2. Cosine dynamics and external addresses

We start by revising basic properties of cosine maps. We refer to [ Reference Rottenfusser, Schleicher, Rippon and Stallard28 , Reference Schleicher29 ] for extensive work on their dynamics. Note also that cosine maps arise as lifts of holomorphic self-maps of $\mathbb{C}^*$ ; see [ Reference Fagella and Martí–Pete10 , corollary 1·5]. Recall that for a holomorphic map $f\;:\;\widetilde{S}\rightarrow S$ between Riemann surfaces, the local degree of f at a point $z_0\in \widetilde{S}$ , denoted by $\deg(\,f,z_0)$ , is the unique integer $n\geq 1$ such that the local power series development of f is of the form

\begin{equation*}f(z)=f(z_0) + a_n (z-z_0)^n + \text{(higher terms)},\end{equation*}

where $a_n\neq 0$ . Thus, $z_0\in \mathbb{C}$ is a critical point of f if and only if $\deg(\,f,z_0)>1$ . We say that f has bounded criticality on a set A if $\textrm{AV}(\,f) \cap A=\emptyset$ and there exists a constant $M<\infty$ such that $\deg(\,f,z)<M \text{ for all } z \in A.$

2·1 (Basic properties of cosine maps). Each cosine map $f(z)\;:\!=\; ae^z+be^{-z}$ with $a,\,b \in \mathbb{C}^\ast $ is $2\pi i$ -periodic and has exactly two critical values, namely $\pm 2\sqrt{ab}$ . Furthermore, any preimage of a critical value is a critical point of local degree 2, and hence both critical values are totally ramified. More specifically,

\begin{equation*}\textrm{Crit}(\,f)=\left\{\frac{1}{2}\log\left(\frac{a}{b}\right)+ \pi i n\;:\; n\in \mathbb{Z} \right\},\end{equation*}

where the branch of the logarithm is chosen such that $\vert \textrm{Im} \; ({1}/{2})\log({a}/{b})\vert\leq \pi/2$ . It is easy to check that f has no asymptotic values, and thus, $S(\,f)\;=\!:\;\{v_1, v_2\}$ , with $v_i=\pm 2\sqrt{ab}$ and signs chosen so that $v_1$ is the image of $({1}/{2})\log({a}/{b})+2\pi i\mathbb{Z}$ , while $v_2$ is the image of $({1}/{2})\log({a}/{b})+\pi i+2\pi i\mathbb{Z}$ . In particular, since S(f) is bounded and f is of order of growth one, the Julia set of any disjoint type cosine map is a Cantor bouquet. Roughly speaking, a Cantor bouquet consists of an uncountable collection of curves to infinity satisfying a certain density condition; see [ Reference Barański, Jarque and Rempe6 , definition 2·1]. Moreover, by the Denjoy–Carleman–Ahlfors theorem, for any choice of bounded domain $D\supset S(\,f)$ , the number of connected components of $f^{-1}(\mathbb{C} \setminus D)$ , called tracts, is at most two. Note that for any such domain D, f maps points for which the absolute value of their real part is sufficiently large, to $\mathbb{C} \setminus D$ . Hence, a left and a right half plane are contained in the union of tracts, which implies that f has at least two, and hence exactly two, tracts for any choice of D.

Observation 2·2 (Parameter space of cosine maps). All cosine maps belong to the same parameter space; that is, any two cosine maps are quasiconformally equivalent. To see this, let $f(z)\;:\!=\; ae^z+be^{-z}$ and $g(z)\;:\!=\; ce^z+de^{-z}$ for $a,\,b,\,c,\,d \in \mathbb{C}^\ast$ . Consider the linear maps $\psi(z)\;:\!=\; z+ \log \sqrt{{bc}/{ad}}$ and $\varphi(z)\;:\!=\; \sqrt{({bc}/{ad})} z$ , which are clearly quasiconformal. Then, for all $z\in \mathbb{C}$ ,

\begin{equation*} (\,f\circ \psi)(z)=a e^z\sqrt{\frac{bc}{ad}}+be^{-z} \sqrt{\frac{ad}{bc}}=c e^z\sqrt{\frac{ab}{cd}}+de^{-z} \sqrt{\frac{ab}{cd}}=(\varphi\circ g)(z).\end{equation*}

Consequently, in order to prove the second part of Theorem 1·3, by [ Reference Rempe24 , theorem 3·1], it suffices to construct a conjugacy between $\mathcal{F}\colon J(\mathcal{F})\rightarrow J(\mathcal{F})$ and any specific disjoint type cosine map. We could have followed this approach, but for the sake of generality, our proof will relate to any disjoint-type cosine map.

Let us fix a cosine function $f(z)\;:\!=\; ae^z+be^{-z}$ with $a,\,b\in \mathbb{C}^\ast$ . We will make use of some estimates from [ Reference Rottenfusser, Schleicher, Rippon and Stallard28 ] that require of constants related to the following:

\begin{equation*} \mathcal{K}(\,f) \;:\!=\; \!\max \! \left\{\!\left(\sqrt{\left\vert \frac{2b}{a} \right\vert} +\!\sqrt{\left\vert\frac{2a}{b} \right\vert}\right)\!(\vert a \vert + \vert b \vert), 8\vert ab \vert, 1, \frac{1}{2}\ln \left\vert \frac{2b}{a}\right\vert , \frac{1}{2}\ln\left\vert \frac{2a}{b}\right\vert, \ln \frac{16}{\vert a b \vert} \right\}\!. \end{equation*}

More precisely, recall from Section that for any cosine map f and for any choice of Jordan domain $D\supset S(\,f)$ , $f^{-1}(\mathbb{C}\setminus D)$ has two connected components, that is, two tracts. We will choose D large enough as to guarantee Euclidean expansion within tracts, that is, that the modulus of the derivative of any point in the tracts is large enough. Let us choose $R>\mathcal{K}(\,f)$ big enough such that

\begin{equation*} \bigcup_{k\in \mathbb{Z}} \{z+2\pi k i \colon z \in \mathbb{D}_R \} \supset \{z \in \mathbb{C} \colon \vert \textrm{Re} \; z \vert\leq \mathcal{K}(\,f)\} \end{equation*}

and $f(\mathbb{D}_R)\supset \mathbb{D}_{\mathcal{K}(\,f)}$ . In particular, the domain $D\;:\!=\; f(\mathbb{D}_R)$ contains S(f), and so we can define a pair of tracts $\mathcal{T}_f$ as the connected components of $f^{-1}(\mathbb{C}\setminus D)$ . Then, by definition,

(2·1) \begin{equation} \mathcal{T}_f \subseteq \{z\in \mathbb{C} \colon \vert \textrm{Re} \; (z)\vert >\mathcal{K}(\,f) \} \quad \text{ and } \quad S(\,f) \subset \mathbb{D}_{\mathcal{K}(\,f)} \subset \mathbb{C} \setminus f(\mathcal{T}_f).\end{equation}

A simple calculation shows that for any $z\in \mathbb{C}$ such that $\vert \textrm{Re} \; z\vert >\mathcal{K}(\,f)$ ,

(2·2) \begin{equation}\vert f' (z) \vert >2;\end{equation}

see [ Reference Rottenfusser, Schleicher, Rippon and Stallard28 , lemma 3·6]. Hence, we say that $\mathcal{T}_f$ are expansion tracts.

2·3. (Fundamental domains and inverse branches). Let us fix a cosine map $f(z)\;:\!=\; ae^z+be^{-z}$ for some $a,\,b\in \mathbb{C}^\ast$ , and let $\mathcal{T}_f$ be a pair of expansion tracts. Let $S(\,f)\;=\!:\;\{v_1, v_2\}$ , with $v_1$ and $v_2$ labelled according to Section 2·1. In particular, by (2·1), $S(\,f)\subset D\;:\!=\; \mathbb{C} \setminus f(\mathcal{T}_f)$ and $\{z\in \mathbb{C} \colon \vert \textrm{Re} \; (z)\vert <\max(\vert \textrm{Re} \; v_1\vert , \vert \textrm{Re} \; v_2 \vert )\}\subset \mathbb{C}\setminus \mathcal{T}_f$ . If $\textrm{Im} \; (v_1)> \textrm{Im} \; (v_2)$ , we define $\delta$ as the vertical straight line starting at $v_1$ in upwards direction restricted to $\mathbb{C}\setminus D$ . If on the contrary $\textrm{Im} \; (v_1)< \textrm{Im} \; (v_2)$ , $\delta$ is the downwards vertical line joining $v_2$ to infinity restricted to $\mathbb{C}\setminus D$ . In any case, $\delta \subset \mathbb{C}\setminus (\mathcal{T}_f\cup D)$ , and so we can define fundamental domains for f as the connected components of $\mathcal{T}_f\setminus f^{-1}(\delta)$ ; see [ Reference Pardo–Simón18 , section 2] for the definition of fundamental domains in a more general setting. Since f is in the cosine family, by definition, the image of points in $\mathbb{R}$ whose modulus is large enough have uniformly large modulus, and so they must be totally contained in a fundamental domain. By $2\pi i$ -periodicity of f, the same holds for all their $2\pi i$ -translates. Hence, for each $n\in \mathbb{Z}$ , we denote by $F_{(n,R)}$ the fundamental domain that contains an unbounded subset of $2\pi n i + \mathbb{R}^+$ , and by $F_{(n,\,L)}$ the fundamental domain that contains an unbounded subset of $2\pi ni + \mathbb{R}^-$ . Since f maps each fundamental domain to its image $f(\mathcal{T}_f) \setminus \delta$ as a conformal isomorphism, see [ Reference Pardo Simón17 , proposition 2·19], we can define for each $(n, \ast) \in \mathbb{Z}\times\{L, R\}$ the inverse branch

(2·3) \begin{equation} f^{-1}_{(n, \ast)}\colon f(\mathcal{T}_f)\setminus \delta \rightarrow F_{(n, \ast)}, \end{equation}

which in particular is a bijection.

