Last 15th October 2014, the “flat seminar” coorganized by Anton Zorich, Jean-Christophe Yoccoz and myself restarted in a new format: instead of one talk per week, we shifted to one talk per month.
The first talk of this seminar in this new format was given by Alba Málaga, and the next two talks (on next November 12th and December 10th) will be given by Giovanni Forni (on the ergodicity for billiards in irrational polygons) and James Tanis (on equidistribution for horocycle maps): the details can be found here.
In this blog post, we will discuss Alba’s talk about some of the results in her PhD thesis (under the supervision of J.-C. Yoccoz) concerning a family of maps preserving the measure of (as hinted by the title of this post). Of course, any mistakes/errors in what follows are my entire responsibility.
In her PhD thesis, Alba studies the following family of dynamical systems (“cylinder flows”).
The phase space is where is the unit circle. We call the circle of level in the phase space.
The parameter space is .
Given a parameter , we can define a transformation of the phase space by rotating the elements of the circle of level by , and then by putting them at the level (one level up) or (one level down) depending on whether they fall in the first or second half of the circle of level . In other terms,
where is the rotation by on the unit circle .
Note that we have left undefined at the points such that or . Of course, one can complete the definition of by sending each of the points in this countable family to a level up or down in an arbitrary way. However, we prefer not do so because this countable family of points will play no role in our discuss of typical orbits of . Instead, we will think of the set of points where is undefined as a (very mild) singular set.
Alba’s initial motivation for studying this family comes from billiards in irrational polygons. Indeed, our current knowledge of the dynamics of billiard maps on irrational polygons (i.e., polygons whose angles are not all rational multiples of ) is very poor, and, as Alba explained very well in her talk (with the aid of computer-made figures), she has a good heuristic argument suggesting that the billiard map on an irrational lozenge obtained by small perturbation of an unit square can be thought as a small perturbation of some members of the family . However, we will not pursue further this direction today and we will focus exclusively on the features of from now on.
It is an easy exercise to check that, for any parameter , the corresponding dynamical system preserves the infinite product measure , where is the counting measure on and is the Lebesgue measure on .
In this setting, Alba’s thesis is concerned with the dynamics of for a typical parameter (in both Baire-category and measure-theoretical senses).
Before stating some of Alba’s results, let us quickly discuss the dynamical behavior of for some particular choices of the parameter .
Example 1 Consider the constant sequence . By definition, acts by a translation by on the -coordinate of all points of the phase space. In particular, the second iterate of any point has the form where . Furthermore, the function is not difficult to compute: since , we see that if , resp. , then resp. and, hence, . In other words, for all , and, thus, is a periodic transformation (of period two).
Example 2 Consider the constant sequence . Similarly to the previous example, acts periodically (with period ) on the -coordinate in the sense that where . Again, the function is not difficult to compute: by dividing the unit circle into the six intervals , , one can easily check that
In particular, we see that systematically moves the copy of an interval with even, resp. odd, at the circle of level to the corresponding copy , resp. , of the interval at level , resp. . In other terms, has wandering domains (i.e., domains which are disjoint from all its non-trivial iterates under the map) of positive -measure and, hence, is not conservative in the sense that it does not satisfy Poincaré’s recurrence theorem with respect to the infinite invariant measure : for example, for each , sends the subset of -measure always “upstairs” to its copy at the -th level, so that the orbits of points in escape to (one of the “ends”) in the phase space .
Remark 1 The reader can easily generalize the previous two examples to obtain that the transformation associated to the constant sequence with (a rational number written in lowest terms) is periodic or it has wandering domains of positive measure depending on whether the denominator is even or odd.
Today, we will give sketches of the proofs of the following two results: