Invariant Measures for Continuous Maps
4.4 Measure Rigidity and Equidistribution
4.4.3 Equidistribution for Irrational Polynomials
Example 4.18 may be thought of as a statement in number theory: for an irrational α, the values of the polynomial p(n) = x+αn, when reduced modulo 1, form an equidistributed sequence for any value of x. Weyl [381]
generalized this to more general polynomials, and Furstenberg [98] found that this result could also be understood using ergodic theory. We recall the statement of Weyl’s polynomial equidistribution Theorem (Theorem 1.4 on p. 4): Letp(n) =aknk+· · ·+a0be a real polynomial with at least one coeffi- cient amonga1, . . . , akirrational. Then the sequence (p(n)) is equidistributed modulo 1.
As indicated in Example 4.18, the unique ergodicity of irrational circle rotations proves Theorem 1.4 fork= 1. More generally, Theorem4.10shows that the orbits of any transformation of the circle for which the Lebesgue mea- sure is the unique invariant measure are equidistributed. In order to apply this to the case of polynomials, we turn to a structural result of Fursten- berg [99] that allows more complicated transformations to be built up from simpler ones while preserving a dynamical property (in Chap. 7 a similar approach will be used for another application of ergodic theory).
Notice that by Theorem4.10, orbits of a uniquely ergodic transformation are equidistributed with respect to the unique invariant measure.
Theorem 4.21 (Furstenberg).LetT :X→X be a uniquely ergodic home- omorphism of a compact metric space with unique invariant measureμ. LetG be a compact group∗ with Haar measuremG, and letc:X→Gbe a contin- uous map. Define the skew-product mapS onY =X×Gby
S(x, g) = (T(x), c(x)g).
If S is ergodic with respect to μ×mG, then it is uniquely ergodic.
Proof.To see that S preserves μ×mG, let f ∈C(Y). Then, by Fubini’s theorem,
∗The reader may replaceGby a torusTkwith group operation written additively, together with Lebesgue measuremTk. Notice that in any case the Haar measure is invariant under multiplication on the right or the left sinceGis compact (see Sect. C.2).
4.4 Measure Rigidity and Equidistribution 115
Y
f◦Sd(μ×mG) =
X
G
f(T x, c(x)g) dmG(g) dμ(x)
=
X
G
f(T x, g) dmG(g) dμ(x)
=
X
G
f(x, g) dmG(g) dμ(x) =
Y
fd(μ×mG).
Assume thatS is ergodic. Let
E={(x, g)|(x, g) is generic w.r.t.μ×mG}.
By Corollary4.20,μ×mG(E) = 1. We claim thatE is invariant under the map (x, g)→(x, gh). To see this, notice that (x, g)∈Emeans that
1 N
N−1 n=0
f(Sn(x, g))−→
fd(μ×mG) for allf ∈C(X×G). Writingfh(·, g) =f(·, gh), it follows that
1 N
N−1 n=0
f(Sn(x, gh)) = 1 N
N−1 n=0
fh(Sn(x, g))
−→
fhd(μ×mG) =
fd(μ×mG)
sincemG is invariant under multiplication on the right, so (x, gh)∈E also.
It follows thatE =E1×G for some set E1 ⊆X, μ(E1) = 1. Now assume that ν is an S-invariant ergodic measure on Y. Write π : Y → X for the projection π(x, g) = x. Then π∗ν is a T-invariant measure, so by unique ergodicity π∗ν = μ. In particular, ν(E) = ν(E1 ×G) = μ(E1) = 1. By Corollary4.20,ν-almost every point is generic with respect toν. Thus there must be a point (x, g)∈E generic with respect toν. By definition of E, it
follows thatν =μ×mG.
Corollary 4.22.Let αbe an irrational number. Then the mapS:Tk →Tk defined by
S:
⎛
⎜⎜
⎜⎝ x1 x2
... xk
⎞
⎟⎟
⎟⎠−→
⎛
⎜⎜
⎜⎝
x1+α x2+x1
... xk+xk−1
⎞
⎟⎟
⎟⎠
is uniquely ergodic.
Proof. Notice that the transformation S is built up from the irrational circle map by taking (k−1) skew-product extensions as in Theorem 4.21.
By Theorem 4.21, it is sufficient to prove thatS is ergodic with respect to
Lebesgue measure on Tk. Let f ∈ L2(Tk) be an S-invariant function, and write
f(x) =
n∈Zk
cne2πin·x
for the Fourier expansion off. Then, sincef(x) =f(Sx), we have
n∈Zk
cne2πin·Sx=
n∈Zk
cne2πin1αe2πiSn·x where
S :
⎛
⎜⎜
⎜⎜
⎜⎝ n1 n2 ... nk−1
nk
⎞
⎟⎟
⎟⎟
⎟⎠−→
⎛
⎜⎜
⎜⎜
⎜⎝
n1+n2 n2+n3
... nk−1+nk
nk
⎞
⎟⎟
⎟⎟
⎟⎠
is an automorphism ofZk. By the uniqueness of Fourier coefficients,
cSn= e2πiαn1cn, (4.8)
and in particular |cSn| = |cn| for all n. Thus for each n ∈ Zk we either haven, Sn,(S)2n, . . . all distinct (in which casecn= 0 since
n|cn|2<∞) or (S)pn= (S)qn for somep > q, so n2=n3=· · ·=nk= 0 (by downward induction on k, for example). Now for n = (n1,0, . . . ,0), (4.8) simplifies tocn= e2πin1αcn, son1= 0 orcn= 0. We deduce thatf is constant, soSis
ergodic.
