2.1 The inducing scheme

For $N \in \mathbb {N}$ to be chosen below, we will take a collection of cylinders $w_1, \ldots , w_K\in \{ 1, \ldots , M \}^N $ containing z and consider the first return map $F=\sigma ^\tau $ to $\Sigma \setminus \{w_1, \ldots , w_K\}$ , where $\tau $ is the first return time. We consider the domains of this map to be cylinders $w\in \bigcup _{n\geqslant 1} \bigcup _{w^{\prime }\in \{ 1, \ldots , M\}^n }w^{\prime }$ , rather than unions of these. Let $\Sigma _{N,K}$ denote the set of 1-cylinders for F.

We will use the above partly to ensure our inducing scheme has exponential tails (see Proposition 2.5). We will also choose the scheme to be compatible with the periodic structure at z, see Remark 2.8.

We will make our final choice of Y once we choose $\varepsilon _0$ before Remark 2.8. We will abuse notation and let $F = f^\tau $ denote the first return map to Y as well.

Let $\{I_i\}_i$ be the domains (intervals) of monotonicity of the first return map and, for brevity, write $\tau _i=\tau |_{I_i}$ . Note that each $I_i$ is associated to a cylinder as in the symbolic model.

Proof. Property (a) follows from the SFT structure and property (b) follows from the Koebe lemma added to the fact that we removed a neighbourhood of the orbit of the critical value to create Y.

To prove property (c), by Lemma 2.4, given $I_i$ , there is a periodic point $x\in I_i$ of period $\tau _i+k$ for $0\leqslant k \leqslant P$ . Thus, by Proposition 2.1(1),

(2.1)

Since we also have bounded distortion, we see that $|I_i|=O( \unicode{x3bb} _{\mathrm {per}}^{-\tau _i})$ and hence applying Lemma 2.2, $|\{\tau = n\}| = O(e^{-n(\log \unicode{x3bb} _{\mathrm {per}}-\eta )})$ , as required.

In the light of this result, we will assume $\eta \in (0,\frac 12\log \unicode{x3bb} _{\mathrm {per}})$ from here on (we make a final choice for $\eta $ before Remark 2.8).

Recall that $\mu $ is the unique acip for f according to Proposition 2.1. Define $\mu _Y:={\mu |_Y}/{\mu (Y)}$ , which by Kac’s lemma is an F-invariant probability measure. We can recover $\mu $ from $\mu _Y$ by the well-known formula,

(2.2) $$ \begin{align} \mu(A) = \mu(Y) \sum_i \sum_{j=0}^{\tau_i-1}\mu_Y(I_i \cap f^{-j}(A)) \quad \mbox{for } A \subset I. \end{align} $$

Note that $f^{k_1}$ in a neighbourhood of c is a 2-to-1 map composed with a diffeomorphism. The Hartman–Grobman theorem implies that $f^p$ restricted to any small enough neighbourhood of z, the linearization domain, is conjugated to the transformation $x\mapsto \unicode{x3bb} _z x$ and is indeed asymptotic to this for x close to z. Let $\varepsilon _0>0$ be such that this theorem holds in $(z-\varepsilon _0, z+\varepsilon _0)$ .

For $\varepsilon \leqslant \varepsilon _0$ , let $\delta = \delta (\varepsilon )$ be such that $f^{k_1}|_{(c-\delta , c+\delta )}$ is 2-to-1 onto either $(z-\varepsilon , z]$ or $[z, z+\varepsilon )$ . Set $\unicode{x3bb} _\delta := \unicode{x3bb} _{(c-\delta (\varepsilon _0), c+\delta (\varepsilon _0))}$ and $K_\delta := K_{(c-\delta (\varepsilon _0), c+\delta (\varepsilon _0))}$ from Proposition 2.1. Then choose $\eta> 0$ such that $\eta < \min \{ \frac 12 \log \unicode{x3bb} _{\mathrm {per}}, \log \unicode{x3bb} _\delta \}$ . Now with $\varepsilon _0$ and $\eta $ fixed, we choose $ N_1 \geqslant N(\eta , K)$ so that the cylinders making up $Z_{\tilde N_1}$ from the proof of Lemma 2.3 are contained in $\bigcup _{i=1}^{p}f^i(z-\varepsilon _0, z+\varepsilon _0)$ and adjust our inducing domain to $\Sigma ^{\prime }=\Sigma ^{\prime }(N_1)$ , thus also adjusting Y.

With Y fixed, we choose $\varepsilon _1\in (0, \varepsilon _0]$ so that $Y\cap ( \bigcup _{i=1}^{p}f^i(z-\varepsilon _1, z+\varepsilon _1)) = {\emptyset }$ .

It will be convenient to treat the left and right neighbourhoods of z separately. For this purpose, we introduce the following notation for asymmetric holes. Given $\varepsilon _L, \varepsilon _R>0$ , we write $\boldsymbol {\varepsilon }= \{\varepsilon _L, \varepsilon _R\}$ and define the corresponding hole at z by $H_{\boldsymbol {\varepsilon }}(z) = (z-\varepsilon _L, z+\varepsilon _R)$ . We will abuse notation in the following ways: we write $\boldsymbol {\varepsilon }=0$ to mean $\varepsilon _L=\varepsilon _R=0$ , and we write $\boldsymbol {\varepsilon }<\varepsilon $ (also $\boldsymbol {\varepsilon } \in (0, \varepsilon )$ ) to mean $\varepsilon _L, \varepsilon _R<\varepsilon $ (and $\varepsilon _L, \varepsilon _R\in (0, \varepsilon )$ ). We extend the definition of $\delta (\varepsilon )$ to $\delta (\boldsymbol {\varepsilon })$ in the natural way, noting that, for example, in the orientation preserving case, $\delta (\boldsymbol {\varepsilon })$ will only depend on one of $\varepsilon _L$ and $\varepsilon _R$ .

For notation, we will denote by $(z-\varepsilon _L)_i$ , $(z+\varepsilon _R)_i$ the local inverse of $f^{ip}$ applied to $z-\varepsilon _L$ , $z+\varepsilon _R$ .

For a hole $H_{\boldsymbol {\varepsilon }}(z)$ , we denote by $H_{\boldsymbol {\varepsilon }}^{\prime }$ the set of intervals J in Y such that $f^s(J)\subset H_{\boldsymbol {\varepsilon }}$ for $s<\tau |_J$ . Thus, $H^{\prime }_{\boldsymbol {\varepsilon }}$ represents the ‘induced hole’ for the first return map F. In fact, due to the construction here, each such J will be one of the domains $I_i$ . The main expression we must estimate is ${\mu (H_{\boldsymbol {\varepsilon }}^{\prime })}/{\mu (H_{\boldsymbol {\varepsilon }})}$ for appropriately chosen $\varepsilon _L, \varepsilon _R$ : we will show how this allows us to estimate ${\mu (H_\varepsilon ^{\prime })}/{\mu (H_\varepsilon )}$ for all sufficiently small $\varepsilon>0$ in §5.

2.2 Chain structure

Let the interval of $I\setminus Y$ containing z be denoted by $[a, b]$ . Now suppose $I_i$ is a domain of Y with $f^{s}(I_i)\subset [a, b]$ with $1\leqslant s\leqslant \tau _i$ . Since there is an interval U closer to z than $f^{s}(I_i)$ with $f^p(U) = f^{s}(I_i)$ , then by the first return structure, there must be some $I_{i^{\prime }}$ with $f^{s}(I_{i^{\prime }})=U$ . Iterating this, we call the resulting domains a chain. We consider $I_i$ here to have some depth k and the $I_{i^{\prime }}$ above to have depth $k+1$ , and write these as $I_i^k$ and $I_i^{k+1}$ , respectively. For each chain, there is an ‘outermost’ interval which we consider to have depth 1: this is the interval $I_{i^{\prime \prime }} =I_i^1$ , a domain in Y where $f^s(I_{i^{\prime \prime }}) \subset [a, b]$ , but $f^{s+p}(I_{i^{\prime \prime }})$ not contained in $[a, b]$ . Thus, chains are denoted $\{I_i^k\}_{k=1}^\infty $ and $f^{s+kp}(I_{i}^k) = f^s(I^1_i)$ .

Note that the cylinder structure implies $f^p(a), f^p(b)\notin (a, b)$ .

See Figure 2 for a sketch of case (1) of the lemma.

Figure 2

The chain structure mapped near z. We are assuming that $f^p$ is locally orientation preserving and focussing attention on the left-hand side of z. We sketch some intervals of the images (by $f^s$ ) of the chain $\{I_{i}^k\}_{k\geqslant 1}$ . Note that here, $\tau |_{I_i^k} = s+k$ .

In case (2) above, we call $\{I_i^k\}_k$ and $\{I_j^k\}_k$ alternating chains. We also call any $\{I_i^k\}_{k\geqslant j}$ , for some $j\geqslant 1$ , a subchain.

Adding the result of this lemma to the first return structure, we see that the images in $[a, b]$ of any chain must be of the same fixed form, that is, if $\{I_i^k\}_k$ , $\{I_j^k\}_k$ are two chains which map by $f^s$ and $f^t$ respectively into $[a, b]$ , then assuming $f^s(I_i^k)$ and $f^t(I_j^k)$ , for some depth $k\geqslant 1$ , are on the same side of z, then these images coincide. Moreover, $\overline {\bigcup _k f^s(I_i^k)}$ will be either $[a, z]$ or $[z, b]$ if $f^p$ is orientation preserving around z, and $\overline {\bigcup _k f^s(I_i^k\cup I_j^k)} = [a, b]$ if $f^p$ is orientation reversing around z, and the alternating chains are as in the statement of the lemma.

2.3 Extra cuts for a ‘Markovian’ property

Suppose that $f^p$ is orientation preserving around z. Let $a_0= a$ and then, inductively, let $f^p(a_{i+1})=a_i$ , so that $a_i\to z$ as $i\to \infty $ . We similarly define $(b_i)_i$ to the right of z, accumulating on z. Observe that domains $I_i^k$ of F which map into $[a, b]$ by some $f^s$ with $1\leqslant s<\tau _i$ must map onto some $(a_{k-1}, a_k)$ or $(b_k, b_{k-1})$ .

In the orientation-reversing case, we define $a_0=a$ and $a_1$ such that $f^p(b)=a_1$ , and inductively define $a_{i}$ by $f^{2p}(a_{i+2})= a_{i}$ . Similarly, we define $(b_i)_i$ . Here, if $f^s(I_i^k)\subset [a, b]$ for $1\leqslant s<\tau _i$ , then $f^s(I_i^k) \in \{(a_{k-1}, a_k), (b_k, b_{k-1})\}$ .

Note that by bounded distortion, the ratios $ (z-a_i)/(b_i-z)$ will be comparable to $(z-a)/(b-z)$ .

The above implies that if we were to take our holes around z as, say $(a_i, b_j)$ , these would be Markov holes in the sense that either an interval of the inducing scheme enters the hole before returning to Y, or it never intersects it. Note that by topological transitivity, the only way z cannot be accumulated by domains on both the left and the right as above is if z is a boundary point of the dynamical core $[f^2(c), f(c)]$ , a simpler case which needs only obvious modifications.

For generic $\varepsilon _L, \varepsilon _R\in (0, \varepsilon _1)$ , the corresponding hole will not have this Markov property, so we make an extra cut here, namely, if $z-\varepsilon _L\in f^s(I_i)$ (in the sketch in Figure 2, ${I_i=I_i^{k-2}}$ ), then we split $I_i$ into two so that one of the resulting intervals maps by $f^s$ into the hole and one maps outside, and similarly for $z+\varepsilon _R$ , thus defining $F_{\boldsymbol {\varepsilon }}$ from $F=F_0$ . For this map, in the generic case, the chains no longer have common boundary points, but now, in the orientation-preserving case, they come in pairs $\{I_i^k\}_k, \{\hat I_i^k\}$ , where $I_i^{k+1}\bar {\cup } \hat I_i^k$ is a domain of F and $\bar {\cup }$ denotes the union of the open intervals along with their common boundary point.

In the case where $f^p$ is orientation reversing around z, we may need to cut a domain of F twice to produce $F_{\boldsymbol {\varepsilon }}$ . For simplicity, we will not address this case here, but all the ideas below go through similarly. From now on, unless indicated otherwise, when we discuss our induced map, we mean $F_{\boldsymbol {\varepsilon }}$ .

Remark that due to the extra cuts, although the domains of $F_{\boldsymbol {\varepsilon }}$ have the property that each one either maps into $H_{\boldsymbol {\varepsilon }}(z)$ or never intersects it, for generic holes, this does not define a countable Markov partition for $F_{\boldsymbol {\varepsilon }}$ . This is because $F_{\boldsymbol {\varepsilon }}(I_i)$ may not be a union of 1-cylinders if a boundary point of $I_i$ is created by one of the extra cuts.

