The Gottschalk-Hedlund theorem, cocycles, and small divisors

Jordan Bell
July 23, 2014

1 Introduction

This note consists of my working through details in the paper Resonances and small divisors by Étienne Ghys.11 1 Aside from containing mathematics, Ghys makes thoughtful remarks about the history of physics, unlike the typically thoughtless statements people make about the Ptolemaic system. He insightfully states “Kepler’s zeroth law”: “If the orbit of a planet is bounded, then it is periodic.” I can certainly draw a three dimensional bounded curve that is not closed, but that curve is not the orbit of a planet. It is also intellectually lazy to scorn Kepler’s correspondence between orbits and the Platonic solids (“Kepler’s fourth law”).

2 Almost periodic functions

Suppose that f: is continuous. For ϵ>0, we call T an ϵ-period of f if


T is a period of f if and only if it is an ϵ-period for all ϵ>0.

We say that f is almost periodic if for every ϵ>0 there is some Mϵ>0 such that if I is an interval of length >Mϵ then there is an ϵ-period in I.

If f is periodic, then there is some M>0 such that if I an interval of length >M then at least one multiple T of M lies in I, and hence for any t we have f(t+T)-f(t)=f(t)-f(t)=0. Thus, for every ϵ>0, if I is an interval of length >M then there is an ϵ-period in I. Therefore, with a periodic function, the length of the intervals I need not depend on ϵ, while for an almost periodic function they may.

3 The Gottschalk-Hedlund theorem

The Gottschalk-Hedlund theorem is stated and proved in Katok and Hasselblatt.22 2 Anatole Katok and Boris Hasselblat, Introduction to the Modern Theory of Dynamical Systems, p. 102, Theorem 2.9.4. The following case of the Gottschalk-Hedlund theorem is from Ghys. We denote by


the projection maps.

Theorem 1 (Gottschalk-Hedlund theorem).

Suppose that u:R/ZR is continuous, that


that x0R/Z, and that α is irrational. If there is some C such that

|k=0nu(x0+kα)|C,n0, (1)

then there is a continuous function v:R/ZR such that


Say there is some C>0 satisfying (1). Define g:/×/× by


For n0,


The set {gn(x0,0):n0}, namely the orbit of (x0,0) under g, is contained in /×[-C,C]. Let K be the closure of this orbit. Because K is a metrizable topological space, for (x,y)K there is a sequence a(n) such that ga(n)(x0,0)(x,y). As g is continuous we get ga(n)+1(x0,0)g(x,y), which implies that g(x,y)K. This shows that K is invariant under g. Let 𝒦 be the collection of nonempty compact sets contained in K and invariant under g. Thus K𝒦, so 𝒦 is nonempty. We order 𝒦 by AB when AB. If 𝒞𝒦 is a chain, let C0=C𝒞C. It follows from K being compact that C0 is nonempty, hence C0𝒦 and is a lower bound for the chain 𝒞. Since every chain in 𝒦 has a lower bound in 𝒦, by Zorn’s lemma there exists a minimal element M in 𝒦: for every A𝒦 we have MA, i.e. MA. To say that M is invariant under g means that g(M)M, and M being a nonempty compact set contained in K implies that g(M) is a nonempty compact set contained in K, hence by the minimality of M we obtain g(M)=M.

The set M is nonempty, so take (x,y)M. Because M is invariant under g, {gn(x,y):n0}M. The set


is dense in /, hence π1(M) is dense in /. Moreover, M being compact implies that π1(M) is closed, so π1(M)=/.

For t, define τt:/×/× by τt(x,y)=(x,y+t). For any t,


so τtg=gτt. Hence, if A/× and g(A)A, then g(τt(A))=τtg(A)τt(A), namely, if A is invariant under g then τt(A) is invariant under g. Therefore τt(M) is invariant under g, and so Mτt(M) is invariant under g. This intersection is compact and is contained in K, so either Mτt(M)= or by the minimality of M, Mτt(M)=M. Suppose by contradiction that for some nonzero t, Mτt(M)=M. Then using g(M)=M we get τt(M)=M, and hence for any positive integer k we have τkt(M)=τtk(M)=M. But because M is compact, π2(M) is contained in some compact interval I, and then there is some positive integer k such that π2(τkt(M)) is not contained in I, a contradiction. Therefore, when t0 we have Mτt(M)=. Let x/. If there were distinct y1,y2 such that (x,y1),(x,y2)M, then with t=y2-y10 we get τt(x,y1)=(x,y2)M, contradicting Mτt(M)=. This shows that for each x/ there is a unique y such that (x,y)M, and we denote this y by v(x), thus defining a function v:/. Then M is the graph of v, and because M is compact, it follows that the function v is continuous. Let (x,v(x))M. As M is invariant under g,


and as M is the graph of v we get v(x)+u(x)=v(x+α) and hence v(x+α)-v(x)=u(x), completing the proof. ∎

4 Cohomology

In this section I am following Tao.33 3 Terence Tao, Cohomology for dynamical systems, Suppose that a group (G,) acts on a set X and that (A,+) is an abelian group. A cocycle is a function ρ:G×XA such that

ρ(gh,x)=ρ(h,x)+ρ(g,hx),g,hG,xX. (2)

If F:XA is a function, we call the function ρ(g,x)=F(gx)-F(x) a coboundary. This satisfies


showing that a coboundary is a cocycle. We now show how to fit the notions of cocycle and coboundary into a general sitting of cohomology. We show that they correspond respectively to a 1-cocycle and a 1-coboundary.

