On deformation space analogies between Kleinian reflection groups and antiholomorphic rational maps

Abstract

In a previous paper (Lodge et al. in Forum Math Sigma 10:e3, 2022), we constructed an explicit dynamical correspondence between certain Kleinian reflection groups and certain anti-holomorphic rational maps on the Riemann sphere. In this paper, we show that their deformation spaces share many striking similarities. We establish an analogue of Thurston’s compactness theorem for critically fixed anti-rational maps. We also characterize how deformation spaces interact with each other and study the monodromy representations of the union of all deformation spaces.

Appendices

Appendix A. Laminations, Automorphisms and Accesses

In this appendix, we show how the Julia sets of critically fixed anti-rational maps are related via laminations (Proposition A.4).

Laminations appear naturally as dual objects of enrichments of Tischler graphs. We study the action of the automorphism groups of critically fixed anti-rational maps on the set of laminations. This allows us to count the number of inequivalent semiconjugacies between Julia sets (Proposition A.8), and the number of accesses from one pared deformation space to another (Proposition A.10).

We remark that degenerations of kissing reflection groups are achieved by pinching suitable multi-curves on the conformal boundaries. By lifting, the multi-curves give a lamination on each component of the domain of discontinuity, and the limit set of the limiting kissing reflection group is obtained by simultaneously pinching these laminations on the domain of discontinuity. We shall see that a similar phenomenon also holds in the anti-rational map setting.

1.1 Lamination.

A lamination ${\mathcal {L}}$ is a family of disjoint hyperbolic geodesics in ${\mathbb {D}}$ together with the two endpoints in ${\mathbb {S}}^1\cong {\mathbb {R}}/{\mathbb {Z}}$, whose union is closed. The hyperbolic geodesics are called the leaves of the lamination.

A lamination ${\mathcal {L}}$ generates an equivalence relation $\sim _{\mathcal {L}}$ on ${\mathbb {S}}^1$ given by $a\sim _{{\mathcal {L}}} b$ if $a,b \in {\mathbb {S}}^1$ are connected by a finite chain of leaves in ${\mathcal {L}}$.

Let $d\ge 2$, and $m_{- d}: {\mathbb {S}}^1\longrightarrow {\mathbb {S}}^1$ be the map $m_{- d}(t) = - d\cdot t$. A lamination ${\mathcal {L}}$ is said to be $m_{-d}$-invariant if it satisfies the following two conditions.

• If there is a leaf with endpoints x and y, then either $m_{- d}(x) = m_{- d}(y)$ or there is a leaf with end-points $m_{- d}(x)$ and $m_{- d}(y)$;

• If there is a leaf with endpoints x and y, then there is a set of d disjoint leaves with one endpoint in $m_{- d}^{-1}(x)$ and the other endpoint in $m_{- d}^{-1}(y)$.

Note that there are exactly $d+1$ fixed points $0, \frac{1}{d+1}, \ldots , \frac{d}{d+1} \in {\mathbb {S}}^1$ of $m_{-d}$. These $d+1$ fixed points partition the circle into $d+1$ intervals, which form a Markov partition for $m_{-d}$ with associated transition matrix

$$\begin{aligned} M = \begin{bmatrix} 0 &{}\quad 1 &{}\quad 1 &{}\quad \dots &{}\quad 1 \\ 1 &{}\quad 0 &{}\quad 1 &{}\quad \dots &{}\quad 1 \\ \vdots &{}\quad \vdots &{}\quad \vdots &{}\quad \ddots &{}\quad \vdots \\ 1 &{}\quad 1 &{}\quad 1 &{}\quad \dots &{}\quad 0 \end{bmatrix}. \end{aligned}$$

Thus the 2-cycles of $m_{-d}$ are in bijective correspondence with pairs of non-adjacent Markov pieces.

Given a 2-cycle ${\mathcal {C}} = \{x,y\}$, we denote the closure (in ${\mathbb {D}}\cup {\mathbb {S}}^1$) of the hyperbolic geodesic of ${\mathbb {D}}$ connecting x, y by $l_{{\mathcal {C}}}$. We call a collection of 2-cycles ${\mathcal {C}}_1,\ldots , {\mathcal {C}}_k$ simple if the corresponding (compactified) geodesics $l_{{\mathcal {C}}_1},\ldots , l_{{\mathcal {C}}_k}$ are pairwise disjoint.