We next observe that in the orientation-preserving case, there is a unique pair of subchains we denote $\{I_{L}^k\}_{k\geqslant k_{L, \boldsymbol {\varepsilon }}}, \{\hat I_{L}^k\}_{k\geqslant k_{L, \boldsymbol {\varepsilon }}} \subset (c-\delta (\boldsymbol {\varepsilon }), c)$ and a unique pair of subchains $\{I_{R}^k\}_{k\geqslant k_{R, \boldsymbol {\varepsilon }}}, \{\hat I_{R}^k\}_{k\geqslant k_{R, \boldsymbol {\varepsilon }}}\subset ( c, c+\delta (\boldsymbol {\varepsilon }))$ . We call these the principal subchains. Note, $\overline {\bigcup _{k\geqslant k_{L, \boldsymbol {\varepsilon }}}(I_{L}^k\cup \hat I_{ I_L}^k) \cup (\bigcup _{k\geqslant k_{R, \boldsymbol {\varepsilon }}}I_{R}^k\cup \hat I_{I_R}^k)}= [c-\delta (\boldsymbol {\varepsilon }), c+\delta (\boldsymbol {\varepsilon })]$ . We can naturally extend this idea to the orientation-reversing case. In either case, $\delta (\boldsymbol {\varepsilon })$ is determined by only one of $\varepsilon _L$ or $\varepsilon _R$ , depending on whether $f^{k_1}(c - \delta (\boldsymbol {\varepsilon }), c+\delta (\boldsymbol {\varepsilon })) = (z - \varepsilon _L, z]$ or $f^{k_1}(c - \delta (\boldsymbol {\varepsilon }), c+\delta (\boldsymbol {\varepsilon })) = [z, z + \varepsilon _R)$ . Let us call $\varepsilon _*$ this value in $\{ \varepsilon _L, \varepsilon _R \}$ .

3.1 The contribution from principal subchains

To estimate $\mu (H_{\boldsymbol {\varepsilon }}^{\prime }) $ and $\mu (H_{\boldsymbol {\varepsilon }})$ , we first estimate the contribution from the principal subchains.

Lemma 3.1. For x close to c,

$$ \begin{align*}|x-c|\sim \bigg(\frac{|f^{k_1}(x)-z|}{A |Df^{k_1-1}(f(c))|}\bigg)^{1/\ell}.\end{align*} $$

Proof. By the mean value theorem, there exists $\theta \in (f(x), f(c))$ such that $|f^{k_1}(x) - f^{k_1}(c)| = Df^{k_1-1}(\theta ) |f(x) - f(c)|$ . Then,

$$ \begin{align*} \frac{|f^{k_1}(x) - f^{k_1}(c)|}{|x-c|} = \frac{|f(x) - f(c)|}{|x-c|} \frac{|f^{k_1}(x) - f^{k_1}(c)|}{|f(x) - f(c)|} = \frac{A|x-c|^\ell}{|x-c|} |Df^{k_1-1}(\theta)| , \end{align*} $$

where A is from §1. The result follows using $|Df^{k_1-1}(\theta )| \sim |Df^{k_1-1}(f(c))|$ by bounded distortion.

We can see from Lemma 3.1 that $|\overline {\bigcup _{k\geqslant k_{L, \boldsymbol {\varepsilon }}}(I_{L}^k\cup \hat I_{L}^k)}| \sim C_p (\varepsilon _*)^{1/\ell }$ , where $C_p:={1}/{(A|Df^{k_1-1}(f(c))|)^{1/\ell }}$ , that is, $\delta (\boldsymbol {\varepsilon }) \sim C_p (\varepsilon _*)^{1/\ell }$ . Since c is bounded away from the critical orbit, as in [Reference MisiurewiczM, Theorem 6.1] (see also [Reference NowickiN, Theorem A]), the invariant density is continuous at c and defining $\rho :={d\mu }/{dm}(c)$ , the $\mu _Y$ -measure of the set can be estimated by

(3.1) $$ \begin{align}\sum_{k\geqslant k_{L, \boldsymbol{\varepsilon}}} \mu_Y(I_{L}^k\cup\hat I_L^k) & \sim \frac{C_p \rho}{\mu(Y)} (\varepsilon_*)^{1/\ell}. \end{align} $$

Similarly for $ \sum _{k\geqslant k_{L, \boldsymbol {\varepsilon }}} \mu _{Y}(I_{R}^k\cup \hat I_R^k) $ . As in equation (2.2), the sum of these two quantities scaled by $\mu (Y)$ is the relevant contribution to $\mu (H_{\boldsymbol {\varepsilon }}^{\prime })$ .

Now, for the contribution to $\mu (H_{\boldsymbol {\varepsilon }})$ , let us suppose that $f^p$ is locally orientation preserving at z and $f^{k_1-1}$ is also orientation preserving at $f(c)$ (which implies that ${\varepsilon _*=\varepsilon _L}$ ), the other cases follow similarly. Denote $J^k = \bigcup _{k^{\prime } \geqslant k} I_L^{k^{\prime }}\bar \cup \hat I_L^{k^{\prime }}$ . Then, $f^{k_1}(J^{k_{L, \boldsymbol {\varepsilon }}+k}) = ( (z-\varepsilon _L)_{k-1}, z)$ , so using Lemma 3.1 again, we have

Then, using equation (2.2), we estimate the (scaled) contribution to $\mu (H_{\boldsymbol {\varepsilon }})$ from the left of c by

(3.2)

An identical estimate follows for the domains $\{I_R^{k^{\prime }}, \hat I_R^{k^{\prime }}\}_{k^{\prime }\geqslant k}$ from the right of c.

3.2 The contribution from non-principal subchains

Now, for the subchains that are not in $(c-\delta (\boldsymbol {\varepsilon }), c+\delta (\boldsymbol {\varepsilon }))$ , but which map into $H_{\boldsymbol {\varepsilon }}$ before returning to Y, we will use a different, rougher, type of estimate. First notice that such subchains cannot be contained in $(c-\delta (\varepsilon _0), c+\delta (\varepsilon _0))$ since if $I_i\subset (c-\delta (\varepsilon _0), c+\delta (\varepsilon _0))\setminus (c-\delta (\boldsymbol {\varepsilon }), c+\delta (\boldsymbol {\varepsilon }))$ , then the repelling structure of our map around z means that $I_i$ will return to Y before mapping into $H_{\boldsymbol {\varepsilon }}$ . With this in mind, recall $\unicode{x3bb} _\delta := \unicode{x3bb} _{(c-\delta (\varepsilon _0), c+\delta (\varepsilon _0))}$ and $K_\delta := K_{(c-\delta (\varepsilon _0), c+\delta (\varepsilon _0))}$ defined after Remark 2.6 when we fixed the definition of Y.

We will deal with the orientation-preserving case here, the orientation-reversing case is similar. Suppose that $\{I_i^k\}_{k\geqslant k_i}$ , $\{\hat I_i^k\}_{k\geqslant k_i}$ is a pair of non-principal subchains such that $f^s(I_i^k\cup \hat I_i^k)\subset H_{\boldsymbol {\varepsilon }}$ for any $k\geqslant k_i$ (so this is the first time that any element of the subchain enters $H_{\boldsymbol {\varepsilon }}$ ). For $x\in I_i^k$ , by the mean value theorem, there is $x\in I_i^k$ such that

by Proposition 2.1. So since $f^s(\bigcup _{k\geqslant k_i} (I_i^k\cup \hat I_i^k))$ covers either $(z-\varepsilon _L, z)$ or $(z, z+\varepsilon _R)$ , we thus obtain an analogue of equation (3.1): for $\varepsilon ^{\prime }= \max \{\varepsilon _L, \varepsilon _R\}$ ,

We know that there are $O(e^{s\eta })$ domains $I_j$ which have $f^s(I_j)=f^j(I_i)$ , so, also accounting for the intervals which map to the other side of z, the total measure of non-principal chains contributing to $\mu (H_{\boldsymbol {\varepsilon }}^{\prime })$ can be estimated by

(3.3)

and recall we have chosen $\eta \in (0, \log \unicode{x3bb} _\delta )$ : in our definition of Y. Similarly to equation (3.2), we can obtain an analogous estimate, also $O(1)\varepsilon ^{\prime }$ , for the contribution to $\mu (H_{\boldsymbol {\varepsilon }})$ .

3.3 The limiting ratio in the periodic postcritical case

We will assume that ${\varepsilon _L}/{\varepsilon _R}$ is uniformly bounded away from 0 and infinity (so in particular, $\varepsilon _L, \varepsilon _R = O(\varepsilon _*)$ ), so that the density spike dominates the measure of our holes. Now recalling equations (2.2), (3.1) together with equation (3.3) imply

$$ \begin{align*}\mu(H_{\boldsymbol{\varepsilon}}^{\prime}) \sim 2C_p \rho(\varepsilon_*)^{1/\ell} + O(1)\varepsilon_* \end{align*} $$

while equation (3.2) and the analogue of equation (3.3) imply

so the two principal chains dominate this estimate and the limiting ratio is

(3.4)

3.4 The preperiodic postcritical case

For the case where $z=f^{k_1}(c)$ for $1\leqslant k_1< k_0$ , we choose the inducing scheme Y via a minor adaptation of the method described in the proof of Lemma 2.3: properties (a) and (d) of that lemma will follow here along with the condition (b’): if $x\in I\setminus Y$ and i is minimal such that $f^i(x)\in Y$ , then $f^i(x) \in (f^{k_0+p-1}(c)-\varepsilon _0, f^{k_0+p-1}(c)+\varepsilon _0)$ for some small $\varepsilon _0>0$ . In the notation of the proof of that lemma, we choose $w_L$ and $w_R$ adjacent to z (or just one of these if $z\in \{f(c), f^2(c)\}$ is one of the endpoints of I) and then choose the rest of the cylinders around orb $(f(c))$ , $Z_{\tilde N}:=\{w_{j, L}, w_{j, R}: j= 0,\ldots , k_1-1\}\cup \{f^i(w_L), f^i(w_R): i=0, \ldots , k_0+p-k_1-1\}$ to be the sets we remove from $\Sigma $ , where $f^{k_1-j}(w_{j, L}) =w_L$ and $f^{k_1-j}(w_{j, R}) =w_R$ . If $w_L$ and $w_R$ are chosen small enough then properties (a), (c) and (d) of Lemma 2.3 hold, as does Proposition 2.5.

This construction also guarantees, for small enough $\boldsymbol {\varepsilon }>0$ , each $I_i$ passes at most once through $(z-\varepsilon _L, z+\varepsilon _R)$ before returning to Y, ensuring that ${\mu (H_{\boldsymbol {\varepsilon }}^{\prime })}={\mu (H_{\boldsymbol {\varepsilon }})}$ . Moreover, note that an $I_i$ which does pass through $(z-\varepsilon _L, z+\varepsilon _R)$ must also pass through a neighbourhood of $\{f^{k_1+1}(c), \ldots , f^{ k_0+p-1}(c)\}$ before returning to Y.

4.1 A uniform spectral gap for $\beta $ -allowable holes

In this section, we show that for each fixed $\beta>0$ and in any set of $\varepsilon _L, \varepsilon _R>0$ such that $H_{\boldsymbol {\varepsilon }}$ are $\beta $ -allowable, the transfer operators associated to $\mathring {F}_{\boldsymbol {\varepsilon }}$ and its punctured counterpart have a spectral gap that is uniform in $\beta $ when acting on functions of bounded variation (BV) in Y.

Given a measurable function $\psi : Y \to \mathbb {R}$ , define the variation of $\psi $ on an interval ${J \subset Y}$ (or a finite collection of intervals $J \subset Y$ ) by

(4.3) $$ \begin{align} \bigvee_J \psi = \sup_{x_0 < x_1 < \cdots < x_N} \sum_{k = 1}^N |\psi(x_k) - \psi(x_{k-1})| , \end{align} $$

where $\{ x_k \}_{k=0}^N$ is the set of endpoints of a partition of J into N intervals, and the supremum ranges over all finite partitions of J. Define the BV norm of $\psi $ by

$$ \begin{align*} \| \psi \|_{BV} = \bigvee_Y \psi + |\psi|_{L^1(m)} , \end{align*} $$

where m denotes Lebesgue measure on Y. Let $BV(Y)$ denote the set of functions $\{ \psi \in L^1(m) : \| \psi \|_{BV} < \infty \}$ . (By the variation of $\psi \in L^1(m)$ , we mean the essential variation, that is $\bigvee _Y \psi = \inf _g \bigvee _Y g$ , where the infimum ranges over all functions g in the equivalence class of $\psi $ .)

