Critical points of the multiplier map for the quadratic family
Anna Belova Igors Gorbovickis




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Ghostly Galaxies as Solitons of Bose-Einstein Dark Matter 
    S.-H. Henry Tye 
    
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The multiplier λ_n of a periodic orbit of period n can be viewed as a (multiple-valued) algebraic function on the space of all complex quadratic polynomials p_c(z)=z^2+c. 
We provide a numerical algorithm for computing critical points of this function (i.e., points where the derivative of the multiplier with respect to the complex parameter c vanishes). We use this algorithm to compute critical points of λ_n up to period n=10.






§ INTRODUCTION



It has been known since the works of Fatou and Julia that multipliers of periodic orbits can carry not only local, but also global information about the holomorphic dynamical system at hand.
In <cit.> J. Milnor used the multipliers of the fixed points to parameterize the moduli space of degree 2 rational maps. Using this parameterization he proved that this moduli space is isomorphic to ℂ^2. 
In the attempt to generalize this approach, it was observed by the second author <cit.> that the multipliers of any m-1 distinct periodic orbits provide a local parameterization of the moduli space of degree m polynomials in a neighborhood of its generic point. It is then a natural question to describe the set of polynomials at which this local parameterization fails, 
that is, to describe the set of all critical points of the multiplier map, defined as the map which assigns to each degree m polynomial the (m-1)-tuple of multipliers at the chosen periodic orbits.
The goal of the current paper is to collect numerical data for this problem in the most basic case m=2, i.e., the case of the quadratic family 

    p_c(z)=z^2+c.

Even in this case the general problem seems to be quite complicated.

In the current paper we provide a numerical algorithm, that computes critical points of the multiplier map on the space of quadratic polynomials p_c. More specifically, given n∈ℕ, the algorithm finds the values of the parameter c, for which the map p_c has a periodic orbit of period n, whose multiplier, viewed as a locally analytic function of c, has a vanishing derivative. Using this algorithm, we compute critical points of the multiplier map together with the corresponding periodic orbits, for periods up to n=10. In particular, we find a complete list of all critical points of the multiplier map, for periods up to n=8.

Last but not least, let us mention another important motivation for the current study – the connection between the critical points of the multiplier map and the hyperbolic components of the famous Mandelbrot set. The argument of quasiconformal surgery implies that appropriate inverse branches of the multiplier map are Riemann mappings[A Riemann mapping of a simply connected domain is a conformal diffeomorphism of the unit disk onto that domain.] of the hyperbolic components <cit.>. Possible existence of analytic extensions of these Riemann mappings to larger domains might allow to estimate the geometry of the hyperbolic components <cit.> <cit.>, which in turn, might shed light on one of the central questions in one-dimensional holomorphic dynamics, the question whether the Mandelbrot set is locally connected. Critical values of the multiplier map are the only obstructions for the above mentioned analytic extensions to exist.






§ NOTATION AND TERMINOLOGY


Let p_c(z) = z^2 + c and denote its n-th iteration by p^∘ n_c(z).

A point z is a periodic point of p_c, if there exists a positive integer n, such that p^∘ n_c(z)=z. The minimal such n is called the period of z.



Given n, let the period n curve Per_n ⊂ℂ×ℂ be the closure of the locus of points (c,z) such that z is a periodic point of p_c of period n (see  <cit.> for more details).
Observe that each pair (c,z)∈Per_n determines a periodic orbit

    z=z_0↦ z_1 ↦⋯↦ z_n=z_0.



Let ℤ_n denote the cyclic group of order n. This group acts on Per_n by cyclicly permuting points of the same periodic orbits for each fixed value of c. 
Then the factor space Per_n/ℤ_n consists of pairs (c, 𝒪) such that 𝒪 is a periodic orbit of p_c.
Note that according to <cit.>, the space Per_n/ℤ_n (as well as Per_n) has a structure of a smooth algebraic curve.
(Note that there is a natural projection from Per_n to Per_n/ℤ_n.)

