1 Introduction
In 1965, Brolin [Reference Brolin3] studied asymptotic properties of polynomials $P(z)\in \mathbb {C}[z]$ of degree bigger than or equal to $2$ . He proved the existence of a probability measure for which the preimages $P^{n}(z_0)$ at time n of any point $z_0\in \mathbb {C}$ (with at most one exception) asymptotically equidistribute, as n tends to infinity. In 1983, Freire, Lopes, and Mañé [Reference Freire, Lopes and Mañé20] and Ljubich [Reference Ljubich24] independently proved the generalization to rational maps of degree at least $2$ on the Riemann sphere. These results have been generalized to different settings. For instance, see [Reference Dinh, Sibony, Gentili, Guenot and Patrizio18, §1.4] and the references therein for higher dimensions, and [Reference Favre and RiveraLetelier19, Reference Gignac22] for the nonArchimedian setting.
The equidistribution properties of holomorphic correspondences have also attracted considerable interest. Roughly speaking, a holomorphic correspondence on a complex manifold X is a multivalued map induced by a formal sum $\Gamma =\sum n_i\Gamma (i)$ of complex varieties $\Gamma (i)\subset X\times X$ of the same dimension. The multivalued map sends z to w if $(z,w)$ belongs to some $\Gamma (i)$ (see §2.1). Let $d(F)$ denote the number of preimages of a generic point under F. We call this number the topological degree of F, just as in the case of rational maps. We study the existence of a Borel probability measure $\mu $ on X that has the property that for all but at most finitely many $z_0\in X$ ,
as $n\to \infty $ . Here, $(F^n)_*$ denotes the pushforward operator associated to $F^n$ . In [Reference Dinh13], Dinh studied the case of polynomial correspondences whose Lojasiewicz exponent is strictly bigger than $1$ , in which case we always have that $d(F^{1})<d(F)$ . The case where $d(F)=d(F^{1})$ is open but some subcases are known. For instance, Clozel, Oh, and Ullmo [Reference Clozel, Oh and Ullmo9] proved equidistribution for irreducible modular correspondences, Clozel and Otal [Reference Clozel and Otal10] proved it for exterior modular correspondences, and Clozel and Ullmo [Reference Clozel and Ullmo11] for those that are selfadjoint. On the other hand, Dinh, Kaufmann, and Wu proved in [Reference Dinh, Kaufmann and Wu15] that if such F is not weakly modular, then the statement holds for both F and $F^{1}$ . Modular correspondences are weakly modular, but the reverse containment does not hold. On a different classification, Bharali and Sridharan [Reference Bharali and Sridharan1] proved equidistribution for correspondences with $d(F)\geq d(F^{1})$ having a repeller in the sense of [Reference McGehee26]. We will study a $1$ parameter family in the gap between weakmodularity and modularity, and for which the result in [Reference Bharali and Sridharan1] does not apply.
Our object of study is the family of correspondences $\lbrace \mathcal {F}_a\rbrace _{a\in \mathcal {K}}$ on the Riemann sphere $\widehat {\mathbb {C}}$ , where $\mathcal {F}_a$ is given in affine coordinates by
and $\mathcal {K}$ is the Klein combination locus defined in §2.2. This family was studied by Bullett et al in [Reference Bullett and Lomonaco4–Reference Bullett and Penrose6, Reference Bullett and Harvey8]. In [Reference Bullett and Lomonaco4], Bullett and Lomonaco proved that there is a two sided restriction $f_a$ of $\mathcal {F}_a$ that is hybrid equivalent to a quadratic rational map P that has a fixed point with multiplier $1$ . We refer the reader to [Reference Lomonaco25] for conjugacy of paraboliclike mappings. Moreover, for the parameters for which the Julia set of P is connected, we have that $\mathcal {F}_a$ is a mating between the rational map P and the modular group $\operatorname {PSL}_2(\mathbb {Z})$ . This generalizes a previous result by Bullett and Penrose [Reference Bullett and Penrose6]. The correspondence $\mathcal {F}_a$ has two homeomorphic copies of the filled Julia set $K$ of $P_A$ , denoted $\Lambda _{a,}$ and $\Lambda _{a,+}$ , and they satisfy that $\mathcal {F}_a^{1}(\Lambda _{a,})=\Lambda _{a,}$ and $\mathcal {F}_a(\Lambda _{a,+})=\Lambda _{a,+}$ . These are called the backward and forward limit set, respectively (see §3.2).
The following theorem states that this family does not fit the conditions for any of the equidistribution results listed above (see §1.1).
Theorem 1.1. For every $a\in \mathcal {K}$ , we have that:

