arXiv:1710.10150v7 [math.DS] 1 Jul 2020

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LOCAL LIMIT THEOREMS FOR SUSPENDEDSEMIFLOWS

JON. AARONSON & DALIA TERHESIU

Dedicated to the memory of Nat Friedman

Abstract. We prove local limit theorems for a cocycle over asemiflow by establishing topological, mixing properties of the as-sociated skew product semiflow. We also establish conditional ra-tional weak mixing of certain skew product semiflows and vari-ous mixing properties including order 2 rational weak mixing ofhyperbolic geodesic flows on the tangent spaces of cyclic covers.19/6/2020

§0 Introduction

Semiflows.We denote a standard σ-finite, measure space by (X,m). The space

X is Polish (a complete, separable metric space) and the measure m isdefined on B = B(X) the Borel subsets of X .Let

MPT (X,m) ∶= measure preserving transformations of (X,m).

The collection MPT (X,m) is a Polish semigroup under compositionwith respect to the weak operator topology defined by Tn → T if

1A TnmÐÐ→

n→∞1A T ∀ A ∈ B.

A semiflow on (X,m) is a continuous (semigroup) homomorphismΦ ∶ R+ → MPT (X,m).A flow on (X,m) is a continuous (group) homomorphism Φ ∶ R →

MPT (X,m).

2010 Mathematics Subject Classification. 37A40, 60F05.Key words and phrases. Local limit mixing, rational weak mixing, operator local

limit theorem, suspended semiflow, cocycle, continuous time random walk, fiberedsystem, transfer operator, quasicompact action, G-M map, AFU map, infinite er-godic theory.

Aaronson’s research was partially supported by ISF grant No. 1289/17.1

http://arxiv.org/abs/1710.10150v7

2 Local limits via mixing

The semiflow Φ ∶ R+ → MPT (X,m) is called invertible if each trans-formation Φt is invertible. In this case, Φ embeds in a flow Ψ ∶ R →MPT (X,m) defined by Ψt ∶= Φ

sgn(t)

∣t∣(t ≠ 0) & Ψ0 ∶= Id.

Cocycles.Let G be a locally compact, Polish Abelian group and let (X,m,T )

be a measure preserving transformation.Given a measurable function c ∶X → G, we may define C ∶ N×X → G

by C(n,x) = cn(x) ∶= ∑n−1k=0 c(Tkx) and obtain the cocycle equation

C(n + n′, x) = C(n,x) +C(n′, T n(x)).Analogously, for (X,m,Φ) a semiflow, a G-valued Φ-cocycle is a mea-

surable function C ∶ R+ × X → G satisfying C(t + t′, x) = C(t, x) +C(t′,Φt(x)).A Φ-cocycle is a Φ-coboundary if it has the form (t, x) ↦ H(x) −

H(Φt(x)) where H ∶ X → G is measurable and two Φ-cocycles arecohomologous if they differ by a Φ-coboundary.

Local limit theorems. In this paper we study the local limit theoryof cocycles over a semiflow. Let G ≤ Rκ be a closed subgroup of full di-mension, let (X,m,Φ) be a probability preserving (abbr. PP) semiflowand let C ∶ R+×X → G be a Φ-cocycle. For 0 < p ≤ 2, let S be a globallysupported, strictly p-stable random variable on Rκ and let b ∶ R+ → R+

be 1p-regularly varying. Fix a ring R ⊂ B(X).

The Φ-cocycle C ∶ R+ ×X → G is said to satisfy:– the integrated (S , b)-local limit theorem (with respect to R) if

b(t)κm(A ∩Φ−1t B ∩ [C(t, ⋅) ∈ x(t) +U])(Int-LLT)

ÐÐÐÐÐÐÐ→t→∞,

x(t)b(t)→z

fS(z) ⋅m(A)m(B)mG(U)∀ A,B ∈R & U ⊂ G precompact, mG(∂U) = 0;

where fS ∶ Rd → [0,∞) is the probability density function of S and– the conditional (S , b)-local limit theorem (with respect to R) if

b(t)κΦt(1A∩[C(t,⋅)∈x(t)+U])ÐÐÐÐÐÐÐ→t→∞,

x(t)b(t)→z

fS(z) ⋅m(A)mG(U)(Con-LLT)

a.e. ∀ A ∈ R & U ⊂ G precompact, mG(∂U) = 0.Here and throughout, for T ∈ MPT (X,m), T ∶ L1(m) → L1(m) denotesthe transfer operator of T defined by

∫XT f ⋅ gdm = ∫

Xf ⋅ g Tdm.

Jon. Aaronson & Dalia Terhesiu 3

The convergence (in both (Con-LLT) and (Int-LLT)) is often uniformin z in compact subsets of G, and the convergence in (Con-LLT) is oftenalso uniform on subsets in the ring R.Versions of (Int-LLT) in the case where p = 2 (i.e. S is Gaussian)

can be found in [Wad96], [Iwa08] and [DN17].

What’s new.In this paper, we give conditions for (Con-LLT) and for (Int-LLT)

(theorem 1).Our approach is via mixing properties of the associated skew product

semiflow.Conditions are obtained for conditional rational weak mixing

of skew product semiflows (theorem 2).Applying all this, we establish various mixing properties for the ge-

odesic flow of a cyclic cover of a compact hyperbolic surface (theorem3).

§1 Background

Skew product semiflows. Let (Y, ν,Φ) be a measure preservingsemiflow and let G ∶ R+ × Y → G ≤ Rκ (a closed subgroup of full dimen-sion) be a Φ-cocycle.The skew product semiflow over the base (Y, ν,Φ) with displacement

G is the measure preserving semiflow (Y ×G, ν ×mG,Φ(G)) defined by

Φ(G)t (y, z) ∶= (Φt(y), z +G(t, y)) (t > 0).

The deck holonomies are defined by

Qh(y, z) ∶= (y, z + h) (h ∈ G)and commute with the semiflow: Qh Φ

(G)t = Φ(G)t Qh.

Suspended semiflow and renewal process.Let (X,m,T ) be an ergodic MPT (EMPT). Let r ∶ X → R+ be measur-

able (aka the roof function).The renewal process generated is

N(y)(x) ∶=# (Γ(x)∩(0, y]) where Γ(x) ∶= rn(x) ∶= n−1∑k=0

r(T kx) ∶ n ≥ 1.In case m(X) = 1 and E(r) < ∞, E(N(t))

tÐÐ→t→∞

1E(r) by the ergodic

theorem. The limit λ ∶= 1E(r) > 0 is called the asymptotic intensity of

the renewal process.


The suspended semiflow with section (X,m,T ) is the semiflow (Y, ν,Φ)where

Y = (x, y) ∈ X ×R+ ∶ 0 ≤ y < r(x),ν(A ×C) =m(A)Leb (C) (A ×C ∈ B(Y )),Φt(x, y) = (T nx, y + t − rn(x))where n = N (t)(x, y) is such that

rn(x) ≤ y + t < rn+1(x).i.e. N (t)(x, y) =# (Γ(x) ∩ (0, y + t]) = N(t + y)(x).

It follows from the definitions that ν(Y ) = ∫X rdm and that for t > 0,

t↦ Φt ∈ MPT(Y, ν)is continuous.Next it is routine to check that

N (t + t′) = N (t) +N (t′) Φt ∀ (x, y) ∈ Y, t, t′ > 0;()

whence Φt+t′ = Φt Φt′ , showing that Φ ∶ R → MPT(Y, ν) is indeed asemiflow and N ∶ R+ × Y → R is a Φ-cocycle.

Accordingly, we’ll denote the suspended semiflow with section (X,m,T )by (Y, ν,Φ) =∶ (X,m,T )r and call N ∶ R+ ×Y → R the renewal cocycle.Note that Y ≅ X ×R+/ ∼ with (x, y) ∼ (x′, y′) iff ∃n ≥ 0 so that either

T nr (x, y) = (x′, y′) or T nr (x′, y′) = (x, y) where Tr(x, y) ∶= (Tx, y − r(x)).Topological suspended flows. Let T be a bi-Lipschitz, homeomor-phism of a Polish space (X,d) and let r ∶X → R+ be uniformly contin-uous and bounded away from 0,∞.There is a metric on Y so that Y equipped with the topology gener-

ated is locally homeomorphic with neighbourhoods in X × R+, whenceY is Polish; and so that Φ is a continuous flow on Y . Moreover, if Xis [locally] compact then so is Y . See [BW72, §4].

Jump cocycles over a suspended semiflow.Let (Y, ν,Φ) = (X,m,T )r be a suspended semiflow as above, let G ≤

Rκ be a subgroup of full dimension and let ϕ ∶ X → G be measurable.Now define the jump cocycle J = J(ϕ) ∶ R+ × Y → G by

J(t, (x, y)) ∶= ϕN (t)(x,y)(x)(o)

where ϕN ∶=∑N−1k=0 ϕ Tk.

By (), J is a Φ-cocycle.


We refer to both ϕ ∶ X → G and J(ϕ) ∶ R+×Y → G as the displacement.

By a simple change of variables, the skew product semiflow of thejump cocycle over a suspended semiflow is isomorphic with the sus-pended semiflow over the displacement skew product. That is: if(Y, ν,Φ) = (X,m,T )r and J ∶= J(ϕ), then

(X ×G, ν ×mG,Φ(J)) ≅ (X ×G,m ×mG, Tϕ)r(±)

where r(x, z) ∶= r(x) and Tϕ(x, y) ∶= (Tx, y +ϕ(x)). See [AN17].

Example: Continuous time random walks.A random walk (RW) on the locally compact, Polish, Abelian group

G is a sequence of G-valued random variables (Sn ∶ n ≥ 1) defined bySn ∶= ∑nk=1Xk with (Xn ∶ n ≥ 1) independent, identically distributedrandom variables on G. Their mutual distribution f ∈ P(G) is knownas the jump distribution of the RW.

A continuous time random walk (abbr. CTRW) (in [MW65], jumpprocess in [Bre68]) on G is a one-parameter family (S(t) ∶ t > 0) ofrandom variables on G defined by S(t) ∶= SN(t) where Sn = ∑nk=1Xk

is as above and N(t) ∶= #Γ ∩ (0, t] (t > 0) where Γ is a Poissonpoint process on R+ which is independent of (Xn ∶ n ≥ 1) and whereE(N(t)) = λt ∀ t > 0 where λ > 0 (the intensity of the process).We’ll exhibit a jump cocycle over a suspended semiflow with the

same distribution as (S(t) ∶ t > 0).Let (X,m,T ) be given by X ∶= (G × R+)N, T = shift and m = µN

where

µ(U ×A) ∶= f(U)λ∫Ae−

xλdx.

Denote x ∈ X by x = (k, r) = ((k1, k2, . . . ), (r1, r2, . . . )) and define r ∶X → R+ by r(x) ∶= r1 and ϕ ∶X → G by ϕ(x) ∶= k1.Let (X,m,T )r = (Y, ν,Φ) and define J(ϕ) ∶ R+ × Y → G as above.We claim that the ν-distribution of (J(ϕ)(t, ⋅) ∶ t > 0) coincides with

the distribution of (S(t) ∶ t > 0).Indeed, considered with respect to m, rn(⋅) ∶ n ≥ 1 =∶ Γ(⋅) is a

Poisson point process on R+ with intensity λ. For y > 0, so is Γy(⋅) ∶=(Γ(⋅) − y) ∩R+.For (x, y) ∈ Y , let N (t)(x, y) ∶=#(Γ(⋅)∩(y, y+t]) =#(Γy(⋅)∩(0, t]),

then the distribution of (N (t)(⋅, y) ∶ t > 0) does not depend on y > 0.Consequently, for fixed y > 0

(J(ϕ)(t, (⋅, y)) ∶ t > 0) = (N (t)(x,y)∑j=1

kj ∶ t > 0) dist.= (S(t) ∶ t > 0). 2


Radon measures.

