On the derivative of the Hausdorff dimension

On the derivative of the Hausdorff dimension of the Julia sets for

Ludwik Jaksztas, Michel Zinsmeister Faculty of Mathematics and Information Sciences, Warsaw University of Technology, Pl. Politechniki 1, 00-661 Warsaw, Poland jaksztas@impan.gov.pl Institut Denis Poisson, COST, Université d’Orléans, BP6749 45067 Orléans Cedex, France zins@univ-orleans.fr
Abstract.

Let denote the Hausdorff dimension of the Julia set of the polynomial . We will investigate behavior of the function when real parameter tends to a parabolic parameter .

2000 Mathematics Subject Classification:
Primary 37F45, Secondary 37F35
partially supported by Polish NCN grants NN201 607 640 and UMO-2014/13/B/ST1/01033

1. Introduction

For a polynomial of degree at least 2, we define the filled-in Julia set as the set of all points that do not escape to infinity under iteration of . The Julia set is the boundary of , i.e.

We will consider the family of quadratic polynomials of the form

As usual, and abbreviate and respectively.

We define the Mandelbrot set as the set of all parameters for which the Julia set is connected, or equivalently,

We are interested in the function , where denotes the Hausdorff dimension of the Julia set .

Recall that a polynomial (or more generally a rational function) is called hyperbolic (expanding) if

The function is real-analytic on each hyperbolic component of (consisting of parameters related to hyperbolic maps) as well as on the exterior of (see [20])

M. Shishikura proved in [21] that there exists a residual (hence dense) set of parameters such that . Therefore is not continuous at if . In particular, it follows from [22] that is not continuous at any parabolic parameter.

Nevertheless the Hausdorff dimension is continuous along some paths. This was first proved by O. Bodart and M. Zinsmeister (see [2]) when the real parameter tends to from the left. Later, the continuity was proved (see [18]) when approaches other parabolic parameters in a "good way". In particular the function is continuous on the interval (included in ), where is the Feigenbaum parameter. Note that is not right-continuous at , i.e. when approaches from outside of the Mandelbrot set (see [6]). For results concerning other parameters see [8] and [19].

In this paper we restrict to and investigate the derivative of the Hausdorff dimension with respect converging to a parabolic parameter. This derivative has been studied in several papers:

First, G. Havard and M. Zinsmeister proved in [10] that
Theorem I. There exist and such that for every

We know from [11] that . Thus, when (tends from the left). A similar problem in the case of the exponential family, i.e. parabolic map with one petal, was solved in [9].

Next, it was proven, under the assumption (see [13] and [14]), that
Theorem II. There exist and such that for every

In particular , when (tends from the right).
Theorem III. There exists such that,

provided .

Figure 1. The bifurcation at :

The main goal of this paper is to prove the following generalization of Theorem II and III:

Theorem 1.1.

If has a parabolic fixed point with two petals then

  1. if then there exist such that

  2. if then there exists such that

  3. if then there exists such that

In particular, if then is differentiable at . Whereas, if then , as .

Finally, concerning Hausdorff dimensions of connected parabolic Julia sets, we will use the important theorem, due to Anna Zdunik (see [24]), which asserts that they greater than .

The one petal case of Theorem I can also be generalized to small copies of the Mandelbrot set, but this renormalizable case will be considered in a future paper.

Notation. means that , where constant does not depend on and under consideration.

We will write if for every there exist and a neighborhood of the fixed point under consideration, such that , where and .

2. Thermodynamical formalism

The goal of this section is to establish the formula (2.2) which is the starting point of this work.

Let be a holomorphic family of hyperbolic polynomials of degree with connected Julia sets, where is an open subset of . Note that we will be interested in families which are conjugated to , (after reparametrization). Write

If , then there exist a holomorphic motion of parametrized by (see [12]), such that and conjugates to (i.e. ). Thus, the function is holomorphic for every .

Now we use the thermodynamical formalism, which holds for hyperbolic rational maps. We will consider only such maps. Let , , and let be a Hölder continuous function, to be often called a potential function. We will consider potentials of the form for .

The topological pressure can be defined as follows:

where , and the limit exists and does not depend on . If and , then . Hence

The function is decreasing from to . In particular, there exists a unique such that . By Bowen’s Theorem (see [16, Corollary 9.1.7] or [25, Theorem 5.12]) we have

(2.1)

Thus, we have . Put .

The Ruelle or transfer operator is defined as

The Perron-Frobenius-Ruelle theorem [25, Theorem 4.1] asserts that is a single eigenvalue of associated with an eigenfunction . Moreover, there exists a unique probability measure such that , where is conjugated to .

For we have , and then is a -invariant measure called an equilibrium state after normalization. But in the present work it will be more convenient not to normalize, contrarily to the tradition. We denote by and the measures and respectively (measures supported on ). Next, we take , and (image measures supported on ).

So, the measure is -invariant, whereas the measure is called -conformal with exponent d(c), i.e. is a Borel probability measure such that for every Borel subset ,

provided is injective on .

