# On the Continuity of Hausdorff Dimension of Julia Sets Concerning Potts Models

Authors: Gang Liu; Junyang Gao
Journal: ISRN Mathematical Analysis
 (2013)

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## Abstract

Considering the Julia sets of a family of rational maps concerning two-dimensional diamond hierarchical Potts models in statistical mechanics, we show the continuity of their Hausdorff dimension.


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## Body

## 1. Introduction

The continuity of Hausdorff dimension of Julia sets is an important and interesting problem for rational maps with degreed≥2. In general, this problem adheres to the continuity of Julia sets which is response to the stability of system. It is well known that both the Julia set J(R) and its Hausdorff dimension of a rational map R vary continuously in the parameter space Ratd if R is hyperbolic [1, 2]. However, as we know, there are no direct relationship between them when R is not hyperbolic though there are many works devoted to the two problems [1, 3, 4].In this paper, we discuss a family of rational mapsTnλ:ℂ¯→ℂ¯  for n=3; here
(1)Tnλ(z)=(z2+λ-12z+λ-2)n
with two parameters n∈ℕ and λ∈ℝ. Tnλ is a renormalization transformation of λ-state Potts models on the two-dimensional diamond-like hierarchical lattice with bifurcation number n in statistical mechanics [5]. In turn, the zeros of the partition function for the model with bifurcation number n condense to the Julia sets of Tnλ [6]. It has been shown that there exists some relationship between the critical temperatures, the critical amplitudes, and the structures of the Julia sets [7]. Therefore, much interest has been devoted to these physical models, since they exhibit a connection between statistical mechanics and complex dynamics [6, 8–15].We have known that, for any givenn∈ℕ, the Julia set J(Tnλ) of Tnλ is continuous in the Hausdorff distance for any λ∈ℝ except two points [11]. Whether the Hausdorff dimension of J(Tnλ)   is also continuous for any λ∈R except two points? From the proof of the main result in [10, 11], for even integer n, it is easy to see that Tnλ is hyperbolic in the real axis ℝ except countable points. Except at most three points from those countable points, Tnλ is subhyperbolic but not hyperbolic; though the dynamical property of Tnλ is simple, it is difficult to compute all the iteration number of critical points which are eventually equal to the repelling fixed points in the iteration of Tnλ. Therefore, we cannot give a quantitative analysis for the corresponding critical points when the parameter is close to the above points. For any odd integer n≥5, there exist at least two real numbers λ1,λ∈(1,2) such that Tnλ1 and Tnλ2 are Feigenbaum-like maps [15]. As we have seen, for the simplest Feigenbaum quadratic polynomials, the continuity of Hausdorff dimension of its Julia sets is unknown. Based on the above reason, we just consider the case for n=3.We define the following constants:(2)α=2+min0≤t≤1t6-2t4+1t-1,β=2+max-2≤t≤0t6-2t4+1t-1.We have the following result.Theorem 1.
T
3
λ is defined in (1) and λ∈ℝ. Let HD(J(T3λ)) be the Hausdorff dimension of J(T3λ). Then HD(J(T3λ)) is continuous at λ∈ℝ∖{α,0,β}.


## 2. Some Notations and Preliminary Results

LetR:ℂ¯→ℂ¯ be a rational map with degree deg(R)≥2. We denote by  Rn the nth iteration of R. A point z is called critical point if R′(z)=0. A point z is called periodic point if Rk(z)=z for some k≥1; the minimal of such k is called the period of z. For a periodic point z0, denote the multiplier of z0 by (Rk)′(z0); the periodic point z0 is either attracting, indifferent, or repelling according to |(Rk)′(z0)|<1, |(Rk)′(z0)|=1 or |(Rk)′(z0)|>1. In the indifferent case, we say z0 is parabolic if (Rk)′(z0) is a root of unity.TheJulia set, denoted by J(R), is the closure of repelling periodic points. Its complement is called Fatou set, denoted by F(R); a connected component of F(R) is called a Fatou component. A rational map R is called hyperbolic, if P(R)∩J(R)=∅, and geometrically finite, if the set P(R)∩J(R) is finite; here the postcritical setP(R) of R is the closure of the forward orbits of critical points. A geometrically finite map is subhyperbolic (resp. parabolic) if it has no (resp. some) parabolic periodic points. It is called critically nonrecurrent if c∉ω(c) for each critical point c∈J(R), where ω(c) is the ω-limit set of c. A critically nonrecurrent map is semihyperbolic if it has no parabolic periodic points. For the classical results in complex dynamics, see [12, 16, 17].Definition 2.
A domainD⊂ℂ is called a John domain if there exists c>0 such that, for any z0∈D, there is an arc γ joining z0 to some fixed reference point w0∈D satisfying
(3)dist(z,∂D)≥c|z-z0|,z∈γ.
If ∞∈∂D, we use the spherical metric to measure the distance.Lemma 3 (see [18]).
SupposeR is semihyperbolic rational map, then every Fatou component of F(R) is a John domain.Definition 4.
A probability measureμ on the Julia set J(R) is called t-conformal measure for a rational map R:ℂ¯→ℂ¯ if μ(R(A))=∫A‍|R′|tdμ for every Borel set A⊂J(R) such that R|A is injective; t is called the conformal exponent about μ.Lemma 5 (see [19]).
Leth denote the Hausdorff dimension of J(R) of a subhyperbolic rational map R, then there exists a unique invariant probability measure μ equivalent to the h-conformal measure;  moreover, the normalized h-dimension Hausdorff measure is the only h-dimension conformal measure for R.Lemma 6 (see [1]).
Any normalized invariant conformal probability measureμ supported on the Julia set of a geometrically finite rational map R  is either the conformal measure of Hausdorff dimension of J(R), or an atomic measure supported on the inverse orbits of parabolic points and critical points.For simplicity,Tλ=T3λ, and A~B (A,B∈ℝ) means that C-1B<A<CB for some implicit constant C. By (1), for λ≠0, we have
(4)Tλ′(z)=6(z-1)(z+λ-1)(z2+λ-1)2(2z+λ-2)4.
So, Tλ has ten critical points: 1, 1-λ, ±λ-1i (with the multiplicity 2), (1-λ)/2 (with the multiplicity 3), ∞. It is easy to see that z=1 and ∞ are two superattracting fixed points.Lemma 7 (see [6]).
α
∈
(
-
2,0
), β∈(2,3), and(1)
Tλ has only two real fixed points q, 1 (q<-1) for λ∈(-∞,α);(2)
Tλ has only two real fixed points 1, q(q>1) for λ∈(β,+∞);(3)
Tλ has only three real fixed points q1, q2, 1 (q1<-1, 0<q2<1) for λ=α or λ=0;(4)
Tλ has only three real fixed points q1, 1, q2(q1<-1, q2>1) for λ=β;(5)
Tλ has only four real fixed points q1, 0, 1, q2(q1<-1,q2>1) for λ=1;(6)
Tλ has only four real fixed points q1, q2, 1, q3(q1<q2<0,q3>1) for λ∈(1,β);(7)
Tλ has only four real fixed points q1, q2, 1, q3(q1<-1,q2∈(0,1),q3>1) for λ∈(0,1);(8)
Tλ has only four real fixed points q1, 0, 1, q3(q1<-1,q2,q3∈(0,1)) for λ∈(α,0).Lemma 8 (see [10]).
T
λ is hyperbolic for λ∈ℝ∖{α,β,3±2}, T3±2 is subhyperbolic, and Tα and Tβ are parabolic.


