LOCAL LIMIT THEOREMS FOR GIBBS–MARKOV MAPS

Jon. Aaronson and Manfred Denker

Abstract. We prove conditional local limit theorems for Gibbs-Markov processes whose marginals are in the domain of attraction of a stable law with order in (0, 2).

Introduction

Given a ℝ-valued stationary stochastic sequence X_{1}, X_{2}, … defined on a probability space (Ω, F, P), we consider local limits of the partial sums S_{n} := X_{1} + ⋯ + X_{n}, that is the existence of constants A_{n}, B_{n} ∈ ℝ, B_{n} → +∞ such that ∀ κ ∈ ℝ and I ⊂ ℝ ( an interval),

(LL)  B_{n}P (S_{n} − k_{n} ∈ I) → |I|g(κ) as \frac{k_{n} − A_{n}}{B_{n}} → κ

where g is a continuous probability density on ℝ.

These local limits are closely connected to distributional limits where

(DL)  \frac{S_{n} − A_{n}}{B_{n}} ⟶^{D} Y

for some limit random variable Y, indeed it can be shown that if the convergence in (LL) is uniform in κ ∈ compact subsets of ℝ and g = f_{Y} (the density of Y) then (DL) is satisfied with the same constants A_{n}, B_{n} (see the De Moivre- Laplace proof of the classical central limit theorem).

In the case where X_{1}, X_{2}, … are independent, (identically distributed) the forms of the possible limit random variables Y are known ([Le], [G-K], [I-L]) and are distributed according to the well known stable laws which have smooth densities.

The local limits for non-lattice independent random variables satisfying (DL) can be found in [Bre].

Analogous local limits for suitable lattice random variables also exist including the celebrated de Moivre-Laplace local limit theorem (see [Bre], [Fel], [Gne]).

Typeset by AMS-TEX

1980 Mathematics Subject Classification (1985 Revision). 28D05, 58F11, 58F15, 60F05.

This research was supported by a grant from G.I.F., the German-Israel Foundation for Scientific Research and Development. .

The extension of this limit theory to Markov chains began with the central limit theorem of Markov ([Mar]) for Markov chains with 3 states. This was extended to certain chains with countably many states by Doeblin in [Doe].

Local limits were obtained by Kolmogorov ([Kol], see also [Gne]) for Markov chains with finite state space using the methods of [Doe].

Nagaev ([Nag]) obtained both distributional, and local limits for a wide class of stationary Markov chains with infinite state space.

In general, a stationary stochastic sequence (X_{1}, X_{2}, …) is generated by a dynamical system and a measurable function: X_{n} = f ∘ T^{n} where T is a probability preserving transformation of some probability space Ω, and f : Ω → ℝ is measurable.

Local (and distributional) limits were obtained in the normal case for stationary sequences generated by Lasota-Yorke maps of the interval and functions of bounded variation by Rousseau-Egele ([Rou], see also [Mor1], [Mor2]), and for Lipschitz continuous functions of mixing finite state topological Markov shifts under Gibbs measures ([G-H], see also [G] and [Bro] for multidimensional extensions).

One feature of Nagaev’s results which interested us, is that the constants A_{n} and B_{n} appearing in (LL) and (DL) (in [Nag]) are completely determined by the marginal distributions of the X_{n} in case the limit stable random variable Y is not normal.

We show here that this phenomenon persists for stationary sequences generated by mixing Gibbs-Markov maps (see §1) together with aperiodic (see §3), Lipschitz continuous functions. Local (and distributional) limit theorems are given in §6.

In fact, the results of §6 are established with respect to the sequence of conditional measures on the fibres of T^{n} given by the Frobenius-Perron operators (defined in §1). This enables application to infinite ergodic theory in §7 where we establish pointwise dual ergodicity of certain skew products (examples of which are given in §8).

The methods of this paper closely follow those of [Nag] (as do those of [G-H], [Mor1],

[Mor2] and [Rou]).

The main idea underlying both (DL) and (LL) is that a stationary sequence (X_{1}, X_{2}, …) generated by a mixing Gibbs-Markov map together with a Lipschitz continuous function has the property that ∃ 0 < r < 1, ϵ > 0 and a function λ : (−ϵ, ϵ) → B_{ℂ}(0, 1) such that

(N)  sup_{|t|<ϵ} ess-sup_{y∈X}|E(e^{itS_{n}}|T^{n}(·) = y) − λ(t)^{n}p_{t}(y)| = O(r^{n}) as n → ∞

where p_{t} → 1 uniformly as t → 0.

This property is established in §4 using the spectral theory of Ionescu-Tulcea and Marinescu (see [I-T-M]) and characteristic function operators (see §2).

The asymptotic expansion of the function λ : (−ϵ, ϵ) → B_{ℂ}(0, 1) is obtained in §5 when the distribution of X_{1} is in the domain of attraction of a p-stable law (0 < p < 2). It is the same as that of the characteristic function of X_{1} (see remark 1 after theorem 5.1). Both (DL) and (LL) are then established as in the independent case and the constants A_{n} and B_{n} appearing in (LL) and (DL) are completely determined by the marginal distributions of the X_{n} as remarked above.

The classical case where distribution of X_{1} is in the classical domain of attraction of the Gaussian law (i.e. E(X_{1}^{2}) < ∞) can be obtained by straightforward modificattion of [G-H] or [Rou] and is not included here.

There is some discussion of stationary sequences generated by mixing Gibbs-Markov maps together with periodic (= non-aperiodic) Lipschitz continuous functions in §3 and §7.

§1 Preliminaries on Markov maps

Let (X, B, m, T) denote a nonsingular transformation of a standard probability space. It is called a Markov map if there is a measurable partition α such that T a ∈ σ(α) mod m ∀ a ∈ α and which generates B under T in the sense that σ({T^{−n}α : n ≥ 0}) = B. In this case, T |_{a} is invertible for a ∈ α. Markov maps are called Markov fibred systems in [A-D-U].

Write α = {a_{s} : s ∈ S} and endow S^{ℕ} with its canonical (Polish) product topology. Let

Σ = {s = (s_{1}, s_{2}, …) ∈ S^{ℕ} : m( ⋂_{k=1}^{n} T^{−k}a_{s_{k}}) > 0 ∀ n ≥ 1},

then Σ is a closed, shift invariant subset of S^{ℕ}, and there is a measurable map ϕ : Σ → X defined by {ϕ(s_{1}, s_{2}, …)} := ⋂_{k=1}^{∞} T^{−(k−1)}a_{s_{k}}. If m′ = m ∘ ϕ^{−1} ∈ P(S^{ℕ}) then Σ is the closed support of m′, and ϕ is a conjugacy of (X, B, m, T) with (Σ, B(Σ), m′, shift). Thus there is no loss of generality in assuming that X = Σ, T is the shift, and α = {[s] : s ∈ S}.

Throughout this paper, we fix r ∈ (0, 1) and define the metric d = d_{r} on X by d(x, y) = r^{t(x,y)} where t(x, y) = min{n ≥ 1 : x_{n} ≠ y_{n}} ≤ ∞, then (X, d) is a Polish space and T : X → X is Lipschitz continuous on each a ∈ α.

Remark. A function f : X → ℝ is called Hölder continuous on X if ∃ θ ∈ (0, 1) and M > 0 such that

|f(x) − f(y)| ≤ Mθ^{t(x,y)} ∀ x, y ∈ X.

Thus Hölder continuous functions are in fact Lipschitz continuous with respect to the appropriate d = d_{θ}. Note that for s > 0, d(x, y)^{s} = (r^{s})^{t(x,y)}, so

There are m-nonsingular inverse branches of T denoted v_{a} : T a → a (a ∈ α) with Radon Nikodym derivatives

v′_{a} := \frac{dm ∘ v_{a}}{dm} = (\frac{dm ∘ T}{dm} ∘ v_{a})^{−1}.

Indeed for n ≥ 1, there are m-nonsingular inverse branches of T^{n} v_{a} : T^{n}a → a (a ∈ α_{0}^{n−1}) with analogously defined Radon Nikodym derivatives.

Since T α ⊂ σ(α), T^{n}α_{0}^{n−1} = T α, and ∃ a (finite or countable ) partition β ≻ α so that σ(T α) = σ(β).

We’ll assume throughout that T is mixing in the sense that

∀ a, b ∈ α, ∃ n_{a,b} ∋ T^{n}a ⊃ b ∀ n ≥ n_{a,b}.

The Frobenius-Perron operators P_{T^{n}} : L^{1}(m) → L^{1}(m) defined by

∫_{X} P_{T^{n}}f · gdm = ∫_{X} f · g ∘ T^{n}dm

have the form

P_{T^{n}}f = ∑_{b∈β} 1_{b} ∑_{a∈α_{0}^{n−1}, T^{n}a⊃b} v′_{a} · f ∘ v_{a}.

As mentioned in the introduction, they give the conditional probabilities

E(f|T^{n}(·) = y) = P_{T^{n}}f(y).

Definitions.

1) Let C, L be Banach spaces such that C ⊃ L and ∥ · ∥_{C} ≤ ∥ · ∥_{L}. We say that the pair (C, L) is adapted if L-bounded sets are precompact in C.

2) Let (C, L) be an adapted pair of Banach spaces. A linear operator P : C → C is said to be an ITM operator on (C, L) if ∃ θ ∈ (0, 1), M > 0, n ∈ ℕ such that

∥P^{n}f∥_{L} ≤ θ∥f∥_{L} + M∥f∥_{C} ∀ f ∈ L.

We’ll call this latter inequality an ITM inequality.

The celebrated theorem of Ionescu-Tulcea and Marinescu (proved in [I-T-M]) states that an ITM operator on (C, L) is quasi compact on L.

Definitions.

3) A function f : X → ℝ^{d} is Lipschitz continuous on A ⊂ X if

D_{A}f := sup_{x,y∈A} \frac{|f(x) − f(y)|}{d(x, y)} < ∞,

and Lipschitz continuous at x ∈ X if it is Lipschitz continuous on some neighbourhood of x.

4) A function is locally Lipschitz continuous on A ⊂ X if it is Lipschitz continuous at every point of A

Given a partition ρ of X into open-closed sets and 1 ≤ q ≤ ∞, we consider Lip_{q,ρ} ⊂ L^{q}(m), consisting of functions X → ℂ which are Lipschitz continuous on each a ∈ ρ. The norm on Lip_{q,ρ} is defined by

∥f∥_{Lip_{q,ρ}} := ∥f∥_{q} + D_{ρ}f

where

D_{ρ}f := sup_{a∈ρ} D_{a}f.

If ρ is infinite and q < ∞, then not all functions in Lip_{q,ρ} are bounded, but if f ∈ Lip_{q,ρ} and a ∈ ρ, then

sup_{x∈a} |f(x)| ≤ \frac{1}{m(a)} ∫_{a} |f|dm + rD_{a}f.

This is because |f(x) − f(y)| ≤ rD_{a}f ∀ x, y ∈ a.

