Proof. If the number of $1$ s in a word w equals n, then for any letter a, the number of $1$ s in ${\textrm {Pre}}_{2|w|+1}(\mathfrak {D}(w\cdot a))$ equals $2|w|+1-n$ . It follows that if n is odd, $w\cdot 1 <_E w \cdot 0$ and $\mathfrak {D}(w\cdot 1) <_E \mathfrak {D}(w\cdot 0)$ ; if n is even, $w\cdot 0 <_E w\cdot 1$ and $\mathfrak {D}(w\cdot 0) <_E \mathfrak {D}(w\cdot 1)$ . Thus, $\mathfrak {D}$ preserves $\leq _E$ . Furthermore, if a word w has positive cumulative sign, then the number, n, of $1$ s in w is even, implying that $\mathfrak {D}(w)$ , which contains $2w-n 1$ s, also has positive cumulative sign.

Proof. By induction, it is easy to see that we only need to prove it for $k=1$ , that is, $\mathfrak {D}(\mathrm {{It}}_{\unicode{x3bb} ^2})=\mathrm {{It}}_\unicode{x3bb} $ . For any $\unicode{x3bb} \leq \sqrt {2}$ , the tent map $f_{\unicode{x3bb} }$ sends the interval $[2/(\unicode{x3bb} +1), 1]$ to $[2-\unicode{x3bb} , 2/(\unicode{x3bb} +1)]$ and vice versa. Hence, $f^2_{\unicode{x3bb} }$ is a tent map from $[2/(\unicode{x3bb} +1), 1]$ of slope $\unicode{x3bb} ^2$ , and any $x=f^{2k}_{\unicode{x3bb} }(1)$ lies on the left-hand side of the critical point of $f^2_{\unicode{x3bb} }$ if and only if x and $f_{\unicode{x3bb} }(x)$ are both to the right of $1/\unicode{x3bb} $ , while $f^{2k}_{\unicode{x3bb} }(1)$ lies on the right-hand side of the critical point of $f^2_{\unicode{x3bb} }$ if and only if x is to the right of $1/\unicode{x3bb} $ and $f_{\unicode{x3bb} }(x)$ is to the left of $1/\unicode{x3bb} $ , and this finishes the proof for the case when $\mathrm {{It}}_{\unicode{x3bb} ^2}$ is not periodic. The case when $\mathrm {{It}}_{\unicode{x3bb} ^2}$ is periodic follows from this argument together with Lemma 3.1.

Proof. Suppose w and $w'$ are words satisfying $\mathfrak {D}(w')=w$ . It is easy to see that if $w=w_1w_2\ldots w_{2n}$ has a positive cumulative sign, then $w'=w^{\prime }_1w^{\prime }_2\ldots w^{\prime }_n$ also has a positive cumulative sign. So, the proposition follows from the following more general statement: if $w'$ is any word, w is the doubling of $w'$ , then

(3.1) $$ \begin{align} F(w, z)-1={z-1\over z+1}(F(w', z^2)-1). \end{align} $$

We will prove (3.1) using induction on $|w'|$ . In the base case, $|w'|=1$ , $w'=1$ , or $w'=0$ , and the statement is true by calculation. Now assume the statement is true for all words $w'$ such that $|w'| \leq n-1$ . Let $w'$ and w be words with $|w'| =n$ and $\mathfrak {D}(w')=w$ . Let $w^{\prime }_0$ be $w'$ with the last letter removed, and let $w_0$ be w with the last two letters removed. Then, by the inductive hypothesis,

$$ \begin{align*}F(w_0, z)-1={z-1\over z+1}(F(w^{\prime}_0, z^2)-1).\end{align*} $$

We divide the inductive step into two cases.

Case 1: $w^{\prime }_n=0$ . This implies $w=w_0\cdot 11$ , so

$$ \begin{align*} F(w, z)-1 &=2-z (2-z(F(w_0, z)))-1\\ &=2-z \bigg(2-z \bigg({z-1\over z+1}(F(w^{\prime}_0, z^2)-1)+1 \bigg)\bigg)-1\\ &={z-1\over z+1}(z^2F(w^{\prime}_0, z^2)-1)\\ &={z-1\over z+1}(F(w', z^2)-1). \end{align*} $$

Case 2: $w^{\prime }_n=1$ . This implies $w=w_0\cdot 10$ , so

$$ \begin{align*} F(w, z)-1 &=z(2-z(F(w_0, z)) )-1\\ &=z \bigg(2-z \bigg({z-1\over z+1}(F(w^{\prime}_0, z^2)-1)+1\bigg)\bigg)-1\\ &={z-1\over z+1} (2-z^2F(w^{\prime}_0, z^2)-1)\\ &={z-1\over z+1}(F(w', z^2)-1).\\[-3.4pc] \end{align*} $$

Proof. First, it is easy to see that a sequence is renormalizable if and only if all its odd index letters are $1$ , and a word is renormalizable if and only if it has even length and all its odd indexed letters are $1$ . Because any admissible word starts with $10$ , an admissible word w is renormalizable if and only if $w^\infty $ is admissible and renormalizable.

