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

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.

1. Introduction

The continuity of Hausdorff dimension of Julia sets is an important and interesting problem for rational maps with degree 𝑑≥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 Rat_𝑑 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 maps 𝑇_𝑛𝜆∶ℂ→ℂ for 𝑛=3; here 𝑇_𝑛𝜆(𝑧)=𝑧²+𝜆−12𝑧+𝜆−2^𝑛 (1) with two parameters 𝑛∈ℕ and 𝜆∈ℝ. 𝑇_𝑛𝜆 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 𝑇_𝑛𝜆 [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 given 𝑛∈ℕ, the Julia set 𝐽(𝑇_𝑛𝜆) of 𝑇_𝑛𝜆 is continuous in the Hausdorff distance for any 𝜆∈ℝ except two points [11]. Whether the Hausdorff dimension of 𝐽(𝑇_𝑛𝜆) 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 𝑇_𝑛𝜆 is hyperbolic in the real axis ℝ except countable points. Except at most three points from those countable points, 𝑇_𝑛𝜆 is subhyperbolic but not hyperbolic; though the dynamical property of 𝑇_𝑛𝜆 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 𝑇_𝑛𝜆. 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 𝑇_𝑛𝜆1 and 𝑇_𝑛𝜆2 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: 𝛼=2+min_0≤𝑡≤1𝑡⁶−2𝑡⁴+1,𝑡−1𝛽=2+max_{−2≤𝑡≤0}𝑡⁶−2𝑡⁴+1.𝑡−1 (2)

We have the following result.

Theorem 1. 𝑇_3𝜆 is defined in (1) and 𝜆∈ℝ. Let HD(𝐽(𝑇_3𝜆)) be the Hausdorff dimension of 𝐽(𝑇_3𝜆). Then HD(𝐽(𝑇_3𝜆)) is continuous at 𝜆∈ℝ⧵{𝛼,0,𝛽}.

2. Some Notations and Preliminary Results

Let 𝑅∶ℂ→ℂ be a rational map with degree deg(𝑅)≥2. We denote by  𝑅^𝑛 the 𝑛th iteration of 𝑅. A point 𝑧 is called critical point if 𝑅^′(𝑧)=0. A point 𝑧 is called periodic point if 𝑅^𝑘(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 (𝑅^𝑘)^′(𝑧₀); the periodic point 𝑧₀ is either attracting, indifferent, or repelling according to |(𝑅^𝑘)^′(𝑧₀)|<1, |(𝑅^𝑘)^′(𝑧₀)|=1 or |(𝑅^𝑘)^′(𝑧₀)|>1. In the indifferent case, we say 𝑧₀ is parabolic if (𝑅^𝑘)^′(𝑧₀) is a root of unity.

The Julia 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 set 𝑃(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 domain 𝐷⊂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 dist(𝑧,𝜕𝐷)≥𝑐𝑧−𝑧₀,𝑧∈𝛾. (3) If ∞∈𝜕𝐷, we use the spherical metric to measure the distance.

Lemma 3 (see [18]). Suppose 𝑅 is semihyperbolic rational map, then every Fatou component of 𝐹(𝑅) is a John domain.

Definition 4. A probability measure 𝜇 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]). Let ℎ 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 measure 𝜇 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, 𝑇_𝜆=𝑇_3𝜆, and 𝐴∼𝐵 (𝐴,𝐵∈ℝ) means that 𝐶⁻¹𝐵<𝐴<𝐶𝐵 for some implicit constant 𝐶. By (1), for 𝜆≠0, we have 𝑇^′_𝜆(𝑧)=6(𝑧−1)(𝑧+𝜆−1)𝑧²+𝜆−1²(2𝑧+𝜆−2)⁴. (4) So, 𝑇_𝜆 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)𝑇_𝜆 has only two real fixed points 𝑞, 1 (𝑞<−1) for 𝜆∈(−∞,𝛼);(2)𝑇_𝜆 has only two real fixed points 1, 𝑞(𝑞>1) for 𝜆∈(𝛽,+∞);(3)𝑇_𝜆 has only three real fixed points 𝑞₁, 𝑞₂, 1 (𝑞₁<−1, 0<𝑞₂<1) for 𝜆=𝛼 or 𝜆=0;(4)𝑇_𝜆 has only three real fixed points 𝑞₁, 1, 𝑞₂(𝑞₁<−1, 𝑞₂>1) for 𝜆=𝛽;(5)𝑇_𝜆 has only four real fixed points 𝑞₁, 0, 1, 𝑞₂(𝑞₁<−1,𝑞₂>1) for 𝜆=1;(6)𝑇_𝜆 has only four real fixed points 𝑞₁, 𝑞₂, 1, 𝑞₃(𝑞₁<𝑞₂<0,𝑞₃>1) for 𝜆∈(1,𝛽);(7)𝑇_𝜆 has only four real fixed points 𝑞₁, 𝑞₂, 1, 𝑞₃(𝑞₁<−1,𝑞₂∈(0,1),𝑞₃>1) for 𝜆∈(0,1);(8)𝑇_𝜆 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}, 𝑇_√3±2 is subhyperbolic, and 𝑇_𝛼 and 𝑇_𝛽 are parabolic.