We show that for large enough n ≥ 1, P_{T^{n}} is an ITM operator on the adapted pair (L^{q′}(m), Lip_{q,ρ}) for 1 ≤ q′ < q. The following standard theorem is a version of the classical Arzela-Ascoli theorem:

Theorem. If f_{n} ∈ Lip_{q,ρ}, and sup_{n≥1} ∥f_{n}∥_{Lip_{q,ρ}} < ∞, then ∃ n_{k} → ∞ and g ∈ Lip_{q,ρ} such that

f_{n_{k}}(x) → g(x) as k → ∞ ∀ x ∈ X,

∥g∥_{Lip_{q,ρ}} ≤ lim inf_{n→∞} ∥f_{n}∥_{Lip_{q,ρ}},

and

∥f_{n_{k}} − g∥_{q′} → 0 as k → ∞ ∀ 1 ≤ q′ < q.

A Markov map (X, B, m, T, α) is Gibbs (Gibbs-Markov) if the two additional assumptions are satisfied:

inf_{a∈α} m(T a) > 0

(we call this the big image property), and

(𝔊)  ∃ M > 0 ∋ |\frac{v′_{a}(x)}{v′_{a}(y)} − 1| ≤ Md(x, y) ∀ n ≥ 1, a ∈ α_{0}^{n−1}, x, y ∈ T^{n}a.

Example 1. Markov chains.

Let S be a countable set, P : S × S → [0, 1] be an aperiodic, irrreducible stochastic matrix and π ∈ P(S), π_{s} > 0 ∀ s ∈ S. Let T : S^{ℕ} → S^{ℕ} be the shift and define a Markovian probability m ∈ P(S^{ℕ}) by

m([s_{1}, s_{2}, …, s_{n}]) := π_{s_{1}}p_{s_{1},s_{2}} … p_{s_{n−1},s_{n}}.

Let X = {x ∈ S^{ℕ} : m([x_{1}, …, x_{n}]) > 0 ∀ n ≥ 1} and let α = {[s] : s ∈ S}. Evidently (X, B, m, T, α) is a Markov map.

It can be shown that (X, B, m, T, α) is Gibbs-Markov iff ∃ M > 1 such that

\frac{1}{M} ≤ \frac{p_{s,t}}{π_{t}} ≤ M whenever s, t ∈ S and p_{s,t} > 0.

Example 2. Markov interval maps.

Let I := [0, 1] and let m be Lebesgue measure on I. Let α be a partition of I mod m into open intervals and let T : I → I a non-singular transformation so that T : a → T a is a homeomorphism ∀ a ∈ α.

Clearly (I, B(I), m, T, α) is a Markov map iff T a ∈ σ(α) ∀ a ∈ α.

Now suppose that ∀ a ∈ α, there is a C^{2}-diffeomorphism v_{a} : \overline{T a} → \overline{a} such that T |_{a} = v_{a}^{−1}. Assume in addition that T is uniformly expanding in the sense that ∃ c > 1 such that |T′| ≥ c, and that T has the so-called Adler property (see [Ad])

sup_{x∈a∈α} \frac{|T′′(x)|}{T′(x)^{2}} := M < ∞.

A calculation (see R. Adler’s afterword in [Bo]) shows that

\frac{|T^{n′′}(x)|}{T^{n′}(x)^{2}} ≤ M_{1} := \frac{Mc}{c − 1} ∀ n ≥ 1, x ∈ a ∈ α_{0}^{n−1}

whence

|\frac{v′_{a}(x)}{v′_{a}(y)} − 1| ≤ M_{1}|x − y| ∀ n ≥ 1, a ∈ α_{0}^{n−1}, x, y ∈ T^{n}a.

We claim that (𝔊) holds with d = d_{r}, r = \frac{1}{c}. This is because if x, y ∈ I and r^{n+1} < d(x, y) ≤ r^{n}, then ∃ b ∈ α_{0}^{n−1} such that x, y ∈ b, whence

|x − y| ≤ diam. b = diam. v_{b}(T^{n}b) ≤ ∥v′_{b}∥_{∞} ≤ r^{n} ≤ r^{−1}d(x, y),

and

|\frac{v′_{a}(x)}{v′_{a}(y)} − 1| ≤ r^{−1}M_{1}d(x, y).

Thus if T is a uniformly expanding piecewise C^{2} Markov map having the big image property and the Adler property, then (I, B(I), m, T, α) is a Gibbs-Markov map.

The following shows when the Gibbs-Markov property is preserved under passage to an equivalent measure.

Proposition 1.1. Suppose that (X, B, m, T, α) is a Gibbs-Markov map, and that μ ∼ m, log \frac{dμ}{dm} ∈ Lip_{∞,α};

then, (X, B, μ, T, α) is a Gibbs-Markov map.

Henceforth, unless stated otherwise, (X, B, m, T, α) will denote a mixing Gibbs-Markov map.

Proposition 1.2 (Renyi’s property).

∀ n ≥ 1, a ∈ α_{0}^{n−1}, v′_{a} = M^{′′±1}m(a) a.e. on T^{n}a.

Here, and throughout a = c^{±1}b (where a, b, c > 0) means c^{−1}b ≤ a ≤ cb.

Proofs of proposition 1.2 and the next corollary (a descendant of Renyi’s theorem [Ren]) can be found in §2 and §3 of [A-D-U].

LOCAL LIMIT THEOREMS FOR GIBBS–MARKOV MAPS 7

Corollary.

1) T is exact, and ∃ h ∈ L^{∞}(m) such that h > 0 a.e., and P_{T} h = h.

2) If |T α| < ∞, then log h ∈ L^{∞}(m).

We now consider P_{T} acting on the space L := Lip_{∞,β}. Lemma 1.3. Suppose that g ∈ Lip_{1,β} and a ∈ α_{0}^{n−1}, then

|v′_{a}(x)g(v_{a}x) − v′_{a}(y)g(v_{a}y)| ≤ M′′d(x, y)(M ∫_{a} |g|dm + (M + 1)m(a)r^{n}D_{a}g).

Proof.

|v′_{a}(x)g(v_{a}(x)) − v′_{a}(y)g(v_{a}(y))|

≤ v′_{a}(x)|g(v_{a}(x))||\frac{v′_{a}(y)}{v′_{a}(x)} − 1| + v′_{a}(y)|g(v_{a}(x)) − g(v_{a}(y))|

= I + II.

We have that

|g(v_{a}x)| ≤ \frac{1}{m(a)} ∫_{a} |g|dm + r^{n}D_{a}g ∀ x ∈ X.

Hence, by Renyi’s property, and (𝔊),

I ≤ MM′′d(x, y)m(a)|g(v_{a}(x))|

≤ MM′′d(x, y)( ∫_{a} |g|dm + m(a)r^{n}D_{a}g).

Also by Renyi’s property,

II ≤ M′′m(a)d(v_{a}(x), v_{a}(y))D_{a}g = M′′d(x, y)m(a)r^{n}D_{a}g.

The result is that

I + II ≤ M′′d(x, y)(M ∫_{a} |g|dm + (M + 1)m(a)r^{n}D_{a}g).

□ Proposition 1.4 (ITM Inequality). For f ∈ Lip_{1,β},

∥P_{T^{n}}f∥_{L} ≤ M′′ ((M + 2)r^{n}D_{β}f + (M + 1)∥f∥_{1})

Remark Note that P_{T} : Lip_{1,β} → L. Proof. Let g ∈ L, then

P_{T^{n}}g = ∑_{a∈α_{0}^{n−1}} 1_{T^{n}a}v′_{a} · g ∘ v_{a}.

For each n ≥ 1, and a ∈ α_{0}^{n−1} we have

|g(v_{a}x) − \frac{1}{m(a)} ∫_{a} gdm| ≤ D_{a}gr^{n} ∀ x ∈ T^{n}a

whence, using Renyi’s property and D_{a}g ≤ D_{β}g,

|P_{T^{n}}g(x)| ≤ ∑_{a∈α_{0}^{n−1}} 1_{T^{n}a}(x)v′_{a}(x)(\frac{1}{m(a)} ∫_{a} |g|dm + D_{β}gr^{n})

≤ M′′ ∑_{a∈α_{0}^{n−1}} m(a)(\frac{1}{m(a)} ∫_{a} |g|dm + D_{β}gr^{n})

= M′′D_{β}gr^{n} + M′′∥g∥_{1}.

It follows that

∥P_{T^{n}}g∥_{∞} ≤ M′′D_{β}gr^{n} + M′′∥g∥_{1}.

For g ∈ L, x, y ∈ b ∈ β, |P_{T^{n}}g(x) − P_{T^{n}}g(y)|

≤ ∑_{a∈α_{0}^{n−1}, T^{n}a⊃b} |(v′_{a}(x)g(v_{a}(x)) − v′_{a}(y)g(v_{a}(y)))|

= M′′d(x, y) ∑_{a∈α_{0}^{n−1}, T^{n}a⊃b} (M ∫_{a} |g|dm + (M + 1)m(a)r^{n}D_{β}g) by lemma 1.3

≤ M′′d(x, y)(M∥g∥_{1} + r^{n}(M + 1)D_{β}g). □

Corollary 1.5 ([Hal]). Let h ∈ L^{∞}(m) satisfy P_{T} h = h, then h ∈ L.

This is an application of the ITM inequality, which implies sup_{n} ∥P_{T^{n}}1∥_{L} < ∞. Theorem 1.6.

P_{T} = μ + Q

where μf = ∫_{X} fdm · h, Qμ = μQ = 0, and the spectral radius 𝔯(Q) < 1. Proof. By the ITM theorem,

P_{T} = μ + Q

where Qμ = μQ = 0, 𝔯(Q) < 1 and ∃ N ∈ ℕ, λ_{1}, …, λ_{N} ∈ ℂ, |λ_{k}| = 1 and finite dimensional subspaces E_{1}, …, E_{N} ⊂ L such that μ = ∑_{k=1}^{N} λ_{k}P_{E_{k}} where P_{E_{k}} is a projection onto E_{k} (1 ≤ k ≤ N).

By the corollary, T is exact and so E_{k} = {0} whenever λ_{k} ≠ 1. It follows that μf = ∫_{X} fdm · h. □

Henceforth, unless stated otherwise, we shall assume that the mixing Gibbs-Markov map (X, B, m, T, α) is probability preserving.

It is now possible to obtain a (well-known) strengthening of the exactness part of Renyi’s theorem.

By theorem 1.6, ∃ θ ∈ (0, 1) and K > 1 such that

∥P_{T^{n}}f − ∫_{X} fdm∥_{L} ≤ Kθ^{n}∥P f∥_{L} ∀ n ≥ 1, f ∈ Lip_{1,β}.

Since ∀ n ≥ 1, a ∈ α_{0}^{n−1} we have P_{T^{n}}1_{a} = v′_{a} and ∥v′_{a}∥_{L} ≤ (M + 1)M′′m(a), it follows that (T, α) is continued fraction mixing (see [A2], [A-D-U]):-

for some K′ > 0,

|m(a ∩ T^{−(n+k)}B) − m(a)m(B)| ≤ Kθ^{n}m(a)m(B) ∀ n, k ≥ 1, a ∈ α_{0}^{k−1}, B ∈ B.