Now suppose $w^\infty $ is admissible and renormalizable. Suppose the second $0$ in $w^\infty $ is at the $k\textrm {th}$ location. It suffices to show that ${\textrm {Pre}}_{k-1}(w^\infty )$ has positive cumulative sign, which is equivalent to showing that k is even, because the $(k-1)$ -prefix of $w^\infty $ and $10\cdot 1^\infty $ are the same. This is an immediate consequence of the admissibility of $w^\infty $ .

Now we prove the other direction. The sequence $w^\infty $ being admissible implies that the first $0$ in $w^\infty $ is at the second location. If we can further prove that the distance between any two consecutive $0$ s is even, then all $0$ s are at even locations, and hence $w^\infty $ is admissible. Denote by $i_k$ the location of the $k\textrm {th} 0$ . Let $k_m$ be the smallest number such that $i_{k_m}-i_{k_{m-1}}$ is odd. Then by definition of $<_E$ ,

$$ \begin{align*}\sigma^{i_{k_{m-1}}-1}(w^\infty)>_E10\cdot 1^\infty.\\[-36pt]\end{align*} $$

Proof. For any $1 < \unicode{x3bb} < 2$ , let

$$ \begin{align*}{\Xi}^{\prime}_\unicode{x3bb} = \bigcap_{\unicode{x3bb}'> \unicode{x3bb}} Y_{\unicode{x3bb}'}.\end{align*} $$

We will first prove ${\Xi }_{\unicode{x3bb} } \subseteq {\Xi }_{\unicode{x3bb} }'$ . For any $\unicode{x3bb} '$ , define the set $Z_{\unicode{x3bb} '}$ to be the closure of the set of Galois conjugates of critically periodic growth rates that are at most $\unicode{x3bb} '$ , union with $S^1$ . By the persistence theorem, $\unicode{x3bb} _1 < \unicode{x3bb} _2$ implies $Z_{\unicode{x3bb} _1} \subseteq Z_{\unicode{x3bb} _2}$ . So if any point $x \in \bigcap _{\unicode{x3bb} '> \unicode{x3bb} } Z_{\unicode{x3bb} '}$ , then $x \in {\Xi }_{\unicode{x3bb} }$ since $\Upsilon _2^{cp}$ is closed; similarly, if $x \not \in \bigcap _{\unicode{x3bb} '> \unicode{x3bb} } Z_{\unicode{x3bb} '}$ , then $x \not \in {\Xi }_{\unicode{x3bb} }$ . Hence,

$$ \begin{align*}{\Xi}_{\unicode{x3bb}}\cap\overline{\mathbb{D}} = \bigcap_{\unicode{x3bb}'> \unicode{x3bb}} Z_{\unicode{x3bb}'}.\end{align*} $$

The conclusion will now follow from the statement that $Z_{\unicode{x3bb} '} \subseteq Y_{\unicode{x3bb} '}$ for all $\unicode{x3bb} '$ . If z is a Galois conjugate of a critically periodic growth rate $\unicode{x3bb} "$ that is at most $\unicode{x3bb} '$ , then z is a root of the Parry polynomial $P_w$ such that $w^{\infty }=\mathrm {{It}}_{\unicode{x3bb} "}$ , and $\mathrm {{It}}_{\unicode{x3bb} "} \leq _E \mathrm {{It}}_{\unicode{x3bb} '}$ by Corollary 2.4. Thus, $Z_{\unicode{x3bb} '} \subseteq Y_{\unicode{x3bb} '}$ for all $\unicode{x3bb} '$ .

We will now prove ${\Xi }^{\prime }_\unicode{x3bb} \subseteq {\Xi }_\unicode{x3bb} $ . To do this, it suffices to show

$$ \begin{align*}Y_{\unicode{x3bb}'} \subseteq \bigcap_{\unicode{x3bb}"> \unicode{x3bb}'} Z_{\unicode{x3bb}"}.\end{align*} $$

We first consider the case $\unicode{x3bb} '\geq \sqrt {2}$ . Suppose z is the root of some $P_w$ , where w is admissible and the leading root of $P_w$ is no larger than $\unicode{x3bb} '$ . ( $Y_{\unicode{x3bb} '}$ is the closure of all such z terms.) For any $\epsilon>0$ , Proposition 4.4 guarantees the existence of a dominant word $w_1$ such that for any $w_2$ , the leading root of $P_{w_1w_2}$ is in $[\unicode{x3bb} ', \unicode{x3bb} '+\epsilon )$ and $w_1>_E10\cdot 1^{|w_1|-2}$ . By monotonicity (Corollary 2.4), $w_1^{\infty }>_E w^{\infty }$ . Without loss of generality, we may choose $w_1$ so that its length, $|w_1|$ , is arbitrarily big (this is because as we let $\epsilon \to 0$ , we get arbitrarily many such dominant strings, and there are finitely many strings of at most any given length). Thus we may assume that $w_1$ and w satisfy the assumptions of Proposition 4.2 with the m of Proposition 4.2 being arbitrarily large, and in particular, m is $\geq $ the N of Proposition 4.3 using w for $w_2$ . Let $w_3$ be the word constructed by Proposition 4.2. Because $w_3$ is admissible, has positive cumulative sign, and $P_{w_3}(z)/(z-1)$ is irreducible, $w_3^{\infty } = \mathrm {{It}}_{\unicode{x3bb} _3}$ for some $\unicode{x3bb} _3$ by Proposition 2.3. We know $\unicode{x3bb} _3 \in [\unicode{x3bb} ', \unicode{x3bb} '+\epsilon ]$ because $w_3$ has the prefix $w_1$ . Also, any root of $P_{w_3}$ in $\mathbb {D}$ will be a Galois conjugate of $\unicode{x3bb} _3$ , and by construction, $P_{w_3}$ has a root close to z. The containment now follows from letting $\epsilon \rightarrow 0$ .