3. The Proof of Theorem 1

In the following, we denote 𝑇²_𝜆(±√𝜆−1𝑖)=𝑇_𝜆(0)=𝑣_𝜆, 𝑞_𝜆=𝑞₁ is the repelling fixed point for 𝜆 close but not equal to 3−√2, and 𝑞_𝜆=𝑞 is also the repelling fixed point for 𝜆 close but not equal to 3+√2. It is easy to see that 𝑣_𝜆−𝑞_𝜆→𝑣_𝜆0−𝑞_𝜆0=0 when 𝜆→𝜆₀, 𝜆₀∈{3−√2,3+√2}.

Proposition 9. Consider𝑞_𝜆=𝜆₀−1𝜆₀−2³+𝑘𝜆−𝜆₀+𝑂𝜆−𝜆₀² (5) as 𝜆→𝜆₀; here 𝑘=(78+36√2)/97 for 𝜆₀=3−√2 and 𝑘=(78−36√2)/97 for 𝜆₀=3+√2.

Proof. Considering the real fixed points of 𝑇_𝜆 and taking 𝑡=√³𝑥, from the equation 𝑇_𝜆(𝑥)=𝑥, it follows that 𝑡𝜆=2+⁶−2𝑡⁴+1.𝑡−1 (6)
When 𝜆 is close but not equal to 𝜆₀, denote that 𝑞_𝜆=𝜆₀−1𝜆₀−2³+𝑘𝜆−𝜆₀+𝑂𝜆−𝜆₀². (7)(1) If 𝜆₀=3−√2, 𝑞_𝜆0=−2√2. By the continuity, 𝑞_𝜆<0. By (6) and 𝜆∈ℝ, it satisfies 𝜆−𝜆₀+𝜆₀−2√³𝑞_𝜆−1=𝑞²_𝜆−2𝑞√_𝜆³𝑞_𝜆+1. (8) Substituting (8) with (7), by a calculation, we can deduce that 𝜆−𝜆₀+1−√2−√𝑘2−1+𝜆−𝜆₀6+𝑂𝜆−𝜆₀²=−2√2+𝑘𝜆−𝜆₀²−2−2√2+𝑘𝜆−𝜆₀×−√𝑘2+𝜆−𝜆₀6+1+𝑂𝜆−𝜆₀², (9) then 𝑘=(78+36√2)/97.(2) If 𝜆₀=3+√2, 𝑞_𝜆0=2√2. By the similar method used in Case (1), we can deduce that 𝑘=(78−36√2)/97.

Proposition 10. HD(𝐽(𝑇_𝜆)) is continuous for 𝜆∈{3+√2,3−√2}.