## 3. The Proof of Theorem1

In the following, we denoteTλ2(±λ-1i)=Tλ(0)=vλ, qλ=q1 is the repelling fixed point for λ close but not equal to 3-2, and  qλ=q is also the repelling fixed point for λ close but not equal to 3+2. It is easy to see that vλ-qλ→vλ0-qλ0=0 when λ→λ0, λ0∈{3-2,3+2}.Proposition 9.
Consider(5)qλ=(λ0-1λ0-2)3+k(λ-λ0)+O((λ-λ0)2)
as λ→λ0; here k=(78+362)/97 for λ0=3-2 and k=(78-362)/97 for λ0=3+2.Proof.
Considering the real fixed points ofTλ and taking t=x3, from the equation Tλ(x)=x, it follows that
(6)λ=2+t6-2t4+1t-1.
Whenλ is close but not equal to λ0, denote that
(7)qλ=(λ0-1λ0-2)3+k(λ-λ0)+O((λ-λ0)2).(1)
Ifλ0=3-2, qλ0=-22. By the continuity, qλ<0. By (6) and λ∈ℝ, it satisfies
(8)(λ-λ0+λ0-2)(qλ3-1)=qλ2-2qλqλ3+1.Substituting (8) with (7), by a calculation, we can deduce that
(9)(λ-λ0+1-2)(-2-1+k(λ-λ0)6)+O((λ-λ0)2)=(-22+k(λ-λ0))2-2(-22+k(λ-λ0))×(-2+k(λ-λ0)6)+1+O((λ-λ0)2),thenk=(78+362)/97.(2)
Ifλ0=3+2, qλ0=22. By the similar method used in Case (1), we can deduce that k=(78-362)/97.Proposition 10.
HD
(
J
(
T
λ
)
) is continuous for λ∈{3+2,3-2}.Proof.
By Lemma8, Tλ is hyperbolic for λ close but not equal to λ0. Then there exists a unique conformal probability measure μλ for Tλ supported in J(Tλ); μλ has exponent dλ=HD(J(Tλ)). This means that, for every measurable set V⊂J(Tλ) where Tλ is injective, μλ(Tλ(V))=∫V‍|(Tλ)′|dλdμλ. Furthermore the measure of a point is zero for μλ; that is, μλ is not atomic.
SinceTλ0 is subhyperbolic, by Lemma 5, there exists a unique conformal probability measure for Tλ0 supported in J(Tλ0). By cases (6) and (10) in the proof of Theorem 1 of the paper [10], we know that 1-λ0∈F(Tλ0) for λ0=3±2. By Lemma 6, the unique conformal probability measure has exponent dλ0=HD(J(Tλ0)) or is atomic, supported in {Tλ0-k(±λ0-1i)}k≥0. By a similar discussion used in [4], in order to prove that
(10)limλ→λ0HD(J(Tλ))=HD(J(Tλ0)),
it is enough to prove that
(11)limr→0limλ→λ0μλ(Br(±λ0-1i))=0;
here Br(x)={z∣|z-x|<r}. Noting that J(Tλ) and F(Tλ)(λ∈ℝ) are symmetry with the real axis, it suffices to prove that
(12)limr→0limλ→λ0μλ(Br(λ0-1i))=0.
In fact, if μλ0 is any weak limit of {μλ}, then μλ0 is a conformal probability measure for Tλ0 supported in J(Tλ0). The previous limit implies that the measure μλ0 is not atomic at λ0-1i, so, it has exponent dλ0=HD(J(Tλ0)). Noting that μλ0(Tλ0(V))=∫V‍|(Tλ0)′|dλ0dμλ0 and μλ(Tλ(V))→μλ0(Tλ0(V)) as λ→λ0 for any measurable set V, it follows that dλ→dλ0. Next we set that λ is close but not equal to λ0.