Let λ̃_n: Per_n →ℂ be the map defined by

    λ̃_n (c,z)↦∂ p^∘ n_c/∂ z(z) = 2^n z_1 ⋯ z_n.

Observe that for all regular points of the projection (c,z)↦ c, the value λ̃_n(c,z) is the multiplier of the periodic point z. Furthermore, if z_1 and z_2 belong to the same periodic orbit of p_c, then λ̃_n(c,z_1)=λ̃_n(c,z_2), hence the map λ̃_n projects to a well defined map λ_nPer_n/ℤ_n→ℂ that assigns to each pair (c,𝒪) the multiplier of the periodic orbit 𝒪.

Both λ_n and λ̃_n are proper algebraic maps (c.f. <cit.>). 
The goal of this work is to study (compute) critical points of the multiplier map λ_n.





§ ALGORITHM FOR COMPUTING CRITICAL POINTS OF THE MULTIPLIER MAP




 §.§ Computing derivatives


Observe that all points (c,z)∈Per_n satisfy the following equation

    p^∘ n_c (z) = z.

Together with the Implicit Function Theorem this implies that the parameter c can serve as a local chart on Per_n at all points (c,z)∈Per_n, such that λ̃_n(c,z)≠ 1. Hence, in a neighborhood of any such point, one can implicitly define a map z(c), so that (c,z(c))∈Per_n for all nearby values of c. Then one can express the multiplier map λ̃_n in the above local chart as

    λ̃_n(c)=λ̃_n(c,z(c)).


According to Lemma 4.5 in <cit.>, if λ̃_n(c,z)=1, then (c,z) cannot be a critical point of the multiplier map λ̃_n. Thus, in order to study all critical points of this map, it is sufficient to work in local charts associated with the parameter c (i.e. the critical points of the multiplier map λ̃_n correspond to those points (c,z)∈Per_n, in a neighborhood of which the map λ̃_n(c)=λ̃_n(c,z(c)) is defined and λ̃_n'(c)=0).

We will use the following notation for the partial derivatives


    ∂ p^∘ n_c/∂ c := ∂ f_n/∂ c,

where f_n(c,z) = p^∘ n_c(z). 
By differentiating both left and right sides of  (<ref>) we get

    dz/dc=∂ p^∘ n_c/∂ c(z)+∂ p^∘ n_c/∂ z(z)dz/dc.

Therefore

    z'=dz/dc = ∂ p^∘ n_c/∂ c(z)(1-∂ p^∘ n_c/∂ z(z))^-1 = ∂ p^∘ n_c/∂ c(z)(1-λ_n(c))^-1.

Observe that ∂ p^∘ n_c/∂ c(z) satisfies the recurrence relation 

    ∂ p^∘ n_c/∂ c(z) 
    =
    ∂/∂ cp_c(p^∘ n-1_c(z))
    =
    1+2 · p^∘ n-1_c(z)∂ p^∘ n-1_c/∂ c(z).

We therefore get an expression for the derivative of the multiplier map

    dλ_n/dc   = dλ_n(c)/dc
       = 
    2^n[ z' · p_c(z) · p^∘ 2_c(z)⋯ p^∘ n-1_c(z) 
    . 
           . + z ·(∂ p_c/∂ c(z) + z' ∂ p_c/∂ z(z) )· p^∘ 2_c(z)⋯ p^∘ n-1_c(z)
    . 
                       . + ⋯ + z· p_c(z)⋯ p^∘ n-2_c(z)·(∂ p^∘ n-1_c/∂ c(z) + z' ∂ p^∘ n-1_c/∂ z(z))
    ] 
       =
    2^n[z'∏_i=1^n-1p^∘ i_c(z) + z ∑_i=1^n-1(dp^∘ i_c/dc∏_j=1
     j≠ i^n-1p^∘ j_c(z))],

where for i=1,…, n-1, we denote

    dp^∘ i_c/dc = ∂ p^∘ i_c/∂ c + z' ∂ p^∘ i_c/∂ z.

