Limit Theorems for toral translationshomepages.warwick.ac.uk/~masdbl/TorusTalk.pdf · Limit...

Limit Theorems for toral translations

Dmitry Dolgopyat

Dmitry Dolgopyat Limit Theorems for toral translations

Limit theorems for dynamical systems.

Goal of the conference: given a map T : (X , µ)→ (X , µ)understand the statistical behaviour of SN(x) =

∑Nn=1 A(f nx) for a

large class of observables A.

In this talk X = Td , µ-Haar measure and f (x) = x + α.

The case of smooth observables is well understood. Namely if A issufficiently smooth then for almost all α there exists B(x , α) :

A(x) = B(x +α, α)−B(x , α)⇒ SN(x) = B(x +Nα, α)−B(x , α).


Non smooth observables.

(I) Meromorphic functiomns.

Theorem 1. (Sinai-Ulcigrai, 2008) If A has one simple pole onT1 and (x , α) is uniformly distributed on T2 then SN has limitingdistribution as N →∞.

Thus the result is the same as for smooth observable.

Question 1. More general meromorphic functions such assin 2πx/(sin 2πx + 3 cos 2πy).The case d = 1 seems easier. For example we have

Theorem 2. If A =1

sin2 πxand (x , α) is uniformly distributed on

T2 thenSNN2

has limiting distribution as N →∞.

(II) Indicator functions. Today we concentrate on A = χΩ. Let

DN(Ω, x , α) =N−1∑n=0

χΩ(x + nα)− NVol(Ω)

Vol(Td).





Thus the result is the same as for smooth observable.Question 1. More general meromorphic functions such assin 2πx/(sin 2πx + 3 cos 2πy).

The case d = 1 seems easier. For example we have

Theorem 2. If A =1


T2 thenSNN2



DN(Ω, x , α) =N−1∑n=0


Vol(Td).





Thus the result is the same as for smooth observable.Question 1. More general meromorphic functions such assin 2πx/(sin 2πx + 3 cos 2πy).The case d = 1 seems easier. For example we have

Theorem 2. If A =1


T2 thenSNN2



DN(Ω, x , α) =N−1∑n=0


Vol(Td).


Worst case scenario.

Theorem 3. (Ostrowski 1922, Khintchine 1923, Beck 1994)Let

DN(α) = supΩ−box

DN(Ω, x , α).

and φ(t) be a positive increasing function. Then

DN(α)

(lnN)dφ(ln lnN)

is bounded for almost every α iff∑

n φ(n) <∞.Question 2. Is it true that lim inf DN(α)

lnd N> 0 for all α ∈ T d?

Question 3. Is it true that DN(α)

lnd Nconverges in distribution as

N →∞?


Why randomize α?

For fixed α the discrepancy does not have a limit distribution.For example for d = 1 the Denjoy-Koksma inequality says that

|Sqn − qn

∫A(x)dx | < Var(A).

In particular Dqn(I , x , α) can take at most 3 values.

In higher dimensions one can show that if Ω is either a box or aconvex set then for almost all α and almost all tori

DN(Ω,Rd/L, α, ·)aN

does not converge to a non-trivial limiting distribution for anychoice of aN = aN(α, L).Question 4. Is this true for all α, L?


Why randomize α?


|Sqn − qn


In particular Dqn(I , x , α) can take at most 3 values.In higher dimensions one can show that if Ω is either a box or aconvex set then for almost all α and almost all tori


does not converge to a non-trivial limiting distribution for anychoice of aN = aN(α, L).

Question 4. Is this true for all α, L?


Why randomize α?


|Sqn − qn


In particular Dqn(I , x , α) can take at most 3 values.In higher dimensions one can show that if Ω is either a box or aconvex set then for almost all α and almost all tori


does not converge to a non-trivial limiting distribution for anychoice of aN = aN(α, L).Question 4. Is this true for all α, L?


Limit points. d = 1

Question 5. Study the distributions which can appear as weak

limits ofDN(I , α, ·)

aN, in particular their relation with number

theoretic properties of α?

For example all limit distributions are atomic for all I iff α ∈ Q.