Let X be a locally compact, Polish space. We’ll call a measurem ∶ B(X)→ [0,∞] Radon if m(C) <∞ ∀ C ⊂X compact.Let µn, m be Radon measures on X . As in [Bre68, Chapter 10], we

say that µn converges weakly to m and write µnwÐÐ→

n→∞m if

µn(U) ÐÐ→n→∞

m(U) ∀ U ∈ Cc(X)where Cc(X) denotes the collection of continuous, R-valued, compactlysupported functions on X , and where µ(U) ∶= ∫X Udµ for U ∶ X → R

µ-integrable. This abuse of notation will be useful when, in a dearth ofclopen sets, we need to replace 1A by U ∈ Cc(X) with µ(A) = µ(1A) ≈µ(U).We’ll call set A ∈ B(X) almost open if m(∂A) = 0 and a function

f ∶ X →M (a metric space) Riemann integrable (RI) if

m(y ∈ X ∶ f not continuous at y) = 0.We’ll denote the collections of almost open sets and precompact,

almost open sets by J (m) and Jc(m) respectively.As is well known µn

wÐÐ→n→∞

m iff

µn(A)ÐÐ→n→∞

m(A) ∀ A ∈ Jc(m).Let X be a locally compact, Polish space.A collection H ⊂ Cc(X) is called measure determining if for α, β ∶B(X)→ [0,∞] Radon measures on X :α(C) = β(C) ∀ C ∈ H implies α ≡ β.For example, if H ⊂ Cc(X) is closed under multiplication, separates

points of X and∀ U ⊂ X open, precompact, ∃ f ∈ H, f ≥ 1U , then by the Stone-Weierstrass theorem, H is measure determining.

Weak Convergence Lemma Let (Z,µ) be a locally compact measurespace.Suppose that H ⊂ Cc(X) is a measure determining collection.If µn (n ≥ 1) & µ are Radon measures on X and

µn(A)ÐÐ→n→∞

µ(A) ∀ A ∈ H,then µn

wÐÐ→n→∞

µ.


Proof Fix a sequence (Hk ∶ k ≥ 1) of compact subsets of X so thatHk ⊂ Ho

k+1 ↑ Z. By compactness, ∀ U ∈ Cc(X), ∃ k ≥ 1 so that[U ≠ 0] ⊂ Hk.Using Helly’s theorem and diagonalization, we obtain tj →∞ and

a Radon measure m on X so that ∀ k ≥ 1,

µtj(U)ÐÐ→j→∞

m(U) ∀ U ∈ Cc(X), [U ≠ 0] ⊂ Hk,

and in particular ∀U ∈ H.By assumption,

µtj(A)ÐÐ→j→∞

µ(A) ∀ A ∈ H.Since H is measure determining, m ≡ µ.Thus, the only accumulation point of µt ∶ t ≥ 1 is µ and µt

wÐÐ→t→∞

µ..

2

Riemann integrable transformations of locally compact mea-sure spaces.A σ-finite, Polish measure space (X,m) is locally compact if X is

locally compact and m ∶ B(X) → [0,∞] is so that m(C) <∞ ∀ C ⊂ Xcompact.

Let (Y, ν) & (Y ′, ν′) be locally compact, separable measure spaces.It is standard that if π ∶ Y → Y ′ is nonsingular in the sense that for

A ∈ B(Y ′), ν(π−1A) = 0 ⇐⇒ ν′(A) = 0, then π is RI if and only if π−1J (ν) ⊂ J (ν).Consequently, the composition of nonsingular, RI maps is RI.

RI semiflows. Let (Y, ν) be a locally compact, measure space. Asemiflow Φ ∶ R+ → MPT (Y, ν) is a RI semiflow if ∀ R > 0, (t, x) ↦ Φt(x)is a RI map (0,R) × Y → Y with respect to the measure Leb × ν.It is not hard to show that if (Y, ν,Φ) is a RI semiflow and G ∶

R+ ×Y → G is a RI, Φ-cocycle where G ≤ Rκ is a closed subgroup of fulldimension, then the skew product semiflow (Z,µ,Φ(G)) is also RI.Let (Z,µ,Ψ) & (Z ′, µ′,Ψ′) be RI semiflows. We’ll say that (Z,µ,Ψ)

is a RI extension of (Z ′, µ′,Ψ′) if there is a measure preserving, RI factormap π ∶ Z → Z ′ with π Ψt = Ψ′t π and that (Z,µ,Ψ) & (Z ′, µ′,Ψ′)are RI-conjugate if there is an invertible RI-factor map π ∶ Z → Z ′ withπ−1 RI.

(S , b)-mixing on a locally compact measure space.Local limit mixing properties are defined for RI semiflows on locally

compact measure spaces.


Let 0 < p ≤ 2, let S be a symmetric, p-stable, globally supported,random variable on Rκ and let b(t)→∞ be 1

p-regularly varying.

We call the RI skew product semiflow (Z,µ,Φ(G))– (integrated) (S , b)-mixing if ∀ A, B ∈ Jc(µ),

b(t)κµ(B ∩Φ(G)−1t Qk(t)A)ÐÐÐÐÐÐÐÐÐÐÐ→t→∞, k(t)∈G, k(t)

b(t)→zfS(z)µ(A)µ(B);(ý)

– conditionally (S , b)-mixing if for A ∈ Jc(µ),b(t)κΦ(G)t (1Qk(t)A)(x)ÐÐÐÐÐÐÐÐÐÐÐ→

t→∞, k(t)∈G, k(t)b(t)→z

fS(z)µ(A) a.e..(a)

Here Qk(t) denotes a deck holonomy as defined on p. 3.In order to use weak convergence of Radon measures, we’ll need

following standard (equivalent) reformulations of (ý) and (a) in termsof Cc(Z). For U, V ∈ Cc(Z),

b(t)κµ(U ⋅ V Qk(t) Φ(G)t ) ÐÐÐÐÐÐÐÐÐÐÐÐ→

t→∞, k(t)∈G, k(t)b(t)→z

fS(z)µ(U)µ(V );(ýC)

b(t)κΦ(G)t (U Qk(t))(x) ÐÐÐÐÐÐÐÐÐÐÐÐ→

t→∞, k(t)∈G, k(t)b(t)→z

fS(z)µ(U) a.e..(aC)

Let H ⊂ Cc(Z). We’ll say that the conditional (S , b)-mixing of theRI skew product semiflow (Z,µ,Φ(G)) is H-uniform if for U ∈ H, M >0 & E ⊂ Z compact, we have

supx∈E

∣b(t)κΦ(G)t (UQk(t))(x)−fS(z)µ(U)∣ ÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐ→t→∞, k(t)∈G, ∥k(t)∥≤Mb(t), k(t)

b(t)→z0.

Theorem 1 (below) gives sufficient conditions for (S , b)-mixing (bothintegrated and conditional).

Remarks.(o) Both conditional and integrated local limit mixing are invariant

under RI conjugacy.(i) By taking k(t) ∶= 0 in (ý) we see that (S , b)-mixing implies

Krickeberg mixing as in [Kri69] (Jc(µ)-mixing in [Fri78]):

b(t)κµ(B ∩Φ(G)−1t A)ÐÐ→t→∞

µ(A)µ(B) ∀ A, B ∈ Jc(µ).(ii) The local limit theorems (Int-LLT) and (Con-LLT) for G with

respect to R = Jc(ν) (as on p. 2) are consequences of (ý) and (a)(respectively).

The mixing lemma shows that the reverse implications also hold.


Mixing Lemma Let (Z,µ,Φ(G)) be a RI skew product semiflow ona locally compact measure space. Let 0 < p ≤ 2, let S be a symmetric,p-stable, globally supported, random variable on Rκ and let b(t)→∞ be1p-regularly varying.

Suppose that H ⊂ Cc(X) is a measure determining collection.

(i) If (ý)C holds ∀ A, B ∈ H, then Φ(G) is (S , b)-mixing.

(ii) If (a)C holds ∀ A ∈H, then Φ(G) is conditionally (S , b)-mixing.

Proof We’ll only prove (ii), the proof of (i) being similar and simpler.

Fix k ∶ R+ → G measurable, k(t)b(t) → z as t → ∞. We’ll apply the

weak convergence lemma to the transition measures µ(x)t defined (using

standard disintegration theory) for a.e. x ∈X & t > 0 by

µ(x)t (U) ∶= b(t)κΦ(G)t (U Qk(t))(x) (U ∈ Cc(X)).

These are finite measures and the map

(t, x) ∈ R+ ×Z0 ↦ µ(x)t ∈ Radon measures on Z,

is measurable.By assumption, for a.e. x ∈ Z,

µ(x)t (U) ÐÐ→

t→∞fS(z)µ(U) ∀ U ∈ H

whence by the weak convergence lemma µ(x)t

wÐÐ→t→∞

fS(z)µand (Z,µ,Φ(G)) is conditionally (S , b)-mixing. V

Smooth cocycles over suspended semiflows.Given f ∶ Y → Rκ measurable and flow integrable in the sense that

∫ r(x)0∣f(x, t)∣dt < ∞ a.e., we can define (as in [Wad96], [Iwa08] and

[DN17]) the smooth cocycle F (f) ∶ R+ × Y → G by

F (f)(t, (x, y)) ∶= ∫ t

0f Φs(x, y)ds.

This is a Φ-cocycle. It is cohomologous to the jump cocycle J(ϕ) where

ϕ ∶X → Rκ is defined by ϕ(x) ∶= ∫ r(x)0

f(x, y)dy:F (f)(t, (x,u)) = J(ϕ)(t, (x,u)) − E(x,u) + E Φt(x,u)

where E(x,u) ∶= ∫ u0 f(x, s)ds.In case the Φ & f are RI, the skew product semiflows Φ(F ) and Φ(J)

on (Y ×Rκ, ν ×mRκ) are RI conjugate and share the same (S , b)-mixingproperties.


Conditional rational weak mixing.We call the measure preserving transformation (Z,µ,R) condition-

ally rationally weakly mixing if there exist constants un > 0 so that witha(n) ∶=∑nk=1 uk,

1

a(N)N

∑k=1

∣Rk1C − ukµ(C)∣(P)

ÐÐÐ→N→∞

0 a.e. ∀ C ∈ B(Z), 0 < µ(C) <∞.This property was considered in [AN17].

Let 0 < p ≤ 2, let S be a symmetric, p-stable, globally supported,random variable on Rκ and let b(t)→∞ be 1

p-regularly varying so that

a(N) ∶= N

∑n=1

1

b(n)κ ÐÐÐ→N→∞∞

(i.e. κ = 1 & 1 ≤ p ≤ 2 or κ = 2 & p = 2).We’ll see that if (Z,µ,Φ(G)) is an ergodic, (S , b)-mixing, skew prod-

uct semiflow then each Φ(G)t is conditionally rationally weakly mixing.

Theorem 2 gives sufficient conditions for this which are more generalthan those of theorem 1.

Transfer operator of a suspended semiflow.Let (Y, ν,Φ) = (X,m,T )r be a suspended semiflow. For t > 0, the

transfer operator of Φt is the operator Φt ∶ L1(ν)→ L1(ν) defined by

∫AΦtfdν = ∫

Φ−1t Afdν (f ∈ L1(ν), A ∈ C).

For f = 1B×I with B × I ∈ C, it is given by

Φt(1B×I)(ω, y) = ∞∑n=0

T n(1[N (t)=n]∩B×I(⋅, y))(ω)=∞

∑n=0

T n(1B∩[rn(⋅)∈I+t−y])(ω).

