Many problems in several areas of Mathematics (including Dynamical Systems and Number Theory) can “reduced” to the analysis of *dynamical Cantor sets*: for instance, the theorems of Newhouse, Palis and Takens on homoclinic bifurcations of surfaces diffeomorphisms, and the theorems of Hall, Freiman and Moreira on the structure of the classical Lagrange and Markov spectra rely on the study of dynamical Cantor sets of the real line.

The wide range of applicability of dynamical Cantor sets is partly explained by the fact that several natural examples of Cantor sets are defined in terms of Dynamical Systems: for example, Cantor’s ternary set is

where is .

In some applications of dynamical Cantor sets, it is important to dispose of estimates on their Hausdorff dimensions: for instance, the celebrated work of Bourgain and Kontorovich on Zaremba’s conjecture needs particular types of dynamical Cantor sets with Hausdorff dimension close to one.

For this reason, a considerable literature on this topic was developed. Among the diverse settings covered by many authors, one finds the articles of Bumby, Hensley, …, Jenkinson-Pollicott, Falk-Nussbaum, where the so-called *thermodynamical methods* are exploited to produce approximations for the Hausdorff dimension of Cantor sets defined in terms of continued fraction expansions (i.e., Cantor sets of number-theoretical nature).

In general, the thermodynamical methods quoted above provide a sequence of fast-converging approximations for the Hausdorff dimension of dynamical Cantor sets: for instance, the algorithm described by Jenkinson-Pollicott here gives a sequence converging to at *super-exponential speed*, i.e., for some constants and , where is the Cantor set of real numbers whose continued fraction expansions contain only and .

In particular, the thermodynamical methods give good *heuristics* for the first several digits of the Hausdorff dimension of dynamical Cantor sets (e.g., if we list for and the first three digits of coincide for all , then it is likely that one has found the first three digits of ).

The heuristic bounds provided by the thermodynamical methods can be turned into rigorous estimates: indeed, one of the goals of the recent work of Jenkinson-Pollicott consists into rigorously computing the first 100 digits of .

However, the conversion of heuristic bounds into rigorous estimates is not always easy, and, for this reason, sometimes a *slowly* converging method producing two sequences and of *rigorous* bounds (i.e., for all ) might be interesting for practical purposes.

In this post, we explain a method described in pages 68 to 70 of Palis-Takens book giving explicit sequences converging slowly (e.g., for some constant and all ) towards , and, for the sake of comparison, we apply it to exhibit crude bounds on the Hausdorff dimensions of some Cantor sets defined in terms of continued fraction expansions.

**1. Dynamical Cantor sets of the real line **

Definition 1A -dynamical Cantor set is

where:

- is an expanding -map (i.e., for every ) from a finite union of pairwise disjoint closed intervals to the convex hull of ;
- is a Markov partition, that is, is the convex hull of the union of some of the intervals , and
- is topologically mixing, i.e., for some , for all .

Example 1As we already mentioned, Cantor’s ternary set is a dynamical Cantor set:

where is the affine map .

Example 2Let be a finite alphabet of finite words . The Cantor set

of real numbers whose continued fraction expansions are given by concatenations of the words in is a dynamical Cantor set.In fact, it is possible to construct intervals such that

where and is the Gauss map: see, e.g., this paper here for more explanations.

By definition, a dynamical Cantor set can be inductively constructed as follows. We start with the Markov partition given by the connected components of the domain of the expanding map defining . For each , we define as the collection of connected components of , .

For later use, for each , we denote by

** 1.1. Hausdorff dimension and box counting dimension **

Recall that the Hausdorff dimension and the box-counting dimension of a compact set are defined as follows.

The Hausdorff -measure of is

and the Hausdorff dimension of is

The box-counting dimension is

where is the smallest number of intervals of lengths needed to cover .

Exercise 1Show that .

** 1.2. Upper bound on the dimension of dynamical Cantor sets **

Let be a -dynamical Cantor set associated to an expanding map . Consider the quantities defined by

*Proof:* Let and fix . By definition, there exists such that

for all .

In other words, given , we can cover using a collection of intervals with such that every has length .

It follows from the definitions that, for each , the pre-images of the intervals under form a covering of by intervals of length . Therefore,

for all and , and, *a fortiori*,

for all and .

Hence, if we define , then

for all .

By iterating this argument times, we conclude that

for all .