We shall study the action of the transfer operators associated with $F_{\boldsymbol {\varepsilon }}$ and $\mathring {F}_{\boldsymbol {\varepsilon }}$ acting on $BV(Y)$ . For $\psi \in L^1(m)$ , define

$$ \begin{align*} \mathcal{L}_{\boldsymbol{\varepsilon}} \psi(x) = \sum_{y \in F_{\boldsymbol{\varepsilon}}^{-1}x} \frac{\psi(y)}{|DF_{\boldsymbol{\varepsilon}}(y)|} \quad\text{and for each}\ n \geqslant 0, \; \mathring{\mathcal{L}}_{\boldsymbol{\varepsilon}}^n \psi = \mathcal{L}_{\boldsymbol{\varepsilon}}^n( 1_{\mathring{Y}^n_{\boldsymbol{\varepsilon}}} \psi). \end{align*} $$

We do not claim, or need, that $1_{\mathring {Y}^n_{\boldsymbol {\varepsilon }}} \in BV(Y)$ .

Our first result proves a uniform set of Lasota–Yorke inequalities for $\mathring {\mathcal {L}}_{\boldsymbol {\varepsilon }}^{n_0}$ , depending only on $\beta $ .

Let $\mathring {{\mathcal I}}_{\boldsymbol {\varepsilon }}$ denote the (countable) collection of 1-cylinders for $\mathring {F}_{\boldsymbol {\varepsilon }}^{n_0}$ , and let $\mathring {{\mathcal J}}_{\boldsymbol {\varepsilon }}$ denote the finite set of images of elements of $\mathring {{\mathcal I}}_{\boldsymbol {\varepsilon }}$ .

Proposition 4.5. For any $\beta>0$ and any $\beta $ -allowable hole $H_{\boldsymbol {\varepsilon }}(z)$ , for all $\psi \in BV(Y)$ and all $k \geqslant 0$ ,

(4.4) $$ \begin{align} \bigvee \mathring{\mathcal{L}}_{\boldsymbol{\varepsilon}}^{kn_0} \psi & \leqslant \bigg(\frac{2}{3}\bigg)^k \bigvee_Y \psi + (1+C_d) (C_d + 2 C_\beta^{-1} ) \sum_{j=0}^{k-1} \bigg(\frac 23 \bigg)^j \int_{\mathring{Y}^{n_0(k-j)}_{\boldsymbol{\varepsilon}}} |\psi| \, dm , \end{align} $$

(4.5) $$ \begin{align} \int_Y |\mathring{\mathcal{L}}_{\boldsymbol{\varepsilon}}^k \psi | \, dm & \leqslant \int_{\mathring{Y}^k_{\boldsymbol{\varepsilon}}} |\psi| \, dm. \end{align} $$

Proof. The second inequality follows immediately from the definition of $\mathring {\mathcal {L}}_{\boldsymbol {\varepsilon }}$ , so we will prove the first. In fact, we will prove the inequality for $k=1$ , which can then be iterated trivially to produce equation (4.4).

For $\psi \in BV(Y)$ , letting $\{ \bar u_j, \bar v_j \}_j$ denote the endpoints of elements of $\mathring {{\mathcal J}}_{\boldsymbol {\varepsilon }}$ and $\{ u_i, v_i \}_i$ denote the endpoints of elements of $\mathring {{\mathcal I}}_{\boldsymbol {\varepsilon }}$ , we estimate

(4.6) $$ \begin{align} \bigvee_Y \mathring{\mathcal{L}}_{\boldsymbol{\varepsilon}}^{n_0} \psi & \leqslant \sum_{J_j \in \mathring{{\mathcal J}}_{\boldsymbol{\varepsilon}}} \bigvee_{J_j} \mathring{\mathcal{L}}_{\boldsymbol{\varepsilon}}^{n_0} \psi + \mathring{\mathcal{L}}_{\boldsymbol{\varepsilon}}^{n_0} \psi(\bar u_j) + \mathring{\mathcal{L}}_{\boldsymbol{\varepsilon}}^{n_0} \psi(\bar v_j) \nonumber \\ & \leqslant \sum_{I_i \in \mathring{{\mathcal I}}_{\boldsymbol{\varepsilon}}} \bigvee_{I_i} \frac{\psi}{|DF^{n_0}_{\boldsymbol{\varepsilon}}|} + \sum_{I_i \in \mathring{{\mathcal I}}_{\boldsymbol{\varepsilon}}} \frac{|\psi(u_i)|}{|DF_{\boldsymbol{\varepsilon}}^{n_0}(u_i)|} + \frac{|\psi(v_i)|}{|DF_{\boldsymbol{\varepsilon}}^{n_0}(v_i)|}. \end{align} $$

For the first term above, given a finite partition $\{ x_k \}_{k=0}^N$ of $I_i$ , we split the relevant expression into two terms.

$$ \begin{align*} &\sum_k \bigg| \frac{\psi(x_k)}{|DF_{\boldsymbol{\varepsilon}}^{n_0}(x_k)|} - \frac{\psi(x_{k-1})}{|DF_{\boldsymbol{\varepsilon}}^{n_0}(x_{k-1})|} \bigg|\\ &\quad\leqslant \frac{1}{3} \bigvee_{I_i} \psi+ \sum_k |\psi(x_k)| \bigg| \frac{1}{|DF_{\boldsymbol{\varepsilon}}^{n_0}(x_k)|} - \frac{1}{|DF_{\boldsymbol{\varepsilon}}^{n_0}(x_{k-1})|} \bigg|, \end{align*} $$

where we have used equation (4.2) in the first term. For the second term, we use bounded distortion, Proposition 2.5(b), to estimate

$$ \begin{align*} \bigg| \frac{1}{|DF_{\boldsymbol{\varepsilon}}^{n_0}(x_k)|} - \frac{1}{|DF_{\boldsymbol{\varepsilon}}^{n_0}(x_{k-1})|} \bigg| \leqslant C_d \frac{|F_{\boldsymbol{\varepsilon}}^{n_0}(x_k) - F_{\boldsymbol{\varepsilon}}^{n_0}(x_{k-1})|}{|DF_{\boldsymbol{\varepsilon}}^{n_0}(x_k)|} \leqslant (1+C_d)C_d |x_k - x_{k-1}|, \end{align*} $$

where we have applied the mean value theorem to $F_{\boldsymbol {\varepsilon }}^{n_0}$ on $[x_{k-1}, x_k]$ . Putting these estimates together yields

$$ \begin{align*} \begin{split} \sum_k \bigg| \frac{\psi(x_k)}{|DF_{\boldsymbol{\varepsilon}}^{n_0}(x_k)|} - \frac{\psi(x_{k-1})}{|DF_{\boldsymbol{\varepsilon}}^{n_0}(x_{k-1})|} \bigg| & \leqslant \frac 13 \bigvee_{I_i} \psi + C_d(1+C_d) \sum_{k=1}^N |\psi(x_k)| (x_k - x_{k-1}) \\ & \leqslant \frac 13 \bigvee_{I_i} \psi + C_d(1+C_d) \int_{I_i} |\psi| + \kappa_N(\psi), \end{split} \end{align*} $$

where we have recognized the second term as a Riemann sum, and the error term ${\kappa _N(\psi ) \to 0}$ as $N \to \infty $ . Since the variation is attained in the limit of partitions as $N \to \infty $ , we have the following bound on the first term from equation (4.6):

(4.7) $$ \begin{align} \bigvee_{I_i} \frac{\psi}{|DF_{\boldsymbol{\varepsilon}}^{n_0}|} & \leqslant \frac 13 \bigvee_{I_i} \psi + C_d(1+C_d) \int_{I_i} |\psi| \, dm. \end{align} $$

Next, for the second term in equation (4.6), we use the bound,

(4.8) $$ \begin{align} |\psi(u_i)| + |\psi(v_i)| \leqslant 2 \inf_{I_i} |\psi| + \bigvee_{I_i} \psi \leqslant \frac{2}{|I_i|} \int_{I_i} |\psi| + \bigvee_{I_i} \psi. \end{align} $$

Then, using again bounded distortion together with Lemma 4.3, we have

(4.9) $$ \begin{align} |I_i| \inf_{I_i} |DF_{\boldsymbol{\varepsilon}}^{n_0}| \geqslant (1+C_d)^{-1} |F_{\boldsymbol{\varepsilon}}^{n_0}(I_i)| \geqslant (1+C_d)^{-1} C_\beta. \end{align} $$

Putting these estimates together with equation (4.7) into equation (4.6), and using again equation (4.2), we conclude

$$ \begin{align*} \bigvee_Y \mathring{\mathcal{L}}_{\boldsymbol{\varepsilon}}^{n_0} \psi &\leqslant \frac 23 \sum_i \bigvee_{I_i} \psi + (C_2(1+C_d) + 2(1+C_d)C_\beta^{-1}) \int_{I_i} |\psi| \leqslant \frac 23 \bigvee_Y \psi\\ &\quad+ (1+C_d)(C_d + 2C_\beta^{-1}) \int_{\mathring{Y}^{n_0}_{\boldsymbol{\varepsilon}}} |\psi| , \end{align*} $$

which is the required inequality for $k=1$ .

Next, to show that $\mathring {\mathcal {L}}_{\boldsymbol {\varepsilon }}$ has a uniform spectral gap (depending on $\beta $ and for $\boldsymbol {\varepsilon }$ sufficiently small), we will apply the perturbative framework of Keller and Liverani [Reference Keller and LiveraniKL1]. To this end, define the norm of an operator ${\mathcal P} : BV(Y) \to L^1(m)$ by

(4.10) $$ \begin{align} ||| {\mathcal P} ||| = \sup \{ |{\mathcal P} \psi |_{L^1(m)} : \| \psi \|_{BV} \leqslant 1 \}. \end{align} $$

Our next lemma is standard: $||| \mathcal {L}_0 - \mathring {\mathcal {L}}_{\boldsymbol {\varepsilon }} |||$ is small as a function of $m(H_{\boldsymbol {\varepsilon }}^{\prime })$ .

Proof. Let $\psi \in BV(Y)$ . Then,

$$ \begin{align*} \int_Y |(\mathring{\mathcal{L}}_{{\boldsymbol{\varepsilon} ^{\prime}}} - \mathring{\mathcal{L}}_{\boldsymbol{\varepsilon}}) \psi | \, dm \leqslant \int_Y |\psi 1_{H_{\boldsymbol{\varepsilon}}^{\prime} \setminus H^{\prime}_{{\boldsymbol{\varepsilon} ^{\prime}}}} | \, dm \leqslant \| \psi \|_{BV} m(H_{\boldsymbol{\varepsilon}}^{\prime} \setminus H^{\prime}_{{\boldsymbol{\varepsilon} ^{\prime}}}).\\[-42pt] \end{align*} $$

Theorem 4.7. For any $\beta>0$ , there exists $\varepsilon _\beta>0$ such that for any $\beta $ -allowable hole $H_{\boldsymbol {\varepsilon }}(z)$ with $\boldsymbol {\varepsilon } < \varepsilon _\beta $ , $\mathring {\mathcal {L}}_{\boldsymbol {\varepsilon }}$ is a continuous perturbation of $\mathcal {L}_0$ . Indeed, it is Hölder continuous in $m(H_{\boldsymbol {\varepsilon }}^{\prime })$ .

As a consequence, $\mathring {\mathcal {L}}_{\boldsymbol {\varepsilon }}$ has a spectral gap on $BV(Y)$ . In particular, there exist ${\eta _\beta , B_\beta>0}$ , such that for all $\boldsymbol {\varepsilon }<\varepsilon _\beta $ , the spectral radius of $\mathring {\mathcal {L}}_{\boldsymbol {\varepsilon }}$ is $\Lambda _{\boldsymbol {\varepsilon }} <1$ and there exist operators $\Pi _{\boldsymbol {\varepsilon }}, {\mathcal R}_{\boldsymbol {\varepsilon }}: BV(Y) \circlearrowleft $ satisfying $\Pi _{\boldsymbol {\varepsilon }}^2 = \Pi _{\boldsymbol {\varepsilon }}$ , $\Pi _{\boldsymbol {\varepsilon }} {\mathcal R}_{\boldsymbol {\varepsilon }} = {\mathcal R}_{\boldsymbol {\varepsilon }} \Pi _{\boldsymbol {\varepsilon }} = 0$ and $\| {\mathcal R}_{\boldsymbol {\varepsilon }}^n \|_{BV} \leqslant B_\beta \Lambda _{\boldsymbol {\varepsilon }}^n e^{-\eta _\beta n}$ such that

(4.11) $$ \begin{align} \mathring{\mathcal{L}}_{\boldsymbol{\varepsilon}} \psi = \Lambda_{\boldsymbol{\varepsilon}} \Pi_{\boldsymbol{\varepsilon}} \psi + {\mathcal R}_{\boldsymbol{\varepsilon}} \psi. \end{align} $$

Moreover, $\Pi _{\boldsymbol {\varepsilon }} = \mathring {e}_{\boldsymbol {\varepsilon }} \otimes \mathring {g}_{\boldsymbol {\varepsilon }}$ for some $\mathring {e}_{\boldsymbol {\varepsilon }} \in BV(Y)^*$ and $\mathring {g}_{\boldsymbol {\varepsilon }} \in BV(Y)$ satisfying $\mathring {\mathcal {L}}_{\boldsymbol {\varepsilon }} \mathring {g}_{\boldsymbol {\varepsilon }} = \Lambda _{\boldsymbol {\varepsilon }} \mathring {g}_{\boldsymbol {\varepsilon }}$ with $\int _Y \mathring {g}_{\boldsymbol {\varepsilon }} \, dm = 1$ .