Observation 2·4 (Horizontal straight lines contained in fundamental domains). Following 2.3, by construction, there is a constant $A>\mathcal{K}(\,f)$ so that for all $n\in \mathbb{Z}$ ,

\begin{align*} \{z:\; \textrm{Re} \; z<- A \text{ and } \textrm{Im} \; z=2\pi n\} &\subset F_{(n, L)}\qquad\text{and} \\ \{z:\; \textrm{Re} \; z>A \text{ and } \textrm{Im} \; z=2\pi n\} &\subset F_{(n, R)}. \end{align*}

We note that our choice of fundamental domains in Section agrees with the partition defined in [ Reference Rottenfusser, Schleicher, Rippon and Stallard28 , sections 1 and 2], where the maps “ $f^{-1}_{(n, \ast)}$ ” are labelled as “ $L_s$ ”. Then, the estimates appearing in [ Reference Rottenfusser, Schleicher, Rippon and Stallard28 ] regarding this partition and the maps from (2·3) apply to our setting. In particular, we will use the following:

Definition 2·6 (External addresses). Let f be a cosine map. An external address is an infinite sequence ${\underline s}=F_0F_1F_2\ldots$ of the fundamental domains specified in Section 2·3. If ${\underline s}$ is such an external address, we denote

\begin{equation*} J_{\underline s}= \{z\in J(\,f) \colon f^n(z)\in F_n \text{ for all } n\geq 0\}. \end{equation*}

We let $\textrm{Addr}(\,f)$ be the set of all ${\underline s}$ for which $J_{\underline s}$ is not empty, that we endow with the usual lexicographic cyclic order topology; see [ Reference Pardo–Simón18 , 2·13] for details.

Observation 2·7. If g is of disjoint type, then $J(g)= \bigcap_{k\geq 0}g^{-k}( \mathcal{T}_g)$ and $g^{-n}(\mathcal{T}_g) \subset \mathcal{T}_g$ for all $n\geq 0$ ; see [ Reference Rempe–Gillen25 , proposition 3·2]. In particular,

\begin{equation*}J(g)=\bigcup_{{\underline s}\in \textrm{Addr}(g)} J_{\underline s}. \end{equation*}

Notation. For each element $(n, \ast)\in \mathbb{Z}\times\{L, R\}$ , we denote $\vert (n, \ast)\vert \;:\!=\; \vert n\vert $ and $\{(n, \ast)\}\;:\!=\; n$ .

3. A model for cosine dynamics

3.1 (Topological space $(\mathcal{M}, \tau_\mathcal{M})$ ). Consider the set

\begin{equation*}\mathcal{M}\;:\!=\; [0,\infty)\times(\mathbb{Z}\times\{L, R\})^\mathbb{N}.\end{equation*}

Let “ $<_{_\mathbb{Z}}$ ” be the usual linear order on integers. We define a total order in the set $\mathbb{Z}\times\{L, R\}$ as follows:

(3·1) \begin{align} (n,\ast)<(m,\star)& \iff \left\{\begin{array}{@{}l@{\quad}l@{}} \ast=R=\star & \text{and} \quad n<_{_\mathbb{Z}}m, \quad \text{ or }\\ \ast=L=\star & \text{and} \quad m<_{_\mathbb{Z}}n, \quad \text{ or }\\ \ast=L &\text{and} \quad \star=R. \end{array}\right. \end{align}

Then, $<$ induces a lexicographic order “ $<_{\ell}$ ” in $(\mathbb{Z}\times\{L, R\})^\mathbb{N}$ . In turn, we define a cyclic order induced by $<_{\ell}$ in the usual way: for ${\underline s},\underline{\alpha},\underline{\tau} \in (\mathbb{Z}\times\{L, R\})^\mathbb{N}$ ,

\begin{equation*}[{\underline s},\underline{\alpha},\underline{\tau}]_{\ell} \quad \text{if and only if} \quad {\underline s} <_{_\ell}\underline{\alpha}<_{_\ell}\underline{\tau} \quad \text{ or } \quad \underline{\alpha}<_{_\ell} \underline{\tau} <_{_\ell} {\underline s} \quad \text{ or } \quad \underline{\tau} <_{_\ell} {\underline s} <_{_\ell} \underline{\alpha}.\end{equation*}

Moreover, given two different elements ${\underline s},\underline{\tau} \in (\mathbb{Z}\times\{L, R\})^\mathbb{N}$ , we define the open interval from ${\underline s}$ to $\underline{\tau}$ , denoted by $({\underline s},\underline{\tau})$ , as the set of all points $x\in (\mathbb{Z}\times\{L, R\})^\mathbb{N}$ such that $[{\underline s},x,\underline{\tau}]$ . The collection of all such open intervals forms a base for the cyclic order topology. We then provide the space $\mathcal{M}$ with the topology $\tau_\mathcal{M}$ defined as the product topology of $[0, \infty)$ with the usual topology, and $(\mathbb{Z}\times\{L, R\})^\mathbb{N}$ with the just described cyclic order topology.

Observation 3·2 (Correspondence between topological spaces). Let g be any disjoint type cosine map, and let $\textrm{Addr}(g)$ be the set of external addresses, see Definition 2·6. In particular, $\textrm{Addr}(g)$ is endowed with a cyclic order topology. We note that there exists a one-to-one correspondence between $(\mathbb{Z}\times\{L, R\})^\mathbb{N}$ and $\textrm{Addr}(g)$ that preserves their topologies, namely, the one that converts sequences as follows

\begin{equation*}(m,\star)(n,\ast)\cdots \leftrightsquigarrow F_{(m,\star)} F_{(n,\ast)}\cdots.\end{equation*}

Since the curve $\delta$ chosen in Section is a vertical straight line, the linear order in fundamental domains chosen to define the cyclic order topology in $\textrm{Addr}(g)$ agrees with the linear order (3·1) that determines the topology in $(\mathbb{Z}\times\{L, R\})^\mathbb{N}$ , up to the specified correspondence. Hence, from now on we omit the specification of the correspondence, and ${\underline s}$ might denote either an element $(m,\star)(n,\ast)\ldots$ of $(\mathbb{Z}\times\{L, R\})^\mathbb{N}$ , or its corresponding element $F_{(m,\star)} F_{(n,\ast)}\cdots$ in $\textrm{Addr}(g)$ .

Definition 3·3 (A topological model for cosine dynamics). Let $(\mathcal{M}, \tau_\mathcal{M})$ be defined as in Section 3·1. Define $\mathcal{F} \colon (\mathcal{M},\tau_\mathcal{M}) \to (\mathcal{M},\tau_\mathcal{M})$ as

\begin{equation*}\mathcal{F}(t,{\underline s})\;:\!=\;(\,F(t) -2\pi |s_1| , \sigma({\underline s})), \end{equation*}

where $\sigma$ is the shift map on one-sided infinite sequences of $(\mathbb{Z}\times\{L, R\})^\mathbb{N}$ , and $F(t)\;:\!=\; e^t-1$ is the standard map that codes exponential growth. Let $T\colon \mathcal{M} \rightarrow [0, \infty)$ be given by $T(t, {\underline s})\;:\!=\; t $ . We set

\begin{align*} J(\mathcal{F}) &\;:\!=\; \{ x\in \mathcal{M} \colon \; T(\mathcal{F}^n(x)) \geq 0 \text{ for all $n\geq 0$}\}, \qquad\text{and} \\ I(\mathcal{F})&\;:\!=\; \{ x \in J(\mathcal{F})\colon T(\mathcal{F}^n(x)) \to \infty \text{ as }n\to\infty \}. \end{align*}

We say that ${\underline s} \in (\mathbb{Z}\times\{L, R\})^{\mathbb{N}}$ is exponentially bounded if $(t, {\underline s}) \in J(\mathcal{F})$ for some $ t>0$ , and denote by $\mathscr{S}^{\,\mathbb{N}}$ the set of exponentially bounded elements. We moreover let

\begin{align*} t_{\underline s}&\;:\!=\; \left\{\begin{array}{@{}l@{\quad}l@{}} \min\{t\geq 0 \;:\; (t, {\underline s}) \in J(\mathcal{F})\} & \text{if } {\underline s} \text{ is exponentially bounded}, \\ \infty & \text{otherwise}. \end{array}\right. \end{align*}

In other words, $J(\mathcal{F})$ is the set of all points that stay in the space $\mathcal{M}$ under $\mathcal{F}^n$ for all $n\geq 0$ .

We shall use the relation specified above between the exponential and cosine models to prove properties of the latter:

In order to prove continuity of $\mathcal{F}\vert_{J(\mathcal{F})}$ , let us fix an arbitrary $(t,{\underline s}) \in J(\mathcal{F})$ and let V be an open neighbourhood of $\mathcal{F}(t, {\underline s})$ . Without loss of generality, we may assume that $V=((t_1, t_2)\times \mathcal{I}) \cap J(\mathcal{F})$ for some open interval $\mathcal{I} \in (\mathbb{Z}\times \{L, R\})^\mathbb{N}$ and $t_1, t_2\in \mathbb{R}^+$ so that $t_1\leq T(\mathcal{F}(t,{\underline s}))< t_2$ . Suppose that ${\underline s}=s_0s_1\cdots$ and denote $\tilde{\mathcal{I}}\;:\!=\; \{s_0\underline{\tau} \;:\; \underline{\tau} \in \mathcal{I}\}$ . In particular, ${\underline s} \in \tilde{\mathcal{I}}$ , and since by definition of $\mathcal{F}$ , $t=\log(T(\mathcal{F}(t, {\underline s}))+1+2\pi\{s_1\})$ and the function $\log$ is increasing,

\begin{equation*}U\;:\!=\;(\log(t_1+1+2\pi \{s_1\} ), \log(t_2+1+2\pi \{s_1\}))\times \tilde{\mathcal{I}})\cap J(\mathcal{F})\end{equation*}

is an open neighbourhood of $(t,{\underline s})$ such that $\mathcal{F}(U)\subset V.$

In order to prove Theorem 1·3, our main task will be to construct for each disjoint type cosine map g, a continuous map $\Phi\;:\; J(\mathcal{F}) \rightarrow J(g)$ that conjugates the dynamics of $\mathcal{F}$ to those of $g\vert_{ J(g)}$ . Then, the result for any cosine map f will follow using Rempe’s conjugacy between f and g “near infinity”; [ Reference Rempe24 ]. In the disjoint type case, the map $\Phi$ will send each point $(t,\underline{s})\in J(\mathcal{F})$ to a point $z\in J(g)$ such that $z\in J_{\underline s}$ and $ \vert \textrm{Re} \; z\vert \approx\;t$ , see Observation 3·2. We will obtain the map $\Phi$ as the limit of a series of approximations $\{ \Phi_n\}_{n\in \mathbb{N}}$ ; compare [ Reference Mihaljević–Brandt15 , Reference Pardo–Simón21 , Reference Rempe22 , Reference Rempe24 ] for similar arguments. The first approximation should be a projection from the space $J(\mathcal{F})$ to the dynamical plane of g.