The following easy lemma can be proved by induction on the backward orbits of the endpoints of the geodesics.

Lemma A.1

Let ${\mathcal {C}}_1,\ldots , {\mathcal {C}}_k$ be a simple collection of 2-cycles of $m_{-d}$. Then there exists a unique $m_{-d}$-invariant lamination ${\mathcal {L}}$ so that

• every leaf l of ${\mathcal {L}}$ is eventually mapped to $l_{{\mathcal {C}}_i}$ for some i; and

• no leaves other than $l_{{\mathcal {C}}_i}$ separate the fixed points of $m_{-d}$.

We call the lamination ${\mathcal {L}}$ in Lemma A.1 the lamination generated by the 2-cycles ${\mathcal {C}}_1,\ldots , {\mathcal {C}}_k$.

1.2 Lamination as dual of $(d+1)$-ended ribbon trees.

Let ${\mathcal {L}}$ be a lamination generated by the 2-cycles ${\mathcal {C}}_1,\ldots , {\mathcal {C}}_k$. The leaves $l_{{\mathcal {C}}_1}, \ldots , l_{{\mathcal {C}}_k}$ cut the closed disk ${\overline{{\mathbb {D}}}}$ into finitely many regions. We can construct the dual tree ${\mathcal {T}}$ by introducing a vertex for each complementary component of

$$\begin{aligned} {\overline{{\mathbb {D}}}} \setminus \bigcup _{i=1}^k l_{{\mathcal {C}}_i}, \end{aligned}$$

connecting two vertices if the corresponding regions share a common leaf as boundary, and attaching a ray to a vertex for each fixed point of $m_{-d}$ on the boundary of the corresponding region (see Figure 7).

Fig. 7

The leaves of the laminations (generated by 2-cycles of $m_{-5}$) are drawn in blue, the fixed points (of $m_{-5}$) are marked in red, and the corresponding dual trees are drawn in green (color figure online).

The following lemma can be verified directly from the definitions.

Lemma A.2

Let ${\mathcal {L}}$ be a lamination generated by 2-cycles. The dual tree ${\mathcal {T}}$ of ${\mathcal {L}}$ is a $(d+1)$-ended ribbon tree.

Conversely, every $(d+1)$-ended ribbon tree arises as the dual tree for some lamination generated by 2-cycles.

Remark A.3

We regard the dual tree ${\mathcal {T}}$ of ${\mathcal {L}}$ as a marked $(d+1)$-ended ribbon tree by declaring the fixed point $0\in {\mathbb {S}}^1$ of $m_{-d}$ as the marked endpoint of ${\mathcal {T}}$.

1.3 Quotient of Julia set by lamination.

Invariant laminations naturally give quotient maps of Julia sets. Such a construction works for general (anti-)rational maps. For simplicity, we will restrict ourselves to the current setting.

Let ${\mathcal {R}}_\Gamma$ be a critically fixed anti-rational map with a boundary marking q (see Section 4.1). Let $U_1,\ldots , U_m$ be an enumeration of the critical Fatou components. Then the induced dynamics on the ideal boundary ${\mathbb {S}}^1 = I(U_i)$ is conjugate to $m_{-d_i}$, where $d_i$ is the degree of the map $f: U_i \longrightarrow U_i$.

Let ${\mathcal {L}}_i$ be an $m_{-d}$-invariant lamination generated by a simple collection of 2-cycles ${\mathcal {C}}_{i,1},\ldots , {\mathcal {C}}_{i, k_i}$ of $m_{-d_i}$. Such a collection of laminations ${\mathcal {L}}_1,\ldots , {\mathcal {L}}_m$ defines an equivalence relation $\sim$ on the Julia set ${\mathcal {J}}({\mathcal {R}}_\Gamma )$ as follows.

Let $x, y\in \partial U_i$. We define $x\sim y$ if there exists ${\tilde{x}} \in \phi _i^{-1}(x)$ and ${\tilde{y}} \in \phi _i^{-1}(y)$ with ${\tilde{x}} \sim _{{\mathcal {L}}_i} {\tilde{y}}$, where $\phi _i: {\mathbb {S}}^1 \longrightarrow \partial U_i$ is the unique semiconjugacy compatible with the boundary marking q (more precisely, $\phi _i$ maps $0\in {\mathbb {R}}/{\mathbb {Z}}$ to the marked fixed point on the ideal boundary of $U_i$).