Now we deal with the case $1<\unicode{x3bb} '<\sqrt {2}$ . Let k be the unique natural number such that $(\unicode{x3bb} ')^{2^k}\in [\sqrt {2}, 2)$ . Remark 3.5 implies that w has a $k\textrm {th}$ renormalization $w_0$ , and $z^{2^k}$ is a root of $P_{w_0}$ . Using $w_0$ in place of w in the argument in the previous paragraph, we get a critically periodic growth rate $\unicode{x3bb} _4$ close to $(\unicode{x3bb} ')^{2^k}$ , such that one of its Galois conjugates $z_2$ is close to $z^{2^k}$ . The conclusion in Proposition 4.2 further implies that any $(2^k)\textrm {th}$ root of $z_2$ must be a Galois conjugate of the $(2^k)\textrm {th}$ root of $\unicode{x3bb} _4$ as well, which implies that there is a Galois conjugate of $\unicode{x3bb} _4^{2^{-k}}$ which is close to z, which finishes the proof of the proposition.

Proof. The growth rates $\unicode{x3bb} _3$ , as well as the growth rates $\unicode{x3bb} _3^{2^{-k}}$ , $k \in \mathbb {N}$ , constructed in the proof of Theorem 4.5 all satisfy conditions (1)–(3).

Proof. We will show that the set of all sequences that are not $\unicode{x3bb} $ -suitable is open. To do this, it suffices to show that given any sequence w which is not $\unicode{x3bb} $ -suitable, we can find a prefix of w such that every sequence that shares this prefix is not $\unicode{x3bb} $ -suitable. It is clear that conditions (1) and (2) are closed conditions. For condition (3), we choose the prefix to be one that contains the first $k+1$ consecutive $0$ s. Condition (4) is similar.

Proof. Observe that for any $n \in \mathbb {N}$ , if we pick some $k\in \mathbb {N}$ such that $n+k$ is some multiple of $|w|$ , then we have

(5.1) $$ \begin{align} {\textrm{Rev}} ( {\textrm{Pre}}_n({\textrm{Rev}}(w)^{\infty}))) = {\textrm{Pre}}_n(\sigma^{k}(w^{\infty})). \end{align} $$

Since w is admissible, ${\textrm {Pre}}_n(\sigma ^k(w^{\infty })) \leq _E {\textrm {Pre}}_n(w^{\infty })$ for all $k,n \in \mathbb {N}$ . By Corollary 2.4, for any $\unicode{x3bb} '> \unicode{x3bb} $ ,

(5.2) $$ \begin{align} \mathrm{{It}}_{\unicode{x3bb}} <_E \mathrm{{It}}_{\unicode{x3bb}'}. \end{align} $$

We thus have that for any $n \in \mathbb {N}$ ,

$$ \begin{align*} {\textrm{Rev}} ( {\textrm{Pre}}_n({\textrm{Rev}}(w)^{\infty})) = {\textrm{Pre}}_n(\sigma^{k}(w^{\infty})) \leq_E {\textrm{Pre}}_n(w^{\infty}) \leq_E {\textrm{Pre}}_n(\mathrm{{It}}_{\unicode{x3bb}}) \leq_E {\textrm{Pre}}_n(\mathrm{{It}}_{\unicode{x3bb}'}), \end{align*} $$

which is condition (1) of the definition of $\unicode{x3bb} $ -suitability.

Now suppose that for some $\unicode{x3bb} '> \unicode{x3bb} $ ,

$$ \begin{align*}{\textrm{Rev}} ( {\textrm{Pre}}_n({\textrm{Rev}}(w)^{\infty}) = {\textrm{Pre}}_n(\mathrm{{It}}_{\unicode{x3bb}'})\end{align*} $$

and ${\textrm {Pre}}_n(\mathrm {{It}}_{\unicode{x3bb} '})$ has a positive cumulative sign. Then from (5.1), we have

$$ \begin{align*}{\textrm{Pre}}_n(\sigma^{k}(w^{\infty})) = {\textrm{Pre}}_n(\mathrm{{It}}_{\unicode{x3bb}'}).\end{align*} $$

Admissibility of w and (5.2) together imply that

(5.3) $$ \begin{align} \sigma^{k}(w^{\infty})\leq_E w^{\infty}<_E\mathrm{{It}}_{\unicode{x3bb}'}. \end{align} $$