Fibered systems. A fibered system is a quadruple (X,m,T,α) where(X,m,T ) is a non-singular transformation and α is a countable, mea-surable partition which generates B under T (σ(T −nα ∶ n ≥ 0) = B)such that T ∣a invertible and nonsingular for a ∈ α.If (X,m,T,α) is a fibered system, then for n ≥ 1, so are(X,m,T,αn) & (X,m,T n, αn) where αn = αn−10 ∶= ⋁n−1k=0 T

−kα.


The collection of α-cylinders of (X,m,T,α) isCα ∶= ⋃

n≥1

αn.

For n ≥ 1, there are m-nonsingular inverse branches of T n denotedva ∶ T na→ a (a ∈ αn) with Radon Nikodym derivatives

v′a ∶=dm vadm

∶ T na → R+.

Fibered systems (X,m,T,α) and (X ′,m′, T ′, α′) are isomorphic ifthere is a measure theoretic isomorphism π ∶ (X,m,T ) → (X ′,m′, T ′)so that π(α) = α′.Any fibered system (X,m,T,α) is isomorphic to a subshift fibered

system, that is one where X ⊂ SN is T -invariant and closed with respectto the Polish topology on SN (where S ⊂ N); T ∶ SN → SN is the shiftand α = [s] = (x1, x2, . . . ) ∈ X ∶ x1 = s ∶ s ∈ S.Thus, in a subshift fibered system (X,m,T,α), X is a Polish space

and T ∶ X →X is continuous with α a collection of clopen sets in X .If S is finite, then X is compact, but if not, X may not even be

locally compact.

Locally compact fibered systems.We’ll need these in order to consider (S , b)-mixing properties of their

skew product extensions and suspensions.The fibered system (X,m,T,α) is locally compact if (X,m,T ) is

locally compact, T is RI and Cα ⊂ Jc(m) where Cα denotes the collectionof α-cylinders as on p. 11.The fibered system (X,m,T,α) is Markov if Ta ∈ σ(α) ∀ a ∈ α. In

this case, so are (X,m,T,αn) & (X,m,T n, αn).An interval map is a fibered system (X,m,T,α) with X ⊂ R a

bounded interval, m Lebesgue measure and α a partition mod m ofX into open intervals so that for each a ∈ α, T ∣a is (the restriction of)a bi-absolutely continuous homeomorphism. It is locally compact.Next, Markov fibered systems have locally compact representations:

Interval map lemma Let (X,m,T,α) be a Markov fibered system,then (X,m,T,α2) is isomorphic to a Markov interval map.

Proof WLOG (X,m,T,α) is a subshift fibered system as above withX ⊂ SN T -invariant and closed and so that s, t ∈ S ⊂ N, s < t Ô⇒m([s]) ≥m([t]).Let Y ∶= [0,1] & µ ∶= Leb.


Construct a partition β = Bs ∶ s ∈ S of Y into open intervals sothat

(i) t, u ∈ S, t < u Ô⇒ Bt < Bu (i.e. x < y ∀ x ∈ Bt, y ∈ Bu)

(ii) µ(Bs) =m([s]) ∀ s ∈ S;Continue, constructing partitions βn (n ≥ 1) of Y into open intervals

of form

βn ∶= B(k1,k2,...,kn) ∶ k = (k1, k2, . . . , kn) ∈ Snsatisfying

(a) k ≺ ℓ (lexicographically) ⇒ Bk < Bℓ;

(b) µ(Bk) =m([k]).(c) ∀ n ≥ 1, k = (k1, k2, . . . , kn) ∈ Sn,

Bk =⊍i≥1

Bk,i.

Define a map π ∶X → [0,1] by⋂n≥1

B(x1,x2,...,xn) = π(x).The map is continuous and m π−1 = µ. It is injective on π−1(Y0) with

Y0 ∶= Y ∖ ⋃n≥1, b∈βn

∂b

– a dense Gδ set of full measure and, indeed, π ∶ π−1(Y0) → Y0 is anhomeomorphism.Thus τ ∶ Y0 → Y0 defined by τ(y) ∶= π(T (π−1(y)) is continuous and

π ∶ (X,m,T,α2)→ (Y,µ, τ, β2) is an isomorphism of fibered systemsIt remains to show that (Y,µ, τ, β2) is an interval map.To this end,

¶ For k, ℓ ∈ S so that m([k, ℓ]) = µ(Bk,ℓ) > 0, τ ∶ Bk,ℓ ∩ Y0 → Bℓ ∩ Y0extends to an increasing, continuous map τk,ℓ ∶ Bk,ℓ → Bℓ.

Proof For such k, ℓ ∈ S, by the Markov property, T ∶ [k, ℓ] → [ℓ] issurjective, whence also τ ∶ Bk,ℓ ∩ Y0 → Bℓ ∩ Y0.First, τ ∶ Bk,ℓ ∩ Y0 → Bℓ ∩ Y0 is strictly increasing because:if x, y ∈ Bk,ℓ∩Y0, x < y, then ∃ n ≥ 1 and x, y ∈ Sn with xj = yj ∀ 1 ≤

j < n & xn < yn so thatx ∈ Bk,ℓ,x & y ∈ Bk,ℓ,y, whence τ(x) ∈ Bℓ,x & τ(y) ∈ Bℓ,y and τ(x) <

τ(y).Next, for z ∈ Bk,ℓ, set

τk,ℓ(z) ∶= limx→z−, x∈Bk,ℓ∩Y0

τ(x).We claim that τk,ℓ ∶ Bk,ℓ → Bℓ is continuous and increasing.


Proof of increasing:

x, y ∈ Bk,ℓ, x < y ⇒ ∃ u, v ∈ (x, y) ∩ Y0, u < vÔ⇒ τk,ℓ(x) ≤ τk,ℓ(u) < τk,ℓ(v) ≤ τk,ℓ(y).

Proof of continuity: Being monotone, either τk,ℓ ∶ Bk,ℓ → Bℓ iscontinuous or there is a nonempty, open interval J ⊂ Bℓ with τk,ℓ(Bk,ℓ) ⊂Bℓ ∖ J contradicting τk,ℓ(Bk,ℓ) ⊃ Bℓ ∩ Y0. V ¶

It is now standard that (Y,µ, τ, β2) is a Markov interval map. V

Quasicompact action.Let L ⊂ L∞ be a Banach space which is admissible in the sense that∥ ⋅ ∥L ≥ ∥ ⋅ ∥∞. We say that T acts quasicompactly on L if TL ⊂ L and∃ M ≥ 1, ρ ∈ (0,1) so that

∥T nf − ∫Xfdm∥L ≤Mρn∥f∥L ∀ n ≥ 1, f ∈ L.

This definition of quasicompact action differs from the one in [HH01],but coincides with it when the fibered system is weakly mixing.The fibered systems (X,m,T,α) have transfer operators which act

quasicompactly on an associated admissible Banach space denoted LT .

Example 1: AFU maps.An AFU map is an interval map (X,m,T,α) where for each a ∈ α,

T ∣a is (the restriction of) a C2 diffeomorphism T ∶ a→ Ta satisfying inaddition:

supX

∣T ′′∣(T ′)2 <∞,(A)

Tα ∶= TA ∶ A ∈ α is finite,(F)

infx∈a∈α∣T ′(x)∣ > 1.(U)

An AFUmap with finite α is known as a Lasota-Yorkemap after [LY73].It is known that

[Ryc83], [Zwe98]: an AFU map has a Lebesgue-equivalent, invariantprobability whose density has bounded variation and which is uniformlybounded below;

[Ryc83]: for T a mixing, AFU map, T acts quasicompactly on LT =BV, the space of functions on the interval X with bounded variation;this quasicomptacness persists when m is replaced by the absolutelycontinuous, T -invariant probability;


[Ryc83]: a topologically mixing AFU map is exact and ([AN05]) thegenerated stochastic process (α T n ∶ n ≥ 1) is reverse φ-mixing in thesense that

φ−(n) ∶= supk≥1

m(A∣B) −m(A)∣ ∶ A ∈ αk−10 , B ∈ α∞k+nÐÐ→n→∞0

where αKJ ∶= σ(α T j ∶ J ≤ j <K + 1) for 0 ≤ J <K ≤∞.Indeed, here, φ−(n)ÐÐ→

n→∞0 exponentially.

Example 2: Gibbs-Markov maps.A Gibbs-Markov (G-M) map (X,m,T,α) is a fibered system which is

Markov in the sense that

Ta ∈ σ(α) mod m ∀ a ∈ α(a)

and which satisfies

infa∈α

m(Ta) > 0(b)

and, for some θ ∈ (0,1)sup

n≥1, a∈αn

DTna,θ(log v′a) <∞(gθ)

where, for A ⊆ X, H ∶ A→ R

DA,θ(H) ∶= supx,y∈A

∣H(x) −H(y)∣θt(x,y)

<∞

with t(x, y) =minn ≥ 1 ∶ αn(x) ≠ αn(y) ≤∞.A G-M map on a finite state space is a subshift of finite type (SFT).It is known (see [AD01]) that

a G-M map has a m-equivalent, invariant probability whose densityh is θ-Holder continuous on X in the sense that DX,θ(h) < ∞ where,for A ⊆ X

DA,θ(h) ∶= supx,y∈A

∣h(x) − h(y)∣θt(x,y)

<∞

where t(x, y) =minn ≥ 1 ∶ xn ≠ yn ≤∞;

for T a mixing, G-M map, T acts quasicompactly on LT = Hθ,the space of θ-Holder continuous functions X → R with θ as in (gθ)equipped with the norm

∥f∥Hθ∶= ∥f∥∞ +DX,θ(f).

if the invariant density is bounded below, the quasicompactnesspersists when m is replaced by the absolutely continuous, T -invariantprobability;


a topologically mixing G-M map is exact and the stochastic process(α T n ∶ n ≥ 1) is exponentially continued fraction mixing,

a Markov AFU map is also a G-M map.

Aperiodicity and non-arithmeticity.Let H be a locally compact, Abelian, Polish group.We call the measurable G ∶ X → H :

non-arithmetic if ∄ solution to

G = k + g − g T, g ∶ X → H measurable,

k ∶ X → K measurable, where K is a proper subgroup of H.

and

aperiodic if ∄ solution to

G = k + g − g T, g ∶ X → H measurable,

k ∶ X → K measurable, where K is a proper coset of H.

Geodesic flows on cyclic covers.We denote by (U(M),Λ, g), the geodesic flow on the unit tangent

bundle U(M) of the hyperbolic surface M equipped with the g-invariant Liouville measure Λ (for definitions see [Hop71], [Ree81],[AN17]).The hyperbolic surface V is a cyclic cover of the compact hyperbolic

surface M if there is a covering map p ∶ V → M and a monomor-phism γ ∶ Z → Isom (V ) (hyperbolic isometries of V ), so that fory ∈ V, p−1p(y) = γ(n)y ∶ n ∈ Z.It was shown in [AN17] that such (U(V ),Λ, g) is rationally weakly

mixing, and rationally ergodic of order 2 (as defined there). This isstrengthened in theorem 3 (below) where we establish (S , b)-mixing(with S centered Gaussian and b(t) ∝√n) along with multiple Kricke-berg mixing and rational weak mixing of order 2.

§2 Results

Set-up.We’ll prove local limit mixing theorems (Theorems 1 and 2 below)

for J(ϕ) (defined above in (o) on p. 4) with respect to the semiflow(Y, ν,Φ) = (X,T,µ,α)r. with (X,T,µ,α) a topologically mixing, prob-ability preserving fibered system, r ∶ X → R+ and ϕ ∶ X → G measurablesatisfying certain conditions listed below.


General assumptions.For all results, we assume that

(X,m,T,α) is probability preserving, mixing, locally compact andeither a G-M map or an AFU map;

r ∈ LT (see p. 13) & r > 0;

D(T )α (ϕ) <∞ wherefor (X,m,T,α) a G-M map,

D(T )α (ϕ) ∶= supa∈α

Da,θ(ϕ) <∞and for (X,m,T,α) an AFU map,

D(T )α (ϕ) ∶= supa∈α

Lipa(ϕ) <∞.Here, for A ⊂X ,

LipA(ϕ) ∶= sup ∥ϕ(x) − ϕ(y)∥∣x − y∣ ∶ x, y ∈ A.Note that for A ⊂X an interval,

⋁A

ϕ ∶= supn≥1, t0<⋅⋅⋅<tn, t0,...,tn∈A

n−1

∑k=0

∥ϕ(tk+1) −ϕ(tk)∥ ≤m(A)LipA(ϕ).Also, by (U), if ϑ ∈ (0,1) is sufficiently large, then

Da,ϑ(ϕ) ≤ Lipa(ϕ) ∀ a ∈ α.

Weak Independence.Our weak independence assumption is that for some c > 0 we have

m([ϕn ∈ U, rn ∈ J]) ≤ cm([ϕn ∈ U])m([rn ∈ J])(î)

∀ n ≥ 1, U ⊂ G, J ∈ R+ open subsets.

Under the general assumptions, a restricted version of “weak in-dependence” holds whenever ϕn

b(n) converges in distribution to a non-

normal stable random variable. See (á) on p. 33. But we do not knowif this suffices for (î).On the other hand, this “weak independence” cannot hold for the

whole roof- and dispacement- processes ((r T n ∶ n ≥ 0) and (ϕ T n ∶n ≥ 0) respectively) without their independence. Indeed:, If, for some M > 0,m(A∩B) ≤Mm(A)m(B) ∀ A ∈ σ(rT n ∶ n ≥ 0), B ∈ σ(ϕT n ∶ n ≥ 0),


then

m(A∩B) =m(A)m(B) ∀A ∈ σ(rT n ∶ n ≥ 0), B ∈ σ(ϕT n ∶ n ≥ 0).To see ,, write Ar ∶= σ(r T n ∶ n ≥ 0) & Aϕ ∶= σ(ϕ T n ∶ n ≥ 0)

define the probability µ ∈ P(X,Ar ∨Aϕ) by µ(A ∩B) ∶=m(A)m(B).The mixing properties of (X,m,T ) ensure that (X,Ar ∨Aϕ, µ, T ) is

an EPPT where Ar ∨Aϕ ∶= σ(Ar ∪Aϕ)).So is (X,Ar ∨Aϕ,m∣Ar∨Aϕ

, T ) (as a factor of (X,m,T )).Thus µ & m∣Ar∨Aϕ

are either disjoint or identical; the former beingeliminated by the assumption. V ,

Aperiodicity. We’ll mainly consider the joint aperiodicity assump-tion:

(J-Ap) (r, ϕ) ∶ X → R ×G is aperiodic

In the sequel, we’ll prove the semiflow LLT under (J-Ap).

Distributional assumptions.For some 0 < p ≤ 2 and some 1

p-regularly varying sequence (b(n) ∶

n ≥ 1),ϕn

b(n)dist.ÐÐ→n→∞

S(L)

where S is a globally supported, symmetric p-stable (SpS) random variableon Rκ.We’ll consider the following separate cases of (L):

(CN) the classical normal case where E(∥ϕ∥2) <∞, b(n) = √n and Sis normal,

(NNS) the non-normal, stable case where p ≠ 2.Remark We consider it likely that our results in the NNS case ex-tend to the case where the displacement limit random variable S isstrictly stable. However, the methods in the present paper and theirdependencies rely on the (stronger) symmetry assumption.

Statements.

Theorem 1: semiflow local limit mixing

(i) Under the general assumptions, (J-Ap), and (CN) with E(∥ϕ∥4) <∞, the skew product semiflow is conditionally (S , b)-mixing and this isH-uniform with

H ∶= LT ×Cc(R+) ×Cc(G) ⊂ Cc(Z).


(ii) Under the general assumptions, (J-Ap), weak independence and(NNS) or (CN) with E(∥ϕ∥4) =∞ the skew product semiflow is (S , b)-mixing.

CTRW – a toy model for theorem 1.

LLT for CTRWs Let (S(t) ∶ t > 0) be a CTRW on G ≤ Rκ, a subgroup offull dimension, with jump random variables (Xk ∶ k ≥ 1) and intensityλ > 0.

Suppose that Sn

b(n)dist.ÐÐ→t→∞

S where p ∈ (0,2], S is a globally supported,

symmetric, p-stable random variable on Rκ and b(t) is 1p-regularly vary-

ing as t → ∞, then for U ⊂ G a compact neighborhood of 0 withmG(∂U) = 0

b(λt)κP (S(t) ∈ k(t) +U)ÐÐÐÐÐÐÐÐÐ→t→∞,

k(t)b(λt)→z∈R

κ

mG(U)fS(z)uniformly in z ∈ K a compact subset of Rκ where fS is the probabilitydensity function of S.

For simple continuous time random walk, this is theorem 2.5.6 of[LL10]. The general case uses the same methods. In view of thediscussion on p. 5, the stronger statement in Theorem 1(ii) can beobtained.

Assuming “recurrence” (but not “weak independence”), we have

Theorem 2Under the general assumptions, (J-Ap), and (CN) or (NNS), with

a(n) ∶= n

∑k=1

1

b(k)κ ÐÐ→n→∞∞,

the skew product semiflow is conditionally, rationally weakly mixing.

Theorem 3For V a cyclic cover of a compact, hyperbolic surface, and (U(V ),Λ, g)

its geodesic flow, then for S ∈ RV(R) centered Gaussian and b(t) = c√t(some c > 0),

(o) (U(V ),Λ, g) is (S , b) mixing i.e. for A, B ∈ Jc(Λ), k ∶ R+ → Z so

that k(t)√tÐÐ→t→∞

z,

b(t)Λ(A ∩ gtQk(t)B) ÐÐ→t→∞

Λ(A)Λ(B)fS(z);(f)


(i) (U(V ),Λ, g) is multiply Krickeberg mixing i.e.

for N ≥ 1, A, A1, . . . ,AN ∈ Jc(Λ),Λ(A ∩ N

⋂i=1

gtiAi) ∼ Λ(A) N∏i=1

Λ(Ai)b(∆ti)(v)

as t = 0 = t0 < t1 < ⋅ ⋅ ⋅ < tN, ∆ti ∶= ti − ti−1 →∞, (1 ≤ i ≤ N).

(ii) (U(V ),Λ, g) is rationally weakly mixing of order 2 i.e.

for each τ > 0 fixed,

1

logN

N

∑k=1

∣Λ(A ∩ gτkB ∩ g2τkC) − Λ(A)Λ(B)Λ(C)b(τk)2 ∣(Y)

ÐÐÐ→N→∞

0 ∀ A,B,C ⊂ U(M) bounded, measurable.

Methods.The proofs of theorems 1 & 2 use an operator LLT (Op-LLT) for(r, ϕ) ∶X → R×G with respect to T . In the (CN) case, this boils down

to the classical multidimensional LLT (as in [GH88]) . See [AD01] and[ADSZ04] for the versions considered here.The Op-LLT is proved here in the (NNS) case (below). For more

information about operator stability & limits, see [JM93] and for anOp-LLT in the independent case, see [Don91].The proof of theorem 1 also uses an extended LLT estimate for the

renewal cocycle based on proposition 4 in [P09].Theorem 3 is an application of theorem 1(i).

Plan of the paper.

The proofs of theorems 1 and 2 both use a splitting of the LHS of(Con-LLT) (see p. 2) according to the value of the renewal cocycle. See(a) on p. 21.The term I in (a) where the renewal cocycle takes moderate values

is estimated using the the Op-LLT in §3. The remaining term II is alsotreated in §3 using a deviation lemma which is proved in the appendix(§6) by means of an extended LLT estimate also established there.In §4, we prove the Op-LLT, given an infinite divisibility lemma (p.

33), which in turn is established in §5.

§3 Proofs


Notations. In the sequel, we’ll use the following notations:

For sequences a, b ∶ N→ R, a(n) ≈ b(n) means a(n)− b(n) ÐÐ→n→∞

0.

For positive sequences a, b ∶ N→ R,

a(n) ≪ b(n) means that ∃ M > 0 so that a(n) ≤ b(n) ∀ n ≥ 1 large;

a(n) ≫ b(n) means that b(n) ≪ a(n);a(n) ≍ b(n) means that a(n) ≪ b(n) and a(n) ≫ b(n);a(n) ∼ b(n) means that a(n)

b(n) ÐÐ→n→∞1;

a(n) ∝ b(n) means that ∃ limn→∞a(n)b(n) ∈ R+.

For a sequence of Rd-valued, measurable functions (Xn ∶ n ≥ 0)defined on the probability space (Ω,F , P ):Xn

dist.ÐÐ→n→∞

Y (where Y is a random variable on Rd) means

∫Ωg(Xn)dP ÐÐ→

n→∞E(g(Y )) ∀ g ∶ Rd → R, bounded, continuous.

random variables

Proof of theorem 1. We consider the skew product semiflow (Z,µ,Φ(J))over the suspended semiflow (Y, ν,Φ) ≅ (X,m,T,α)r with J = J(ϕ), ϕ ∶X → G & r ∶X → R+. We’ll use (±) on p. 5:

(Z,µ,Φ(J)) ≅ (X ×G,m ×mG, Tϕ)rwith r(x, z) ∶= r(x).It suffices to establish convergence for the measure determining col-

lection

H ∶= Cα ×Jc(mR) × Jc(mG) ⊂ Jc(µ)where Cα denotes the collection of α-cylinders as on p. 11.

Fix x ∶ R+ → G so that x(t)b(λt) ÐÐ→t→∞

z ∈ Rκ where

λ ∶=1

E(r) ∈ R+

and let A × I ×U ∈H.We have

Φ(J)t(1A×I×(x(t)+U))(x, z, y) = Φt(1(A×I)∩[−J(ϕ)(t,⋅)∈x(t)−z+U])(ω, y)= Φt(1(A×I)∩[J(ϕ)(t,⋅)∈z−x(t)+V ])(ω, y)

where V ∶= −U .For each M > 0, we have (as in [AN17]) the following splitting:


Φt(1(A×I)∩[J(ϕ)(t,⋅)∈z−x(t)+V ])(ω, y)(a)

= Φt(1(A×I)∩[J(ϕ)(t,⋅)∈z−x(t)+V ]1[∣N (t)−λt∣≤M√t])(ω, y)++ Φt(1(A×I)∩[J(ϕ)(t,⋅)∈z−x(t)+V ]1[∣N (t)−λt∣>M√t])(ω, y)

=∶ I(M,t,ω) + II(M,t,ω).The rest of the proof consists of establishing the following claims (I)

and (II):

limM→∞

limt→∞

b(λt)κI(M,t,ω) = fS(z)µ(A)Leb (I)mG(U).(I)

uniformly on compact subsets of Z; which holds in all cases and wherein case (i) (CN), we write b(t) ∶=√t;and

limt→∞

b(t)κII(M,t, ⋅) ÐÐÐ→M→∞

0 in case (i);(II)

limt→∞

b(t)κE(II(M,t, ⋅)) ÐÐÐ→M→∞

0 in the cases (ii)

where in cases (o) & (i), the convergence is uniform on compact subsetsof Z.Using (Y, ν,Φ) ≅ (X,m,T,α)r we have for K ⊂ N,

Φt(1(A×I)∩[J(ϕ)(t,⋅)∈z−x(t)+V ]1[N (t)∈K])(ω, y)= ∑n∈K

Φt(1(A×I)∩[J(ϕ)(t,⋅)∈z−x(t)+V ]1[N (t)=n])(ω, y)= ∑n∈K

T n(1A∩[ϕn∈z−x(t)+V, rn∈I+t−y])(ω)whence

I(M,t,ω) = ∑n∈N, n=λt±M

√t

T n(1A∩[ϕn∈z−x(t)+V, rn∈I+t−y])(ω)II(M,t,ω) = ∑

n∈N, ∣n−λt∣>M√t

T n(1A∩[ϕn∈z−x(t)+V, rn∈I+t−y])(ω).To prove (I), the independence assumptions are not needed. We shall

need:

in the (CN) case, the local limit theorem for aperiodic (r, ϕ) as in[GH88],and

in the (NNS) case, an operator local limit theorem (Op-LLT) foraperiodic (r, ϕ)which we state now and prove in the sequel:


The operator local limit theorem statement.