§2 characteristic function operators

Let (X, B, m, T, α) be a mixing, probability preserving Gibbs-Markov map.

For ω : X → 𝕋 := {z ∈ ℂ : |z| = 1} measurable, define

P_{ω}f := P (ωf) (f ∈ L^{1}(m))

where P = P_{T}, and for ϕ : X → ℝ^{d} measurable, t ∈ ℝ^{d} set P_{t} := P_{χ_{t}(ϕ)} where χ_{t}(y) := e^{i⟨t,y⟩}.

In the independent case where ϕ is α-measurable and α, T^{−1}α, … are independent,

∫ P_{t}1dm = E(e^{i⟨t,ϕ⟩})

which is why the P_{t} are sometimes called characteristic function operators.

We’ll use these operators in §5 to obtain local (and distributional) limits. The relevant spectral theory is developed in §4 (along the lines of [Nag]). In this section, we establish the necessary basic properties of these operators as perturbations of P.

Proposition 2.1 (ITM inequality). Suppose that ω : X → 𝕋 is Lipschitz continuous on each a ∈ α, and that D_{α}ω < ∞, then for f ∈ L,

∥P_{ω}^{n}f∥_{L} ≤ M′′((M + \frac{rD_{α}ω}{1 − r} + 1)∥f∥_{1} + (M + \frac{rD_{α}ω}{1 − r} + 2)r^{n}D_{β}f).

Proof. Note that

P_{ω}^{n}(f) = P_{T^{n}}(ω_{n}f) where ω_{n}(x) := ∏_{k=0}^{n−1} ω(T^{k}x).

10 JON. AARONSON AND MANFRED DENKER

It follows that

P_{ω}^{n}(f)(x) = ∑_{b∈β} 1_{b}(x) ∑_{a∈α_{0}^{n−1}, T^{n}a⊃b} v′_{a}(x)ω_{n}(v_{a}(x))f(v_{a}(x)).

We have that |P_{ω}^{n}(f)(x)| ≤ P_{T^{n}}(|f|)(x) ≤ M′′ (∥f∥_{1} + r^{n}D_{β}f) as in proposition 1.3.

For x, y ∈ b ∈ β, |P_{ω}^{n}(f)(x) − P_{ω}^{n}(f)(y)|

≤ ∑_{a∈α_{0}^{n−1}, T^{n}a⊃b} |v′_{a}(x)ω_{n}(v_{a}(x))f(v_{a}(x)) − v′_{a}(y)ω_{n}(v_{a}(y))f(v_{a}(y))|

≤ ∑_{a∈α_{0}^{n−1}, T^{n}a⊃b} (I_{a} + II_{a})

where

I_{a} = |ω_{n}(v_{a}x)f(v_{a}x)v′_{a}(x) − f(v_{a}y)v′_{a}(y))|
≤ M′′d(x, y)(M ∫_{a} |f|dm + m(a)(M + 1)r^{n}D_{β}f)

by lemma 1.3, and

II_{a} = v′_{a}(y)|ω_{n}(v_{a}(x))f(v_{a}(y)) − ω_{n}(v_{a}(y))f(v_{a}(y))|
= v′_{a}(y)|f(v_{a}(y))||ω_{n}(v_{a}(x)) − ω_{n}(v_{a}(y))|

≤ M′′(∫_{a} |f|dm + r^{n}m(a)D_{β}f)|ω_{n}(v_{a}(x)) − ω_{n}(v_{a}(y))|

by the Renyi property, and Lipschitz continuity.

Now

|ω_{n}(v_{a}(x)) − ω_{n}(v_{a}(y))| = ∑_{k=0}^{n−1} |ω(T^{k}v_{a}x) − ω(T^{k}v_{a}y)|

≤ ∑_{k=0}^{n−1} r^{n−k}D_{α}ω d(x, y)

≤ \frac{rD_{α}ω}{1 − r} d(x, y),

so

II_{a} ≤ M′′ \frac{rD_{α}ω}{1 − r} d(x, y)(∫_{a} |f|dm + r^{n}m(a)D_{β}f);

and the conclusion is

D_{β}(P_{ω}^{n}(f))

≤ ∑_{a∈α_{0}^{n−1}} M′′((M + \frac{rD_{α}ω}{1 − r} ) ∫_{a} |f|dm + (M + \frac{rD_{α}ω}{1 − r} + 1)r^{n}m(a)D_{β}f)

= M′′((M + \frac{rD_{α}ω}{1 − r} )∥f∥_{1} + (M + \frac{rD_{α}ω}{1 − r} + 1)r^{n}D_{β}f).

□

Corollary 2.2. Suppose that ω : X → 𝕋 is Lipschitz continuous on each a ∈ α, and that D_{α}ω < ∞.

If g : X → ℂ is measurable, and g ∘ T = λωg for some λ ∈ ℂ, then g ∈ L.

Proof. Since T is conservative (being probability preserving) and ergodic, we have that λ ∈ 𝕋, and |g| is constant. Also

P_{ω}g = P (ωg) = P (\overline{λ}g ∘ T) = \overline{λ}g.

It follows from [I-T-M] that g ∈ L. □

Corollary 2.3. Suppose that ϕ : X → ℝ^{d} is Lipschitz continuous on each a ∈ α, and that D_{α}ϕ < ∞. If g : X → ℝ^{d} is measurable, and ϕ = g ∘ T − g, then g is locally Lipschitz continuous.

Proof. By corollary 2.2, χ_{t}(g) ∈ L ∀ t ∈ ℝ^{d}, and so g is locally Lipschitz continuous. □

Theorem 2.4 (Continuity). Suppose that ϕ : X → ℝ^{d} is Lipschitz continuous on each a ∈ α, and that D_{α}ϕ < ∞, then

∥P_{s} − P_{t}∥_{Hom(L,L)} ≤

M′′((1 + 2M + 2|t|D_{α}ϕ)E|1 − χ_{t−s}(ϕ)| + (2 + 2M + |t|D_{α}ϕ)2|s − t|D_{α}ϕ).

Proof.

For g ∈ L and t ∈ ℝ^{d} we have

P_{t}g = P (e^{i⟨t,ϕ⟩}g) = ∑_{a∈α} χ_{t}(ϕ ∘ v_{a})1_{T a}v′_{a} · g ∘ v_{a},

whence

(P_{t} − P_{s})g = ∑_{a∈α} 1_{T a}χ_{t}(ϕ_{a})(1 − χ_{s−t}(ϕ ∘ v_{a}))v′_{a} · g ∘ v_{a}.

12 JON. AARONSON AND MANFRED DENKER

We’ll use that for x, y ∈ a ∈ α,

|χ_{t}(ϕ(x)) − χ_{t}(ϕ(y))| ≤ |t|D_{a}ϕ

whence

|χ_{t}(ϕ(v_{a}x)) − \frac{1}{m(a)} ∫_{a} χ_{t}(ϕ)dm| ≤ |t|D_{a}ϕ ∀ x ∈ T a,

and

(1)  |1 − χ_{t}(ϕ ∘ v_{a})| ≤ \frac{1}{m(a)} ∫_{a} |1 − χ_{t}(ϕ)|dm + |t|D_{a}ϕ on T a.

Also, using (1), ∀ x, y ∈ a ∈ α, |(χ_{t}(ϕ(x)) − χ_{s}(ϕ(x))) − (χ_{t}(ϕ(y)) − χ_{s}(ϕ(y)))| (2)

≤ |χ_{s−t}(ϕ(y)) − χ_{s−t}(ϕ(x))| + |1 − χ_{s−t}(ϕ(y))||χ_{t}(ϕ(x)) − χ_{t}(ϕ(y))|

≤ d(x, y)(|s − t|D_{a}ϕ(1 + |t|D_{a}ϕ) + |t|D_{a}ϕ\frac{1}{m(a)} ∫_{a} |1 − χ_{s−t}(ϕ)|dm).

In particular, we have that, ∀ x, y ∈ T a, a ∈ α,

|(P_{s} − P_{t})g(x)|

≤ ∑_{a∈α} 1_{T a}(x)|1 − χ_{s−t}(ϕ(v_{a}x))|v′_{a}(x)|g(x)|

≤ M′′ ∑_{a∈α} 1_{T a}(x)( ∫_{a} |1 − χ_{s−t}(ϕ)|dm + m(a)|s − t|D_{a}ϕ)∥g∥_{∞} by (1)

≤ M′′(E(|1 − χ_{s−t}(ϕ)|) + |s − t|D_{a}ϕ)∥g∥_{∞}.

For x, y ∈ b ∈ β, |(P_{s} − P_{t})g(x) − (P_{s} − P_{t})g(y)| =

| ∑_{{a∈α \atop T a⊃b}}(χ_{t}(ϕ(v_{a}x)) − χ_{s}(ϕ(v_{a}x)))v′_{a}(x)g(v_{a}x)−

(χ_{t}(ϕ(v_{a}y)) − χ_{s}(ϕ(v_{a}y)))v′_{a}(y)g(v_{a}y)|

≤ ∑_{{a∈α \atop T a⊃b}} (v′_{a}(x)|g(v_{a}x)||(χ_{t}(ϕ(v_{a}x)) − χ_{s}(ϕ(v_{a}x))) − (χ_{t}(ϕ(v_{a}y)) − χ_{s}(ϕ(v_{a}y)))|

+ |1 − χ_{s−t}(ϕ(v_{a}y))||v′_{a}(x)g(v_{a}x) − v′_{a}(y)g(v_{a}y)|)

= ∑_{a∈α, T a⊃b}(I_{a} + II_{a}).

By lemma 1.3 and (1)

II_{a} = |v′_{a}(x)g(v_{a}(x)) − v′_{a}(y)g(v_{a}(y))||1 − χ_{s−t}(ϕ(v_{a}x))|
≤ M′′d(x, y)(D_{β}g + M∥g∥_{∞})( ∫_{a} |1 − χ_{s−t}(ϕ)|dm + m(a)|s − t|D_{α}ϕ)

whence

∑_{a∈α, T a⊃b} II_{a} ≤ M′′d(x, y)(D_{β}g + M∥g∥_{∞})(E|1 − χ_{s−t}(ϕ)| + |s − t|D_{α}ϕ).

Using Renyi’s property and (2),

I_{a} = v′_{a}(y)|g(v_{a}(y))||(χ_{t}(ϕ(v_{a}x)) − χ_{s}(ϕ(v_{a}x))) − (χ_{t}(ϕ(v_{a}y)) − χ_{s}(ϕ(v_{a}y)))|
≤ M′′∥g∥_{∞}d(x, y)(m(a)|s − t|D_{α}ϕ(1 + |t|D_{α}ϕ) + |t|D_{α}ϕ ∫_{a} |1 − χ_{s−t}(ϕ)|dm)

and

∑_{a∈α, T a⊃b} I_{a} ≤ M′′∥g∥_{∞}d(x, y)(|s − t|D_{α}ϕ(1 + |t|D_{α}ϕ) + |t|D_{α}ϕE(|1 − χ_{s−t}(ϕ)|)).