Because ${\textrm {Pre}}_n(\mathrm {{It}}_{\unicode{x3bb} '})$ is the common prefix of $\sigma ^{k}(w^{\infty })$ and $\mathrm {{It}}_{\unicode{x3bb} '}$ , (5.3) implies it must also be a prefix of $w^\infty $ . Removing this common n-prefix with positive cumulative sign from both sides of the inequality (by applying $\sigma ^n$ ) yields

$$ \begin{align*}w^\infty\leq_E\sigma^{n}(w^\infty).\end{align*} $$

However, admissibility also implies that $\sigma ^{n}(w^\infty )\leq w^\infty $ , so in fact

$$ \begin{align*}w^\infty = \sigma^{n}(w^\infty).\end{align*} $$

Therefore,

(5.4) $$ \begin{align} w^\infty=({\textrm{Pre}}_n(\mathrm{{It}}_{\unicode{x3bb}'}))^{\infty}. \end{align} $$

Let j be the index of the first place $w^{\infty }$ differs from $\mathrm {{It}}_{\unicode{x3bb} '}$ . Clearly, $j>n$ . Pick $m \in \mathbb {N}$ such that $mn<j\leq (m+1)n$ . Then, after removing the common prefix of length $mn$ and positive cumulative sign from both $w^{\infty }$ and $\mathrm {{It}}_{\unicode{x3bb} '}$ , we get from (5.4) and (5.2) that

$$ \begin{align*}\sigma^{mn}(w^{\infty}) = w^{\infty}<_E \mathrm{{It}}_{\unicode{x3bb}'},\end{align*} $$

and hence,

$$ \begin{align*}{\textrm{Pre}}_n(\mathrm{{It}}_{\unicode{x3bb}'})={\textrm{Pre}}_n(w^\infty)<_E{\textrm{Pre}}_n(\sigma^{mn}(\mathrm{{It}}_{\unicode{x3bb}'})),\end{align*} $$

which contradicts with the fact that $\mathrm {{It}}_{\unicode{x3bb} '}$ is admissible (by Theorem 2.2). Thus, condition (2) of the definition of $\unicode{x3bb} $ -suitability holds.

Now condition (3) of the definition of $\unicode{x3bb} $ -suitability follows from the assumption that $w^{\infty } \leq _E \mathrm {{It}}_{\unicode{x3bb} }$ .

For condition (4), suppose for some $\unicode{x3bb} '>\unicode{x3bb} $ , $\sqrt {2}\leq (\unicode{x3bb} ')^{2^n}<2$ . Then $\unicode{x3bb} ^{2^n}<2$ , so by Lemma 3.4, $w=\mathfrak {D}^n(w')$ for some $w'$ . Hence,

$$ \begin{align*}({\textrm{Rev}}(w))^\infty=(\mathfrak{D}^{\prime{n}}({\textrm{Rev}}(w'))^\infty.\end{align*} $$

Because $\mathfrak {D}$ preserves $\leq _E$ and sends itineraries to itineraries (Lemma 3.2), the number of consecutive $0$ s in $({\textrm {Rev}}(w'))^\infty $ , which is the number of consecutive $0$ s in $w^{\prime \infty }$ , cannot be more than the number of consecutive $0$ s in $\mathrm {{It}}_{\unicode{x3bb} ^{\prime 2^n}}$ .

Proof. Let $\alpha '={\textrm {Rev}}(\alpha )$ . It suffices to show that the admissibility criterion

$$ \begin{align*}\sigma^k((w_0\alpha')^\infty)\leq_E (w_0\alpha')^\infty\end{align*} $$

holds for all $1\leq k< |\alpha |+|w_0|$ .

Case 1: $k<|w_0|$ . This implies that the comparison between $\sigma ^k((w_0\alpha ')^\infty )$ and $(w_0\alpha ')^\infty $ is equivalent to the comparison of a proper suffix of $w_0$ concatenated with $1$ with a prefix of $w_0$ of the same length. Hence,

$$ \begin{align*}\sigma^k((w_0\alpha')^\infty)\leq_E (w_0\alpha')^\infty\end{align*} $$

because $w_0$ is dominant.

Case 2: $|w_0|\leq k < |\alpha |+|w_0|$ . Suppose the first place $\sigma ^k((w_0\alpha ')^\infty )$ and $(w_0\alpha ')^\infty $ differ is at the $j\textrm {th}$ position. It is evident that $1\leq j\leq |\alpha |+|w_0|$ . We divide this into two subcases.