(1) $\mathcal {F}_a$ is a weakly modular correspondence that is not modular; and

(2) $\partial \Lambda _{a,}$ is not a repeller for $\mathcal {F}_a$ .
Furthermore, we prove that $\mathcal {F}_a$ satisfies a property that is stronger than weakmodularity (see Remark 3.4).
The purpose of this paper is to show that equidistribution holds for the family $\lbrace \mathcal {F}_a\rbrace _{a\in \mathcal {K}}$ . Put
We prove the following equidistribution theorem.
Theorem 1.2. Let $a\in \mathcal {K}$ . There exist two Borel probability measures $\mu _+$ and $\mu _$ on $\widehat {\mathbb {C}}$ , with $\operatorname {supp}(\mu _+)=\partial \Lambda _{a,+}$ and $\operatorname {supp}(\mu _)=\partial \Lambda _{a,}$ , such that for every $z_0\in \widehat {\mathbb {C}}\setminus \mathcal {E}_a$ ,
weakly, as $n\to \infty $ .
In later work [Reference Matus de la Parra12], we prove that the measures $\mu _+$ and $\mu _$ maximize entropy: the metric entropy in [Reference Vivas and Sirvent31] yields equality for the halfvariational principle with the topological entropy in [Reference Dinh and Sibony17].
Let F be a holomorphic correspondence on X with graph $\Gamma $ . Denote by $\Gamma ^{(n)}$ the graph of $F^n$ and by
the diagonal in $X\times X$ . Then the set of periodic points of F of period n is defined as the set
and for $z\in \operatorname {Per}_n(F)$ , we define the multiplicity of z as a periodic point of F of order n to be the number
, defined in §2.1.
Another source of motivation is whether or not periodic points equidistribute. In [Reference Ljubich24], Ljubich showed that this is the case for rational maps of degree bigger than or equal to $2$ , where periodic points are counted either with or without multiplicity. The equidistribution of periodic points is also studied in [Reference Briend and Duval2, Reference Dinh, Nguyen and Truong16, Reference Favre and RiveraLetelier19] in the case of maps, and in [Reference Dinh13, Reference Dinh14] in the case of correspondences. We prove this holds for the family $\lbrace \mathcal {F}_a\rbrace _{a\in \mathcal {K}}$ as well.
Theorem 1.3. For $a\in \mathcal {K}$ ,
are both weakly convergent to $\tfrac 12(\mu _+\mu _+)$ , as $n\to \infty $ .
In §5, we define the set $\hat {P}^{\Gamma }_n$ of superstable parameters of order n. Combining the main results of [Reference Bullett and Lomonaco5, Reference Petersen and Roesch29], we obtain a homeomorphism $\Psi :\mathcal {M}\to \mathcal {M}_{\Gamma }$ between the Mandelbrot set $\mathcal {M}$ and the connectedness locus $\mathcal {M}_{\Gamma }$ of the family $\lbrace \mathcal {F}_a\rbrace _{a\in \mathcal {K}}$ . These results, together with the equidistribution result in [Reference Levin23], yield the following theorem.
Theorem 1.4. In $\mathcal {M}_{\Gamma }$ , superstable parameters equidistribute with respect to $\Psi ^*m_{\operatorname {BIF}}$ , that is,
1.1 Notes and references
There is a bigger family studied by Bullett and Harvey in [Reference Bullett and Harvey8], given by replacing the righthand side of equation (1) by $3k$ , where $k\in \mathbb {C}$ . For these correspondences, it is also possible to define limit sets $\Lambda _{a,k,}$ and $\Lambda _{a,k,+}$ analogous to those in the case where $k=1$ . In [Reference Bharali and Sridharan1], Bharali and Sridharan show how their equidistribution result applies to these correspondences in the case where $\Lambda _{a,k,}$ is a repeller. Parameters for which this is the case exist from the results in [Reference Bullett and Harvey8]. However, we prove in Theorem 1.1 part $(2)$ that this is never the case when $k=1$ and $a\in \mathcal {K}$ .
1.2 Organization
The structure of this paper is as follows. In §2.1, we give an introduction to holomorphic correspondences and their action on Borel measures. In §2.2, we introduce the correspondences $\mathcal {F}_a$ given by equation (1). We define critical values and find those of $\mathcal {F}_a$ in §2.3, and define Klein combination pair and the Klein combination locus $\mathcal {K}$ in §2.4. In §3.1, we define modular and weakly modular correspondences, and prove part $(1)$ of Theorem 1.1. To prove that $\mathcal {F}_a$ is weakly modular, we use the decomposition $\mathcal {F}_a=\operatorname {J}_a\circ \operatorname {Cov^Q_0}$ , given in [Reference Bullett and Lomonaco4], into a certain involution $\operatorname {J}_a$ composed with the deleted covering correspondence $\operatorname {Cov^Q_0}$ described in §2.2, and construct the measures in the definition of weakly modular using the symmetry of the graph of $\operatorname {Cov^Q_0}$ . The fact that $\mathcal {F}_a$ is not modular follows from the fact that Borel measures assigning positive measure to nonempty open sets are not invariant by $\mathcal {F}_a$ . In §3.2, we define the limit sets $\Lambda _{a,}$ and $\Lambda _{a,+}$ , and prove part $(2)$ of Theorem 1.1 by showing that the parabolic fixed point in $\partial \Lambda _{a,}$ violates the definition of a repeller. In §4, we describe the exceptional set of the twosided restriction $f_a$ and the set $\operatorname {Per}_n(\mathcal {F}_a)$ of periodic points of period n. Finally, in §5, we use the description of $f_a$ given in [Reference Bullett and Lomonaco4] together with the results in [Reference Freire, Lopes and Mañé20, Reference Ljubich24] to prove Theorems 1.2 and 1.3 about asymptotic equidistribution of images, preimages, and periodic points. We finish with the proof of Theorem 1.4 about equidistribution of special points in the modular Mandelbrot set $\mathcal {M}_{\Gamma }$ .
2 Preliminaries
2.1 Holomorphic correspondences
Let X be a compact Riemann surface and let $\pi _j:X\times X\to X$ be the canonical projection to the jth coordinate, $j=1,2$ . We say that a formal sum $\Gamma =\sum _i n_i\Gamma (i)$ is a holomorphic 1chain on $X\times X$ if its support is a subvariety of $X\times X$ of pure dimension $1$ whose irreducible components are exactly the $\Gamma (i)$ , and the $n_i$ are nonnegative integers. We say that the $\Gamma (i)$ are the irreducible components of $\Gamma $ .
Let $\Gamma =\sum _in_i\Gamma (i)$ be a holomorphic 1chain satisfying that for $j=1,2$ and every i such that $n_i>0$ , the restriction
of the canonical projection $X\times X\to X$ to the irreducible component $\Gamma (i)$ is surjective. The chain $\Gamma $ induces a multivalued map F from X to itself by
The multivalued map F is called a holomorphic correspondence and it is said to be irreducible if $\sum _i n_i=1$ . We say that
is the graph of the holomorphic correspondence F. Let $\iota :X\times X\to X\times X$ be the involution $(z,w)\mapsto (w,z)$ . We can define the adjoint correspondence $F^{1}$ of F by the relation
, which is a holomorphic correspondence, whose graph is the holomorphic 1chain $\Gamma _F^{1}=\sum _in_i\iota (\Gamma (i))$ .
In [Reference Stoll30], Stoll introduced a notion of multiplicity that will be useful for this paper. Let M be a quasiprojective variety and N a smooth quasiprojective variety. If $g:M\to N$ is regular, and $a\in M$ , then we say that a neighborhood U of a is distinguished with respect to g and a if $\overline {U}$ is compact and $g^{1}(g(a))\cap \overline {U}=\lbrace a\rbrace $ . Such neighborhoods exist if and only if $\dim _a g^{1}(g(a))=0$ and, in this case, they form a base of neighborhoods. If U is distinguished with respect to g and a, then put . It can be shown that does not depend on the distinguished neighborhood U, and the maps $n_b$ defined on N by $a\mapsto \sum \nolimits _{b\in g^{1}(a)}\hat {\nu }_g(b)$ are constant in each component.
Suppose that $g:M\to N$ is a finite and surjective regular map, with M and N as above. Stoll proved in [Reference Stoll30] that $\hat {\nu }_g(a)$ generalizes the notion of multiplicity of g at a and whenever $\varphi $ is a continuous function with compact support in M, the map
is continuous.
To study dynamics, we proceed to define the composition of two holomorphic correspondences F and G with associated holomorphic 1chains $\Gamma _F=\sum _i n_i\Gamma _F(i)$ and $\Gamma _G=\sum _j m_j\Gamma _G(j)$ , respectively. For each i and j, let $A_{i,j}$ be the image of the projection $p_{i,j}:(\Gamma _G(j)\times \Gamma _F(i))\cap \lbrace x_2=x_3\rbrace \hookrightarrow X\times X$ that forgets the second and third coordinates, i.e.,