Lastly, the spectra and spectral projectors vary (Hölder) continuously as functions of $\boldsymbol {\varepsilon }$ in the $||| \cdot |||$ -norm, that is, as operators from $BV(Y)$ to $L^1(m)$ .

Proof. The Lasota–Yorke inequalities of Proposition 4.5 apply also to the unpunctured operator $\mathcal {L}_{\boldsymbol {\varepsilon }} = \mathcal {L}_0$ with $\mathring {Y}^n_{\boldsymbol {\varepsilon }}$ replaced by Y. Thus, $\mathcal {L}_0^{n_0}$ is quasi-compact on $BV(Y)$ , and since $\mathcal {L}_0$ is also bounded as an operator on $BV(Y)$ (although we do not obtain the same contraction for one iterate of $\mathcal {L}_0$ , the norm estimate as in the proof of Proposition 4.5 is finite), then $\mathcal {L}_0$ is also quasi-compact on $BV(Y)$ . Since $F_0$ is mixing by Lemma 2.3(e) and has finite images by Proposition 2.5(a), then $F_0$ is covering in the sense of [Reference Liverani, Saussol and VaientiLSV]. It follows that $\mathcal {L}_0$ has a spectral gap. Then, so does $\mathcal {L}_0^{n_0}$ . Moreover, if $g_0$ is the unique element of $BV(Y)$ such that $\mathcal {L}_0 g_0 = g_0$ and $\int _Y g_0 \, dm = 1$ , then [Reference Liverani, Saussol and VaientiLSV, Theorem 3.1] implies

(4.12) $$ \begin{align} c_g := \inf_Y g_0> 0. \end{align} $$

Next, due to the uniform Lasota–Yorke inequalities (for fixed $\beta>0$ ) of Proposition 4.5 together with Lemma 4.6 for ${\boldsymbol {\varepsilon } ^{\prime }}=0$ , [Reference Keller and LiveraniKL1, Corollary 1] implies that the spectra and spectral projectors of $\mathring {\mathcal {L}}_{\boldsymbol {\varepsilon }}^{n_0}$ outside the disk of radius $2/3$ vary continuously in $\boldsymbol {\varepsilon }$ for $\boldsymbol {\varepsilon }$ sufficiently small (depending on $\beta $ ). The spectral gap and the rest of the spectral decomposition for $\mathring {\mathcal {L}}_{\boldsymbol {\varepsilon }}$ then follow from the analogous decomposition for $\mathcal {L}_0$ .

Lastly, fixing $\boldsymbol {\varepsilon } < \varepsilon _\beta $ and using Lemma 4.6, we apply again [Reference Keller and LiveraniKL1, Corollary 1] to $\mathring {\mathcal {L}}_{\boldsymbol {\varepsilon }}$ to conclude that the spectra and spectral projectors of $\mathring {\mathcal {L}}_{{\boldsymbol {\varepsilon } ^{\prime }}}$ vary Hölder continuously as functions of $|{\boldsymbol {\varepsilon } ^{\prime }} - \boldsymbol {\varepsilon }|$ whenever $\boldsymbol {\varepsilon }^{\prime } < \varepsilon _\beta $ .

The above theorem implies in particular that the size of the spectral gap is at least $\eta _\beta $ for all $\beta $ -allowable holes $H_{\boldsymbol {\varepsilon }}(z)$ with $\boldsymbol {\varepsilon } < \varepsilon _\beta $ .

4.2 Local escape rate

In this section, we will set up the estimates necessary to prove Theorem 1.1 via the induced map $F_{\boldsymbol {\varepsilon }}$ . The strategy is essentially the same as that carried out in [Reference Demers and ToddDT2, §7], but of course now we are interested in the case in which z lies in the critical orbit, which was not allowed for geometric potentials in [Reference Demers and ToddDT2].

We fix $\beta>0$ and consider the zero-hole limit as $\varepsilon \to 0$ for $\beta $ -allowable holes only. As a first step, we use the spectral gap for $\mathring {\mathcal {L}}_{\boldsymbol {\varepsilon }}$ given by Theorem 4.7 to construct an invariant measure $\nu _{\boldsymbol {\varepsilon }}$ for the induced open map $\mathring {F}_{\boldsymbol {\varepsilon }}$ supported on the survivor set $\mathring {Y}^\infty _{\boldsymbol {\varepsilon }} := \bigcap _{n=1}^\infty \mathring {Y}^n_{\boldsymbol {\varepsilon }} = \bigcap _{n=0}^\infty \mathring {F}_{\boldsymbol {\varepsilon }}^{-n}(Y)$ .

Define for $\psi \in BV(Y)$ ,

(4.13) $$ \begin{align} \nu_{\boldsymbol{\varepsilon}}(\psi) := \lim_{n\to\infty} \Lambda_{\boldsymbol{\varepsilon}}^{-n} \int_{\mathring{Y}^n_{\boldsymbol{\varepsilon}}} \psi \, \mathring{g}_{\boldsymbol{\varepsilon}} \, dm. \end{align} $$

Lemma 4.8. Fix $\beta>0$ and let $\boldsymbol {\varepsilon } < \varepsilon _\beta $ be such that $H_{\boldsymbol {\varepsilon }}$ is $\beta $ -allowable. The limit in equation (4.13) exists and defines a Borel probability measure, supported on $\mathring {Y}^\infty _{\boldsymbol {\varepsilon }}$ and invariant for $\mathring {F}_{\boldsymbol {\varepsilon }}$ . Moreover, $\nu _{\boldsymbol {\varepsilon }}$ varies continuously as a function of $\boldsymbol {\varepsilon }$ (for fixed $\beta $ ) and

(4.14) $$ \begin{align} -\log \Lambda_{\boldsymbol{\varepsilon}} = \bigg( \int \tau \, d\nu_{\boldsymbol{\varepsilon}} \bigg) \mathfrak{e}(H_{\boldsymbol{\varepsilon}}(z)), \end{align} $$

where $\tau $ is the inducing time for $F_{\boldsymbol {\varepsilon }}$ and $\mathfrak {e}(H_{\boldsymbol {\varepsilon }}(z))$ is the escape rate for f from equation (1.1). (Indeed, we show that $\nu _{\boldsymbol {\varepsilon }}(\tau )$ is continuous in $\boldsymbol {\varepsilon }$ although $\tau \notin BV(Y)$ .)

Proof. The limit in equation (4.13) exists due to the spectral decomposition from Theorem 4.7 and the conformality of m:

$$ \begin{align*} \lim_{n \to \infty} \Lambda_{\boldsymbol{\varepsilon}}^{-n} \int_{\mathring{Y}^n_{\boldsymbol{\varepsilon}}} \psi \, \mathring{g}_{\boldsymbol{\varepsilon}} \, dm = \lim_{n \to \infty} \int_Y \Lambda_{\boldsymbol{\varepsilon}}^{-n} \mathring{\mathcal{L}}_{\boldsymbol{\varepsilon}}^n(\psi \mathring{g}_{\boldsymbol{\varepsilon}}) \, dm = \mathring{e}_{\boldsymbol{\varepsilon}}(\psi \mathring{g}_{\boldsymbol{\varepsilon}}) \end{align*} $$

for any $\psi \in BV(Y)$ since if $\psi \in BV(Y)$ , then also $\psi \mathring {g}_{\boldsymbol {\varepsilon }} \in BV(Y)$ . From equation (4.13), $| \nu _{\boldsymbol {\varepsilon }}(\psi ) | \leqslant \nu _{\boldsymbol {\varepsilon }}(1) |\psi |_\infty $ , so that $\nu _{\boldsymbol {\varepsilon }}$ extends to a bounded linear functional on $C^0(Y)$ , that is, $\nu _{\boldsymbol {\varepsilon }}$ is a Borel measure, clearly supported on $\mathring {Y}_{\boldsymbol {\varepsilon }}^\infty $ . Since $\nu _{\boldsymbol {\varepsilon }}(1)=1$ , $\nu _{\boldsymbol {\varepsilon }}$ is a probability measure.

Next, we prove that $\nu _{\boldsymbol {\varepsilon }}$ is continuous as an element of $BV(Y)^*$ . Remark that by the above calculation, $\nu _{\boldsymbol {\varepsilon }}(\psi ) = \mathring {e}_{\boldsymbol {\varepsilon }}(\mathring {g}_{\boldsymbol {\varepsilon }} \psi )$ for $\psi \in BV(Y)$ , and when $\varepsilon =0$ , $\mu _Y(\psi ) = e_0(g_0 \psi ) = \int g_0 \psi \, dm$ , since m is conformal for $\mathcal {L}_0$ . Indeed, $\mathring {e}_{\boldsymbol {\varepsilon }}$ defines a conformal measure $\mathring {m}_{\boldsymbol {\varepsilon }}$ for $\mathring {\mathcal {L}}_{\boldsymbol {\varepsilon }}$ , so that $\mathring {e}_{\boldsymbol {\varepsilon }}(\psi ) = \int _Y \psi \, d\mathring {m}_{\boldsymbol {\varepsilon }}$ and $d\nu _{\boldsymbol {\varepsilon }} = \mathring {g}_{\boldsymbol {\varepsilon }} \,d\mathring {m}_{\boldsymbol {\varepsilon }}$ . Thus, for $\psi \in BV(Y)$ and $\boldsymbol {\varepsilon }, \boldsymbol {\varepsilon }^{\prime } < \varepsilon _\beta $ ,

(4.15) $$ \begin{align} | \nu_{\boldsymbol{\varepsilon}}(\psi) - \nu_{{\boldsymbol{\varepsilon} ^{\prime}}}(\psi) | & \leqslant | \mathring{e}_{\boldsymbol{\varepsilon}}(\mathring{g}_{\boldsymbol{\varepsilon}} \psi - \mathring{g}_{\boldsymbol{\varepsilon}^{\prime}} \psi) | + | \mathring{e}_{\boldsymbol{\varepsilon}}(\mathring{g}_{\boldsymbol{\varepsilon}^{\prime}} \psi) - e_{\boldsymbol{\varepsilon}^{\prime}}(\mathring{g}_{\boldsymbol{\varepsilon}^{\prime}} \psi)| \nonumber \\& \leqslant \bigg|\kern-2pt \int (\mathring{g}_{\boldsymbol{\varepsilon}} - \mathring{g}_{\boldsymbol{\varepsilon}^{\prime}} ) \psi \, d\mathring{m}_{\boldsymbol{\varepsilon}} \bigg| + \bigg|\kern-2pt \int ( \Pi_{\boldsymbol{\varepsilon}}(\mathring{g}_{\boldsymbol{\varepsilon}^{\prime}} \psi) - \Pi_{\boldsymbol{\varepsilon}^{\prime}}(\mathring{g}_{\boldsymbol{\varepsilon}^{\prime}} \psi) ) \, dm \bigg| \nonumber\\& \leqslant |\psi|_\infty | \mathring{g}_{\boldsymbol{\varepsilon}} - \mathring{g}_{\boldsymbol{\varepsilon}^{\prime}} |_{L^1(\mathring{m}_{\boldsymbol{\varepsilon}})} + ||| \Pi_{\boldsymbol{\varepsilon}} - \Pi_{\boldsymbol{\varepsilon}^{\prime}} ||| \, \| \mathring{g}_{\boldsymbol{\varepsilon}^{\prime}} \psi \|_{BV}. \end{align} $$

Both differences go to 0 as $\boldsymbol {\varepsilon }^{\prime } \to \boldsymbol {\varepsilon }$ by Theorem 4.7, while $\| \mathring {g}_{\boldsymbol {\varepsilon }^{\prime }} \|_{BV}$ is uniformly bounded in $\boldsymbol {\varepsilon }^{\prime }$ by Proposition 4.5. We conclude that $\nu _{\boldsymbol {\varepsilon }}$ is continuous in $\boldsymbol {\varepsilon }$ for fixed $\beta $ when acting on $BV$ functions.

It remains to prove equation (4.14). Unfortunately, $\tau \notin BV(Y)$ , so first we must show that $\nu _{\boldsymbol {\varepsilon }}(\tau )$ is well defined. Indeed, it is easy to check that $\mathring {\mathcal {L}}_{\boldsymbol {\varepsilon }} \tau \in BV$ . This holds since $\tau $ is constant on each 1-cylinder $Y_{i,\boldsymbol {\varepsilon }}$ for $F_{\boldsymbol {\varepsilon }}$ . Thus, $\mathring {\mathcal {L}}_{\boldsymbol {\varepsilon }} \tau $ has discontinuities only at the endpoints of $\{ \mathring {F}_{\boldsymbol {\varepsilon }}(Y_{i, \boldsymbol {\varepsilon }}) \}_i $ , which is a finite collection of intervals.

