2. Preliminaries

2·1. Hardy spaces

We denote by $H^2$ the usual Hardy space consisting of functions $ f(z)=\sum_{n=0}^{\infty}\widehat{f}_nz^n$ holomorphic in ${\mathbb{D}}$ , for which $\sum_{n=0}^{\infty}\rvert \widehat{f}_n\lvert^2\lt+\infty$ . The norm in $H^2$ is defined as

(2·1) \begin{equation}\lVert\, f \rVert_2\;:\!=\;\lim_{r\to1^-}\left(\frac{1}{2\pi}\int_{{\mathbb{T}}} \lvert\,f(r\zeta) \rvert^2 d\ell(\zeta)\right)^{1/2}\lt+\infty,\end{equation}

where $\ell$ is the Lebesgue probability measure in the unit circle ${\mathbb{T}}$ .

It is known [ Reference Duren9 , theorem 2·2] that for functions $f\in H^2$ , the radial limits $f(\zeta)\;:\!=\;\lim_{r\to1}f(r\zeta)$ exist for Lebesgue almost every $\zeta\in{\mathbb{T}}$ . In fact, the norm in $H^2$ can be obtained by the boundary integral

(2·2) \begin{equation}\lVert\, f \rVert_2=\left(\frac{1}{2\pi}\int_{{\mathbb{T}}} \lvert\,f(\zeta) \rvert^2d\ell(\zeta)\right)^{1/2}.\end{equation}

Another way of computing this norm involving an area integral is given by the Littlewood–Paley Identity [ Reference Shapiro14 , p. 178]

(2·3) \begin{equation}\lVert\, f \rVert_2^2 = \lvert\,f(0) \rvert^2 +2\int_{{\mathbb{D}}} \lvert\,f'(z) \rvert^2 \log \frac{1}{\lvert z\rvert} dA(z),\end{equation}

where A is the Lebesgue probability measure in ${\mathbb{D}}$ .

2·2. Composition operators

For a holomorphic function $f\colon{\mathbb{D}}\to{\mathbb{D}}$ , we define the linear composition operator $C_f\colon H^2\to H^2$ with $C_f(g)=g\circ f$ .

Littlewodd’s Subordination Principle [ Reference Duren9 , theorem 1·7] shows that this operator is well-defined and bounded. In particular, we have $\lVert C_f \rVert \leq \sqrt{({1+\lvert \,f(0) \rvert})/({1-\lvert \,f(0) \rvert})}$ (see [ Reference Shapiro14 , p. 16]). Note that in the special case where $f(0)=0$ this inequality becomes

(2·4) \begin{equation}\lVert C_f \rVert \leq 1.\end{equation}

At this point we need to make a remark about the notation of norms. The notation $\lVert C_f \rVert$ used above denotes the operator norm induced by the $H^2$ norm. We shall follow this convention throughout the article; i.e. $\lVert \cdot \rVert_2$ will denote the norm of a function in $H^2$ , whereas $\lVert \cdot \rVert$ will denote the operator norm of a composition operator.

The main tool for our proof of Theorem 1·1 is evaluating the norms of composition operators in $H^2$ . In order to do so, we require the so-called “Change of variables formula” that appears in [ Reference Shapiro14 , p. 179]. To present this, let us first define the Nevanlinna counting function of a holomorphic function $f\colon{\mathbb{D}}\to{\mathbb{D}}$ , as

\[N_f(w)=\begin{cases}\displaystyle \sum_{z\in f^{-1}(\{w\})}\log \frac{1}{\lvert z \rvert}, \quad \text{for all}, & w\in f({\mathbb{D}})\setminus\{\,f(0)\},\\ {} {} {} {} {}0, &w\notin f({\mathbb{D}})\setminus\{\,f(0)\}, \end{cases}\]

where points in the preimage $f^{-1}(\{w\})$ are repeated according to their multiplicity. The change of variables formula we promised is the following:

(2·5) \begin{equation}\lVert C_f(g) \rVert_2^2 = \lvert g(\,f(0)) \rvert^2 + 2 \int_{{\mathbb{D}}}\lvert g'(w)\rvert^2 N_f(w) dA(w).\end{equation}

We now present some properties of the Nevanlinna counting function that will be important in our proofs. First, an application of Jensen’s formula yields an integral interpretation of the Nevanlinna counting function, as follows (see [ Reference Shapiro14 , p. 187]):

\[N_f(w) = \lim_{r\to1^-}\int_{{\mathbb{T}}}\log\left\lvert \frac{w-f(r\zeta)}{1-\overline{w}f(r\zeta)}\right\rvert d\ell(\zeta) + \log\left\lvert\frac{1-\overline{w}f(0)}{w-f(0)}\right\rvert.\]

Since functions in $H^2$ have radial limits almost everywhere in ${\mathbb{T}}$ , we can use Fatou’s Lemma to obtain that whenever $f\in H^2$ , we have

(2·6) \begin{equation}N_f(w) \leq \int_{{\mathbb{T}}}\log\left\lvert \frac{w-f(\zeta)}{1-\overline{w}f(\zeta)}\right\rvert d\ell(\zeta) + \log\left\lvert\frac{1-\overline{w}f(0)}{w-f(0)}\right\rvert.\end{equation}

Another estimate of the Nevanlinna counting function is the famous Littlewood’s Inequality [ Reference Shapiro14 , p.187], which states that for any $f\colon {\mathbb{D}}\to{\mathbb{D}}$ holomorphic and any $w\in{\mathbb{D}}\setminus \{\,f(0)\}$ , we have

(2·7) \begin{equation}N_f(w)\leq \log\left\lvert \frac{ 1-\overline{w}f(0)}{w-f(0)}\right\rvert.\end{equation}

In fact, (2·6) is simply an intermediate step in the proof of Littlewood’s Inequality in [ Reference Shapiro14 , p.187].

2·3. Hyperbolic geometry

We finish the preliminary material by presenting some basic facts for the hyperbolic geometry of the unit disc. For a complete treatise on hyperbolic geometry, we refer to [ Reference Abate1 , chapter 1].

To start with, the hyperbolic metric d of ${\mathbb{D}}$ is defined as

\[d(z,w)=\frac{1}{2}\log \frac{1+\left\lvert\frac{z-w}{1-\overline{w}z}\right\rvert}{1-\left\lvert\frac{z-w}{1-\overline{w}z}\right\rvert}, \quad \text{ for all}\quad z,w\in{\mathbb{D}}.\]

It is easy to see that d is indeed a metric, which turns ${\mathbb{D}}$ into a complete metric space. The geodesics of this metric space are arcs of circles orthogonal to the unit circe ${\mathbb{T}}$ , and the space in uniquely geodesic; i.e. any two points $z,w\in{\mathbb{D}}$ can be joined by a unique geodesic of $({\mathbb{D}},d)$ .

Moreover, if $\gamma\colon[0,1]\to{\mathbb{D}}$ is a geodesic joining $z,w\in{\mathbb{D}}$ , then there exists a unique point $a\in\gamma([0,1])$ so that $d(z,a)=d(w,a)=d(z,w)/2$ . We call this point a the hyperbolic midpoint of z and w.

The main advantage of working with the hyperbolic metric of ${\mathbb{D}}$ , instead of the Euclidean, is the Schwarz–Pick Lemma, which can be stated as follows:

If $f\colon{\mathbb{D}}\to{\mathbb{D}}$ is holomorphic, then

(2·8) \begin{equation}d(\,f(z),f(w))\leq d(z,w),\quad\text{for all}\quad z,w\in {\mathbb{D}}.\end{equation}

Equality in (2·8) is achieved for some (and hence any) $z\neq w$ if and only if

\[f(z)=e^{i\theta}\frac{z-a}{1-\overline{a}z}, \quad \text{for some}\ \theta\in{\mathbb{R}} \ \text{and}\ a\in{\mathbb{D}}.\]

The Möbius transformations of this form will be called automorphisms of ${\mathbb{D}}$ , and are exactly the orientation preserving isometries of d.

3. Proof of the main result

This section is dedicated to proving Theorem 1·1. For the convenience of the reader, let us recall the assumptions of Theorem 1·1. Suppose $(\,f_n)$ is a sequence of holomorphic self-maps of ${\mathbb{D}}$ such that the left iterated function system $F_n=f_n\circ f_{n-1}\circ \cdots\circ f_1$ converges locally uniformly to a constant $z_0\in{\mathbb{D}}$ . We assume that there exists a subsequence $(\,f_{n_k})\subseteq (\,f_n)$ and $\ell$ -measurable sets $E_k\subset {\mathbb{T}}$ such that $\lvert\,f_{n_k}(\zeta)\rvert\lt1$ for all $\zeta\in E_k$ and $\limsup_k \ell(E_k)\gt0$ .