Question 6. Is it true that all limit distributions are either atomicor Gaussian for almost all I iff α is of bounded type?Theorem 4. (Huveneers 2009) If α 6∈ Q and I = [0, 1/2] then

there is a sequence Nj such thatDNj

(I , α, ·)j

converges to N(0, 1).


d = 1

Theorem 5. (Beck 1997, Beck 2011) Let α be a quadratic surd.

(a) If (x , a, l) is uniformly distributed on T3 thenDaN([0, l ], α, x)√

lnNconverges to N(0, σ).(b) If a is uniformly distributed on T1 thenDaN([0, 1/2], α, 0)− C (α) lnN√

lnNconverges to N(0, σ(α)).

Theorem 6. (Kesten 1961) If (x , α) is uniformly distributed on

T2 then DN([0,l ],x ,α)c(l) lnN converges to the Cauchy distribution. Here

c(l) ≡ c0 if l 6∈ Q and takes different values for l ∈ Q.


d > 1.

Question 7. Suppose that Ω is semialgebraic (given by a finitenumber of algebraic inequalities) then ∃aN = aN(Ω) such that for

random translation of a random torusDN(Ω,Rd/L, x , α)

aNconverges in distribution as N →∞.Random translation of a random torus means that L = AZd and(x , α,A) has a smooth density.

Theorem 7. (Fayad-D) If Ω is real analytic and strictly convex

thenDN

N(d−1)/(2d)converges in distribution as N →∞.

Theorem 8. (Fayad-D) If Ω is the cube thenDN

c lnd Nconverges

to Cauchy distribution as N →∞.Question 8. Describe (quenched and annealed) large deviationsfor this system.Question 9. Does local limit theorem hold?


Poisson regime

Let Ω = ΩN shrink so that E (DN(ΩN , x , α)) = C .Theorem 9. (Marklof 2000) Suppose that Ω has piecewisesmooth boudary.If (x , α) is uniformly distributed on Td × Td then bothDN(N−1/dΩ, α, x) and DN(N−1/dΩ, α, 0) converge in distribution.

In fact one can also handle several sets in the same time.Theorem 10. (Marklof 2000)(a) If (x , α) is uniformly distributed on Td × Td thenN1/d ||x + nα|| converges in distribution to(X ,Y ) ∈ L : Y ∈ [0, 1]where L ∈ Rd+1 is a random affine lattice.(b) If α is uniformly distributed on Td N1/d ||nα|| converges indistribution to(X ,Y ) ∈ L : Y ∈ [0, 1]where L ∈ Rd+1 is a random lattice centered at 0.


Poisson regime

Let Ω = ΩN shrink so that E (DN(ΩN , x , α)) = C .Theorem 9. (Marklof 2000) Suppose that Ω has piecewisesmooth boudary.If (x , α) is uniformly distributed on Td × Td then bothDN(N−1/dΩ, α, x) and DN(N−1/dΩ, α, 0) converge in distribution.In fact one can also handle several sets in the same time.Theorem 10. (Marklof 2000)(a) If (x , α) is uniformly distributed on Td × Td thenN1/d ||x + nα|| converges in distribution to(X ,Y ) ∈ L : Y ∈ [0, 1]where L ∈ Rd+1 is a random affine lattice.(b) If α is uniformly distributed on Td N1/d ||nα|| converges indistribution to(X ,Y ) ∈ L : Y ∈ [0, 1]where L ∈ Rd+1 is a random lattice centered at 0.


Theorem 2: 1N2

∑Nn=1

1sin2(2π(x+nα))

converges in law.

Proof: Let S ′N be the sum of terms with ||x + nα|| > R/N,S ′′N be the sum of terms with ||x + nα|| < R/N.

E (S ′N/N2) =

1

NE (sin−2 ξχ| sin ξ|>R/N) = O(1/R).

S ′′N/N2 ∼

∑||nα||<1/(εN)

1

(N||x + nα||)2,

Hence

S ′′N/N2 ⇒

∑(X ,Y )∈L:Y∈[0,1],|X |<R

1

X 2.

Leting R →∞ we get

S ′′N/N2 ⇒

∑(X ,Y )∈L:Y∈[0,1]

1

X 2.


Shrinking targets.

