Hi! A few months ago, my friend Carlos Gustavo (Gugu) Moreira posted at the IMPA’s preprint server an article entitled “Geometric properties of the Markov and Lagrange spectra” explaining the proofs of his results on the Markov and Lagrange spectrum (see this previous post for an introduction of these spectra and the statements of Gugu’s results). Today, we’ll discuss the dynamical aspects of Gugu’s results. However, before starting the discussion of this preprint, let me take the opportunity to congratulate Gugu: he managed to post the first version of his interesting article at the same time of his first son’s birth! Moreover, let me thank my wife Aline Gomes Cerqueira whose nice comments helped me to clarify my thoughts about Gugu’s argument (and also helped the improvement of this work by her PhD advisor) .
[Update: Ops, I forgot to congratulate Gugu also for his recent UMALCA Prize 2009!]
[Update (August 11, 2009): The mistakes pointed out by Yuri are now fixed.]
During this section, we review the discussion of the first section of the this post. Given an irrational number , Dirichlet’s theorem claims that the inequality
has infinitely many rational solutions (in fact, it is a simple exercise to the reader to check that this is a direct consequence of Dirichlet’s pigeonhole principle). Furthermore, Markov and Hurwitz proved that the inequality
has infinitely many rational solutions for all irrational and is the biggest constant with this property, namely, for every ,
has only a finite number of rational solutions.
Nevertheless, during the study of Diophantine properties of specific irrational numbers , it is interesting to introduce the function
assigning to each irrational number its best rational approximation constant . Observe that Khintchine’s theorem says that the set of irrational numbers with has full Lebesgue measure.
In any case, we can consider the Lagrange spectrum
formed by the collection of finite best constants .
Concerning the structure of , Markov (1879) showed that
where are explicit quadratic irrationals (i.e., , , i.e., by Lagrange theorem, has periodic continued fraction expansion). In particular, the beginning of is discrete. On the other hand, M. Hall (1947) showed that and G. Freiman (1975) determined the biggest half-line contained , namely, he proved that is
Therefore, it remains only to understand the structure of middle of . This is the content of Gugu’s preprint.
However, before entering this issue, let me mention that the arguments of Hall and Freiman use the study of arithmetic sums of dynamically defined Cantor sets. The interaction between the dynamically defined (regular) Cantor sets and the Lagrange spectrum occurs via the continued fraction expansion. We’ll explain this relationship later (when we introduce the dynamically defined Cantor sets associated to the Gauss map). In any case, we can reformulate the Lagrange spectrum in the language of Dynamical Systems as follows. We observe that, for any irrational number (where () is the usual continued fraction notation), it holds
where and . Indeed, this fact follows from the identity (which can be checked by induction). Keeping this notation in mind, we can give an alternative definition of the Lagrange spectrum:
and we introduce the function given by
In this setting, the Lagrange spectrum is
where is the shift dynamics .
Of course, this alternative definition of the Lagrange spectrum motivates the introduction of the Markov spectrum :
Remark 1 admits the following arithmetic characterization:
Remark 2 It is possible to prove that and are closed subsets of such that .
A nice reference for the theory of continued fractions and the Markov and Lagrange spectrum containing the proofs of all facts quoted above is the book of Cusick and Flahive.
At this point, let us state the main theorems of Gugu’s preprint (concerning about the middle part of ).
The main three results of Gugu’s preprint are:
Theorem 3 Given , we have (where stands for the Hausdorff dimension of the set ). Furthermore, is a continuous and surjective function from to . Moreover,
- where is a continuous function from to ;
Remark 3 A very interesting consequence of theorem 3 is the fact that is not Hölder continuous (with respect to any exponent ). Indeed, sends the subset of Hausdorff dimension onto the interval for every . Thus, if is -Hölder continuous for some , one would have for all , a contradiction (for a sufficiently small ).
In the sequel, we’ll content ourselves with the discussion of the first part of theorem 3 only (i.e., the continuity of ). The starting point of Gugu’s argument is inspired by the proofs of the theorems of M. Hall and G. Freiman, namely, one consider the dynamically defined Cantor sets related to the Gauss map. Then, the basic idea during the proof of the continuity properties of the function is the combination of some approximation arguments of the Lagrange spectrum by the arithmetic sum of two regular Cantor sets with the fact that the Hausdorff dimension of these arithmetic sums are well-behaved due to the so-called dimension formula. Now, let’s turn to the details.
A fundamental tool in Gugu’s article is the so-called dimension formula (which will appear in a forthcoming paper by Gugu). Basically, it says that the Hausdorff dimension of the arithmetic sum of generic Cantor sets , is . In order to properly state this theorem, we introduce the notion of non essentially affine regular Cantor sets. We recall that a -regular (i.e., dynamically defined) Cantor set is the maximal invariant set of a transitive expanding map from a disjoint finite union of compact intervals to the real line verifying the Markov partition property (see this post for more details). Observe that, for every periodic point of period of , one can find a diffeomorphism of the convex hull of such that is affine in , where is the connected component of the domain of containing .