Our goal is to show that $F_n(\zeta)\to z_0$ , as $n\to+\infty$ , for $\ell$ -almost all $\zeta\in{\mathbb{T}}$ . Observe that if $f_k$ is constant, for some $k\in{\mathbb{N}}$ , then the result is trivial due to the convergence of $(F_n)$ to $z_0$ . So, for the rest of the proof, we assume that all $f_n$ are non-constant self-maps of ${\mathbb{D}}$ . Also, consider the automorphism

\[\psi_0(z)=\frac{z-z_0}{1-\overline{z_0}z},\]

and the functions $g_n=\psi_0 \circ f_n\circ \psi_0^{-1}$ . We can see that $(F_n(\zeta))$ converges to $z_0$ if and only if the sequence $G_n(\zeta)=g_n\circ\cdots\circ g_1(\zeta)=\psi_0\circ F_n\circ \psi_0^{-1}(\zeta)$ converges to 0. Therefore, it suffices to assume that $z_0=0$ .

Throughout the proof we use D(z,r) to denote the Euclidean disc of radius $r\gt0$ , centred at $z\in{\mathbb{C}}$ . Before we proceed with the main body of the proof, we require the following result from [ Reference Christodoulou and Short7 , theorem 3·3] (see, also, [ Reference Abate and Christodoulou2 , theorem 4·1 (ii)]).

The main idea for our proof is to find an appropriate normalisation for the compositions $F_n$ that will allow us to employ estimates on certain Hardy norms. Such a normalisation can be obtained as follows:

By passing to a subsequence of $(\,f_{n_k})$ , we can assume that there exists $c\in(0,1)$ such that $\ell(E_k)\gt c$ , for all $k\in{\mathbb{N}}$ . Moreover, for every $\epsilon\gt0$ , and all $k\in{\mathbb{N}}$ , there exists a Lebesgue measurable set $E_k'\subseteq E_k$ and a constant $\delta_{k,\epsilon}\in(0,1)$ so that $\lvert\,f_{n_k}(\zeta) \rvert\lt\delta_{k,\epsilon}$ , for every $\zeta\in E_k'$ , and $\ell(E_k')\geq \ell(E_k)-\epsilon$ . So, without loss of generality, we also assume that for every $k\in{\mathbb{N}}$ , there exists $\delta_k\in(0,1)$ , so that

(3·1) \begin{equation}\lvert\,f_{n_k}(\zeta) \rvert\lt\delta_k,\quad \text{for every} \quad \zeta\in E_k\end{equation}

and $\ell(E_k)\gt c$ , for some $c\in(0,1)$ .

Next, we define the compositions $\phi_k=f_{n_{k+1}-1}\circ\cdots\circ f_{n_k}$ for $k\geq 1$ . Then, if $m\in{\mathbb{N}}$ and k is the largest integer such that $m\gt n_{k+1}$ , we have that

\[F_m= f_m\circ\cdots\circ f_{n_{k+1}}\circ \phi_k\circ\phi_{k-1}\circ\cdots\circ \phi_1\circ f_{n_1-1}\circ\cdots \circ f_1, \quad \text{for all}\ m \ \text{large enough}.\]

We would first like to examine the convergence properties of $(\phi_k)$ . Since $(F_n)$ converges locally uniformly to 0 and all $f_n$ are non-constant (as assumed in the beginning of this section), for any fixed $n_0\in{\mathbb{N}}$ , the sequence $(\,f_n\circ\cdots\circ f_{n_0})$ converges to 0, pointwise on the open set $f_{n_0-1}\circ\cdots \circ f_1\left(D(0,{1}/{2})\right)$ . Then, the Vitali–Porter Theorem [ Reference Schiff16 , p.75] implies that $(\,f_n\circ\cdots\circ f_{n_0})$ converges to 0, locally uniformly as $n\to+\infty$ . So, by passing to a further subsequence if necessary, we can assume that the integers $n_k$ are sparse enough, so that for all $k\in{\mathbb{N}}$ , the following hold:

(3·2) \begin{align}&\phi_k\left(\overline{D\left(0,\frac{1}{2}\right)}\right)\subset D\left(0,\frac{1}{2}\right), \end{align}

(3·3) \begin{align}&\phi_k(z)\xrightarrow{k\to\infty}0,\quad\text{for all}\ z\in{\mathbb{D}},\quad \text{and}\end{align}

(3·4) \begin{align}&\lvert \phi_k(\zeta) \rvert\lt\frac{1}{2}, \quad \text{for all} \ \zeta\in E_k.\end{align}

This last inequality, (3·4), is possible due to (3·1).

The sequence $(\phi_k)$ is the key ingredient of our normalisation scheme. However, our work is not done, as we would like each $\phi_k$ to fix the point 0, which might not happen in principle.

To overcome this obstacle, we consider the automorphism $e_1(z)=({\phi_1(0)-z})/({1-\overline{\phi_1(0)}z})$ and recursively define the automorphism $e_k$ with

\[e_k(z)=\frac{\phi_k(e_{k-1}(0))-z}{1-\overline{\phi_k(e_{k-1}(0))}z}, \quad\text{for}\quad k\geq2.\]

Let us examine some properties of the automorphisms $e_k$ . Firstly, observe that each $e_k$ is self-inverse. Moreover, $e_k$ interchanges the points 0 and $\phi_k(e_{k-1}(0))$ (or 0 and $\phi_1(0)$ for the case of $e_1$ ). Therefore, every $e_k$ has a fixed point $w_k\in{\mathbb{D}}$ , which is the hyperbolic midpoint of 0 and $\phi_k(e_{k-1}(0))$ (and analogously for $e_1$ ). Now, observe that

(3·5) \begin{equation}e_k(0)=\phi_k(e_{k-1}(0))=\phi_k(\phi_{k-1}(e_{k-2}(0)))=\cdots=\phi_k\circ\cdots\circ \phi_1(0)\in D\left(0,\frac{1}{2}\right),\end{equation}

due to (3·2). Therefore, we have that

(3·6) \begin{equation}w_k\in D\left(0,\frac{1}{2}\right), \quad \text{for all}\ k\in{\mathbb{N}}.\end{equation}

We can in fact say more about the sequence of points $e_k(0)=\phi_k\circ\cdots\circ\phi_1(0)$ . Note that (3·2) and (3·3) imply that $(\phi_k)$ converges locally uniformly to the constant function 0 (again by the Vitali–Porter Theorem). So, we can use Theorem 3·1 in order to deduce that the left iterated function system $\Phi_k=\phi_k\circ \phi_{k-1}\circ\cdots \circ \phi_1$ converges locally uniformly to 0. This means that

(3·7) \begin{equation}e_k(0)=\Phi_k(0)\xrightarrow{k\to+\infty}0.\end{equation}

Next, we define the sequence of functions $\widetilde{\phi}_1=e_1\circ\phi_1$ and $\widetilde{\phi}_k=e_k\circ \phi_k\circ e_{k-1}$ , for $k\geq 2$ . Notice that $\widetilde{\phi}_k(0)=0$ for all $k\geq1$ , as required for our technique. Also, since each $e_k$ is self-inverse, we can rewrite the composition $F_m$ as follows,

(3·8) \begin{align}F_m&=f_m\circ \cdots \circ f_{n_{k+1}}\circ e_k\circ e_k \circ \phi_k \circ e_{k-1}\circ e_{k-1}\circ \phi_{k-1}\circ\cdots \circ e_1\circ \phi_1\circ f_{n_1-1}\circ\cdots \circ f_1\nonumber\\ {}&= f_m\circ\cdots \circ f_{n_{k+1}}\circ e_k\circ \widetilde{\phi}_k\circ \widetilde{\phi}_{k-1}\circ \cdots\circ \widetilde{\phi}_1\circ f_{n_1-1}\circ\cdots \circ f_1,\end{align}

which is valid for all m large.

Rewriting the left iterated function system $(F_n)$ in the form (3·8) will turn out to be exactly the normalisation needed for our purposes. So, with this endeavour completed, we move on to obtaining estimates on the norms of the composition operators induced by the functions $\widetilde{\phi_k}$ .

Observe that (3·4) is equivalent to $\lvert \phi_k\circ e_{k-1}(\zeta) \rvert \lt {1}/{2}$ , for all $\zeta\in e_{k-1}(E_k)$ . Since each $e_k$ is an automorphism of ${\mathbb{D}}$ with fixed point $w_k\in D\left(0,{1}/{2}\right)$ (see (3·6)), there exists a universal constant $r_0\in(0,1)$ , so that $e_k\left(D\left(0,{1}/{2}\right)\right)\subset D(0,r_0)$ , for all $k\in{\mathbb{N}}$ . Therefore,