Case 2A: $j\leq |w_0|+|\alpha |-k$ . The fact that $|w_0|>|\alpha |$ and $k\geq |w_0|$ implies that $j\leq |w_0|$ . Hence, the comparison between $\sigma ^k((w_0\alpha ')^\infty )$ and $(w_0\alpha ')^\infty $ is equivalent to the comparison of a proper suffix of $\alpha '$ with a prefix of $w_0$ . Hence, item (1) of Definition 1.5 gives us

$$ \begin{align*}\sigma^k((w_0\alpha')^\infty)\leq_E (w_0\alpha')^\infty.\end{align*} $$

Case 2B: $j>|w_0|+|\alpha |-k$ . The word $\beta :={\textrm {Suf}}_{|w_0|+|\alpha |-k}(\alpha ')$ , which is a common prefix of $\sigma ^k((w_0\alpha ')^\infty )$ and $(w_0\alpha ')^\infty $ , is identical to a prefix of $w_0$ , which is $\ge _E$ than a prefix of $\mathrm {{It}}_{\unicode{x3bb} '}$ for some $\unicode{x3bb} '>\unicode{x3bb} $ . Hence, due to item (2) of Definition 1.5, $\beta $ has a negative cumulative sign. Now, using the conclusion of Case 1, we have

$$ \begin{align*}\sigma_k^{|w_0|+|\alpha|-k}(\sigma^k((w_0\alpha')^\infty))=(w_0\alpha')^\infty>_E \sigma_k^{|w_0|+|\alpha|-k}((w_0\alpha')^\infty).\end{align*} $$

Hence,

$$ \begin{align*}\sigma^k((w_0\alpha')^\infty)\leq_E (w_0\alpha')^\infty\end{align*} $$

because $\beta $ has a negative cumulative sign.

Proof. By induction, we only need to prove it for $k=1$ . Assume $w=\mathfrak {D'}(w')$ is $\unicode{x3bb} $ -suitable, we will now show that $w'$ satisfies conditions (1)–(4) of Definition 1.5. By definition,

$$ \begin{align*}{\textrm{Rev}}(\mathfrak{D'}(v))=\mathfrak{D}({\textrm{Rev}}(v))\end{align*} $$

for any word v, so for any $\unicode{x3bb} '>\unicode{x3bb} $ ,

$$ \begin{align*}{\textrm{Rev}}({\textrm{Pre}}_{2n}(\mathfrak{D'}(w'))={\textrm{Rev}}(\mathfrak{D'}({\textrm{Pre}}_n(w')))\end{align*} $$

$$ \begin{align*}=\mathfrak{D}({\textrm{Rev}}({\textrm{Pre}}_n(w')))\leq {\textrm{Pre}}_{2n}(\mathrm{{It}}_{\unicode{x3bb}'})=\mathfrak{D}({\textrm{Pre}}_n(\mathrm{{It}}_{\unicode{x3bb}^{\prime2}})).\end{align*} $$

Hence condition (1) is true for $w'$ because of Lemmas 3.1 and 3.2. Condition (2) of Definition 1.5 can be verified similarly. It is easy to see that w satisfies condition (4) implies that $w'$ satisfies condition (4). Lastly, we will now show that w satisfies condition (4) will imply $w'$ satisfies condition (3): if $\unicode{x3bb} ^2\geq \sqrt {2}$ , this follows from the statement of condition (4). If $\unicode{x3bb} ^2<\sqrt {2}$ , condition (4) implies that $w'=\mathfrak {D'}(w")$ for some $w"$ , which implies that $w'$ can never have more than one consecutive 0, and hence it also satisfies condition (3).

Proof of Theorem 1.7

For the reader’s convenience, we reproduce here the statement of Theorem 1.7. For any $\unicode{x3bb} \in (1,2]$ ,

$$ \begin{align*}{\Xi}_\unicode{x3bb} \cap \overline{\mathbb{D}} = S^1\cup \{z\in\mathbb{D}: G(w, z)=1\text{ for some }\unicode{x3bb}\text{-suitable sequence }w \}.\end{align*} $$

By Remark 1.11, the result holds for $\unicode{x3bb} =2$ . So fix $\unicode{x3bb} \in (1,2)$ . For brevity, let

$$ \begin{align*}Z_\unicode{x3bb}:=\{z\in \mathbb{D}: G(w, z)=1\text{ for some }\unicode{x3bb}\text{-suitable sequence }w\}.\end{align*} $$

First, we show that $S^1\cup Z_\unicode{x3bb} $ is compact. For each sequence w, the function from $\mathbb {D}$ to $\mathbb {C}$ given by $z \mapsto G(w, \cdot )-1$ is a power series with bounded coefficients. Furthermore, the map from the set of sequences w (with the product topology) to the set of power series (with the product topology on coefficients) given by $w \mapsto G(w,\cdot )-1$ is continuous. Therefore, Lemma 6.2 implies that the map $\rho $ from the set of sequences with the product topology to $\mathcal {C}$ , the set of compact subsets of $\mathbb {C}$ with the Hausdorff topology, given by

$$ \begin{align*}\rho(w) = S^1 \cup \{z \in \mathbb{D} : G(w,z) = 1\},\end{align*} $$

is continuous. By Lemma 5.4, the set of all $\unicode{x3bb} $ -suitable sequences is closed (in the product topology on the set of sequences), and hence compact. Therefore, since $\rho $ is continuous,

$$ \begin{align*}\{\rho(w) : w \textrm{ is } \unicode{x3bb}-\textrm{suitable}\}\end{align*} $$

is a compact subset of $\mathcal {C}$ . Hence, Lemma 6.1 implies that

$$ \begin{align*}\bigcup_{w\text{ is }\unicode{x3bb}\text{-suitable}}\rho(w)\end{align*} $$

is compact. However, this set is precisely $S^1\cup Z_\unicode{x3bb} $ , so we have shown $S^1\cup Z_\unicode{x3bb} $ is compact for any $\unicode{x3bb} \in [1,2]$ .