The conclusion is that

∥(P_{s} − P_{t})g∥_{L}

≤ M′′(E(|1 − χ_{s−t}(ϕ)|) + |s − t|D_{α}ϕ)∥g∥_{∞}

+ M′′(D_{β}g + M∥g∥_{∞})(E|1 − χ_{s−t}(ϕ)| + |s − t|D_{α}ϕ)

+ M′′∥g∥_{∞}(|s − t|D_{α}ϕ(1 + |t|D_{α}ϕ) + |t|D_{α}ϕE(|1 − χ_{s−t}(ϕ)|))

≤ M′′∥g∥_{L}((2 + M + |t|D_{α}ϕ)E|1 − χ_{s−t}(ϕ)| + (3 + M + |t|D_{α}ϕ)|s − t|D_{α}ϕ).

□

§3 Periodic and aperiodic cocycles

Suppose that (X, B, m, T) is an ergodic probability preserving transformation, G is a locally compact, Abelian, second countable topological group, and ϕ : X → G is measurable.

Define the skew product transformation T_{ϕ} : X×G → X×G by T_{ϕ}(x, y) = (T x, y+ϕ(x)). Evidently T_{ϕ} preserves the measure m × m_{G} on X × G where m_{G} is Haar measure on G.

We say that ϕ is aperiodic if there is no character γ ∈ Ĝ so that γ ∘ϕ is T -cohomologous to a constant, i.e. the equation

γ ∘ ϕ = \frac{λg(x)}{g(T x)}

has no solution λ ∈ 𝕋, g : X → 𝕋 measurable other than λ = 1, g ≡ 1. We say that ϕ is periodic if it is not aperiodic.

In case T_{ϕ} is ergodic, aperiodicity is the same as demanding that any eigenfunction for T_{ϕ} is the lifting of an eigenfunction for T (defined on X).

In this section, we consider a probability preserving, mixing Gibbs-Markov map (X, B, m, T, α) and we discuss the aperiodicity of Lipschitz continuous functions ϕ : X → G. We begin with sufficient conditions for aperiodicity. The results are analogous to those of [Kow1] and [Kow2].

Theorem 3.1.

Let (X, B, m, T, α) be a probability preserving, mixing Gibbs-Markov map, and suppose that ϕ : X → 𝕋 is α-measurable.

If g : X → 𝕋 is measurable, λ ∈ 𝕋 and ϕ = λ · g · \overline{g} ∘ T, then g is α_{∗}-measurable where α_{∗} is the finest partition with the property that each T a is contained in an atom of α_{∗}.

We’ll say that a Gibbs-Markov map (X, B, m, T, α) is almost onto if α_{∗} = {X} mod m; equivalently ∀ b, c ∈ α, ∃ n ≥ 1, b = a_{0}, a_{1}, …, a_{n} = c ∈ α such that T a_{k} ∩ T a_{k+1} ≠ ∅ (0 ≤ k ≤ n − 1).

Corollary 3.2. Let T : X → X be a probability preserving, almost onto Gibbs-Markov map with respect to the partition α.

1) Suppose that ϕ : X → 𝕋 is α-measurable and λ ∈ 𝕋. If g : X → 𝕋 is measurable, and ϕ = λ\overline{g} · g ∘ T, then g is constant.

2) Suppose that ϕ : X → G is α-measurable, then either ϕ is aperiodic, or ∃ γ ∈ Ĝ, λ ∈ 𝕋 such that

γ ∘ ϕ ≡ λ.

To prove this theorem, we consider skew products over (X, B, m, T, α) as in [Kow1] and [Kow2].

Let (Y, C, μ) be a standard σ-finite measure space and suppose that {T_{a} : a ∈ α} are non-singular transformations of Y (i.e. μ ∘ T_{a}^{−1} ∼ μ).

Define the Y -skew product over α τ : X × Y → X × Y by

τ(x, y) = (T (x), T_{α(x)}y)

where x ↦ α(x) is the so-called α-name of x (i.e. x ∈ α(x) ∈ α).

We have that

τ^{n}(x, y) = (T^{n}x, T_{a_{n}(x)}y)

where x ↦ a_{n}(x) ∈ α_{0}^{n−1} is the α_{0}^{n−1}-name of x, and T_{(a_{0},...,a_{n−1})} := T_{a_{n−1}} ∘ ⋯ ∘ T_{a_{0}}.

The Frobenius-Perron operators P_{τ^{n}} : L^{1}(m × μ) → L^{1}(m × μ) satisfy

P_{τ^{n}}f(x, y) = ∑_{b∈β} 1_{b}(x) ∑_{a∈α_{0}^{n−1}, τ^{n}a⊃b} v′_{a}(x)P_{T_{a}}(f(v_{a}x, ·))(y)

where P_{T_{a}} : L^{1}(μ) → L^{1}(μ) is the Frobenius-Perron operator corresponding to T_{a}.

Definition ([Kow2]). A Markov map (X, B, m, T, α) is called quasi-Markov if whenever τ is a Y -skew product over α, g ∈ L^{1}(m × μ)_{+} satisfies P_{τ} g = g, then [g > 0] is α × Cmeasurable.

We prove

Theorem 3.3. Any probability preserving, mixing Gibbs-Markov map is quasi-Markov

and deduce theorem 3.1 from it.

For f : X ×Y → ℝ measurable and y ∈ Y let f_{y} : X → ℝ be defined by f_{y}(x) := f(x, y). Set

∥|f∥|_{L} := ∫_{Y} ∥f_{·}∥_{L}dμ = ∫_{Y} ∥f_{·}∥_{∞}dμ + ∫_{Y} ( sup_{x,x′∈b∈β} \frac{|f(x, ·) − f(x′, ·)|}{d(x, x′)})dμ,

∥f∥_{L} := inf {∥|g∥|_{L} : g = f m × μ − a.e.},

and let

L = {f : X × Y → ℝ : ∥f∥_{L} < ∞}.

Clearly f ∈ L entails the existence of g = f a.e. such that g_{y} ∈ L ∀ y ∈ Y.

Proposition 3.4. If f_{n} ∈ L, f ∈ L^{1}(m × μ), and f_{n} ⟶^{m×μ} f,

then

lim inf_{n→∞} ∥f_{n}∥_{L} ≥ ∥f∥_{L}.

Proof. Suppose without loss of generality that

∥f_{n}∥_{L} = ∫_{Y} ∥(f_{n})_{·}∥_{L}dμ ≤ M ∀ n ≥ 1.

It follows that for lim inf_{n→∞} ∥(f_{n})_{·}∥_{L} =: M(·) < ∞ and that for μ-a.e. y ∈ Y, ∃ a subsequence n_{k}(y) → ∞ so that ∥(f_{n_{k}(y)})_{y}∥_{L} → M(y) as k → ∞.

By the version of the Arzela-Ascoli theorem proved in §1, there are further subsequences n′_{k}(y) → ∞ (of {n_{k}(y)}_{k≥1}), and functions g_{y} ∈ L so that

f_{n′_{k}(y)}(x, y) = (f_{n′_{k}(y)})_{y}(x) → g_{y}(x) ∀ x ∈ X, & ∥g_{y}∥_{L} ≤ M(y).

Since f(x, y) = g_{y}(x) a.e., we have that

∥f∥_{L} ≤ ∫_{Y} ∥g_{·}∥_{L}dμ

≤ ∫_{Y} lim inf_{n→∞} ∥(f_{n})_{·}∥_{L}dμ

≤ lim inf_{n→∞} ∫_{Y} ∥(f_{n})_{·}∥_{L}dμ by Fatou’s lemma

= lim inf_{n→∞} ∥f_{n}∥_{L}.

□ Now let T : X → X be a Gibbs-Markov map with respect to the partition α, and let τ

be a skew product over (T, α). Lemma 3.5.

∥P_{τ^{n}}f∥_{L} ≤ M′′((M + 1)∥f∥_{1} + (M + 2)r^{n}∥f∥_{L}) ∀ f ∈ L, n ≥ 1

where M and M′′ are as in §1. Proof.

|P_{τ^{n}}f(x, y)| ≤ ∑_{b∈β} 1_{b}(x) ∑_{a∈α_{0}^{n−1}, τ^{n}a⊃b} v′_{a}(x)P_{T_{a}}(|f(v_{a}x, ·)|)(y)

≤ ∑_{b∈β} 1_{b}(x) ∑_{a∈α_{0}^{n−1}, τ^{n}a⊃b} v′_{a}(x)P_{T_{a}}(\frac{1}{m(a)} ∫_{a} |f_{·}|dm + r^{n}∥f_{·}∥_{L})(y)

≤ M′′ ∑_{a∈α_{0}^{n−1}} P_{T_{a}}(∫_{a} |f_{·}|dm + r^{n}m(a)∥f_{·}∥_{L})(y).

Hence, using ∫_{Y} P_{T_{a}}fdμ = ∫_{Y} fdμ,

∫_{Y} ∥(P_{T^{n}}f)_{·}∥_{∞}dμ ≤ M′′(∥f∥_{1} + r^{n}∥f∥_{L}).

Next, we have for x, x′ ∈ b ∈ β,

|P_{τ^{n}}f(x, ·) − P_{τ^{n}}f(x′, ·)|

= | ∑_{a∈α_{0}^{n−1}, T^{n}a⊃b} P_{T_{a}}(v′_{a}(x)f(v_{a}x, ·) − v′_{a}(x′)f(v_{a}x′, ·))|

≤ ∑_{a∈α_{0}^{n−1}, T^{n}a⊃b} P_{T_{a}}(|v′_{a}(x)f(v_{a}x, ·) − v′_{a}(x′)f(v_{a}x′, ·)|)

≤ ∑_{a∈α_{0}^{n−1}} P_{T_{a}}(M′′d(x, x′)(M ∫_{a} |f_{·}|dm + (M + 1)r^{n}m(a)∥f_{·}∥_{L})) by lemma 1.3,

and it follows that

∫_{Y} ( sup_{x,x′∈b∈β} \frac{|P_{τ^{n}}f(x, ·) − P_{τ^{n}}f(x′, ·)|}{d(x, x′)})dμ ≤ M′′(M∥f∥_{1} + (M + 1)r^{n}∥f∥_{L}).

□

Proposition 3.6. If h ∈ L^{1}(m × μ)_{+} satisfies P_{τ} h = h, then h ∈ L.

Proof. It follows from lemma 3.5 that ∃ M′′′ > 0 such that

∥P_{τ^{n}}f∥_{L} ≤ M′′′∥f∥_{L} ∀ f ∈ L.