Finally, in order to find the critical points of the multiplier map λ̃_n, we combine (<ref>), (<ref>) and (<ref>) into the following system of three algebraic equations

    {[                                 p^∘ n_c(z)-z                                            =                                            0; z' - ∂ p^∘ n_c/∂ c(z)(1-∂ p^∘ n_c/∂ z(z))^-1                                            =                                            0;                                      dλ_n/dc                                            =                                           0, ].

with three unknowns c, z, z'. Any critical point of the multiplier map λ̃_n corresponds to a solution of the above system, thus, the problem of finding critical points of the map λ̃_n can be reduced to the problem of solving the above system.



 §.§ The number of critical points of the multiplier map λ_n


A question that naturally arises using the numerical methods, as for example Newton method, is how to make sure that all solutions are found. In this section we derive an upper bound for the number of critical points of the multiplier map which will be later used in the numerical algorithm. 

Let ν(n) be the number of periodic points of p_c of period n for a generic value of c. One can observe that the numbers ν(n) satisfy the recursive relation

    ν(1)=2    and    ν(n)=2^n-∑_m divides n,
     m≠ nν(m).

In particular, this implies that ν(n)∼ 2^n as n→∞.

It was shown in  <cit.> that c can be used as a local uniformizing parameter at any non-parabolic point of Per_n/ℤ_n (i.e., where λ_n(c,𝒪) ≠ 1). Moreover, the projection map π_n:Per_n/ℤ_n →ℂ, defined as

    π_n (c, 𝒪)↦ c,

is proper of degree deg π_n = ν(n)/n. Conversely, in a neighbourhood of a point (c,𝒪) with λ_n(c,𝒪)=1, the multiplier λ_n serves as a local uniformizing parameter for the curve Per_n/ℤ_n and λ_n: Per_n/ℤ_n →ℂ is a proper map of degree deg λ_n = ν(n)/2.

In order to derive the number of critical points of λ_n recall the Riemann-Hurwitz formula (e.g. <cit.>). Let X, Y be Riemann surfaces, where X is connected with finite-dimensional homology, and χ(X), χ(Y) denote the corresponding Euler characteristics. Suppose f:Y→ X is a proper analytic map of degree deg  f with finitely many critical values. Then f has finitely many critical points and 

    χ(Y) = deg  f ·χ(X) - ∑_y∈ Y(deg_y f-1),

where deg_y f ≥ 1 is called a ramification index (or a local degree) of f at y and is defined in the following way. There exists an open neighbourhood U⊂ Y of y, such that x=f(y) has only one preimage in U, i.e. f^-1(x) ∩ U ={y}, and for all other points x̂∈ f(U) the number of preimages is deg_y f. Observe that deg_y f≠ 1 only at the critical points of f and hence the sum in  (<ref>) is finite.


Denote the number of (finite) critical points of π_n and λ_n by N_π_n and N_λ_n respectively. Let Y be the Riemann surface obtained from Per_n/ℤ_n by smooth compactification (i.e., compactification in ℂℙ^n, possibly followed by resolution of singularities at infinity). Then Y = Per_n/ℤ_n ∪ Z, where Z is a finite set of points at infinity.



We continuously extend π_n to the map π_n: Y →ℂℙ^1 of the whole surface Y by setting π_n(z) = ∞, for all z∈ Z.


In order to continuously extend the multiplier map in the similar way we need the following Proposition.

	The following relation holds:
	
    lim_(c,𝒪) ∈Per_n/ℤ_n,
     c →∞λ_n(c,𝒪)=∞.




Assume that c≠ 0 and denote by 𝔻_R:=𝔻_R(0) the disc of radius R = |c|/10. Then for any z∈ℂ∖𝔻_R we have

    |p_c(z)| = |z^2+c|> R|z|-|c|> (R-10)|z|.
 
