The present post will focus on the paper Law of large numbers for certain cylinder flows in collaboration with Patricia Cirilo and Enrique Pujals, in which we construct a class of cylinder flows that are rationally ergodic along a subsequence of iterates (and thus have explicit law of large numbers). See CF3 for the definitions.
Firstly, we remind what is known: let be the Haar function, defined as
As we proved in CF2, the associated cylinder flow is ergodic, for any irrational . With respect to rational ergodicity, the only result is from J. Aaronson and M. Keane. They proved in The visits to zero of some deterministic random walks that if is quadratic surd, then is rationally ergodic. An irrational number is quadratic surd if it satisfies a quadratic equation with integer coefficients or, equivalently, if its continued fraction expansion is pre-periodic. Specifically, they proved that is rationally ergodic along the sequence of denominators of . When is quadratic surd, the jumps between two consecutive `s are not dramatic and then one can interpolate the rational ergodicity along to every positive integer.
It is not known to what extent their result can be extended to every irrational basis. Indeed, for a generic , the jumps between the `s are large and so it is natural first to ask about rational ergodicity along a subsequence of iterates, for some roof function (not necessarily ). Our goal is to prove that such phenomenon happens for almost every .
Our class of will satisfy the following properties: there exists a sequence of (non consecutive) denominators of such that
- converges to zero as goes to infinity, and
- divides .
Appendix B of the paper is devoted to prove that the above properties define a set of full Lebesgue measure.
Throughout this post we will eventually change the basis rotation and for this reason we will denote the th Birkhoff sum of a function with respect to the irrational rotation by . We remind that denotes the Lebesgue measure on .
1. The roof function
The roof function we construct is different in nature from the others used in this context. Consider a sequence satisfying properties 1 and 2 above and let
One can see as the limit of worser and worser coboundaries
On one hand, the telescoping character of allows to easily calculate its Birkhoff sums. On the other hand, the increasing bad feature of is what will guarantee that has the required properties. In some sense, our construction resembles Anosov-Katok method of fast approximations developed in New examples in smooth ergodic theory, in which the authors construct differentiable maps sufficiently close to fibered maps of the torus (and, more generally, of any manifold that admits a -action) with exotic dynamical properties. Indeed, the referred maps are obtained as limits of periodic maps and here we will also use this perspective to prove Theorem 1.
Observe that the bigger is, the closer the points and are. In particular, the (at least) exponential growth of guarantees that for every . Furthermore, the sequence converges pointwise to .
More than this, condition 2 implies that the Birkhoff sums of and agree up to iterate in a large subset of . To this purpose, we need the
has Lebesgue measure equal to .
Proof: First, observe that changing by , we can assume that . The function is -periodic, with
For each interval , is different from if and only one of the discontinuities belong to the interval in defined by the points and . This happens for an interval of length and so, multiplying by the number of these intervals, the desired assertion is proved.
whenever and . This implies that
The `s form an ascending chain of subsets of and has, by Lemma 2, Lebesgue measure at most , which by property 1 can be assumed to be small.
In this section we prove that, with the above definition, the cylinder flow is ergodic. We proceed in two steps.
Step 1. For any of positive measure, the union contains .
Step 2. and have positive measure.
Step 2 is easy: by Lemma 2, for the set of points such that
- , and
has Lebesgue measure at least , which can be assumed to be positive if we pass to a subsequence of .
We now give the idea to prove Step 1. The formal proof requires care with some technicalities which I think are not worth in a first reading. The interested reader might check the details in the paper. The main observation is the following: assuming the divisibility condition on the `s, the sequence of partial sums given by
defines a simple random walk in . This is easy to see: each map is equal to 1 in intervals of length each, and in other intervals of length each. We call each of these intervals a plateau of order and denote by the one that contains . By the divisibility property, each divides itself into plateaux of order , half of them on which and half on which . This proves the random walk character of . In particular, it has the level-crossing property: almost every two random walks intersect infinitely often. Before going into the proof of ergodicity, we make a further remark: by the very definition of a plateau,
Now let be a subset of positive measure, and let and be points of density of and , respectively. The goal is to find such that the th iterate of a neighborhood of is mapped into a neighborhood of . This is equivalent to having, for every ,
- is close to , and
As the `s converges to , we might assume that is a point of density of for large and, whenever , the second condition is equivalent to the simpler one
This, in turn, means that . But, because of the first condition, very likely we will have and thus . This hints us to do the following: let large such that
Then let such that is close enough to to guarantee that is mapped inside . In this way, for any ,
and so is mapped, under , to a neighborhood of . This concludes the proof of Step 1 and thus of ergodicity.
Observe that, because we can always restrict to a subsequence, it is no loss of generality to assume that each orbit is -dense in , where depends only on , and so the term “close enough” in the previous paragraph makes sense.
3. Counting the number of returns to zero
The goal of this section is to calculate the number of returns of an arbitrary point to , under successive iterations of . More specifically, we want to calculate the distribution of the map given by
Once we have such information, we will be able to prove in the next section that is rationally ergodic along the sequence of iterates , by showing that is a sweep-out set for which the Renyi inequality holds (see Definition 6 in CF3).
Firstly, to the purpose of iterates up to order , we can work with instead of . Better than this, we consider the “rational” truncated versions of defined by
where is the good rational approximation associated to . This reduction is, again, similar in spirit to the Anosov-Katok method of fast approximations, in which the authors define transformations as the limit of coboundaries, not from the proper irrational rotation, but from good rational approximations of it. We claim that and coincide in a large set for . Just define, similarly to (1), the set by the relations
- for and , and
- for .
By Lemma 2, the complement of has -measure at most
which goes to zero as goes to infinity (by again, if necessary, passing to a subsequence of ). So the proof boils down to understanding the map given by
For each , the value of is defined by the vector
We claim that the possible combinatorics of this vector, as runs over the set , are equally distributed, for almost every .
The reader should interpret the above result as a binary tree: if we let, for , the set
and each of the sets has half of the cardinality of .
Once the lemma is established, we know exactly what is .
Corollary 4 For each with the same parity of ,
for a set of measure equal to .
We leave the proof of the corollary as an exercise: just use the random walk character of . We now proceed to prove Proposition 3: the idea is to interpret, for each , the map as a function in . Define
Each is a periodic function with period
Dilating to given by
then each is periodic with period and to analyse along is the same as analysing along . Observing that is a complete residue system modulo , Proposition 3 is thus a consequence of the following
Lemma 5 Let be a periodic function with period , . Assume that
- is even and is a multiple of for , and
- there are such that
Let be a complete residue system modulo . Then, for any sequence ,
The proof of the above lemma is by induction on . Here we prove the case and refer the reader to the paper for the full proof. Firstly, let us give an idea of why this must be true. If, instead of being interested in the behavior of along integers, we want to calcule the Lebesgue measure of a set with a specific combinatorics and , then the result is clear. In our case, once we understand how the residue classes of modulo are related, the induction argument holds without further problems, whenever the values , , are not discontinuities of (this is guaranteed by the assumption that , and it holds for almost every ).
We now prove the case : let be a function with period such that
- is even and
- there is such that
and let be a complete residue system modulo . We claim
To this matter, consider the sets
It is clear that and that . Also, if and only if for some . Because is a complete residue system module , (2) is proved.
4. Renyi inequality
It is a simple task to check that satisfies a Renyi inequality. Here, we denote the asymptotic relation by . On one hand,
where in the fifth passage we used Stirling`s approximation formula. On the other hand,
Finally, we prove the Renyi inequality for . Note that
and thus . Similarly, and so
which concludes the proof.