Definition 3·6 (Projection function). For each $A\geq 0$ , we define a projection function $\mathcal{C}_A\;:\; J(\mathcal{F}) \to \mathbb{C}$ as

\begin{align*} \mathcal{C}_A(t, {\underline s})&\;:\!=\; \left\{\begin{array}{@{}l@{\quad}l@{}} t+A + 2\pi \{s_0\}i & \text{if} \quad s_0=(n,R) \text{ for some }n\in \mathbb{Z}, \\ -t-A+ 2\pi \{s_0\}i & \text{otherwise}, \end{array}\right. \end{align*}

where ${\underline s}=s_0s_1\cdots$ , and if $s_0=(n, \ast)$ , then $\{s_0\}=\{(n, \ast)\}=n$ .

Recall that cosine maps behave like the exponential map for points with modulus large enough and sufficiently far from the imaginary axis. In particular, all such points are contained in fundamental domains. An essential characteristic of our model for cosine dynamics is that, as occurs for the exponential model, for each $(t,{\underline s})\in J(\mathcal{F})$ , $\vert \mathcal{C}_A(\mathcal{F}(\underline{s},t))\vert$ is roughly the exponential of its real part. More precisely:

Proposition 3·8 (Model acts similar to the exponential). If $(t,{\underline s}) \in J(\mathcal{F})$ and $A>0$ ,

(3·2) \begin{equation} \frac{F(t)+A}{\sqrt{2}} \leq \vert \mathcal{C}_A(\mathcal{F}(t,{\underline s}))\vert \leq F(t)+A. \end{equation}

Proof. Suppose that ${\underline s}=s_0s_1\cdots $ and let $b\;:\!=\; 2\pi \{s_1\} $ . Then,

(3·3) \begin{equation} \begin{split} \vert \mathcal{C}_A(\mathcal{F}(t,{\underline s}))\vert &= \vert \pm(\,F(t)-b+A) + ib \vert=\sqrt{(\,F(t)+A-b)^{2}+ b^2} \\ &=\sqrt{(\,F(t)+A)^{2}-2(\,F(t)+A)b +2b^{2}}. \end{split} \end{equation}

The second inequality in (3·2) follows from the assumption $T(\mathcal{F}(t,{\underline s}))\geq 0$ , that is, $F(t)-b \geq \;0$ , because by (3·3),

\begin{equation*}\vert \mathcal{C}_A(\mathcal{F}(t,{\underline s}))\vert\leq \sqrt{(\,F(t)+A)^{2}} \iff -2(\,F(t)+A)b +2b^{2}\leq 0 \iff b \leq F(t)+A,\end{equation*}

where we have used that $A,\,b,\,F(t)\geq 0$ . For the first inequality in (3·2), we have

\begin{equation*} \begin{split} \sqrt{(\,F(t)+A)^{2}} \leq \sqrt{2} \vert \mathcal{C}_A(\mathcal{F}(t,{\underline s}))\vert & \iff (\,F(t)+A)^{2}-4(\,F(t)+A)b +4b^{2} \geq 0 \\ &\iff(\,F(t)+A-2b)^{2}\geq 0. \end{split} \end{equation*}

We describe the underlying idea in the construction of the map $\Phi$ that conjugates $\mathcal{F}$ to any disjoint type map $g\vert_{J(g)}$ . For each $n\geq 0$ , a function $\Phi_n\colon J(\mathcal{F}) \rightarrow \mathbb{C}$ will be defined the following way: we iterate each point $x=(t, {\underline s})\in J(\mathcal{F})$ , with ${\underline s}=s_0s_1\cdots$ , under the model function $\mathcal{F}$ a number n of times. In particular, $\mathcal{F}^n(t, {\underline s})=(t', \sigma^n({\underline s}))$ for some $t'>0$ . Next, we move to the dynamical plane of g using the function $\mathcal{C}_A$ for some constant A big enough such that $(\mathcal{C}_A\circ\mathcal{F}^n)(t, {\underline s})\in F_{s_n}$ . Then, we use the composition of n inverse branches of g specified in (2·3) to obtain a point in $F_{s_0}$ , which will define $\Phi_n(x)$ ; see Figure 1. Finally, we use (Euclidean) expansion of g on its tracts to show that $\{\Phi_n\}_{n\geq 0}$ is a uniformly convergent sequence. We now formalise these ideas:

Fig. 1. A schematic of the functions and curves involved in the definition of $\{\Phi_n\}_{n\in \mathbb{N}}$ .

Definition 3·9 (Functions $\Phi_n$ ). Let g be a disjoint-type cosine map, and let A be a constant provided by Observation 2·4. Then, for each $n\geq 0$ we define the function $\Phi_{n}\;:\; J(\mathcal{F})\rightarrow \mathbb{C}$ as

\begin{equation*}\Phi_0(x)\;:\!=\; \mathcal{C}_A(x)\qquad \text{ and }\qquad \Phi_{n+1}(x)\;:\!=\; g^{-1}_{s_0}(\Phi_{n}(\mathcal{F}(x)), \end{equation*}

for $x=(t,{\underline s})$ and ${\underline s}=s_0s_1\cdots$ .

The function $\Phi_0$ is clearly well-defined. In order to see that for all $n\geq 1$ the function $\Phi_n$ is also well-defined, fix $x=(t, {\underline s})\in J(\mathcal{F})$ and suppose that ${\underline s}=s_0s_1\cdots$ . Then, expanding definitions

(3·4) \begin{equation}\Phi_{n}(x)= \left(g^{-1}_{s_0}\circ g^{-1}_{s_1} \circ \cdots \circ g^{-1}_{s_{n-1}} \circ \mathcal{C}_A \circ \mathcal{F}^n\right)(x).\end{equation}

By Observations 3·7 and 2·7, the composition of the inverse branches $\{g^{-1}_{s_i}\}_{i< n}$ is well-defined on $\mathcal{C}_A(\mathcal{F}^n(x))\in F_{s_n}$ . Moreover, by construction, for all $n\geq 0$ ,

(3·5) \begin{equation}\Phi_{n}\circ \mathcal{F}= g \circ \Phi_{n+1}.\end{equation}

Proof. Let us fix an arbitrary $(t,{\underline s})\in J(\mathcal{F})$ with ${\underline s}=s_0s_1\cdots$ , as well as some $\epsilon>0$ . To see that $\Phi_0\equiv\mathcal{C}_A$ is continuous, let $\mathcal{I} \subset (\mathbb{Z}\times\{L, R\})^\mathbb{N}$ be any open interval containing ${\underline s}$ and such that if $\underline{\tau}=\tau_0\tau_1\cdots \in \mathcal{I}$ , then $s_0=\tau_0$ . Then, $U\;:\!=\; \left((t-\epsilon, t+\epsilon) \times \mathcal{I}\right) \cap J(\mathcal{F})$ is an open neighbourhood of $(t, {\underline s})$ such that

\begin{equation*}\mathcal{C}_A(U)\subset (\pm t-\epsilon, \pm t+\epsilon) \pm A + 2\pi i \{s_0\} \subset \mathbb{D}_\epsilon(\pm t\pm A + 2\pi i \{s_0\})= \mathbb{D}_\epsilon(\mathcal{C}_A(t,{\underline s})),\end{equation*}

where $\pm$ equals “ $+$ ” or “ minus;” depending on whether $s_0=(n,R)$ or $s_0=(n,\,L)$ for some $n\in \mathbb{Z}$ . Hence, we have shown continuity of $\Phi_0$ . For each $n\geq 1$ , let

\begin{equation*}L_n\;:\!=\; g_{s_0}^{-1} \circ g_{s_1}^{-1} \circ \cdots \circ g^{-1}_{s_{n-1}}\circ \Phi_0 \circ \mathcal{F}^n\end{equation*}

and note that for any subset $U\subset J(\mathcal{F})$ such that $\Phi_0(\mathcal{F}^n(U))\subset F_{s_n}$ , by Proposition 3·5 and the definition of the maps $\{g^{-1}_{s_i}\}_{i< n}$ , $L_n\vert_{U}$ is a continuous function, as it is a composition of continuous functions. By (3·4), $L_n(t,{\underline s})=\Phi_n(t,{\underline s})$ . Hence, in order to prove continuity of $\Phi_n$ at $(t,{\underline s})$ , since by Observation 3·7 $\Phi_0(\mathcal{F}^n(t,{\underline s}))\subset F_{s_n}$ , it suffices to find a neighbourhood $V \ni (t,{\underline s})$ such that $\Phi_{n}\vert_{V} \equiv L_{n}\vert_{V}$ . Let $J_n \subset (\mathbb{Z}\times\{L, R\})^\mathbb{N}$ be any open interval containing ${\underline s}$ and such that if $\underline{\tau}=\tau_0\tau_1\cdots \in J_n$ , then $s_i=\tau_i$ for all $0\leq i \leq n$ , and choose $t_1,t_2\in \mathbb{R}^+$ so that $t_1\leq t \leq t_2$ . Then, $V\;:\!=\; ((t_1, t_2)\times J_n) \cap J(\mathcal{F})$ satisfies the properties required and continuity of $\Phi_n$ follows.

We are now ready to prove Theorem 1·3. We will do so by first showing that for any given disjoint-type cosine map g, the functions $\{\Phi_n\}_{n\in \mathbb{N}}$ converge to a continuous function $\Phi\colon J(\mathcal{F})\to J(g)$ satisfying the properties listed on the second part of the statement. Then, for the first part, given any cosine map f, we will use [ Reference Pardo–Simón19 , theorem 4·6] to relate its dynamics to those of a disjoint-type cosine map, and then apply the second part. We note that [ Reference Pardo–Simón19 , theorem 4·6] relies heavily on Rempe’s analogs of Bottcher’s map for transcendental maps, and more specifically on [ Reference Rempe24 , theorem 3·2].

