Rational ergodicity of Step function Skew Products

We study rational step function skew products over certain rotations of the circle proving ergodicity and bounded rational ergodicity when rotation number is a quadratic irrational. The latter arises from a consideration of the asymptotic temporal statistics of an orbit as modelled by an associated affine random walk.


and of quadratic irrationals
It is known that QUAD ⊂ BAD and that BAD has Lebesgue measure zero (see e.g. [8]).
Denominator of a rational step function.
Fix d, Q ∈ N, Q ≥ 2 and Φ ∶ Z Q → R d . The rational step function with denominator Q and values Φ is the step function ϕ = ϕ (Φ) ∶ T → R d defined by where κ ∶ [0, 1) → Z Q is defined by κ(x) ∶= ⌊Qx⌋. Every rational step function is of this form for some Q ≥ 2 and Φ ∶ Z Q → R d .
If ϕ ∶ T → R d is a rational step function with denominator Q, then We prove, for ϕ ∶ T → R d a rational step function with denominator Q and values Φ and which is centered in the sense that ∫ T ϕ(x)dx = 0:
Here and throughout, CEMPT means conservative, ergodic, measure preserving transformation, m G denotes Haar measure on the locally compact, Polish, Abelian group G, normalized if G is compact. Also, ϕ ∶ T → R d is always going to mean a centered rational step function.
Theorem 1' will follow from the stronger theorem 1 (see below). The technique of the proof of theorem 1 is not new. For older, related results, see [6], [16] and references therein.

Theorem 2: Temporal CLT
If α ∈ QUAD and dim span R ϕ(T) = d, then ∃ ℓ k ∈ N, ℓ k ↑ & ℓ k ∝ λ k for some λ > 1 and µ (0) ∈ R d so that for any box I ⊂ R d , where Z is a globally supported, centered, normal random variable on R d and f Z is its probability density function.
For an introduction to temporal statistics in dynamics see [7]. Theorem 2 here is a generalization of a subsequence version of theorem 1.1 in [3], which in turn has been recently strengthened in [4].
Notations. Here and throughout, for a n , b n > 0: a n ≪ b n means ∃ M > 0 so that a n ≤ Mb n for each n ≥ 1, a n ≍ b n means a n ≪ b n and b n ≪ a n , a n ∝ b n means ∃ lim n→∞ an bn ∈ R + ∶= (0, ∞) and a n ∼ b n means an bn → n→∞ 1.
Outline of the rest of the paper.
In §1 we prove theorem 1, a stronger version of theorem 1'. The rest of the paper is devoted to the proofs of theorems 2 and 3.
To study the temporal statistics of these tuples, we consider the "temporal random variables" x k ∶ {1, ..., ℓ k } → R d , defined by x k (n) = ϕ n (0), where n is a uniformly distributed random variable with values in {1, ..., ℓ k }. In other words, The recursive properties of the tuples (see §2), allow us to construct an associated affine random walk (ARW) which models the distribution of the "temporal random variables" (see §3).
In §4 we show that when α is quadratic, the sequence of expectations E (x k ) is asymptotically linear. This culminates in the approximation of the distribution of x k − E (x k ) by an affine random walk generated by a sequence of centered, independent, identically distributed affine transformations (see the ARW centering lemma).
This enables proof in §5 of theorem 2 which is a central limit theorem for (x k ∶ k ≥ 1). The proof of theorem 3 in §6 is based on a "weak, rough local limit theorem" for (x k ∶ k ≥ 1). Both proofs use a spectral theory of ARWs based on perturbation theory of stochastic matrices (as in [9]). §1 Ergodicity
Here, and throughout, for k ∈ Z, k ≥ 0, we denote The principal denominators q n of α are given by the numerators p n are given by p 0 = 0, p 1 = 1, p n+1 = a n+1 p n + p n−1 and the principal convergents pn qn satisfy 1 We'll also need theorems 16, 17 and 19 in [13]: The following is also well known (see e.g. [8], [13]): For Q ≥ 2, we'll also need the collection Evidently, BAD ⊂ SBAD ∶= ⋂ Q≥2 SBAD Q and it is not hard to show that SBAD has full Lebesgue measure.
In the following, ϕ = ϕ (Φ) ∶ T → R d is a rational step function with denominator Q ≥ 2.