Operator local limit theoremSuppose (J-Ap), and (NNS) with 0 < p < 2, then there is a 1

p-regularly

varying sequence (b(n) ∶ n ≥ 1), a random variable Z = (S ,N ) on Rκ×Rwith a positive, continuous, probability density function, where S is aglobally supported, symmetric p-stable random variable on Rκ, N iscentered, Gaussian on R, and independent of S so that for A ∈ Cα andI ⊂ R an interval so that A × I ⊂ Y, & U ⊂ G a compact neighborhoodof 0 with mG(∂U) = 0,√

nb(n)κT n(1A∩[ϕn∈zn+U, rn∈I+nE(r)+rn])(Op-LLT)

≈n→∞

fZ(ζn, ρn)m(A)∣I ∣mG(U).where (zn, rn) ∈ G × R with zn = b(n) ⋅ ζn, rn =

√n ⋅ ρn with (ζn, ρn) =

O(1).For fixed (zn, rn) ∈ G×R as above, the convergence is uniform on X.

In the sequel, we’ll refer to (Op-LLT) as the Op-LLT corresponding tothe distributional convergence

( rn−nE(r)√n

, ϕn

b(n)) dist.ÐÐ→n→∞

Z

Proof of (I) (as on p. 21) given the Op-LLT.By the LLT for (r, ϕ) in case (CN) and (Op-LLT) in case (NNS),

√nb(n)κT n(1A∩[ϕn∈zn+U, rn∈I+nE(r)+rn]) ≈

n→∞fZ(ζn, ρn)m(A)∣I ∣mG(U).

for A ∈ Cα and I ⊂ R an interval so that A × I ⊂ Y, & U ⊂ G acompact neighborhood of 0 with mG(∂U) = 0, where (zn, rn) ∈ G × R

with zn = b(n) ⋅ ζn, rn =√n ⋅ ρn with (ζn, ρn) = O(1).

Fix t, M > 0, then for n ≥ 1,

t = E(r)n + x√n with x = xn,t, ∣x∣ ≤ME(r) 32 ⇐⇒n = λt −

x′

E(r) 32√t with ∣x − x′∣ = O( 1√

t) & in this case

T n(1A∩[ϕn∈x(t)+U, rn∈I+t−y]) ∼ m(A)∣I ∣b(n)κ√nfZ(z, xn,t)

as t, n→∞, ∣xn,t∣ ≤M.


It follows that for fixed M > 0,b(λt)κI(M,t) = b(λt)κ ∑

n=λt±M√t

T n(1A∩[ϕn∈x(t)+U, rn∈I+t−y])∼t→∞

∑n=λt±M

√t

b(n)κT n(1A∩[ϕn∈x(t)+U, rn∈I+t−y])∼t→∞

m(A)∣I ∣mG(U) ∑n=λt±M

√t

fZ(z, xn,t)√n

Now,1√n−

1√n + 1

∼ 1

2n√n

so

xn,t − xn+1,t =t −E(r)n√

n−t −E(r)(n + 1)√

n + 1

=E(r)√n + 1

+ (t −E(r)n)( 1√n−

1√n + 1

)=E(r)√n(1 +O( 1√

n)).

Thus

b(λt)κ ∑n=λt±M

√t

T n(1A∩[ϕn∈x(t)+U, rn∈I+t−y])≈t→∞

m(A)∣I ∣mG(U) ∑n=λt±M

√t

(xn+1,t − xn,t)fZ(z, xn,t)ÐÐ→t→∞

m(A)∣I ∣mG(U)∫[−M,M]fZ(z, x)dx

ÐÐÐ→M→∞

m(A)∣I ∣mG(U)fS(z). 2 (I)

Proof of (II) (as on p. 21) & the deviation lemma.

Fix x ∶ R+ → G so that x(t)b(λt) ÐÐ→t→∞

z ∈ Rκ and I, U as on p. 20, and a

compact subset K ∶=X ×K × J ⊂ Z, then for (x, z, y) ∈ K,II(M,t, (x, z, y)) = ∑

n≥1, ∣n−λt∣>M√t

T n(1A∩[ϕn∈x(t)−z+V, rn∈I+t−y])(x)= ∑n∈A(M,t)

T n(1A∩[ϕn∈x(t)−z+V, rn∈I+t−y])(x)where

A(M,t) ∶= n ≥ 1 ∶ m([rn ∈ t + I − J]) > 0 & ∣n − λt∣ >M√t.


Since r is uniformly bounded away from 0,∞, ∃ K = K(I, J) > 1 sothat

m([rn ∈ t + I − J]) > 0 Ô⇒ n ∈ [t/K, tK].(R)

In particular

A(M,t) ⊂ BK(M,t)(Ï)

∶= A(MK, t) ∩ [ t

K, tK]

= n ∈ [ tK, tK] ∶ ∣n − λt∣ > M

√n

K.

Thus

II(M,t, (x, z, y)) ≤ ∑n∈BK(M,t)

T n(1A∩[ϕn∈x(t)−z+V, rn∈I+t−y])(x).(J)

We’ll need the following lemma which follows from a multidimen-sional version of proposition 4 in [P09] (“extended LLT estimate”).Both the lemma and the estimate are proved in the appendix (§6).

Deviation Lemma(i) There are constants Γ, η > 0 so that ∀ M > 0 ∃ t0(M) > 0 such

that

T n1A∩[rn∈I+t−y] ≤ Γ(exp[−η∣n−λt∣2

t]√

t+

1

n3

2

)∀ M > 0, t > t0(M) & n ∈ BK(M,t).

(ii) If in addition, E(∥ϕ∥4) <∞, then, possibly changing Γ, η, t0(M),we have for M > 0, t > t0(M), x(t) ∈ Rκ, ∥x(t)∥ ≤ M

√t

2λK2 :

T n1A∩[ϕn∈x(t)−z+V, rn∈I+t−y] ≤ Γ

tκ2

⋅ (exp[−η∣n−λt∣2

t]√

t+

1

n3

2

)∀ n ∈ BK(M,t).

In the deviation lemma, (y, z) ∈ R+ ×G &I × V ⊂ R+ ×G are consideredfixed, but may vary inside a compact subset of R+ ×G.

Proof of (II) (as on p. 21) given the Deviation Lemma.Evidently,

∑n∈BK(M,t)

1

n32

≤ ∑n≥ t

K

1

n32

≤ 4√K√t

(w)


Next we claim that ∃ C > 0, ζ > 0 so that ∀ R, t > 0,1√t∑

n∈BK(R,t)exp[−η∣n−λt∣2

t] ≤ Ce−ζR2

.(B)

Proof of (B) Let Nt ∶= [λt] & rt ∶= λt, then

∑n∈BK(R,t)

exp[−η∣n−λt∣2t] ≤ ∑

n∈BK(R,t)exp[−η∣(n−Nt)−rt∣2

t]

≪ ∑k∈Z, ∣k∣≥R

√t

K

exp[−η∣k−rt∣2t]

≪ 2 ∑k≥R

√t

K

exp[−ηk2t]

≪ 2∫∞

R√

tK

exp[−ηy2t]dy

= 2√t∫

∞

RK

exp[−ηx2]dy≍√t exp[−ηR2

K2 ]≪ √t exp[−ζR2]

with ζ ∶= η2K2 . V (B).

Proof in case (i) For (x, z, y) ∈ K ∶=X ×K × J ⊂ Z (compact), using(ii) in the deviation lemma,

tκ2 II(M,t, (x, z, y))= t

κ2 Φt(1A×(x(t)+U)×I]∩[∣N (t)−λt∣≥M√t])(x, z, y)

= tκ2 ∑n≥1, ∣n−λt∣>M

√t

T n(1A∩[−ϕn∈x(t)−z+U, rn∈I+t−y])(x)≤ tκ2 ∑

n∈BK(M,t)T n(1A∩[−ϕn∈x(t)−z+U, rn∈I+t−y])(x)

≤ Γ√t∑

n∈BK(M,t)exp[−η∣n−λt∣2

t] + Γ ∑

n∈BK(M,t)

1

n32

.

Using (w) and (B), we obtain

tκ2 II(M,t, (x, z, y)) ≪ e−ζM

2

+ 1√t

which suffices for (II). V


Proof of (II) in case (ii) Define y ∶ Y → R+, y(x, y) ∶= y then using(J) on p. 24 and (î) on p. 16

E(II(M,t, (x, z, ⋅)∥y)) ≤ ∑n∈BK(M,t)

m(A ∩ [ϕn ∈ x(t) − z + V, rn ∈ I + t − y])≤ c ∑

n∈BK(M,t)m([ϕn ∈ x(t) − z + V ])m([rn ∈ I + t − y])

with c as in the weak independence assumption and where E(⋅∥y)denotes conditional expectation on (Y, ν) with respect to y.As t → ∞ and n = n(t) ∈ BK(M,t) (as on p. 24), n(t) = K±1t and

b(λn(t)) ≍ b(t).Thus by the LLT for (X,m,T,ϕ), ∃ k > 1 so that

b(t)κm([ϕn ∈ x(t) − z + V ]) ≤ k ∀ t > 0 large, n ∈ BK(M,t).By part (i) of the deviation lemma,

m([rn ∈ I + t − y]) ≤ Γ(exp[−γ∣n−λt∣2

t]√

t+

1

n3

2

) ∀ t > 0, n ∈ BK(M,t).It follows that

b(t)κE(II(M,t,(x, z, ⋅)∥y)) ≤ cΓk ∑n∈BK(M,t)

( exp[− γ∣n−λt∣2t

]√n

+ 1

n32

)≪ e−ζM

2

+ 1√t

as before by (B) & (w). This suffices for (II). V

Proof of theorem 2.Under the assumptions for theorem 2, we have by [AD01] that the

measure preserving transformation (X × G,m × mG, Tϕ) is pointwisedual ergodic with

an(Tϕ) ∼ n

∑k=1

fS(0)b(k)κ .

Since (X × G,m ×mG, Tϕ) is a good section for the skew productsemiflow (Z,µ,Φ(J)) in the sense of [AD99], we have (as in [AN17])

that each Φ(J)t is pointwise dual ergodic with

an(Φ(J)t ) ∼n

∑k=1

fS(0)b(λtk)κ .

Now let A ∈ Cα, I ⊂ R be an interval and U ⊂ G be a compactneighborhood with mG(∂U) = 0.


By (I), as t→∞,

Φ(J)t(1A×U×I)(ω, z, y) == Φt(1(A×I)∩[J(ϕ)(t,⋅)∈z−U])(ω, y)≳ fS(0)µ(A)Leb (I)mG(U) ⋅ 1

b(λt)κ .By pointwise dual ergodicity, for fixed t > 0, as N →∞,

N

∑k=1

Φkt(1(A×I)∩[J(ϕ)(t,⋅)∈U])(ω, y)∼ fS(z)m(A)Leb (I)mG(U) N

∑k=1

1

b(λkt)κ .The rest of the proof follows the proof of proposition 4.2 in [AN17]. VThe proof of theorem 2 actually establishes the (possibly) more gen-

eral

Abstract PropositionIf (X,m,T,α) is probability preserving, mixing, G-M– or AFU– map

and (ϕ, r) ∶ X → G × R+, with E(r) < ∞, satisfies D(T )α ((ϕ, r)) < ∞and the operator local limit theorem corresponding to the distributionalconvergence

( rn−nE(r)β(n) , ϕn

b(n)) dist.ÐÐ→n→∞

Z

(as on p. 22) where b(n) is p-regularly varying with 0 < p ≤ 2, β(n)is q-regularly varying with 1 < p ≤ 2 and Z is a globally supportedrandom variable on Rκ+1 admitting a smooth probability density func-tion; then the skew product semiflow is conditionally, rationally weaklymixing.

