Hi! Today I would like to start my series of posts on some interesting lectures held during the School and Workshop of Dynamical Systems at ICTP (Trieste, Italy) with C. (Gugu) Moreira‘s talk about his recent work around -stable intersections of dynamically defined Cantor sets.
In order to motivate Moreira’s lecture, we will reserve this post just for the historical aspects of the theory of stable intersections of Cantor sets (and its applications), so that we will only discuss the proof of Moreira’s main result (to be stated below) in a nearby future.
The organization of this post is as follows. We begin by recalling two settings where the dynamically defined (a.k.a. regular) Cantor sets and their intersections arise naturally: firstly, we briefly outline some ideas behind M. Hall and G. Freiman theorems about the biggest half-line inside the Lagrange spectrum, and secondly we sketch the construction of open sets of dynamical systems exhibiting the so-called Newhouse phenomena. After that, we will state the main result of Gugu’s talk.
Let be an irrational number. Dirichlet’s theorem (based on the pigeonhole principle) says that there are infinitely many rational numbers such that
This result was later improved by Markov and Hurwitz (independently): they proved that, for any irrational , there are infinitely many rational numbers verifying
and is the best constant with this property: it is possible to show that for any , there are only finitely many rational numbers satisfying
where is the golden ratio. On the other hand, it is reasonable to expect that the constant can be improved for other specific irrational numbers . This lead us to the definition of the function
associating to each irrational its best constant .
Remark 1. In terms of the function , our previous discussion can be translated into the facts for all and .
Remark 2. It is not hard to prove that for almost every (in the sense of the Lebesgue measure).
The study of fine Diophantine properties of real numbers is encoded by the set
We call the Lagrange spectrum. In 1879, Markov showed that
is a countable set accumulating exactly at and any is a quadratic irrationality (i.e., with ). In other words, Markov was able to completely describe the ”beginning” of However, this structure (of a countable set) does not happens in the whole set . Indeed, this is the content of the following theorems:
Theorem 1(M. Hall, 1947). contains an entire half-line (e.g., ).
Theorem 2(G. Freiman, 1975). The biggest half-line contained in is
Of course it is impossible to give a complete proof of these results in this post, but we can provide some hints. The basic idea is the following: let be the set of real numbers whose continued fraction involves only the coefficients 1, 2, 3 and 4. It is possible to check that is a Cantor set (basically by the same reason that the ternary Cantor set – composed of number whose decimal on basis 3 involves only the coefficients 0 and 2 – is a Cantor set, i.e., a closed subset of the line with empty interior and without isolated points). Moreover, one can show (following M. Hall) that
Using this result, it is quite easy to show that the Lagrange spectrum contains a half-line (such as ). Observe that the set is formed by the parameters such that . In other words, the fact that contains an interval means that the Cantor sets and intersect in a “stable” way (i.e., any small translation of the Cantor set still meet ). For more details see the excelent book of Cusick and Flahive.
Therefore, this shows that the number-theoretical problem about the structure of the Lagrange spectrum can be handled by the investigation of sums and intersections of Cantor sets. This ends our first application (of number-theoretical nature) of Cantor sets.
Remark 3. After a closer look at the statements of the theorem of Markov, Hall and Freiman, the curious reader might be asking: what’s the structure of Lagrange spectrum in the “middle” part (i.e., the region of comprised between the half-lines and )? It turns out that this is the most exciting part of . Indeed, this is the content of the following result (whose final version is still in preparation by Moreira):
- is a continuous and surjective function;
- and ;
- (where stands for the Hausdorff dimension);
Moreover, the set of accumulation points of is a perfect set.
In other words, this theorem says that the Lagrange spectrum in the intermediary region is a Cantor set with a very intricate structure: by moving the parameter from to , we see that is a Cantor set whose Hausdorff dimension increases from 0 to 1. In particular, this means that is not a self-similar fractal (like the ternary Cantor set, the Sierpinski gasket and the Mandelbrot set) where every small piece resembles the whole set.
Of course it is hard to explain the proof of this profound result in a few lines, but I can give some key words appearing in the argument. Roughly speaking, we consider the Lagrange spectrum and we approximate some conveniently choosen piece of both from inside and outside by (regular) Cantor sets whose Hausdorff dimensions are sufficiently close (in order to perform this step one should investigate the Lagrange spectrum using continued fractions and the dynamics of the Gauss map so that the approximation problem becomes a question about the approximation of certain subshifts of finite type by complete shifts). After that, one applies a formula for the Hausdorff dimension of the arithmetic sum of two Cantors (which is derived via the so-called “Scale Recurrence Lemma” of Moreira and Yoccoz).
