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 1 A -dynamical Cantor set is
- 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 1 As we already mentioned, Cantor’s ternary set is a dynamical Cantor set:
where is the affine map .
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 1 Show 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 .
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 1 If 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 :
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 .
Note that the functions and defined on the interval are the inverse branches of .
Remark 2 Here, 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
Hence, the sequences and defined in the previous section are the solutions of the equations
(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
Remark 3 The approximations and of (obtained from computing with the words in ) are very poor in comparison with the approximation
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
Thus, and are the solutions of
Once more, we can calculate the first terms of the sequences and with some computer-aid: this Mathematica routine here reveals that
Remark 4 After 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).