begin{align*} \hat{P}_{iM-1}\cdots \hat{P}_0(z)\in (F^{iM})_x(y)+(0,\epsilon_2) \cup (1-\epsilon_2,1). \end{align*} $$

Similarly, for any $z\in (y,y+1)$ , there exists $0\leq i\leq m_1$ such that

$$ \begin{align*}\hat{P}^{-1}_{-iM}\cdots \hat{P}^{-1}_{-1}(z) \in (F^{-iM})_x(y)+ (0,\epsilon_2)\cup (1-\epsilon_2,1). \end{align*} $$

Proof. We will only detail the first statement, since the second one follows from a similar argument. If the first statement is false, let $z_i = \hat {P}_{iM -1}\cdots \hat {P}_0(z)$ for all $0 \leq i \leq m_1$ , then we would have

$$ \begin{align*} z_i \not\in\mathbb{Z}+ (F^{i M})_x(y)+ (-\epsilon_2,\epsilon_2)\quad \text{for all } 0\leq i\leq m_1. \end{align*} $$

Then by equation (7.12), we have

(7.15) $$ \begin{align} \hat{P}_{(i+1)M - 1} \circ \cdots \circ \hat{P}_{iM}(z_i)& > \check{P}_{(i+1)M - 1} \circ \cdots \circ \check{P}_{iM}(z_i) + \epsilon_1 \nonumber \\ & \geq (\check{P}_{(i+1)M - 1})_{\epsilon^{\prime}_1} \circ \cdots \circ (\check{P}_{iM})_{\epsilon^{\prime}_1}(z_i). \end{align} $$

By Corollary 7.2, and equations (7.13) and (7.15), we would have

$$ \begin{align*} & \hat{P}_{m_1 M -1}\cdots \hat{P}_0(y+1)\\ &\quad \geq \hat{P}_{m_1 M -1} \cdots \hat{P}_0(z) \geq (\check{P}_{m_1M - 1})_{\epsilon^{\prime}_1} \circ \cdots \circ (\check{P}_{0})_{\epsilon^{\prime}_1}(y)> (F^{m_1 M})_x(y)+2. \end{align*} $$

This is a contradiction.

Let

$$ \begin{align*} U_- &=\{z\in (y,y+1)\mid \mbox{there exists } 0\leq i\leq m_1\ \mbox{such that}\\&\quad\ \ \hat{P}_{iM-1}\cdots \hat{P}_0(z)\in (F^{iM})_x(y)+(0,\epsilon_2)\}, \\U_+ &=\{z\in (y,y+1)\mid \mbox{there exists } 0\leq i\leq m_1\ \mbox{such that}\\&\quad\ \ \hat{P}_{iM-1}\cdots \hat{P}_0(z)\in (F^{iM})_x(y)+(1-\epsilon_2,1)\}. \end{align*} $$

Then by our claim, we have

$$ \begin{align*}U_- \cup U_+ = (y,y+1).\end{align*} $$

It is direct to see that $U_-, U_+$ are both non-empty connected open sets. Since $(y, y + 1)$ is connected, we conclude that $U_- \cap U_+\neq \emptyset $ . We take $y'$ to be an arbitrary element in $U_- \cap U_+$ .

Again by Claim 7.5, there exists $0 \leq n^{\prime }_- \leq m_1$ such that for $N^{\prime }_- = M n^{\prime }_-$ , we have

$$ \begin{align*}\hat{P}^{-1}_{-N^{\prime}_{-}} \cdots \hat{P}^{-1}_{-1}(y')\in (F^{-N^{\prime}_{-}})_x(y)+(0,\epsilon_2)\cup(1-\epsilon_2,1).\end{align*} $$

Without loss of generality, we may assume that

$$ \begin{align*}\hat{P}^{-1}_{-N^{\prime}_{-}} \cdots \hat{P}^{-1}_{-1}(y')\in (F^{-N^{\prime}_{-}})_x(y)+(0,\epsilon_2),\end{align*} $$

as the other case can be dealt with by a similar argument. Then we have

$$ \begin{align*}\hat{P}^{-1}_{-N^{\prime}_{-}} \cdots \hat{P}^{-1}_{-1}[y,y')\subset (F^{-N^{\prime}_{-}})_x(y)+[0,\epsilon_2).\end{align*} $$