Suppose that h ∈ L^{1}(m × μ)_{+} satisfies P_{τ} h = h. Given 0 < ϵ < 1, choose f_{ϵ} ∈ L with ∥f_{ϵ} − h∥_{1} < ϵ. By the stochastic ergodic theorem ([Kre]),

\frac{1}{n} ∑_{k=0}^{n−1} P_{τ^{k}}f_{ϵ} ⟶^{m×μ} h_{ϵ}

where P_{τ} h_{ϵ} = h_{ϵ} ∈ L^{1}(m × μ).

It follows from Fatou’s lemma that

∥h − h_{ϵ}∥_{1} ≤ lim inf_{n→∞} ∥h − \frac{1}{n} ∑_{k=0}^{n−1} P_{τ^{k}}f_{ϵ}∥_{1} ≤ ∥f_{ϵ} − h∥_{1} < ϵ.

We claim that h_{ϵ} ∈ L. This follows from proposition 3.4, as

∥h_{ϵ}∥_{L} ≤ lim inf_{n→∞} ∥\frac{1}{n} ∑_{k=0}^{n−1} P_{τ^{k}}f_{ϵ}∥_{L} ≤ M′′′∥f_{ϵ}∥_{L}.

Again, by lemma 3.5, we have

∥h_{ϵ}∥_{L} = ∥P_{τ^{n}}h_{ϵ}∥_{L} ∀ n ≥ 1
≤ M′′(r^{n}∥h_{ϵ}∥_{L} + ∥h_{ϵ}∥_{1})
⟶ M′′∥h_{ϵ}∥_{1} as n → ∞
≤ M′′(∥h∥_{1} + 1).

To finish, set H_{n} := h_{\frac{1}{n}}, then H_{n} ∈ L and

H_{n} ⟶^{m×μ} h and sup_{n} ∥H_{n}∥_{L} < ∞. By proposition 3.4, h ∈ L.

□

Proof of theorem 3.3. Let (X, B, m, T, α) be a probability preserving, mixing Gibbs-Markov map. Suppose that τ is a Y -skew product over α and that h ∈ L^{1}(m×μ)_{+} satisfies P_{τ} h = h.

Let A := [h > 0]. Clearly τ^{−1}A ⊃ A mod m × μ. We show that A is α × C-measurable.

To this end, let

A_{1} := {(x, y) ∈ A : m([x_{1}] \ A_{y}) = 0},

then A_{1} = ⋃_{a∈α} a × B_{a} where B_{a} = {y ∈ Y : m(a \ A_{y}) = 0}. We show that A_{1} = A mod m × μ.

We first claim that if this is not the case, then ∃ N ≥ 2, s = [s_{0}, …, s_{N}] ∈ α_{0}^{N−1} and C ∈ C, μ(C) > 0 such that s × C ⊂ A \ A_{1}.

To see this we note that by proposition 3.6, h ∈ L and so for μ-a.e. y ∈ Y, x ↦ h(x, y) is continuous X → ℝ.

It follows that for m × μ-a.e. (x, y) ∈ A, ∃ n ≥ 1 x ∈ a ∈ α_{0}^{n−1} such that m(a \ A_{y}) = 0 whence A = ⋃_{n≥1, a∈α_{0}^{n−1}} a × C_{a} where C_{a} = {y ∈ Y : m(a \ A_{y}) = 0}; and our claim is established.

Let p ≪ m × μ be that finite (T -invariant) measure with density \frac{d p}{dm×μ} = h then (A, B × C, p, τ) is conservative and so ∃ n > N such that p((s × C) ∩ τ^{−n}(s × C)) > 0. It follows that also p((s × C) ∩ τ^{n}(s × C)) > 0.

Since n > N, T^{n}s ∈ σ(α) and (s_{0} × Y) ∩ τ^{n}(s × C) = s_{0} × C′ for some C′ ∈ C. We have that μ(C ∩ C′) > 0 because (s × C) ∩ τ^{n}(s × C) = s × (C ∩ C′).

The conclusion is that s_{0} × C′ ⊂ τ^{n}A ⊂ A whence s_{0} × C′ ⊂ A_{1}; and p(A_{1} ∩ (s × C)) ≥ p((s_{0} × C′) ∩ (s × C)) > 0 contradicting s × C ⊂ A \ A_{1}. □

Proof of theorem 3.1. This proof is that of lemma 3 in [Kow2]. Let Y = [0, 1) equipped with μ = Lebesgue measure. Define the transformations T_{a} : Y → Y by T_{a}y = y + ψ(a) mod 1 where ϕ = λe^{2πiψ} and let τ be the corresponding skew product. The probability p = m × μ is τ-invariant. If h(x, y) = g(x)e^{2πiy} then

h ∘ τ(x, y) = h(T x, y + ψ(x)) = g(T x)\overline{λ}ϕ(x)e^{2πiy} = h(x, y).

We claim that h is α×C-measurable. To see this, note that if A ∈ B×C is τ-invariant, then P_{τ} 1_{A} = 1_{A} and by the quasi Markov property, A ∈ α × C. It follows that any measurable, τ-invariant function is α × C-measurable; for example h.

It now follows that g is α-measurable and an easy computation using that g ∘ T is also α-measurable shows that g is indeed α_{∗}-measurable. □

To complete this section, we now turn to the periodic case.

Let (X, B, m, T, α) be a mixing Gibbs-Markov map with respect to the partition α and invariant probability m, and let ϕ : X → ℝ^{d} be Lipschitz continuous on each a ∈ α with Dϕ := sup_{a∈α} D_{a}ϕ < ∞.

Proposition 3.7.

Let t ∈ ℝ^{d}, the following are equivalent:

1) ∥P_{t}∥_{Hom(L,L)} = 1,

2) ∃ g : X → 𝕋 Lipschitz continuous and z ∈ 𝕋 such that P_{t}(g) = zg,

3) χ_{t}(ϕ) is cohomologous to a constant.

Proof. The equivalence of 1) and 2) follows from the ITM theorem. Suppose that χ_{t}(ϕ) = z\overline{g}g ∘ T where z ∈ 𝕋 and g : X → 𝕋 is measurable. It follows that P_{t}(g) = P (gχ_{t}(ϕ)) = P (zg∘T) = zg, whence by corollary 2.2 g is Lipschitz continuous and |λ(t)| = 1. Conversely, suppose that ∥P_{t}∥_{Hom(L,L)} = 1, then ∃ g : X → 𝕋 Lipschitz continuous and z ∈ 𝕋 such that P_{t}(g) = zg whence | < χ_{t} ∘ ϕg, zg ∘ T > | = | < P_{t}g, zg > | = ∥g∥_{2}. By the Cauchy-Schwartz inequality, χ_{t} ∘ ϕg and g ∘ T are linearly dependent and χ_{t}(ϕ) is cohomologous to the constant z. □

Suppose that ϕ : X → ℝ^{d}. Let

𝔔 = 𝔔(ϕ) := {t ∈ ℝ^{d} : χ_{t}(ϕ) is cohomologous to a constant}.

Proposition 3.8. 𝔔 is a closed subgroup of ℝ^{d} and ∃ z ∈ 𝔔̂ such that χ_{t}(ϕ) is cohomologous to, and only to z(t).

Proof. Evidently 𝔔 is a subgroup of ℝ^{d}. Also, for each t ∈ 𝔔, ∃ ! z(t) ∈ ℝ^{d} such that χ_{t}(ϕ) is cohomologous to z(t) (as non-unicity would imply ∃ z ∈ 𝕋, z ≠ 1 such that z is cohomologous to 1, an impossibility due to the weak mixing of T).

To see that 𝔔 is closed suppose that t_{n} ∈ 𝔔 satisfy P_{t_{n}}g_{n} = z(t_{n})g_{n} where g_{n} ∈ L, |g_{n}| = 1 a.e.. Suppose also that t_{n} → t ∈ ℝ^{d} and z(t_{n}) → ζ ∈ 𝕋.

By proposition 2.1 ∃ M, k ≥ 1, 0 < θ < 1 such that ∥P_{t_{n}}^{k}h∥_{L} ≤ θ∥h∥_{L} + M∥h∥_{1} ∀ n ≥ 1, h ∈ L

and it follows that ∥g_{n}∥_{L} = ∥P_{t_{n}}^{k}g_{n}∥_{L} ≤ θ∥g_{n}∥_{L} + M ∀ n ≥ 1 whence ∥g_{n}∥_{L} ≤ \frac{M}{1−θ}.

By the Arzela-Ascoli theorem ∃ n_{k} → ∞, g ∈ L such that ∥g_{n_{k}} − g∥_{1} → 0. Clearly |g| = 1 a.e., and to see that t ∈ 𝔔, we must show that P_{t}g = ζg.

By theorem 2.4, P_{t_{n}} ⟶^{Hom(L,L)} P_{t}, whence

∥P_{t}g − ζg∥_{1} ≤ ∥P_{t}g − P_{t}g_{n}∥_{1} + ∥P_{t}g_{n} − P_{t_{n}}g_{n}∥_{L} + ∥z(t_{n})g_{n} − ζg∥_{1}

≤ ∥g − g_{n}∥_{1} + \frac{M}{1 − θ} ∥P_{t} − P_{t_{n}}∥_{Hom(L,L)} + |z(t_{n}) − ζ| + ∥g − g_{n}∥_{1}

→ 0

as n → ∞. □

If 𝔔 = ℝ^{d} then by [H-O-O] and [M-S], ϕ = h∘T −h+c for some c ∈ ℝ^{d} and h : X → ℝ^{d} measurable (and hence Lipschitz continuous on each a ∈ α by corollary 2.2).

More generally,

Proposition 3.9. If 𝔔 is not a discrete subgroup of ℝ^{d} then ∃ a vector subspace V ⊂ 𝔔, v ∈ V, and Lipschitz continuous functions g : X → V, ψ : X → V^{⊥}, Lipschitz continuous on each a ∈ α, such that ϕ = g ∘ T − g + v + ψ and such that 𝔔(ψ) ∩ V^{⊥} is a discrete subgroup of V^{⊥}.

Proof. For each t ∈ ℝ^{d}, {u ∈ ℝ : ut ∈ 𝔔} is a closed subgroup of ℝ and hence either discrete or = ℝ. Set V := {t ∈ ℝ^{d} : ut ∈ 𝔔 ∀ u ∈ ℝ}. It follows that V ⊂ 𝔔 is a vector subspace of ℝ^{d} and that 𝔔(ϕ) ∩ V^{⊥} is a discrete subgroup of V^{⊥}.

Taking projections we obtain functions ϕ_{0} : X → V, ψ : X → V^{⊥} Lipschitz continuous on each a ∈ α such that ϕ = ϕ_{0} + ψ.

We check that 𝔔(ψ) ∩ V^{⊥} = 𝔔(ϕ) ∩ V^{⊥} and hence is a discrete subgroup of V^{⊥}. Also 𝔔(ϕ_{0}) ∩ V = 𝔔(ϕ) ∩ V = V whence by the above ∃ v ∈ V and g : X → V Lipschitz continuous on each a ∈ α, such that ϕ_{0} = g ∘ T − g + v. □

§4 A Nagaev-type Spectral theorem

Let (X, B, m, T, α) be a mixing Gibbs-Markov map with respect to the partition α and invariant probability m, and ϕ : X → ℝ^{d} be Lipschitz continuous on each a ∈ α, and D_{α}ϕ := sup_{a∈α} D_{a}ϕ < ∞.