Next, if U is a strictly pre-fixed Fatou component with pre-period l, then $f^{\circ l}: \partial U \longrightarrow \partial U_i$ is a homeomorphism for some i. We use this identification to pull back the equivalence relation to $\partial U$.

Since the equivalence relation induced by ${\mathcal {L}}_i$ in each Fatou component is closed, and the diameters of the Fatou components of ${\mathcal {R}}_\Gamma$ shrink to 0 as the pre-period $l \rightarrow \infty$, it is easy to check that the equivalence relation $\sim$ is closed. We say that such an equivalence relation $\sim$ is generated by pinching 2-cycles.

Let $\Gamma ' \ge \Gamma$. Let $i: \Gamma \longrightarrow \Gamma '$ be an embedding. By Lemma 4.8, the embedding i makes the dual graph ${\mathscr {T}}'$ of $\Gamma '$ an enrichment of ${\mathscr {T}}$. Thus, on each invariant Fatou component of ${\mathcal {R}}_\Gamma$, ${\mathscr {T}}'$ gives a lamination generated by 2-cycles. Let $\sim _{i}$ denote the equivalence relation generated by these laminations on invariant Fatou components of ${\mathcal {R}}_\Gamma$.

We remark that there may be many different embeddings $i: \Gamma \longrightarrow \Gamma '$, giving rise to different equivalence relations. This phenomenon is related to the automorphism group of $\Gamma$ and $\Gamma '$ and is discussed in Appendices A.2 and A.3.

The following proposition, which can be verified directly, relates the topology of the Julia sets.

Proposition A.4

Let $\Gamma ' \ge \Gamma$ and let $i: \Gamma \longrightarrow \Gamma '$ be an embedding. Then the Julia sets of the critically fixed anti-rational maps ${\mathcal {R}}_{\Gamma }$ and ${\mathcal {R}}_{\Gamma '}$ satisfy

$$\begin{aligned} {\mathcal {J}}({\mathcal {R}}_{\Gamma '}) = {\mathcal {J}}({\mathcal {R}}_\Gamma )/\sim _{i}. \end{aligned}$$

Moreover, the quotient map $\Psi _i: {\mathcal {J}}({\mathcal {R}}_{\Gamma '}) \longrightarrow {\mathcal {J}}({\mathcal {R}}_\Gamma )$ is a semi-conjugacy.

Remark A.5

For the principal hyperbolic component of ${\mathcal {M}}_d^-$ (i.e., the one containing $[{\overline{z}}^d]$), degenerations in the corresponding pared deformation space are parametrized by a pair of $m_{-d}$-invariant laminations generated by 2-cycles. The corresponding enriched Tischler graphs are admissible if and only if the associated pair of laminations are non-parallel; i.e., they share no common leaf under the natural identification of the two copies of the circle. In this case, a simple extension of Proposition A.4 shows that an accumulation point of such a degeneration is a geometric mating of a pair of parabolic anti-polynomials (cf. [LLM22, Theorem 1.3]).

1.4 Automorphisms.

Let f be an anti-rational map. The automorphism group ${{\,\mathrm{Aut}\,}}(f)$ is the subgroup of ${{\,\mathrm{PSL}\,}}_2({\mathbb {C}})$ consisting of Möbius maps that commute with f. Let ${{\,\mathrm{QC}\,}}(f)$ denote the group of all quasiconformal maps $\phi : {\widehat{{\mathbb {C}}}} \longrightarrow {\widehat{{\mathbb {C}}}}$ that commute with f. The modular group of f, denoted by ${{\,\mathrm{Mod}\,}}(f)$, is the group of isotopy classes of such $\phi$: in other words, it is the quotient of ${{\,\mathrm{QC}\,}}(f)$ by the path component of the identity. Note that there is a natural inclusion ${{\,\mathrm{Aut}\,}}(f) \subseteq {{\,\mathrm{Mod}\,}}(f)$.

For an f-invariant closed subset $K_f\subset {\widehat{{\mathbb {C}}}}$, we define ${{\,\mathrm{QC}\,}}(K_f,f)$ as the group of all quasiconformal homeomorphisms $\phi : {\widehat{{\mathbb {C}}}} \rightarrow {\widehat{{\mathbb {C}}}}$ that commute with f on $K_f$. The modular group of f on $K_f$, denoted by ${{\,\mathrm{Mod}\,}}(K_f, f)$ is the group of isotopy classes of such $\phi$ (see [McM88a] for details).