Remark. It seems reasonable to expect the rational weak mixing of: the geodesic flow on the unit tangent bundle of C∖Z (as in [AD99]):and the planar Lorentz systems considered in [SV04] & [SV07];

as these flows are (measure theoretic) natural extensions of skewproduct semiflows for which the above abstract proposition might hold.

Proof of theorem 3. Let V be the Z-cover of the compact hyperbolicsurface M (as on p. 15) equipped with a covering map p ∶ V → M sothat there exists a monomorphism γ ∶ Z → Isom (V ), such that fory ∈ V, p−1p(y) = γ(n)y ∶ n ∈ Z.


The corresponding tangent map, also denoted p ∶ U(V ) → U(M) isequivariant with the geodesic flows.As gM is an Anosov flow, by Bowen’s theorem ([Bow73], see also

[PP90]), there is a suspended flow (X,m,T )r (see p. 5) and a contin-uous, measure theoretic isomorphism π ∶ (X,m,T )r → (U(M),Λ, g).Here, X ⊂ SZ with S finite, is a mixing, two-sided SFT with T the

shift and m a Gibbs measure.Denote the corresponding one-sided SFT by (X+,m+, T+) with X+ ∶=(x) ∶ x ∈ X where ∶X → SN, (x) ∶= x∣N; m+ ∶=m −1 & T+ is

the shift.The roof function r ∶ X → R+ is bounded away from 0,∞ and

one-sided in the sense that r = r+ where log r+ ∶ X+ → R is Holdercontinuous:

∣ log r+(x) − log r+(y)∣ ≤Mθt(x,y)

where M > 1, θ ∈ (0,1) & t(x, y) ∶=min n ≥ 1 ∶ xn ≠ yn ≤∞.As in [Ree81], ∃ a block map φ ∶ X → Z and a continuous, measure

theoretic isomorphism

P ∶ (X ×Z,m ×#, Tφ)r → (U(V ),Λ, g)so that P Qn = γ(n) P. Here (X ×Z,m ×#, Tφ)r is as in (±) on p.5.Recall that, as mentioned on p. 4, (X,m,T )r = (Y, ν,ψ) is a con-

tinuous flow on a compact metric space and (X × Z,m × #, Tφ)r isa continuous flow on the locally compact, Polish, metric space. Thecontinuity of P above is with respect to this topology.It suffices to prove (f), (v) & (Y) for (X ×Z,m ×#, Tφ)r.To this end, we show first

¶ φ ∶ X → Z is centered and (r, φ) ∶X → R ×Z is aperiodic.

Proof By [Ree81], (U(V ),Λ, g) is conservative, whence ([Hop37]), er-godic. These properties pass to (X ×Z,m ×#, Tφ)r, whence E(φ) = 0.

As in [Sha93], V is homologically full in the sense that as t→∞, ∃exponentially many closed geodesics of length ≤ t in each homologyclass.

Therefore, by the lemma in [Sol01] the function (r, φ) ∶ X → R × Z isaperiodic. V ¶

Next, by Sinai’s theorem ([Sin72], see also [Sar09]) there are blockfunctions ξ ∶X → Z & , φ+ ∶X+ → Z so that φ = ξ − ξ T + φ+.It follows that

(X ×Z,m ×#, Tφ)r & (X ×Z,m ×#, Tφ+)r+


are RI conjugate.Thus, it suffices to prove (f), (v) & (Y) for

(Z, µ,Ψ) ∶= (X ×Z,m ×#, Tφ+)r+.The continuous natural extension ∶ (X,m,T ) → (X+,m+, T+) in-

duces a continuous natural extension

Π ∶ (Z, µ,Ψ)→ (Z,µ,Φ) ∶= (X+ ×Z,m+ ×#, Tφ+)r+Let

H ∶= LT ⊗Cc(R+)⊗Cc(Z) ⊂ Cc(Z).By theorem 1(i), with b(t) ∝ √t and some centered Gaussian randomvariable S on R, for each A ∈ H and

k ∶ R+ → G,k(t)b(t) ÐÐ→t→∞

z, supt>0

∥k(t)∥b(t) <∞,

b(t)Φt(A Qk(t))(x)ÐÐ→t→∞

fS(z)µ(A)(÷)

uniformly on compact subsets of Z.

Applying this with k ≡ 0 repeatedly, it follows that for J ≥ 1, Ai ∈ H,(1 ≤ i ≤ J), as t1, . . . , tJ →∞:

J

∏j=1

b(tj)ΦtJ (AJΦtJ−1(AJ−1⋯Φt2(A2Φt1(A1)⋯)(F)

→J

∏j=1

fS(zj)µ(Aj) uniformly on compact subsets of Z.

From (÷), we calculate that

b(t)µ(U ⋅ V Qk(t) Ψt)ÐÐÐÐÐÐÐ→t→∞,

k(t)b(t)→z

fS(z)µ(U)µ(V )()

∀ U, V ∈ H ∶= G π Ψ−s ∶ G ∈ H, s > 0and from (F) we see that for N ≥ 1, A, A1, . . . ,AN ∈ H,

Λ(A ∩ N

∏i=1

Ai Ψti) ∼ Λ(A) N

∏i=1

Λ(Ai)b(∆ti)(t)

as t = 0 = t0 < t1 < ⋅ ⋅ ⋅ < tN, ∆ti ∶= ti − ti−1 →∞, (1 ≤ i ≤ N).

Evidently, H ⊂ Cc(Z) is closed under multiplication. Moreover Hseparates points of Z. Thus by the Stone-Weierstrass theorem, Span His locally dense in Cc(Z) and H ⊂ Cc(Z) is indeed a measure determin-

ing collection on Z.


Standard, repeated application of the weak convergence lemma nowestablishes

() ∀ U, V ∈ Cc(Z) and (t) ∀ N ≥ 1, A, A1, . . . ,AN ∈ Cc(Z).As remarked above, this proves theorem 3(o) & (i). VTo conclude, it follows from (÷) with k(t) = 0 and proposition 3.1

in [AN17] that (Z,µ,Φ(J)) is rationally, weakly mixing of order 2 as in(Y) (on p. 19).This property passes to its measure theoretic natural extension (U(V ),Λ, g).

this proves theorem 3(ii). V

§4 Operator Local limit theorem

We’ll use the spectral and perturbation theories of the transfer op-erator of a fibered system (X,m,T,α) which is assumed to be mixing,probability preserving and either a G-M map, or an AFU map. Detailscan be found in §2 in [AD01] for G-M maps and §5 in [ADSZ04] for AFUmaps.Let G ≤ Rκ, , a subgroup of full dimension and let ϕ = (ϕ(1), . . . , ϕ(κ)) ∶

X → G satisfy

D(T )α (ϕ) <∞where D(T )α is as on p. 16.For t ∈ G define Pt ∶ L1(m)→ L1(m) by

Pt(f) ∶= T (ei⟨t,ϕ⟩f),then, each Ps ∶ LT → LT acts quasicompactly where as above LT ∶=

Hθ in the G-M case & LT = BV in the AFU case.Moreover s↦ Ps is continuous (G→ Hom(LT ,LT )).For small t ∈ G, the characteristic function operator Pt has a simple,

dominant eigenvalue and indeed, by Nagaev’s theorem:

1) There are constants ǫ > 0, K > 0 and θ ∈ (0,1); and continuousfunctions λ ∶ B(0, ǫ)→ BC(0,1), N ∶ B(0, ǫ) → Hom(L,L) such that

∥P nt h − λ(t)nN(t)h∥L ≤Kθn∥h∥L ∀ ∣t∣ < ǫ, n ≥ 1, h ∈ L

where ∀∣t∣ < ǫ, N(t) is a projection onto a one-dimensional subspace(spanned by g(t) ∶= N(t)1).2) If ϕ is aperiodic, then ∀ M > 0, ǫ > 0, ∃ K ′ > 0 and θ′ ∈ (0,1) suchthat

∥P nt h∥L ≤K ′θ′n∥h∥L ∀ ǫ ≤ ∣t∣ ≤ M, h ∈ L.


Operator Local limits. Now suppose that 0 < p < 2 and that thedistribution of ϕ is in the strict domain of attraction of a symmetricp-stable random variable S ; equivalently for some 1

p-regularly varying

sequence (b(n) ∶ n ≥ 1),E(exp[i⟨ ϕ

b(n) , u⟩])n ÐÐ→n→∞E(ei⟨S,u⟩).

As shown in [AD01], and [ADSZ04],

E(exp[i⟨ ϕn

b(n) , u⟩])n ≈ λ( ub(n))n ÐÐ→n→∞

E(ei⟨S,u⟩)(J)

Using this, one obtains (by the methods of [Bre68] & [Sto67]) theconditional LLT

b(n)κT n(1A∩[ϕn∈zn+U]) ≈n→∞

fS(ζn)µ(A)mG(U).for A ∈ Cα & U ⊂ G a compact neighborhood of 0 with mG(∂U) = 0,where zn ∈ G with zn = b(n) ⋅ ζn, ζn = O(1).Proof of the Op-LLT as on p. 22 given the Infinite Divisibility

Lemma

We have

rn−nE(r)√n

dist.ÐÐ→n→∞

G & ϕn

b(n)dist.ÐÐ→n→∞

S

where G is centered Gaussian on R and S is symmetric, p-stable onRκ

It follows that (( rn−nE(r)√n

, ϕn

b(n)) ∶ n ≥ 1) is a tight sequence of random

variables on R × Rκ.Let Z = (G,S) ∈ RV (R×Rκ) be a weak limit point of the sequence. By

the Infinite Divisibility Lemma (see p. 33) Z is infinitely divisible.By the Levy-Ito decomposition (see [Sat99]), Z = G + C + c whereG, C ∈ RV (R × Rκ) with G centered Gaussian, C compound Poisson(e.g. p-stable with p < 2) and c is constant. This decomposition isunique and, moreover, G & C are independent.Now, Z = (G,0) + (0,S), whence the Levy-Ito decomposition is

G = (G,0), C = (0,S) and c = 0, with the conclusion that G & Sare independent.This determines Z uniquely and so

( rn−nE(r)√n

, ϕn

b(n)) ÐÐ→n→∞Z.

It follows from the analog of (J) (on p. 31) that if λ(x, y) is thedominant eigenvalue of

P(x,y)f = T(exp[ix(r −E(r)) + i⟨y,ϕ⟩]f),


then

λ( x√n, yb(n))n ÐÐ→n→∞

E(eixG)E(ei⟨y,S⟩)

uniformly on compact subsets.Now, the stable characteristic functions have the forms

E(eixG) = e−ax2 & E(ei⟨y,S⟩) = e−cp,ν(y).