By $y' \in U_+$ , we see that there exists $n^{\prime }_+ \in \{0, \ldots , m_1 \}$ such that for $N^{\prime }_+ = M n^{\prime }_+$ , we have

$$ \begin{align*} \hat{P}_{N^{\prime}_{+}-1} \cdots \hat{P}_0(y') \in (F^{N^{\prime}_+})_x(y)+(1-\epsilon_2,1). \end{align*} $$

Hence,

(7.16) $$ \begin{align} \hat{P}_{N^{\prime}_{+}-1} \cdots \hat{P}_0[y',y+1)\subset (F^{N^{\prime}_+})_x(y)+(1-\epsilon_2,1). \end{align} $$

We set

$$ \begin{align*} M_+=N^{\prime}_+ +m_0\quad\mbox{and}\quad p_i = \hat{p}_i,\quad P_i=\hat{P}_i\quad \text{for all } 0 \leq i \leq N^{\prime}_+ -1. \end{align*} $$

By Lemma 2.5, equation (7.8), and the fact that

$$ \begin{align*} (F^{m_0})_{g^{N^{\prime}_+}(x)}\hat{P}_{N^{\prime}_{+}-1} \cdots \hat{P}_0(y+1) = (F^{M_+})_x(y+1)\in z_+ +(-3,3), \end{align*} $$

there exists $(p_{N^{\prime }_+}, \ldots , p_{M_+ -1} )\in {\mathcal {H}}^{m_0}$ such that:

1. (1) $d_{\mathcal {H}}(p_{N^{\prime }_+ +i}, H(g^{N^{\prime }_+ +i}(x)) )<\epsilon $ for any $0\leq i\leq m_0-1$ ;

2. (2) $P_{M_+ -1}\cdots P_{N^{\prime }_+} ( (F^{N^{\prime }_+})_x (y)+1)=z_+$ ,

where $P_i$ is a lift of $\iota _{\mathcal {H}}(p_i)$ close to $F_{g^{i}(x)}$ for each $N^{\prime }_+ \leq i \leq M_+ - 1$ . Then by equations (7.16) and (7.9), we have

$$ \begin{align*} P_{M_+-1}\cdots P_0[y',y+1) &\subset P_{M_+-1}\cdots P_{N^{\prime}_+}( (F^{N^{\prime}_+})_x(y)+(1-\epsilon_2,1])\\ &\subset (z_+-\epsilon, z_++\epsilon). \end{align*} $$

By a similar method, we may set

$$ \begin{align*} M_- = - N^{\prime}_- - m_0, \end{align*} $$

and define $(p_{M_-}, \ldots, p_{-1})$ and a lift $P_i$ of $\iota _{\mathcal {H}}(p_i)$ for each $M_- \leq i \leq - 1$ . It is then direct to verify items (1) and (2).

Proof. We will detail the proof of the case (Forward contraction). The other case follows from a similar argument.

We fix a small constant $\sigma _0> 0$ such that for every $x \in X$ and $w \in {\mathbb T}$ , we have

$$ \begin{align*} E(\sigma_0, w, x, H) \in \prod_{i=0}^{k-1} \mathcal B_{\mathcal{H}}(H(g^{i}(x)), \epsilon), \end{align*} $$

where the map E is given by Lemma 4.3. We let $r_0, \epsilon _2$ be given by Lemma 4.3(3) for $\sigma _0$ .

Fix an arbitrary $x \in X$ . For any integer $n\geq 1$ , we define

$$ \begin{align*} A_{n,\epsilon_2} = \{ w \in \mathbb{T}\mid D(f^n)_x(w)> 1000 k \epsilon_2^{-1} e^{n\epsilon_2/(100k)}\}.\end{align*} $$

By the identity in equation (2.2) and Markov’s inequality, we have

$$ \begin{align*}\bigg| \bigcup_{n\geq 1}A_{n,\epsilon_2}\bigg| \leq \sum_{n\geq 1} |A_{n,\epsilon_2}|<\sum_{n\geq 1} \frac{\epsilon_2}{1000 k}e^{-n\epsilon_2/(100k)}<1.\end{align*} $$

We fix an arbitrary $\bar {y}_1\in \mathbb {T}\setminus (\bigcup _{n\geq 1} A_{n,\epsilon _2})$ .