Definition 6 We say that is non essentially affine whenever we can find a periodic point (as above) such that the map verifies for some .
In other words, is non essentially affine if it is not possible to perform a conjugation of the dynamics so that it becomes affine near the corresponding Cantor set. Using this notion, Gugu will show (in a forthcoming article) the following theorem:
Remark 4 The proof of this result uses in a crucial way the previous work of Gugu and J.C. Yoccoz about the stable intersections of -Cantor sets. More precisely, Gugu plans to use the so-called Scale Recurrence Lemma of this paper to prove the dimension formula. In fact, the curious reader can consult the article of P. Shmerink (15 pages) for a complete proof (along these lines) of Gugu’s dimension formula under slightly stronger assumptions (of non-resonance).
Remark 5 The appearance of the term on the right-hand side of the dimension formula has a natural explanation: as we saw in a previous post about Marstrand’s theorem, the arithmetic sum is (essentially) the orthogonal projection of the product set into the diagonal of (making angle of with the -axis). Hence, the dimension formula says that, for generic Cantor sets, this projection has the correct dimension, i.e., (the Hausdorff dimension of ).
Assuming the validity of the dimension formula, we can begin the discussion of the proof of theorem 3.
In Number Theory, the Gauss map is the fractional part of (this should not be confused with the Gauss map from the Differential Geometry). Of course, the iterates of are intimately related to the continued fraction algorithm. A central role in the subsequent arguments will be played by the regular Cantor sets associated to complete shifts via the Gauss map. More precisely, given a finite set () of finite sequences () of positive integers such that doesn’t start with for every , we introduce
Definition 8 The complete shift associated to is the subset of formed by the sequences obtained by concatenations of the elements for every .
Using the complete shift and the Gauss map, we can construct a regular Cantor set:
Definition 9 We denote by the Cantor set of real numbers whose continued fractions are sequences of , that is,
Exercise 1 Show that is a regular Cantor set. (Hint: For each , , consider the interval where and and define )
In view of the shape of the graph of the Gauss map, the following proposition is very natural:
Proof: For each (say ), let us denote by be the fixed point of . From the theory of continued fractions (see Cusick and Flahive), we know that
where is the -th rational approximation of (for ). It follows that is a positive solution of .
On the other hand, since is a Möbius transformation with an expanding fixed point , we can find a Möbius function such that , and is affine. This reduces our task to show that is not affine (because the second derivative of a non-affine Möbius transformation never vanishes).
Assuming that is affine, we see that is a common fixed point of and (since the point at infinity is a common fixed point of the affine maps and ) and, a fortiori, it is a common solution of the quadratic equations for . Because these two quadratic polynomials of are irreducible (since and are irrational numbers), they must coincide. In particular, this forces , a contradiction with .
Once we know that the regular Cantor sets are non essentially affine, we’ll try to combine this information with the dimension formula (for the arithmetic sum of Cantor sets) in order to show the continuity of . Note that this is a natural approach because the Lagrange and Markov spectra are naturally related to the values of a certain (sum) function over the orbits of bi-infinite sequences by the shift map. Before starting the discussion of this topic, let us close this section with some remarks and interesting facts.
Remark 6 Given a sequence , we define its transposition by . Of course, this notion extends to finite sets of sequences (via . In this notation, we have the classical fact about the continuants of the continued fractions associated to and (see the appendix of Cusick and Flahive). Using this fact, it is not hard to prove that .
Remark 7 This corollary is related to the fact that (see the theorem 3).
Denote by where and and let be the shift operator. Recall that the Lagrange spectrum is and the Markov spectrum is , where and (see the definition 1 for the definition of and ).
Remark 8 If , then . In particular, if , we have that belongs to M. Hall’s ray (since ).
Therefore, since we are interested in the continuity properties of the function , the previous remark allows us to make the following assumption:
From now on, we’ll always assume that for every , and .
In our approach to and , it is convenient to introduce the Lagrange and Markov spectrum associated to a complete shift :
Definition 12 Let be a complete shift. Its Lagrange spectrum is and its Markov spectrum is .
A nice feature of the Lagrange and Markov spectrum of complete shifts is:
Lemma 13 (lemma 2 of Gugu’s paper) Let be a complete shift. Then, it holds
Proof: The reader clearly sees that , where $R$ is the length of the biggest word of $B$, so that
On the other hand, given , it suffices to construct two regular Cantor sets and such that and . In this direction, we note that, up to replacing by for a large , one can assume that for any with . Next, we order the elements of (resp. ) as follows: given (resp. ), we say that if and only if . Using this total order, we can select and (resp. and ) the minimal and maximal elements of (resp. ). We define
Observe that and (here we used the remark 6). Of course, since we removed the minimal and maximal elements of and , we expect that the values of on decrease in view of the following classical comparison result of the theory of continued fractions:
if and only if , where is the smallest integer such that (this parity issue during the comparison of two continued fractions justifies the exclusion of the minimal and maximal elements of ).