(3·9) \begin{equation}\left\lvert\widetilde{\phi}_k(\zeta)\right\rvert\lt r_0,\quad \text{for all}\quad \zeta\in e_{k-1}(E_k)\quad \text{and every}\quad k\in{\mathbb{N}}.\end{equation}

Now, simple computations show that for all $\zeta\in{\mathbb{T}}$ , we have

(3·10) \begin{equation}\lvert e_k'(\zeta)\rvert = \frac{1-\lvert \phi_k(e_{k-1}(0))\rvert^2}{\left\lvert 1-\overline{\phi_k(e_{k-1}(0))}\zeta\right\rvert^2}\geq \frac{3}{16},\end{equation}

where to obtain the inequality we used the triangle inequality and (3·5). Since each $e_k$ is a conformal automorphism of ${\mathbb{D}}$ , [ Reference Pommerenke13 , theorem 6·8] implies that

\[\ell(e_{k-1}(E_k))=\int_{E_k}\lvert e_k'(\zeta)\rvert d\ell(\zeta),\]

which, when combined with (3·10) yields

(3·11) \begin{equation}\ell\left(e_{k-1}(E_k)\right)\geq \frac{3}{16}\ell(E_k)\geq \frac{3c}{16}.\end{equation}

Our first claim is the analogue of [ Reference Shapiro15 , theorem 3·2] to our setting.

Claim 3·1. There exist constants $r_1,\delta\in(0,1)$ so that for all $k\in{\mathbb{N}}$

\[N_{\widetilde{\phi}_k}(w)\leq \delta\ \log\frac{1}{\lvert w \rvert}, \quad \text{for all} \quad r_1\leq \lvert w \rvert\lt 1.\]

Proof of Claim 3·1. Let $w\in{\mathbb{D}}$ , $k\in{\mathbb{N}}$ and $\zeta\in e_{k-1}(E_k)$ . Recall that by inequality (3·9) we have that $\lvert\widetilde{\phi}_k(\zeta)\rvert \lt r_0$ . So,

(3·12) \begin{align}1-\left\lvert \frac{w-\widetilde{\phi}_k(\zeta)}{1-\overline{w}\widetilde{\phi}_k(\zeta)} \right\rvert^2&=\frac{(1-\lvert w \rvert^2)(1-\lvert \widetilde{\phi}_k(\zeta) \rvert^2)}{\lvert 1- \overline{w}\widetilde{\phi}_k(\zeta)\rvert^2}\geq \frac{(1-\lvert w \rvert^2)(1-{r_0}^2)}{(1+\lvert\widetilde{\phi}_k(\zeta)\rvert)^2}\nonumber\\[5pt] {}&\geq \frac{(1-\lvert w \rvert^2)(1-{r_0}^2)}{(1+r_0)^2}=\frac{(1-\lvert w \rvert^2)(1-{r_0})}{1+r_0}. \end{align}

Moreover, $1-x\leq \log({1}/{x})$ , for all $x\in(0,1)$ , and there exists $r'\in(0,1)$ so that $\log({1}/{x})\leq 2(1-x)$ , for all $x\in[r',1)$ . Write $r_1\;:\!=\;\sqrt{r'}$ . Then, using these inequalities and (3·12), we get that for all $r_1\leq\lvert w \rvert \lt1$ ,

(3·13) \begin{align}\log \left\lvert \frac{1-\overline{w}\widetilde{\phi}_k(\zeta)}{w-\widetilde{\phi}_k(\zeta)} \right\rvert^2&\geq 1-\left\lvert \frac{w-\widetilde{\phi}_k(\zeta)}{1-\overline{w}\widetilde{\phi}_k(\zeta)} \right\rvert^2\geq \frac{(1-\lvert w \rvert^2)(1-{r_0})}{1+r_0}\nonumber\\[5pt] {}&\geq \frac{1-{r_0}}{2(1+r_0)}\log\frac{1}{\lvert w \rvert^2}=\frac{1-{r_0}}{1+r_0}\log\frac{1}{\lvert w \rvert}.\end{align}

Observe that

\[\left\lvert \frac{w-\widetilde{\phi}_k(\zeta)}{1-\overline{w}\widetilde{\phi}_k(\zeta)} \right\rvert\leq 1,\]

and so

(3·14) \begin{equation}\log\left\lvert \frac{w-\widetilde{\phi}_k(\zeta)}{1-\overline{w}\widetilde{\phi}_k(\zeta)} \right\rvert\leq 0, \quad\text{for all}\ \zeta\in e_{k-1}(E_k).\end{equation}