Let $\lbrace \Gamma (i,j,k)\rbrace _{k=1}^{N(i,j)}$ be the irreducible components of $A_{i,j}$ . Observe that since $\Gamma _G(j)$ and $\Gamma _F(i)$ are both quasiprojective, and so is $\lbrace x_2=x_4\rbrace \subset X^4$ , then $p_{i,j}$ is a regular map from the quasiprojective variety $(\Gamma _G(j)\times \Gamma _F(i))\cap \lbrace x_2=x_3\rbrace $ to the smooth quasiprojective variety $X\times X$ . Then we have that the map
defined on $\Gamma (i,j,k)$ is constant. Therefore,
denotes the number of $x\in X$ such that $((z,x),(x,w))\in \Gamma _G(j)\times \Gamma _F(i)$ , for a generic point $(z,w)\in \Gamma (i,j,k)$ . Define the composition $F\circ G$ as the holomorphic correspondence determined by the holomorphic 1chain
Note that the $\operatorname {supp}\Gamma _{F\circ G}=\bigcup _{i,j}A_{i,j}$ .
Set . We have that $d(F\circ G)=d(F)d(G)$ . Thus, in particular, for every integer $n\geq 1$ , we have that $d(F^n)=(d(F))^n$ . We call $d(F)$ the topological degree of F, and it corresponds to the number of preimages of a generic point under F.
If F is an irreducible holomorphic correspondence over X with graph $\Gamma $ and $\varphi :X\to \mathbb {C}$ is a continuous function, then
is continuous as well, see [Reference Clozel and Ullmo11, Lemma 1.1]. Now let F be a holomorphic correspondence that is not necessarily irreducible, with graph $\Gamma _F=\sum _i n_i\Gamma (i)$ . We denote by $F_i$ the holomorphic correspondence induced by $\Gamma (i)$ and we put
and
. Then for every continuous function $\varphi :X\to \mathbb {C}$ , the map
is also continuous.
The holomorphic correspondence F induces an action $F_*$ on finite Borel measures $\mu $ by duality, namely
, called the pushforward operator and the resultant measure $F_*\mu $ is the pushforward measure of $\mu $ under F. We define as well the action
, called the pullback operator and the resultant measure $F^*\mu $ is called the pullback measure of $\mu $ under F. This action on measures agrees with the action on points
where $\delta _z$ is the Dirac delta at z.
To see this, note that for every continuous function $\varphi :X\to \mathbb {C}$ ,
2.2 The family $\lbrace \mathcal {F}_a\rbrace _a$
Let $Q(z)\in \mathbb {C}[z]$ be a nonlinear polynomial. The deleted covering relation of Q on $\mathbb {C}\times \mathbb {C}$ is defined by $w\in \operatorname {Cov^Q_0}(z)$ if and only if
Note that the denominator ‘deletes’ the obvious association of z with itself in the equation $Q(z)=Q(w)$ .
In this section, we will identify $\widehat {\mathbb {C}}$ with the complex projective line when it is convenient to work with homogeneous coordinates $(z:w)$ .
Proposition 2.1. Put . The closure of the relation in equation (2) is an irreducible quasiprojective complex variety $\Gamma _0$ of $\widehat {\mathbb {C}}\times \widehat {\mathbb {C}}$ of dimension $1$ . Moreover, the projections and are both surjective and of degree $2$ .
Proof. Note that $P_Q(z,w)=z^2+zw+w^23$ and consider $P_Q(z,w)$ as a single variable polynomial in $(\mathbb {C}[z])[w]$ . Then its discriminant $3z^2+12$ is not a square in $\mathbb {C}[z]$ . Therefore, $P_Q(z,w)$ is an irreducible polynomial, and hence
is an irreducible subvariety of $\mathbb {C}\times \mathbb {C}$ .
Now we want to describe the closure $\overline {\mathcal {Z}}$ of $\mathcal {Z}$ in $\widehat {\mathbb {C}}\times \widehat {\mathbb {C}}$ . Observe that if we fix $z\in \mathbb {C}$ , then $\lim \nolimits _{w\to \infty }P_Q(z,w)=\infty $ , and if we fix $w\in \mathbb {C}$ , then $\lim \nolimits _{z\to \infty }P_Q(z,w)=\infty $ . Therefore, there are no points of the form $(z,\infty )$ or $(\infty , w)$ in $\overline {\mathcal {Z}}$ . Given $R>0$ , let $z\in \mathbb {C}$ be such that $z=R$ . Observe that $P_Q(z,\cdot )\in \mathbb {C}[w]$ is nonconstant and therefore has at least one root in $\mathbb {C}$ . Let $w\in \mathbb {C}$ be a root. Then $(z,w)\in \mathcal {Z}$ and we have that $w^33w=Q(w)=Q(z)=z^33z\geq R^33R$ . By taking $R\to \infty $ , we get that $(z,w)\to (\infty ,\infty )$ . Therefore, $\overline {\mathcal {Z}}=\mathcal {Z}\cup \lbrace (\infty ,\infty )\rbrace $ . In particular, $\overline {\mathcal {Z}}$ extends the relation given by equation (2) from $\mathbb {C}\times \mathbb {C}$ to $\widehat {\mathbb {C}}\times \widehat {\mathbb {C}}$ .
Take the homogenization
of $P_Q(z,w)$ , so $P_Q(z,w)=T(z,1,w,1)$ , and note that $T(\unicode{x3bb} _1z,\unicode{x3bb} _1x,\unicode{x3bb} _2w,\unicode{x3bb} _2y)=\unicode{x3bb} _1^2\unicode{x3bb} _2^2T(z,x,w,y)$ . Thus, for the closed subvariety
of $\widehat {\mathbb {C}}\times \widehat {\mathbb {C}}$ , we have that $\Gamma _0=\overline {\mathcal {Z}}$ . To prove that $\Gamma _0$ is irreducible, note that each of its irreducible components intersecting $\mathbb {C}\times \mathbb {C}$ must be a closed subset of $\Gamma _0$ containing $\Gamma _0\cap (\mathbb {C}\times \mathbb {C})=\mathcal {Z}$ , and therefore it is $\Gamma _0$ itself. Thus, $\Gamma _0$ has only one irreducible component and hence it is irreducible.
We proceed to show $\Gamma _0$ has dimension 1. Observe that the polynomial $T(z,x,w,y)$ is irreducible in $\mathbb {C}[z,x,w,y]$ , as whenever $S(z,x,w,y)T(z,x,w,y)$ in $\mathbb {C}[z,x,w,y]$ , then $S(z,1,w,1)P_Q(z,w)$ in $\mathbb {C}[z,w]$ . Therefore, the zero set $Z(T)\subset \mathbb {C}^2\times \mathbb {C}^2$ of T is an irreducible hypersurface of $\mathbb {C}^2\times \mathbb {C}^2$ , and hence it has codimension 1. Now let $p:\mathbb {C}^2\setminus \lbrace (0,0)\rbrace \to \widehat {\mathbb {C}}$ be the projection sending $(z,x)\mapsto (z:x)$ . Note that in the chart
the map p is simply $(z,w)\mapsto (z/w:1)$ , and in the chart
it becomes $(z,w)\mapsto (1:w/z)$ . Let $\hat {p}:(\mathbb {C}^2\setminus \lbrace (0,0)\rbrace )\times (\mathbb {C}^2\setminus \lbrace (0,0)\rbrace )\to \widehat {\mathbb {C}}\times \widehat {\mathbb {C}}$ be the map defined by
. Then
is a regular map between irreducible varieties, and
has constant fiber dimension equal to 2, as T is homogeneous in $(z,x)$ and in $(w,y)$ . Therefore,
Finally, observe that the polynomial equation (2) has at least one and at most two solutions for every $z\in \mathbb {C}$ , and by symmetry, the same holds for $w\in \mathbb {C}$ . Note as well that $\infty $ is in correspondence with and only with itself. Thus, the projections and are both surjective. Moreover, $P_Q(1,1)=P_Q(1,2)=0$ , and hence $P_Q(1,w)$ has exactly two solutions. Therefore, .
Remark 2.2. Proposition 2.1 holds for a large class of polynomials $Q(z)$ . Observe that no polynomial can have $Q(z)=Q(\infty )$ for a finite number z. Therefore, following the proof of Proposition 2.1, we conclude that to get an irreducible holomorphic correspondence, it suffices to prove that $P_Q(z,w)$ is irreducible over $\mathbb {C}$ . This holds under fairly general conditions. For instance, this is the case when Q is indecomposable and not linearly related to either $z^n$ or a Chebyshev polynomial (see [Reference Fried21]).
On the other hand, note that whenever $Q=R\circ S$ with R and S of degree greater than $1$ , then $P_S(z,w)$ divides $P_Q(z,w)$ , and therefore $P_Q(z,w)$ is reducible.
Proposition 2.1 says that $\Gamma _0$ is the graph of an irreducible holomorphic correspondence, where $\Gamma _0$ is a quasiprojective variety and is a finite and surjective morphism over $\mathbb {C}$ , and hence we can use our definition of pullback and pushforward operators induced by the correspondence on finite measures. We call this correspondence the deleted covering correspondence of Q, denoted by $\operatorname {Cov^Q_0}$ as well. That is, $\operatorname {Cov^Q_0}$ is the holomorphic correspondence on $\widehat {\mathbb {C}}\times \widehat {\mathbb {C}}$ such that $\Gamma _{\operatorname {Cov^Q_0}}=\Gamma _0$ . From now on, we always consider $Q(z)=z^33z$ .
Now take $a\in \mathbb {C}\setminus \lbrace 1\rbrace $ and let $\operatorname {J}_a:\widehat {\mathbb {C}}\to \widehat {\mathbb {C}}$ be the involution
The composition of $\operatorname {Cov^Q_0}$ with the involution $\operatorname {J}_a$ is again an irreducible holomorphic correspondence