Let $\Gamma$ be a plane graph. We define ${{\,\mathrm{Aut}\,}}(\Gamma )$ as the group of planar automorphisms of the graph $\Gamma$. The first isomorphism in the following proposition is an anti-holomorphic counterpart of [McM88a, Proposition 6.10], while the second isomorphism is a consequence of Thurston rigidity of postcritically finite maps.

Proposition A.6

Let ${\mathcal {R}}_{\Gamma }$ be a critical fixed anti-rational map. Then

$$\begin{aligned} {{\,\mathrm{Mod}\,}}({\mathcal {J}}({\mathcal {R}}_\Gamma ), {\mathcal {R}}_\Gamma ) \cong {{\,\mathrm{Aut}\,}}({\mathcal {R}}_\Gamma ) \cong {{\,\mathrm{Aut}\,}}(\Gamma ). \end{aligned}$$

1.5 Enumeration of the quotient maps.

Let ${\mathcal {H}}_\Gamma$ be a hyperbolic component, and $c_i$ be the distinct critical points of the critically fixed map ${\mathcal {R}}_\Gamma \in {\mathcal {H}}_\Gamma$ with local degree $d_i$ at $c_i$, $i\in \{1,\ldots , m\}$.

Recall that

$$\begin{aligned} \widetilde{{\mathcal {H}}}_\Gamma \cong {{\,\mathrm{{\mathcal {B}}}\,}}^-_{d_1} \times {{\,\mathrm{{\mathcal {B}}}\,}}^-_{d_2} \times \cdots \times {{\,\mathrm{{\mathcal {B}}}\,}}^-_{d_m} \end{aligned}$$

is the lift of ${\mathcal {H}}_\Gamma$ consisting of elements of ${\mathcal {H}}_\Gamma$ equipped with boundary markings. As an extension of terminology, we say that an m-tuple of laminations ${\mathcal {L}}:= ({\mathcal {L}}_1,\ldots , {\mathcal {L}}_m)$ is a lamination for $\widetilde{{\mathcal {H}}}_\Gamma$ if each ${\mathcal {L}}_i$ is an $m_{-d_i}$-invariant lamination generated by 2-cycles. By Lemma A.2, each lamination ${\mathcal {L}}_i$ is dual to some $(d_i+1)$-ended ribbon tree.

We call ${\mathcal {L}}$ an admissible lamination for $\widetilde{{\mathcal {H}}}_\Gamma$ if the corresponding enrichment of ${\mathscr {T}}$ (obtained by blowing up the vertex $c_i$ of the Tischler graph ${\mathscr {T}}$ of ${\mathcal {R}}_\Gamma$ to the marked $(d_i+1)$-ended ribbon tree respecting markings) is admissible. We will denote the set of admissible laminations (for $\widetilde{{\mathcal {H}}}_\Gamma$) by $\widetilde{{{\mathcal {A}}}{{\mathcal {L}}}}(\Gamma )$.

The group ${{\,\mathrm{Aut}\,}}({\mathcal {R}}_\Gamma )$ acts on $\widetilde{{\mathcal {H}}}_\Gamma$ in the following way. Fix a representative $({\mathcal {R}}_\Gamma , (x_1^0,\ldots , x_m^0))$ of the ‘center’ of $\widetilde{{\mathcal {H}}}_\Gamma$. Then, for a representative $(R, (x_1,\ldots , x_m))$ of any element in $\widetilde{{\mathcal {H}}}_\Gamma$, there exists a unique orientation preserving homeomorphism $\kappa$ of the sphere that conjugates ${\mathcal {R}}_\Gamma \vert _{{\mathcal {J}}({\mathcal {R}}_\Gamma )}$ to $R\vert _{{\mathcal {J}}(R)}$ preserving the marking. For $M\in {{\,\mathrm{Aut}\,}}({\mathcal {R}}_\Gamma )$, we define its action on $\widetilde{{\mathcal {H}}}_\Gamma$ as

$$\begin{aligned} M\cdot [R, (x_1,\ldots , x_m)]= [R, (\kappa (\widetilde{M}(x_1^0)),\ldots , \kappa (\widetilde{M}(x_m^0)))], \end{aligned}$$

where $\widetilde{M}$ is the action on the ideal boundaries of the fixed Fatou components of R induced by M. (It is trivial to check that the action does not depend on the choice of the representative $(R, (x_1,\ldots , x_m))$.) We note that this action is not free as each element of ${{\,\mathrm{Aut}\,}}({\mathcal {R}}_\Gamma )$ fixes the ‘center’ of $\widetilde{{\mathcal {H}}}_\Gamma$. Identifying ${{\,\mathrm{Aut}\,}}(\Gamma )$ with ${{\,\mathrm{Aut}\,}}({\mathcal {R}}_\Gamma )$ (using Proposition A.6), we have that