Proof of Theorem 1·3. We start by showing the second part of the statement. Let g be a disjoint type cosine map and let $\mathcal{T}_g$ be a pair of expansion tracts for g. Let $\{\Phi_{n}\}_{n\geq 0}$ be the sequence of functions from Definition 3·9, and suppose that $g(z)=ae^z+ be^{-z}$ for some $a,\,b\in \mathbb{C}^\ast$ . If $M\;:\!=\; \max\{\vert a \vert, \vert b \vert\}$ , then by Propositions 2·5 and 3·8, for each $x=(t, {\underline s}) \in J(\mathcal{F})$ with ${\underline s}=s_0s_1s_2\cdots$ ,

\begin{align*} \vert \textrm{Re} \; ( \Phi_1(x)) \vert & =\vert \textrm{Re} \; ( (g_{s_0}^{-1} \circ \mathcal{C}_A \circ\mathcal{F})(x))\vert \leq \ln\vert \mathcal{C}_A(\mathcal{F}(x)) \vert \\[4pt] & \quad +\vert \ln(M) \vert +1 \leq t+A +\vert\ln(M) \vert+2. \end{align*}

Similarly, $\vert \textrm{Re} \; ( \Phi_1(x)) \vert \geq t-\ln(\sqrt{2})-\vert\ln(M) \vert-2$ . By the definition of $\Phi_1$ and Observation 2·4, both $\Phi_0(x)$ and $\Phi_1(x)$ lie in the same fundamental domain $F_{s_0}$ . Thus, either both points have positive real part, of both have negative real part. By this and using that $\vert \textrm{Re} \; (\Phi_0(x))\vert = t+A$ , we have

(3·6) \begin{equation} \vert \textrm{Re} \; (\Phi_0(x)) - \textrm{Re} \; (\Phi_1(x))\vert\leq A+\ln(\sqrt{2}) +\vert \ln(M) \vert+2. \end{equation}

Moreover, by (3·6) and Proposition 2·5,

(3·7) \begin{equation} \vert \Phi_0(x) -\Phi_1(x)\vert \leq A +\ln(\sqrt{2}) + \vert \ln(M)\vert +2 + 3\pi \;=\!:\;\mu, \end{equation}

where we note that the constant $\mu$ does not depend on the point x. In particular, $\Phi_0(x)$ and $\Phi_1(x)$ lie in the same tract, and hence the straight segment joining these two points is totally contained in a connected component of $\{z\;:\; \vert \textrm{Re} \; z\vert >\mathcal{K}(g)\}$ , which is a convex set. Moreover, by (3·4), if $\mathsf{g}^{-1}_{{\underline s},n}\;:\!=\; g^{-1}_{s_0}\circ g^{-1}_{s_1} \circ \cdots \circ g^{-1}_{s_{n-1}}$ for some $n\geq 1$ , then

\begin{equation*}\Phi_n(x)=(\mathsf{g}^{-1}_{{\underline s},n}\circ \Phi_0 \circ \mathcal{F}^n)(x) \quad \text{ and } \quad \Phi_{n+1}(x)=(\mathsf{g}^{-1}_{{\underline s},n}\circ\Phi_1 \circ\mathcal{F}^n)(x).\end{equation*}

Note that if $\gamma$ is the straight segment connecting $\Phi_0(\mathcal{F}^n(x))$ and $\Phi_1(\mathcal{F}^n(x))$ , then since the map $\mathsf{g}^{-1}_{{\underline s},n}$ is a bijection to its image as it is a composition of bijections, $\mathsf{g}^{-1}_{{\underline s},n} (\gamma)$ is a curve with endpoints $\Phi_n(x)$ and $\Phi_{n+1}(x)$ . Thus, using (3·7) and (2·2),

(3·8) \begin{equation} \vert \Phi_{n+1}(x)-\Phi_n(x)\vert \leq \frac{\vert \Phi_0(\mathcal{F}^n(x)) -\Phi_1(\mathcal{F}^n(x))\vert} {2^n} \leq \frac {\mu} {2^n}. \end{equation}

Hence, $\{\Phi_n\}_{n\geq 0}$ is a uniformly Cauchy sequence of continuous functions, and so they converge uniformly to a continuous limit function $\Phi\;:\;J(\mathcal{F}) \rightarrow \mathbb{C}$ , that by (3·5) satisfies

(3·9) \begin{equation} \Phi\circ \mathcal{F}=g \circ \Phi. \end{equation}

Note that for each $x \in J(\mathcal{F})$ , $\Phi(x)$ is the limit of the backward orbit of a point in $\mathcal{T}_g$ , see (3·4). Hence, by Observation 2·7, $\Phi(x)\in J(g)$ and thus, $\Phi(J(\mathcal{F}))\subset J(g)$ . Moreover, since $\mathcal{C}_A \equiv \Phi_0$ , for each $x\in J(\mathcal{F})$ ,

(3·10) \begin{equation} \vert \Phi(x)-\mathcal{C}_A(x)\vert \leq \sum_{n=0} ^{\infty} \vert\Phi_{n+1}(x)-\Phi_n(x)\vert \leq\sum_{j=0} ^{\infty}\frac{\mu}{2^{n}} =2\mu. \end{equation}

This means for any sequence $\lbrace x_n \rbrace_{n\in \mathbb{N}} \subset J(\mathcal{F})$ that $\Phi(x_n)\rightarrow\infty$ if and only if $\mathcal{C}_A(x_n)\rightarrow\infty$ as $n\rightarrow \infty$ . By this, the definition of $I(\mathcal{F})$ and Proposition 3·8,

(3·11) \begin{equation} \begin{split} x \in I(\mathcal{F})& \Leftrightarrow \lim_{n \to \infty} T(\mathcal{F}^n(x))=\infty \Leftrightarrow \lim_{n \to \infty}\vert \mathcal{C}_A(\mathcal{F}^n(x)) \vert =\infty \\ & \Leftrightarrow \lim_{n \to \infty} \Phi(\mathcal{F}^n(x))= \lim_{n \to \infty} g^{n}(\Phi(x))=\infty \Leftrightarrow \Phi(x) \in I(g). \end{split} \end{equation}

Equivalently, $\Phi(I(\mathcal{F})) \subseteq I(g)$ and $\Phi(J(\mathcal{F}) \setminus I(\mathcal{F})) \subseteq J(g)\setminus I(g)$ . Consequently, surjectivity of $\Phi$ would imply $\Phi(I(\mathcal{F})) = I(g).$

Claim. The function $\Phi\;:\; J(\mathcal{F}) \rightarrow J(g)$ is surjective.

Proof of claim. Fix an arbitrary $z\in J(g)$ . Then, $z\in J_{\underline s}$ for some ${\underline s}=s_0s_1s_2\cdots \in \textrm{Addr}(g)$ , see Observation 2·7. Note that by definition, the function $\mathcal{F}$ is injective on its first coordinate. That is, for each fixed ${\underline s}\in (\mathbb{Z}\times \{L,R\})^\mathbb{N}$ , $\mathcal{F}_{{\underline s}}\;:\!=\; \mathcal{F}(\cdot, {\underline s})\colon \mathbb{R}^+\rightarrow \mathbb{C}$ given by $t\mapsto\mathcal{F}(t, {\underline s})$ is injective. Hence, we can consider the sequence of real positive numbers $\{t_k\}_{k\geq 0}$ uniquely determined by the equations

\begin{equation*} \mathcal{F}^{k}(t_k,{\underline s})=(\vert \textrm{Re} \; ( g^{k}(z))\vert, \sigma^{k}({\underline s})). \end{equation*}

In particular, $T(\mathcal{F}^{k}(t_k,{\underline s}))=\vert\textrm{Re} \; (g^{k}(z))\vert>0$ , and hence one can see using a recursive argument that for all $0\leq j\leq k$ ,

(3·12) \begin{equation} T(\mathcal{F}^{j}(t_k,{\underline s}))=\log \left( T(\mathcal{F}^{j+1}(t_k,{\underline s}))+2\pi \{ s_{j+1}\} +1 \right)>0, \end{equation}

and so $\mathcal{F}^j(t_k, {\underline s})$ is indeed well-defined for all $j\leq k$ . By definition of the map $\mathcal{C}_A$ , it holds that $\textrm{Re} \; ( \mathcal{C}_A(\mathcal{F}^{k}(t_k,{\underline s})))=\pm\textrm{Re} \; (g^{k}(z)) \pm A$ , where “ $\pm$ ” equals “ $+$ ” or “ minus;” depending on whether $s_k$ belongs to $\mathbb{Z}\times \{R\}$ or $\mathbb{Z}\times \{L\}$ . Moreover, by Observation 3·7, both $g^{k}(z),\mathcal{C}_A(\mathcal{F}^{k}(t_k,{\underline s})) \in F_{s_k}$ , and hence by Proposition 2·5,

(3·13) \begin{equation} \vert \mathcal{C}_A(\mathcal{F}^{k}(t_k,{\underline s})) -g^{k}(z) \vert< 3\pi + A. \end{equation}

Note that for all $j\leq k$ , since the second coordinate of $\mathcal{F}^{j}(t_k,{\underline s})$ equals $\sigma^j({\underline s})$ , we have that $\Phi_{k-j}(\mathcal{F}^{j}(t_k,{\underline s}))=(g^{-1}_{s_{j}}\circ \cdots \circ g^{-1}_{s_{k-1}}\circ \mathcal{C}_A \circ\mathcal{F}^{k})(t_k,{\underline s})$ and $g^{j}(z)=(g^{-1}_{s_{j}}\circ \cdots \circ g^{-1}_{s_{k-1}}\circ g^k)(z)$ . Hence, using (2·2), (3·10) and (3·13), by the same contraction argument as when showing (3·8), for any $j\leq k$ ,

\begin{equation*} \begin{split} \vert \mathcal{C}_A(\mathcal{F}^{j}(t_k,{\underline s})) -g^{j}(z) \vert & \! \leq \! \vert \mathcal{C}_A(\mathcal{F}^{j}(t_k,{\underline s})) \!-\!\Phi_{k-j}(\mathcal{F}^{j}(t_k,{\underline s})) \vert\! + \!\vert \Phi_{k-j}(\mathcal{F}^{j}(t_k,{\underline s}))\!-\!g^j(z)\vert \\ &<2\mu + \frac{3\pi + A}{2^{k-j}}< 2(\mu +3\pi + A)\;=\!:\;\eta. \end{split} \end{equation*}

We note that the constant $\eta$ does not depend on k. In particular, by taking $j=0$ we see that $t_k$ is uniformly bounded from above by a constant independent of k, and thus $t_k\nrightarrow \infty$ as $k \rightarrow \infty$ . This means that there exists at least one finite limit point for the sequence $\{t_k\}_{k\geq 0}$ , say $t\geq 0$ , that by (3·12) satisfies $(t,{\underline s})\in J(\mathcal{F})$ . Since by (3·9), for each $j\geq 0$ , we have $g^j(\Phi(t,{\underline s}))=\Phi (\mathcal{F}^j(t,{\underline s}))$ ,

\begin{equation*} \begin{split} \vert g^j(\Phi(t,{\underline s})) -g^{j}(z) \vert \! \leq \!\vert \Phi(\mathcal{F}^j(t,{\underline s}))\!- \!\mathcal{C}_A(\mathcal{F}^{j}(t,{\underline s})) \vert \!+\! \vert \mathcal{C}_A(\mathcal{F}^{j}(t,{\underline s}))-g^j(z)\vert \!< \!2\mu + \eta, \end{split} \end{equation*}

and this upper bound does not depend on j. Since $g^j(\Phi(t,{\underline s}))$ and $g^{j}(z)$ belong to the same fundamental domain $F_{s_j}$ for each $j\geq 0$ , we can once more use the same contraction argument to conclude that the points $\Phi(t,{\underline s})$ and z are equal.

To prove injectivity of $\Phi$ , note that if $(t,{\underline s}), (t', {\underline s}) \in J(\mathcal{F})$ for some $t\neq t'$ , the orbits of $(t,{\underline s})$ and $(t', {\underline s})$ under $\mathcal{F}$ will eventually be far apart by definition of $\mathcal{F}$ . Then, by (3·9) and (3·10), so will be the orbits under g of $\Phi(t, {\underline s})$ and $\Phi(t', {\underline s})$ , and injectivity follows.

Since the compactification $J(\mathcal{F})\cup \{\tilde{\infty}\}$ is by Proposition 3·5 a sequential space, and so is $\widehat{\mathbb{C}}\;:\!=\; \mathbb{C} \cup \{\infty\} $ , the notions of continuity and sequential continuity for functions between these spaces are equivalent. Thus, using (3·11), we can extend $\Phi$ to a continuous map $\tilde{\Phi}\;:\; J(\mathcal{F}) \cup \lbrace \tilde{\infty} \rbrace \rightarrow J(g) \cup \lbrace \infty \rbrace$ by defining $\tilde{\Phi}(\tilde{\infty})\;:\!=\;\infty$ . Then, $\tilde{\Phi}^{-1}$ is continuous as it is the inverse of a continuous bijective map on a compact space, and consequently, by respectively removing $\tilde{\infty}$ and $\infty$ from the domain and codomain of $\tilde{\Phi}^{-1}$ , it follows that $\Phi^{-1}$ is also continuous.

We have shown that for any cosine map g of disjoint type, $\Phi \colon J(\mathcal{F}) \rightarrow J(g)$ is a homeomorphism and $\Phi(I(\,F))=I(g)$ .

Suppose now that f is any cosine map. If follows from [ Reference Pardo–Simón19 , theorem 4·6] that there is a disjoint type cosine map g and a continuous function $\theta\colon J(g)\to J(\,f)$ with the following properties:

1. (i) the restriction $\theta\colon \theta^{-1}(J_K(\,f)) \to J_K(\,f)$ is a homeomorphism for some $K>0$ ;

2. (ii) $\theta \circ g=f\circ \theta$ in $\theta^{-1}(J_K(\,f))$ ;

3. (iii) $\theta(I(g))\subset I(\,f)$ .

More specifically, the map g in [ Reference Pardo–Simón19 , theorem 4·6] is of the form $\lambda f$ for some $\lambda \in \mathbb{C}\setminus \{0\}$ , and so g is a cosine map. Then, [ Reference Pardo–Simón19 , theorem 4·6] states that J(g) contains a so-called strongly absorbing Cantor bouquet X, [ Reference Pardo–Simón19 , definition 1·5], such that $\theta\vert_X$ is a homeomorphism to its image, [ Reference Pardo–Simón19 , theorem 4·6(d)], and so that $\theta(X)\supset J_K(\,f)$ for some $K>0$ , [ Reference Pardo–Simón19 , line 18 in the proof of theorem 4·6]. Thus, (1) follows. Item (2) is a consequence of this together with [ Reference Pardo–Simón19 , theorem 4·6(b)], and (3) is part of [ Reference Pardo–Simón19 , theorem 4·6(g)]. Then, if $\tilde{\Phi}\colon J(\mathcal{F}) \rightarrow J(g)$ is the homeomorphism provided by the second part of the statement, the composition $\Phi \;:\!=\; \theta\circ \tilde{\Phi} \colon J(\mathcal{F})\to J(\,f)$ satisfies the desired properties.

4. A model for strongly postcritically separated cosine maps

We defined in the introduction what it means for a cosine map to be strongly postcritically separated. This notion, introduced in [ Reference Pardo–Simón20 ], generalizes to some functions in class $\mathcal{B}$ :

1. (i) $P_{F}\;:\!=\; P(\,f)\cap F(\,f)$ is compact;

2. (ii) f has bounded criticality on J(f);

3. (iii) for each $z\in J(\,f)$ , $\#(\textrm{Orb}^+(z)\cap \textrm{Crit}(\,f)) \leq c$ ;

4. (iv) for all distinct $z,w\in P_J\;:\!=\; P(\,f)\cap J(\,f)$ , $\vert z-w\vert\geq \epsilon \max\{\vert z \vert, \vert w \vert\}$ .

For cosine maps, some conditions in the definition of strongly postcritically separated maps are trivially satisfied, and thus they can be characterised the following way:

In this section we prove our main results on sps cosine maps; namely, Theorems 1·2 and Section 1·4. We start by providing a formal definition of dynamic rays:

1. (i) for each $n\geq 1$ , $t \mapsto f^{n}(\gamma(t))$ is injective with $\lim_{t \rightarrow \infty} f^{n}(\gamma(t))=\infty$ ;

2. (ii) $f^{n}(\gamma(t))\rightarrow \infty$ uniformly in t as $n\rightarrow \infty$ .

A dynamic ray of f is a maximal injective curve $\gamma\;:\;(0,\infty)\rightarrow I(\,f)$ such that the restriction $\gamma_{|[t,\infty)}$ is a ray tail for all $t > 0$ . We say that $\gamma$ lands at z if $\lim_{t \rightarrow 0^+} \gamma(t)=z$ , and we call z the endpoint of $\gamma$ . We denote the set of endpoints of dynamic rays of f by E(f).

Recall from the introduction that whenever a ray tail contains a critical value, its preimages can be interpreted as several ray tails that split or break at critical points, and that may be extended to overlap pairwise. In [ Reference Pardo–Simón18 ], combinatorics for maps exhibiting this phenomenon are developed, where the extensions of ray tails at critical points, as described for the map $z \mapsto \cosh(z)$ , are formalised.

More precisely, let f be a cosine map, let $\textrm{Addr}(\,f)$ be the set of external addresses as specified in Definition 2·6 and denote

(4·2) \begin{equation}\textrm{Addr}(\,f)_\pm\;:\!=\; \textrm{Addr}(\,f)\times \{-,+\}.\end{equation}

We call each of its elements $({\underline s}, \ast) \in \textrm{Addr}(\,f)_\pm$ a signed (external) address.

It is shown in [ Reference Pardo–Simón18 , section 3] that $\textrm{Addr}(\,f)_\pm$ can be endowed with a topology such that each $z\in I(\,f)$ has at least two signed addresses that depend continuously on z. This leads to a folliation of I(f) as a collection of rays, that we call canonical, indexed by signed external addresses. More precisely, we shall use the following results from [ Reference Pardo–Simón18 ].

Definition and Theorem 4.4 (Canonical rays). Let f be a sps cosine map. Then, for each $({\underline s}, \ast) \in \textrm{Addr}(\,f)_\pm$ there is a curve $\Gamma({\underline s}, \ast)$ , that is either a ray tail or a dynamic ray possibly with its endpoint, such that

(4·3) \begin{equation}I(\,f)\subset \bigcup_{({\underline s},\ast)\in \textrm{Addr}(\,f)_\pm} \Gamma({\underline s}, \ast). \end{equation}

We say that $\Gamma({\underline s}, \ast)$ is a canonical ray, and we call any ray tail contained in a canonical ray a canonical tail. Landing of all canonical rays implies landing of all dynamic rays in J(f). Each $z\in I(\,f)$ belongs to exactly

(4·4) \begin{equation}\textrm{Addr}(z)_\pm \;:\!=\; 2\prod^{\infty}_{j=0}\deg(\,f,f^{j}(z))\end{equation}

different canonical rays, for $\textrm{Addr}(z)_\pm \in \{2,4,8\}.$

Next, we define our model for sps cosine maps. We start by setting the topology we shall use in the ambient space $\mathcal{M}\times \{-,+\}$ that contains $J(\mathcal{F})\times \{-,+\}$ :

4·5 (Definition of topologies). Consider the set

(4·5) \begin{equation} \mathcal{M}_\pm \;:\!=\; \mathcal{M}\times \{-,+\}= [0,\infty)\times(\mathbb{Z}\times\{L, R\})^\mathbb{N} \times \{-,+\}, \end{equation}

and let $<_{\ell}$ denote the lexicographic order in $(\mathbb{Z}\times\{L, R\})^\mathbb{N}$ defined in 3.1. Let us give the set $\lbrace -,+\rbrace$ the order $\lbrace -\rbrace \prec\lbrace +\rbrace$ , and define the linear order in $(\mathbb{Z}\times\{L, R\})^\mathbb{N} \times \{-,+\}$

(4·6) \begin{equation} (\underline{s}, \ast )<_{_A} (\underline{\tau}, \star) \qquad \text{ if and only if } \qquad \underline{s} <_{_\ell} \underline{\tau} \quad \text{ or } \quad \underline{s} =_{_\ell} \underline{\tau}\: \text{ and } \: \ast \prec \star,\end{equation}

where the symbols “ $\ast, \star$ ” denote generic elements of $\lbrace-, +\rbrace.$ This linear order gives rise to a cyclic order: for $a,\,x,\,b \in (\mathbb{Z}\times\{L, R\})^\mathbb{N} \times \{-,+\}$ ,

\begin{equation*} [a,\,x,\,b]_{_A} \quad \text{if and only if} \quad a<_{_A} x<_{_A} b \quad \text{ or } \quad x <_{_A}b <_{_A} a \quad \text{ or }\quad b <_{_A} a <_{_A} x.\end{equation*}

This cyclic order allows us to provide the set $(\mathbb{Z}\times\{L, R\})^\mathbb{N} \times \{-,+\}$ with the corresponding cyclic order topology $\tau_I$ .

We define the topological space $(\mathcal{M}_\pm, \tau_{\mathcal{M}_\pm})$ , with $\tau_{\mathcal{M}_\pm}$ being the product topology of $[0,\infty)$ with the usual topology, and $(\mathbb{Z}\times\{L, R\})^\mathbb{N} \times \{-,+\}$ with the topology $\tau_I$ .