on $\widehat {\mathbb {C}}\times \widehat {\mathbb {C}}$ with graph $\Gamma _a$ for which we can use the pullback and pushforward operators above as well. Note that $d(\mathcal {F}_a)=d(\mathcal {F}_a^{1})=2$ and $\mathcal {F}_a^{1}=\operatorname {Cov^Q_0}\circ \operatorname {J}_a=\operatorname {J}_a\circ \mathcal {F}_a\circ \operatorname {J}_a$ , since $\operatorname {Cov^Q_0}^{1}=\operatorname {Cov^Q_0}$ and $\operatorname {J}_a^{1}=\operatorname {J}_a$ .
Set . Then $\phi _a^{1}\circ \mathcal {F}_a\circ \phi _a$ is a holomorphic correspondence on $\widehat {\mathbb {C}}\times \widehat {\mathbb {C}}$ that, restricted to $(\widehat {\mathbb {C}}\setminus \lbrace 1\rbrace )\times (\widehat {\mathbb {C}}\setminus \lbrace 1\rbrace )$ , induces the relation given by equation (1). Thus, $(z,w)\in (\widehat {\mathbb {C}}\setminus \lbrace 1\rbrace )\times (\widehat {\mathbb {C}}\setminus \lbrace 1\rbrace )$ satisfies equation (1) if and only if $w\in \mathcal {F}_a(z)$ [Reference Bullett and Lomonaco4, Lemma 3.1].
Observe that $\mathcal {F}_a^{1}(1)=\operatorname {Cov^Q_0}(\operatorname {J}_a(1))=\operatorname {Cov^Q_0}(1)=\lbrace 1,2\rbrace $ , independent of $a\in \mathbb {C}\setminus \lbrace 1\rbrace $ . In particular, $1\in \operatorname {Per}_1(\mathcal {F}_a)$ and we say that $1$ is a fixed point of $\mathcal {F}_a$ .
2.3 Critical values
In this section, we will discuss what parts of the graph of $\mathcal {F}_a$ are locally the graph of a holomorphic function, by defining and finding all critical values and ramification points of $\Gamma _a$ . This will be used in §4 to find the exceptional set.
Definition 2.3. Let $\Gamma $ be the graph of an irreducible holomorphic correspondence on $\widehat {\mathbb {C}}$ , and put
for $j=1,2$ and
We extend the definition to holomorphic 1chains $\Gamma =\sum _i n_i\Gamma (i)$ by
We call $A_2(\Gamma )$ the set of ramification points of the holomorphic correspondence associated to $\Gamma $ , and $B_2(\Gamma )$ the set of its critical values.
Note that $A_1(\Gamma )=\iota (A_2(\Gamma ^{1}))$ and $A_2(\Gamma )=\iota (A_1(\Gamma ^{1}))$ , where $\iota :\widehat {\mathbb {C}}\times \widehat {\mathbb {C}}\to \widehat {\mathbb {C}}\times \widehat {\mathbb {C}}$ is the involution $\iota (z,w)=(w,z)$ .
Suppose $\Gamma $ is the graph of a holomorphic correspondence on $\widehat {\mathbb {C}}$ . Let $g:\Omega \to \widehat {\mathbb {C}}$ be a holomorphic function defined on a domain $\Omega $ of $\widehat {\mathbb {C}}$ , whose graph $\operatorname {Gr}(g)$ is contained in one of the irreducible components $\Gamma (i)$ of $\Gamma $ . If $a\in \Omega $ is a critical point for g, then $(a,g(a))\in A_2(\Gamma (i))\subset A_2(\Gamma )$ and therefore $g(a)\in B_2(\Gamma )$ , i.e., the critical value of the function g is a critical value for the holomorphic correspondence associated to $\Gamma $ , as well.
On the other hand, if $\alpha \notin A_1(\Gamma )$ , then there exists a holomorphic function $g:\Omega \to \widehat {\mathbb {C}}$ defined on a neighborhood $\Omega $ of $a=\pi (\alpha )$ , such that $(a,g(a))=\alpha $ , and $\operatorname {Gr}(g)\subset \Gamma (i)$ for some i. If in addition $\alpha \in A_2(\Gamma )$ , then g is not locally injective at a, and therefore $g'(a)=0$ . Therefore, a is a critical point of g and $\pi _2(\alpha )=g(a)$ is a critical value of g.
If we denote by $\operatorname {CritPt}(g)$ the set of critical points of g, and by
the set of critical values of g, then we get a motivation for the name ‘critical values’ in Definition 2.3 by the containment
where the first union runs over the irreducible components of $\Gamma $ and the second union runs over all the holomorphic functions $g:\Omega \to \widehat {\mathbb {C}}$ whose graph $\operatorname {Gr}(g)$ is contained in $\Gamma (i)$ .
Proposition 2.4. For every $a\in \mathbb {C}\setminus \lbrace 1\rbrace $ , we have that
and
As a consequence, $B_1(\Gamma _a)=\lbrace \infty ,2,2\rbrace $ and $B_2(\Gamma _a)=\lbrace ({a+1})/{2},({4a+2})/({a+5}), {2}/({3a})\rbrace $ . Moreover, $\Gamma _a$ is smooth at all points except $(\infty ,({a+1})/{2})$ .
To prove this proposition, we first prove the following lemma.
Lemma 2.5. For each $a\in \mathbb {C}\setminus \lbrace 1\rbrace $ ,
and
Thus, $B_1(\Gamma _{\operatorname {Cov^Q_0}})=\lbrace \infty , 2,2\rbrace $ and $B_2(\Gamma _{\operatorname {Cov^Q_0}})=\lbrace \infty , 1,1\rbrace $ .
Proof. Differentiating the equation $P_Q(z,w)=0$ with respect to w, we get that $\partial _w P_Q(z,w)=z+2w$ vanishes if and only if $w={z}/{2}$ . Also, $P_Q(z,{z}/{2})=0$ if and only if $z=\pm 2$ . Therefore, ${dw}/{dz}$ exists on $\mathbb {C}\setminus \lbrace 2, 2\rbrace $ . Thus, by the implicit function theorem, for every $(z,w)\in \Gamma _{\operatorname {Cov^Q_0}}$ such that $z\in \mathbb {C}\setminus \lbrace 2,2\rbrace $ , there exists a domain $\Omega $ containing z and a holomorphic function $g:\Omega \to \widehat {\mathbb {C}}$ such that $g(z)=w$ and $\operatorname {Gr}(g)=\Gamma _{\operatorname {Cov^Q_0}}\cap U$ , for some open neighborhood U of $(z,w)$ . In addition, the function g will be locally injective at z if $\partial _z P_Q(z,w)=2z+w$ is nonzero. Therefore, if $(z,w)\in \Gamma _{\operatorname {Cov^Q_0}}$ satisfies that both z and w are different from $\pm 2$ , then $(z,w)\notin A_2(\Gamma _{\operatorname {Cov^Q_0}})$ .
On the other hand, observe that the only points $(z,w)\in \Gamma _{\operatorname {Cov^Q_0}}$ with $z=\pm 2$ are $(2,1)$ and $(2,1)$ . In particular, $w\neq \pm 2$ , and by the symmetry of the above argument, we can use the implicit function theorem to obtain a neighborhood U of w and a function $g:U\to \widehat {\mathbb {C}}$ satisfying $\operatorname {Gr}(g)\subset \Gamma _{\operatorname {Cov^Q_0}}$ and $g(w)=z$ , and such that is injective in the neighborhood $(g(U)\times U)\cap \Gamma _{\operatorname {Cov^Q_0}}$ of $(z,w)$ . This proves that neither $(2,1)$ or $(2,1)$ belong to $A_2(\Gamma _{\operatorname {Cov^Q_0}})$ . Since the only points $(z,w)\in \Gamma _{\operatorname {Cov^Q_0}}$ with $z=\pm 1$ are $(1,2)$ and $(1,2)$ , and since $\Gamma _{\operatorname {Cov^Q_0}}\setminus (\mathbb {C}\times \mathbb {C})=\lbrace (\infty ,\infty )\rbrace $ , we have that $A_2(\Gamma _{\operatorname {Cov^Q_0}})$ is contained in $\lbrace (\infty ,\infty ),(1,2),(1,2)\rbrace $ .
We will check that for every neighborhood W of $(\infty ,\infty )$ , $(1,2)$ , and $(1,2)$ , we have that is not injective. Let $W_1,W_2$ , and $W_3$ be open neighborhoods of $(\infty ,\infty )$ , $(1,2)$ , and $(1,2)$ , respectively. Then there exists $T>0$ such that for every $0<t<T$ ,
and
We conclude that $A_2(\Gamma _{\operatorname {Cov^Q_0}})=\lbrace (\infty ,\infty ),(1,2),(1,2)\rbrace $ , and by the symmetry of $\operatorname {Cov^Q_0}$ , $A_1(\Gamma _{\operatorname {Cov^Q_0}})=\lbrace (\infty ,\infty ),(2,1),(2,1)\rbrace $ . We conclude that
Proof of Proposition 2.4.
Since $\mathcal {F}_a=\operatorname {J}_a\circ \operatorname {Cov^Q_0}$ and $\operatorname {J}_a$ is an involution, we have that
for $j=1,2$ . Using Lemma 2.5, this gives us equations (3) and (4), and thus $B_1(\Gamma _a)=\lbrace \infty , 2,2\rbrace $ and
In addition, observe that locally, $\Gamma _a$ is either a function on z or on w for all $(z,w)\in \Gamma _a$ with $z\neq \infty $ , then the only point that can be irregular is $(\infty ,({a+1})/{2})$ . Indeed, this point is irregular, as the curve given by the points
selfintersects at $(\infty ,\infty )$ with an angle of ${2\pi }/{3}$ . In other words, there are two functions $z(w)$ which intersect with different derivatives, which makes $(\infty ,\infty )$ an irregular point of $\Gamma _{\operatorname {Cov^Q_0}}$ . Thus, passing through the involution $\operatorname {J}_a$ , we get that $(\infty ,({a+1})/{2})$ is an irregular point of $\Gamma _a$ .
Remark 2.6. The correspondence $\operatorname {Cov^Q_0}$ , and hence $\mathcal {F}_a$ , sends open sets to open sets. Indeed, let $U\subset \widehat {\mathbb {C}}$ be open and take $w_0\in \operatorname {Cov^Q_0}(U)$ . Then there exists $z_0\in U$ for which $(z_0,w_0)\in \Gamma _{\operatorname {Cov^Q_0}}$ . We will prove that $\operatorname {Cov^Q_0}(U)$ is open by showing that in all the cases, $w_0\in \operatorname {int}(\operatorname {Cov^Q_0}(U))$ .

