# A new perspective on the Sullivan dictionary via Assouad type dimensions and spectra

## Abstract

The Sullivan dictionary provides a beautiful correspondence between Kleinian groups acting on hyperbolic space and rational maps of the extended complex plane. We focus on the setting of geometrically finite Kleinian groups with parabolic elements and parabolic rational maps. In this context an especially direct correspondence exists concerning the dimension theory of the associated limit sets and Julia sets. In recent work we established formulae for the Assouad type dimensions and spectra for these fractal sets and certain conformal measures they support. This allows a rather more nuanced comparison of the two families in the context of dimension. In this expository article we discuss how these results provide new entries in the Sullivan dictionary, as well as revealing striking differences between the two families.

## 1. Introduction

Seminal work of Sullivan in the 1980s Reference 39 resolved a long-standing problem in complex dynamics by proving that the Fatou set of a rational map has no wandering domains. This work served to establish remarkable connections between the dynamics of rational maps and the actions of Kleinian groups. This connection subsequently stimulated activity in both the complex dynamics and hyperbolic geometry communities and led to what is now known as the *Sullivan dictionary*; see, for example, Reference 30. The Sullivan dictionary provides a framework to study the relationships between Kleinian groups and rational maps; see Table 1. In many cases there are analogous results, even with similar proofs, albeit expressed in a different language; see Reference 12, Table 1 and also Reference 41 and references therein.

Both Kleinian groups and rational maps generate important examples of dynamically invariant fractal sets: *limit sets* in the Kleinian case, and *Julia sets* in the rational map case; see Figure 1. The Sullivan dictionary is very well suited to understanding the connections between these two families of fractals, and the correspondence is especially strong in the context of dimension theory: in both settings there is a *critical exponent* which, for certain classes of Kleinian groups and rational maps, describes all of the most commonly used notions of fractal dimension. For Kleinian groups the critical exponent is the Poincaré exponent, denoted by and for rational maps the critical exponent is the smallest zero of the topological pressure, denoted by , For both nonelementary geometrically finite Kleinian groups and parabolic (or hyperbolic) rational maps, the critical exponent coincides with the Hausdorff, packing, and box dimensions of the associated fractal as well as the Hausdorff, packing, and entropy dimensions of the associated ergodic conformal measure of maximal dimension. .

There has been a recent increase in interest in the *Assouad type dimensions*, and these dimensions (and associated dimension spectra) do not behave in such a straightforward manner in the presence of parabolicity. In particular, the critical exponent does *not necessarily* give the Assouad dimension of the associated fractals. As we shall see, by slightly expanding the family of dimensions considered, a much richer and more varied tapestry of results emerges. In this expository paper we discuss recent work from Reference 19Reference 21Reference 22 and show how this can be used to provide a new perspective on the Sullivan dictionary.

## 2. Definitions and background

### 2.1. Dimensions of sets and measures and dimension interpolation

We recall and motivate the key notions from fractal geometry and dimension theory which we use. For a more in-depth treatment see the books Reference 6Reference 18 for background on Hausdorff and box dimensions, and Reference 20 for Assouad type dimensions. We work with fractals in two distinct settings. Kleinian limit sets will be compact subsets of the sphere -dimensional which we view as a subset of On the other hand, Julia sets will be compact subsets of the Riemann sphere . However, by a standard reduction we will assume that the Julia sets are compact subsets of the complex plane . which we identify with see Section ;2.3. Therefore, it is convenient to recall dimension theory for nonempty compact subsets of Euclidean space only.