Next, we show that

$$ \begin{align*}{\Xi}_\unicode{x3bb}\cap \overline{\mathbb{D}}\subseteq S^1\cup Z_\unicode{x3bb}.\end{align*} $$

Theorem 4.5 shows that

(6.1) $$ \begin{align} {\Xi}_\unicode{x3bb}\cap \overline{\mathbb{D}}=\bigcap_{\unicode{x3bb}'>\unicode{x3bb}}Y_{\unicode{x3bb}'}, \end{align} $$

where $Y_{\unicode{x3bb} '}$ is defined to be the closure of the set of roots in $\overline {\mathbb {D}}$ of all Parry polynomials $P_w$ such that w is admissible and $w^\infty \leq _E\mathrm {{It}}_{\unicode{x3bb} '}$ , union with $S^1$ . For each such w, let $w_r$ be the sequence

$$ \begin{align*}w_r :=({\textrm{Rev}}(w))^\infty.\end{align*} $$

So fix $\unicode{x3bb} '> \unicode{x3bb} $ and consider any admissible word w such that $w^{\infty } \leq _E \mathrm {{It}}_{\unicode{x3bb} '}$ . By Lemma 2.1,

$$ \begin{align*}P_w(z) = (1-z^{|w|})G(w_r,z).\end{align*} $$

By Lemma 5.6, $w_r$ is $\unicode{x3bb} '$ -suitable. Hence, all roots in $\overline {\mathbb {D}}$ of $P_w$ are in $S^1 \cup Z_{\unicode{x3bb} '}$ . Then, since $Z_{\unicode{x3bb} '}$ is closed, we have that

(6.2) $$ \begin{align} Y_{\unicode{x3bb}'}\subseteq S^1\cup Z_{\unicode{x3bb}'}. \end{align} $$

Now, combining (6.1) and (6.2) shows that for any point $z\in {\Xi }_\unicode{x3bb} \cap \mathbb {D}$ , for each $n\in \mathbb {N}$ , there exists a $(\unicode{x3bb} +{1/n})$ -suitable sequence $v_n$ such that $G(v_n, z)=1$ . Let $v_\infty $ be an accumulation point of the set $\{v_n : n \in \mathbb {N}\}$ . By Lemma 5.5, the sequence $v_\infty $ is $\unicode{x3bb} $ -suitable. The continuity of $w\mapsto G(w, \cdot )$ implies that $G(v_\infty , z)=1$ . Hence, ${\Xi }_{\unicode{x3bb} }\cap \overline {\mathbb {D}}\subseteq S^1\cup Z_{\unicode{x3bb} }$ .

Lastly, we show that $S^1\cup Z_\unicode{x3bb} \subseteq {\Xi }_\unicode{x3bb} \cap \overline {\mathbb {D}}.$ We know from [Reference Bray, Davis, Lindsey and WuBDLW19] that $S^1 \times [1,2] \subset \Upsilon _2^{cp}$ . Thus, $S^1\subset {\Xi }_\unicode{x3bb} $ , so it suffices to show that $Z_\unicode{x3bb} \subset {\Xi }_{\unicode{x3bb} }$ . Fix a point $z\in Z_\unicode{x3bb} $ and let w be a $\unicode{x3bb} $ -suitable sequence such that $G(w, z)=1$ . By condition (4) of Definition 1.5, there exists a sequence $w'$ such that $w=\mathfrak {D'}^k(w')$ , and by Lemma 5.8, $w'$ is $\unicode{x3bb} ^{2^k}$ -suitable and $\unicode{x3bb} ^{2^k}\geq \sqrt {2}$ . In particular, if $\unicode{x3bb} \geq \sqrt {2}$ , we can let $k=0$ and $w'=w$ . As a consequence, there are infinitely many prefixes of $w'$ that end with $1$ and have a positive cumulative sign.

For any $m \in \mathbb {N}$ such that ${\textrm {Pre}}_m(w')$ has a positive cumulative sign and any word $w"$ with a positive cumulative sign, it follows immediately from the definitions of a Parry polynomial and of G that the first m terms of the power series $G(w', z)-1$ and $P_{w" \cdot {\textrm {Rev}}({\textrm {Pre}}_m(w'))}(z)$ agree. Therefore, for any fixed $\epsilon _1>0$ , by Lemma 6.3, there exists $N \in \mathbb {N}$ such that ${\textrm {Pre}}_N(w')$ ends with $1$ and has a positive cumulative sign, and for any word $w"$ with a positive cumulative sign, there exists a point $z' \in B_{\epsilon _1}(z^{2^k})$ such that

(6.3) $$ \begin{align} P_{w"\cdot {\textrm{Rev}}({\textrm{Pre}}_N(w'))}(z')=0. \end{align} $$