• Suppose $(z_0,w_0)\notin A_1(\Gamma _{\operatorname {Cov^Q_0}})$ . By Lemma 2.5 and since $(\operatorname {Cov^Q_0})^{1}(\infty )=\lbrace \infty \rbrace $ , we have that $w_0\neq \infty $ . Moreover, there exists a holomorphic function $g:\Omega \to \mathbb {C}$ on an open subset $\Omega \subset U$ , and $(z_0,w_0)\in \operatorname {Gr}(g)\subset \Gamma _{\operatorname {Cov^Q_0}}$ . Furthermore, $\Gamma _{\operatorname {Cov^Q_0}}$ is irreducible and $\operatorname {Cov^Q_0}$ is not constant, so g is not constant. Thus, g is open and then $w_0\in g(\Omega )\subset \operatorname {int}(\operatorname {Cov^Q_0}(U))$ .

• Now suppose $(z_0,w_0)\notin A_2(\Gamma _{\operatorname {Cov^Q_0}})$ . By Lemma 2.5 and since $\operatorname {Cov^Q_0}(\infty )=\lbrace \infty \rbrace $ , then $z_0\neq \infty $ and there exists a holomorphic function $\tilde {g}:\widetilde {\Omega }\to \mathbb {C}$ on an open subset $\widetilde {\Omega }\subset \mathbb {C}$ so that $(z_0,w_0)\in \iota (\operatorname {Gr}(\tilde {g}))\subset \Gamma _{\operatorname {Cov^Q_0}}$ . Since $\tilde {g}$ is continuous, then $\tilde {g}^{1}(U)$ is open and $w_0\in \tilde {g}^{1}(U)\subset \operatorname {int}(\operatorname {Cov^Q_0}(U))$ .

• Finally, if $(z_0,w_0)\in A_1(\Gamma _{\operatorname {Cov^Q_0}})\cap A_2(\Gamma _{\operatorname {Cov^Q_0}})$ , then Lemma 2.5 implies that $(z_0,w_0)=(\infty ,\infty )$ . For each $r>0$ , put . To show that $\infty \in \operatorname {int}(\operatorname {Cov^Q_0}(U))$ , we will show that for every $R>\sqrt {3}$ , $U_{2R}\subset \operatorname {Cov^Q_0}(U_R)$ . We proceed by the contrapositive. If $P_Q(z,w)=0$ and $w\leq R$ , then
$$ \begin{align*} z^2=3zww^2\leq 3+zw+w^2\leq 2R^2+zR. \end{align*} $$Hence, $z^2Rz2R^2\leq 0$ and $z\leq 2R$ . This implies that for every $R>\sqrt {3}$ , we have that $\operatorname {Cov^Q_0}(U_{2R})\subset U_R$ . By the symmetry of $\operatorname {Cov^Q_0}$ ,$$ \begin{align*}U_{2R}\subset\operatorname{Cov^Q_0}(\operatorname{Cov^Q_0}(U_{2R}))\subset\operatorname{Cov^Q_0}(U_R).\end{align*} $$Moreover, if $R>\sqrt {3}$ is large enough so that $U_R\subset U$ , then$$ \begin{align*} \infty\in U_{2R}\subset \operatorname{int}(\operatorname{Cov^Q_0}(U)).\end{align*} $$
Therefore, $\operatorname {Cov^Q_0}$ sends open sets to open sets. Since the involution $\operatorname {J}_a$ also sends open sets to open sets, then so does $\mathcal {F}_a$ .
2.4 Klein combination pairs
In this section, we will define the set $\mathcal {K}$ of parameters we will consider for our family.
Definition 2.7. A fundamental domain for an irreducible holomorphic correspondence F is an open set $\Delta _F$ that is maximal with the property that $\Delta _F\cap F(\Delta _F)=\emptyset $ .
Definition 2.8. We say that a pair $(\Delta _{\operatorname {Cov^Q_0}},\Delta _{\operatorname {J}_a})$ of fundamental domains for $\operatorname {Cov^Q_0}$ and $\operatorname {J}_a$ , respectively, is a Klein combination pair for $\mathcal {F}_a$ if both $\Delta _{\operatorname {Cov^Q_0}}$ and $\Delta _{\operatorname {J}_a}$ are simply connected domains, bounded by Jordan curves, and satisfy
We define as well the Klein combination locus $\mathcal {K}$ to be the set of parameters $a\in \mathbb {C}\setminus \lbrace 1\rbrace $ for which there exist a Klein combination pair.
In [Reference Bullett and Lomonaco4], for $a4\leq 3$ , $a\neq 1$ , the authors found a Klein combination pair for $\mathcal {F}_a$ , where $\Delta _{\operatorname {Cov^Q_0}}$ is given by the right side of the curve
and $\Delta _{\operatorname {J}_a}$ is given by the exterior of the circle passing through $z=1$ and $z=a$ with diameter contained in the real line.
This pair $(\Delta _{\operatorname {Cov^Q_0}},\Delta _{\operatorname {J}_a})$ is composed by simply connected fundamental domains for $\operatorname {Cov^Q_0}$ and $\operatorname {J}_a$ , respectively, whose boundaries are Jordan curves, smooth except from $\partial \Delta _{\operatorname {Cov^Q_0}}$ at $(\infty ,\infty )$ (see Figure 1 and [Reference Bullett and Lomonaco4, Proposition 3.3]). In particular,
From now on, whenever $a\in \mathcal {K}$ , we denote by $(\Delta _{\operatorname {Cov^Q_0}},\Delta _{\operatorname {J}_a})$ a Klein combination pair for $\mathcal {F}_a$ .
The following remark will be useful to prove that $\mathcal {F}_a$ is not modular, and later to analyze the asymptotic behavior of $\mathcal {F}_a$ .
Remark 2.9. Let $a\in \mathcal {K}$ .

(1) Note that $\mathcal {F}_a(\widehat {\mathbb {C}}\setminus \Delta _{\operatorname {J}_a})\subset \widehat {\mathbb {C}}\setminus \Delta _{\operatorname {J}_a}$ and ${\mathcal {F}_a((\widehat {\mathbb {C}}\setminus \Delta _{\operatorname {J}_a})\setminus \lbrace 1\rbrace )\subset \widehat {\mathbb {C}}\setminus \overline {\Delta _{\operatorname {J}_a}}}$ . Indeed,
$$ \begin{align*}\mathcal{F}_a(1)=\operatorname{J}_a(\operatorname{Cov^Q_0} (1))=\operatorname{J}_a(\lbrace 1,2\rbrace)=\bigg\lbrace 1,\frac{4a+2}{a+5}\bigg\rbrace.\end{align*} $$From Remark 2.6, we have that $\operatorname {Cov^Q_0}$ sends open sets to open sets. In particular, $$ 2 cannot belong to $\Delta _{\operatorname {Cov^Q_0}}$ , as $1\notin \operatorname {Cov^Q_0}(\Delta _{\operatorname {Cov^Q_0}})=\widehat {\mathbb {C}}\setminus \overline {\Delta _{\operatorname {Cov^Q_0}}}$ . By the Klein combination pair condition, this implies that $2\in \Delta _{\operatorname {J}_a}$ , and thus $\operatorname {J}_a(2)=({4a+2})/({a+5})\notin \Delta _{\operatorname {J}_a}$ . Since we also have that $1\notin \Delta _{\operatorname {J}_a}$ , we have that
$$ \begin{align*}\mathcal{F}_a(1)\subset\widehat{\mathbb{C}}\setminus\Delta_{\operatorname{J}_a}.\end{align*} $$However, for $z\in \widehat {\mathbb {C}}\setminus \Delta _{\operatorname {J}_a}$ , $z\neq 1$ , we have that $z\in \Delta _{\operatorname {Cov^Q_0}}$ . Therefore,
$$ \begin{align*}\operatorname{Cov^Q_0}(z)\subset\widehat{\mathbb{C}}\setminus\overline{\Delta_{\operatorname{Cov^Q_0}}}\subset\Delta_{\operatorname{J}_a},\end{align*} $$and thus, since $\operatorname {J}_a$ is an involution fixing $\partial \Delta _{\operatorname {J}_a}$ ,
$$ \begin{align*}\mathcal{F}_a(z)=\operatorname{J}_a\circ\operatorname{Cov^Q_0}(z)\subset\operatorname{J}_a(\Delta_{\operatorname{J}_a})=\widehat{\mathbb{C}}\setminus\overline{\Delta_{\operatorname{J}_a}}.\end{align*} $$In particular, $\Delta _{\operatorname {J}_a}\setminus \mathcal {F}_a(\widehat {\mathbb {C}}\setminus \Delta _{\operatorname {J}_a})\neq \emptyset $ .

