This first post discusses some uniform distribution results of the orbits of on . It is a classical result that they are all uniformly distributed. Moreover, Denjoy-Koksma’s inequality states that, for iterates equal to the denominators of the continued fraction expansion of , the orbits are highly equidistributed. We give a proof of this result for the classical assumption of bounded variation and its strengthening for differentiable functions. The last part is devoted to describe the basic properties of a cylinder flow and how its ergodic properties are related to the uniform distribution of orbits of the respective irrational rotation. More specifically, we want to convince the reader that the ergodicity of a cylinder flow over implies – on the contrary to Denjoy-Koksma’s inequality – that the orbits of the irrational rotation have discrepancy, in the sense that uniform distribution is not regular. This gives another reason, beyond their intrinsic ergodic theoretical importance, to investigate cylinder flows.
1. Uniform distribution
It is a classical result that the only -invariant probability measure is the Lebesgue measure on . This is equivalent to
for any continuous function . By approximation, this is equivalent to, for any interval , the frequency of visits to of any orbit of being equal to :
The unique ergodicity of thus says that, for any , the sequence is uniformly distributed. We can identify equidistribution in a quantitative fashion by considering the discrepancy of the sequence.
Definition 2 Let be a finite sequence of real numbers. The number
is called the discrepancy of the sequence. For an infinite sequence, the discrepancy is equal to the discrepancy of the first terms.
It is direct, and we leave the proof as an exercise to the reader, that
Proposition 3 The sequence is uniformly distributed iff
In other words, the theorem states the following (in principle stronger yet equivalent) interesting fact: whenever the sequence is equidistributed, then the convergence in (1) is uniform along all intervals .
The distinction between different types of equidistribution is made by the analysis of the non-averaged expressions
If a sequence has good equidistribution, one would hope for boundedness of . It turns out that, by the following result of W.M. Schmidt, this is never the case: is always infinite.
Theorem 4 (W.M. Schmidt) There is an absolute constant with the following property: for any infinite sequence of , we have the inequality
for infinitely many positive integers .
The interested reader may check the proof in the book Uniform distribution of sequences of Kuipers and Niederreiter.
2. Denjoy-Koksma’s inequality
The above theorem guarantees the existence of infinitely many times for which the uniform distribution behaves in an irregular fashion. Nevertheless, for the special case of the irrational rotation, there are indeed iterates for which the sequence is highly equidistributed in the sense that is bounded, and this is given by Denjoy-Koksma’s inequality. The iterates exhibiting such behavior are the denominators of the continued fraction expansion of .
Given irrational, consider its continued fraction expansion
The are called the partial quotient denominators or just denominators of . They give the best rational approximations of . More precisely, the approximation is equal to
and is always at most . On these denominators, the dynamics of the rotation is partially described by the rational rotation , and that’s why the sequence is highly equidistributed. This is the content of the next
Proof: The argument is based on the article Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations of Michel Herman. Let . Firstly, assume
Then, for ,
where is the residue of modulo . As runs overs all residues modulo , every such interval contains at least (and then exactly) one element of the set .
The case is similar, in which case .
Exercise 1 Show that the above lemma implies that for every .
Translating to the setup of functions, we arrive at the
for every .
Proof: Let and . By Lemma 5, each interval
has exactly one element of the set . Thus
Denjoy-Koksma’s inequality can be improved if is of class . Let denote the -norm.
Proof: Again, we can assume that belongs to the set of functions of zero mean. The main property we’ll use is that the -closure of the set
of coboundary functions is equal to . For this, recall the set of trigonometric polynomials
of zero mean is -dense in . Hence, it suffices to prove that each trigonometric polynomial is a coboundary. This is a simple task: given ,
The theorem now follows: given and , let be a coboundary for which . In particular, . Then
where in the second equality we used Denjoy-Koksma’s inequality and in the third the mean value inequality. For large , the last expression is smaller than .
Remark 1 Given any , the trigonometric polynomials are also dense in and then the same argument above proves that the -closure of the set
is equal to , the set of functions of zero mean.
3. Cylinder flows
We now focus our attention on cylinder flows. As defined in CF0, for any irrational number and function , where or , we define the cylinder flow by
The dynamics of is intimately connected to the Birkhoff sums of under : denoting these by , we have
By Birkhoff’s theorem, converges to almost surely. In particular, if has drift, i.e. , then
and so almost every orbit of diverges to infinity. From now on, we assume has no drift.
There is a natural invariant measure for : it is , where and are the Lebesgue measure of and , respectively. It is an infinite sigma-finite measure. Measures of this type often behave in very strange ways and fail to comply with rules forming the basis of finite ergodic theory but, regardless this, are worth studying and open the possibility for very different statistical behaviors.
Let be a measure-preserving system: is a measure space, a (possibly infinite) -finite measure and is a measurable map on invariant under .
Definition 8 We say is conservative if whenever is such that are pairwise disjoint. We say is ergodic if it has only trivial invariant sets, that is, if or whenever is a measurable set invariant under .
If is finite, then it is necessarily conservative. This is Poincaré’s recurrence theorem (see ERT1). In the infinite measure situation, this is not generally the case. For example, the translation , , is -invariant but non-conservative. Even more weird, it is ergodic! This is a pathological example that indeed is global in the sense of the
Exercise 2 Let be an invertible measure-preserving system. If it is ergodic and non-conservative, then it is isomorphic to the translation , .
In the non-invertible situation, there are many other examples. See the end of this paper of J. Aaronson and T. Meyerovitch. As cylinder flows are invertible and clearly not isomorphic to the translation in , conservativity follows if we prove ergodicity.
Coming back to our specific skew product, the mere assumption of zero drift guarantees that is conservative, according to the following
Theorem 9 (Atkinson) Let be an ergodic probability measure-preserving system and a measurable function such that . Then -almost every has the following property: for any set containing with and any , the set
The proof is in the paper Recurrence of co-cycles and random walks. A classical result of W. Gottschalk and G. Hedlund characterizes transitivity.
Theorem 10 (Gottschalk – Hedlund) Let be a compact metric space, a minimal homeomorphism and continuous. Then defined by
is transitive if and only if
Another theorem of Gottschalk and Hedlund indeed shows that, under the assumptions of Theorem 10, the two conditions above are equivalent to not being a coboundary. See the book Topological dynamics for a proof.
Not much can be obtained in the topological perspective. By this we mean that, when is continuous, is never minimal, whose proof is the content of this paper of P. Le Calvez and J.-C. Yoccoz.
Theorem 11 (Le Calvez – Yoccoz) There are no minimal homeomorphisms on an open annulus.
The next post will discuss ergodicity of cylinder flows. Before finishing the present one, let’s emphasize the relation between ergodicity and irregularity of equidistribution: Denjoy-Koksma’s inequality only gives information on the partial quotient denominators of . It might happen that, for different iterates, the Birkhoff sums become unbounded. The analysis of this, at least in the qualitative perspective, can be made via cylinder flows: if is ergodic, then must assume arbitrarily large values, because the orbit of a generic point has to visit sets far from the fiber . In particular no such high equidistribution as that in Denjoy-Koksma’s inequality might happen. It is because of this that
“Ergodicity of cylinder flows are related to the irregularity of equidistribution”.
Previous posts: CF0.