Limit Theorems for toral translationshomepages.warwick.ac.uk/~masdbl/TorusTalk.pdf · Limit...
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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).
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).
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).
Dmitry Dolgopyat Limit Theorems for toral translations
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).
Dmitry Dolgopyat Limit Theorems for toral translations
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).
Dmitry Dolgopyat Limit Theorems for toral translations
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).
Dmitry Dolgopyat Limit Theorems for toral translations
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).
Dmitry Dolgopyat Limit Theorems for toral translations
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 →∞?
Dmitry Dolgopyat Limit Theorems for toral translations
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 →∞?
Dmitry Dolgopyat Limit Theorems for toral translations
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?
Dmitry Dolgopyat Limit Theorems for toral translations
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?
Dmitry Dolgopyat Limit Theorems for toral translations
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?
Dmitry Dolgopyat Limit Theorems for toral translations
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).
Dmitry Dolgopyat Limit Theorems for toral translations
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).
Dmitry Dolgopyat Limit Theorems for toral translations
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.
Dmitry Dolgopyat Limit Theorems for toral translations
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.
Dmitry Dolgopyat Limit Theorems for toral translations
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?
Dmitry Dolgopyat Limit Theorems for toral translations
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?
Dmitry Dolgopyat Limit Theorems for toral translations
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.
Dmitry Dolgopyat Limit Theorems for toral translations
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.
Dmitry Dolgopyat Limit Theorems for toral translations
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.
Dmitry Dolgopyat Limit Theorems for toral translations
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.
Dmitry Dolgopyat Limit Theorems for toral translations
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.
Dmitry Dolgopyat Limit Theorems for toral translations
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.
Dmitry Dolgopyat Limit Theorems for toral translations
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.
Dmitry Dolgopyat Limit Theorems for toral translations
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?
Dmitry Dolgopyat Limit Theorems for toral translations
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.
Dmitry Dolgopyat Limit Theorems for toral translations
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.
Dmitry Dolgopyat Limit Theorems for toral translations
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?
Dmitry Dolgopyat Limit Theorems for toral translations
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.
Dmitry Dolgopyat Limit Theorems for toral translations
Proof.
Theorem 22. (Hooper-Hubert-Weiss 2013) S is a Veechsurface. Its Veech group has finite index in SL2(Z).
Dmitry Dolgopyat Limit Theorems for toral translations
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).
Dmitry Dolgopyat Limit Theorems for toral translations
Thank you!
Dmitry Dolgopyat Limit Theorems for toral translations