We let $N_3> 0$ be a large integer to be determined depending only on H and $\epsilon $ , and let

(7.17) $$ \begin{align} \epsilon_3 = \epsilon_2/(100 k). \end{align} $$

By the choice of $\bar {y}_1$ and by letting $N_3$ be sufficiently large, we have

(7.18) $$ \begin{align} D(f^n)_x(\bar{y}_1)\leq 1000 k \epsilon_2^{-1}e^{n\epsilon_2/(100k)} < e^{n\epsilon_2/(50k)} = e^{2n\epsilon_3}\quad \text{for all } n\geq N_3. \end{align} $$

For any $N\geq N_3$ , we define

(7.19) $$ \begin{align} (q_{ik+j} )_{j=0}^{k-1} = E(\sigma_0, (f^{ik})_x(\bar{y}_1), g^{ik}(x), H)\quad \text{for all } 0 \leq i \leq \lfloor (N - 1)/k \rfloor - 1 \end{align} $$

and define

$$ \begin{align*} q_j = H(g^{j}(x))\quad \text{for all } k \lfloor (N - 1)/k \rfloor \leq j < N. \end{align*} $$

Then item (1) follows from our choice of $\sigma _0$ . It is direct to verify item (4) by Lemma 4.3(2).

We let $r'>0$ be a small constant depending only on $H, {{\mathcal {H}}}, k$ , and $\epsilon $ such that

(7.20) $$ \begin{align} \frac{D(f^{k})_{x'}(\bar y")}{D(f^{k})_{x'}(\bar y')} < e^{\min( \epsilon_3, \eta) / 2}\quad \text{for all } x' \in X, |\bar{y}'- \bar{y}"| < 2r' \end{align} $$

and

(7.21) $$ \begin{align} \frac{DE(\sigma_0, w, x', H)(\bar y")}{DE(\sigma_0, w, x', H)(\bar y')} < e^{\min( \epsilon_3, \eta) / 2}\quad \text{for all } x' \in X, |\bar y'- \bar y"| < 2r'. \end{align} $$

By equations (7.17), (7.18), (7.19), and Lemma 4.3(3), we have

(7.22) $$ \begin{align} D(\iota_{\mathcal{H}}((q_i)_{i=0}^{n-1}))(\bar{y}_1)< e^{- n\epsilon_2/2k} = e^{ - 50 n \epsilon_3}\quad \text{for all } N_3 \leq n \leq N. \end{align} $$

This proves item (2). We choose $r_1 = r_1(H, \epsilon , \eta ) \in ( 0, r')$ to be sufficiently small, so that we have

$$ \begin{align*} \iota_{\mathcal{H}}((q_i)_{i=0}^{n-1})(\bar{y}_1-r_1, \bar{y}_1 +r_1)\subset (f^n)_x(\bar{y}_1) + (-r',r')\quad \text{for all } 0\leq n\leq N_3. \end{align*} $$

By equations (7.21), (7.22) and a simple induction, we obtain item (3).

Proof. By equation (2.2), we can choose $\bar {y}_3 \in \mathbb {R}$ so that $D(f^N)_x(\bar {y}_3) = 1$ . Then by letting $r_2$ be sufficiently small, and by continuity, we can verify (Forward expansion). The proof of (Backward expansion) is similar.