Proof. By Lemma 8, 𝑇_𝜆 is hyperbolic for 𝜆 close but not equal to 𝜆₀. Then there exists a unique conformal probability measure 𝜇_𝜆 for 𝑇_𝜆 supported in 𝐽(𝑇_𝜆); 𝜇_𝜆 has exponent 𝑑_𝜆=HD(𝐽(𝑇_𝜆)). This means that, for every measurable set 𝑉⊂𝐽(𝑇_𝜆) where 𝑇_𝜆 is injective, 𝜇_𝜆(𝑇_𝜆(𝑉))=∫_𝑉|(𝑇_𝜆)^′|^𝑑𝜆𝑑𝜇_𝜆. Furthermore the measure of a point is zero for 𝜇_𝜆; that is, 𝜇_𝜆 is not atomic.
Since 𝑇_𝜆0 is subhyperbolic, by Lemma 5, there exists a unique conformal probability measure for 𝑇_𝜆0 supported in 𝐽(𝑇_𝜆0). By cases (6) and (10) in the proof of Theorem 1 of the paper [10], we know that 1−𝜆₀∈𝐹(𝑇_𝜆0) for 𝜆₀=3±√2. By Lemma 6, the unique conformal probability measure has exponent 𝑑_𝜆0=HD(𝐽(𝑇_𝜆0)) or is atomic, supported in {𝑇_𝜆^−𝑘₀(±𝜆₀−1𝑖)}_𝑘≥0. By a similar discussion used in [4], in order to prove that lim_𝜆→𝜆0HD𝐽𝑇_𝜆=HD𝐽𝑇_𝜆0, (10) it is enough to prove that lim_𝑟→0lim_𝜆→𝜆0𝜇_𝜆𝐵_𝑟±𝜆₀−1𝑖=0; (11) here 𝐵_𝑟(𝑥)={𝑧∣|𝑧−𝑥|<𝑟}. Noting that 𝐽(𝑇_𝜆) and 𝐹(𝑇_𝜆)(𝜆∈ℝ) are symmetry with the real axis, it suffices to prove that lim_𝑟→0lim_𝜆→𝜆0𝜇_𝜆𝐵_𝑟𝜆₀−1𝑖=0. (12) In fact, if 𝜇_𝜆0 is any weak limit of {𝜇_𝜆}, then 𝜇_𝜆0 is a conformal probability measure for 𝑇_𝜆0 supported in 𝐽(𝑇_𝜆0). The previous limit implies that the measure 𝜇_𝜆0 is not atomic at 𝜆₀−1𝑖, so, it has exponent 𝑑_𝜆0=HD(𝐽(𝑇_𝜆0)). Noting that 𝜇_𝜆0(𝑇_𝜆0(𝑉))=∫_𝑉|(𝑇_𝜆0)^′|^𝑑𝜆0𝑑𝜇_𝜆0 and 𝜇_𝜆(𝑇_𝜆(𝑉))→𝜇_𝜆0(𝑇_𝜆0(𝑉)) as 𝜆→𝜆₀ for any measurable set 𝑉, it follows that 𝑑_𝜆→𝑑_𝜆0. Next we set that 𝜆 is close but not equal to 𝜆₀.
Since 𝑞_𝜆0 and 𝑞_𝜆 are the real repelling fixed points of 𝑇_𝜆0 and 𝑇_𝜆, respectively, by the continuity, 𝑞_𝜆→𝑞_𝜆0 as 𝜆→𝜆₀. By the Koenig’s Theorem [16], there exist a neighborhood 𝑈₀ of 𝑞_𝜆0 with diameter not more than a 𝛿>0 and a conformal map 𝜙_𝜆0∶𝑈₀→𝐵_𝛿1(0) for some 𝛿₁>0 such that 𝜙_𝜆0 conjugates 𝑇_𝜆0 on 𝑈₀ to the scaling function 