The basic reference for this section is the excelent book of Palis and Takens. We start with a -diffeomorphism of a manifold . For a fixed point of , we denote by the stable manifold of and the unstable manifold of . We say that is a homoclinic point associated to whenever . Moreover, if and are tangent at a homoclinic point , we say that is a homoclinic tangency associated to .
Perhaps one of the first mathematicians to notice the dynamical richness behind the existence of homoclinic points was H. Poincaré during his studies of the stability of the solar system: indeed, Poincaré observed that the presence of homoclinic orbits implies a divergence of series associated to the Hamiltonian equations of the -body problem.
Nowadays, we have several instances of confirmation of Poincaré’s intuition: for example, after the works of Birkhoff and Smale, we know that a transverse homoclinic intersection (i.e., not a homoclinic tangency) leads to the existence of infinitely many periodic points (of different periods) nearby the homoclinic point (this is far from obvious from the local picture). In fact, the modern (and geometric) proof of this result uses the so-called Smale’s horseshoe (found by Smale in the beaches of Rio 🙂 ). But, for this section, it is more interesting to see that homoclinic tangencies also lead to rich dynamics. To describe how one can extract rich dynamics from tangencies, let me introduce a few definitions:
Definition 1. A compact -invariant set is called hyperbolic whenever there is a decomposition such that is a uniform contraction and is a uniform expansion. For a point of a hyperbolic set , we define the stable and unstable manifolds of as
It is possible to show that these stable and unstable manifolds are immersed manifolds whose union over the points of give two -invariant (continuous) foliations and .
Example 0. The orbits of hyperbolic fixed (or periodic) points are trivial examples of hyperbolic sets.
Example 1. Smale’s horseshoe is a (very representative) non-trivial hyperbolic set.
Definition 2. The non-wandering set of is formed by such that for every neighborhood of one can find a positive integer with . We say that is hyperbolic if is a hyperbolic set.
Remark 4. The importance of the non-wandering set can be explained in the philosophical level as follows: one of the basic questions in the theory of dynamical systems is the long-time behaviour of all (or most orbits). Since it is easy to check that any orbit accumulates a point inside the non-wandering set, it “suffices” to understand the orbits of .
Remark 5. In the sequel, we will start with hyperbolic diffeomorphisms and we will perform perturbations in order to create homoclinic tangencies. The basic reason to begin with hyperbolic systems is quite simple: since the pioneering works of Anosov, Smale, Sinai, Bowen (among several others), we have a global complete picture of the dynamics of such systems (both from the topological and the statistical points of view). Therefore, the idea is deform the dynamics of a well-behaved systems to investigate what’s happening outside the hyperbolic realm.
After these preliminaries, we are ready to describe the Newhouse phenomena. We begin with a -diffeomorphism of a surface such that has a periodic point inside a hyperbolic set (e.g., a horseshoe) whose stable and unstable thickness and are sufficiently high (the precise condition is ). Here, the thickness is a real number measuring how “thick” is a given Cantor set of the real line (basically you compute the ratios between the bridges and the gaps in the construction of your Cantor set; see the book of Palis and Takens for the details). Also, the stable (resp. unstable) thickness of is the thickness of the Cantor set (resp. ) contained into the line (resp. ).
Assume that there is a homoclinic tangency associated to . The following picture (extracted from page 746 of this article of Moreira) contains a geometric description of the main features of :
Here is the line of tangencies (i.e., the line of tangent intersections between the extension of the foliations and to a neighborhood of the horseshoe ) and the Cantor sets , are the images of the Cantor sets , under the holonomy of the stable and unstable foliations (i.e., the intersections of the stable and unstable leaves of points of and with the line ).
Under these assumptions on , one can show that belongs to the closure of an open set of -diffeomorphisms with persistent tangencies: any can be -approximated by a diffeomorphism exhibiting a homoclinic tangency (between the stable and unstable manifolds of the horseshoe obtained by continuation of the horseshoe of ). In order to see why should be located at the boundary of such an open set , we begin with the following remark: given a diffeomorphism -close to denote by , the associated Cantor sets and suppose that and intersect. Then, by definition of these sets, has a homoclinic tangency. That is, the existence of homoclinic tangencies can be detected by the intersections of the Cantor sets and .
On the other hand, it is possible (using an important tool in dynamical systems called bounded distortion) to show that the Cantor sets and are close (in an appropriate topology) to the Cantor sets and (of ) whenever is -close to . In particular, since the Cantor sets and of are assumed to be very “thick”, one can prove that and have big thickness as well. At this point, we are ready to use the following lemma:
Gap Lemma. Let and be two thick Cantor sets (i.e., ). Then, one of the following possibilities occur:
- is contained in a gap of (i.e., a connected component of the complement of ),
- is contained in a gap of ,
For a clear proof of this lemma see the book of Palis and Takens. In view of this lemma, we can get the open set by perfoming some perturbations of such that the Cantor sets and moves in the “correct” direction. More precisely, in the picture above, we would like to make perturbations so that moves upwards and moves downwards on the line . Doing so, we can apply the gap lemma to obtain that (since their relative positions after these perturbations do not allow that one of them is completely inside the gap of the other). From our previous remark, this says that has a homoclinic tangency so that the desired is constructed.