Theorem 4.1.

1) There are constants ϵ > 0, K > 0 and θ ∈ (0, 1); and functions λ : B(0, ϵ) → B_{ℂ}(0, 1), g : B(0, ϵ) → L such that

∥P_{t}^{n}h − λ(t)^{n}g(t) ∫_{X} hdm∥_{L} ≤ Kθ^{n}∥h∥_{L} ∀ |t| < ϵ, n ≥ 1, h ∈ L

and

∥g(t) − 1∥_{L} ≤ K(|t| + E(|e^{itϕ} − 1|)).

2) If ϕ is aperiodic, then ∀ M̃ > 0, ϵ > 0, ∃ K′ > 0 and θ′ ∈ (0, 1) such that

∥P_{t}^{n}h∥_{L} ≤ K′θ^{′n}∥h∥_{L} ∀ ϵ ≤ |t| ≤ M̃, h ∈ L.

By theorem 2.4, t ↦ P_{t} is continuous ℝ^{d} → Hom (L, L), and by proposition 2.1 P_{t} is an ITM operator ∀ t ∈ ℝ^{d}. The proof of the theorem is established by two lemmas about ITM operators.

In the next two lemmas, suppose that (C, L) is adapted, and write ∥P ∥ := ∥P ∥_{Hom (L,L)} for P ∈ Hom (L, L).

Lemma 4.2 (c.f. [Nag]).

Suppose that P_{0} ∈ Hom(L, L) is a ITM operator, and suppose that dim{g ∈ L : P_{0}g = g} = 1, then ∃ ϵ > 0, K ∈ ℝ_{+}, λ : B(P_{0}, ϵ) → B_{ℂ}(0, 1) and θ ∈ (0, 1) such that ∀ P ∈ Hom(L, L), ∥P − P_{0}∥ < ϵ,

P^{n} = λ(P)^{n}N_{1}(P) + Q(P)^{n}

where Q(P) ∈ Hom (L, L), N_{1}(P) is a projection onto a 1-dimensional subspace,

|λ(P) − λ(P′)|, ∥N_{1}(P) − N_{1}(P′)∥ = O(∥P − P′∥)

as ∥P − P′∥ → 0, P, P′ ∈ B(P_{0}, ϵ), and

∥Q(P)^{n}∥ ≤ Kθ^{n} ∀ n ≥ 1, P ∈ B(P_{0}, ϵ).

Proof. Recall that the resolvent set of P is ρ(P) := {z ∈ ℂ : ∃ (zI −P)^{−1}}, and the spectral radius of P is

𝔯(P) = sup\{|z| : z ∉ ρ(P)\} ← ∥P^{n}∥^{\frac{1}{n}} ,

and for |z| > 𝔯(P),

R_{P} (z) := (zI − P)^{−1} = ∑_{n=0}^{∞} \frac{1}{z^{n+1}} P^{n}.

By the ITM theorem,

P_{0} = μ_{0} + Q_{0}

where μ_{0}P_{0} = P_{0}μ_{0} = μ_{0} = μ_{0}^{2}, μ_{0}Q = Qμ_{0} = 0, and 𝔯(Q_{0}) < 1. Let

B_{1} = B(1, \frac{1 − 𝔯(Q_{0})}{3}), B_{2} = B(0, \frac{1 + 2𝔯(Q_{0})}{3}),

then \overline{B}_{1}, \overline{B_{2}^{c}} ⊂ ρ(P_{0}).

It follows that

M̃:= sup_{z∈\overline{B}_{1}∪\overline{B_{2}^{c}}} ∥R_{P_{0}}(z)∥ < ∞,

whence if P ∈ Hom(L, L) and ∥P − P_{0}∥ < ϵ_{1} := M̃^{−1} then ∀ z ∈ B_{1} ∪ \overline{B_{2}^{c}},

∑_{n=1}^{∞} ∥((P − P_{0})R_{P_{0}}(z))^{n}∥ < ∞,

whence

R_{P_{0}}(z) ∑_{n=0}^{N}((P − P_{0})R_{P_{0}}(z))^{n} → (zI − P)^{−1}

in Hom(L, L) as N → ∞. This shows that

\overline{B}_{1} ∪ \overline{B_{2}^{c}} ⊂ ρ(P) ∀ P ∈ B(P_{0}, ϵ_{1}).

For P ∈ B(P_{0}, ϵ_{1}), define

N_{1}(P) := \frac{1}{2πi} ∮_{∂B_{1}} R_{P} (z)dz, & N_{2}(P) := \frac{1}{2πi} ∮_{∂B_{2}} R_{P} (z)dz.

To check continuity of P ↦ N_{i}(P),

N_{i}(P) − N_{i}(P′) = \frac{1}{2πi} ∮_{∂B_{i}}(R_{P} (z) − R_{P′}(z))dz

= \frac{1}{2πi} ∮_{∂B_{i}} R_{P} (z)(P − P′)R_{P′}(z))dz

whence

∥N_{i}(P) − N_{i}(P′)∥ ≤ ∥\frac{1}{2πi} ∮_{∂B_{i}} R_{P} (z)(P − P′)R_{P′}(z))dz∥ ≤ M̃^{2}∥P − P′∥.

As well-known, N_{i}(P) (i = 1, 2) is a projection, and

P^{n}N_{i}(P) = N_{i}(P)P^{n} = \frac{1}{2πi} ∮_{∂B_{i}} z^{n}R_{P} (z)dz.

Since N_{1}(P_{0}) = μ_{0} and P ↦ N_{1}(P) is continuous, ∃ ϵ_{2} > 0, ϵ_{2} < ϵ_{1} such that

∥N_{1}(P) − μ_{0}∥ < 1 ∀ P ∈ B(P_{0}, ϵ_{2}).

We now claim that for such P, dim N_{1}(P) = 1. To see this, suppose otherwise, then either N_{1}(P) = 0 or ∃ h_{1}, h_{2} ∈ N_{1}(P)L which are linearly independent.

Suppose that μ_{0}L =span ψ, then μ_{0}ψ = ψ and

∥N_{1}(P)ψ∥ ≥ ∥μ_{0}ψ∥ − ∥(μ_{0} − N_{1}(P))ψ∥ ≥ (1 − ∥(μ_{0} − N_{1}(P))∥)∥ψ∥ > 0,

so N_{1}(P) ≠ 0.

Suppose that h_{1}, h_{2} ∈ N_{1}(P)L are linearly independent. Clearly ∃ 0 ≠ (a, b) ∈ ℝ^{2} such that if h = ah_{1} + bh_{2} then h ≠ 0 but μ_{0}h = 0. It follows that

(N_{1}(P) − μ_{0})h = N_{1}(P)h = h

contradicting ∥N_{1}(P) − μ_{0}∥ < 1.

In consequence, ∃ λ : B(P_{0}, ϵ_{2}) → B_{ℂ}(0, 1) such that

P^{n}N_{1}(P) = λ(P)^{n}N_{1}(P).

To check continuity of P ↦ λ(P),

|λ(P) − λ(P′)|
≤ ∥λ(P)N_{1}(P) − λ(P′)N_{1}(P)∥
≤ ∥λ(P)N_{1}(P) − λ(P′)N_{1}(P′)∥ + ∥N_{1}(P) − N_{1}(P′)∥
= ∥P N_{1}(P) − P′N_{1}(P′)∥ + ∥N_{1}(P) − N_{1}(P′)∥
≤ ∥P N_{1}(P) − P′N_{1}(P)∥ + ∥P′N_{1}(P) − P′N_{1}(P′)∥ + ∥N_{1}(P) − N_{1}(P′)∥
≤ ∥P − P′∥ + (∥P′∥ + 1)∥N_{1}(P) − N_{1}(P′)∥
= O(∥P − P′∥).

It follows from \overline{B}_{1} ∪ \overline{B_{2}^{c}} ⊂ ρ(P) for P ∈ B(P_{0}, ϵ_{2}), that

P^{n} = \frac{1}{2πi}( ∮_{∂B_{1}} + ∮_{∂B_{2}} )z^{n}R_{P} (z)dz
= P^{n}N_{1}(P) + P^{n}N_{2}(P)
= λ(P)^{n}N_{1}(P) + Q(P)^{n}

where Q(P) := P N_{2}(P), and

∥Q(P)^{n}∥ = \frac{1}{2π} ∥ ∮_{∂B(0, \frac{1+2𝔯(Q_{0})}{3} )} z^{n}R_{P} (z)dz∥ ≤ M̃(\frac{1 + 2𝔯(Q_{0})}{3})^{n+1}.

□

Lemma 4.3.

Suppose that K ⊂ Hom(L, L) is a compact set of ITM operators, none of which has an L-eigenvalue on 𝕋 (the unit circle), then ∃ K ∈ ℝ_{+} and θ ∈ (0, 1) such that

∥P^{n}∥ ≤ Kθ^{n} ∀ n ≥ 1, P ∈ K.

Proof.

We first show that max_{P ∈K}𝔯(P) < 1.

For P ∈ K and z ∈ ρ(P)

R_{P} (z) = (zI − P)^{−1}.

For b > 𝔯(P),

M(P, b) := sup_{|z|≥b} ∥R_{P} (z)∥ < ∞.

If P′ ∈ Hom(L, L) and ∥P − P′∥ < M(P, b)^{−1} then ∀ |z| > b

∑_{n=1}^{∞} ∥((P′ − P)R_{P} (z))^{n}∥ < ∞,

whence

R_{P} (z) ∑_{n=0}^{N}((P′ − P)R_{P} (z))^{n} → (zI − P′)^{−1}

in Hom(L, L) as N → ∞ and B(0, b)^{c} ⊂ ρ(P′) which implies 𝔯(P′) ≤ b.

For each P ∈ K, choose r_{P} ∈ (𝔯(P), 1). As above, for each P ∈ K, ∃ ϵ_{P} = M(P, r_{P})^{−1} such that

𝔯(Q) ≤ r_{P} ∀ Q ∈ B(P, ϵ_{P}).

By compactness of K, ∃ P_{1}, …, P_{N} ∈ K such that

K ⊂ ⋃_{k=1}^{N} B(P_{k}, ϵ_{P_{k}})

with the consequence that

𝔯(P) ≤ r_{0} := max_{1≤k≤N} r_{P_{k}} < 1 ∀ P ∈ K.

To complete the proof choose

max_{P ∈K} 𝔯(P) < b < 1.

We have that (z, P) ↦ (zI − P)^{−1} is continuous {|z| = b} × K → Hom(L, L).

Therefore

sup_{|z|=b, P ∈K} ∥(zI − P)^{−1}∥ =: K < ∞.