Using inequality (3·14) and the inequality (2·6) for the Nevanlinna counting function, we obtain that for all $w\in{\mathbb{D}}$ and all $k\in{\mathbb{N}}$ ,

\[N_{\widetilde{\phi}_k}(w)\leq \int_{{\mathbb{T}}}\log\left\lvert \frac{w-\widetilde{\phi}_k(\zeta)}{1-\overline{w}\widetilde{\phi}_k(\zeta)} \right\rvert d\ell(\zeta)+\log\frac{1}{\lvert w \rvert}\leq \int_{e_{k-1}(E_k)} \log\!\left\lvert \frac{w-\widetilde{\phi}_k(\zeta)}{1-\overline{w}\widetilde{\phi}_k(\zeta)} \right\rvert d\ell(\zeta)+\log\frac{1}{\lvert w \rvert}.\]

Therefore, restricting to $r_1\leq \lvert w\rvert \lt1$ and using (3·13) yields

\[N_{\widetilde{\phi}_k}(w)\leq -\frac{1}{2}\frac{1-{r_0}}{1+r_0}\ell(e_{k-1}(E_k))\log\frac{1}{\lvert w \rvert} + \log\frac{1}{\lvert w \rvert} = \left(1-\frac{1}{2}\frac{1-{r_0}}{1+r_0}\ell(e_{k-1}(E_k))\right) \log\frac{1}{\lvert w \rvert}.\]

Finally, using the estimate $\ell(e_{k-1}(E_k))\geq {3c}/{16}$ from (3·11), we obtain

\[N_{\widetilde{\phi}_k}(w)\leq \left(1-\frac{3c(1-{r_0})}{32(1+r_0)}\right)\log\frac{1}{\lvert w \rvert},\]

which is the desired conclusion for $\delta = 1-{3c(1-{r_0})}/{32(1+r_0)}\in (0,1)$ .

As promised, we are now ready to prove an upper bound for the norm of the composition operator $C_{\widetilde{\phi}_k}$ , similar to [ Reference Shapiro15 , theorem 5·1]. Let us denote by $H_0^2$ the linear subspace of $H^2$ consisting of all functions $f\in H^2$ with $f(0)=0$ . Since $\widetilde{\phi}_k(0)=0$ , the restriction $C_{\widetilde{\phi}_k}\arrowvert_{H_0^2}$ is a well-defined operator on $H_0^2$ .

For the proof of Claim 3·2, we require the following simple estimate from measure theory (for a proof, see [ Reference Shapiro15 , lemma 2·7]).

Lemma 3·3. Suppose $\mu$ is a positive, finite Borel measure on [0, 1) that has the point 1 is in its closed support. For every $r\in(0,1)$ , there exists a constant $\gamma\in(0,1)$ , depending only on r and $\mu$ , such that for every non-negative, increasing function g on [0, 1), we have

\[\int_{[0,r)}g d \mu\leq \gamma \int_{[0,1)} g d \mu.\]

Proof of Claim 3·2. Suppose $g\in H_0^2$ . By the change of variables formula (2·5), we have that

\[\lVert C_{\widetilde{\phi}_k}(g) \rVert_2^2 = 2\int_{{\mathbb{D}}} \lvert g'(w) \rvert N_{\widetilde{\phi}_k}(w)d A(w).\]

From Claim 3·1, there exist constants $r_1,\delta\in(0,1)$ so that for all $w\in {\mathbb{D}}$ , with $\lvert w \rvert\in[r_1,1)$ , we have $N_{\widetilde{\phi}_k}(w)\leq \delta\log({1}/{\lvert w \rvert})$ . Also, Littlewood’s Inequality (2·7) applied to $\widetilde{\phi}_k$ implies that $N_{\widetilde{\phi}_k}(w)\leq \log({1}/{\lvert w\rvert})$ , for all $w\in{\mathbb{D}}\setminus\{\widetilde{\phi}_k(0)\}$ . So,