It follows also that $\mathring {\mathcal {L}}_{\boldsymbol {\varepsilon }} (\tau \mathring {g}_{\boldsymbol {\varepsilon }}) \in BV(Y)$ . Thus, using equation (4.11),

$$ \begin{align*} \begin{split} \lim_{n \to \infty} \Lambda_{\boldsymbol{\varepsilon}}^{-n} \int_{\mathring{Y}^n_{\boldsymbol{\varepsilon}}} \tau \mathring{g}_{\boldsymbol{\varepsilon}} \, dm &= \lim_{n \to \infty} \Lambda_{\boldsymbol{\varepsilon}}^{-n} \int_Y \mathring{\mathcal{L}}_{\boldsymbol{\varepsilon}}^{n-1}(\mathring{\mathcal{L}}_{\boldsymbol{\varepsilon}}(\tau \mathring{g}_{\boldsymbol{\varepsilon}})) \, dm \\ & = \Lambda_{\boldsymbol{\varepsilon}}^{-1} \int \Pi_{\boldsymbol{\varepsilon}} (\mathring{\mathcal{L}}_{\boldsymbol{\varepsilon}}(\tau \mathring{g}_{\boldsymbol{\varepsilon}})) \, dm + \lim_{n \to \infty} \int \Lambda_{\boldsymbol{\varepsilon}}^{-n} {\mathcal R}_{\boldsymbol{\varepsilon}}^{n-1} ( \mathring{\mathcal{L}}_{\boldsymbol{\varepsilon}}(\tau \mathring{g}_{\boldsymbol{\varepsilon}}) ) \, dm , \end{split} \end{align*} $$

and the second term converges to 0 by Theorem 4.7. Thus, the limit defining $\nu _{\boldsymbol {\varepsilon }}(\tau )$ exists and is uniformly bounded in $\boldsymbol {\varepsilon }$ for fixed $\beta $ . More than this, the above calculation can be improved to show that $\tau $ is uniformly (in $\boldsymbol {\varepsilon }$ ) integrable with respect to $\nu _{\boldsymbol {\varepsilon }}$ , as follows. For each $N>0$ , we use the above to estimate

$$ \begin{align*} \nu_{\boldsymbol{\varepsilon}}(1_{\tau> N} \cdot \tau) = \lim_{n \to \infty} \Lambda_{\boldsymbol{\varepsilon}}^{-n} \int \mathring{\mathcal{L}}_{\boldsymbol{\varepsilon}}^{n-1}(\mathring{\mathcal{L}}_{\boldsymbol{\varepsilon}}( 1_{\tau>N} \cdot \tau \mathring{g}_{\boldsymbol{\varepsilon}})) \, dm \leqslant \Lambda_{\boldsymbol{\varepsilon}}^{-1} | \mathring{\mathcal{L}}_{\boldsymbol{\varepsilon}}( 1_{\tau>N} \cdot \tau \mathring{g}_{\boldsymbol{\varepsilon}})|_\infty. \end{align*} $$

Then, using bounded distortion as in equation (4.19), one has for $x \in Y$ ,

where we have used Proposition 2.5(c).

It follows that for each $\kappa\kern1.4pt{>}\kern1.4pt0$ , there exists $N\kern1.4pt{>}\kern1.4pt0$ such that ${\sup _{\boldsymbol {\varepsilon } \in [0, \varepsilon _\beta )} \{ \nu _{\boldsymbol {\varepsilon }}(1_{\tau> N} \cdot \tau ) \} \kern1.4pt{<}\kern1.4pt \kappa} $ , where the sup is restricted to values of $\boldsymbol {\varepsilon }$ such that $H_{\boldsymbol {\varepsilon }}$ is $\beta $ -allowable. Let $\tau ^{(N)} = \min \{ \tau , N \}$ and note that $\tau ^{(N)} \in BV(Y)$ . Then, taking a limit along $\boldsymbol {\varepsilon }^{\prime }$ corresponding to $\beta $ -allowable $H_{\boldsymbol {\varepsilon }^{\prime }}$ yields

$$ \begin{align*} \lim_{\boldsymbol{\varepsilon}^{\prime} \to \boldsymbol{\varepsilon}} | \nu_{\boldsymbol{\varepsilon}}(\tau) - \nu_{{\boldsymbol{\varepsilon} ^{\prime}}}(\tau)| &\leqslant \lim_{\boldsymbol{\varepsilon}^{\prime} \to \boldsymbol{\varepsilon}} | \nu_{\boldsymbol{\varepsilon}}(\tau^{(N)}) - \nu_{{\boldsymbol{\varepsilon} ^{\prime}}}(\tau^{(N)})|\\ &\quad+ |\nu_{\boldsymbol{\varepsilon}}(1_{\tau>N} \cdot \tau)| + |\nu_{{\boldsymbol{\varepsilon} ^{\prime}}}(1_{\tau>N} \cdot \tau)| \leqslant 2\kappa , \end{align*} $$

since we have shown that $\nu _{\boldsymbol {\varepsilon }^{\prime }} \to \nu _{\boldsymbol {\varepsilon }}$ as elements of $BV(Y)^*$ . Since $\kappa>0$ was arbitrary, this proves that $\nu _{\boldsymbol {\varepsilon }}(\tau )$ varies continuously in $\boldsymbol {\varepsilon }$ .

Recall that $\{ Y_i \}_i$ denotes the set of 1-cylinders for $F_0$ and let ${\mathcal J}_0 = \{ F_0(Y_i) \}_i$ denote the finite set of images. The covering property of f implies that the set of preimages of endpoints of elements of ${\mathcal J}_0$ is dense in I. Thus, there is a dense set of $\boldsymbol {\varepsilon } < \varepsilon _\beta $ such that $F_{\boldsymbol {\varepsilon }}$ admits a countable Markov partition with finite images. For such $\boldsymbol {\varepsilon }$ , [Reference Demers and ToddDT1, §6.4.1] implies that $\nu _{\boldsymbol {\varepsilon }}$ is an equilibrium state for the potential $- \Xi _{\boldsymbol {\varepsilon }} \cdot \log (DF_{\boldsymbol {\varepsilon }}) - \log \Lambda _{\boldsymbol {\varepsilon }}$ , where $\Xi _{\boldsymbol {\varepsilon }}(x) = 1$ if $x \in Y \setminus H_{\boldsymbol {\varepsilon }}^{\prime }$ and $\Xi _{\boldsymbol {\varepsilon }} = \infty $ if $x \in H_{\boldsymbol {\varepsilon }}^{\prime }$ . Similarly, [Reference Bruin, Demers and MelbourneBDM, Lemma 5.3] implies that $\nu _{\boldsymbol {\varepsilon }}$ is a Gibbs measure for the potential $- \Xi _{\boldsymbol {\varepsilon }} \cdot \log (DF_{\boldsymbol {\varepsilon }}) - \tau \mathfrak {e}(H_{\boldsymbol {\varepsilon }}(z))$ with pressure equal to 0. Putting these together yields equation (4.14) for such ‘Markov holes’.

Finally, we extend the relation to all $\boldsymbol {\varepsilon }$ via the continuity of $\Lambda _{\boldsymbol {\varepsilon }}$ and $\nu _{\boldsymbol {\varepsilon }}(\tau )$ . Since $\mathfrak {e}(H_{\boldsymbol {\varepsilon }}(z))$ is monotonic in $\boldsymbol {\varepsilon }$ and equals $-\log \Lambda _{\boldsymbol {\varepsilon }}/\nu _{\boldsymbol {\varepsilon }}(\tau )$ on a dense set of $\boldsymbol {\varepsilon }$ , it must also be continuous in $\boldsymbol {\varepsilon }$ . Thus, the relation in equation (4.14) holds for all $\boldsymbol {\varepsilon } < \varepsilon _\beta $ for which $H_{\boldsymbol {\varepsilon }}$ is $\beta $ -allowable.

With Lemma 4.8 in hand, we see that the limit we would like to compute to prove Theorem 1.1 can be expressed as follows:

(4.16) $$ \begin{align} \frac{\mathfrak{e}(H_{\boldsymbol{\varepsilon}}(z))}{\mu(H_{\boldsymbol{\varepsilon}}(z))} = \frac{- \log \Lambda_{\boldsymbol{\varepsilon}}}{\mu_Y(H_{\boldsymbol{\varepsilon}}^{\prime})} \cdot \frac{\int \tau \, d\mu_Y}{\int \tau \, d\nu_{\boldsymbol{\varepsilon}} } \cdot \frac{\mu(H_{\boldsymbol{\varepsilon}}^{\prime})}{\mu(H_{\boldsymbol{\varepsilon}}(z))} , \end{align} $$

where, as before, $\mu _Y = {\mu |_Y}/{\mu (Y)}$ , and $1/\mu (Y) = \int _Y \tau \, d\mu _Y$ by Kac’s lemma since $F_\varepsilon $ is a first return map to Y. Theorem 1.1 will follow once we show that as $\boldsymbol {\varepsilon } \to 0$ ,

(4.17)

The third limit above is precisely equation (3.4) when $z \in \mbox {orb}(f(c))$ is periodic (and is simply 1 by §3.4 when $z \in \mbox {orb}(f(c))$ is preperiodic), so we proceed to prove the first two. We first prove these limits for $\beta $ -allowable holes with a fixed $\beta>0$ in Lemmas 4.9 and 4.10. Then, in §5, we show how to obtain the general limit as $\boldsymbol {\varepsilon } \to 0$ .

Lemma 4.9. For fixed $\beta>0$ and any sequence of $\beta $ -allowable holes $H_{\boldsymbol {\varepsilon }}(z)$ ,

$$ \begin{align*} \lim_{\boldsymbol{\varepsilon} \to 0} \frac{- \log \Lambda_{\boldsymbol{\varepsilon}}}{\mu_Y(H_{\boldsymbol{\varepsilon}}^{\prime})} = 1. \end{align*} $$

Proof. The lemma could follow using the results of [Reference Keller and LiveraniKL2], yet since $z \notin Y$ in our setting, it is not clear how to verify the aperiodicity of $H_{\boldsymbol {\varepsilon }}^{\prime }$ without imposing an additional condition on the reentry of points to Y which have spent some time in a neighbourhood of z. To avoid this, we will argue directly, as in [Reference Demers and ToddDT2, Proof of Theorem 7.2], yet our argument is simpler since our operators $\mathring {\mathcal {L}}_{\boldsymbol {\varepsilon }}$ approach a fixed operator $\mathcal {L}_0$ via Lemma 4.6 in contrast to the situation in [Reference Demers and ToddDT2].

We assume that $\boldsymbol {\varepsilon } < \varepsilon _\beta $ so that we are in the setting of Theorem 4.7. Then, since ${g_0 \in BV(Y)}$ , iterating equation (4.11) yields for any $n \geqslant 1$ ,

$$ \begin{align*} \mathring{\mathcal{L}}_{\boldsymbol{\varepsilon}}^n g_0 = \Lambda_{\boldsymbol{\varepsilon}}^n \mathring{e}_{\boldsymbol{\varepsilon}}(g_0) \mathring{g}_{\boldsymbol{\varepsilon}} + {\mathcal R}_{\boldsymbol{\varepsilon}} g_0 \implies \mathring{g}_{\boldsymbol{\varepsilon}} = \frac{1}{\mathring{e}_{\boldsymbol{\varepsilon}}(g_0)} ( \Lambda_{\boldsymbol{\varepsilon}}^{-n} \mathring{\mathcal{L}}_{\boldsymbol{\varepsilon}}^n g_0 - \Lambda_{\boldsymbol{\varepsilon}}^{-n} {\mathcal R}_{\boldsymbol{\varepsilon}}^n g_0 ). \end{align*} $$

Using this relation and Lemma 4.6 yields

(4.18) $$ \begin{align} 1 - \Lambda_{\boldsymbol{\varepsilon}} & = \int \mathring{g}_{\boldsymbol{\varepsilon}} \, dm - \int \mathring{\mathcal{L}}_{\boldsymbol{\varepsilon}} \mathring{g}_{\boldsymbol{\varepsilon}} \, dm = \int (\mathcal{L}_0 - \mathring{\mathcal{L}}_{\boldsymbol{\varepsilon}}) \mathring{g}_{\boldsymbol{\varepsilon}} \, dm = \int_{H_{\boldsymbol{\varepsilon}}^{\prime}} \mathring{g}_{\boldsymbol{\varepsilon}} \, dm \nonumber \\ & = \frac{1}{\mathring{e}_{\boldsymbol{\varepsilon}}(g_0)} \int_{H_{\boldsymbol{\varepsilon}}^{\prime}} ( \Lambda_{\boldsymbol{\varepsilon}}^{-n} \mathring{\mathcal{L}}_{\boldsymbol{\varepsilon}}^n g_0 - \Lambda_{\boldsymbol{\varepsilon}}^{-n} {\mathcal R}_{\boldsymbol{\varepsilon}}^n g_0 ) \, dm \nonumber\\ & = \frac{1}{\mathring{e}_{\boldsymbol{\varepsilon}}(g_0)} \bigg( \int_{H_{\boldsymbol{\varepsilon}}^{\prime}} g_0 \, dm - \int_{H_{\boldsymbol{\varepsilon}}^{\prime}} (1-\Lambda_{\boldsymbol{\varepsilon}}^{-n} \mathring{\mathcal{L}}_{\boldsymbol{\varepsilon}}) g_0 \, dm - \int_{H_{\boldsymbol{\varepsilon}}^{\prime}} \Lambda_{\boldsymbol{\varepsilon}}^{-n} {\mathcal R}_{\boldsymbol{\varepsilon}}^n g_0 \, dm \bigg). \end{align} $$

By Theorem 4.7, the third term on the right-hand side of equation (4.18) is bounded by

$$ \begin{align*} \int_{H_{\boldsymbol{\varepsilon}}^{\prime}} \Lambda_{\boldsymbol{\varepsilon}}^{-n} {\mathcal R}_{\boldsymbol{\varepsilon}}^n g_0 \, dm \leqslant B_\beta e^{- \eta_\beta n} \| g_0 \|_{BV} c_g^{-1} \int_{H_{\boldsymbol{\varepsilon}}^{\prime}} g_0 \, dm , \end{align*} $$

where $c_g = \inf _Y g_0> 0$ according to equation (4.12).