For any fixed $\unicode{x3bb} '$ satisfying $2> \unicode{x3bb} ' > \unicode{x3bb} ^{2^k}$ , pick a critically periodic growth rate $\unicode{x3bb} " \in (\unicode{x3bb} ^{2^k},\unicode{x3bb} ')$ and word $w_0$ with a positive cumulative sign such that $\mathrm {{It}}_{\unicode{x3bb} "}=w_0^{\infty }$ . Since $\unicode{x3bb} " < \unicode{x3bb} '$ , for sufficiently large n,

$$ \begin{align*}w_0^n <_E {\textrm{Pre}}_{n|w_0|}(\mathrm{{It}}_{\unicode{x3bb}'}).\end{align*} $$

Hence, by Proposition 2.5, there exists $n \in \mathbb {N}$ and a word $w_1'$ such that the word

$$ \begin{align*}w_1:= w_0^nw^{\prime}_1\end{align*} $$

is dominant, $|w_1|> |w'|$ , and

(6.4) $$ \begin{align} w_1<_E{\textrm{Pre}}_{|w_1|}(\mathrm{{It}}_{\unicode{x3bb}'}). \end{align} $$

By Lemma 5.7,

$$ \begin{align*}w_1 \cdot {\textrm{Rev}}({\textrm{Pre}}_N(w))'\end{align*} $$

is admissible. By (6.3),

$$ \begin{align*}P_{w_1\cdot {\textrm{Rev}}({\textrm{Pre}}_N(w'))}\end{align*} $$

has a root within distance $\epsilon _1$ of $z^{2^k}$ . By (6.4),

$$ \begin{align*}(w_1\cdot {\textrm{Rev}}({\textrm{Pre}}_N(w')))^\infty<_E \mathrm{{It}}_{\unicode{x3bb}'}.\end{align*} $$

Hence, the $k\textrm {th}$ doubling of $w_1\cdot {\textrm {Rev}}({\textrm {Pre}}_N(w'))$ , denoted as $w_d$ , satisfies

$$ \begin{align*}w_d^\infty<_E\mathrm{{It}}_{(\unicode{x3bb}')^{1/2^k}}\end{align*} $$

and $P_{w_d}$ has leading root in $[\unicode{x3bb} , (\unicode{x3bb} ')^{1/2^k}]$ and a root in $B_{\epsilon ^{\prime }_1}(z)$ , where $\epsilon ^{\prime }_1$ is the diameter of the preimage of $B_{\epsilon '}(z^{2^k})$ under the map $z\mapsto z^{2^k}$ .

Now, since $\epsilon _1>0$ and $\unicode{x3bb} '>\unicode{x3bb} $ were arbitrary, and since ${\Upsilon _2^{cp}}$ is closed, we obtain that $(z, \unicode{x3bb} )\in {\Upsilon _2^{cp}}$ , and hence $z\in {\Xi }_\unicode{x3bb} $ .

Proof of Theorem 1.8

For any $\unicode{x3bb} \in (1, 2)$ , denote by $f_\unicode{x3bb} $ the $\unicode{x3bb} $ -tent map, and define the set

Next, define

We will show that

$$ \begin{align*}R = {\Upsilon_2^{cp}} \cap (\{z : |z| \geq 1\} \times (1,2)).\end{align*} $$

Our argument will almost immediately follow from results in [Reference TiozzoTio18], though these results must be translated from the context of the Mandelbrot set to the context of tent maps. The Milnor–Thurston kneading theory [Reference Milnor and ThurstonMT88] yields an entropy-preserving semiconjugacy from the set of real quadratic polynomials $f_c$ , $c \in [-2,1/4]$ , to the set of tent maps. If the quadratic $f_c$ is sent by this semiconjugacy to a tent map $f_{\unicode{x3bb} }$ , then $(1-z^{-1})K_c(z^{-1}) = H(\mathrm {It}_{\unicode{x3bb} },z)$ , and hence the roots (in $\mathbb {D}$ ) of the kneading determinant $K_c$ coincide with the reciprocals of the roots (in $\mathbb {C} \setminus \overline {\mathbb {D}}$ ) of $H(\mathrm {It}_{\unicode{x3bb} },z)$ . By [Reference JungJun14, Theorem 4.7], on the real slice of the Mandelbrot set, core entropy of $f_c:z \mapsto z^2+c$ is strictly increasing except on ‘small Mandelbrot sets,’ where the core entropy is constant. Consequently, the entropy-preserving semiconjugacy from real quadratics to tent maps is injective except for collapsing each small Mandelbrot set to a single tent map. Furthermore, the semiconjugacy induces a surjection from the set of critically periodic quadratics $f_c$ to the set of critically periodic tent maps, and each small Mandelbrot set contains a critically periodic quadratic.