Proof of Lemma 4.8

As before, we denote $f = \Phi (H)$ . We set

$$ \begin{align*} D= \Vert f\Vert_{C^{0,1}} + 3 \ (\mbox{so that}\ \log D \geq 1), \quad \eta = \bigg(\frac{L_+(f)-L_-(f)}{100 \log D }\bigg)^2> 0. \end{align*} $$

We denote $n_1 = N^{\prime }_0(H, \eta )> 0$ , where $N^{\prime }_0$ is given by Lemma 2.8. Then for any $n\geq n_1$ , we have

(7.23) $$ \begin{align} e^{(L_-(f)-\eta)n} < D(f^n)_x( w ) < e^{(L_+(f)+\eta)n}\quad \text{for all } (x, w)\in X\times\mathbb{T}. \end{align} $$

By continuity and subadditivity, it is direct to see that there exists $\epsilon '=\epsilon '(H, \eta )>0$ such that for any $n\geq n_1$ , for any $(h_0,\ldots , h_{n-1})\in {\mathcal {H}}^n$ satisfying $d_{\mathcal {H}}(h_i,H(g^i(x)))<\epsilon '$ for all $0\leq i\leq n-1$ , we have

$$ \begin{align*} e^{(L_-(f)-2\eta)n} < D(\iota_{\mathcal{H}}((h_i)_{i=0}^{n-1}))(w) < e^{(L_+(f)+2\eta)n}\quad \text{for all } (x, w) \in X \times \mathbb{T}. \end{align*} $$

Without loss of generality, we can assume that $\epsilon \in (0, \epsilon ')$ . We let $\epsilon _3 =\epsilon _3(H,\epsilon ), r_1 =r_1(H,\epsilon , \eta ), N_3 =N_3(H,\epsilon )$ be given by Lemma 7.6. We let $r_2 = r_2(H, \eta )> 0$ be given by Lemma 7.7. Denote

$$ \begin{align*} r_0 = r_0(H, \epsilon) = \min( r_1, r_2,\epsilon)/2. \end{align*} $$

Let $N_2 = N_2(H, r_0)$ be given by Lemma 7.4. It is clear that, ultimately, $N_2$ depends only on H and $\epsilon $ .

We let $\epsilon _0 = \epsilon _0(H, \epsilon /2)$ be given by Lemma 2.5. Define

(7.24) $$ \begin{align} m_2 = \lceil 2\kappa_1(\Phi(H), \epsilon_0)^{-1}N_1(\Phi(H), \epsilon_0)\rceil+1, \end{align} $$

where $\kappa _1, N_1$ are given Lemma 2.5.

We let $N_4> 0$ be a large integer to be determined depending only on H and $\epsilon $ . Taking an arbitrary integer $N>N_4$ , we set

$$ \begin{align*}x_0=g^{\lceil {N}/{2}\rceil}(x).\end{align*} $$

We apply Lemma 7.4 for $(r_0, x_0, 0)$ in place of $(\epsilon , x, y)$ to obtain $(M_1, M_2)$ in place of $(M_+,M_-)$ . We have $-N_2 \leq M_2 \leq 0\leq M_1 \leq N_2$ .

By letting $N_4$ be sufficiently large, we have $N -\lceil {N}/{2}\rceil - M_1>N_3$ . Then by Lemma 7.6 (Forward contraction) for $(g^{M_1}(x_0),N-\lceil {N}/{2} \rceil -M_1)$ in place of $(x,N)$ , we obtain $\bar {y}_1\in {\mathbb T}$ and $\tilde {p}_0,\ldots ,\tilde {p}_{N-\lceil {N}/{2} \rceil -M_1-1} \in {\mathcal {H}}$ , such that:

1. (g1) $d_{\mathcal {H}}(\tilde {p}_i,H(g^{M_1+i}(x_0)) ) <\epsilon $ for all $0\leq i\leq N-\lceil {N}/{2}\rceil -M_1-1$ ;

2. (g2) $D\iota _{\mathcal {H}}((\tilde {p}_{j})_{j = 0}^{i-1})(\bar {y}_1) < e^{-2\epsilon _3 i}$ for all $N_3\leq i\leq N-\lceil {N}/{2}\rceil -M_1-1$ ;

3. (g3) $({D \iota _{\mathcal {H}}((\tilde {p}_j)_{j=0}^{i-1})(\bar {y}') }/{D \iota _{\mathcal {H}}((\tilde {p}_j)_{j=0}^{i-1})(\bar {y}_1) }) \in (e^{-\min (\eta ,\epsilon _3) i}, e^{\min (\eta ,\epsilon _3) i})$ for any $0 \leq i\leq N-\lceil {N}/{2}\rceil -M_1-1$ and any $\bar {y}'\in (\bar {y}_1-r_1, \bar {y} + r_1)$ ;

4. (g4) $\iota _{\mathcal {H}}((\tilde {p}_j)_{j=0}^{N-\lceil {N}/{2}\rceil -M_1-1})(\bar {y}_1) = ( f^{N-\lceil {N}/{2}\rceil -M_1})_{g^{M_1}(x_0)}(\bar {y}_1)$ .