𝑧→𝑇^′_𝜆0(𝑞_𝜆0)𝑧 on 𝐵_𝛿1(0). Similarly, there exists a conformal map 𝜙_𝜆∶𝑈^𝜆₀→𝐵_𝛿′1(0) which conjugates 𝑇_𝜆 to the scaling function 𝑧→𝑇^′_𝜆(𝑞_𝜆)𝑧. It is easy to construct a quasiconformal map 𝜙∶𝐴_𝛿1={𝑧∣𝛿₁<|𝑧|<𝛿₂}→𝐴_𝛿2={𝑧∣𝛿^′₁<|𝑧|<𝛿^′₂}; here 𝛿₂=|𝑇^′_𝜆0(𝑞_𝜆0)|𝛿₁ and 𝛿^′₂=|𝑇^′_𝜆(𝑞_𝜆)|𝛿^′₁, such that 𝜙(𝑇^′_𝜆0(𝑞_𝜆0)𝑧)=𝑇^′_𝜆(𝑞_𝜆)𝜙(𝑧) for |𝑧|=𝛿₁. Pull back by the scaling function; we can extend 𝜙 to a quasiconformal map 𝜙∶𝐵_𝛿2(0)→𝐵_𝛿′2(0) which conjugates 𝑧→𝑇^′_𝜆0(𝑞_𝜆0)𝑧 to 𝑧→𝑇^′_𝜆(𝑞_𝜆)𝑧. For every 𝜆∈(𝜆₀−𝜖,𝜆₀+𝜖), define 𝑗_𝜆=𝜙_𝜆⁻¹∘𝜙∘𝜙_𝜆0∶𝑈₀→𝑈^𝜆₀. (13) Hence, 𝑗_𝜆 is a conjugation between 𝑇_𝜆0 on 𝑈₀ and 𝑇_𝜆 on 𝑈^𝜆₀. Let 𝑧(𝜆)=𝑗_𝜆(𝑞_𝜆0), by definition, 𝑧(𝜆₀)=𝑞_𝜆0 and 𝑧(𝜆)=𝑞_𝜆.
Reducing 𝜖>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 𝑞_𝜆, 𝑇^𝑚_𝜆^′𝑞_𝜆⁻¹≤𝐶₀𝜃^𝑚₀. (14) On the other hand, for every 𝑘≥1, let 𝑈_𝑘 be the preimage of 𝐵_𝛿(𝑞_𝜆0) under 𝑇^𝑘_𝜆0 containing 𝑞_𝜆0, and let 𝑉_𝑘 be the pullback of 𝑈_𝑘 by 𝑇²_𝜆0 containing 𝜆₀−1𝑖. Moreover, we denote 𝑗_𝜆(𝑈_𝑘) by 𝑈^𝜆_𝑘 containing 𝑞_𝜆0 and let 𝑉^𝜆_𝑘 be the pullback of 𝑈^𝜆_𝑘 by 𝑇²_𝜆 containing 𝜆₀−1𝑖. By Koebe Distortion Theorem, reducing 𝛿>0 if necessary, there is an implicit constant 𝐾>1 such that, for all 𝜔∈𝑈^𝜆_𝑘⊂𝐵_𝛿(𝑞_𝜆0) and all 𝜆∈(𝜆₀−𝜖,𝜆₀+𝜖), 1𝐾≤𝑇^𝑘_𝜆^′𝑞_𝜆𝑇^𝑘_𝜆^′(𝜔)≤𝐾. (15) So, |(𝑇^𝑚_𝜆)^′(𝜔)|⁻¹≤𝐾𝐶₀𝜃^𝑚₀; that is, the distortion of 𝑇^𝑘_𝜆 in 𝑈_𝑘 is bounded by 𝐾; denote this property as the uniform Bounded Distortion Property.
We also denote the largest 𝑘=𝑝 such that 𝐵_𝑟(𝜆₀−1𝑖)⊂𝑉^𝜆_𝑘 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 lim_𝑝→∞lim_𝜆→𝜆0𝜇_𝜆𝑉^𝜆_𝑝=0. (16)Step 1. Let 𝐷 be a disc containing 𝜆₀−1𝑖, small enough such that deg𝑇_𝜆|_𝐷=3, since 𝜆₀−1𝑖 is a critical point with the multiplicity 2. Reducing 𝜖>0 if necessary, such that 𝑈^𝜆₁⊂𝑇²_𝜆(𝐷). 