Proof of Theorem1.4.Assume that Theorem 1.4 holds for all polynomials of degree strictly less thank. Ifakis rational, thenqak ∈Zfor some integerq.
Then the quantitiesp(qn+j) modulo 1 for varyingnand fixedj = 0, . . . , q−1, coincide with the values of polynomials of degree strictly less thanksatisfying the hypothesis of the theorem. It follows that the values of each of those polynomials are equidistributed, so the values of the original polynomial are equidistributed modulo 1 by induction. Therefore, we may assume without loss of generality that the leading coefficientak is irrational.
A convenient description of the transformationS in Corollary4.22comes from viewingTk as{α} ×Tk with a map defined by
⎛
⎜⎜
⎜⎜
⎜⎝ 1 1 1
1 1 . ..
1 1
⎞
⎟⎟
⎟⎟
⎟⎠
⎛
⎜⎜
⎜⎜
⎜⎝ α x1
x2 ... xk
⎞
⎟⎟
⎟⎟
⎟⎠
=
⎛
⎜⎜
⎜⎜
⎜⎝ α x1+α x2+x1
... xk+xk−1
⎞
⎟⎟
⎟⎟
⎟⎠ .
Iterating this map gives
4.4 Measure Rigidity and Equidistribution 117
⎛
⎜⎜
⎜⎜
⎜⎝ 1 1 1
1 1 . ..
1 1
⎞
⎟⎟
⎟⎟
⎟⎠
n⎛
⎜⎜
⎜⎜
⎜⎝ α x1 x2
... xk
⎞
⎟⎟
⎟⎟
⎟⎠
=
⎛
⎜⎜
⎜⎜
⎜⎝ 1 n 1 n
2
n 1 ... . .. . .. n
k
. . . n 1
⎞
⎟⎟
⎟⎟
⎟⎠
⎛
⎜⎜
⎜⎜
⎜⎝ α x1 x2
... xk
⎞
⎟⎟
⎟⎟
⎟⎠
=
⎛
⎜⎜
⎜⎜
⎜⎝
α nα+x1
n
2
α+nx1+x2
... n
k
α+ n
k−1
x1+· · ·+nxk−1+xk
⎞
⎟⎟
⎟⎟
⎟⎠ .
Now defineα=k!ak, and choose points x1, . . . , xk so that p(n) =
n k
α+ n
k−1
x1+· · ·+nxk−1+xk.
Then by Corollary4.22, the orbits of this map are equidistributed onTk, so the same holds for its last component, which coincides with the sequence of
values ofp(n) reduced modulo 1 inT.
An alternative approach in the quadratic case will be described in Exer- cise 7.4.2.
Exercises for Sect. 4.4
Exercise 4.4.1.Consider the circle-doubling mapT2:x→2x (mod 1) onT with Lebesgue measuremT.
(a) Construct a point that is generic formT.
(b) Construct a point that is generic for aT2-invariant ergodic measure other thanmT.
(c) Construct a point that is generic for a non-ergodicT2-invariant measure.
(d) Construct a point that is not generic for anyT2-invariant measure.
Exercise 4.4.2.Extend Lemma 4.17to show that (4.7) holds for Riemann- integrable functions (cf. Exercise4.3.4). Could it hold for Lebesgue-integrable functions?
Exercise 4.4.3.Use Exercise 4.3.4to show that the fractional parts of the sequence (nα) are uniformly distributed in [0,1]. That is,
|{n|0n < N, nα− nα ∈[a, b)}|
N →(b−a)
asN → ∞, for any 0a < b1.
Exercise 4.4.4.Carry out the procedure used in the proof of Theorem 1.4 to prove that the sequence (xn) defined by xn = α1n
α2n2
is equidistributed inT2 if and only ifα1, α2∈/ Q.
Exercise 4.4.5.A numberαis called aLiouville number if there is an infi- nite sequence (pqn
n)n1 of rationals with the property that pn
qn −α < 1
qnn
for all n 1. Notice that Exercise 3.3.3 shows that algebraic numbers are not Liouville numbers.
(a) Assuming thatαis not a Liouville number, prove the following error rate in the equidistribution of the sequence (x+nα)n1 modulo 1:
1 N
N−1 n=0
f(x+nα)− 1
0
f(x)dx
S(α, f)1 N,
forf ∈C∞(T) and some constantS(α, f) depending onαandf. (b) Formulate and prove a generalization to rotations ofTd.
Exercise 4.4.6.Use the ideas from Exercise 2.8.4 to prove a mean ergodic theorem along the squares: for a measure-preserving system (X,B, μ, T) andf ∈L2μ, show that
1 N
N−1 n=0
UTn2f
converges inL2μ. Under the assumption thatT is totally ergodic (see Exer- cise 2.5.6), show that the limit is
fdμ.
Exercise 4.4.7.LetX be a compact metric space, and assume thatνn→μ in the weak*-topology onM(X). Show that for a Borel setBwithμ(∂B) = 0,
nlim→∞νn(B) =μ(B).