Lemma A.7

The hyperbolic component ${\mathcal {H}}_\Gamma = \widetilde{{\mathcal {H}}}_\Gamma / {{\,\mathrm{Aut}\,}}(\Gamma )$.

The automorphism group ${{\,\mathrm{Aut}\,}}(\Gamma )$ also acts naturally on $\widetilde{{{\mathcal {A}}}{{\mathcal {L}}}}(\Gamma )$. Indeed, given ${\mathcal {L}}= ({\mathcal {L}}_1,\ldots , {\mathcal {L}}_m)$, think of ${\mathcal {L}}_i$ as a lamination sitting inside the Fatou component $U_i\ni c_i$ of ${\mathcal {R}}_\Gamma$ such that the marked endpoint of the dual tree corresponds to the marked fixed point on the (ideal) boundary of $U_i$. Any $M\in {{\,\mathrm{Aut}\,}}({\mathcal {R}}_\Gamma )$ permutes the invariant Fatou components of ${\mathcal {R}}_\Gamma$. Thus, if $c_s=M(c_r)$, the image of ${\mathcal {L}}_r$ under M is a lamination in $U_s$, but the M-image of the marked endpoint of the corresponding dual tree is not necessarily the marked fixed point on the (ideal) boundary of $U_s$. Hence, the images of various ${\mathcal {L}}_i$ under M yield a new m-tuple of laminations, denoted by $M\cdot {\mathcal {L}}$, whose s-th coordinate is a rotate of the lamination ${\mathcal {L}}_r$. It is easy to see that this m-tuple of laminations is an admissible lamination for $\widetilde{{\mathcal {H}}}_\Gamma$. We denote the quotient by

$$\begin{aligned} {{\mathcal {A}}}{{\mathcal {L}}}(\Gamma ) := \widetilde{{{\mathcal {A}}}{{\mathcal {L}}}}(\Gamma )/{{\,\mathrm{Aut}\,}}(\Gamma ). \end{aligned}$$

The automorphism groups ${{\,\mathrm{Aut}\,}}(\Gamma )$ and ${{\,\mathrm{Aut}\,}}(\Gamma ')$ act naturally on the set of embeddings $i: \Gamma \longrightarrow \Gamma '$ by pre-composition and post-composition. Let

be the number of embeddings of $\Gamma$ into $\Gamma '$ up to the action of ${{\,\mathrm{Aut}\,}}(\Gamma ) \times {{\,\mathrm{Aut}\,}}(\Gamma ')$. Since each embedding realizes the Julia set ${\mathcal {J}}({\mathcal {R}}_{\Gamma '})$ as a quotient of ${\mathcal {J}}({\mathcal {R}}_{\Gamma })$ (see Proposition A.4), using Proposition A.6, we have

Proposition A.8

Let $\Gamma ' \ge \Gamma$. Then there are inequivalent semiconjugacies between ${\mathcal {J}}({\mathcal {R}}_{\Gamma })$ and ${\mathcal {J}}({\mathcal {R}}_{\Gamma '})$.

Here two semiconjugacies $\Psi _1$ and $\Psi _2$ are said to be equivalent if they differ by pre and post-composing with automorphisms of ${\mathcal {J}}({\mathcal {R}}_{\Gamma })$ and ${\mathcal {J}}({\mathcal {R}}_{\Gamma '})$.

1.6 Accesses to $\partial {\mathcal {H}}_\Gamma (K)$.

The cardinality of the double quotient space is also related to the number of accesses from ${\mathcal {H}}_\Gamma (K)$ to ${\mathcal {H}}_{\Gamma '}(K)$.