Sinceqλ0 and qλ are the real repelling fixed points of Tλ0 and Tλ, respectively, by the continuity, qλ→qλ0 as λ→λ0. By the Koenig’s Theorem [16], there exist a neighborhood U0 of qλ0 with diameter not more than a δ>0 and a conformal map ϕλ0:U0→Bδ1(0) for some δ1>0 such that ϕλ0 conjugates Tλ0 on U0 to the scaling function z→Tλ0′(qλ0)z on Bδ1(0). Similarly, there exists a conformal map ϕλ:U0λ→Bδ1′(0) which conjugates Tλ to the scaling function z→Tλ′(qλ)z. It is easy to construct a quasiconformal map ϕ:Aδ1={z∣δ1<|z|<δ2}→Aδ2={z∣δ1′<|z|<δ2′}; here δ2=|Tλ0′(qλ0)|δ1 and δ2′=|Tλ′(qλ)|δ1′, such that ϕ(Tλ0′(qλ0)z)=Tλ′(qλ)ϕ(z) for |z|=δ1. Pull back by the scaling function; we can extend ϕ to a quasiconformal map ϕ:Bδ2(0)→Bδ2′(0) which conjugates z→Tλ0′(qλ0)z to z→Tλ′(qλ)z. For every λ∈(λ0-ϵ,λ0+ϵ), define
(13)jλ=ϕλ-1∘ϕ∘ϕλ0:U0→U0λ.
Hence, jλ is a conjugation between Tλ0 on U0 and Tλ on U0λ. Let z(λ)=jλ(qλ0), by definition, z(λ0)=qλ0 and z(λ)=qλ.
Reducingϵ>0 if necessary, there are constants C0>0 and θ0∈(0,1) such that, for all m≥1, all λ∈(λ0-ϵ,λ0+ϵ), and all qλ,
(14)|(Tλm)′(qλ)|-1≤C0θ0m.
On the other hand, for every k≥1, let Uk be the preimage of Bδ(qλ0) under Tλ0k containing qλ0, and let Vk be the pullback of Uk by Tλ02 containing λ0-1i. Moreover, we denote jλ(Uk) by Ukλ containing qλ0 and let Vkλ be the pullback of Ukλ by Tλ2 containing λ0-1i. By Koebe Distortion Theorem, reducing δ>0 if necessary, there is an implicit constant K>1 such that, for all ω∈Ukλ⊂Bδ(qλ0) and all λ∈(λ0-ϵ,λ0+ϵ),
(15)1K≤|(Tλk)′(qλ)(Tλk)′(ω)|≤K.
So, |(Tλm)′(ω)|-1≤KC0θ0m; that is, the distortion of Tλk in Uk is bounded by K; denote this property as the uniform Bounded Distortion Property.
We also denote the largestk=p such that Br(λ0-1i)⊂Vkλ for r>0 small enough and all λ sufficiently close to λ0. It follows that, for r→0,  p=p(r)→∞. The following suffices to prove that
(16)limp→∞limλ→λ0μλ(Vpλ)=0.Step 1. Let D be a disc containing λ0-1i, small enough such that degTλ|D=3, since λ0-1i is a critical point with the multiplicity 2. Reducing ϵ>0 if necessary, such that U1λ⊂Tλ2(D). Tλ is hyperbolic when λ is close to λ0, then the probability measure μλ is not atomic; we have
(17)μλ(Vpλ)=∑m≥p‍μλ(Vmλ-Vm+1λ)
for all p≥1. By the construction of the t-conformal measure μ of a rational map R([2]), we know that μ(A-1)=∫A-1‍|(R-1)′|tdμ for every Borel set A⊂J(R) such that R:A-1→A is conformal. For λ∈(λ0-ϵ,λ0+ϵ), we have
(18)μλ(Vmλ-Vm+1λ)≤3μλ(Umλ-Um+1λ)×infz∈(Vmλ-Vm+1λ)∩J(Tλ)|(Tλ2)′(z)|-dλ.
By the uniform Bounded Distortion Property, note that z(λ)=qλ and μλ is a probability measure, then
(19)μλ(Umλ-Um+1λ)≤Kdλ|(Tλm)′(z(λ))|-dλ.
Furthermore, we claim that there exists C1>0 such that, for all λ∈(λ0-ϵ,λ0+ϵ) and z∈V1λ,
(20)|(Tλ2)′(z)|≥C1|Tλ2(z)-vλ|2/3.