Unfortunately, the exclusion of the minimal and maximal elements of and is not sufficient to ensure that (in fact, although the values of on decrease, this doesn’t guarantees that they belong to . However, this technical problem can be solved by considering some smaller replicas of and . Pick , , a point of where the maximum of is attained (in a position associated to ). Using , for each , we construct the subsets formed of the sequences
where for every and for every . By a compactness argument (involving the fact that we can eventually increase the value of by replacing some of its elements by or ), Gugu shows that there exists and a large such that, for every , the value is attained at a position associated to the central
and at any position outside . Next, we pick an arbitrary such that and we associated to each the sequence
Observe that, for each position of the central block, doesn’t depend on , so that , the function is a Möbius transformation (for such positions ). Moreover, these functions are mutually distinct. In particular, the values (for the positions of ) are distinct except for a finitely many choices of . This allows us to take such that for every two distinct positions and of . Let’s denote by the position of where the value is maximum.
Now, we take large such that the enlarged block verifies the following property: we have
for any with and (), where
The existence of is a consequence of the choice of (so that the is attained at some position of ), the fact that the values of are mutually distinct at the positions associated to (so that for every position of ) and the fact that the values of are very close to for a sufficiently large .
Finally, we define
Observe that is a (diffeomorphic) copy of and is a (diffeomorphic) copy of , so that and .
We claim that . Indeed, given
it is not hard to check that where
and . This ends the proof of the lemma.
During the study of the function , it is interesting to introduce the subshifts given by
and the associated Cantor sets
where is the natural projection. For later use (in the arguments related to Hausdorff dimension), we’ll say that the size of a finite sequence of strictly positive integers is the length of the interval
Note that the extremities of are and , so that .
We also introduce ,
for each , and
The cardinatily of is denoted by . In this setting, it is possible to show that the box dimension of is
This is expected since the collection corresponds to the natural intervals (with respect to the continued fraction algorithm) of scale covering the set . For more details see Gugu’s preprint.
Proof: In fact, from the theory of continued fractions, we have that for any finite sequences and any . It follows that one can cover using the intervals of the form , where , , . Furthermore, these intervals verifies . In particular, , so that
In other words, the sequence is sub-aditive. Hence,
Using this information, we can conclude that is continuous: otherwise, one can find some and such that for every (since is monotone). In view of the previous identity, it follows that we can select some large integer such that
for every and . On the other hand, because for any and (by a simple compactness argument), by taking the limit when , we see that the previous estimate implies
a contraction. This completes the proof of the proposition.
Now, we are able to state the following lemma about the approximation of the Lagrange/Markov spectrum by regular Cantor sets associated to complete shifts:
Lemma 15 (lemma 1 of Gugu’s paper) Given and , we can find and a regular Cantor set associated to a complete shift such that
Proof: Of course, it is not hard to approximate by a complete shift (in order to ensure is close to ). Indeed, consider and fix large so that
for any . Now, define and put . Take and we introduce
It is not hard to check that the Hausdorff dimension of is close to : firstly, a simple counting argument shows that ; in particular, since the intervals , are a covering of such that (exercise), we see that , but the main problem is the fact that we don’t have any control on the values of on (since we would like to decrease them from to ). In order to overcome this difficulty, Gugu introduces the notion of lef-good and right-good positions of a given .
More precisely, we say that is a right-good position of whenever one can find two elements () of such that the following inequality holds
Similarly, is a left-good position of whenever one can find two elements () of such that
Finally, we say that is a good position of if it is both a left-good and a right-good position.
By a crude bound on the number of bad positions (namely, given a position, there are only two choices of making a bad position: ), it is possible to prove that the quantity of words of with good positions is . Such words are called excellent words by Gugu. Of course, we can hope to decrease the values of on any excellent words (by some fixed amount ), but a new problem emerges: it may happen that the subset of formed by excellent words doesn’t give a complete subshift (e.g., it may be not possible to concatenate excellent words). At this stage, Gugu employs a exclusion procedure combined with a double counting argument to show that we can build a complete shift by focusing at a large portion of the good positions of and fixing the words of appearing into these good positions. Namely, Gugu shows that there are special good positions (i.e., they are good positions such that are also good positions) such that the subset of excellent words whose entries at these good positions are equal to conveniently chosen words (i.e., and verifies
Furthermore, Gugu proves that there are two integers such that the image of the projection of the form
is the desired subset of words such that (essentially by the definition of good position) and (since , , is a covering of composed of intervals verifying ).
This completes the proof of the lemma.
–End of the proof of theorem 3–
On the other hand,
Putting these four estimates together and letting , we conclude the proof of the claim. Observe that this argument also shows that .
Finally, the continuity of follows from the formula and the fact that is continuous (see the proposition 14).
Remark 9 From the identity , we see that the first item of theorem 3 is equivalent to (where ). Although this is not terribly difficult to show, we are not going to prove this fact here.