Accumulation set of critical points of the multipliers in the quadratic family

Fix $c\in\mathbb{C}\setminus\partial\mathbb{M}$ and a neighborhood $U$ as in the statement of the lemma. Since $U\cap\partial\mathbb{M}=\varnothing$, all repelling periodic orbits of $f_{c_{0}}$ remain to be repelling after analytic continuation in $c\in U$. This implies that all maps from the family

are defined in the neighborhood $U$ and are analytic in it. (We recall that the maps $g_{\mathcal{O}}$ were defined in (2) and are appropriate branches of the roots of the multipliers.) Furthermore, since all functions from $\mathcal{G}_{c}$ are locally uniformly bounded, the family $\mathcal{G}_{c}$ is normal in $U$. Together with Proposition 2.2, this implies normality of the family $\{\nu_{\mathcal{O}}\mid\mathcal{O}\in\Omega_{c}\}$.

If $c\in\mathbb{C}\setminus\mathbb{M}$, then without loss of generality we may assume that the domain $U$ is simply connected and unbounded. Since for all $\tilde{c}\in\mathbb{C}$ sufficiently close to $\infty$, the Julia set $J_{\tilde{c}}$ is contained in the annulus centered at zero with inner and outer radii being equal to $\sqrt{|\tilde{c}|}\pm 1$, it follows that for every $\tilde{c}\in U$ sufficiently close to $\infty$ and for any $\mathcal{O}\in\Omega_{c}$, we have

which implies that none of the limiting maps of the family $\mathcal{G}_{c}$ is a constant map. Then it follows that the identical zero is not a limiting map of the normal family $\{\nu_{\mathcal{O}}\mid\mathcal{O}\in\Omega_{c}\}$. ∎

First, we observe that property i of Theorem B is an immediate corollary from the Averaging Lemma (Lemma 2.1). Indeed, if $c\neq-2$, then convexity of $\mathcal{Y}_{c}$ is obvious from Lemma 2.1. On the other hand, if $c=-2$, then according to the same lemma, the set $\mathcal{Y}_{-2}$ is the union of a convex set and a single point $\nu_{\{2\}}(-2)$, corresponding to the periodic orbit $\mathcal{O}=\{2\}$. A direct computation shows that

hence, $\nu_{\{2\}}(-2)=-1/6$.

We proceed with the proof of part ii as follows: for $c\in\mathbb{C}\setminus\mathbb{M}$, let $U$ be a neighborhood of $c$ that satisfies the conditions of Lemma 3.1. First, we observe that according to Lemma 3.1, the family $\{\nu_{\mathcal{O}}\mid\mathcal{O}\in\Omega_{c}\}$, defined on $U$, is locally uniformly bounded, hence, the set $\mathcal{Y}_{c}$ is bounded.

Necessary condition for $c\in\mathcal{X}$: If $c\in\mathcal{X}$, then there exists a sequence of points $\{c_{k}\}_{k=1}^{\infty}$ and a sequence of periodic orbits $\{\mathcal{O}_{k}\}_{k=1}^{\infty}\subset\Omega_{c}$, such that

According to Lemma 3.1, after extracting a subsequence, we may assume that the sequence of maps $\nu_{\mathcal{O}_{k}}$ converges to some holomorphic map $\nu\colon U\to\mathbb{C}$ uniformly on compact subsets of $U$. Since for any $k\in\mathbb{N}$, we have $\nu_{\mathcal{O}_{k}}(c_{k})=0$, it follows by continuity that $\nu(c)=0$. Finally, convergence of the maps $\nu_{\mathcal{O}_{k}}$ to $\nu$ implies that

hence, $0\in\mathcal{Y}_{c}$.