Here a > 0 & cp,ν(y) ∶= ∫Sκ−1 ∣⟨y, s⟩∣pν(ds) where ν is a symmetric mea-sure on Sκ−1. In other words,

−n logλ( x√n, yb(n))ÐÐ→n→∞

ax2 + cp,ν(y)(®)

uniformly on compact subsets.As in [Sto67], fix f ∈ L1(R × G) so that f ∶ R ×G → C is continuous,

compactly supported, A ∈ Cα and (rn, zn) ∈ R×G with zn = b(n)⋅ζn, rn =√n ⋅ ρn with (ζn, ρn)→ (ζ, ρ).By Nagaev’s theorem, (®) (p. 32) and (J-Ap) as on p. 17 (respec-

tively), there exist ϑ ∈ (0,1) & δ > 0 so that

sup(x,y)∈B(0,δ)

∥Pn(x,y)1A − λ(x, y)

nN(x, y)1A∥Hom (L,L) = O(ϑn) as n→∞,(i)

− nRe logλ( x√n,

yb(n)) ≥ θ(ax

2 + cp,ν(y)) ∀ n ≥ 1, (x, y) ∈ B(0, δ),(ii)

sup(x,y)∈supp f∖B(0,δ)

∥Pn(x,y)1A∥L = O(ϑ

n) as n→∞.(iii)

We have, using (i) and (iii):

√nb(n)κT n(1Af(rn −E(r)n − rn, ϕn − zn))=√nb(n)κ ∫

R×G

e−irnx−i⟨y,zn⟩f(x, y)P n(x,y)1Adxdy

=√nb(n)κ(∫

B(0,δ)+∫

R×G∖B(0,δ))e−irnx−i⟨y,zn⟩f(x, y)P n

(x,y)1Adxdy

=√nb(n)κ ∫

B(0,δ)e−irnx−i⟨y,zn⟩f(x, y)P n

(x,y)1Adxdy +O(√nb(n)κϑn)=√nb(n)κ ∫

B(0,δ)e−irnx−i⟨y,zn⟩f(x, y)λ(x, y)nN(x, y)1Adxdy +O(√nb(n)κϑn).


Writing ∆n(x, y) ∶= ( x√n, 1b(n) ⋅ y) and changing variables,

√nb(n)κ ∫

B(0,δ)e−irnx−i⟨y,zn⟩f(x, y)λ(x, y)nN(x, y)1Adxdy

= ∫Rκ+1

1∆−1n B(0,δ)(x, y)e−iρnx−i⟨y,ζn⟩f(∆n(x, y))λ(∆n(x, y))nN(∆n(x, y))1AdxdyÐÐ→n→∞

f(0) ⋅N(0)1A ⋅ ∫R×Rκ

e−iρx−i⟨y,ζ⟩e−ax2−cν,p(y)dxdy

= ∫G

f(y)dy ⋅m(A) ⋅ fZ(ρ, ζ).The convergence here is by Lebesgue’s dominated convergence theorem,since by (ii):

1∆nB(0,δ)(x, y)∣λ(∆n(x, y))∣n ≤ exp[−θ(ax2 + cp,ν(y))].(Op-LLT) follows from this as in [Bre68]. VRemarkAs in the proof of lemma 6.4 in [AD01], “changing variables” in (®),

we see that

− logλ(su, tv) ∼ as2 + cp,ν(v)b−1(1

t) as ∥(s, t)∥ → 0, b(1

t) ∝ 1

s2(á)

uniformly in u = ±1, v ∈ Sκ−1. c.f. lemma 2.4 in [Tho16].

§5 Infinite divisibility lemma

In this section, we complete the proof of the Op-LLT by establishing:

Infinite divisibility lemmaLet (X,m,T,α) be a reverse φ-mixing (as on p. 14) fibered system

and let ϕ ∶ X → G be θ-Holder continuous on each a ∈ α with Dα,θ(ϕ) <∞.Suppose that ∆n = diag(δ(n)1 , . . . , δ

(n)κ ) are κ × κ diagonal matrices

satisfying

(i) each δ(n)j = 1

bj(n) with bj(n), γj-regularly varying with γj > 0

and

(ii) ∆nϕn ∶ n ≥ 1 is uniformly tight.Let nk →∞ and suppose that

∆nkϕnk

dist.ÐÐ→k→∞

Z

where Z is a random variable on Rκ, then Z is infinitely divisible.

ProofBy tightness, c(t) ∶= supn≥1m([∥∆nϕn∥ > t])ÐÐ→

t→∞0.


Suppose that 1 ≤ r ≤ n, then∥∆nϕr∥ = ∥∆n∆

−1r ∆rϕr∥ ≤ ∥∆n∆

−1r ∥ ⋅ ∥∆rϕr∥

whence for ǫ > 0,m([∥∆nϕr∥ > ǫ]) ≤m([∥∆rϕr∥ > ǫ

∥∆n∆−1r ∥])()

≤ c( ǫ∥∆n∆−1r ∥])

→ 0 as n ≥ r →∞, ∥∆n∆−1r ∥→ 0.

Next, for Z a random variable, let

σ(Z) ∶= E ( ∥Z∥1 + ∥Z∥) ,

then for Zn (n ≥ 1) random variables

σ( n∑k=1

Zk) ≤ n

∑k=1

σ(Zk),Zn

PÐÐ→n→∞

0 ⇐⇒ σ(Zn)ÐÐ→n→∞

0.

Similar to (), we have

C(t) ∶= sup σ(∆nϕr) ∶ n ≥ r ≥ 1, ∥∆n∆−1r ∥ ≤ tÐÐ→

t→0+0.(♠)

Proof of (♠) If not, then ∃ νk ≥ rk ≥ 1 and ǫ > 0 so that

∥∆νk∆−1rk∥→ 0 & σ(∆νkϕrk) ≥ ǫ > 0.

But in this case, by () ∆νkϕrkPÐÐ→

k→∞0, whence σ(∆νkϕrk) ÐÐ→

k→∞0.

Contradiction. V(♠)

Using the regular variation of each bj and possibly passing to a sub-sequence of nk →∞, we can ensure that

bj(nk) = 2±1kγjbj(mk) ∀ 1 ≤ j ≤ κ where mk ∶= ⌊nk

k⌋(X)

in addition to

k max1≤J≤2k

∥∆nk∆−1J ∥→ 0(M)

and

∆nkϕnk

dist.ÐÐ→k→∞

Z.

Write nk = kmk + rk with 0 ≤ rk <mk.


Let b ∶= minj γj > 0. By (X), we have ∥∆nk∆−1mk∥ ≤ 2

kb→ 0 and hence

by (♠), that

∆nkϕmk

PÐÐ→k→∞

0.

Next, we claim that

∆nk

k−1

∑j=1

ϕ2k Tjmk−2k

PÐÐ→k→∞

0.(j)

Proof of (j)

σ(∆nk

k−1

∑j=1

ϕ2k T(j+1)mk−2k) ≤ kσ(∆nk

ϕ2k)ÐÐ→k→∞

0 by (M). V(j)

Next, we’ll use (j) to show :

∆nk

k−1

∑j=0

(ϕmk−2k Tjmk + ϕrk T

jmk) dist.ÐÐ→k→∞

Z.(C)

Proof

∆nk

k−1

∑j=0

(ϕmk−2k Tjmk + ϕrk T

jmk) −∆nkϕnk

=∆nk

k−1

∑j=1

ϕ2k Tjmk−2k

PÐÐ→k→∞

0 by (j). This proves (C). V

For n ≥ 1, a ∈ αn, fix ζn(a) ∈ T na so that ∃ zn(a) ∈ a satisfyingT nzn(a) = ζn(a).Now define πn ∶X →X by

πn(x) ∶= zn(αn(x)).It follows that for n, k ≥ 1,

∣ϕn − (ϕ πn+k)n∣ ≤ Dα,θ(ϕ)θk1 − θ

.(T)

Set

Wk,j ∶=∆nkϕmk−2k T

jmk , 0 ≤ j ≤ k − 1 & Wk,k ∶=∆nkϕrk T

kmk .

Yk,j ∶=∆nkϕmk−2k πmk−k T

jmk , 0 ≤ j ≤ k − 1 & Yk,k ∶=∆nkϕrk T

kmk .


By (T), for each 0 ≤ j ≤ k,∥Yk,j −Wk,j∥ ≤ ∥∆nk

∥Dα,θ(ϕ)kθk1 − θ

whencek

∑j=0

∥Yk,j −Wk,j∥ ≤ ∥∆nk∥Dα,θ(ϕ)kθk1 − θ

ÐÐ→k→∞

0.

It follows thatk

∑j=0

Yk,jdist.ÐÐ→k→∞

Z &(i)

maxj

σ(Yk,j)ÐÐ→k→∞

0.(ii)

Now (Yk,k−j ∶ 0 ≤ j ≤ k) ∶ k ≥ 1) is a φ-mixing, triangular array asin [Sam84] and as shown there, Z is infinitely divisible. V

§6 Appendix

In this appendix, we state and prove an Extended LLT estimate anddeduce the the Deviation lemma from it.The estimate is a multidimensional version of proposition 4 in [P09].

For earlier work on the iid case, see [Pet75, Chapter VII,§3].

Let

(X,m,T,α) be a mixing fibered system, either GM or AFU;

d ≥ 1, H ≤ Rd be a closed subgroup of full dimension;

Ψ ∶X → H be aperiodic, D(T )α (Ψ) <∞ and E(∥Ψ∥4) <∞.

Extended LLT estimate ([P09])Fix U ⊂ H, mH(∂U) = 0 precompact. There exist Γ > 0, η > 0 so that

T n1[Ψn∈x+U] ≤ Γ

nd2

⋅ (e− η∥x∥2n + 1

n) ∀ n ≥ 1, x ∈ Rd.(Z)

Proof We’ll show that

T n1[Ψn∈x+U] ≤ Γ

nd2

⋅ (b( ∥x∥√n)e− η∥x∥2

n + 1n) ∀ n ≥ 1, x ∈ R

d(-)

where b ∶ R → R is a polynomial of degree 3 with non-negative coeffi-cients and b(0) = 1.This suffices for (Z) with suitably adjusted constants Γ > 0, η > 0.


As before, consider the operators P (x) ∈ Hom (L,L) (x ∈ H) definedby

P (x)f = T (exp[i⟨x,Ψ⟩]f).As shown in [HH01], since E(∥Ψ∥4) < ∞, the function P ∶ H →

Hom (L,L) is 4-times continuously differentiable with

(( ∂k1∂xj1

∂k2

∂xj2. . . ∂

kq

∂xjq)P (x))(f) = ipT( q

∏i=1

Ψkijiexp[i⟨x,Ψ⟩]f)

∀ 1 ≤ j1, j2, . . . , jq ≤ d, k1, k2, . . . , kq, ∑i

ki = p, 1 ≤ p ≤ 4.

Moreover, by Nagaev’s theorem, there are constants ǫ > 0, K > 0and θ ∈ (0,1); and continuous functions λ ∶ B(0, ǫ) → BC(0,1), N ∶B(0, ǫ)→ Hom(L,L) such that for x ∈ B(0, ǫ),

∥P (x)nf − λ(x)nN(x)f∥L ≤Kθn∥f∥L ∀ n ≥ 1, f ∈ L(c)

whereN(x) is a projection onto a one-dimensional subspace (spannedby N(x)1).It follows that x ↦ N(x) is also C4 B(0, ǫ) → Hom(L,L) as is x ↦

λ(x) (B(0, ǫ) → BC(0,1)).In particular, there is a symmetric, positive definite matrix γ, and

M > 0 so that for x ∈ B(0, ǫ),Logλ(x) = −⟨γx,x⟩(1 + E(x)) where ∣E(x)∣ ≤M∥x∥.()

Next, fix h ∈ L1(H)+ so that 1U ≤ h and h ∈ C∞C (H). 1

Note that if hz(y) = h(y − z), then 1U+z ≤ hz and hz(x) = ei⟨x,z⟩h(x)has the same bounded support as h(x).Possibly increasing θ ∈ (0,1) & K > 0 in (c) (p. 37), we see by

aperiodicity, that

supx∈supp h∖B(0,δ)

∥P (x)n1∥L ≤Kϑn.(H)

1e.g. h = g with g(x) = ∫H f(y)f(y−x)dmH(y) for some f ∈ C∞

C(H) with f ≥ 1U .