Without loss of generality, we may assume that the lifts of $\bar {y}_1$ and $\bar {y}^{\prime }_1$ , denoted by ${y_1, y^{\prime }_1 \in {\mathbb R}}$ , respectively, satisfy that

$$ \begin{align*} \vert (F^{M_1})_{x_0}(0)-y_1\vert, \vert (F^{M_2})_{x_0}(0)-y^{\prime}_1\vert <2. \end{align*} $$

We denote by $\tilde {P}_i$ the unique lift of $\iota _{\mathcal {H}}(\tilde {p}_i)$ close to $F_{g^{M_1 + i}(x_0)}$ for every $0 \leq i \leq N -\lceil {N}/{2}\rceil - M_1- 1$ . Then by $r_0 < r_1$ and item (g3), for any $1\leq i\leq N -\lceil {N}/{2}\rceil - M_1$ , we have

(7.25) $$ \begin{align} \frac{D(\tilde{P}_{i-1}\cdots\tilde{P}_0)(y')}{D(\tilde{P}_{i-1}\cdots\tilde{P}_0)(y_1)} \in (e^{-i\min(\eta,\epsilon_3)},e^{i\min(\eta,\epsilon_3)})\quad \text{for all } y'\in(y_1-r_0,y_1+r_0). \end{align} $$

By Lemma 7.4 for $(r_0, y_1, y^{\prime }_1)$ in place of $(\epsilon , z_+, z_-)$ , we obtain $y^{\prime }_0\in (0,1)$ and $(\hat {p}_{M_2},\ldots , \hat {p}_{M_1-1})\in {{\mathcal {H}}}^{M_1-M_2}$ such that $d_{\mathcal {H}}(\hat {p}_i,H( g^i(x_0)) )<r_0\leq \epsilon $ for all $M_2\leq i\leq M_1-1$ , and, denote by $\hat {P}_i$ the lift of $\iota _{\mathcal {H}}(\hat {p}_i)$ close to $F_{g^{i}(x_0)}$ , we have:

1. Case I. Either

   (7.26) $$ \begin{align} \hat{p}_{M_1-1}\cdots \hat{p}_0[0,y^{\prime}_0) &\subset (y_1-r_0,y_1 + r_0), \end{align} $$
   (7.27) $$ \begin{align} \hat{p}^{-1}_{M_2}\cdots \hat{p}_{-1}^{-1}[y^{\prime}_0,1) &\subset (y^{\prime}_1-r_0, y^{\prime}_1+ r_0); \end{align} $$

2. Case II. Or

   (7.28) $$ \begin{align} \hat{p}_{M_1-1}\cdots \hat{p}_0[y^{\prime}_0,1) &\subset (y_1-r_0, y_1+r_0), \end{align} $$
   (7.29) $$ \begin{align} \hat{p}^{-1}_{M_2}\cdots \hat{p}_{-1}^{-1}[0,y^{\prime}_0) &\subset (y^{\prime}_1-r_0, y^{\prime}_1 +r_0). \end{align} $$

We will only detail the proof for Case I, as the other case follows from a similar argument.

We now define $p_{\lceil {N}/{2}\rceil }, \ldots ,p_{N-1}$ . Take an arbitrary integer

(7.30) $$ \begin{align} m_1\in \bigg(\frac{9\log D (N - \lceil N/2 \rceil)}{9\log D+L_+(f)-L_-(f)}, \frac{(9\log D+\eta) (N - \lceil N/2 \rceil)}{9\log D+L_+(f)-L_-(f)}\bigg). \end{align} $$

Define

$$ \begin{align*} &x_1 = g^{M_1}(x_0), \quad x_2=g^{m_1}(x_1),\\ &x_3 =g^{m_2}(x_2), \quad m_3=N-\bigg\lceil \frac{N}{2}\bigg\rceil -M_1-m_1-m_2. \end{align*} $$