Thus,

Since is arbitrary, we deduce from the previous inequality that

Because , we get that , and, *a fortiori*, .

** 1.3. Lower bound on the dimension of dynamical Cantor sets **

Let be a -dynamical Cantor set associated to an expanding map . Consider the sequence defined in the previous subsection and fix be a constant such that for all (e.g., certainly works).

Take such that for all and set

Remark 1If is a full Markov map, i.e., for all , then we can choose and .

Consider the quantities given by

*Proof:* Suppose by contradiction that and take .

By definition, , so that for every there is a finite cover of with

Note that any interval of length strictly smaller than

intersects at most one .

Thus, if we define , then each element of the cover of intersects at most one element of . Hence, if we define

then, given any , one has that has *fewer* elements than .

Consider such that for all . From the definitions, for each , we see that is a well-defined cover of such that

Since , we get that

We want to exploit this estimate to prove that

for some . In this direction, suppose that

In this case, the discussion above would imply

a contradiction because on one hand (by definition of ) and on the other hand (by our choice of ).

In summary, we assumed that , we considered an arbitrary cover of with

and we found a cover of with *fewer* elements than such that

By iterating this argument, we would end up with a cover of containing no elements, a contradiction. This proves that .

** 1.4. Slow convergence of towards the Hausdorff dimension **

Let be a -dynamical Cantor set associated to an expanding map . In general, the sequences discussed above converge slowly towards :

*Proof:* The so-called *bounded distortion property* (see Theorem 1 in Chapter 4 of Palis-Takens book or this blog post here) ensures the existence of a constant such that

for all and .

Let and, for all , define

In this setting, we have that

Therefore, , that is,

This proves the proposition.

**2. Hausdorff dimension of Gauss-Cantor sets **

In this section, we apply the previous discussion in the context of Gauss-Cantor sets , that is, the dynamical Cantor sets from Example 2 above.

For the sake of simplicity of exposition, we will not describe the calculation of the sequences and approaching for a general , but we shall focus on two particular examples.

** 2.1. Some bounds on the Hausdorff dimension of **

Consider the alphabet consisting of the words and . The corresponding Cantor set is the Cantor set — denoted by in the beginning of this post — of real numbers whose continued fraction expansions contain only and .

The theory of continued fractions (see Moreira’s paper and Cusick-Flahive book) says that is the dynamical Cantor set associated to the restriction of the Gauss map to , where

Note that the functions and defined on the interval are the inverse branches of .

Remark 2Here, denotes the infinite word obtained by periodic repetition of the block .

By applying to the Markov partition , we deduce that consists of the intervals , , with extremities

The quantities and are not hard to compute using the following remarks. First, is monotone on each (because are Möbius transformations induced by integral matrices with determinant ), so that the values and are attained at the extremities of . Secondly, the derivative of the Gauss map is . By combining these facts, we get that

and

Hence, the sequences and defined in the previous section are the solutions of the equations

and

(Here, we used Remark 1 and the fact that is a full Markov map in order to get the equation of .)

Of course, these equations allow to find the first few terms of the sequences and approaching with some computer-aid: for example, this Mathematica routine here shows that

and

In particular,

Remark 3The approximations and of (obtained from computing with the words in ) are very poor in comparison with the approximation

provided by the thermodynamical method of Jenkinson-Pollicott (also based on calculations with the words in ). In fact, Jenkinson-Pollicott showed in this paper here that the first 100 digits of are

Thus, the first 18 digits of accurately describe (while only the first two digits of and the first digit of are accurate).

** 2.2. Some bounds on the Hausdorff dimension of **

Let us consider now the alphabet consisting of the words and . The theory of continued fractions says that the convex hull of is the interval with extremities and . The images and of under the inverse branches

of the first two iterates of the Gauss map provide the first step of the construction of the Cantor set . In general, given , the collection of intervals of the th step of the construction of is given by

Hence, the interval associated to a string has extremities and .

Similarly as in the previous subsection, we conclude that the quantities and are given by

and

Thus, and are the solutions of

and

Once more, we can calculate the first terms of the sequences and with some computer-aid: this Mathematica routine here reveals that

and

In particular,

Remark 4After implementing of the Jenkinson-Pollicott method with this Mathematica routine here, we get the following approximations for :

The super-exponential convergence of Jenkinson-Pollicott method suggests that

Note that the first two digits of this approximation are accurate (because the first two digits of and coincide).

## Leave a Reply