Now, for n ≥ 1,

P^{n} = \frac{1}{2πi} ∮_{|z|=b}(zI − P)^{−1}z^{n}dz = \frac{1}{2π} ∫_{0}^{2π}(be^{it}I − P)^{−1}b^{n+1}e^{i(n+1)t}dt

whence

∥P^{n}∥ ≤ \frac{1}{2π} ∫_{0}^{2π} ∥(be^{it}I − P)^{−1}∥b^{n+1}dt ≤ Kθ^{n+1}.

□

Proof of theorem 4.1.

The first statement follows from lemma 4.2, and theorem 2.4.

The aperiodicity of ϕ implies that for t ≠ 0, P_{t} has no eigenvalue on 𝕋. To see this, suppose that P_{t}h = λh with λ ∈ 𝕋, then | < γ∘ϕh, λh∘T > | = | < P_{γ}h, λh > | = ∥h∥_{2} and by the Cauchy-Schwartz inequality, γ ∘ ϕh and h ∘ T are linearly dependent, contradicting the aperiodicity of ϕ.

By theorem 2.4, {P_{t} : ϵ ≤ |t| ≤ M̃} is compact in Hom(L, L), and so by lemma 4.3, ∃ θ ∈ (0, 1) and K > 0 so that ∥P_{t}^{n}∥_{Hom(L,L)} ≤ Kθ^{n} for n ≥ 1, ϵ ≤ |t| ≤ M̃. □

§5 Expansion of the eigenvalue

Definition.

A random variable X on ℝ is called stable if for all a, b > 0 there are c > 0 and v ∈ ℝ such that aX + bX′ =^{d} cX + v where X′ is an independent copy of X and Y =^{d} Z means that the random variables Y and Z have the same distribution.

In this case necessarily a^{p} + b^{p} = c^{p} for some 0 < p ≤ 2, and p is called the order of X.

We’ll denote a stable random variable with order p ∈ (0, 2] by X_{p}. It is known that (up to translation) X_{p} has a characteristic function of form

g_{X_{p}}(t) := E (e^{itX_{p}}) = \{ \begin{matrix}e^{−c|t|^{p}(1−iβsgn(t) tan( \frac{pπ}{2} ))} p ≠ 1, \\ e^{−c|t|+i \frac{2βc}{π} t log \frac{1}{|t|}} p = 1 \end{matrix}

where c > 0, β ∈ ℝ are constants, and an absolutely continuous distribution function with smooth, positive density f_{X_{p}}.

Throughout this section, we let (X, B, m, T, α) be a mixing, probability preserving Gibbs-Markov map, and let

ϕ : X → ℝ

be Lipschitz continuous on each a ∈ α, with D_{α}ϕ := sup_{a∈α} D_{a}ϕ < ∞ and distribution G in the domain of attraction of a stable law with order 0 < p < 2 equivalently ([Fel], [G-K], [I-L], [Le]):

L_{1}(x) := x^{p}(1 − G(x)) = (c_{1} + o(1))L(x), L_{2}(x) := x^{p}G(−x) = (c_{2} + o(1))L(x)

as x → +∞ where L is a slowly varying function on ℝ_{+} and where c_{1}, c_{2} ≥ 0, c_{1} + c_{2} > 0.

Let the operator P_{t} : L → L be defined (as in §4) by P_{t}f = P_{T} (χ_{t}(ϕ)f), let ϵ > 0 and λ(t) := λ(P_{t}) (|t| < ϵ) be as in theorem 4.1, and let E(e^{itϕ}) = Ĝ(t).

Theorem 5.1 (Expansion of the eigenvalue).

Re log λ(t) = −c|t|^{p}L(|t|^{−1})(1 + o(1)),

and

Im log λ(t)

= \{ \begin{matrix}tγ + cβ|t|^{p}sgn(t) tan(\frac{pπ}{2} ) + o(|t|^{p}L(1/|t|)) (p ≠ 1), \\ tγ + \frac{2βc}{π}CtL(1/|t|)) + t(H_{1}(1/|t|) − H_{2}(1/|t|)) + o(|t|L(1/|t|)) (p = 1) \end{matrix}

as t → 0, where

H_{j}(λ) = ∫_{0}^{λ} \frac{xL_{j}(x)dx}{1 + x^{2}} + o(L(λ)) as λ → ∞ (j = 1, 2),

C = ∫_{0}^{∞} ( cos y − \frac{1}{1 + y^{2}} )\frac{dy}{y} ,

and the constants c > 0, β, γ ∈ ℝ are defined by

β = \frac{c_{1} − c_{2}}{c_{1} + c_{2}} , c = \{ \begin{matrix}(c_{1} + c_{2})Γ(1 − p) cos(\frac{pπ}{2} ) p ≠ 1, \\ \frac{(c_{1}+c_{2})π}{2} p = 1; \end{matrix}

γ = \{ \begin{matrix}0 p < 1, \\ ∫_{−∞}^{∞} (\frac{x}{1+x^{2}} + sgn(x) ∫_{0}^{|x|} \frac{2u^{2}}{(1+u^{2})^{2}} du)G(dx) p = 1, \\ ∫_{−∞}^{∞} xG(dx) p > 1. \end{matrix}

Remark.

The expansion of Ĝ is given by theorem 2.6.5 in [I-L] in case p ≠ 1, and by theorem 2 in [A-D1] in case p = 1. As a corollary, we obtain that under the conditions of theorem 5.1

| log λ(t) − log Ĝ(t)| = o(|t|^{p}L(1/|t|)) as t → 0.

Lemma 5.2.

E(|1 − e^{itϕ}|) = \{ \begin{matrix}O(|t|) 1 < p < 2, \\ O (\sqrt{|t|L(\frac{1}{|t|})}) p = 1 \end{matrix}

as t → 0.

Proof. These estimates follow from the expansion of Ĝ (see theorem 2.6.5 in [I-L] in case p ≠ 1 and theorem 2 in [A-D1] in case p = 1). □

In the next 5 lemmata, h : ℝ_{+} → ℝ_{+} is locally integrable and slowly varying at ∞; η > 0, and g : [−η, η] × ℝ_{+} → ℝ satisfies

lim_{t→0} sup_{x∈ℝ} |g(t, x) − K| = 0

for some constant K ≥ 0. We’ll denote ∥g∥ := sup_{x∈ℝ, |t|≤η} |g(t, x)|.‘

Lemmas 5.3-5 with g constant follow from lemmas 2.6.1-2 in [I-L], and lemma 5.6 with g constant corresponds to lemma 2 in [A-D1].

The proofs of the lemmas can be easily extracted from their corresponding prototypes (with g constant) and so are not given.

Lemma 5.3. For 0 < p < 2, if the function u → u^{−p}h(u)g(t, u) is decreasing for every fixed t, then

∫_{0}^{∞} \frac{sin(y)}{y^{p}}h(\frac{y}{t})g(t, \frac{y}{t})dy = \{ \begin{matrix}(K + o(1))h(\frac{1}{t})Γ(1 − p) cos(\frac{pπ}{2} ) & if p ≠ 1 \\ (K + o(1))h(\frac{1}{t})\frac{π}{2} & if p = 1. \end{matrix}

Lemma 5.4. For 0 < p < 1, if the function u → u^{−p}h(u)g(t, u) is decreasing for every fixed t, then

∫_{0}^{∞} \frac{cos(y)}{y^{p}}h(\frac{y}{t})g(t, \frac{y}{t})dy = (K + o(1))h(\frac{1}{t})Γ(1 − p) sin(\frac{pπ}{2} ).

LOCAL LIMIT THEOREMS FOR GIBBS–MARKOV MAPS 27

Lemma 5.5. Let 1 < p < 2,

∫_{0}^{∞} \frac{cos(y) − 1}{y^{p}}h(\frac{y}{t})g(t, \frac{y}{t})dy = (K + o(1))h(\frac{1}{t})Γ(1 − p) sin(\frac{pπ}{2} ).

Lemma 5.6. Suppose that the function u → u^{−1}h(u)g(t, u) is decreasing for every fixed t, then

∫_{0}^{∞} [cos y − \frac{1}{1 + y^{2}} ] \frac{1}{y} h(\frac{y}{t})g(t, \frac{y}{t})dy = (K + o(1))h(\frac{1}{t}) ∫_{0}^{∞} [cos y − \frac{1}{1 + y^{2}} ] \frac{1}{y} dy.

Lemma 5.7. Let

H̃_{g}(λ) := ∫_{0}^{∞} \frac{xh(x)g(\frac{1}{λ}, x)dx}{(1 + x^{2})(1 + \frac{x^{2}}{λ^{2}} )}, H(λ) := ∫_{0}^{λ} \frac{xh(x)dx}{1 + x^{2}},

and suppose that ∃ ϵ > 0 such that sup_{x∈ℝ_{+}} |g(\frac{1}{λ}, x) − 1| = O(λ^{−ϵ}) as λ → ∞, then

H̃_{g}(λ) = H(λ) + o(h(λ)) as λ → ∞.

Proof. This corresponds to lemma 3 in [A-D1], and is deduced from it. Set

H̃(λ) := ∫_{0}^{∞} \frac{xh(x)dx}{(1 + x^{2})(1 + \frac{x^{2}}{λ^{2}} )}.

It is shown in lemma 3 of [A-D1] that

H̃(λ) = H(λ) + o(h(λ)) as λ → ∞.

The lemma is therefore a consequence of

|H̃_{g}(λ) − H̃(λ)| = o(h(λ)) as λ → ∞.

To see this

|H̃_{g}(λ) − H̃(λ)| ≤ ∫_{0}^{∞} |g(\frac{1}{λ}, x) − 1|\frac{xh(x)}{(1 + x^{2})(1 + \frac{x^{2}}{λ^{2}} )}

≤ sup_{x∈ℝ_{+}} |g(\frac{1}{λ}, x) − 1| H(λ)

= O(H(λ)λ^{−ϵ})

= o(h(λ)) as λ → ∞

because both h and H are slowly varying at ∞. □

28 JON. AARONSON AND MANFRED DENKER

Proof of theorem 5.1. Let g̃_{t} = (∫_{X} g(t)dm)^{−1}g(t) - the eigenfunction of P_{t} with eigenvalue λ(t) satisfying ∫_{X} g̃_{t} dm = 1, then

λ(t) = λ(t) ∫_{X} g̃_{t} dm = ∫_{X} P_{t}g̃_{t} dm = ∫_{X} P (g̃_{t}e^{itϕ}) dm = ∫_{X} g̃_{t}e^{itϕ} dm.

By theorem 4.1

∥g(t) − 1∥_{L} = O(∥P_{t} − P ∥_{Hom (L,L)}) as t → 0,

whence

∥g̃_{t} − 1∥_{∞} = O(∥P_{t} − P ∥_{Hom (L,L)}) as t → 0.

By theorem 2.4,

∥P − P_{t}∥_{Hom (L,L)} = O(|t| + E|1 − e^{itϕ}|) as t → 0,

whence from lemma 5.2,

∥g̃_{t} − 1∥_{∞} = \{ \begin{matrix}O(|t|) 1 < p < 2, \\ O (\sqrt{|t|L(\frac{1}{|t|}}) p = 1 \end{matrix} as t → 0.