If |c| is sufficiently large, then R-10>2, and the above inequality implies that the orbit of any point z∈ℂ∖𝔻_R converges to ∞ under the dynamics of the map p_c. In particular, this means that all periodic points of the map p_c lie in the disc 𝔻_R.

We now consider the disc 𝔻_r:=𝔻_r(0) of radius r=1/2√(|c|). Let z∈ D_r, i.e. 0 ≤ |z| ≤ r. Observe that

    |p_c(z)|=|z^2+c|≥ |c|-|c|/4 > R,





i.e. any point z from the disc 𝔻_r is mapped outside of the disc 𝔻_R under one iteration of the map p_c and tends to infinity under further iterations, provided that |c| is sufficiently large. 
Hence all periodic points of the map p_c lie inside the annulus  𝔻_R∖𝔻_r. 

Since R→∞ and r→∞ as c→∞, for all periodic points z of p_c it follows that |z|→∞ as c→∞. 

Recall that the multiplier λ_n of the periodic point z satisfies 

    λ_n = ∂ p^∘ n_c/∂ z(z) = 2^n z_1 ⋯ z_n,

where z_1,…, z_n denotes the points of the orbit of z under the map p_c. Combining the above observations, it follows that 
	
    lim_(c,𝒪) ∈Per_n/ℤ_n,
     c →∞λ_n(c,𝒪)=∞.



According to Proposition <ref>, we continuously extend λ_n to the map λ_n: Y →ℂℙ^1 of the whole surface Y by setting λ_n(z)=∞ for all z∈Z.



For further reference, let us state the following propositions:

	For any y∈Per_n/ℤ_n, we have deg_yπ_n ≤ 2.


	If y=(c,𝒪) and deg_yπ_n > 1, then by the Implicit Function Theorem we have λ_n(y)=1, which means that 𝒪 is a parabolic periodic orbit. According to the Fatou-Shishikura inequality (c.f. <cit.>),  the ramification index deg_yπ_n is not greater than 1 plus the number of critical points of p_c, lying in the basin of attraction of the orbit 𝒪. Since the polynomial p_c has only one critical point, the statement of the proposition follows.



	If λ_n(c,𝒪)=1, for some point (c,𝒪)∈Per_n/ℤ_n, then either (c,𝒪) is a critical point of π_n, or 𝒪 is a periodic orbit of period p<n, n=pr, for some integer r>1, and λ_p(c,𝒪) is a primitive root of unity of degree r.


	The proposition follows from the Fatou-Shishikura inequality in a similar way as Proposition <ref>.



We recall that Z⊂ Y is the inverse image of infinity under the map π_n. Applying the Riemann-Hurwitz formula (<ref>) to the projection π_n and using Proposition <ref>, we get

    χ(Y)    = deg π_n ·χ(ℂℙ^1)-N_π_n-∑_y∈ Z(deg_y π_n - 1).

Therefore, assuming that Z consists of κ points, we have

    ∑_y∈ Z(deg_y π_n - 1) = deg π_n-κ,

and 

    χ(Y) = deg π_n-N_π_n+κ.


Observe that if (c,𝒪)∈Per_n/ℤ_n is a critical point of the projection π_n, then λ_n(c,𝒪)=1. 
Since all points in the closure of the unit disc 𝔻 are regular values of λ_k for all k∈ℕ, (c.f. <cit.>), Proposition <ref> implies the following formula for the number of critical points of π_n:


    N_π_n = deg λ_n-∑_∀ r,p  s.t.
     n=rp
    p<ndeg λ_p·φ(r),

where φ(r) is the Euler's function that counts the positive integers up to r that are relatively prime with r.

Analogous computations for λ_n using the Riemann-Hurwitz formula (<ref>) show that

    χ(Y)    ≤deg λ_n ·χ(ℂℙ^1)-N_λ_n-∑_y∈ Z(deg_y λ_n - 1)
           =
    deg λ_n - N_λ_n + κ.