Theorem 1
Suppose that either The rest of this section is devoted to the proof of theorem 1.
Essential values and Periods. Let (X, B, m) be a standard probability space, and let T ∶ X → X be an invertible, ergodic, probability preserving transformation and B + ∶= {A ∈ B ∶ m(A) > 0}.
Suppose that G is a locally compact, Polish, Abelian group equipped with the translation invariant metric ρ (e.g. ρ(x, y) = x − y if G ≤ R d ).
Let ϕ ∶ X → G be measurable and define ϕ n ∶ X → G by The collection of essential values of ϕ (as in [17]) is and preserves the measure m × m G .
Define the collection of periods for T ϕ -invariant functions: It is not hard to see that T ϕ is ergodic iff T is ergodic & Per (ϕ) = G.
In view of this, the conclusion of theorem 1 is equivalent to We prove this first in the case that Φ(Z Q ) is countable and then deduce the uncountable case. Let Denjoy-Koksma inequality ([11], [10]) where ⋁ T ϕ denotes the total variation of ϕ.

Remark. Consequently, when Γ
Given an Abelian group G and g 0 ∈ G, let r g 0 ∶ G → G denote the group rotation on G given by r g 0 (g) ∶= g + g 0 .
Proof of theorem 1 in the countable case Sublemma 1 For theorem 1 in the countable case, it suffices that We claim moreover that #Γ Γ 0 ≤ Q. To see this, using which is ergodic, being a product of two ergodic group rotations with disjoint spectra.
Thus H is constant a.e., whence also h, and T α,ϕ is ergodic.
We'll prove the sublemma using Oren's Lemma [16] If there exist n k ∈ N and A k ⊂ T such that ϕ n k is constant on A k and ϕ n k A k → a, inf m T (A k ) > 0 and lim k →∞ n k α = 0 then a ∈ Per (ϕ).
Here and and throughout, x ∶= min k∈Z x − k .
Note that a version of Oren's lemma is implicit in [5].
Next, we claim that for ( ), it suffices to show For any ǫ ∈ Z Q there are sequences of measurable sets (A k ), (B k ) ⊂ T and positive integers n k ∈ N such that ϕ qn k is constant on A k and B k , Indeed, by the remark after Denjoy-Koksma inequality, there is a finite set F so that Finally, we construct the sequences of measurable sets A k , B k ⊂ T as in .
To this end, we prove first that the discontinuities of ϕ are "dynamically separated".
Let q ∈ N. Since ϕ is a step function with the set of discontinuities Hence the distance between the discontinuities is bounded below by Proof of ( ) when α ∈ SBAD By definition of SBAD, there exist a sequence (m k ) ∈ N, ν ∈ N and ε > 0 such that Proof of ( ) when Q is prime This further splits into two separate cases.
(i) There are only finitely many n's such that q n = 0 mod Q: Choose N large enough such that q n ≠ 0 mod Q for n ≥ N. For n > N, As before, we have min 0<j<qn jα > 1 2q n .
Since q h is prime to Q for all h ≥ n, for 0 < j < q n , jQ is not a multiple of q n+r for r ≥ 0. Thus by [13, theorem 19] if then Qj is a multiple of q r for some r < n; in this case Therefore min 0<j<qn Qjα ≥ min q n−1 α , (ii)There are infinitely many n's such that q n = 0 mod Q: Let (n k ) be the subsequence such that q n k = 0 mod Q. Let the ν-th term of the continued fraction expansion of α be given by a ν ; we know a n 1 1 0 a n−1 1 1 0 ⋯ a 1 1 1 0 Since det ( aν 1 1 0 ) = −1 for all ν, there determinant of the product is either 1 or −1. Thus q n k +1 ≠ 0 mod Q.
By the recursion formula for the principal denominators, we have that q n k +r+1 = M(r)q n k + S(r)q n k +1 for some M(r), S(r) ∈ N. Again, for some 0 < j < q n k +1 then by [13, theorem 19], jQ is a multiple of q ν for some ν.
Since q n k +1 ≠ 0 mod Q, if jQ = iq n k +1 for some i ∈ N then j ≥ q n k +1 ; it follows that jQ is not a multiple of q n k +1 . Therefore i ≠ n k + 1. Since q n k = 0 mod Q if jQ = iq n k +r+1 = iM(r)q n k + iS(r)q n k +1 = 0 mod Q for some i then iS(r) is multiple of Q implying j ≥ q n k +1 ; thus the number jQ cannot be a multiple of q n k +r+1 for any r ∈ N and it follows that i ≤ n k .
Construction of measurable sets as in By ( ) there exist a subsequence (n k ) ↑ ∞ and θ > 0 such that disc(q n k ) > θ qn k and such that q n k is sufficiently large compared to F 2 , where F is the finite set of values taken by ϕ qn k as in the remark after the Denjoy-Koksma inequality.
Fix k and let ∂ be the partition of T by the discontinuities { ǫ Q − hα ∶ 0 ≤ h ≤ q n k − 1} of the step function ϕ qn k .
For 0 ≤ h < q n k , let I − h ∈ ∂ be the interval with right endpoint ǫ Q − hα and I + h ∈ ∂ be the interval with left endpoint ǫ Q − hα. We and by ( ) These sets are as in and the proof of theorem 1 in the countable case is now complete.

Proof of theorem 1 in the uncountable case
Let Consider the cocycle Ψ ∶ T → Z K defined by It follows that ⟨Ψ(T)⟩ = Z K . We claim that To see this,

By linear independence of {e
Thus, by ( ) as on page 6 in the countable case, and Schmidt's theorem, By linearity of L, The orbit sequence Theorems 2 and 3 both depend on the modeling of the orbit sequence (ϕ n (0) ∶ n ≥ 1) by an associated affine random walk. To extract this affine random walk we first obtain a sequential substitution construction of the jump sequence (ϕ({nα}) ∶ n ≥ 1) for α ∈ (0, 1) ∖ Q.
The quadratic case. If α ∈ QUAD, then so does β = {Qα} and there exist (n 1 , n 2 , . . . , Here and throughout, • 1 denotes a vector all of whose coordinates are 1, if the last symbol in b k (1) is changed from "1" to "0".
, then Parities, Jumps & Orbits. Next, we compute the jump blocks. We call κ(x) = ⌊Qx⌋ ∈ Z Q the parity of x and we begin by calculating the parity blocks with a generalization of [2, theorem 2.2]. For then by ( ) as on page 13 and [2, theorem 2.1] respectively, where the addition is mod Q and ∑ ν∈∅ ∶= 0. Note that B 0 (i, ǫ) = (ǫ).

Theorem 5.1 (Parity recursions)
Here (as before) the addition is mod Q.

It follows that
Using [2, theorem 2.1] and ( ) as on page 14, we have mod Q,
The parity states are given by ǫ 0 (i) = P + i mod Q and
It follows from ( ) that for i = 0, 1: where addition is mod Q and that the jump block Orbit blocks.
Define the auxiliary orbit blocks In particular by (☼) Our goal here is to obtain the transition between auxiliary orbit blocks.

Orbit block transitions.
The simple displacement over the auxiliary jump block J k (i, ǫ) is The cumulative displacements over the concatenation jump blocks Generating functions of orbit blocks.
and their generating functions Here and throughout, i ∶= √ −1.
Transition matrices. Noting that

It follows that
The random affine model Probabilities. Here, we consider the probabilities and each x k (i, ǫ) ∈ R d as a random variable with sample space (Ω k (i), P (i) k ), and understand the transitions of the resulting stochastic processes

Random variables.
We denote by RV (Z), for Z a measurable space the collection of Zvalued random variables.
Consider any sequence of independent, random vectors Note that when n k+1 = 2, then L is defined when and only when P (L (k+1)

Random affine transformations.
Given a random vector This RAT is of flip-type in the sense of [1]. Throughout this paper we'll often denote a flip-type RAT Given a sequence (L random vectors as before, consider the associated RAT sequence RAT characteristic function.
The characteristic function of the RAT F = (L,

RAT lemma
For each k ≥ 1, s ∈ S: is as in ( ) as on 16.
Here and throughout for K ≤ L & RATs (F j ∶ K ≤ j ≤ L) Proof For k ≥ 1, define By (J) as on page 18, The result follows by induction since Ξ 0 =Ξ 0 ≡ 1.

Associated affine random walks.
We associate to a sequence

Elementary presentation.
We now split the random vectors (L i are Z Q -valued random variables; the latter is given by It is not hard to see that In the sequel, we'll have recourse to the elementary random vector We next establish that the centered RAT sequence (as in [1]) corresponding to a quadratic irrational and a rational step function is "asymptotically eventually periodic".
The proofs of theorems 2 & 3 rely on this fact.
This asymptotic eventual periodicity is obtained via centering. We'll see that elementary random vector sequence is always asymptotically eventually periodic, however, the cumulative displacements may have linear growth. The centering is needed to offset this possibility.
In this section, we'll often "possibly extend the period in " to demonstrate eventual periodicity of related sequences. Recall (( ) on page 15) that the parity state transitions are given by In the quadratic case, these transitions form an eventually periodic sequence, whence is also eventually periodic.
Thus there exist matrices M (k+1) ∶ S × S → Z such that Displacement lemma Suppose that α ∈ QUAD, then there exist K, L ∈ N and c, d ∈ (R d ) S so that σ K+Ln = c + nd.
For α ∈ QUAD, the simple displacement transitions are eventually periodic and the proof of the displacement lemma rests on the Denjoy-Koksma inequality and a spectral analysis of the simple displacement transformations on C S over a period (as in the "eigenvalue lemma" below).

Subspace decomposition & eigenvalues.
For α ∈ QUAD, the parity sequence (ǫ k ∶ k ≥ 1) is eventually periodic, whence the above sequence of matrices (M (k) ∶ k ≥ 1) giving the displacement transitions is also eventually periodic.
and let ⃗ e r ∈ C Q be given by (⃗ e r ) s ∶= γ rs for 0 ≤ r ≤ Q − 1 and 1 ≤ s ≤ Q.
Since ⃗ e s ⊥ ⃗ e t ∀ s, t ∈ Z Q , s ≠ t, we have that (⃗ e r ∶ 0 ≤ r ≤ Q − 1) form an orthogonal basis for C Q and Next, define the bracket

It follows from (T) that
To summarize, letting for 0 ≤ r ≤ Q − 1,

Eigenvalue lemma
For 1 ≤ r ≤ Q − 1, all the eigenvalues of B Vr are roots of unity.

Proof
We have that B V 0 is a product of integer matrices of the form N 1 N − 1 1 with N ∈ N; we have N ≥ 2 for at least one of these matrices. Therefore B V 0 is a positive matrix with integer coefficients and unit determinant. It follows that the characteristic polynomial of B V 0 is an integer polynomial of form z 2 − Jz + 1 for some J ∈ N (and that We claim first that no B Vr (1 ≤ r ≤ Q − 1) is hyperbolic. If this were not the case for 1 ≤ r ≤ Q − 1, there would be λ > 1 and a rational cocycle Φ(≠ 0) ⊥ 1 with ⟨Φ, ⃗ e r ⟩ ≠ 0 giving rise to either • σ K+Ln ≫ λ n which is impossible by the Denjoy-Koksma estimate; or • σ K+Ln ≪ 1 λ n which is impossible by theorem 1.
To continue, since B is an integer matrix, det(B − zId) is a polynomial with integer coefficients.
It follows that is also a polynomial with integer coefficients. As shown above, all its roots are of unit modulus. By Kronecker's theorem ( [15]), all these roots are roots of unity.

Proof of the displacement lemma
Let {γ j ∶ j ∈ J } be the collection of eigenvalues of B V ⊥ 0 counting multiplicity which are all roots of unity. Let V j be the corresponding Jordan subspace, then by the above, We may extend the period in as on page 21 so that γ j = 1 ∀ j ∈ J . For each j ∈ J let (e j (j) ∶ j = 1, 2) be the Jordan basis of V j . For j ∈ J , x = x 1 e 1 (j) + x 2 e 2 (j) and N ≥ 1, we have that This proves the displacement lemma.
In the sequel, we'll also need the following.
Positivity proposition By possibly extending the period in as on Page 21, we may ensure that B s,t > 0 ∀ s, t ∈ S.

Proof
It follows from ( ) as on page 23 that where D 1 , D 2 , D 3 ∈ M Q×Q (N 0 ) are matrices where each row and column has at least one non-zero entry. By the parity proposition, η k and η k+1 generate the group Z Q . Applying this to k = r −1, we get that there exists an N such that for all n ≥ N, (ρ 0 + ρ ηr + ρ η r−1 ) n > 0, meaning all of its entries are positive. Thus B N +2 > 0. This proves that B is aperiodic and irreducible and that by extending the period, we can ensure that B is a positive matrix.

Asymptotic eventual periodicity & centering.
Let α ∈ QUAD and ϕ be a step function with rational discontinuities with associated RAT sequence (F k ∶ k ≥ 1) and ARW (X (k) ∶ k ≥ 1).

Elementary periodic approximation lemma
There are constants λ, M > 1 and, for each 1 ≤ r ≤ L there is a random vector Proof We have that .
It follows that for i, j = 0, 1 & 1 ≤ r ≤ L, Next, we observe that for n ≥ 1, 1 ≤ r ≤ L, j = 0, 1, the distribution of e Analogously, u (L+Ln+r+1) (i) has a conditional distribution independent of n and we define are as advertised by construction.

RAT periodic approximation lemma
There are random variables a ∈ RV we have by the elementary periodic approximation lemma, sup s,t∈S To study the random variables W (K+Ln+r) s , we'll need formulae for the cumulative displacements.
Using the displacement lemma, for 1 ≤ r ≤ L, It follows that (1 ≤ r ≤ L, n ≥ 1) be the RATs defined by

This has the form
It follows from ( ‡) that H (n) satisfies ( ) and (m).

Coupling.
It follows that there exists a probability space (Ω, A, P ) on which the independent random vectors (H (n) ,F n ) ( n ≥ 1) can be defined so that ARW periodic approximation lemma There is a constant M > 1 so that for all n > J, Proof of (H) For fixed ν ≥ 1 and a measurable function g ∶ Ω → R S , for which g ν is integrable, let then ⋅ ν is a norm. Next, it follows from the RAT periodic approximation lemma that Thus, for some M ′ > 0, Thus possibly increasing M, and (H) follows.
It follows as in [1] that By the positivity proposition, by possibly extending the period in as on page 21, we may ensure that E(a(H (n) )) = Π H (n) (0) is an aperiodic stochastic matrix whence 1 is a simple, dominant eigenvalue with eigenvector 1 ∈ C S . Suppose that π ∈ R S + satisfies ⟨π, 1⟩ = 1 & E(a) * π = π (where A * is the transpose of the matrix A).
We claim next that ∃ µ, ξ J ∈ (R d ) S so that

Proof of ( )
By (C) as on page 31, To obtain the expansion for E(Y (n) J ) from ( ) (with enlarged ρ), it suffices to show that N(E(w)) = 0. This will follow from the Denjoy-Koksma estimate. By (H) as on page 30, we have . Thus, if N(E(w)) ≠ 0, then by ( ), E(X (K+Ln) ) ≍ n 2 contradicting the Denjoy-Koksma estimate that E(X (K+Ln) ) = O(n). The expansion for E(X (K+Ln) ) follows.