We are now ready to prove the main result of this paper, which is a more detailed version of Theorem 1·4. Using the general framework developed in [ Reference Pardo–Simón21 ], its proof will be concise. We remark that, alternatively, we could had provided a more direct but significantly more lengthy and technical proof. More precisely, one could avoid using the results from [ Reference Pardo–Simón21 ] and construct directly the semiconjugacy from $\mathcal{F}$ to $f\vert _{J(\,f)}$ in a similar manner to the proof of [ Reference Rottenfusser, Rückert, Rempe and Schleicher27 , theorem 6·5], using that sps maps expand an orbifold metric in a neighbourhood of their Julia sets; [ Reference Pardo–Simón20 , theorem 1·1]. We have opted for the first approach for clarity of exposition and to show an application of the framework that [ Reference Pardo–Simón21 ] provides.

Recall from Definition 3·3 that $\mathscr{S}^{\,\mathbb{N}}$ denotes the set of exponentially bounded elements associated to $J(\mathcal{F})$ . We define $\mathscr{S}^{\,\mathbb{N}}_\pm \;:\!=\;\mathscr{S}^{\,\mathbb{N}}\times \{-,+\}$ .

Theorem 4·7. Let f be a strongly postcritically separated cosine map and let $J(\mathcal{F})_\pm$ and $\widetilde{\mathcal{F}}$ be the model space and function from Definition 4·6. Then, $\widetilde{\mathcal{F}}$ is continuous, and there exists a continuous surjective function

\begin{equation*} \hat{\varphi} \colon J(\mathcal{F})_{\pm} \longrightarrow J(\,f) \qquad \text{ so that } \qquad f\circ\hat{\varphi} = \hat{\varphi}\circ \widetilde{\mathcal{F}},\end{equation*}

$\hat{\varphi}(I(\mathcal{F})_{\pm})=I(\,f)$ and such that for every $z\in I(\,f)$ , $\# \hat{\varphi}^{-1}(z)\in \{2,4,8\}$ . Moreover, for each $({\underline s}, \ast) \in \mathscr{S}^{\,\mathbb{N}}_\pm$ the restriction map $\hat{\varphi} \colon \{(t, {\underline s}, \ast) \colon t \geq t_{\underline s}\} \rightarrow \overline{\Gamma({\underline s}, \ast)}$ is a bijection, and so $\overline{\Gamma({\underline s}, \ast)}$ is a canonical ray together with its endpoint.

Proof of Theorem 4·7. Let $g\;:\!=\; \lambda f$ be of disjoint type for some $\lambda\in \mathbb{C}^\ast$ and let $\Phi\colon J(\mathcal{F})\to J(g)$ be the homeomorphism from Theorem 1·3. Define $J(g)_\pm\;:\!=\; J(g)\times \{-,+\}.$ In [ Reference Pardo–Simón21 , 5·4, p. 23], $J(g)_\pm$ is induced with a topology $\tau_J$ in an analogous manner as we did in 4.5 for $J(\mathcal{F})_\pm$ , with $(\mathbb{Z}\times\{L, R\})^\mathbb{N}$ replaced by $\mathbb{R}\setminus \mathbb{Q}$ in [ Reference Pardo–Simón21 , equation (5·1)]. In particular,

(4·7) \begin{equation} \hat{\Phi}\colon J(\mathcal{F})_\pm\rightarrow J(g)_\pm; \quad \quad (t,{\underline s},\ast)\mapsto (\Phi(t, {\underline s}), \ast) \end{equation}

is a homeomorphism. We let $I(g)_\pm\;:\!=\; I(g)\times \{-,+\}$ and for each ${\underline s}\in \textrm{Addr}(g)$ , denote $J_{({\underline s}, \ast)}\;:\!=\; J_{\underline s}\times \{\ast\}$ . Let $\tilde{g}\colon J(g)_\pm\rightarrow J(g)_\pm$ be given by $\tilde{g}(z, \ast)\;:\!=\;(g(z), \ast).$ Note that by definition of the functions involved and Theorem 1·3, we have that

(4·8) \begin{equation}\hat{\Phi}\circ \widetilde{\mathcal{F}}=\tilde{g} \circ \hat{\Phi},\end{equation}

and, in particular, $\mathcal{F}$ is continuous as it can be expressed as a composition of continuous functions.

Then, by [ Reference Pardo–Simón21 , theorem 6·5], there exists a continuous, surjective map $\varphi\;:\; J(g)_\pm \to J(\,f)$ such that

(4·9) \begin{equation} f\circ\varphi = \varphi\circ \tilde{g}, \quad \quad \varphi(I(g)_{ \pm})=I(\,f)\end{equation}

and for each $({\underline s}, \ast)\in \textrm{Addr}(g)_\pm$ ,

(4·10) \begin{equation}\varphi \colon J_{({\underline s}, \ast)} \rightarrow \overline{\Gamma({\underline s}, \ast)} \quad \text{is a bijection},\end{equation}

where $\Gamma({\underline s}, \ast)$ is the canonical ray at signed address $({\underline s}, \ast)\in \textrm{Addr}(\,f)_\pm$ . It then follows from Definition and Theorem 4.4 that for each $z\in I(\,f)$ ,

(4·11) \begin{equation} \#\varphi^{-1}(z)=2\prod^{\infty}_{j=0}\deg(\,f,f^{j}(z))\in \{2,4,8\}.\end{equation}

Let $\hat{\varphi}\;:\!=\;\varphi\circ \hat{\Phi}\;:\; J(\mathcal{F})_\pm \rightarrow J(\,f)$ , which is continuous as it is a composition of continuous functions. Then, by (4·9) and (4·8),

\begin{equation*}f \circ \hat{\varphi}=f \circ \varphi \circ \hat{\Phi} =\varphi\circ \tilde{g} \circ \hat{\Phi}= \varphi\circ \hat{\Phi}\circ \widetilde{\mathcal{F}}= \hat{\varphi} \circ \widetilde{\mathcal{F}},\end{equation*}

as shown in the diagram:

Moreover, since, by Theorem 1·3, $\Phi(I(\mathcal{F}))=I(g)$ , we have that $\hat{\Phi}(I(\mathcal{F})_\pm)=I(g)_\pm$ , and consequently $\hat{\varphi}(I(\mathcal{F})_\pm)=I(\,f)$ .

Since $\hat{\Phi}$ is a homeomorphism, using (4·11) we have that $\#\hat{\varphi}^{-1}(z)=\#\varphi^{-1}(z)\in \{2,4,8\}$ . By Observation 3·11, $\Phi\colon \{(t, {\underline s}) \colon t \geq t_{\underline s}\} \to J_{\underline s}$ is a bijection for each ${\underline s}\in \mathscr{S}^{\,\mathbb{N}}$ , with $J_{\underline s}$ being a landing dynamic ray. Hence, by (4·7) and (4·10), $\hat{\varphi} \colon \{(t, {\underline s}, \ast) \colon t \geq t_{\underline s}\} \rightarrow \overline{\Gamma({\underline s}, \ast)}$ is also bijection and $\overline{\Gamma({\underline s}, \ast)}$ is a canonical ray together with its endpoint.

Proof of Theorem 1·2. Let f be a sps cosine map. Then, by Definition and Theorem 4.4, proving that all canonical rays of f land suffices to conclude that all its dynamic rays land. Since, by Theorem 4·7, for each $({\underline s}, \ast) \in \textrm{Addr}(\,f)_\pm$ , $\overline{\Gamma({\underline s}, \ast)}$ is a canonical ray together with its landing point, the result follows. Moreover, by Theorem 4·7,

\begin{equation*}J(\,f)=\hat{\varphi}(J(\mathcal{F})_\pm)= \hat{\varphi}\left(\bigcup_{({\underline s}, \ast) \in \mathscr{S}^{\,\mathbb{N}}_\pm}\{(t, {\underline s}, \ast) \colon t \geq t_{\underline s}\}\right)= \bigcup_{({\underline s}, \ast) \in \textrm{Addr}(\,f)_\pm} \overline{\Gamma({\underline s},\ast)},\end{equation*}

and so every point in J(f) is either on a dynamic ray or it is the landing point of at least one such ray.

5. Combinatorics and landing of rays for $f(z)=\cosh(z)$

In the previous section we provided a semiconjugacy between the restriction of any strongly postcritically separated cosine map f to its Julia set, and a simple dynamical system formed by a map $\widetilde{\mathcal{F}}$ and a model space $J(\mathcal{F})_\pm$ , that reflects the splitting of dynamic rays at critical points. In particular, Theorem 4·7 implies that J(f) can be described as a collection of canonical rays that land, and overlap pairwise.

Given the explicit nature of cosine maps, we claim that it is possible to improve this result by providing a conjugacy rather than a semiconjugacy between a model dynamical system and our maps. Note that if a cosine map f is sps, we let $\hat{\varphi} \colon J(\mathcal{F})_{\pm} \rightarrow J(\,f)$ be the map from Theorem 4·7, and we define the equivalence relation in $J(\mathcal{F})_\pm$

(5·1) \begin{equation}a \sim b \iff \hat{\varphi}(a)=\hat{\varphi}(b),\end{equation}

then, since $\hat{\varphi}$ is continuous, by the Universal Property of Quotient Maps (see for example [ Reference Munkres16 , theorem 2·22]), there exists a unique continuous function $\tilde{\varphi}\colon J(\mathcal{F})_\pm /{\sim} \rightarrow J(\,f)$ such that the diagram

commutes, where $\pi$ is the projection function that takes each element to its equivalence class. In particular, since both $\hat{\varphi}$ and $\pi$ are surjective, $\tilde{\varphi}$ is by definition bijective. By the commutative relation $f\circ \hat{\varphi}= \hat{\varphi} \circ \widetilde{\mathcal{F}}$ from Theorem 4·7, for any $a,\,b\in J(\mathcal{F})_\pm$ ,

\begin{equation*}\pi(a)=\pi(b) \Rightarrow \hat{\varphi}(\widetilde{\mathcal{F}}(a)) =f(\hat{\varphi}(a))=f(\hat{\varphi}(b))=\hat{\varphi}(\widetilde{\mathcal{F}}(b))\Rightarrow \pi(\widetilde{\mathcal{F}}(a))=\pi(\widetilde{\mathcal{F}}(b)),\end{equation*}

and so, the map $h\colon J(\mathcal{F})_\pm /{\sim} \rightarrow J(\mathcal{F})_\pm /{\sim}$ given by $h(\pi(x))\;:\!=\; \pi(\widetilde{\mathcal{F}}(x))$ is a well-defined homeomorphism. In particular, $\tilde{\varphi}$ conjugates h and f as shown in the following diagram:

In this section, we describe “ $\sim$ ” explicitly for the maps $z\mapsto \cosh(z)$ and $z\mapsto \cosh^2(z).$ The following observation is important to us.