(2) Observe that $\mathcal {F}_a^{1}(\Delta _{\operatorname {J}_a})\subset \Delta _{\operatorname {J}_a}$ and $\mathcal {F}_a^{1}(\overline {\Delta _{\operatorname {J}_a}})\subset \Delta _{\operatorname {J}_a}\cup \lbrace 1\rbrace $ . Indeed, for every $z\in \Delta _{\operatorname {J}_a}$ , we have that $\operatorname {J}_a(z)\in \widehat {\mathbb {C}}\setminus \overline {\Delta _{\operatorname {J}_a}}\subset \Delta _{\operatorname {Cov^Q_0}}$ . From Remark 2.6 and since $(\Delta_{\operatorname {Cov^Q_0}},\Delta _{\operatorname {J}_a})$ is a Klein combination pair, then
$$ \begin{align*}\mathcal{F}_a^{1}(z)=\operatorname{Cov^Q_0}(\widehat{\mathbb{C}}\setminus\overline{\Delta_{\operatorname{J}_a}})\subset\operatorname{Cov^Q_0}(\Delta_{\operatorname{Cov^Q_0}})\subset\widehat{\mathbb{C}}\setminus\overline{\Delta_{\operatorname{Cov^Q_0}}}\subset\Delta_{\operatorname{J}_a}.\end{align*} $$Furthermore, for $w\in \overline {\Delta _{\operatorname {J}_a}}$ , we have that
3 Clasification of the family $\lbrace \mathcal {F}_a\rbrace _a$
In this section, we prove that $\mathcal {F}_a$ is weakly modular but not modular, for every $a\in \mathcal {K}$ . In addition, we prove that it does not satisfy the required conditions for the equidistribution result [Reference Bharali and Sridharan1, Theorem 3.5]. All together proving Theorem 1.1.
3.1 Modularity and weak modularity
Definition 3.1. Let G be a connected Lie group, $\Lambda $ a torsion free lattice, and K a compact Lie subgroup. Let $g\in G$ be such that $g\Lambda g^{1}\cap \Lambda $ has finite index in $\Lambda $ . The irreducible modular correspondence induced by g is the multivalued map $F_g$ on $X=\Lambda \setminus G/K$ corresponding to the projection to X of the map $x\mapsto (x,gx)$ on $G\to G\times G$ . Denote by $\Gamma _g$ the graph of $F_g$ . A modular correspondence F is a correspondence whose graph is of the form $\sum _jn_j\Gamma _{g_j}$ , for $\Gamma _{g_j}$ as before.
The following definition was introduced by Dinh, Kaufmann, and Wu in [Reference Dinh, Kaufmann and Wu15].
Definition 3.2. Let X be a compact Riemann surface and let F be a holomorphic correspondence on X with graph $\Gamma $ such that $d(F)=d(F^{1})$ . We say that F is a weakly modular correspondence if there exist Borel probability measures $\mu _1$ and $\mu _2$ on X, such that
Remark 3.3

(1) Let F be a modular correspondence that is also a holomorphic correspondence. Then it is always the case that
$$ \begin{align*} d(F)=\sum_j n_j[\Lambda: g_j\Lambda g_j^{1}\cap\Lambda]=\sum_j n_j[\Lambda:g_j^{1}\Lambda g_j\cap\Lambda]=d(F^{1}). \end{align*} $$ 
(2) With the notation in Definition 3.1, let $\unicode{x3bb} $ be the direct image on X of the finite Haar measure on $\Lambda \setminus G$ . Then, $(1/d){(F_g)}F_g^*\unicode{x3bb} =\unicode{x3bb} $ and if we put $\mu _1=\mu _2=\unicode{x3bb} $ , we get that
Therefore, modular correspondences are weakly modular.

(3) The measure $\unicode{x3bb} $ above is Borel, invariant under $F_g$ , and assigns positive measure to nonempty open sets.
Proof of Theorem 1.1 part 1.
Observe that the graph $\Gamma _{\operatorname {Cov^Q_0}}$ of the correspondence $\operatorname {Cov^Q_0}$ is symmetric with respect to the diagonal $\mathfrak {D}_{\widehat {\mathbb {C}}}=\lbrace (z,z)  z\in \widehat {\mathbb {C}}\rbrace $ , as $z\in \operatorname {Cov^Q_0}(w)$ if and only if $w\in \operatorname {Cov^Q_0}(z)$ . Let m be any positive and finite measure on $\Gamma _{\operatorname {Cov^Q_0}}$ , and $\iota :\Gamma _{\operatorname {Cov^Q_0}}\to \Gamma _{\operatorname {Cov^Q_0}}$ the involution
. Take
The measure $m_0$ is symmetric in $\Gamma _{\operatorname {Cov^Q_0}}$ in the sense that $\iota ^*m'=m'$ and moreover
After normalizing if necessary, this proves that $\operatorname {Cov^Q_0}$ is weakly modular with measures
and
. Our goal is to show there are probability measures $\mu _1$ and $\mu _2$ on $\widehat {\mathbb {C}}$ such that
where $\Gamma _a$ is the graph of the correspondence $\mathcal {F}_a=\mathcal {F}_a\circ \operatorname {Cov^Q_0}$ .
Put and observe that by symmetry of $\operatorname {J}_a$ , we have that
Now let $\widehat {\operatorname {J}_a}:\widehat {\mathbb {C}}\times \widehat {\mathbb {C}}\to \widehat {\mathbb {C}}\times \widehat {\mathbb {C}}$ be given by . Observe that whenever $(z,w)\in \Gamma _{\operatorname {Cov^Q_0}}$ , then $(z,\operatorname {J}_a(w))\in \Gamma _a$ and
On the other hand, with multiplicity $2$ . This, together with equations (5), (7), and (8) yield
Observe that and $\mu _2$ both have $2$ times the mass of $\mu _1$ and $\mu _2$ . After normalizing, equation (9) proves that $\mathcal {F}_a$ is weakly modular, as desired.
To check $\mathcal {F}_a$ is not modular, we will prove that no Borel measure $\unicode{x3bb} $ on $\widehat {\mathbb {C}}$ that gives positive measure to nonempty open sets can be invariant under $\mathcal {F}_a$ . Suppose by contradiction that $\unicode{x3bb} $ is such a measure satisfying $\tfrac 12\mathcal {F}_a^*\unicode{x3bb} =\unicode{x3bb} $ . In particular, we have that $\unicode{x3bb} ((\widehat {\mathbb {C}}\setminus \Delta _{\operatorname {J}_a})\setminus \mathcal {F}_a(\widehat {\mathbb {C}}\setminus \Delta _{J}))=0$ . On the other hand, note that
is closed in $\widehat {\mathbb {C}}$ . By Remark 2.9 part (1), $(\widehat {\mathbb {C}}\setminus \overline {\Delta _{\operatorname {J}_a}})\setminus \mathcal {F}_a(\widehat {\mathbb {C}}\setminus \Delta _{\operatorname {J}_a})$ is open and nonempty, contained in $(\widehat {\mathbb {C}}\setminus \Delta _{\operatorname {J}_a})\setminus \mathcal {F}_a(\widehat {\mathbb {C}}\setminus \Delta _{J})$ . This contradicts part $(3)$ of Remark 3.3, as $\unicode{x3bb} $ cannot assign 0 measure to open sets.
Remark 3.4. For a $(d,d)$ holomorphic correspondence F on compact Riemann surface X, the operator $({1}/{d})F^*$ acts on the space $L^2_{(1,0)}$ of $(1,0)$ forms with $L^2$ coefficients. In [Reference Dinh, Kaufmann and Wu15], the authors showed that the operator norm satisfies $\Vert ({1}/{d})F^*\Vert \leq 1$ , with strict inequality for non weakly modular correspondences. This strict inequality is a key factor of their equidistribution result. However, this is never the case for $F=\mathcal {F}_a$ , $a\neq 1$ . We claim that
and furthermore that the supremum is attained. To prove this, we use that $\({1}/{d})F^*\phi \_{L^2}=\\phi \_{L^2}$ for $\phi \in L^2_{(1,0)}$ if and only if for every $U\subset X\setminus B_1(\Gamma )$ and for every pair of local branches $f_1$ and $f_2$ of F on U, the equality $f_1^*\phi =f_2^*\phi $ holds on U (see [Reference Dinh, Kaufmann and Wu15, Proposition 2.1]).
Observe that the form $\phi (z)=e^{Q(z)}\,dz$ belongs to $L^2_{(1,0)}$ for Q as in §2.2. Let $U\subset \widehat {\mathbb {C}}\setminus B_1(\Gamma _{\operatorname {Cov^Q_0}})$ . Then the deleted covering correspondence $\operatorname {Cov^Q_0}$ sends z to the values w for which $({Q(z)Q(w)})/({zw})=0$ . Hence, any two local branches $f_1$ and $f_2$ of $\operatorname {Cov^Q_0}$ satisfy $f_1^*\phi (z)=Q(z)=f_2^*\phi (z)$ , and thus $\Vert \tfrac 12\operatorname {Cov^Q_0}^*\!\Vert =1$ . Now note that $\operatorname {J}_a$ is an involution, and hence $\operatorname {J}_a^*$ has operator norm $1$ . Thus we can conclude that $\Vert \tfrac 12\mathcal {F}_a^*\Vert =1$ as well, with supremum attained at $\phi $ .
3.2 Limit sets
In this section, we define limit sets and give some properties. We will also prove that the application listed in [Reference Bharali and Sridharan1, §7] does not hold for our family of correspondences, and hence this is a new case to study equidistribution.
Remark 3.5. From Proposition 2.4, observe that $1\notin B_1(\Gamma _a)$ , and hence there is a holomorphic function g whose graph contains $(1,1)$ and is contained in $\Gamma _a$ . After the change of coordinates $\psi (z)=z1$ , the function g has Taylor series expansion
whenever $a\neq 7$ , and
for $a=7$ (see [Reference Bullett and Lomonaco4, Proposition 3.5]). In particular, $g^{\psi }$ has a parabolic fixed point at $0$ with multiplier $1$ .
In [Reference Bullett and Lomonaco4, Proposition 3.8], the authors showed that for each $a\in \mathcal {K}$ , after a small perturbation of $\partial \Delta _{\operatorname {Cov^Q_0}}$ and $\partial \Delta _{\operatorname {J}_a}$ around $z=1$ , we can choose a Klein combination pair $(\Delta ^{\prime }_{\operatorname {Cov^Q_0}},\Delta ^{\prime }_{\operatorname {J}_a})$ so that $\partial \Delta ^{\prime }_{\operatorname {Cov^Q_0}}$ and $\partial \Delta ^{\prime }_{\operatorname {J}_a}$ are both smooth at $1$ , and transverse at $1$ to the line generated by the repelling direction at $z=1$ for $a\neq 7$ , and to the real axis in the case $a=7$ .
For $a\in \mathcal {K}$ and $(\Delta _{\operatorname {Cov^Q_0}},\Delta _{\operatorname {J}_a})$ as above, we define
to be the forward and backward limit set of $\mathcal {F}_a$ , respectively. These sets do not depend on the choice of the Klein combination pair $(\Delta ^{\prime }_{\operatorname {Cov^Q_0}},\Delta ^{\prime }_{\operatorname {J}_a})$ as above.
Lemma 3.6. Let $a\in \mathcal {K}$ . We have that:

(1) $\operatorname {J}_a(\Lambda _{a,\pm })=\Lambda _{a,\mp }$ and $\operatorname {J}_a(\partial \Lambda _{a,\pm })=\partial \Lambda _{a,\mp }$ ;

(2) $\Lambda _{a,}\cap \Lambda _{a,+}=\lbrace 1\rbrace ;$

(3) if $z\notin \Lambda _{a,}$ , then there exists $n\geq 1$ so that $\mathcal {F}_a^{n}(z)\subset \widehat {\mathbb {C}}\setminus \Delta ^{\prime }_{\operatorname {J}_a}$ , and if $z\notin \Lambda _{a,+}$ , then there exists $n\geq 1$ such that $\mathcal {F}_a^{n}(z)\subset \Delta ^{\prime }_{\operatorname {J}_a}$ ;

(4) $\mathcal {F}_a^{1}(\Lambda _{a,})=\Lambda _{a,}$ and $\mathcal {F}_a^{1}(\partial \Lambda _{a,})=\partial \Lambda _{a,}$ ; and

(5) $\mathcal {F}_a(\Lambda _{a,+})=\Lambda _{a,+}$ and $\mathcal {F}_a(\partial \Lambda _{a,+})=\partial \Lambda _{a,+}$ .
Proof. Since $\operatorname {J}_a$ is an involution sending $\overline {\Delta ^{\prime }_{\operatorname {J}_a}}$ and $\widehat {\mathbb {C}}\setminus \Delta ^{\prime }_{\operatorname {J}_a}$ to each other, and since $\operatorname {J}_a\circ \mathcal {F}_a^n=\mathcal {F}_a^{n}\circ \operatorname {J}_a$ , then
and applying $\operatorname {J}_a$ to both sides, we also get $\operatorname {J}_a(\Lambda _{a,})=\Lambda _{a,+}$ . Moreover, since $\operatorname {J}_a$ is continuous, note that
Since $(\Delta ^{\prime }_{\operatorname {Cov^Q_0}},\Delta ^{\prime }_{\operatorname {J}_a})$ is a Klein combination pair, $\widehat {\mathbb {C}}\setminus \Delta ^{\prime }_{\operatorname {J}_a}\subset \Delta ^{\prime }_{\operatorname {Cov^Q_0}}$ , so $\partial \Delta ^{\prime }_{\operatorname {J}_a}\subset \overline {\Delta ^{\prime }_{\operatorname {Cov^Q_0}}}$ .
If $z\in \Delta ^{\prime }_{\operatorname {Cov^Q_0}}$ , then $\operatorname {Cov^Q_0}(z)\subset \widehat {\mathbb {C}}\setminus \Delta ^{\prime }_{\operatorname {J}_a}$ , and $\mathcal {F}_a(z)=\operatorname {J}_a\circ \operatorname {Cov^Q_0}(z)\subset \Delta ^{\prime }_{\operatorname {J}_a}$ . This is a contradiction, as $\mathcal {F}_a(z)$ must belong to $\Lambda _{a,+}\subset \widehat {\mathbb {C}}\setminus \Delta ^{\prime }_{\operatorname {J}_a}$ as well. Again, since $(\Delta ^{\prime }_{\operatorname {Cov^Q_0}},\Delta ^{\prime }_{\operatorname {J}_a})$ is a Klein combination pair, we have that $z\in \partial \Delta ^{\prime }_{\operatorname {Cov^Q_0}}\cap \partial \Delta ^{\prime }_{\operatorname {J}_a}=\lbrace 1\rbrace $ and we conclude that
We prove part $(3)$ by the contrapositive. Suppose that for all n, there exists $w\in \mathcal {F}_a^{n}(z)\cap (\widehat {\mathbb {C}}\setminus \Delta ^{\prime }_{\operatorname {J}_a})$ , then $z\in \mathcal {F}_a^{n}(w)\subset \mathcal {F}_a^{n}(\widehat {\mathbb {C}}\setminus \Delta ^{\prime }_{\operatorname {J}_a})$ for all n. This implies that z must belong to $\Lambda _{a,}$ . The other case is analogous.
It is immediate from the definition of $\Lambda _{a,}$ and from part $(2)$ of Remark 2.9 that
From this and Remark 2.6,
Observe that if $z\in \partial \Lambda _{a,}\setminus \lbrace 1\rbrace \subset \Delta ^{\prime }_{\operatorname {Cov^Q_0}}$ , then
Since $\Lambda _{a,}\subset \overline {\Delta ^{\prime }_{\operatorname {J}_a}}$ , then we conclude that
Put $w\in \partial \Lambda _{a,}$ and $z\in \mathcal {F}_a^{1}(w)\subset \Lambda _{a,}$ . We will show that $z\in \partial \Lambda _{a,}$ by the contrapositive. Suppose $w\neq 1$ and $z\in \operatorname {int}(\Lambda _{a,})$ . Then equation (12) implies that
From Remark 2.6, and the fact that $\Delta ^{\prime }_{\operatorname {J}_a}$ is open, we have that $\mathcal {F}_a(\operatorname {int}(\Lambda _{a,}))\cap \Delta ^{\prime }_{\operatorname {J}_a}$ is open. Moreover, for each $z'\in \operatorname {int}(\Lambda _{a,})$ , the set $\mathcal {F}_a(z')$ consists of a point in $\Delta ^{\prime }_{\operatorname {J}_a}$ and one in ${\widehat {\mathbb {C}}\setminus \Delta ^{\prime }_{\operatorname {J}_a}}$ . Since the one in $\Delta ^{\prime }_{\operatorname {J}_a}$ is actually in $\Lambda _{a,}$ by definition of $\Lambda _{a,}$ , then $\mathcal {F}_a(\operatorname {int}(\Lambda _{a,}))\cap \Delta ^{\prime }_{\operatorname {J}_a}\subset \Lambda _{a,}$ . Then $w\in \operatorname {int}(\Lambda _{a,})$ . This proves that
As for $w=1$ , we have that $z\in \mathcal {F}_a^{1}(1)=\lbrace 2,1\rbrace $ . We know $1\in \partial \Lambda _{a,}$ , so it suffices to show that $2\in \partial \Lambda _{a,}$ as well. Let $U\subset \widehat {\mathbb {C}}$ be an open neighborhood of $2$ . From Remark 2.6, we have that $\mathcal {F}_a(U)$ is an open neighborhood of $1$ . Since $1\in \partial \Delta ^{\prime }_{\operatorname {J}_a}$ , and $\partial \Delta ^{\prime }_{\operatorname {J}_a}$ is a Jordan curve, then there exists a point $w'\in U\cap \partial \Delta ^{\prime }_{\operatorname {J}_a}$ , $w'\neq 1$ . We have that $\mathcal {F}_a(\partial \Delta ^{\prime }_{\operatorname {J}_a}\setminus \lbrace 1\rbrace )\subset \widehat {\mathbb {C}}\setminus \overline {\Delta ^{\prime }_{\operatorname {Cov^Q_0}}}$ , so $\mathcal {F}_a^{1}(w')$ consists of two points, $z',z"\in \widehat {\mathbb {C}}\setminus \overline {\Delta ^{\prime }_{\operatorname {Cov^Q_0}}}$ , with $z'\in U$ . We will prove that $z'\notin \Lambda _{a,}$ by showing that $\mathcal {F}_a(z')\cap \Lambda _{a,}=\emptyset $ , and thus $2\in \Lambda _{a,}\setminus \operatorname {int}(\Lambda _{a,})=\partial \Lambda _{a,}$ . Indeed,
From part $(2)$ of Remark 2.9, we have that $z'\notin \Lambda _{a,}$ , and $2\in \partial \Lambda _{a,}$ as desired. This, together with equations (10), (11), and (13) prove part $(4)$ .