Given f : (X , µ)→ (X , µ) let VN(x) =∑N

n=1 χB(y ,rn)(f nx).f has a shrinking target property (STP) if for any y , rn :∑

n rn =∞ VN(x)→∞ a. s.f has a monotone shrinking target property (MSTP) if for anyy , rn :

∑n rn =∞ and rn is non-increasing VN(x)→∞ a. s.

f has s-(M)STP if for any y , rn :∑

n rsn =∞ (and rn is

non-increasing) VN(x)→∞ almost surely.Theorem 11. (Fayad 2006) Toral translations do not have STP.Let D(σ) = α : ∀k ∈ Zd − 0,m ∈ Z|(k , α)−m| ≥ Ck−(1+σ)/d .Theorem 12. (Kurzweil 1982) Toral translation has the MSTPiff α ∈ D(0).Theorem 13. (Tseng 2008) Toral translation does not have theMSTP if α 6∈ D(0). Circle rotation has the s-MSTP ifα ∈ D(s − 1).Question 10. Does the last result hold for d > 1?


Shrinking targets. Rate of divergence.

Let rn = c/n.Theorem 14. (Beck 1996, Marklof 2003) If α ∈ D(0) thenVN(x)− c lnN√

lnN≈ N(0, σN) with σN ≤ σ.

Question 11. Extend this result for almost all α.

Theorem 15. (Fayad-Vinogradov-D) If (x , α) is unifromly

distributed on Td × Td thenVN(x)− c lnN√

lnN⇒ N(0, σ).

There is an analogous statement for the return times.Theorem 16. (Fayad-Vinogradov-D) If α is unifromly distribted

on Td thenVN(y)− c lnN√

bN⇒ N(0, σ) where

bN =

lnN ln lnN if d = 1

lnN if d ≥ 2.

Question 12. rn = c/nγ , γ < 1.


Dynamics of the skew products.

Let f : (X , µ)→ (X , µ), A : X → Rr , µ(A) = 0. Consider

ZN =N−1∑n=0

A(f nx).

Theorem 17. (Atkinson 1976) If r = 1 then ZN is recurrent.Theorem 18. (Chevallier-Conze 2009) If there existsδN = o(N1/r ) and a sequnce Nk such thatlimk→∞ µ(ZNk

≤ δNk) = 1

Then ZN is recurrent.


Recurrence for toral translations.

X = Td , f (x) = x + α, A =∑

j ajχΩj.

ZN is recurrent for all α ∈ R− Z if d = 1 (by Denjoy-Koksmainequlity).Theorem 19. (Chevallier-Conze 2009)(a) If Ωj are polygons then ZN is reccurrent for almost all α.(b) There are Ωj -polygons and α such that ZN is transient.Corollary 20. Let Ωj be real analytic and strictly convex and(d−1)

2d < 1r . Then ZN is recurrent for almost all α.

Question 13. Is it true that ZN is transient for almost all α if(d−1)

2d > 1r ?

This would follow from Borel-Cantelli Lemma if the Local LimitTheorem is valid.Question 14. Let α be as in Theorem 19 (a) or Theorem 20.Does there exist x such that ZN(x)→∞?Question 15. Let α be as in Theorem 19 (a) or Theorem 20.How often is ||ZN || ≤ R?


Quantitative recurrence.

Let d = r = 1, ZN(x) =∑N−1

n=0

[χ[0,1/2](x + nα)− 1/2

].

Let LN = Card(n ≤ N : Zn = 0).Theorem 21. (Avila-Duriev-Sarig-D) If α is a quadratic surdthen ∃c(α) such that

√lnN

cNLN ⇒ e−G

2/2

where G is a the standard Gaussian.

Similar results have been previously obtained by Ledrappier-Sarigfor abelian covers of compact hyperbolic surfaces.

Question 16. What happens for typical α?LN ∼ N

lnN is expected in view of Kesten’s results.Question 17. Extend the above result to the case when 1/2 isreplaced by(a) any rational number; (b) any irrational number.


Proof.

Theorem 22. (Hooper-Hubert-Weiss 2013) S is a Veechsurface. Its Veech group has finite index in SL2(Z).


The last question.

Question 18. Generalize the results presented here to higherdimensions.

That is, study∑N

n1=1 · · ·∑N

nm=1 A(x +∑

j αjnj).


Thank you!


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