For n0, an n-simplex is an element of Gn×X, i.e., a thing of the form (g1,,gn,x), for g1,,gnG and xX. We denote by Cn(G,X) the free abelian group generated by the collection of all n-simplices, and an element of Cn(G,X) is called an n-chain. In particular, the elements of C0(G,X) are formal -linear combinations of elements of X. For n<0, we define Cn(G,X) to be the trivial group.

For n>0, we define the boundary map :Cn(G,X)Cn-1(G,X) by

(g1,,gn,x) = (g1,,gn-1,gnx)

For n0 we define :Cn(G,X)Cn-1(G,X) to be the trivial map. If n1 then of course 2=0. If n2, one writes out 2(g1,,gn,x) and checks that it is equal to 0, and hence that 2=0. Thus the sequence of abelian groups Cn(G,X) and the boundary maps :Cn(G,X)Cn-1(G,X) are a chain complex.

We denote the kernel of :Cn(G,X)Cn-1(G,X) by Zn(G,X), and elements of Zn(G,X) are called n-cycles. We denote the image of :Cn+1(G,X)Cn(G,X) by Bn(G,X), and elements of Bn(G,X) are called n-boundaries. Because 2=0, an n-boundary is an n-cycle. Zn(G,X) and Bn(G,X) are abelian groups and Bn(G,X) is contained in Zn(G,X), and we write


and call Hn(G,X) the nth homology group.

We define Cn(G,X,A)=Hom(Cn(G,X),A), which is an abelian group. Elements of Cn(G,X,A) are called n-cochains. That is, an n-cochain is a group homomorphism Cn(G,X)A. Because Cn(G,X) is a free abelian group generated by the collection of all n-simplices, an n-cochain is determined by the values it assigns to n-simplices. We thus identity n-cochains with functions Gn×XA.

We define the coboundary map δ:Cn-1(G,X,A)Cn(G,X,A) by


Explicitly, for FCn-1(G,X,A) and for an n-simplex (g1,,gn,x),

(δF)(g1,,gn,x) = F((g1,,gn,x))
= F(g1,,gn-1,gnx)

For FCn-2(G,X,A), write G=δF and take and cCn(G,X). Then,


showing that δ2=0. Thus the sequence of abelian groups Cn(G,X,A) and the coboundary maps δ:Cn-1(G,X,A)Cn(G,X,A) are a cochain complex.

We denote the kernel of δ:Cn(G,X,A)Cn+1(G,X,A) by Zn(G,X,A), and elements of Zn(G,X,A) are called n-cocycles. We denote the image of δ:Cn-1(G,X,A)Cn(G,X,A) by Bn(G,X,A), and elements of Bn(G,X,A) are called n-coboundaries. Because δ2=0, an n-coboundary is an n-cocycle. Zn(G,X,A) and Bn(G,X,A) are abelian groups and Bn(G,X,A) is contained in Zn(G,X,A), and we write


which we call the nth cohomology group.

Take n=1. We identify C1(G,X,A), the group of 1-chains, with functions G×XA. For ρC1(G,X,A), to say that ρ is a 1-cocycle is equivalent to saying that for any (g,h,x)G2×X, (δρ)(g,h,x)=0, i.e. ρ(g,hx)-ρ(gh,x)+ρ(h,x)=0, i.e.


To say that ρ is a 1-coboundary is equivalent to saying that there is a 0-chain F (a function XA) such that ρ=δF, i.e., for any (g,x)G×X,


5 Small divisors

Suppose that u:/ be C and satisfies


For each n, let


We have u^(0)=0. For any x/,


and n|u^(n)|<; for these statements to be true it suffices merely that u be Cβ for some β>12.

Let α be irrational. We shall find conditions under which there exists a continuous function v:/ such that

u(x)=v(x+α)-v(x),x/. (3)

Supposing that for each x, v(x) is equal to its Fourier series evaluated at x and that its Fourier series converges absolutely,


then for each x/,


Then using u(x)=v(x+α)-v(x) we obtain



v^(n)=u^(n)e2πinα-1,n0; (4)

because α is irrational, the denominator of the right-hand side is indeed nonzero for n0. The value of v^(0) is not determined so far. We shall find conditions under which the continuous function v we desire can be defined using (4).

A real number β is said to be Diophantine if there is some r2 and some C>0 such that for all q>0 and p,

|β-pq|>Cq-r. (5)

It is immediate that a Diophantine number is irrational. Suppose that α satisfies (5). Let n0 and let pn be the integer nearest nα. Then

|e2πinα-1| = |e2πi(nα-pn)-1|
= 4|nα-pn|
= 4|n||α-pnn|
> 4|n|C|n|-r
= 4C|n|-r+1.

Because uC, it is straightforward to prove that for each nonnegative integer k there is some Ck>0 such that


Therefore, for each nonnegative integer k, using (4) we have

|v^(n)|=|u^(n)||e2πinα-1|<Ck|n|-k14C|n|-r+1=Ck4C|n|r-k-1,n0. (6)

One can prove that if hn are complex numbers satisfying (6) then the function defined by


is C. Therefore, we have established that if α is Diophantine then there is some v:/ that is C and that satisfies (3).

On the other hand, for α=n=110-n!, Ghys constructs a C function u:/ such that there is no continuous function v:/ satisfying u(x)=v(x+α)-v(x) for all x/.