Next, the second term on the right side of equation (4.18) can be expressed as

$$ \begin{align*} \int_{H_{\boldsymbol{\varepsilon}}^{\prime}} (1-\Lambda_{\boldsymbol{\varepsilon}}^{-n} \mathring{\mathcal{L}}_{\boldsymbol{\varepsilon}}) g_0 \, dm = (1-\Lambda_{\boldsymbol{\varepsilon}}^{-n}) \int_{H_{\boldsymbol{\varepsilon}}^{\prime}} g_0 \, dm + \Lambda_{\boldsymbol{\varepsilon}}^{-n} \int_{H_{\boldsymbol{\varepsilon}}^{\prime}} (\mathcal{L}_0^n - \mathring{\mathcal{L}}_{\boldsymbol{\varepsilon}}^n) g_0 \, dm , \end{align*} $$

using the fact that $\mathcal {L}_0 g_0 = g_0$ . We claim that $(\mathcal {L}_0^n - \mathring {\mathcal {L}}_{\boldsymbol {\varepsilon }}^n) g_0$ can be made small in $L^\infty (Y)$ . To see this, choose $n = kn_0$ and write for $x \in Y$ ,

$$ \begin{align*} (\mathcal{L}_0^n - \mathring{\mathcal{L}}_{\boldsymbol{\varepsilon}}^n) g_0(x) = \sum_{y \in F_0^{-n} x} \frac{g_0(y) 1_{Y \setminus \mathring{Y}^n_{\boldsymbol{\varepsilon}}}(y)}{|DF_0^n(y)|} = \sum_{j=0}^{k-1} \sum_{y \in F_0^{-n}x} \frac{g_0(y) 1_{\mathring{Y}_{\boldsymbol{\varepsilon}}^{jn_0} \setminus \mathring{Y}_{\boldsymbol{\varepsilon}}^{(j+1)n_0}}(y)}{|D F_0^n(y)|} , \end{align*} $$

where we have used the fact that when $n = kn_0$ , $Y \setminus \mathring {Y}_{\boldsymbol {\varepsilon }}^n = \bigcup _{j=0}^{k-1} (\mathring {Y}_{\boldsymbol {\varepsilon }}^{jn_0} \setminus \mathring {Y}_{\boldsymbol {\varepsilon }}^{(j+1)n_0})$ and the union is disjoint. Next, if $Y_{i,j, \varepsilon }(y)$ is the $jn_0$ -cylinder for $F_{\boldsymbol {\varepsilon }}^{jn_0}$ containing y, then by bounded distortion and Lemma 4.3,

(4.19) $$ \begin{align} |DF^{jn_0}(y)| \geqslant \frac{m(F^{jn_0}(Y_{i,j, \varepsilon}(y))}{(1+C_d)m(Y_{i,j, \varepsilon}(y))} \geqslant \frac{C_\beta}{(1+C_d)m(Y_{i,j,\varepsilon}(y))}. \end{align} $$

Then, applying also equation (4.2), we have

$$ \begin{align*} (\mathcal{L}_0^n - \mathring{\mathcal{L}}_{\boldsymbol{\varepsilon}}^n) g_0(x) \leqslant \frac{(1+C_d)|g_0|_\infty}{C_\beta} \sum_{j=0}^{k-1} m(\mathring{Y}^{jn_0}_{\boldsymbol{\varepsilon}} \setminus \mathring{Y}^{(j+1)n_0}_{\boldsymbol{\varepsilon}}) 3^{k-j} =: \rho_n(\boldsymbol{\varepsilon}) , \end{align*} $$

and note that $\rho _n(\boldsymbol {\varepsilon }) \to 0$ as $\boldsymbol {\varepsilon } \to 0$ for fixed $n = kn_0$ . Thus, the second term on the right-hand side of equation (4.18) is estimated by

$$ \begin{align*} \int_{H_{\boldsymbol{\varepsilon}}^{\prime}} (1-\Lambda_{\boldsymbol{\varepsilon}}^{-n} \mathring{\mathcal{L}}_{\boldsymbol{\varepsilon}}) g_0 \, dm \leqslant (1- \Lambda_{\boldsymbol{\varepsilon}}^{-n} + c_g^{-1} \Lambda_{\boldsymbol{\varepsilon}}^{-n} \rho_n(\boldsymbol{\varepsilon}) ) \mu_Y(H_{\boldsymbol{\varepsilon}}^{\prime}). \end{align*} $$

Putting these estimates into equation (4.18) yields

$$ \begin{align*} \frac{1- \Lambda_{\boldsymbol{\varepsilon}}}{\mu_Y(H_{\boldsymbol{\varepsilon}}^{\prime})} = \frac{1}{\mathring{e}_{\boldsymbol{\varepsilon}}(g_0)} ( 1 + \mathcal{O}( e^{-\eta_\beta n} + (1-\Lambda_{\boldsymbol{\varepsilon}}^{-n}) + \Lambda_{\boldsymbol{\varepsilon}}^{-n} \rho_n(\boldsymbol{\varepsilon}) ). \end{align*} $$

Fixing $\kappa>0$ , first choose $n = kn_0$ so that $e^{-\eta _\beta n} < \kappa $ . Next, choose $\boldsymbol {\varepsilon }$ so small that by the continuity of the spectral data from the proof of Theorem 4.7, $|1-\Lambda _{\boldsymbol {\varepsilon }}^{-n}| < \kappa $ , $\Lambda _{\boldsymbol {\varepsilon }}^{-n} \leqslant 2$ , $\rho _n(\boldsymbol {\varepsilon }) < \kappa $ and $|\mathring {e}_{\boldsymbol {\varepsilon }}(g_0) -1| < \kappa $ . This last bound is possible since ${e_0(g_0) = \int g_0 \, dm = 1}$ . Thus, the relevant expression is $1 + \mathcal {O}(\kappa )$ for $\boldsymbol {\varepsilon }$ sufficiently small and $H_{\boldsymbol {\varepsilon }}\ \beta $ -allowable. Since $\kappa $ is arbitrary, the lemma is proved.

Recall that the inducing time $\tau $ for $F_{\boldsymbol {\varepsilon }}$ does not depend on $\boldsymbol {\varepsilon }$ .

Lemma 4.10. For fixed $\beta>0$ and any sequence of $\beta $ -allowable holes $H_{\boldsymbol {\varepsilon }}(z)$ ,

$$ \begin{align*} \lim_{\boldsymbol{\varepsilon} \to 0} \frac{\int \tau \, d\mu_Y}{\int \tau \, d\nu_{\boldsymbol{\varepsilon}} } = 1. \end{align*} $$

6.1 A non-Markovian tower with a hole

Throughout §§6.1–6.3, we will assume $z \in \mbox {orb}(f(c))$ .

Recall the induced map $F_0^{n_0}$ , where $n_0$ is chosen so that equation (4.2) holds. Indeed, by the proof of Proposition 2.5 (see also Lemma 4.1), there exists $\gamma>0$ such that

(6.1) $$ \begin{align} \inf_{x \in Y} |Df^{\tau^{n_0}(x)}(x)| \, e^{-\gamma \tau^{n_0}(x)}> 2.5 , \end{align} $$

where $\tau ^{n_0} = \sum _{k = 0}^{n_0-1} \tau \circ F_0^k$ .

Using $F_0^{n_0}$ , we define an extended system that resembles a Young tower as follows. Define

$$ \begin{align*} \Delta = \{ (y, k) \in Y \times \mathbb{N} : k \leqslant \tau^{n_0} -1 \}. \end{align*} $$

We refer to the kth level of the tower as $\Delta _k = \{ (y, n) \in \Delta : n = k \}$ . Sometimes, we refer to $(x, k) \in \Delta _k$ as x if k is clear from context.

The dynamics on the tower is defined by $f_\Delta (y,k) = (y,k+1)$ if $k < \tau ^{n_0} -1$ and $f_\Delta (y,\tau ^{n_0}(y) -1) = (F_0^{n_0}(y), 0)$ . There is a natural projection $\pi _\Delta : \Delta \to I$ which satisfies $f \circ \pi _\Delta = \pi _\Delta \circ f_\Delta $ . Clearly, $\pi _\Delta (\Delta _0) = Y$ . Remark that $F_0$ , $f_\Delta $ and $\Delta $ depend on z, but not on $\boldsymbol {\varepsilon }$ .

It follows from Lemma 2.3(c) and the fact that f is locally eventually onto that $f_\Delta $ is mixing. Indeed, $f(Y) \supset Y$ and the chain structure around z implies $f^{k_0+p}(Y) \supset I$ . Then, since f is locally eventually onto, for any interval $A \subset Y$ , there exists $n_A \in \mathbb {N}$ such that $f^{n_A}(A) \supset Y$ , and again by Lemma 2.3(c), $f^{n_A+k}(A) \supset Y$ for all $k \geqslant 0$ . Setting $A_0 = (\pi _\Delta |_{\Delta _0})^{-1}(A)$ , this implies $f_\Delta ^{n_A+k}(A_0) \supset \bigcup _{j=0}^k \Delta _j$ for each $k \geqslant 0$ . Thus, $f_\Delta $ is topological mixing. (Indeed, the above argument also implies that there exist $x, y \in A_0$ such that $\tau ^{n_0}(x) = n_A$ and $\tau ^{n_0}(y) = n_A + 1$ so the greatest common divisor g.c.d. $(\tau ^{n_0})=1$ , yet this is not a sufficient condition for $f_\Delta $ to be mixing since our tower has multiple bases.)

We lift Lebesgue measure to $\Delta $ simply by defining $m_\Delta |_{\Delta _0} = m|_Y$ and $m_\Delta |_{\Delta _k} = (f_\Delta ^k)_*(m_\Delta |_{\Delta _0})$ . Note that by Proposition 2.5, $m_{\Delta }(\tau ^{n_0}> n) \leqslant C e^{- \zeta n}$ , where $\zeta =\log \unicode{x3bb} _{\text per} - \eta> 0$ and $\eta $ is chosen before Remark 2.6. Thus, our tower has exponential tails.

Now recall $\alpha \in (0, \infty )$ from the statement of Theorem 1.2. This will determine the scaling at which we consider the hitting time to $H_{\boldsymbol {\varepsilon }}(z)$ . We reduce $\gamma>0$ in equation (6.1) if necessary so that

(6.2) $$ \begin{align} \gamma < \zeta \quad \mbox{and} \quad \gamma/\zeta < \alpha. \end{align} $$

Note that the second condition is relevant only if we consider $\alpha < 1$ .

Given a hole $H_{\boldsymbol {\varepsilon }}(z)$ , define $H_{\Delta _k}(\boldsymbol {\varepsilon }) = (\pi _\Delta |_{\Delta _k})^{-1}(H_{\boldsymbol {\varepsilon }}(z))$ and $H_\Delta (\boldsymbol {\varepsilon }) = \bigcup _{k \geqslant 1} H_{\Delta _k}$ . The corresponding punctured tower is defined analogously to equation (4.1) for $n \geqslant 1$ ,

$$ \begin{align*} \mathring{f}_{\Delta, \boldsymbol{\varepsilon}}^n = f_{\Delta}^n|_{\mathring{\Delta}_{\boldsymbol{\varepsilon}}^n} \quad\mbox{where } \mathring{\Delta}_{\boldsymbol{\varepsilon}}^n := \bigcap_{i=0}^{n-1} f_\Delta^{-i}(\Delta \setminus H_{\Delta}(\boldsymbol{\varepsilon})). \end{align*} $$

The main difference between $f_\Delta : \Delta \circlearrowleft $ and the usual notion of a Young tower is that we do not define a Markov structure on $\Delta $ . Yet as demonstrated above, the tower has the following key properties:

• • exponential tails: $m_{\Delta }(\tau ^{n_0}> n) \leqslant C e^{-\zeta n}$ ;

• • $f_\Delta $ is topologically mixing;

• • large images at returns to $\Delta _0$ : if $H_{\boldsymbol {\varepsilon }}(z)$ is $\beta $ -allowable, then $C_\beta>0$ from Lemma 4.3 provides a lower bound on the length of images in returns to $\Delta _0$ under both $f_\Delta $ and $\mathring {f}_{\Delta , \boldsymbol {\varepsilon }}$ .