(3·15) \begin{align}\lVert C_{\widetilde{\phi}_k}(g) \rVert_2^2 &= 2 \int_{\lvert w \rvert\lt r_1}\lvert g'(w)\rvert^2 N_{\widetilde{\phi}_k}(w)dA(w) + 2\int_{\lvert w \rvert\geq r_1}\lvert g'(w)\rvert^2 N_{\widetilde{\phi}_k}(w)dA(w)\nonumber \\[5pt] {}&\leq 2 \int_{\lvert w \rvert\lt r_1}\lvert g'(w)\rvert^2 \log\frac{1}{\lvert w \rvert} dA(w) + 2\delta \int_{\lvert w \rvert\geq r_1}\lvert g'(w)\rvert^2 \log\frac{1}{\lvert w \rvert}dA(w)\nonumber\\[5pt] {}&=2(1-\delta) \int_{\lvert w \rvert\lt r_1}\lvert g'(w)\rvert^2 \log\frac{1}{\lvert w \rvert} dA(w) + 2\delta \int_{{\mathbb{D}}}\lvert g'(w)\rvert^2 \log\frac{1}{\lvert w \rvert}dA(w).\end{align}

Now, define $h(r)=({1}/{2\pi})\int_0^{2\pi}\lvert g'(re^{i\theta})\rvert^2 d\theta$ , and note that it is an increasing function of $r\in[0,1)$ . Moreover, if we consider the Borel measure $d\mu(t)=2t\log({1}/{t})dt$ , then Lemma 3·3 yields a constant $\gamma\in(0,1)$ , depending only on $r_1$ , so that

\[\int_{[0,r_1)}h(r)d\mu(r)=\int_{\lvert w \rvert\lt r_1}\lvert g'(w) \rvert^2\log\frac{1}{\lvert w \rvert}d A(w)\leq \gamma \int_{{\mathbb{D}}} \lvert g'(w) \rvert^2\log\frac{1}{\lvert w \rvert}d A(w).\]

Therefore, (3·15) implies that

(3·16) \begin{equation}\lVert C_{\widetilde{\phi}_k}(g) \rVert_2^2 \leq (\gamma(1-\delta)+\delta) 2\int_{{\mathbb{D}}} \lvert g'(w) \rvert^2\log\frac{1}{\lvert w \rvert}d A(w).\end{equation}

By the Littlewood–Paley Identity (2·3), the inequality (3·16) can be rewritten as

\[\lVert C_{\widetilde{\phi}_k}(g) \rVert_2^2 \leq (\gamma(1-\delta)+\delta)\lVert g \rVert_2^2.\]

Since $g\in H_0^2$ was arbitrary, we have that $\lVert C_{\widetilde{\phi}_k}\arrowvert_{H_0^2} \rVert\leq \sqrt{\gamma(1-\delta)+\delta} \lt1$ , as required.

Now that the groundwork is laid, we can complete the proof of our main result.

Proof of Theorem 1·1. Since, as mentioned in the beginning of this section, we assume that $z_0=0$ , our goal is to show that the sequence $(F_m(\zeta))$ converges to 0, as $m\to+\infty$ , for Lebesgue almost all $\zeta\in{\mathbb{T}}$ .

Recall that from our normalisation (3·8), the left iterated function system $F_m$ can be rewritten as

\[F_m= f_m\circ\cdots \circ f_{n_{k+1}}\circ e_k\circ \widetilde{\phi}_k\circ \widetilde{\phi}_{k-1}\circ \cdots\circ \widetilde{\phi}_1\circ f_{n_1-1}\circ\cdots \circ f_1,\]

where k is the largest integer so that $m\gt n_{k+1}$ . Therefore,

\[F_{n_{k+1}-1}= e_k\circ \widetilde{\phi}_k\circ \widetilde{\phi}_{k-1}\circ \cdots\circ \widetilde{\phi}_1\circ f_{n_1-1}\circ\cdots \circ f_1.\]

For simplicity, let us also write $f=f_{n_1-1}\circ\cdots \circ f_1$ , so that

\[e_k\circ F_{n_{k+1}-1}= \widetilde{\phi}_k\circ \widetilde{\phi}_{k-1}\circ \cdots\circ \widetilde{\phi}_1\circ f,\]

recalling that the automorphisms $e_k$ are self-inverse.

Our first goal is to show that $(e_k\circ F_{n_{k+1}-1}(\zeta))$ converges to 0 for Lebesgue almost all $\zeta\in{\mathbb{T}}$ .