Let $[R] \in \partial {{\mathcal {H}}}_\Gamma (K)$. Abusing notation, we choose a lift of [R] in $\partial \widetilde{{\mathcal {H}}}_\Gamma (K)$ and also denote it by $[R] \in \partial \widetilde{{\mathcal {H}}}_\Gamma (K)$. Let $[R_n] \in \widetilde{{\mathcal {H}}}_\Gamma (K)$ converge to [R]. After passing to a subsequence, let ${\mathscr {T}}^{En}$ be the enriched Tischler graph for $[R_n]$. By Theorem 4.11, ${\mathscr {T}}^{En}$ is admissible. Let ${\mathcal {L}} \in \widetilde{{{\mathcal {A}}}{{\mathcal {L}}}}(\Gamma )$ be the corresponding admissible lamination. We call it a lamination for [R].

Therefore, we can associate an equivalence class of admissible laminations $[{\mathcal {L}}] \in {{\mathcal {A}}}{{\mathcal {L}}}(\Gamma )$ for any map $[R] \in \partial {\mathcal {H}}_\Gamma (K)$. We remark that by choosing different lifts of [R] in $\partial \widetilde{{\mathcal {H}}}_\Gamma (K)$, one may obtain multiple laminations associated to a map $[R] \in \partial {\mathcal {H}}_\Gamma (K)$, in which case, a self-bump appears on $\partial {\mathcal {H}}_\Gamma (K)$.

This motivates the following definition.

Definition A.9

Let ${\mathcal {U}}$ be a connected component of $\overline{{\mathcal {H}}_\Gamma (K)} \cap \overline{{\mathcal {H}}_{\Gamma '}(K)}$. We say that the equivalence class of admissible laminations $[{\mathcal {L}}] \in {{\mathcal {A}}}{{\mathcal {L}}}(\Gamma )$ gives an access to ${\mathcal {U}}$ if $[{\mathcal {L}}]$ is associated to some $[R] \in {\mathcal {U}}$. We define the multiplicity of ${\mathcal {U}}$ as the number of accesses to ${\mathcal {U}}$.

By Proposition 4.4 and Theorem 4.11, we have that

Proposition A.10

For all large K, there are at least components in $\overline{{\mathcal {H}}_\Gamma (K)} \cap \overline{{\mathcal {H}}_{\Gamma '}(K)}$ counting multiplicity.

We conjecture that the number of components in $\overline{{\mathcal {H}}_\Gamma (K)} \cap \overline{{\mathcal {H}}_{\Gamma '}(K)}$ (counted with multiplicities) is exactly . In “Appendix B”, we shall see how a component of $\overline{{\mathcal {H}}_\Gamma (K)} \cap \overline{{\mathcal {H}}_{\Gamma '}(K)}$ with multiplicity $\ge 2$ is related to the self-bump and shared mating phenomena. We shall also discuss examples where $\overline{{\mathcal {H}}_\Gamma (K)} \cap \overline{{\mathcal {H}}_{\Gamma '}(K)}$ is disconnected.

Appendix B. Shared Matings, Self-bumps and Disconnected Roots

In this appendix, we will briefly discuss some examples of how different pared deformation spaces interact.

1.1 Self-bumps and shared matings.

In this subsection, we will discuss some techniques to construct components of $\overline{{\mathcal {H}}_\Gamma (K)} \cap \overline{{\mathcal {H}}_{\Gamma '}(K)}$ with multiplicity at least 2. We shall also explain how this is related to the shared mating phenomenon using examples.

1.2 An example of self-bump.

Let $\Gamma$ and $\Gamma '$ be the left and right graph in Figure 8. The graph $\Gamma$ has two different embeddings into $\Gamma '$. The images of the embeddings are displayed in black on the right. Note that both ${{\,\mathrm{Aut}\,}}(\Gamma )$ and ${{\,\mathrm{Aut}\,}}(\Gamma ')$ are trivial, so the two embeddings are in different orbits of ${{\,\mathrm{Aut}\,}}(\Gamma ) \times {{\,\mathrm{Aut}\,}}(\Gamma ')$.

Fig. 8

The graph $\Gamma$ on the left has two different embeddings into the graph $\Gamma '$ on the right. The images of the embeddings are depicted in black on the right, where the indices of vertices are preserved under the embeddings.

Fig. 9

The corresponding degeneration diagrams associated to the two embeddings.