We will use these properties to prove the existence of a uniform spectral gap for the punctured transfer operators on the tower acting on a space of functions of weighted bounded variation, as follows.

The inducing domain Y comprises finitely many maximal connected components that are intervals in I. Let $\widetilde {{\mathcal D}}_0$ denote this collection of intervals. Similarly, for each $k \geqslant 1$ , $f^k(Y \cap \{ \tau ^{n_0} \geqslant k+1 \})$ can be decomposed into a set of maximal connected components. We further subdivide these intervals at the first time $j \leqslant k$ that they contain a point in orb $(f(c))$ , $z - \varepsilon _L$ or $z + \varepsilon _R$ . These are the cuts introduced in the definition of $F_\varepsilon $ in §2.3. Let $\widetilde {{\mathcal D}}_k$ denote this collection of intervals and define the collection of lifts by

$$ \begin{align*} {\mathcal D}_k = \{ J = (\pi|_{\Delta_k})^{-1}(\tilde J) : \tilde J \in \widetilde{{\mathcal D}}_k \} \quad\text{for all } k \geqslant 0. \end{align*} $$

For an interval $J \in {\mathcal D}_k$ and a measurable function $\psi : \Delta \to \mathbb {R}$ , define $\bigvee _J \psi $ , the variation of $\psi $ on J as in equation (4.3). On each level $\Delta _k$ , $k \geqslant 1$ , define

$$ \begin{align*} \| \psi \|_{\Delta_k} = e^{-\gamma k} \sum_{J \in {\mathcal D}_k} \bigvee_J \psi + e^{-\gamma k} \sup_{\Delta_k} |\psi|. \end{align*} $$

Finally, define the weighted variation norm on $\Delta $ by

$$ \begin{align*} \| \psi \|_{WV} = \sum_{k \geqslant 0} \| \psi \|_{\Delta_k}. \end{align*} $$

If $\| \psi \|_{WV} < \infty $ , then $| \psi |_{L^1(m_\Delta )} < \infty $ since $\gamma < \zeta $ by equation (6.2). So we denote by $WV(\Delta )$ the set of functions $\psi \in L^1(m_\Delta )$ such that $\| \psi \|_{WV} < \infty $ . Since we consider $WV(\Delta )$ as a subset of $L^1(m_\Delta )$ , we define the variation norm of the equivalence class to be the infimum of variation norms of functions in the equivalence class.

We define the transfer operator $\mathcal {L}_\Delta $ corresponding to $f_\Delta $ acting on $L^1(m_\Delta )$ in the natural way,

$$ \begin{align*} \mathcal{L}_\Delta \psi(x) = \sum_{y \in f_\Delta^{-1}(x)} \frac{f(y)}{Jf_\Delta(y)} , \end{align*} $$

where $Jf_\Delta $ is the Jacobian of $f_\Delta $ with respect to $m_\Delta $ . Then, the transfer operator for the punctured tower is defined for each $n \geqslant 1$ by

$$ \begin{align*} \mathring{\mathcal{L}}_{\Delta, \boldsymbol{\varepsilon}}^n \psi = \mathcal{L}_\Delta^n (1_{\mathring{\Delta}_{\boldsymbol{\varepsilon}}^n} \psi). \end{align*} $$

Our goal is to prove the following proposition, which is the analogue of Theorem 4.7, but for the tower map rather than the induced map.

Proposition 6.1. For any $\beta>0$ , there exists $\varepsilon _\beta (\Delta )$ such that for any $\beta $ -allowable hole $H_{\boldsymbol {\varepsilon }}(z)$ with $\boldsymbol {\varepsilon } < \varepsilon _\beta (\Delta )$ , both $\mathcal {L}_\Delta $ and $\mathring {\mathcal {L}}_{\Delta , \boldsymbol {\varepsilon }}$ have a spectral gap on $WV(\Delta )$ .

Specifically, there exist $\sigma _\beta , A_\beta>0$ such that for all $\beta $ -allowable holes $H_{\boldsymbol {\varepsilon }}$ with $\boldsymbol {\varepsilon } < \varepsilon _\beta (\Delta )$ , the spectral radius of $\mathring {\mathcal {L}}_{\Delta , \boldsymbol {\varepsilon }}$ is $\unicode{x3bb} _{\Delta , \boldsymbol {\varepsilon }} < 1$ and there exist operators $\Pi _{\Delta , \boldsymbol {\varepsilon }}, {\mathcal R}_{\Delta , \boldsymbol {\varepsilon }} : WV(\Delta ) \circlearrowleft $ satisfying $\Pi _{\Delta , \boldsymbol {\varepsilon }}^2 = \Pi _{\Delta , \boldsymbol {\varepsilon }}$ , $\Pi _{\Delta , \boldsymbol {\varepsilon }} {\mathcal R}_{\Delta , \boldsymbol {\varepsilon }} = {\mathcal R}_{\Delta , \boldsymbol {\varepsilon }} \Pi _{\Delta , \boldsymbol {\varepsilon }} = 0$ and $\| {\mathcal R}_{\Delta , \boldsymbol {\varepsilon }}^n \|_{WV} \leqslant A_\beta \unicode{x3bb} _{\Delta , \boldsymbol {\varepsilon }}^n e^{-\sigma _\beta n}$ such that

Indeed, $\Pi _{\Delta , \boldsymbol {\varepsilon }} = \mathring {e}_{\Delta , \boldsymbol {\varepsilon }} \otimes \mathring {g}_{\Delta , \boldsymbol {\varepsilon }}$ for some $\mathring {e}_{\Delta , \boldsymbol {\varepsilon }} \in WV(\Delta )^*$ and $\mathring {g}_{\Delta , \boldsymbol {\varepsilon }} \in WV(\Delta )$ satisfying $\mathring {\mathcal {L}}_{\Delta , \boldsymbol {\varepsilon }} \mathring {g}_{\Delta , \boldsymbol {\varepsilon }} = \unicode{x3bb} _{\Delta , \boldsymbol {\varepsilon }} \mathring {g}_{\Delta , \boldsymbol {\varepsilon }}$ and $\int _{\Delta } \mathring {g}_{\Delta , \boldsymbol {\varepsilon }} \, dm_\Delta = 1$ .

The proof of this proposition is based on the following sequence of lemmas, which prove the compactness of $WV(\Delta )$ in $L^1(m_\Delta )$ , uniform Lasota–Yorke inequalities for $\mathring {\mathcal {L}}_{\Delta , \boldsymbol {\varepsilon }}$ and the smallness of the perturbation when viewing $\mathcal {L}_{\Delta } - \mathring {\mathcal {L}}_{\Delta , \boldsymbol {\varepsilon }}$ as an operator from $WV(\Delta )$ to $L^1(m_\Delta )$ .

Lemma 6.3. Assume $H_{\boldsymbol {\varepsilon }}(z)$ is $\beta $ -allowable. Let $C = (1+C_d)(C_d + 2C_\beta ^{-1})$ . For $k \geqslant 1$ and all $\psi \in WV(Y)$ ,

(6.3) $$ \begin{align} \| \mathring{\mathcal{L}}_{\Delta, \boldsymbol{\varepsilon}} \psi \|_{\Delta_k} & \leqslant e^{-\gamma } \| \psi \|_{\Delta_{k-1}} \end{align} $$

(6.4) $$ \begin{align} \bigvee_{\Delta_0} \mathring{\mathcal{L}}_{\Delta, \boldsymbol{\varepsilon}} \psi & \leqslant \frac{4}{5} \| \psi \|_{WV} + C \int_{\mathring{\Delta}^1_{\boldsymbol{\varepsilon}}} |\psi| \,dm_{\Delta} \end{align} $$

(6.5) $$ \begin{align} | \mathring{\mathcal{L}}_{\Delta, \boldsymbol{\varepsilon}} \psi |_{L^\infty(\Delta_0)} & \leqslant \frac{2}{5} \| \psi \|_{WV} + C_\beta^{-1}(1+C_d) \int_{\mathring{\Delta}_{\boldsymbol{\varepsilon}}^1} |\psi| \end{align} $$

(6.6) $$ \begin{align} \int_{\Delta} |\mathring{\mathcal{L}}_{\Delta, \boldsymbol{\varepsilon}}^n \psi| | \, dm_\Delta & \leqslant \int_{\mathring{\Delta}_{\boldsymbol{\varepsilon}}^n} |\psi | \, dm_\Delta \quad \mbox{for all } n \geqslant 1. \end{align} $$

The same estimates hold for $\mathcal {L}_\Delta $ with $\mathring {\Delta }_{\boldsymbol {\varepsilon }}^n$ replaced by $\Delta $ .

Proof. Note that $Jf_\Delta (x, k) = 1$ if $k < \tau ^{n_0}(x)-1$ . Thus, if $k \geqslant 1$ and $x \in \Delta _k$ , then $\mathring {\mathcal {L}}_{\Delta , \boldsymbol {\varepsilon }} \psi (x) = (1_{\mathring {\Delta }^1_{\boldsymbol {\varepsilon }}} \psi )\circ f_{\Delta }^{-1}(x)$ .

Moreover, for $J \in {\mathcal D}_k$ and $k \geqslant 1$ , we have $f_\Delta ^{-1}(J) \subset J^{\prime } \in {\mathcal D}_{k-1}$ and by definition of ${\mathcal D}_{k-1}$ , $J^{\prime }$ is either disjoint from $H_\Delta $ or contained in it. Thus, $1_{\mathring {\Delta }^1_{\boldsymbol {\varepsilon }}}$ is either identically 0 or 1 on $J^{\prime }$ and so does not affect the variation. With these points noted, equation (6.3) holds trivially. Similarly, equation (6.6) follows immediately from the definition of $\mathring {\mathcal {L}}_{\Delta , \boldsymbol {\varepsilon }}$ .

We proceed to prove equation (6.4). The proof follows closely the proof of Proposition 4.5. As in §4.1, denote by ${\mathcal J}_{\boldsymbol {\varepsilon }}$ the finite set of intervals in Y that are the images of the 1-cylinders for $F_{\boldsymbol {\varepsilon }}^{n_0}$ . Since we identify $\Delta _0$ with Y, we will abuse notation and again denote the finite set of images $(\pi _\Delta |_{\Delta _0})^{-1}({\mathcal J}_{\boldsymbol {\varepsilon }})$ in $\Delta _0$ by simply ${\mathcal J}_{\boldsymbol {\varepsilon }}$ .

Let ${\mathcal I}_{\boldsymbol {\varepsilon }}$ denote the countable collection of intervals in $f_{\Delta }^{-1}(\Delta _0)$ so that for each ${I_i \in {\mathcal I}_{\boldsymbol {\varepsilon }}}$ , $f_\Delta (I_i) = J$ for some $J \in {\mathcal J}_{\boldsymbol {\varepsilon }}$ . We use the notation $I_i = (u_i, v_i)$ and $I_{i,k}$ denotes the intervals in ${\mathcal I}_{\boldsymbol {\varepsilon }}$ that lie in $\Delta _k$ . Remark that each $I_{i,k} \subset J^{\prime } \in {\mathcal D}_k$ so that $I_{i,k}$ is either in $H_\Delta $ or is disjoint from it. (Indeed, since a neighbourhood of z cannot enter Y in one step, every $I_i \in {\mathcal I}_{\boldsymbol {\varepsilon }}$ is disjoint from $H_\Delta $ in this particular tower.) Also, for $(x,k) \in I_{i,k}$ , by definition, $\tau ^{n_0}(\pi _\Delta (x,0)) = k$ , so that

(6.7) $$ \begin{align} Jf_\Delta(x,k) = |DF^{n_0}(\pi_{\Delta}(x,0))| = |Df^{\tau^{n_0}(x)}(x)| = |Df^k(x)| , \end{align} $$

where for brevity, we denote $\pi _\Delta (x,0) = x$ .