𝑇_𝜆 is hyperbolic when 𝜆 is close to 𝜆₀, then the probability measure 𝜇_𝜆 is not atomic; we have 𝜇_𝜆𝑉^𝜆_𝑝=_𝑚≥𝑝𝜇_𝜆𝑉^𝜆_𝑚−𝑉^𝜆_𝑚+1 (17) 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 𝜇_𝜆𝑉^𝜆_𝑚−𝑉^𝜆_𝑚+1≤3𝜇_𝜆𝑈^𝜆_𝑚−𝑈^𝜆_𝑚+1×inf_{𝑉𝑧∈𝜆𝑚−𝑉𝜆𝑚+1𝑇∩𝐽𝜆}𝑇²_𝜆^′(𝑧)^−𝑑𝜆. (18) By the uniform Bounded Distortion Property, note that 𝑧(𝜆)=𝑞_𝜆 and 𝜇_𝜆 is a probability measure, then 𝜇_𝜆𝑈^𝜆_𝑚−𝑈^𝜆_𝑚+1≤𝐾^𝑑𝜆𝑇^𝑚_𝜆^′(𝑧(𝜆))^−𝑑𝜆. (19) Furthermore, we claim that there exists 𝐶₁>0 such that, for all 𝜆∈(𝜆₀−𝜖,𝜆₀+𝜖) and 𝑧∈𝑉^𝜆₁, 𝑇²_𝜆^′(𝑧)≥𝐶₁𝑇²_𝜆(𝑧)−𝑣_𝜆^2/3. (20) In fact, (20) is obvious for 𝑧=√𝜆−1𝑖, since 𝑇^′_𝜆(√𝜆−1𝑖)=0 and 𝑣_𝜆=𝑇²_𝜆(√𝜆−1𝑖). Suppose 𝑧≠√𝜆−1𝑖; by the uniform Bounded Distortion Property and Koebe Distortion Theorem, it follows that dist𝑣_𝜆,𝜕𝑈^𝜆₁∼diam𝑈^𝜆₁,dist√𝜆−1𝑖,𝜕𝑉^𝜆₁∼diam𝑉^𝜆₁∼diam𝑇_𝜆𝑉^𝜆₁^1/3∼diam𝑈^𝜆₁^1/3, (21) since deg𝑇_𝜆|_𝑉𝜆1=3 and deg𝑇_𝜆|_{𝑇𝜆(𝑉𝜆1)}=1. Then 𝑇²_𝜆^′(𝑧)∼diam𝑈^𝜆₁^2/3∼dist𝑣_𝜆,𝜕𝑈^𝜆₁^2/3, (22) so, we get (20).
Step 2. Let 𝑘=𝑘(𝜆) be the largest integer such that 𝑣_𝜆∈𝑈^𝜆_𝑘 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 |(𝑇^𝑚_𝜆)^′(𝑧(𝜆))|⁻¹∼|𝑧(𝜆)−𝑣_𝜆|, since 𝑘→∞ as 𝜆→𝜆₀. By Proposition 9, it follows that 𝑧(𝜆)−𝑣_𝜆∼𝜆−1𝜆−2³−𝜆₀−1𝜆₀−2³∼𝜆−1−𝜆𝜆−2⁰−1𝜆₀∼−2𝜆−𝜆₀, (23) since 𝜆₀≠1+(𝑘/2). So, we get |(𝑇^𝑚_𝜆)^′(𝑧(𝜆))|⁻¹∼|𝜆−𝜆₀| with constant independent of 𝜆; hence, |(𝑇^𝑚_𝜆)^′(𝑧(𝜆))|⁻¹≤𝐶₂|𝜆−𝜆₀| for some constant 𝐶₂>0 independent of 𝜆, but on the other hand, dist𝑣_𝜆,𝑈^𝜆_𝑚−𝑈^𝜆_𝑚+1∩𝐽𝑇_𝜆≥dist𝑣_𝜆,𝐽𝑇_𝜆. (24) Then for all 𝑧∈(𝑉^𝜆_𝑚−𝑉^𝜆_𝑚+1)∩𝐽(𝑇_𝜆), by (20), it follows that 𝑇²_𝜆^′(𝑧)>𝐶₁dist𝑣_𝜆,𝐽𝑇_𝜆^2/3, (25) so, 𝜇_𝜆𝑉^𝜆_𝑚−𝑉^𝜆_𝑚+1≤𝐶₃𝜆−𝜆₀^𝑑𝜆dist𝑣_𝜆,𝐽𝑇_𝜆^{−(2/3)𝑑𝜆}, (26) where 𝐶₃=3(𝐾𝐶₂(𝐶₁)⁻¹)^𝑑𝜆.
Case 2. (𝑚<𝑘−1). Noting that dist𝑣_𝜆,𝑈^𝜆_𝑚−𝑈^𝜆_𝑚+1≥dist𝜕𝑈^𝜆_𝑚+1,𝑈^𝜆_𝑚+2, (27) then by the uniform Bounded Distortion Property, we have dist𝑣_𝜆,𝑈^𝜆_𝑚−𝑈^𝜆_𝑚+1>𝐶₄𝑇^𝑚_𝜆^′(𝑧(𝜆))⁻¹. (28) As in Case 1, we have 𝑇²_𝜆^′(𝑧)>𝐶₁dist𝑣_𝜆,𝑈^𝜆_𝑚−𝑈^𝜆_𝑚+1^2/3≥𝐶₁𝐶₄^2/3𝑇^𝑚_𝜆^′(𝑧(𝜆))^−2/3. (29) It follows that 𝜇_𝜆𝑉^𝜆_𝑚−𝑉^𝜆_𝑚+1≤3𝐾^𝑑𝜆𝑇^𝑚_𝜆^′(𝑧(𝜆))^−𝑑𝜆×𝐶₁𝐶₄^{2/3−𝑑𝜆}𝑇^𝑚_𝜆^′(𝑧(𝜆))^{(2/3)𝑑𝜆}=𝐶₅𝑇^𝑚_𝜆^′(𝑧(𝜆))^{−𝑑𝜆/3}. (30) By (14), 𝜇_𝜆(𝑉^𝜆_𝑚−𝑉^𝜆_𝑚+1)≤𝐶₅𝜃^𝑚𝑑𝜆₀^/3, where 𝐶₅=3𝐾^𝑑𝜆(𝐶₁𝐶₄^2/3)^−𝑑𝜆𝐶^𝑑𝜆₀^/3.