Note that in contrast to the case of the critical points of the projection map π_n, we are not guaranteed that all critical points of the map λ_n are of multiplicity 1 (in fact, we do not know whether this is true or false). Because of this we get an inequality instead of an equality in (<ref>).



Now, combining  (<ref>),  (<ref>),  (<ref>), we derive an upper bound for the number of critical points of the multiplier map

    N_λ_n   ≤deg λ_n - χ(Y) + κ
         =
    deg λ_n - deg π_n+N_π_n - κ + κ
         =
    2 ·deg λ_n - deg π_n -∑_∀ r,p  s.t.
     n=rp
    p<ndeg λ_p·φ(r).

Finally, expressing deg π_n and deg λ_n as ν(n)/n and ν(n)/2 respectively, we get


    N_λ_n≤ν(n)-ν(n)/n -1/2∑_∀ r,p  s.t.
     n=rp
    p<nν(p) ·φ(r).


	We note that the above inequality turns into an equality if the critical points of the multiplier map λ_n are counted with their multiplicities.




 §.§ Algorithm


To solve the system of equations  (<ref>) we use the Newton method. In addition to the critical points c of the multiplier map λ_n, this allows us to determine the corresponding periodic points z and its derivatives z'.
 
We fix the period n. All initial guesses for the Newton method will be randomly chosen on the complex curve defined by the first two equation of (<ref>). Since every solution to the system (<ref>) lies on this curve, we hope that such initial conditions are more likely to belong to the domains of attraction of the solutions. 

More specifically, initial guesses for the Newton method are chosen as follows: 
we first generate a random guess for the parameter c and compute all periodic points z of period n for the polynomial p_c. In order to do this, we apply the algorithm of Hubbard, Schleicher and Sutherland, developed in <cit.>.
For every choice of a periodic point z, obtained this way, we compute the corresponding initial value for z' using (<ref>). The resulting triples (c,z,z') are used as initial guesses for the Newton method.


Note that since each pair (c,z)∈Per_n determines a pair (c,𝒪)∈Per_n/ℤ_n, it is enough to consider only one point z (with corresponding z') from each periodic orbit. 



Since system (<ref>) commutes with complex conjugation, each solution (c,z,z') comes together with its complex conjugate (c,z,z'). Thus once a solution (c,z,z') is found, we can speed up the computation by also including (c,z,z') to the list of solutions. 



 
We can summarize the entire process by the following algorithm. 


  * Input: the period n.
	
	
  0: Set the counter of the critical points of the multiplier map k=0, compute the upper bound on N_λ_n using  (<ref>).
	
  1: Generate randomly c, find all z using the method described in  <cit.>, select one z from each orbit, compute z'. Store triplets of the initial guesses c, z, z' in the set Σ_0.
	
  2: If Σ_0≠∅, then take an initial guess from the set Σ_0, remove it from Σ_0 and proceed to Step 3. If the set Σ_0 was empty, return to Step 1. 
	
  3: Iterate the 3-dimensional Newton operator applied to the system  (<ref>) at the initial guess for maximum 50 times. If the Newton method does not converge after 50 iterations with the desired tolerance, return to Step 2.
	
  4: Test if the point z obtained by the Newton method, is of period n. If this condition is fulfilled, then store the obtained triplet (c,z,z') and its complex conjugate (c,z,z') in the set of solutions of  (<ref>) Σ_n, and set k=k+2.
	
  5: If k < N_λ_n, return to Step 2.     
	

  * Output: the set Σ_n.





§ RESULTS OF THE NUMERICAL EXPERIMENTS


The algorithm described above has been implemented in a C++ program. In this section we present the outcome of the numerical experiments. The complete list of critical points of λ_n found by the program, can be downloaded from <https://www.dropbox.com/sh/nr5847qnhapd8zc/AACdqv2rOxghrBLGQo47zbcma?dl=0>. The tolerance for the Newton's method has been set up at 10^-10.