Finally, assuming that the local dynamics at the periodic point is dissipative, i.e., where is the period of , one can take advantage of the abundance of tangencies in to create rich dynamics in the following sense: there is a residual subset (in the sense of Baire) such that any has infinitely many sinks (i.e., periodic points such that the derivative of along its orbit is contractive). This is the so-called Newhouse phenomena. This result is very important for at least two reasons. Firstly, it shows that for a big set of diffeomorphisms (in the topological sense) the dynamics is so complicated that there are infinitely many attractors. Thus, if we pick at random a point of the manifold , it is hard to decide (from the computational point of view for instance) the future of the orbit of because it can be attracted by any one of the infinitely many sinks. Secondly, it shows that hyperbolic diffeomorphisms are not -dense because hyperbolic diffeomorphisms are stable and they can support only finitely many sinks. This shows that the life is not so easy in the sense that we can’t hope to understand dynamical systems just by approximating a given model by an hyperbolic diffeomorphism and applying the wonderful theory of hyperbolic systems. Now, let us give a rough sketch of the construction of . The basic idea is the following: unfolding the homoclinic tangency and using the dissipativeness condition, one can analyse the dynamics near the almost tangency in order to conclude that the behavior of the orbits will be very close to the dynamics of a certain good model dynamics called tent map. From this fact, it is not hard to make another perturbation to construct a diffeomorphism with one sink. However, this is not sufficient because we need infinitely many at the same time. To accomplish this goal, we use two observations: firstly, a sink is stable (i.e., it is not destroyed by small perturbations) and, secondly, since is an open set of persistent tangencies, one can assume that we created in our previous argument a diffeomorphism with a sink and a new tangency somewhere else. Using these remarks, the reader knows how one can complete the argument: since the sink created is stable and we dispose of an “extra” tangency (after a small perturbation), we can repeat the previous argument to construct two sinks and a tangency. After a new smaller perturbation, it follows that we can create three sinks and a tangency. Continuing in this way, we obtain our diffeomorphism with infinitely many sinks (and in fact the same argument gives the desired residual set ). Of course, there are several details to be checked but this is the idea behind Newhouse phenomena. This completes our second application of Cantor sets.
Remark 6. The careful reader certainly noticed our implicit emphasis on the -regularity assumption of the diffeomorphisms during our discussion of the Newhouse phenomena. It turns out that this assumption is crucial (e.g., for the bounded distortion argument) and in fact this is one of the motivations of Moreira’s theorem as we are going to see below.
–Statement of Moreira’s theorem on -stable intersections–
One of striking conclusions of the Newhouse phenomena discussed above was the fact that hyperbolic diffeomorphisms are not -dense. However, as we pointed out in remark 6, Newhouse’s argument do not extend to the case of -diffeomorphisms. Indeed, it is believed that the opposite is true:
Smale’s conjecture. Hyperbolicity is -dense among surface diffeomorphisms.
Remark 7. Nowadays it is well-known that this conjecture is false in higher dimensions (in any topology) due to the works of Abraham and Smale, Shub and Mañé. However, the explanation for the existence of robust sets of non-hyperbolic diffeomorphisms in high dimensions involves other non-hyperbolic mechanisms than homoclinic tangencies (related to the theory of partial hyperbolicity).
As far as we know, Smale’s conjecture remains open. On the other hand, from Newhouse phenomena we see that any reasonable attack to Smale conjecture should treat the problem of homoclinic tangencies (which constitutes the main obstruction to the hyperbolicity of surface diffeomorphisms as the proof of Palis conjecture by Pujals and Sambarino shows us). Of course, as we learned in the previous section, a toy model of the problem of avoiding tangencies is the question of how frequently two Cantor sets intersect in the -topology. In this direction, Moreira says that there are no obstructions at the (one-dimensional) level of Cantor sets:
Theorem (Moreira). Typically in the -topology, two regular Cantor sets do not intersect.
Note that this is an informal statement because we do not know (at this stage) neither what is a regular Cantor set nor what is the -topology in the space of regular Cantor sets.
In any case, we will postpone these technical details for the next post where we pretend to discuss the proof of this result.
This completes our (long) considerations for today! I hope to see you again at the next post! A bientot!