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
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 1 As we already mentioned, Cantor’s ternary set
is a dynamical Cantor set:
where
is the affine map
.
Example 2 Let
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 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 .
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 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
:
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 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
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 3 The 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 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).
Leave a Reply