By direct computations and by letting $N_4$ be sufficiently large, we see that

(7.31) $$ \begin{align} m_3 \in \bigg( \frac{ (L_+(f)-L_-(f)-2\eta) (N - \lceil N/2 \rceil) }{9\log D+L_+(f)-L_-(f)}, \frac{(L_+(f)-L_-(f) - \eta) (N - \lceil N/2 \rceil)}{9\log D+L_+(f)-L_-(f)}\bigg). \end{align} $$

Then we have

(7.32) $$ \begin{align} N_2 + m_2 = N - \lceil N/2 \rceil - m_1 - m_3 \in \bigg( 0, \frac{ 2\eta(N - \lceil N/2 \rceil) }{9\log D+L_+(f)-L_-(f)}\bigg). \end{align} $$

We define

(7.33) $$ \begin{align} \!\!\qquad p_{\lceil{N}/{2}\rceil +i} &= \hat{p}_i\quad \text{for all } 0\leq i\leq M_1-1, \end{align} $$

(7.34) $$ \begin{align} \!p_{\lceil{N}/{2}\rceil+M_1+i} &= \tilde{p}_i\quad \text{for all } 0\leq i\leq m_1-1, \end{align} $$

(7.35) $$ \begin{align} p_{\lceil{N}/{2}\rceil +M_1+m_1+m_2+i} &= H( g^i(x_3) )\quad \text{for all } 0\leq i\leq m_3-1. \end{align} $$

By equation (7.30) and by letting $N_4$ be sufficiently large, we have $m_1> N_3$ . Then by item (g2) and equation (7.25), we have

(7.36) $$ \begin{align} \tilde{P}_{m_1-1}\cdots\tilde{P}_0(y_1-r_0,y_1+r_0) & \subset\! (F^{m_1})_{x_1}(y_1) + e^{\min(\eta,\epsilon_3)m_1}D(\tilde{P}_{m_1-1}\!\cdots\!\tilde{P}_0)(y_1)(-r_0,r_0) \nonumber \\ & \subset\! (F^{m_1})_{x_1}(y_1) + e^{-\epsilon_3 m_1}(-r_0,r_0). \end{align} $$

By $r_0 \leq r_2$ and Lemma 7.7(Forward expansion) for $(x_3, m_3)$ in place of $(x, N)$ , there exists $y_3\in \mathbb {R}$ , such that

(7.37) $$ \begin{align} D(F^{m_3})_{x_3}(y')> e^{-\eta m_3}\quad \text{for all } y'\in (y_3 -D^{-m_3}r_0, y_3+ D^{-m_3}r_0).\end{align} $$

Without loss of generality, we can choose $y_3$ so that $\vert (F^{m_1+m_2})_{x_1} (y_1) - y_3\vert < 2$ . By equation (7.24), we have $(x_2, (F^{m_1})_{x_1}(y_1) , y_3) \in \Omega _{m_2}(F, \kappa _1(\Phi (H), \epsilon _0) )$ . Then by Lemma 2.5 and $m_2> N_1(\Phi (H), \epsilon _0)$ , we can define $(p_{\lceil {N}/{2}\rceil +M_1+m_1},\ldots , p_{\lceil {N}/{2}\rceil +M_1+m_1+m_2-1})$ so that:

1. (1) $d_{{{\mathcal {H}}}}(p_{\lceil {N}/{2}\rceil +M_1+m_1+i}, H(g^i(x_2)))<\epsilon $ for all $0\leq i\leq m_2-1$ ;

2. (2) we have

   (7.38) $$ \begin{align} P_{\lceil{N}/{2}\rceil+M_1+m_1+m_2-1} \cdots P_{\lceil{N}/{2}\rceil+M_1}(y_1) = y_3. \end{align} $$

Here we denote by $P_{i}$ the unique lift of $\iota _{\mathcal {H}}(p_i)$ close to $F_{g^{i}(x)}$ for every $\lceil {N}/{2}\rceil \leq i \leq N -1$ . Notice that by item (1) above, we have