We will divide our task into two: on one hand, we shall study whether canonical rays share their endpoints, that is, whether some of them land together. On the other hand, one should provide information on the overlaps occuring between canonical rays. For the second, we will use the following result, which is [ Reference Pardo–Simón18 , proposition 3·9]. We recall that $\sigma \colon \textrm{Addr}(\,f)\rightarrow \textrm{Addr}(\,f) $ denotes the one-sided shift map on addresses.

Proposition 5·2 (Overlapping of canonical rays). For each $({\underline s}, \ast) \in {Addr}(\,f)_\pm$ , either $\Gamma({\underline s}, -)=\Gamma({\underline s}, +)$ when ${Orb}^{-}({Crit}(\,f)) \cap \Gamma({\underline s}, \ast) =\emptyset$ , or $\Gamma({\underline s}, \ast)$ can be expressed as a concatenation

(5·2) \begin{equation} \Gamma({\underline s}, \ast)=\cdots {\cdot} \{c_{i+1}\}{\cdot} \gamma^{i+1}_{i} {\cdot} \{c_i\} {\cdot} \cdots {\cdot}\gamma^{1}_{0} {\cdot} \{c_0\} {\cdot} \gamma^{\infty}_{c_0}, \end{equation}

where $\{c_i\}_{i\in I}={Orb}^{-}({Crit}(\,f)) \cap \Gamma({\underline s}, \ast)$ , for each $i\geq 1$ , if it exists, the curve $\gamma^{i+1}_{i}$ is a (bounded) piece of dynamic ray, and $\gamma^{\infty}_{c_0}$ is a piece of dynamic ray joining $c_0$ to infinity. In particular, in the latter case, the following properties hold for $\Gamma({\underline s}, \ast)$ :

1. (i) $\gamma^{\infty}_{c_0} \cup \{c_0\}= \Gamma({\underline s},- ) \cap \Gamma({\underline s}, +)$ and $\gamma^{\infty}_{c_0}$ does not belong to any other canonical ray;

2. (ii) for each $i\geq 0$ , the point $c_i$ belongs to exactly $2 \prod^{\infty}_{j=0}\deg(\,f,f^{j}(c_i))$ canonical rays;

3. (iii) for each $i\geq 0$ , $\gamma^{i+1}_i= \Gamma({\underline s}, \ast)\cap \Gamma(\underline{\tau}, \star)$ , where $\star\neq \ast$ and $\sigma^j(\underline{\tau})=\sigma^j({\underline s})$ for some $j\geq 1$ . Moreover, $\gamma^{i+1}_i$ does not belong to any other canonical ray.

We start our analysis on the map $f(z)=\cosh(z)$ . Following Observation 5·1, we shall provide a combinatorial description of the equivalence classes of $J(\mathcal{F})_\pm /{\sim}$ in terms of the signed addresses of their images under $\tilde{\varphi}$ .

For a function $f\in \mathcal{B}$ , the partition of a neighbourhood of infinity into fundamental domains is commonly regarded as a static partition in the sense that the curve $\delta$ and domain $D\supset S(\,f)$ in its definition do not have dynamical meaning for f. In particular, dynamic rays of f might cross the boundaries of fundamental domains infinitely often. Instead, for our specific example, we can define a dynamical partition, so that the boundaries of the components are ray tails:

5·3 (Dynamical partition for $f(z)=\cosh(z)$ ). For the function $f(z)\;:\!=\; \cosh(z)$ , $J(\,f)=\mathbb{C}$ and $S(\,f)=\textrm{CV}(\,f)=\{-1, 1\}$ . Moreover, the curves $\gamma_1\;:\!=\; \mathbb{R} \setminus (-\infty, 1)$ and $\gamma_{-1}\;:\!=\; \mathbb{R} \setminus (\!-\!1, \infty)$ are ray tails joining 1 and $-1$ to $\infty$ whose forward orbits lie in $\mathbb{R}^+$ . Let

(5·3) \begin{equation}X\;:\!=\;\gamma_1\cup \gamma_{-1}.\end{equation}

Since $\mathbb{C} \setminus X$ is simply-connected and $S(\,f)\subset X$ , by the Monodromy Theorem, each connected component of $f^{-1}(\mathbb{C} \setminus X)$ is a simply-connected domain, and the restriction of f to it is a conformal isomorphism to its image. More specifically, noting that all preimages of the critical values of f are critical points of local degree 2, each connected component of $f^{-1}(\mathbb{C} \setminus X)$ is a horizontal strip of the form

\begin{equation*} U_K\;:\!=\; \{ z\in \mathbb{C} \text{ such that } K\pi < \textrm{Im} \; z <(K+1)\pi) \} \end{equation*}

for some $K\in\mathbb{Z}$ . We denote $\mathcal{U}\;:\!=\; \{U_K\}_{K\in \mathbb{Z}}$ ; see Figure 3.

5·4 (Fixing signed addresses for $f(z)=\cosh(z)$ ). Let us fix any bounded domain $D\supset [-1,1] \supset S(\,f)$ . Then, $f^{-1}(\mathbb{C}\setminus D)$ consists of two unbounded domains that do not contain the imaginary axis, since $f(i\mathbb{R})=[-1,1]$ . Thus, we can choose $\delta\;:\!=\; i\mathbb{R}^+ \setminus D$ and define fundamental domains for f as connected components of $f^{-1}(\mathbb{C} \setminus (\overline{D} \cup \delta))$ . In particular, $f^{-1}(\delta)$ equals the collection of horizontal half-lines

\begin{equation*} \begin{split} \{z\in \mathbb{C} \;:\; \textrm{Re} \; z<0 & \text{ and }\textrm{Im} \; z= (-1/2+2n)\pi \},\\ \{z\in \mathbb{C}\;:\; \textrm{Re} \; z >0 & \text{ and }\textrm{Im} \; z = (1/2+2 n)\pi \} \end{split} \end{equation*}

for all $n \in \mathbb{Z}$ . Thus, each fundamental domain is contained in one of the half-strips

(5·4) \begin{equation} \begin{split} S_{(n,\,L)}& \;:\!=\; \{z\;:\; \textrm{Re} \; z<0, \textrm{Im} \; z\in ((n-1/2)\pi ,(n+3/2)\pi)\},\\ S_{(n,R)}& \;:\!=\; \{z\;:\; \textrm{Re} \; z>0, \textrm{Im} \; z\in ((n-3/2)\pi,(n+1/2)\pi)\}. \end{split} \end{equation}

For each $(n,\ast)\in \mathbb{Z} \times \{-,+\}$ , we denote by $n_\ast$ the unique fundamental domain contained in $S_{(n, \ast)}$ . Using these fundamental domains, we define the set of external addresses $\textrm{Addr}(\,f)$ , and fix the corresponding set of signed external addresses $\textrm{Addr}(\,f)_\pm$ ; see Definition and Theorem 4.4. In particular, for the curves $\gamma_1$ and $\gamma_{-1}$ introduced in Section 5·3, it holds that $f(\gamma_{-1}) \subset \gamma_1$ , $f(\gamma_1)\subset \gamma_1$ , and . Moreover, each of these curves equals two canonical tails with opposite sign, as they do not contain preimages of critical points, see Proposition 5·2. Hence, and for both $\ast \in \{-,+\}$ ; see Figure 2.

Fig. 2. Partition of the plane into fundamental domains and itinerary components of $f(z)=\cosh(z)$ . Each strip of height $\pi$ between two coloured horizontal lines is an itinerary domain. Strips that contain some fundamental domains are indicated by keys. Black horizontal lines are preimages of the imaginary axis, and the rest of curves are the preimages of all horizontal lines.

The dynamical partition from Section will help us determine that no two dynamic rays of f land together. This is because since the boundaries of the elements in $\mathcal{U}$ are ray tails, as we shall see, no other ray tails can cross them. More precisely, in the next proposition, we assign to each curve $\{\overline{\Gamma({\underline s}, \ast)}\}_{({\underline s},\ast)\in \textrm{Addr}(\,f)_\pm}$ a unique element of $\mathcal{U}$ :

Proposition 5·5 (Each canonical ray is in the closure of a unique $U\in \mathcal{U}$ ). For each $({\underline s}, \ast)\in \textrm{Addr}(\,f)_\pm$ , there exists a unique component $U\in \mathcal{U}$ such that $\overline{\Gamma({\underline s}, \ast)} \subset \overline{U}$ . We denote

\begin{equation*}U({\underline s}, \ast)\;:\!=\; U.\end{equation*}

Proof. Since, as described in Section 5·4, the set X defined in (5·3) consists of four canonical tails that overlap pairwise, and, in addition, f is totally ramified, exactly four canonical tails meet at each critical point of $f^{-1}(X)$ , and their union is a connected component of this set. More precisely, following the analysis in Section 5·4, $f^{-1}(\gamma_1)$ is the collection of all the horizontal lines $\{2\pi K i\mathbb{R}\}_{K \in \mathbb{Z}}$ , that in particular contain the critical points $2\pi K i$ for all $K\in \mathbb{Z}$ . Analogously, $f^{-1}(\gamma_{-1})$ is the collection of the horizontal lines $\{2(K+1)\pi i \mathbb{R}\}_{K \in \mathbb{Z}}$ with critical points at $2(K+1)\pi i$ for each $K\in \mathbb{Z}$ . Hence, it follows that for each $K\in \mathbb{Z}$ and $\ast\in \{-,+\}$ ,

(5·5)

We claim that each of the canonical rays displayed in (5·5) belongs to the closure of exactly one component of $\mathcal{U}$ : to see this, let us consider the curves for some $K\in \mathbb{Z}$ and both $\ast\in \{-,+\}$ . Then, since is a nested sequence of left-extended canonical tails, see [ Reference Pardo–Simón18 , definition 3·5 and theorem 3·8], and by Proposition 5·2 it can only intersect the boundaries of the elements of $\mathcal{U}$ in the subcurve $2K\pi i \mathbb{R}^+$ , we conclude that . Similarly, , and arguing analogously for the rest of curves in (5·5), the claim follows. Since no other canonical rays apart from those in (5·5) intersect $\mathbb{C} \setminus \mathcal{U}$ , each of them must be contained in a unique component U, as stated.