In fact, (20) is obvious for z=λ-1i, since Tλ′(λ-1i)=0 and vλ=Tλ2(λ-1i). Suppose z≠λ-1i; by the uniform Bounded Distortion Property and Koebe Distortion Theorem, it follows that
(21)dist(vλ,∂U1λ)~diam(U1λ),dist(λ-1i,∂V1λ)~diam(V1λ)~(diam(Tλ(V1λ)))1/3~(diam(U1λ))1/3,
since degTλ|V1λ=3 and degTλ|Tλ(V1λ)=1. Then
(22)|(Tλ2)′(z)|~(diam(U1λ))2/3~(dist(vλ,∂U1λ))2/3,
so, we get (20).
Step 2. Let k=k(λ) be the largest integer such that vλ∈Ukλ and let m≥1. Then there are three cases.
Case 1. (k-1≤m≤k+1). By the uniform Bounded Distortion Property, it follows that |(Tλm)′(z(λ))|-1~|z(λ)-vλ|, since k→∞ as λ→λ0. By Proposition 9, it follows that
(23)|z(λ)-vλ|~|(λ-1λ-2)3-(λ0-1λ0-2)3|~|λ-1λ-2-λ0-1λ0-2|~|λ-λ0|,
since λ0≠1+(k/2). So, we get |(Tλm)′(z(λ))|-1~|λ-λ0| with constant independent of λ; hence, |(Tλm)′(z(λ))|-1≤C2|λ-λ0| for some constant C2>0 independent of λ, but on the other hand,
(24)dist(vλ,(Umλ-Um+1λ)∩J(Tλ))≥dist(vλ,J(Tλ)).
Then for all z∈(Vmλ-Vm+1λ)∩J(Tλ), by (20), it follows that
(25)|(Tλ2)′(z)|>C1dist(vλ,J(Tλ))2/3,
so,
(26)μλ(Vmλ-Vm+1λ)≤C3|λ-λ0|dλdist(vλ,J(Tλ))-(2/3)dλ,
where C3=3(KC2(C1)-1)dλ.
Case 2.
(
m
<
k
-
1
). Noting that
(27)dist(vλ,(Umλ-Um+1λ))≥dist(∂Um+1λ,Um+2λ),
then by the uniform Bounded Distortion Property, we have
(28)dist(vλ,(Umλ-Um+1λ))>C4|(Tλm)′(z(λ))|-1.
As in Case 1, we have
(29)|(Tλ2)′(z)|>C1(dist(vλ,Umλ-Um+1λ))2/3≥C1C42/3|(Tλm)′(z(λ))|-2/3.
It follows that
(30)μλ(Vmλ-Vm+1λ)≤3Kdλ|(Tλm)′(z(λ))|-dλ×(C1C42/3)-dλ|(Tλm)′(z(λ))|(2/3)dλ=C5|(Tλm)′(z(λ))|-dλ/3.
By (14),  μλ(Vmλ-Vm+1λ)≤C5θ0mdλ/3, where C5=3Kdλ(C1C42/3)-dλC0dλ/3.
Case 3.
(
m
>
k
+
1
). We have
(31)dist(vλ,(Umλ-Um+1λ))≥dist(∂Um-1λ,Umλ).
By a similar discussion as used in Case 2,
(32)dist(vλ,(Umλ-Um+1λ))≥C4|(Tλm)′(z(λ))|-1,
then μλ(Vmλ-Vm+1λ)≤C5θ0mdλ/3.
Step 3.Since Tλ is hyperbolic when λ is close but not equal to λ0, by Lemma 3, every Fatou component of F(Tλ) is a John domain. Noting that F(Tλ) is symmetry with the real axis ℝ and qλ∈J(Tλ), then the angle at qλ of two curves γ1 and γ2 of ∂Aλ(∞) (or ∂Aλ(1)) is positive. Since vλ→qλ as λ→λ0, it follows that dist(vλ,J(Tλ))~dist(vλ,qλ) as λ→λ0. On the other hand, by Proposition 9, it follows that dist(vλ,qλ)~|z(λ)-vλ|~|λ-λ0|. Thus, dist(vλ,J(Tλ))~|λ-λ0| as λ→λ0.
By Steps 1 and 2, forp≥1, we have
(33)μλ(Vpλ)≤3C3|λ-λ0|dλdist(vλ,J(Tλ))-(2/3)dλ+C5∑m≥p,m≠k-1,k,k+1‍θ0mdλ/3.
Since
(34)∑m≥p,‍θ0mdλ/3=(θ0dλ/3)p1-θ0dλ/3,
we conclude that
(35)limp→∞limλ→λ0μλ(Vpλ)=0.
So, HD(J(Tλ)) is continuous at λ∈{3-2,3+2}.The Proof of Theorem1. Since the Hausdorff dimension HD(J(Tλ)) varies continuously in Ratd if Tλ is hyperbolic [1, Theorem 11.1] and deg(T0)≠deg(Tλ)=6 for λ≠0, by Lemma 8 and Proposition 10, HD(J(Tλ)) is continuous for λ∈ℝ∖{α,0,β}.



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