Thus

nd2 T n1[Ψn∈z+U] ≤ n d

2 T n(hz(Ψn))= n

d2 ∫

H

hz(x)T n(exp[i⟨x,Ψn⟩])dx= n

d2 ∫

H

hz(x)P (x)n1dx= n

d2 ∫

B(0,δ)hz(x)λ(x)ng(x)dx + En

where g(x) ∶= N(x)1 and

En

nd2

= ∫B(0,δ)

hz(x)(P (x)n1 − λ(x)ng(x))dx + ∫supp h∖B(0,δ)

hz(x)P (x)n1dx,whence by (c) and (H) (p. 37)

∣En∣ ≤K(1 +mH(supp h)∥hz∥∞)n d2 θn ≤K(1 +mH(supp h)∥h∥1)n d

2 θn.

Next, changing variables,

nd2 ∫

B(0,δ)hz(x)λ(x)ng(x)dx(Ë)

= ∫B(0,√nδ)

hz( x√n)λ( x√

n)ng( x√

n)dx.

Estimation of RHS of (Ë)Since γ in () (p. 37) is positive definite, ∃ η > 0 so that

⟨γx,x⟩ ≥ η∥x∥2 ∀ x ∈ Rd,(*)

whence

∣λ( x√n)∣, exp[− ⟨γx,x⟩

n] ≤ e−2η∥x∥2n ∀ x ∈ B(0,√nδ).

For n ≥ 4, x ∈ B(0,√nδ), setλ = λ( x√

n) & φ = exp[− ⟨γx,x⟩

n],

then φ ∈ (0,1), λ ∈ B(0,1) and some baby algebra shows that

λn = φn + nφn−1(λ − φ) + (λ − φ)2Bn(m)


where Bn = Bn(x) ∶=∑n−1k=1 ∑k−1j=0 φn−1−kλjφk−1−j, whence

∣Bn∣ ≤ n−1∑k=1

∣φn−1−k k−1∑j=0

λjφk−1−j ∣(G)

=n−1

∑k=1

k−1

∑j=0

φn−2+j ∣λj ∣≤ n2e−

2(n−2)η∥x∥2n ≤ n2e−η∥x∥

2

We now estimate the RHS of (Ë) according to (m) (as on p. 38),

∫B(0,√nδ)

hz( x√n)λ( x√

n)ng( x√

n)dx =Mn +Qn +Dn

=∶ ∫B(0,√nδ)

hz( x√n)g( x√

n) ⋅ (φn + nφn−1(λ − φ) + (λ − φ)2Bn) ⋅ dx

where

Mn ∶= ∫B(0,√nδ)

ei⟨x, z√

n⟩h( x√

n)g( x√

n) exp[−⟨γx,x⟩]dx

is the “main term” with “error terms”

Qn = n∫B(0,√nδ)

ei⟨x, z√

n⟩h( x√

n)g( x√

n) exp[−⟨γx,x⟩⋅n−1

n](λ( x√

n)−exp[− ⟨γx,x⟩

n])dx

and

Dn = ∫B(0,√nδ)

ei⟨x, z√

n⟩h( x√

n)g( x√

n) exp[−⟨γx,x⟩⋅n−2

n](λ( x√

n)−exp[− ⟨γx,x⟩

n])2Bn(x)dx.

There is a polynomial q(x) = a0 + a1(x) + a2(x) + a3(x) (where eachai is a sum of monomials of degree i) so that

h(y)g(y) = q(y) +O(∥y∥4),whence

h( x√n)g( x√

n) = q( x√

n) +O( ∥x∥4

n2 ).(Ü)

The term Mn

The following estimations are uniform in z ∈ Rd.By (Ü), we have

Mn = ∫B(0,√nδ)

ei⟨x, z√

n⟩q( x√

n) exp[−⟨γx,x⟩]dx +O( 1

n2 ).For some ϑ ∈ (0,1) we have

∫Rd∖B(0,√nδ)

∥x∥j exp[−⟨γx,x⟩]dx = O(ϑn) (0 ≤ j ≤ 4).(Ý)


Thus

Mn = ∫Rdei⟨x, z√

n⟩q( x√

n) exp[−⟨γx,x⟩]dx +O( 1

n2)and

∫Rdei⟨x, z√

n⟩q( x√

n) exp[−⟨γx,x⟩]dx = 3

∑j=0

1

nj2∫

Rdei⟨x, z√

n⟩aj(x) exp[−⟨γx,x⟩]dx

=3

∑j=0

1

nj2

(aj(∇)φ)( z√n)

=3

∑j=0

1

nj2

bj( z√n)φ( z√

n)

where bj is a sum of degree j monomials, b0 = 1 and φ(y) = ∫Rd ei⟨x,y⟩ exp[−⟨γx,x⟩]dx.Evidently, φ(y) = ce−⟨γy,y⟩ ≤ ce−η∥y∥2 where c > 0 and η > 0 is as in

(*) (as on p. 38).Moreover,

∣ 3

∑j=0

1

nj2

bj( x√n)∣ ≤ b( ∥x∥√

n) ∀ x ∈ Rd

where b ∶ R → R is a polynomial of degree 3 with non-negative coeffi-cients and b(0) = 1.It follows that, uniformly in z:

Mn ≤ b( z√n)e− η∥z∥2

n +O( 1n2 ).

The term DnTo estimate Dn we use (G) as on p. 39 and

∣λ( x√n) − exp[− ⟨γx,x⟩

n]∣ ≤ A ∥x∥3

n32

(A)

obtaining

∣Dn∣ ≤ ∫B(0,√nδ)

exp[−⟨γx,x⟩ ⋅ n−2n]∣λ( x√

n) − exp[− ⟨γx,x⟩

n]∣2∣Bn(x)∣dx

≤ A2

n3 ∫B(0,√nδ)

∥x∥6∣Bn(x)∣ exp[− ⟨γx,x⟩2]dx by (A)

≤ A2

n∫B(0,√nδ)

∥x∥6 exp[− ⟨γx,x⟩2]dx by (G).

The term QnThis needs a Taylor expansion of higher order than (A) (p. 40):For n ≥ 1 and x ∈ B(0,√nδ),

λ( x√n) − exp[− ⟨γx,x⟩

n] = 1

n32

c3(x) + en(x)


where c3 is a sum of degree 3 monomials and ∣en(x)∣ ≤ C∥x∥4n2 .

Thus

Qn = n∫B(0,√nδ)

ei⟨x, z√

n⟩h( x√

n)g( x√

n) exp[−⟨γx,x⟩ ⋅ n−1

n](λ( x√


n])dx

= n [∫B(0,√nδ)

ei⟨x, z√

n⟩q( x√


n](λ( x√


n])dx +O( 1

n2)]

by (Ü) on p. 39

= n∫B(0,√nδ)

ei⟨x, z√

n⟩q( x√


n](λ( x√


n])dx +O(1

n)

= n∫B(0,√nδ)

ei⟨x, z√

n⟩q( x√


n]( 1

n32

c3(x) + en(x))dx +O(1n)

= 1√n ∫

B(0,√nδ)ei⟨x, z√

n⟩q( x√


n]c3(x)dx

+ ∫B(0,√nδ)

ei⟨x, z√

n⟩q( x√


n]en(x)dx +O(1

n)

= 1√n ∫

B(0,√nδ)ei⟨x, z√

n⟩q( x√


n]c3(x)dx +O(1

n).

Now

∫B(0,√nδ)

ei⟨x, z√

n⟩q( x√


n]c3(x)dx

= ∫Rdei⟨x, z√

n⟩q( x√


n]c3(x)dx +O(ϑn) by (Ý) on p. 39

=3

∑j=0

1

nj

2

∫Rdaj(x)c3(x)ei⟨x, z√n ⟩q( x√


n]dx +O(ϑn)

=3

∑j=0

1

nj

2

(aj(∆)c3(∆)φ(n))( z√n) where φ(n)(u) = ∫

Rdexp[i⟨u,x⟩ − ⟨γx,x⟩ ⋅ n−1

n]dx,

=3

∑j=0

1

nj

2

pj( z√n)φ( z√

n)

where the pj are polynomials. 2 ()

This proves (-) and hence (Z). V

Deviation Lemma


(i) There are constants Γ, η > 0 so that ∀ M > 0 ∃ t0(M) > 0 suchthat

T n1A∩[rn∈I+t−y] ≤ Γ(exp[−η∣n−λt∣2

t]√

t+

1

n3

2

)∀ M > 0, t > t0(M) & n ∈ BK(M,t)

where K > 1 & BK(M,t) are as on p. 24.(ii) If in addition, E(∥ϕ∥4) <∞, then, possibly changing Γ, η & t0(M),

we have for M > 0, t > t0(M), x(t) ∈ Rκ, ∥x(t)∥ ≤ M√t

2λK2 :

T n1A∩[ϕn∈x(t)−z+V, rn∈I+t−y] ≤ Γ

tκ2

⋅ (exp[−η∣n−λt∣2

t]√

t+

1

n3

2

)∀ n ∈ BK(M,t).

Proof of (ii) in the deviation lemmaLet ψ ∶= (ϕ, r −κ) ∶ X → G× R where κ ∶= E(r) = 1

λ.

For t > 0, x(t) ∈ Rκ, ∥x(t)∥ ≤ M√t

4λK2 , n ∈ BK(M,t),T n1A∩[ϕn∈x(t)−z+V, rn∈I+t−y] = T

n1A∩[ψn∈xn,t+V ×I]

where xn,t ∶= (x(t) − z, t − nκ − y).By the assumptions, the extended LLT estimate applies and it follows

from (Z) that there exist Γ > 0, η > 0 so that

T n1[ψn∈x+U] ≤ Γ

nκ2

⋅ (exp[−η∥xn,t∥2n]√

n+

Γ

n3

2

) ∀ n ∈ BK(M,t).Next

∥xn,t∥ ≥ ∣t − nκ∣ − ∥x(t)∥ − (∣y∣ + ∥z∥)= ∣n−λt∣

λ− ∥x(t)∥ − (∣y∣ + ∥z∥)

= ∣n−λt∣2λ+ ∣n−λt∣

2λ− ∥x(t)∥ − (∣y∣ + ∥z∥).

Now

∣n−λt∣2λ− ∥x(t)∥ − (∣y∣ + ∥z∥) ≥ M

√n

2λK− M

√t

2λK2 − (∣y∣ + ∥z∥)≥ M

√t

2λK2 − (∣y∣ + ∥z∥) > 0 ∀ M > 0, t > t0(M) ∶= 4λ2K2(∣y∣+∥z∥)2M2 .

Setting yn,t ∶=xn,t√n, we have for M > 0, t > t0(M) and n ∈ BK(M,t),

∥yn,t∥ > ∣n−λt∣2λ√n.


As before, let ξ > 0 be so that

⟨γx,x⟩ ≥ ξ∥x∥2 ∀ x ∈ Rκ+1.It follows that for M > 0, t > t0(M) and n ∈ BK(M,t),

⟨γxn,t,xn,t⟩n

≥ ⟨γyn,t, yn,t⟩ ≥ ξ∥yn,t∥2 ≥ η ∣λt−n∣2t

where η ∶= ξ

4Kλ2.

Thus,

T n1[ψn∈x+U] ≤ K2Γ

tκ2

⋅ ( exp[− η∣n−λt∣2t

]√t

+ Γ

n32

)∀ M > 0, t > t0(M) & n ∈ BK(M,t). 2

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(Aaronson) School of Math. Sciences, Tel Aviv University, 69978

Tel Aviv, Israel.

Webpage : http://www.math.tau.ac.il/∼aaro

E-mail address : [email protected]

(Terhesiu)Department of Mathematics, University of Exeter, Exeter

EX4 4QF, UK

E-mail address, Terhesiu: [email protected]

http://www.math.tau.ac.il/~aaro

arXiv:1710.10150v7 [math.DS] 1 Jul 2020

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