Using Claim 3·2, we can find $\nu\in(0,1)$ so that $\lVert C_{\widetilde{\phi}_k}\arrowvert_{H_0^2}\rVert\leq \nu$ , for all $k\in{\mathbb{N}}$ . Therefore, for the $H^2$ norm of $e_k\circ F_{n_{k+1}-1}$ we get

(3·17) \begin{align}\lVert e_k\circ F_{n_{k+1}-1} \rVert_2 &= \lVert C_{e_k\circ F_{n_{k+1}-1}}(\mathrm{id})\rVert_2 = \lVert C_f\circ C_{\widetilde{\phi}_1}\circ C_{\widetilde{\phi}_2}\circ \cdots \circ C_{\widetilde{\phi}_k}(\mathrm{id})\rVert_2\nonumber\\[5pt] {}&\leq \lVert C_f \rVert \cdot \lVert C_{\widetilde{\phi}_1}\circ C_{\widetilde{\phi}_2}\circ \cdots \circ C_{\widetilde{\phi}_k}(\mathrm{id})\rVert_2\nonumber\\[5pt] {}&\leq \lVert C_f \rVert \cdot \prod_{i=1}^{k}\lVert C_{\widetilde{\phi}_i}\arrowvert_{H_0^2}\rVert\nonumber\\[5pt] {}&\leq\lVert C_f \rVert\cdot \nu^k.\end{align}

Using the Monotone Convergence Theorem, the boundary integral form of the $H^2$ norm in (2·2), and (3·17) we obtain

\begin{align*}\int_{{\mathbb{T}}} \sum_{k=1}^{+\infty} \lvert e_k\circ F_{n_{k+1}-1}(\zeta)\rvert^2 d\ell(\zeta) &= \sum_{k=1}^{+\infty} \int_{{\mathbb{T}}} \lvert e_k\circ F_{n_{k+1}-1}(\zeta)\rvert^2 d\ell(\zeta) \\[5pt] &=\sum_{k=1}^{+\infty} \lVert e_k\circ F_{n_{k+1}-1} \rVert^2 \leq \lVert C_f \rVert^2\cdot \sum_{k=1}^{+\infty} \nu^{2k}\lt+\infty.\end{align*}

This means that $\sum\limits_{k=1}^{+\infty} \lvert e_k\circ F_{n_{k+1}-1}(\zeta)\rvert^2\lt+\infty$ , for almost every $\zeta\in{\mathbb{T}}$ , and so

(3·18) \begin{equation}\lvert e_k\circ F_{n_{k+1}-1}(\zeta) \rvert\xrightarrow{k\to+\infty} 0, \quad \text{for almost every}\ \zeta\in{\mathbb{T}},\end{equation}

as desired.

We finally move on to the general case where

\begin{align*}F_m&= f_m\circ\cdots \circ f_{n_{k+1}}\circ e_k\circ \widetilde{\phi}_k\circ \widetilde{\phi}_{k-1}\circ \cdots\circ \widetilde{\phi}_1\circ f_{n_1-1}\circ\cdots \circ f_1\\[5pt] &=f_m\circ\cdots \circ f_{n_{k+1}}\circ e_k\circ F_{n_{k+1}-1},\end{align*}

and k is the largest integer so that $m\gt n_{k+1}$ . In particular, $k\to+\infty$ whenever $m\to+\infty$ .

Note that (3·18) implies that $e_k\circ F_{n_{k+1}-1}(\zeta)\in{\mathbb{D}}$ , for all k large, and so $F_m(\zeta)$ also lies in ${\mathbb{D}}$ for all m large.

By the triangle inequality for the hyperbolic metric and the Schwarz–Pick Lemma (2·8), we have

(3·19) \begin{align}d(\,f_m\circ\cdots\circ f_{n_{k+1}}(0),0)&\leq d(\,f_m\circ\cdots\circ f_{n_{k+1}} (0),F_m(0))+ d(F_m(0),0)\nonumber\\ {} {}&\leq d(0,F_{n_{k+1}-1}(0)) + d(F_m(0),0).\end{align}

Now, since by assumption $F_m$ converges locally uniformly to the constant 0, we have that the sequences $(F_{n_{k+1}-1}(0)$ and $(F_m(0))$ both converge to 0 as m—and so also as k—tends to $+\infty$ . So, taking limits $m\to+\infty$ in (3·19) and applying these facts yields

(3·20) \begin{equation}f_m\circ\cdots\circ f_{n_{k+1}}(0)\xrightarrow{m\to+\infty}0.\end{equation}

To conclude the proof it suffices to note that for almost all $\zeta\in{\mathbb{D}}$ we have

\begin{align*}d(F_m(\zeta),f_m\circ\cdots\circ f_{n_{k+1}}(0))\leq d(e_k\circ F_{n_{k+1}-1}(\zeta),0),\end{align*}

and apply (3·18) and (3·20).