Throughout this section, let be nonempty and compact. Perhaps the most commonly used notion of fractal dimension is the Hausdorff dimension, but it will be especially important for us to consider several notions of dimension together. We write , , and *Hausdorff, box, upper box*, and *lower box dimensions* of

to denote the diameter of *Assouad dimension* of

The lower dimension is the natural *dual* to the Assouad dimension, and it is particularly useful to consider these notions together. The *lower dimension* of

provided

The Assouad and lower spectra were introduced much more recently in Reference 23 and provide an *interpolation* between the box dimension and the Assouad and lower dimensions, respectively. The motivation for the introduction of these *dimension spectra* was to gain a more nuanced understanding of fractal sets than that provided by the dimensions considered in isolation. This is already proving a fruitful programme with applications emerging in a variety of settings including to problems in harmonic analysis; see work of Anderson, Hughes, Roos, and Seeger Reference 2 and Reference 34. These spectra provide a parametrised family of dimensions by fixing the relationship between the two scales *Assouad spectrum* of

The *lower spectrum* of

In particular,

There is an analogous dimension theory of measures, and the interplay between the dimension theory of fractal sets and the measures they support is fundamental to fractal geometry, especially in the dimension theory of dynamical systems. For example, a problem of interest is to identify dynamical measures witnessing the dimension of the support, e.g., invariant measures of full Hausdorff dimension. Let *support* of *fully supported* on

see Reference 29. The *Assouad dimension* of

and, provided *lower dimension* of

and otherwise it is 0. By convention we assume that

and, furthermore, we have the stronger fact that

and

For *Assouad spectrum* of *lower spectrum* of

It is known (see Reference 17 for example) that

and, if

There are also upper and lower box dimensions for measures, recently introduced in Reference 17. We omit the formal definitions, referring the reader to Reference 17Reference 20. Following Reference 17, it is useful to note that

with an analogous result for the lower box dimension. Furthermore, it was shown that the upper box dimension of

and so

### 2.2. Kleinian groups and limit sets

For a more thorough study of hyperbolic geometry and Kleinian groups, we refer the reader to Reference 4Reference 28. For

equipped with the hyperbolic metric

the *boundary at infinity* of the space *Kleinian* if it is a discrete subgroup of *Fuchsian* in the case when *limit set* of *elementary*, and otherwise it is *nonelementary*. In the nonelementary case, *geometrically finite* Kleinian groups. Roughly speaking, this means that there is a fundamental domain with finitely many sides, but we refer the reader to Reference 8 for a precise definition. We define the *Poincaré exponent* of a Kleinian group

Due to work of Patterson and Sullivan Reference 32Reference 38, it is known that for a nonelementary geometrically finite Kleinian group *infinite* case, *radial* limit set, and therefore always provides a lower bound for the Hausdorff dimension of the limit set; see Reference 5.

From now on we only discuss the nonelementary geometrically finite case. We write *Patterson–Sullivan measure*, which is a measure first constructed by Patterson in Reference 32. Strictly speaking, there is a family of (mutually equivalent) Patterson–Sullivan measures. However, we may fix one for simplicity (and hence talk about *the* Patterson–Sullivan measure since the dimension theory is the same for each measure). The geometry of

If

see Reference 19. Therefore, we assume from now on that

Let *rank* of

It was proven in Reference 38 that

### 2.3. Rational maps and Julia sets

For a more detailed discussion of the dynamics of rational maps, see Reference 11Reference 31. Let *Julia set* of *point at infinity* and noting that the case when the Julia set is the whole of

A periodic point *rationally indifferent* (or *parabolic*) if *parabolic* if

If *hyperbolic* and, analogous to case of geometrically finite Kleinian groups with no parabolic elements,

see Reference 21. Therefore, we assume from now on that

Write

As

We call *petal number* of

It was proven in Reference 1 that

## 3. A new perspective on the Sullivan dictionary

### 3.1. Recent results on Assouad type dimensions and spectra

In this section we state various recent results concerning geometrically finite Kleinian groups with parabolic elements and parabolic Julia sets. These results provide a new perspective on the Sullivan dictionary in the context of dimension theory. We will examine this new perspective more thoroughly in Sections 3.2 and 3.3. The Assouad and lower dimensions of limit sets of geometrically finite Kleinian groups and associated Patterson–Sullivan measures were found in Reference 19. The analogous results for parabolic Julia sets were proved in Reference 21. The results concerning Assouad type spectra were proved in Reference 21Reference 22. Throughout we fix

#### 3.1.1. Patterson–Sullivan measure

#### 3.1.2. Kleinian limit sets

#### 3.1.3. -conformal measures