Definition 5·6 (Itineraries for $f(z)=\cosh(z)$ ). For each $({\underline s}, \ast)\in \textrm{Addr}(\,f)_\pm$ , we define the itinerary of $({\underline s}, \ast)$ as the infinite sequence

\begin{equation*}\textrm{itin}({\underline s}, \ast)\;:\!=\; U({\underline s}, \ast) U(\sigma({\underline s}), \ast)U(\sigma^2({\underline s}), \ast)\cdots. \end{equation*}

Proof. It suffices to show that there are no two canonical rays landing together, since then, by Definition and Theorem 4.4, no two rays would land together. With that aim, let $\Gamma({\underline s}, \ast)$ and $\Gamma(\underline{\tau}, \star)$ be two different canonical rays, that is, $({\underline s}, \ast)\neq (\underline{\tau}, \star)$ , and let $p_{({\underline s}, \ast)}$ and $p_{(\underline{\tau}, \star)}$ be their respective endpoints. If $\Gamma({\underline s}, \ast)$ and $\Gamma(\underline{\tau}, \star)$ land together, i.e., $p_{({\underline s}, \ast)}= p_{(\underline{\tau}, \star)}$ , then by Proposition 5·5 and Observation 5·7, $\textrm{itin}({\underline s}, \ast)=\textrm{itin}(\underline{\tau}, \ast)=U_0U_1\cdots$ . Moreover, for each $i\geq 0$ , $f^i(p_{({\underline s}, \ast)})=f^i(p_{(\underline{\tau}, \star)})$ must belong to the interior of $U_i$ , since by 5.4, the boundaries of the elements of $\mathcal{U}$ are formed by canonical tails that are contained in dynamic rays. For the same reason, $i\mathbb{R}^+$ and $f^{-1}(i\mathbb{R}^+)$ do not contain any endpoints of dynamic rays, as they are formed by pieces of ray tails; see Section for more details. Then, for each $i\geq 0$ , $f^i(p_{({\underline s}, \ast)})=f^i(p_{(\underline{\tau}, \star)})$ belongs to a half-strip of the form

(5·6) \begin{equation} \begin{split} HS^i_{(k,L)}&\;:\!=\; \left\{ z\;:\; \textrm{Re} \; z\leq 0, \textrm{Im} \; z\in \left(\frac{(i+4k)\pi}{4} ,\frac{(i+1+4k)\pi}{4}\right) \right\},\\ HS^i_{(k,R)}&\;:\!=\; \left\{ z\;:\; \textrm{Re} \; z\geq 0, \textrm{Im} \; z\in \left(\frac{(i+4k)\pi}{4} ,\frac{(i+1+4k)\pi}{4}\right) \right\} \end{split} \end{equation}

for some $k\in \mathbb{Z}$ and $0\leq i \leq 3$ . However, each of the half-strips in (5·6) intersects a single fundamental domain, see (5·4) and Figure 2, which contradicts $({\underline s}, \ast)\neq (\underline{\tau}, \star)$ .

Finally, we provide a combinatorial description of the overlaps that occur between canonical rays in terms of their signed addresses, which by Observation 5·1 and Proposition 5·8 suffices to describe the equivalence classes in $J(\mathcal{F})/ {\sim}$ .

5·9 (Overlapping of canonical tails for $ z \mapsto \cosh(z)$ ). Recall that by Proposition 5·2, for all $({\underline s}, \ast)\in \textrm{Addr}(\,f)_\pm$ such that $\Gamma({\underline s}, \ast) \cap \textrm{Orb}^-(\textrm{Crit} (\,f))=\emptyset$ , $\Gamma({\underline s}, -)=\Gamma({\underline s}, +).$ Hence, all other overlap occurs between the preimages of the canonical tails that contain $\textrm{Crit}(\,f)$ . Recall from Section and that $\textrm{Crit}(\,f)=\{\pi i K\;:\;K \in \mathbb{Z}\}$ , and each critical point belongs exactly to four canonical rays. Namely, we saw in (5·5) that

Further identifications between the canonical rays above occur at the connected components of $f^{-1}([0,1])$ and $f^{-1}([-1,0])$ . More precisely, if for each $\pm\in \{-,+\}$ we denote

\begin{equation*}V_K(\pm)\;:\!=\; \left\{z \in \mathbb{C}\;:\; \textrm{Re} \; z=0 \text{ and } \textrm{Im} \; z\in [K\pi i,(K \pm 1/2)\pi i ]\right\},\end{equation*}

then we have the following identifications:

where $\mp=+$ when $\pm=-$ , and $\mp=-$ when $\pm=+$ ; see Figure 3. By Proposition 5·2, any further overlap between canonical rays occur at preimages of the overlap already stated. More specifically, since we have already described all overlap occurring at the boundaries of the elements in $\mathcal{U}$ , all remaining ones must occur between canonical rays whose itinerary agrees on the first N-th elements and that are mapped under $f^N$ to the coordinate axes for some $N\in \mathbb{N}$ . Given the geometry of the fundamental domains, in particular contained in half-strips of height $\pi$ , see (5·4) and Figure 2, providing the identifications at the preimages of the positive imaginary axis allows us to express identifications solely using external addresses. This is because any further identifications must occur at the intersection of a fundamental domain with a component of $\mathcal{U}$ , and hence, the corresponding first entries on the signed addresses of rays that overlap are the same. More specifically, for each $\pm\in \{-,+\}$ , let us denote

Then, for all $K \in \mathbb{Z}^+ \text{ and } P\in \mathbb{Z}$ ,

where $\mp=+$ when $\pm=-$ , and $\mp=-$ when $\pm=+$ ; see Figure 3. Moreover, for all other canonical rays, if $\delta$ is some bounded curve in J(f), then

(5·7) \begin{equation} \Gamma({\underline s}, \ast) =\Gamma(\underline{\tau}, \star) \text { in } \delta \iff \left\{\begin{array}{@{}l@{\quad}l@{}} \exists \, n>0 \;:\; s_j=\tau_j \text{ for all } j\leq n \quad \text{and}\\[5pt] \Gamma(\sigma^n({\underline s}), \ast) = \Gamma(\sigma^n(\underline{\tau}), \star) \text { in } f^n(\delta). \end{array}\right. \end{equation}

Fig. 3. Some canonical tails in the Julia set of the map $z \mapsto \cosh(z)$ that belong to canonical rays. Colour code: the red tail is in , the purple in , the orange in and the dark blue one in . Then, the light blue tail is in , the yellow one in , the green is in and the pink is in .

Fig. 4. Partition of the plane into fundamental domains and itinerary components for $z \mapsto \cosh^2(z)$ . Each strip of height $\pi$ between two coloured lines is an itinerary domain. Some fundamental domains are indicated by keys. Also displayed are the first (coloured lines and imaginary axis), second (other curves that meet at $\{K\pi i\;:\;K\in \mathbb{Z}\}$ ) and third (rest of curves) iterated preimages of the real line.

We have now provided a combinatorial description of all overlappings occurring between canonical rays of $z \mapsto \cosh(z)$ , as well as determined that no two of its dynamic rays land together. Hence, we have concluded our analysis.

Example 5·10 (Overlapping for the map $z \mapsto \cosh^2(z)$ ). Seeking an example where a critical value is mapped to another critical value, we consider the function

\begin{equation*}f(z)\;:\!=\; \cosh^2 (z)\;:\!=\; \cosh (z) \cdot \cosh (z) =\dfrac{e^{2z}+ e^{-2z}}{4}+ \dfrac{1}{2}.\end{equation*}

Even if, strictly speaking, this function is not in the cosine family, it is in the same parameter space, since f is conjugate to $({e}/{2})e^w+ ({e^{-1}}/{2})e^{-w}$ via $z \mapsto 2z-1$ . The dynamics of f have already been explored, namely in [ Reference Rippon and Stallard26 ], where it is shown that I(f) (and in fact its fast escaping set) is connected. The function f is $\pi i$ -periodic, and $S(\,f)= \textrm{CV}(\,f)=\{0,1\}$ , with $f(0)=1 \in I(\,f)$ and ${Orb}^+(1) \subset \mathbb{R}^+$ . The critical points of f are $f^{-1}(1)=\{K\pi i\;:\;K \in \mathbb{Z} \}$ and $f^{-1}(0)=\{(K+1/2)\pi i : K \in \mathbb{Z} \}.$ As for the map $z \mapsto \cosh(z)$ , we can join the critical values to infinity using the ray tails $\gamma_1\;:\!=\; \mathbb{R} \setminus (-\infty, 1)$ and $\gamma_{0}\;:\!=\; \mathbb{R} \setminus (0, \infty)$ . We define a dynamical partition for f as the union of the connected components of $\mathbb{C} \setminus f^{-1}(\gamma_0\cup \gamma_1)$ , which are horizontal half-strips of $\pi/2$ -height that we call itinerary components; see Figure 4. By choosing a bounded domain D containing [0, Reference Aarts and Oversteegen1 ] and $\delta\;:\!=\; i\mathbb{R}^+\setminus D$ , we can define fundamental domains for f as the connected components of $f^{-1}(\mathbb{C}\setminus (\overline{D}\cup \delta))$ . In particular, we label as

the component that contains an unbounded subset of $\mathbb{R}^+$ , and as

the component that contains an unbounded subcurve of $\mathbb{R}^-$ . Additionally, we label as

and

their respective $K\pi i$ -translates; see Figure 4. With this notation, the following identifications between canonical rays occur:

The main difference between the overlap on the canonical rays of $z \mapsto\cosh(z)$ and the overlap on those of f, is that since the critical points in $f^{-1}(0)$ are mapped to a critical point, each of them belongs to eight canonical tails rather than to four, and moreover, both singular values belong to the canonical rays

for both $\ast\in \{-,+\}$ . Compare Figures 3 and 5. Then, we further have the identifications

where $\mp=+$ when $\pm=-$ , and $\mp=-$ when $\pm=+$ . If for each $\pm \in \{-,+\}$ and $K\in \mathbb{Z}$ we let

then for all $K \in \mathbb{Z}^+$ and $P\in \mathbb{Z}$ ,

Any further identifications between canonical rays occur within the intersection of a fundamental domain and an itinerary domain, and hence can be expressed using (5·7). Moreover, arguing as for $z \mapsto \cosh(z)$ , no two dynamic rays of $z \mapsto \cosh^2(z)$ land together, see the proof of Proposition 5·8.

Fig. 5. Picture showing four canonical tails of $z \mapsto \cosh^2(z)$ that contain the critical point 0. These belong to the canonical rays (in blue), (in red), (in green) and (in yellow).