We define an arrow structure on $\Gamma '$ by associating to each face either a dot, or an arrow towards a boundary edge. We call the graph $\Gamma '$ together with an arrow structure an arrowed graph.

The arrow structure is used to indicate the dynamics of a root $[R] \in \partial {\mathcal {H}}_{\Gamma '}$. Note that a face $F'$ of the graph $\Gamma '$ corresponds to a fixed Fatou component $\Omega _{F'}$, and an edge on the boundary of $F'$ corresponds to a fixed point on $\partial \Omega _{F'}$. For each face $F'$ with k edges, a dot indicates the dynamics of R on $\Omega _{F'}$ is conjugate to ${{\bar{z}}}^k$. An arrow towards a boundary edge indicates that $\Omega _{F'}$ is a parabolic Fatou component for R with the corresponding fixed point on $\partial \Omega _{F'}$ being the parabolic fixed point.

The two embeddings of $\Gamma$ into the arrowed graph $\Gamma '$ provide two different accesses to [R], as indicated in Figure 9. We will call the graphs as in Figure 9degeneration diagrams.

We remark that the different perspectives between the arrowed graph and the degeneration diagram (see Figure 8 vs. Figure 9, or Figure 10) is that in the former, $\Gamma '$ is fixed, and the graph $\Gamma$ is represented by different subgraphs of $\Gamma '$; while in the degeneration diagram, the original graph $\Gamma$ is fixed, and $\Gamma '$ is constructed from $\Gamma$ by adding new edges.

Note that the faces of $\Gamma$ are divided into chambers, as indicated by the red boundaries, in the degeneration diagram.

Let $C, C_0$ be two distinct chambers of a face F of $\Gamma$. Then there exists a unique boundary edge of C corresponding to the direction of $C_0$. We say an arrow in C points towards $C_0$ if the arrow points towards this boundary edge. Similarly, let E be an edge of a chamber in F. Then there exists a unique boundary edge of C corresponding to the direction of E. We say an arrow in C points towards E if the arrow points towards this boundary edge. See Figure 9 or Figure 10 for illustrations.

Definition B.1

An arrow structure on a degeneration diagram is said to be compatible if for each face F of $\Gamma$, either all the arrows point towards a unique chamber in F containing the dot, or all the arrows point towards a unique edge on the boundary of a chamber in F.

Let F be a face of $\Gamma$ with e sides. A degeneration diagram partitions the face into finitely many chambers. The associated dual graph is an $(e+1)$-ended tree and the arrow structure gives either a special core vertex or an edge. Thus we can associate either a pointed $(e+1)$-ended tree, or an extended pointed $(e+1)$-ended tree for each face F of $\Gamma$. Theorems 3.5 and 3.15 and the Schwarz lemma now imply that a prescribed dynamical behavior on the fixed Fatou components (given by an arrow structure on $\Gamma '$) is realized in the dynamical plane of an accumulation point of an access determined by a degeneration diagram if and only if the arrow structure is compatible with the degeneration diagram in the sense of Definition B.1.

Let $[R_0] \in \partial {\mathcal {H}}_{\Gamma '}$ be the parabolic map corresponding to the right graph of Figure 8, so that the dynamics on (each of) the invariant parabolic Fatou components is conjugate to a real Blaschke product. This real assumption is required to uniquely specify the map, as unicritical parabolic Blaschke products are not rigid in the anti-rational map setting (cf. [MNS17, Theorem 3.2]).

In our example, each face is divided into at most 2 chambers. Thus, using Proposition 3.16 and Remark 3.17 we can construct two families $[R_{i,t}] \in {\mathcal {H}}_\Gamma (K)$, $i=1,2$, realizing the accesses determined by the two degeneration diagrams with $[R_{i,t}] \rightarrow [R_0]$. Thus, the component U in $\overline{{\mathcal {H}}_\Gamma (K)} \cap \overline{{\mathcal {H}}_{\Gamma '}(K)}$ containing $[R_0]$ has multiplicity $\ge 2$. One can in fact verify that (cf. [Luo21a, Theorem 1.4])

Proposition B.2

For all large K, the pared deformation space ${\mathcal {H}}_\Gamma (K)$ has a self-bump at $[R_0]$, i.e. for all sufficiently small neighborhood $U \subset {\mathcal {M}}_d^-$ containing $[R_0]$, the intersection $U \cap {\mathcal {H}}_\Gamma (K)$ is disconnected.