Note that the multiplier map λ_n does not have critical points for periods n=1,2. We ran the algorithm for several periods n=3,…,10. Table <ref> displays the upper bound for the number of critical points N_λ_n of the multiplier map for each period and the number of critical points computed by the implemented algorithm, i.e. #Σ_n. It is likely that not all critical points have been detected for periods n=9 and n=10, which can be seen in Table <ref>. One of the reasons for the missing points might be that the critical points are lying too close to each other and cannot be distinguished using the standard double precision in the computations. However increasing the precision can significantly change the running time. It could also be that the basins of attraction of some of the points for the Newton's method are very small and are easily missed by the initial guesses. The computations for each period up to n=7 took less than 30 sec in double precision while for period n=8 it took almost 2 hours. Due to the randomness of the initial guesses the running time might be slightly different for the same period for different shots though our experiments showed that this difference is minor.









§ DISCUSSION OF THE RESULTS


In this section we give a basic discussion of the results of our computations and state some questions and conjectures.





Figure <ref> represents critical points of the multiplier map λ_n(c,z) on the parameter space of quadratic polynomials for some periods n. 
The main question is: do the critical points of the multiplier map have any dynamical meaning? We can see from the pictures that they may correspond to quadratic polynomials with both connected and disconnected Julia sets. The pictures also suggest that as n increases, most of the critical points of the multiplier map tend to accumulate on the boundary of the Mandelbrot set. This leads to the following question:
let X_n⊂ℂ be the projection of the set of all critical points of λ_n onto the coordinate c and let ν_n be the probability measure

    ν_n= 1/card (X_n)∑_c∈ X_nδ_c.

Is it true that as n→ +∞, the sequence of measures ν_n converges to a measure μ supported on the boundary of the Mandelbrot set? If yes, is μ the bifurcation measure? Positive answers to such questions have been obtained for various other classes of dynamically significant points for example, in <cit.> and <cit.>.












Next, we can observe that while most of the elements of the sets X_n are strictly complex, for periods n=6, 8 the sets X_n also contain purely real elements. Can we understand this phenomenon? Furthermore, for n=6 one of these purely real critical points lies exactly at c=0. The latter suggests the following:
given a periodic point z_0≠ 0 of the polynomial p_0(z)=z^2, one can compute the derivative of the multiplier dλ̃_n/dc(0, z_0) using the formula

    dλ̃_n/dc(0, z_0) = -2^n∑_j=0^n-1 z_0^-2^j+1,

which was obtained in <cit.>. Using this formula, we can check numerically whether c=0 is a critical point of the multiplier map λ_n for periods n>8. Due to the limited precision, we performed computations up to period n=30 and obtained that the multiplier map λ_n has a critical point at c=0, for periods n=6, 12, 18, 20, 21, 24 and 30. Furthermore, for each of these periods, except n=6, the value dλ_n/dc=0 is obtained at more than one different periodic orbit.














Another important problem is to study the critical values of the multiplier maps λ_n. As it was mentioned in the introduction, the inverse branches of λ_n projected onto the c-coordinate are Riemann mappings of the corresponding hyperbolic components of the Mandelbrot set. In particular, this implies that all critical values of the multiplier maps λ_n lie outside of the open unit disk. The question is: how close can they get to the unit disk? 
Are the critical values of λ_n bounded away from the unit disk uniformly in n? If the answer to this question is positive, then one might use the Koebe Distortion Theorem to get uniform bounds on the geometric shape of the hyperbolic components. 
The results of our computations, summarized in Table <ref> and Figure <ref>, cannot obviously give a definite answer to the stated question. Nevertheless, Figure <ref> suggests that the answer might be positive.







§ ACKNOWLEDGMENTS


The authors would like to thank the Department of Mathematics at Uppsala University, where the main part of this work has been done. The authors would also like to thank Tanya Firsova for some valuable remarks and suggestions.










 





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