We will first show that R is a (relatively) closed subset of $\mathbb {C}\times (1, 2)$ . By Proposition 3.3 of [Reference TiozzoTio18], the set that is the union of $S^1$ and the roots of $K_c$ varies continuously in the Hausdorff topology with the parameter c. Consequently, when $f_\unicode{x3bb} $ is not critically periodic, the function $s\mapsto R_s\cup S_1$ is continuous at $s=\unicode{x3bb} $ . As stated in the proof of Theorem 7.1 of [Reference TiozzoTio18], if $(c_1,c_0)$ is the real section of a small Mandelbrot set of period p, then $K_{c_0}(t) = P_{c_0}(t) / (1-t^p)$ and $K_{c_1}(t) = P_{c_0}(t)(1-2t^p) / (1-t^p)$ . Thus, in the Hausdorff topology,

$$ \begin{align*}\{z \in \mathbb{D} : K_{c_0}(z) = 0 \textrm{ or } z^p = \tfrac{1}{2} \} \cup S^1= \lim_{c_* \nearrow c_1} \{z \in \mathbb{D} : K_{c_*}(z) = 0\} \cup S^1,\end{align*} $$

and so denoting by $\unicode{x3bb} $ the growth rate of $f_{c_0}$ , we have

$$ \begin{align*}\{z \in \mathbb{C} \setminus \overline{\mathbb{D}} : H(\mathrm{It}_{\unicode{x3bb}},z) = 0 \textrm{ or } z^p = 2 \} \cup S^1= \lim_{\unicode{x3bb}_* \searrow \unicode{x3bb}} \{z \in \mathbb{C} \setminus \overline{\mathbb{D}} : H(\mathrm{It}_{\unicode{x3bb}_*},z) = 0\} \cup S^1.\end{align*} $$

Therefore, for any $\unicode{x3bb} $ for which the tent map $f_\unicode{x3bb} $ is critically periodic,

$$ \begin{align*}R_\unicode{x3bb}\cup S^1=\lim_{s\rightarrow \unicode{x3bb}_-}(R_s\cup S^1)\cup \lim_{s\rightarrow \unicode{x3bb}_+}(R_s\cup S^1).\end{align*} $$

Hence, R is a closed subset of $\mathbb {C}\times (1, 2)$ .

When $f_\unicode{x3bb} $ is critically periodic, $H(\mathrm {{It}}_\unicode{x3bb} , z)$ can be written as a rational function with integer coefficients; consequently, $R_{\unicode{x3bb} }$ contains all the Galois conjugates of $\unicode{x3bb} $ in $\mathbb {C} \setminus \mathbb {D}$ . Therefore, for every $\unicode{x3bb} $ such that the tent map $f_{\unicode{x3bb} }$ is critically periodic,

$$ \begin{align*}({\Xi}_\unicode{x3bb}\backslash\mathbb{D})\cup S^1\subseteq R_\unicode{x3bb}\cup S^1.\end{align*} $$

Since ${\Upsilon _2^{cp}} \cap \{z \mid |z| \geq 1\} \times (1,2)$ is formed by taking the closure of the union of all sets $({\Xi }_\unicode{x3bb} \backslash \mathbb {D})\cup S^1$ for $\unicode{x3bb} $ critically periodic, and since R is already closed, this implies

$$ \begin{align*}{\Upsilon_2^{cp}} \cap (\{z : |z| \geq 1\} \times (1,2)) \subseteq R.\end{align*} $$

To show containment in the other direction, Lemma 7.3 in [Reference TiozzoTio18] implies that when $f_\unicode{x3bb} $ is not critically periodic, $R_\unicode{x3bb} \times \{\unicode{x3bb} \}\subset {\Upsilon _2^{cp}}$ . However,

$$ \begin{align*}\bigcup_{f_\unicode{x3bb}\text{ is not critically periodic}}(R_\unicode{x3bb}\cup S^1)\times \{\unicode{x3bb}\}\end{align*} $$

is dense in R, and hence $R\subset {\Upsilon _2^{cp}}$ .

Proof. The set $\Omega _2^{cp} \cap \mathbb {D}$ is invariant under reflection across the real axis because Galois conjugates come in complex conjugate pairs. Tiozzo [Reference TiozzoTio18] showed that $\Omega _2^{cp} \cap \mathbb {D}\backslash S^1$ is the set of all the roots in $\mathbb {D}$ of all power series with all coefficients in $\{\pm 1\}$ . So if $z \in \mathbb {D}$ is a root of a power series S with coefficients in $\{\pm 1\}$ , then $-z$ is a root of the power series formed from S by flipping the sign of the coefficients on all terms of odd degree. Therefore, the complex conjugate, $\overline {-z}$ , is in $\Omega _2^{cp}$ .

Proof of Theorem 1.14

We only need to show that there is some $z\in {\Xi }_{1.82}\cap \mathbb {D}$ such that $-z\not \in {\Xi }_{1.82}\cap \mathbb {D}$ . Consider the tent map with growth rate being the leading root of

$$ \begin{align*}-1 + z^2 - z^4 + z^6 - 2 z^7 + 3 z^8 - 4 z^9 + 3 z^{10} - 2 z^{11} + z^{12} - 2 z^{13} + z^{14},\end{align*} $$

which is approximately $1.8149185987640513$ and is smaller than $1.82$ , and hence any Galois conjugate of this leading root must be in ${\Xi }_{1.82}$ . Let z be the Galois conjugate near the point $-0.5840341196392905+0.4820600149798202 i$ . Applying Algorithm 2 to $-z$ for $N=20$ shows that $-z\not \in {\Xi }_{1.82}$ .