1.3 A shared parabolic mating example.

We will now discuss connections between multiple accesses and shared mating phenomenon.

To construct a parabolic map that can be unmated in two different ways, we consider the arrowed graph $\Gamma '$ in Figure 10. The graph $\Gamma '$ contains two different Hamiltonian cycles that are marked black in Figure 10. It is easy to verify that ${{\,\mathrm{Aut}\,}}(\Gamma ')$ is trivial. Thus, in particular, the two Hamiltonian cycles in Figure 10 are in different orbits of ${{\,\mathrm{Aut}\,}}(\Gamma ')$. Note $\Gamma$ has rotation symmetries, but such a symmetry always preserves the image of $\Gamma$ in $\Gamma '$. Therefore the corresponding two embeddings of the hexagonal graph $\Gamma$ in $\Gamma '$ are in different orbits of ${{\,\mathrm{Aut}\,}}(\Gamma ) \times {{\,\mathrm{Aut}\,}}(\Gamma ')$.

Fig. 10

The top two graphs exhibit a graph $\Gamma '$ with distinct Hamiltonian cycles drawn using black edges. Each Hamiltonian cycle gives a corresponding decomposition of the parabolic rational map as a mating of parabolic polynomials, exhibited below.

The Hamiltonian cycle divides the graph into two sub-graphs, with degeneration diagrams as in the lower part of Figure 10. Note that the corresponding laminations are different, and the arrow structure on $\Gamma '$ is compatible with both degeneration diagrams.

Theorems 3.15 and 4.11 allow us to construct two sequences $[R_{i,n}] \in {\mathcal {H}}_{\Gamma }(K),\, i=1,2$, realizing the accesses determined by the two degeneration diagrams such that the sequences converge to parabolic maps $[R_i],\, i=1,2$. Note that the parabolic maps $[R_1], [R_2]$ correspond to the same arrowed graph, but we cannot guarantee that $[R_1]=[R_2]$ as unicritical parabolic Blaschke products in the anti-rational map setting are not rigid. In fact, we do not know how to prove that the two accesses have a common accumulation point. However, we conjecture that the degeneration diagrams in Figure 10 should also produce a self-bump.

On the other hand, the parabolic map $R_1$ is the shared mating of two different pairs of parabolic anti-polynomials where the unmatings are given by the two distinct Hamiltonian cycles in $\Gamma '$.

1.4 Multiple components in $\overline{{\mathcal {H}}_\Gamma (K)} \cap \overline{{\mathcal {H}}_{\Gamma '}(K)}$.

In this subsection, we explain a technique to construct examples where $\overline{{\mathcal {H}}_\Gamma (K)} \cap \overline{{\mathcal {H}}_{\Gamma '}(K)}$ is disconnected.

Let $\Gamma '$ be the graph with two different Hamiltonian cycles as in Figure 11. One can easily verify that

• ${{\,\mathrm{Aut}\,}}(\Gamma ')$ is trivial, so the corresponding two embeddings are in different orbits; and

• there are exactly two Hamiltonian cycles in $\Gamma '$ indicated as black in Figure 11.

Fig. 11

An example where no arrow structure on $\Gamma '$ is compatible with two different degeneration diagrams.

But the difference between this example and the one considered in Section B.1 is that in this example, there is no arrow structure on $\Gamma '$ compatible with both degeneration diagrams. More precisely, we can not assign an arrow or a dot on each face of $\Gamma '$ so that the arrow structure satisfies the requirements of Definition B.1 for both degeneration diagrams. Therefore, the set of accumulation points of the accesses determined by the two degeneration diagrams in Figure 11 lie in two distinct components of the root locus $\overline{{\mathcal {H}}_\Gamma (K)} \cap \overline{{\mathcal {H}}_{\Gamma '}(K)}$. So the $\overline{{\mathcal {H}}_\Gamma (K)} \cap \overline{{\mathcal {H}}_{\Gamma '}(K)}$ contains at least 2 components.

This example is related to the shared mating phenomenon. Although the existence of two different Hamiltonian cycles shows that the critically fixed anti-rational map $[{\mathcal {R}}_{\Gamma '}]$ is a shared mating of two different pairs of critically fixed anti-polynomials (cf. [LLM22]), no parabolic map $[R] \in \partial {\mathcal {H}}_{\Gamma '}(K)$ can be unmated in two different ways.