We have that part $(1)$ of Remark 2.9 together with the definition of $\Lambda _{a,+}$ prove that $\mathcal {F}_a(\Lambda _{a,+})=\Lambda _{a,+}$ . To prove the rest of part $(5)$ , we use part $(4)$ together with the fact that $\mathcal {F}_a=\operatorname {J}_a\circ \mathcal {F}_a^{1}\circ \operatorname {J}_a$ and $\operatorname {J}_a(\partial \Lambda _{a,\pm })=\partial \Lambda _{a,\mp }$ , as $\operatorname {J}_a$ is a continuous involution. Thus,
The following definition is from [Reference Bharali and Sridharan1, Reference McGehee26].
Definition 3.7. Let F be a holomorphic correspondence on X. We say that $\mathcal {R}\subset X$ is a repeller for F if there exists a set U such that $\mathcal {R}$ is contained in the interior of U, and
where
Bharali and Sridharan proved an equidistribution result similar to that in this paper [Reference Bharali and Sridharan1, Theorem 3.5] for correspondences having a repeller. Moreover, in [Reference Bharali and Sridharan1, §7.2], they showed that there is a set of pairs $(a,k)$ for which their result can be applied to the correspondence
For these correspondences (studied by Bullett and Harvey in [Reference Bullett and Harvey8]), there is a set $\Lambda _{a,}$ analogous to that presented here (see [Reference Bullett and Penrose7] for the general definition of limit sets). For the pair $(a,k)$ to work for their theorem, it is crucial for $\partial \Lambda _{a,}$ to be a repeller for the correspondence. Nevertheless, part $(2)$ of Theorem 1.1 says this never happens for $k=1$ and $a4\leq 3$ .
Proof of Theorem 1.1 part (2).
Let $U\subset \widehat {\mathbb {C}}$ contain $\partial \Lambda _{a,}$ in its interior. We have that $1\in \partial \Lambda _{a,}$ and is a parabolic fixed point of the function g whose graph is contained in $\Gamma _a$ , described in Remark 3.5. Take an attracting petal $\mathcal {P}$ at $1$ so that $\mathcal {P}\subset U$ . We first show that every $K\in \mathfrak {K}(U,\mathcal {F}_a^{1})$ contains $\partial \Lambda _{a,}\cup \mathcal {P}$ . Indeed, for every $K\in \mathfrak {K}(U,\mathcal {F}_a^{1})$ and some integer $n\geq 0$ ,
Moreover, from part (4) of Lemma 3.6, we have that
Putting equations (14) and (15) together, we get that every $K\in \mathfrak {K}(U,\mathcal {F}_a^{1})$ contains the union $\partial \Lambda _{a,}\cup \mathcal {P}$ , and therefore so does the intersection over all K. Since $\operatorname {int}(\partial \Lambda _{a,})=\emptyset $ and $\operatorname {int}(\mathcal {P})\neq \emptyset $ , we have that
Since U is arbitrary, we conclude that $\partial \Lambda _{a,}$ is not a repeller for $\mathcal {F}_a$ .
4 Exceptional set and periodic points
In this section, we will define a twosided restriction of $\mathcal {F}_a$ and prove it is a proper holomorphic map of degree 2. We will find its exceptional set and that of $\mathcal {F}_a$ . This will be important for the next section, as it is the set of all points that may escape from the equidistribution property given in Theorem 1.2.
The following definition is classical.
Definition 4.1. Let $f:U\to V$ be a holomorphic proper map, with $U,V$ open, $U\subset V$ . For $z\in U$ , we denote by $[ z]$ the equivalence class of z by the equivalence relation
We say that z is exceptional for f if $[ z]$ is finite, and we call exceptional set the set $\mathcal {E}$ of all points that are exceptional for f.
Put $a\in \mathcal {K}$ and denote by $f_a$ the twosided restriction , meaning $f_a$ sends each $z\in \mathcal {F}_a^{1}(\Delta ^{\prime }_{\operatorname {J}_a})$ to the unique point in $\mathcal {F}_a(z)\cap \Delta ^{\prime }_{\operatorname {J}_a}$ . This it is a singlevalued, continuous, and holomorphic 2to1 map (see [Reference Bullett and Lomonaco4, Proposition 3.4] and Theorem 4.2 below) that extends on a neighborhood of every point $z\in \partial \mathcal {F}_a^{1}(\Delta _{\operatorname {J}_a})\setminus \lbrace 2\rbrace $ . In particular, $f_a$ extends around $z=1$ and $f_a(1)=1$ . Since $\Delta ^{\prime }_{\operatorname {J}_a}$ is open and $f_a$ is continuous, then $\mathcal {F}_a^{1}(\Delta ^{\prime }_{\operatorname {J}_a})(=f_a^{1}(\Delta ^{\prime }_{\operatorname {J}_a}))$ is open as well. Note that the analogous of part (2) of Remark 2.9 holds for $\Delta ^{\prime }_{\operatorname {J}_a}$ and $\Delta ^{\prime }_{\operatorname {Cov^Q_0}}$ instead, using the definition of Klein pair and the fact that $\operatorname {J}_a$ is open. Thus, $\mathcal {F}_a^{1}(\Delta ^{\prime }_{\operatorname {J}_a})\subset \Delta ^{\prime }_{\operatorname {J}_a}$ .
The following theorem is the main result of this section.
Theorem 4.2. For each $a\in \mathcal {K}$ , we have the following.

(1) The twosided restriction $f_a:\mathcal {F}_a^{1}(\Delta ^{\prime }_{\operatorname {J}_a})\to \Delta ^{\prime }_{\operatorname {J}_a}$ of $\mathcal {F}_a$ is holomorphic and proper, of degree $2$ .

(2) The map $f_a$ has a critical point if and only if $2\in \widehat {\mathbb {C}}\setminus \overline {\Delta ^{\prime }_{\operatorname {J}_a}}$ . Furthermore, in that case, we have that the critical point is $$ 1.

(3) The exceptional set $\mathcal {E}_{a,}$ of $f_a$ is nonempty if and only if $a=5$ . In that case, ${\mathcal {E}_{a,}=\lbrace 1\rbrace }$ .
Computing images and preimages under $\mathcal {F}_a$ , we see that when $a=5$ , we have that $\circlearrowright 1\mapsto 2\circlearrowleft $ . In the following section, it will be useful to use the full orbit of $\mathcal {E}_{a,}$ under $\mathcal {F}_a$ , meaning
Proof of Theorem 4.2.
We first show part (1). Observe that $\infty $ and $\operatorname {J}_a(\infty )$ lie on opposite sides of