It follows from [25, Proposition 6.11] or [16, Theorem 4.6.5] that for every Hölder and at every , we have

The above formula, and implicit function theorem (see [13, Proposition 2.1]) lead to

Proposition 2.1.

[10, Proposition 2.1] Let us consider the family as above, with the further assumptions that:

  • is symmetric with respect to ,

  • .

If is such that is hyperbolic then

(2.2)

3. The two petals case

3.1.

Let us assume that , where , has a parabolic cycle of length , and let be a point in this cycle. Then, has a parabolic fixed point with a multiplier . Since the critical orbit is real, the case corresponds to one petal, and the to two petals. Let us assume, from now on, that .

The parameter lies at the boundary of two hyperbolic components , of . The components , are symmetric with respect to the real axis, is placed from the left side of whereas from the right.

Since , there exists a neighborhood of , and an holomorphic function on , such that for every is an element of a -cycle for . Since the Julia sets move holomorphically in , and in , we may assume that .

This -cycle is attracting for , and repelling for . Note that the attracting cycle for has length . Let

The function is holomorphic on , and its restriction to , is a bijection onto , moreover . Since is smooth at [5] we see that . Thus, from the fact that , where , we obtain

So, the function is increasing in an interval , for some .

3.2.

We will need the following lemma:

Lemma 3.1.

The parabolic cycle of contains at least one point where

Proof.

If has parabolic fixed point (i.e. ), then . So, we can assume that .

We will prove that if , then , where is a point from the parabolic cycle.

Let denote the inverse branch of , such that . Note that at every point from the parabolic cycle, and .

Since , we obtain

(3.1)

Because , we conclude that

So, we have , therefore

Finally, if , then (3.1) combined with the above inequality and the fact that , leads to . ∎

So, we can assume that . Later on, we will see that these estimates do not depend on choice of the point from the parabolic cycle of (see Proposition 9.7 and definitions from Sections 5 and 8‘).

Conjugating by we obtain

Since , we see that . Then, after conjugating by , we get

Because the coefficients , are polynomials, and of course , when , we obtain

Since , . Omitting , , and writing

(i.e. ) we have

Next, we get

(3.2)

and

(3.3)

Note that

(3.4)

Moreover, let us define

(3.5)

So, we have .

3.3.

There exists , such that the Julia set of moves holomorphically on an open and disjoint sets , and .

Let us first fix , (for instance taking the values corresponding to the center of the component). Then there exist two families of injections

conjugating to if , and to if .

Note that for every the function is holomorphic on .

The families , are equicontinuous, and thus, taking uniform limit as from the right or the left, there exist two functions such that

In the sequel, when the context is clear, we will allow ourselves to skip the subscript , and denote by .

3.4.

If then the trajectory of the critical point of is included in the real line. So, the trajectories of the critical points of (which tends to the parabolic points) are included in . Therefore the horizontal directions for the parabolic points are stable, whereas the vertical directions are unstable. Next, because

we conclude that (see the Fatou’s flower theorem [1]). The function is conjugated to by . So, let denote the scaling factor:

Let us assume that the parameter is close to (but ). Then, near the fixed point , there exists a periodic orbit of period two for , such that when . We conclude from (3.4) that satisfies

Since is small, we see that , and then

where in the case , we denote by the principal square root i.e. .

So, if (i.e. ) then the cycle is repelling (hence ) and the periodic points are conjugated. For (i.e. ) the cycle is attracting, and the periodic points are real.

3.5.

If , then we shall assume that . Let us define:

Next, for :

Moreover, later on we will need:

The Fatou’s flower theorem (see [1]) shows that the Julia set approaches the fixed point tangentially to the vertical direction. Now we state the perturbed version of this theorem.

Lemma 3.2.

For every there exist and such that

  1. if , then

  2. if , then

4. Fatou coordinates

In this section we introduce coordinates that we will Fatou-coordinates, even if they do not conjugate to an exact translation. We prove, that in this coordinates the family (after a modification) is close to the translation by 2, on the set near to the fixed point . We shall use results of Buff and Tan Lei (see [3]).

For (i.e. ) we define

If then, as before, we take . Notice that if , then .

Write

Since and , we conclude that and are the fixed points of .

The derivative is close to in a small neighborhood of , whereas the distortion is close to 1. So, using Lemma 3.2, we see that is included in . Moreover, if then .

We define the Fatou coordinates as follows (cf. [3, Example 1]):

The inverse functions are given by

(4.1)

Set , and then (cf. [3])

If then, as in [14] we have , and next we can get , where and is close to . In the case , we similarly obtain and . Thus, we can assume that is included in .

Lemma 4.1.

For every and there exist and such that if , then

Proof.

Let us consider a family of the form

where is holomorphic and . We see from [3, Lemma 5.1] that can be approximated by the flow of differential equation ( is solution of this equation), and then for every , and big enough, we can get

The assumptions of [3, Lemma 5.1] are satisfied because is -stable for every (see [3, Section 2, Example 1 (continued)]), and then the assumptions follows from [3, proof of Lemma 5.3].

Since and are the fixed points of , we conclude that in some neighborhood of 0, can be written in the form