Case 3. (𝑚>𝑘+1). We have dist𝑣_𝜆,𝑈^𝜆_𝑚−𝑈^𝜆_𝑚+1≥dist𝜕𝑈^𝜆_𝑚−1,𝑈^𝜆_𝑚. (31) By a similar discussion as used in Case 2, dist𝑣_𝜆,𝑈^𝜆_𝑚−𝑈^𝜆_𝑚+1≥𝐶₄𝑇^𝑚_𝜆^′(𝑧(𝜆))⁻¹, (32) then 𝜇_𝜆(𝑉^𝜆_𝑚−𝑉^𝜆_𝑚+1)≤𝐶₅𝜃^𝑚𝑑𝜆₀^/3.
Step 3. Since 𝑇_𝜆 is hyperbolic when 𝜆 is close but not equal to 𝜆₀, by Lemma 3, every Fatou component of 𝐹(𝑇_𝜆) is a John domain. Noting that 𝐹(𝑇_𝜆) is symmetry with the real axis ℝ and 𝑞_𝜆∈𝐽(𝑇_𝜆), then the angle at 𝑞_𝜆 of two curves 𝛾₁ and 𝛾₂ of 𝜕𝐴_𝜆(∞) (or 𝜕𝐴_𝜆(1)) is positive. Since 𝑣_𝜆→𝑞_𝜆 as 𝜆→𝜆₀, it follows that dist(𝑣_𝜆,𝐽(𝑇_𝜆))∼dist(𝑣_𝜆,𝑞_𝜆) as 𝜆→𝜆₀. On the other hand, by Proposition 9, it follows that dist(𝑣_𝜆,𝑞_𝜆)∼|𝑧(𝜆)−𝑣_𝜆|∼|𝜆−𝜆₀|. Thus, dist(𝑣_𝜆,𝐽(𝑇_𝜆))∼|𝜆−𝜆₀| as 𝜆→𝜆₀.
By Steps 1 and 2, for 𝑝≥1, we have 𝜇_𝜆𝑉^𝜆_𝑝≤3𝐶₃𝜆−𝜆₀^𝑑𝜆dist𝑣_𝜆,𝐽𝑇_𝜆^{−(2/3)𝑑𝜆}+𝐶_{5𝑚≥𝑝,𝑚≠𝑘−1,𝑘,𝑘+1}𝜃^𝑚𝑑𝜆₀^/3. (33) Since _𝑚≥𝑝,𝜃^𝑚𝑑𝜆₀^/3=𝜃^𝑑𝜆₀^/3𝑝1−𝜃^𝑑𝜆₀^/3, (34) we conclude that lim_𝑝→∞lim_𝜆→𝜆0𝜇_𝜆𝑉^𝜆_𝑝=0. (35) So, HD(𝐽(𝑇_𝜆)) is continuous at 𝜆∈{3−√2,3+√2}.

The Proof of Theorem 1. Since the Hausdorff dimension HD(𝐽(𝑇_𝜆)) varies continuously in Rat_𝑑 if 𝑇_𝜆 is hyperbolic [1, Theorem 11.1] and deg(𝑇₀)≠deg(𝑇_𝜆)=6 for 𝜆≠0, by Lemma 8 and Proposition 10, HD(𝐽(𝑇_𝜆)) is continuous for 𝜆∈ℝ⧵{𝛼,0,𝛽}.

Acknowledgments

The first author was supported by the Construct Program of the Key Discipline in Hunan Province and Science Foundation of Hengyang Normal University of China (no. 12B35). The second author was supported by the Fundamental Research Funds of Central Universities of China (no. 2009QS15). The authors were supported by the NSFs of China (no. 11261002 and no. 11231009). The authors would like to thank the referees for their valuable comments for improving this paper.