Sufficient condition for $c\in\mathcal{X}$: On the other hand, if $0\in\mathcal{Y}_{c}$, then either there exists a periodic orbit $\mathcal{O}\in\Omega_{c}$, such that $\nu_{\mathcal{O}}(c)=0$ or there exists a sequence of periodic orbits $\{\mathcal{O}_{k}\}_{k=1}^{\infty}\subset\Omega_{c}$, such that

In the first case, $\rho_{\mathcal{O}}^{\prime}(c)=0$, so $c\in\mathcal{X}$ according to Theorem 1.2.

In the second case, according to Lemma 3.1, after extracting a subsequence, we may assume that the sequence of maps $\nu_{\mathcal{O}_{k}}$ converges to some holomorphic map $\nu\colon U\to\mathbb{C}$ uniformly on compact subsets of $U$. By continuity, we have $\nu(c)=0$, and, according to Lemma 3.1, $\nu\not\equiv 0$. Then it follows from Rouché’s Theorem that for any sufficiently large $k\in\mathbb{N}$, there exists $c_{k}\in U$, such that $\nu_{\mathcal{O}_{k}}(c_{k})=0$ and $\lim_{k\to\infty}c_{k}=c$. The latter implies that $c\in\mathcal{X}$, and completes the proof of Theorem B. ∎

Since for every $j=1,\ldots,k$, the multipliers $\rho_{\mathcal{O}_{j}}$ are algebraic (multiple-valued) maps, it follows from (1) that the map $\nu$ has a single-valued meromorphic extension to any simply-connected domain $U\subset\mathbb{C}$ that avoids finitely many branching points of the maps $\rho_{\mathcal{O}_{j}}$. Note that none of the branching points lie on the real ray $(-\infty,-3)$, since $(-\infty,-3)\cap\mathbb{M}=\varnothing$. Furthermore, since for any parameter $\tilde{c}\in(-\infty,-3)$, the corresponding Julia set $J_{\tilde{c}}$ lies on the real line, it follows that all maps $\rho_{\mathcal{O}_{j}}$ take real values when restricted to the ray $(-\infty,-3)$. Choose the domain $U$ so that it is unbounded and $(-\infty,-3)\subset U$. Then for any $j=1,\ldots,k$, we have the same asymptotic relation

as $\tilde{c}\to-\infty$ within the domain $U$. A direct computation yields that $\nu_{\mathcal{O}_{j}}(\tilde{c})\sim 1/(2\tilde{c})$, hence,

as $\tilde{c}\to-\infty$ within the domain $U$. Since $\sum_{j=1}^{k}\alpha_{j}\neq 0$, the latter implies that $\nu$ is not a constant map. ∎

Since the convex hull of the points $\nu_{\mathcal{O}_{1}}(c),\ldots,\nu_{\mathcal{O}_{k}}(c)$ contains zero, it follows that there exist real non-negative constants $\alpha_{1},\ldots,\alpha_{k}$, such that $\sum_{j=1}^{k}\alpha_{j}=1$ and the analytic map

defined in some neighborhood of the point $c$, satisfies $\nu(c)=0$.

Since $c\not\in\partial\mathbb{M}$, this means that $c\neq-2$, so it follows from the Averaging Lemma (Lemma 2.1) that there exists a sequence of periodic orbits $\{\mathcal{O}_{m}^{\prime}\}_{m=1}^{\infty}\subset\Omega_{c}$ and a neighborhood $U\subset\mathbb{C}$ of the point $c$, such that all maps $\nu_{\mathcal{O}_{m}^{\prime}}$ are defined and analytic in $U$ and

According to Proposition 3.4, the map $\nu$ is not the identical zero map. Now, since $\nu(c)=0$, it follows from Rouché’s Theorem that for any sufficiently large $m\in\mathbb{N}$, the map $\nu_{\mathcal{O}_{m}^{\prime}}$ has a zero at some point $c_{m}\in U$, and the points $c_{m}$ can be chosen so that $\lim_{m\to\infty}c_{m}=c$. The latter implies that $c\in\mathcal{X}$. ∎