Now for $\psi \in WV(\Delta )$ , following equation (4.6), we obtain

(6.8) $$ \begin{align} \bigvee_{\Delta_0} \mathring{\mathcal{L}}_{\Delta, \boldsymbol{\varepsilon}} \psi \leqslant \sum_{I_i \in {\mathcal I}_{\boldsymbol{\varepsilon}}} \bigvee_{I_i} \frac{\psi}{Jf_\Delta} + \sum_{I_i \in {\mathcal I}_{\boldsymbol{\varepsilon}}} \frac{|\psi(u_i)|}{Jf_\Delta(u_i)} + \frac{|\psi(v_i)|}{Jf_\Delta(v_i)}. \end{align} $$

For $I_i = I_{i,k} \subset \Delta _k$ , using equations (6.1) and (6.7), we estimate the first term above as in equation (4.7),

$$ \begin{align*} \bigvee_{I_{i,k}} \frac{\psi}{Jf_\Delta} \leqslant \frac{1}{2.5} \bigvee_{I_{i,k}} \psi e^{-\gamma k} + C_d(1+C_d) \int_{I_{i,k}} |\psi|. \end{align*} $$

We estimate the second term in equation (6.8) using equation (4.8) and the large images property as in equation (4.9),

(6.9) $$ \begin{align} \frac{|\psi(u_i)|}{Jf_\Delta(u_i)} + \frac{|\psi(v_i)|}{Jf_\Delta(v_i)} \leqslant \frac{1}{2.5} \bigvee_{I_{i,k}} \psi e^{-\gamma k}+ 2 C_\beta^{-1} (1+C_d) \int_{I_{i,k}} |\psi| \, dm_\Delta. \end{align} $$

Putting these estimates together in equation (6.8) completes the proof of equation (6.4),

$$ \begin{align*} \begin{split} \bigvee_{\Delta_0} \mathring{\mathcal{L}}_{\Delta, \boldsymbol{\varepsilon}} \psi & \leqslant \frac{2}{2.5} \sum_{k \geqslant 1} \sum_{I_{i,k}} e^{-\gamma k} \bigvee_{I_{i,k}} \psi + (1+C_d)(C_d + 2C_\beta^{-1}) \int_{I_{i,k}} |\psi| \\ & \leqslant \frac{4}{5} \| \psi\|_{WV} + (1+C_d)(C_d + 2C_\beta^{-1}) \int_{\mathring{\Delta}^1_{\boldsymbol{\varepsilon}}} |\psi|. \end{split} \end{align*} $$

Finally, equation (6.5) follows immediately from equation (6.9) since for $x \in \Delta _0$ ,

$$ \begin{align*} |\mathring{\mathcal{L}}_{\Delta, \boldsymbol{\varepsilon}} \psi(x) | & \leqslant \sum_{I_i \in {\mathcal I}_{\boldsymbol{\varepsilon}}, y \in I_i} \frac{|\psi(y)|}{Jf_\Delta(y)} \leqslant \frac{1}{2.5} \sum_{i,k} \bigvee_{I_{i,k}} \psi e^{-\gamma k} + C_\beta^{-1}(1+C_d) \int_{I_{i,k}} |\psi| \, dm_\Delta \\ & \leqslant \frac{2}{5} \| \psi \|_{WV} + C_\beta^{-1}(1+C_d) \int_{\mathring{\Delta}^1_{\boldsymbol{\varepsilon}}} |\psi| \, dm_{\Delta} .\\[-42pt] \end{align*} $$

Proof. Lemma 6.3 applied to $\mathcal {L}_\Delta $ implies that as an operator on $WV(\Delta )$ , $\mathcal {L}_\Delta $ has spectral radius at most 1 and essential spectral radius at most $\max \{ e^{-\gamma }, 4/5 \}$ . Since $\mathcal {L}_\Delta ^* m_\Delta = m_\Delta $ , 1 is in the spectrum of $\mathcal {L}_\Delta $ so that $\mathcal {L}_\Delta $ is quasi-compact. Indeed, equation (4.5) implies that the peripheral spectrum has no Jordan blocks. Thus, by [Reference KatoKa, III.6.5], $\mathcal {L}_{\Delta }$ admits the following representation: there exist finitely many eigenvalues $e^{i \theta _j}$ , $j = 1, \ldots , N$ and finite-dimensional eigenprojectors $\Pi _j$ corresponding to $\theta _j$ such that

(6.10) $$ \begin{align} \mathcal{L}_\Delta = \sum_{j=1}^N e^{i \theta_j} \Pi_j + {\mathcal R} , \end{align} $$

where ${\mathcal R}$ has spectral radius strictly less than 1 and $\Pi _j \Pi _k = \Pi _j {\mathcal R} = {\mathcal R} \Pi _j = 0$ for all $j \ne k$ .

Note that if $g \in WV(\Delta )$ satisfies $\mathcal {L}_\Delta g = g$ , then $g_0 := g \circ (\pi _{\Delta }|_{\Delta _0})^{-1}$ is in $BV(Y)$ and $\mathcal {L}_0 g_0 = g_0$ . Since $\mathcal {L}_0$ has a spectral gap (see the proof of Theorem 4.7), this implies that there can be at most one (normalized) fixed point for $\mathcal {L}_\Delta $ . Thus, the eigenvalue 1 is simple.

Conversely, if $g_0$ denotes the unique element of $BV(Y)$ with $\int g_0 \, dm =1$ such that $\mathcal {L}_0 g_0 = g_0$ , then we may define $g_\Delta $ such that $\mathcal {L}_\Delta g_\Delta = g_\Delta $ by

$$ \begin{align*} g_\Delta(x,k) = c_0 g_0 \circ \pi_\Delta(x,0) \quad \mbox{for each } (x,k) \in \Delta_k \text{ and } k \geqslant 0, \end{align*} $$

where $c_0$ is chosen so that $\int _\Delta g_0 \, dm_\Delta = 1$ . In particular, for each $(x,k)$ and $k \geqslant 0$ ,

(6.11) $$ \begin{align} c_0 c_g \leqslant g_\Delta(x,k) = g_\Delta(x,0) \leqslant c_0 C_g , \end{align} $$

where $C_g = |g_0|_\infty $ and $c_g>0$ is from equation (4.12). We will use this fact to eliminate the possibility of other eigenvalues of modulus 1.

It is convenient to first establish the following claim.

Claim. The peripheral spectrum of $\mathcal {L}_\Delta $ on $WV(\Delta )$ is cyclic: if $e^{i\theta }$ is an eigenvalue, then so is $e^{i \theta n}$ for each $n \in \mathbb {N}$ .

Proof of Claim

Since $\mathcal {L}_\Delta $ is a positive operator and $\mathcal {L}_\Delta ^* m_\Delta = m_\Delta $ , it follows from Rota’s theorem [Reference Boyarsky and GoraBG, Theorem 2.2.9] (see also [Reference SchaeferS, Theorem 1]) that the peripheral spectrum of $\mathcal {L}_\Delta $ on $L^1(m_\Delta )$ is cyclic. It remains to show that this property holds as well in $WV(\Delta )$ . We follow the strategy in [Reference KellerK, Proof of Theorem 6.1]. For $\theta \in [0, 2\pi )$ , define

$$ \begin{align*} S_{n, \theta} = \frac{1}{n} \sum_{k=0}^{n-1} e^{-i\theta k} \mathcal{L}_{\Delta}^k. \end{align*} $$

It follows from equation (6.10) that $\lim _{n\to \infty } S_{n, \theta } = \Pi _j$ if $\theta = \theta _j$ and $\lim _{n\to \infty } S_{n, \theta } = 0$ otherwise. Indeed, the convergence in both cases is pointwise uniform since $\lim _{n \to \infty } (1/n) \sum _{k=0}^{n-1} e^{i(\theta _j - \theta )k} =0$ whenever $\theta \neq \theta _j$ . Then, $\bigvee c \psi = |c| \bigvee \psi $ for any constant c implies the sequence converges in $\| \cdot \|_{WV}$ .

Since $\mathcal {L}_{\Delta }$ is an $L^1(m_\Delta )$ contraction (that is, $\int |\mathcal {L}_\Delta \psi | \, dm_\Delta \leqslant \int |\psi | \, dm_\Delta $ ), then so is $S_{n,\theta }$ , and since $WV(\Delta )$ is dense in $L^1(m_\Delta )$ , $\lim _{n \to \infty } S_{n,\theta }$ extends to a bounded linear operator on $L^1(m_\Delta )$ , with convergence in $\| \cdot \|_{L^1(m_\Delta )}$ . Taking $\theta = \theta _j$ , we may view $\Pi _j : L^1(m_\Delta ) \to \Pi _j(WV(\Delta ))$ .

Now, if $0 \neq f \in L^1(m_\Delta )$ satisfies $\mathcal {L}_\Delta f = e^{i \theta } f$ , then $0 \ne f = \lim _{n\to \infty } S_{n,\theta }f$ , which implies that $\theta = \theta _j$ for some j and $f \in \Pi _j(WV(\Delta ))$ is an element of $WV(\Delta )$ . Thus, the peripheral spectra of $\mathcal {L}_\Delta $ on $WV(\Delta )$ and $L^1(\Delta )$ coincide.

Returning to the proof of the lemma, suppose for the sake of contradiction that 1 is not the only eigenvalue of modulus 1. Then according to the claim, there exists $h \in WY(\Delta )$ and $p,q \in \mathbb {N}\setminus \{0\}$ such that $\mathcal {L}_\Delta h = e^{i \pi p/q} h$ . It follows that h is complex-valued and that $\int _{\Delta } \text {Re}(h) \, dm_{\Delta } = \int _{\Delta } \text {Im}(h) \, dm_{\Delta } = 0$ .

Since $\mathcal {L}_\Delta ^q h = h$ , h takes on all its possible values in the first q levels, $\bigcup _{k = 0}^{q-1} \Delta _k$ . In particular, $\sup _{\Delta } |\text {Re}(h)| = \sup _{\bigcup _{k=0}^{q-1} \Delta _k} |\text {Re}(h)|$ , and similarly for $\text {Im}(h)$ . Thus, by equation (6.11), we may choose $\kappa> 0$ such that

$$ \begin{align*} \psi := \kappa \text{Re}(h) + g_\Delta \quad \mbox{satisfies } \inf_\Delta \psi> 0. \end{align*} $$

Note $\int _\Delta \psi \, dm_\Delta = 1$ . Next, for $s \in \mathbb {R}$ , define

(6.12) $$ \begin{align} \psi_s = s \psi + (1-s) g_\Delta = s \kappa \text{Re}(h) + g_\Delta. \end{align} $$

Note that $\psi _s$ also takes on all its possible values in $\bigcup _{k=0}^{q-1} \Delta _k$ and $\mathcal {L}_\Delta ^q \psi _s = \psi _s$ .

Let $\mathcal {S} = \{ s \in \mathbb {R} : \operatorname *{\mbox {ess inf}}_{\Delta } \psi _s> 0 \}$ . By construction of $\psi $ and the compactness of $\bigcup _{k=0}^{q-1} \Delta _k$ , $\mathcal {S}$ contains $[0,1]$ and is open.

We will show that $\mathcal {S}$ contains $\mathbb {R}^+$ . Suppose not. Let $t> 1$ be an endpoint of $\mathcal {S}$ that is not in $\mathcal {S}$ . Then, $\operatorname *{\mbox {ess inf}}_\Delta \psi _t = \operatorname *{\mbox {ess inf}}_{\bigcup _{k=0}^{q-1} \Delta _k} \psi _t = 0$ . Without loss of generality, we may work with a representative of $\psi _t$ that is lower semicontinuous. (We use here that any function of bounded variation can be written as the difference of two monotonic functions so that one-sided limits exist at each point [Reference RoydenR, Theorem 5, §5.2].) Since $\int _\Delta \psi _t \, dm_\Delta =1$ , there must exist $(y,j) \in \bigcup _{k=0}^{q-1} \Delta _k$ such that $\psi _t(y)> 0$ . By lower semicontinuity, there exists an interval $A \subset \Delta _j$ such that $\inf _A \psi _t := a> 0$ .

Since $f_\Delta $ is topologically mixing, we can find $N \in \mathbb {N}$ such that $f_\Delta ^{N+i}(A) \supseteq \bigcup _{k=0}^{q-1} \Delta _k$ for $i=0, \ldots , q-1$ . One of these iterates must equal $nq$ for some $n \in \mathbb {N}$ . Thus, for any $x \in \bigcup _{k=0}^{q-1} \Delta _k$ ,

$$ \begin{align*} \psi_t(x) = \mathcal{L}_\Delta^{nq} \psi_t(x) \geqslant \frac{a}{\sup_A |Df^{nq} \circ \pi_\Delta| }> 0 , \end{align*} $$

since $\sup |Df| < \infty $ and n is fixed. This proves that $t \in \mathcal {S}$ so, in fact, $\mathbb {R}^+ \subset \mathcal {S}$ . By equation (6.12), this implies that $\text {Re}(h) \geqslant 0$ , but since $\int _\Delta \text {Re}(h) \, dm_\Delta = 0$ , it must be that ${\text {Re}(h) \equiv 0}$ . A similar argument forces $\text {Im}(h) \equiv 0$ , providing the needed contradiction.