ARW centering lemma
There is a centered, independent, identically distributed RAT sequence (H n ∶ n ≥ 1) and 0 < ρ < r < 1 so that if for J ≥ 1, (Z where (a (n) , v (n) , w (n) ) are independent, identically distributed random variables and I is the identity matrix.
By the remark after the positivity proposition, E(a) is irreducible and aperiodic.
Thus, by the variance lemma in [1], for each s ∈ S, By (H) as on page 30, whence, by the Denjoy-Koksma estimate, E((Ŷ Accordingly, define H n by The lemma follows. §5 Spectral theory and theorem 2 By the Perron-Frobenius theorem, 1 is a simple, dominant eigenvalue of Π H (0) (where H ∶= H 1 and Π H is the RAT-CF as defined by (L) on page 19) with right eigenvector 1 ∈ R S + and left eigenvector π ∈ R S + satisfying ⟨π, 1⟩ = 1. By the implicit function theorem ∃ r = r H > 0 and smooth functions of Π H (θ) with eigenvector v(θ) and left eigenvector π(θ). As in [9], consider the principal projections N(θ) ∶ C S → C S defined by N(θ)x ∶= ⟨π(θ), x⟩v(θ) then possibly reducing r H > 0, we ensure ∃ 0 < ρ < 1 so that The proofs of our limit theorems in the sequel use the following lemma.
Lemma: Taylor expansion of the eigenvalue Under the assumptions of Theroem 2, where d 2 λ(θ) is the matrix of second partial derivatives: and we must show that ∇λ(0) = 0 and that D ∶= −d 2 λ(0) is positive definite.
In view of the assumption that span R ϕ(T) has full dimension, this contradicts theorem 1 and completes the proof of the lemma. (ii) Proof of theorem 2 It suffices to prove that for fixed s ∈ S, θ ∈ R d By asymptotic, eventual periodicity ∃ ρ ∈ (0, 1) so that for any fixed J, s ∈ S and all n ≥ 1, where O n (ρ J ) ≤ Mρ J for a constant M independent of n. By the Taylor expansion of the eigenvalue, To deduce (R) from this, let ε > 0 and choose J = J ε ≥ 1 so that and then choose N = N J,ε > J so that This implies (R). §6 The WRLLT and theorem 3 Visits to zero and RATs. Recall that we assume T (x, z) ∶= ({x + α}, z + ϕ(x)).
Visit lemma Let (X (k) ∶ k ≥ 1) be the associated ARW, then Visit sets.
The visit set to ν ∈ Z d is The auxiliary visit sets to ν ∈ Z d are and the auxiliary visit distributions are the discrete measures V k (i, ǫ) on Z d defined by V k (i, ǫ)(ν) ∶= #(K k (i, ǫ, ν)) (k ≥ 1, i = 0, 1).
Using (4.1) in the sublemma and the Riesz-Fischer theorem, we see that This is (a). To see (b), This is (b).
Thus, F is adapted iff ∀ s ∈ S, ∃ t ∈ S such that P (L s = t) > 0 & Cov (W s,t ) is strictly positive definite.
The following lemma gives a sequence version of adaptedness similar to that in [1].
The proof is in a series of steps, the first two of which are as in [18].
Thus, for s ∈ S, ξv s = t∈S Π s,t v t = t∈S z s P s,t = z s which is (a).
Statement (b) follows from this, whence ¶2 via the Taylor expansion of λ. ¶3 For N ≥ 1 sufficiently large, H N 1 is an adapted RAT with adaptivity group Γ.
It follows that H N 1 is adapted with adaptivity group Γ. To complete the proof of the lemma, fix J ≥ 1 and let (E n ∶= F J+N (n+1) J+N n+1 ∶ n ≥ 1).