For a fixed parameter $c_{0}\in\mathbb{C}\setminus\mathbb{M}$, the Julia set $J_{c_{0}}$ of the map $f_{c_{0}}$ is a Cantor set, and all periodic orbits of $f_{c_{0}}$ are repelling. For any periodic orbit $\mathcal{O}$ of $f_{c_{0}}$, the locally defined map $g_{\mathcal{O}}$ can be extended by analytic continuation to an analytic map of a double cover of the complement of the Mandelbrot set $\mathbb{M}$ (see [Firsova_Gor_equi] for details). This means that if

is a fixed conformal diffeomorphism of $\mathbb{C}\setminus\mathbb{M}$ onto $\mathbb{C}\setminus\overline{\mathbb{D}}$ and $\lambda_{0}\in\mathbb{C}\setminus\overline{\mathbb{D}}$ is a fixed point, such that $\phi_{\mathbb{M}}^{-1}(\lambda_{0}^{2})=c_{0}$, then the map

defined for all $\lambda$ in a neighborhood of $\lambda_{0}$, extends to a global holomorphic map

Now assume that the statement of Lemma 4.1 does not hold. Then there exists a sequence of parameters $\{\lambda_{n}\}_{n\in\mathbb{N}}$ and a corresponding sequence of periodic orbits $\{\mathcal{O}_{n}\}_{n\in\mathbb{N}}$, such that

Since the family of maps $\{\gamma_{\mathcal{O}}\}$ is locally uniformly bounded, hence, normal (c.f. Proposition 5.8 from [Firsova_Gor_equi]), it follows that after extracting a subsequence, we may assume that the sequence of maps $\gamma_{\mathcal{O}_{n}}$ converges to a holomorphic map $\gamma\colon\mathbb{C}\setminus\overline{\mathbb{D}}\to\mathbb{C}\setminus\overline{\mathbb{D}}$ uniformly on compact subsets. Since for any $\tilde{c}\in\mathbb{C}$ sufficiently close to $\infty$ and any $\mathcal{O}\in\Omega_{c_{0}}$, inequality (7) holds, we conclude that $\gamma$, as well as each $\gamma_{\mathcal{O}_{n}}$, are non-constant maps that have a simple pole at infinity. On the other hand, (8) implies that $\gamma$ has at least a double pole at infinity, which provides a contradiction. ∎

When $|\mathcal{O}|=1$, i.e, $\mathcal{O}$ is a fixed point $z$, solving the equation $f_{c}(z)=z$ yields

Then after a direct computation we get

When $|\mathcal{O}|=2$, there is only one periodic orbit of period $2$. Its multiplier is the free term of the polynomial

Now, a direct computation yields the formula for $\nu(c)$ in part ii of the proposition.

Finally, in the case $|\mathcal{O}|=3$, there are two periodic orbits of period $3$ and according to [Stephenson_91], the multiplier $\rho=\rho(c)$ of each of these orbits satisfies the equation

After solving this equation for $\rho$, we obtain

Then a direct computation yields the formula for $\nu(c)$ in part iii of the proposition. ∎

We consider the maps $\nu_{\mathcal{O}}$ in a neighborhood of the point $c=-3/4$ for periodic orbits $\mathcal{O}$ of periods $1$, $2$ and $3$. The parameter $c=-3/4$ is the point at which the hyperbolic component of period $2$ touches the main cardioid of the Mandelbrot set. In particular, all considered functions are defined and analytic in a neighborhood $U$ of that point.

For each $c\in U$, let $H_{c}$ denote the convex hull of the finite set $\{\nu_{\mathcal{O}}(c)\mid|\mathcal{O}|=1,2,3\}$. It follows from Proposition 4.3 that $\nu_{\mathcal{O}}(-3/4)$ is equal to

• •

  $-1$ or $-1/3$, when $|\mathcal{O}|=1$,

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  $2$, when $|\mathcal{O}|=2$,

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  $-\cfrac{10}{183}\pm\cfrac{49}{183}i$, when $|\mathcal{O}|=3$,