Denote by F_{0} the σ-algebra generated by ϕ and let ĝ_{t} ∘ ϕ = E(g̃_{t}|F_{0}), then λ(t) = ∫_{X} ĝ_{t} ∘ ϕ exp[itϕ] dm = ∫_{−∞}^{∞} ĝ_{t}(x) exp[itx]G(dx),

sup_{x∈ℝ} |ĝ_{t}(x) − 1| ≤ ∥g̃_{t} − 1∥_{∞} = \{ \begin{matrix}O(|t|) 1 < p < 2, \\ O (\sqrt{|t|L(\frac{1}{|t|}}) p = 1 \end{matrix} as t → 0,

and

∫_{−∞}^{∞} ĝ_{t}(x) G(dx) = 1 ∀ t ∈ ℝ.

For |t| small enough, g_{t}^{r} := Re ĝ_{t} ≥ 0 and we may write

ĝ_{t} = g_{t}^{r} + ig_{t}^{+} − ig_{t}^{−}

where g_{t}^{±} := (±Im ĝ_{t}) ∧ 0 ≥ 0.

For ∗ = r, +, −, set g_{t} = g_{t}^{∗}, then dG_{t} := g_{t}dG is a (positive) measure on ℝ. Note that

lim_{t→0} sup_{x∈ℝ} |g_{t}(x) − K| = 0

where K = K_{∗} = 1 for ∗ = r and K = 0 otherwise.

LOCAL LIMIT THEOREMS FOR GIBBS–MARKOV MAPS 29

Define distribution functions G^{j}, G_{t}^{j} (j = 1, 2) on ℝ_{+} by G_{t}^{1}(x) := G_{t}(x) − G_{t}(0), G_{t}^{2}(x) := G_{t}(0) − G_{t}(−x),

G^{1}(x) := G(x) − G(0), & G^{2}(x) := G(0) − G(−x).

We have that

G_{t}^{j}(∞) − G_{t}^{j}(x) = \frac{h_{j}(x)}{x^{p}}g_{j}(t, x),

where h_{j}(x) := x^{p}G^{j}(x) = (c_{j} + o(1))L(x) as x → ∞, and

g_{1}(t, x) := \frac{∫_{x}^{∞} g_{t}(u) G(du)}{∫_{x}^{∞} G(du)}, & g_{2}(t, x) := \frac{∫_{−∞}^{−x} g_{t}(u) G(du)}{∫_{−∞}^{−x} G(du)}

and it follows that sup_{x∈ℝ} |g_{j}(t, x) − K| → 0 as t → 0.

First let 0 < p < 1, then ∫_{ℝ}(1 − e^{itx}) G_{t}(dx)

= ∫_{0}^{∞}(1 − e^{itx}) G_{t}^{1}(dx) + ∫_{0}^{∞}(1 − e^{−itx}) G_{t}^{2}(dx).

Integration by parts and substitution of the above gives (for j = 1, 2)

∫_{0}^{∞}(1 − exp[−(−1)^{j}itx])G_{t}^{j}(dx) = i(−1)^{j}t ∫_{0}^{∞} exp[−i(−1)^{j}tx]g_{j}(t, x)\frac{h_{j}(x)}{x^{p}}dx

= i(−1)^{j} sgn(t) ∫_{0}^{∞} exp[−i(−1)^{j}y sgn(t)]g_{j}(t, y/|t|)\frac{h_{j}(y/|t|)}{(y/|t|)^{p}}dy

= ∫_{0}^{∞} sin y g_{j}(t, y/|t|) \frac{h_{j}(y/|t|)}{(y/|t|)^{p}}dy

+ i(−1)^{j} sgn(t) ∫_{0}^{∞} cos[y] g_{j}(t, y/|t|) \frac{h_{j}(y/|t|)}{(y/|t|)^{p}}dy

= |t|^{p} ∫_{0}^{∞} sin[y] g_{j}(t, y/|t|) \frac{h_{j}(y/|t|)}{y^{p}}dy

+ i(−1)^{j}|t|^{p} sgn(t) ∫_{0}^{∞} cos[y] g_{j}(t, y/|t|) \frac{h_{j}(y/|t|)}{y^{p}}dy.

Applying lemmas 5.3 and 5.4 we obtain ∫_{ℝ}(1 − exp[itx])G_{t}(dx) = (K + o(1))h_{1}(1/|t|)|t|^{p}Γ(1 − p) [cos(\frac{pπ}{2} ) − i sgn(t) sin(\frac{pπ}{2} )] + (K + o(1))h_{2}(1/|t|)|t|^{p}Γ(1 − p) [cos(\frac{pπ}{2} ) + i sgn(t) sin(\frac{pπ}{2} )] = (K + o(1))L(\frac{1}{|t|})|t|^{p}Γ(1 − p) [(c_{1} + c_{2}) cos(pπ/2) − i sgn(t) (c_{1} − c_{2}) sin(pπ/2)] = (K + o(1))cL(1/|t|)|t|^{p}(1 − i sgn(t) β tan(pπ/2)).

30 JON. AARONSON AND MANFRED DENKER

Finally

1 − λ(t) = ∫ (1 − exp[itx])g_{t}^{r}(x)G(dx)
+ i ∫ (exp[itx] − 1)g_{t}^{+}(x)G(dx) − i ∫ (exp[itx] − 1)g_{t}^{−}(x)G(dx)
= (1 + o(1))cL(1/|t|)|t|^{p}(1 − i sgn(t) β tan(pπ/2)).

The case 1 < p < 2 is treated analogously using lemmas 5.3 and 5.5, and replacing

∫_{ℝ}(1 − e^{itx})G_{t}(dx) with ∫ (1 − exp[itx] + itx) G_{t}(dx).

We turn to the case p = 1. Write

∫ (1 − e^{itx} + \frac{itx}{1 + x^{2}} ) G_{t}(dx)

= ∫_{0}^{∞}(1 − e^{itx} + \frac{itx}{1 + x^{2}} ) G_{t}^{1}(dx) + ∫_{0}^{∞}(1 − \frac{itx}{1 + x^{2}} − e^{−itx}) G_{t}^{2}(dx).

Integration by parts gives

∫_{0}^{∞}(1 − e^{−(−1)^{j}itx} − (−1)^{j} \frac{itx}{1 + x^{2}} )G_{t}^{j}(dx)

= (−1)^{j}it ∫_{0}^{∞} (e^{−(−1)^{j}itx} − \frac{1 − x^{2}}{(1 + x^{2})^{2}} )\frac{h_{j}(x)g_{j}(t, x)dx}{x} =

|t| ∫_{0}^{∞} sin |t|x\frac{h_{j}(x)g_{j}(t, x)dx}{x} + (−1)^{j}it ∫_{0}^{∞} ( cos tx − \frac{1 − x^{2}}{(1 + x^{2})^{2}} )\frac{h_{j}(x)g_{j}(t, x)dx}{x}.

Now

∫_{0}^{∞} ( cos tx − \frac{1 − x^{2}}{(1 + x^{2})^{2}} )\frac{h_{j}(x)g_{j}(t, x)dx}{x}

= ∫_{0}^{∞} ( cos tx − \frac{1}{1 + (tx)^{2}} )\frac{h_{j}(x)g_{j}(t, x)dx}{x}

+ ∫_{0}^{∞} \frac{x(1 − t^{2})h_{j}(x)g_{j}(t, x)dx}{(1 + x^{2})(1 + (tx)^{2})} + ∫_{0}^{∞} \frac{2xh_{j}(x)g_{j}(t, x)dx}{(1 + x^{2})^{2}}

= ∫_{0}^{∞} ( cos tx − \frac{1}{1 + (tx)^{2}} )\frac{h_{j}(x)g_{j}(t, x)dx}{x} + (1 − t^{2})H̃_{g_{j}}(1/|t|) + γ_{t}^{j}

where

γ_{t}^{j} := ∫_{0}^{∞} \frac{2x^{2}}{(1 + x^{2})^{2}} (G_{t}^{j}(∞) − G_{t}^{j}(x))dx = ∫_{0}^{∞} \frac{2xh_{j}(x)g_{j}(t, x)dx}{(1 + x^{2})^{2}}

LOCAL LIMIT THEOREMS FOR GIBBS–MARKOV MAPS 31

and

H̃_{g_{j}}(λ) := ∫_{0}^{∞} \frac{xh_{j}(x)g_{j}(t, x)dx}{(1 + x^{2})(1 + \frac{x^{2}}{λ^{2}} )}.

Changing variables, and using lemmas 5.3 and 5.6 respectively, we obtain that

∫_{0}^{∞} sin |t|x\frac{h_{j}(x))g_{j}(t, x)dx}{x} = ∫_{0}^{∞} sin x\frac{h_{j}(x/|t|))g_{j}(t, x/|t|)dx}{x}

= \frac{Kc_{j}π}{2} L(\frac{1}{|t|}) + o(L(\frac{1}{|t|})) as t → 0,

and

∫_{0}^{∞} ( cos tx − \frac{1}{1 + (tx)^{2}} )\frac{h_{j}(x)g_{j}(t, x)dx}{x}

= ∫_{0}^{∞} ( cos x − \frac{1}{1 + x^{2}} )\frac{h_{j}(x/|t|)g_{j}(t, x/|t|)dx}{x}

= CKc_{j}L(\frac{1}{|t|}) + o(L(\frac{1}{|t|})) as t → 0.

By lemma 5.7,

H̃_{g_{j}}(λ) = KH_{j}(λ) + o(L(λ)) as λ → ∞

where H_{j}(λ) := ∫_{0}^{λ} \frac{xh_{j}(x)dx}{1+x^{2}}, and t^{2}H_{j}(1/|t|) = o(L(1/|t|)) as t → 0 since H_{j} is slowly varying.

Putting everything together we obtain

∫_{0}^{∞}(1 + \frac{itx}{1 + x^{2}} − e^{itx})G_{t}^{1}(dx) + ∫_{0}^{∞}(1 − \frac{itx}{1 + x^{2}} − e^{−itx})G_{t}^{2}(dx)

= KL(\frac{1}{|t|})|t|(c_{1} + c_{2})π/2 − itL(1/|t|)(c_{1} − c_{2})CK

− itK(H̃_{1}(1/|t|) − H̃_{2}(1/|t|)) − it(γ_{t}^{1} − γ_{t}^{2}) + o(|t|L(\frac{1}{|t|}))

= L(\frac{1}{|t|})|t|(c_{1} + c_{2})Kπ/2 − itL(1/|t|)(c_{1} − c_{2})CK

− itK(H_{1}(1/|t|) − H_{2}(1/|t|)) − it(γ_{t}^{1} − γ_{t}^{2}) + o(|t|L(\frac{1}{|t|})).

Define

γ_{t} := γ̃_{t}^{1} − γ̃_{t}^{2} + ∫_{ℝ} \frac{x}{1 + x^{2}} ĝ_{t}(x)G(dx)

where

γ̃_{t}^{1} := ∫_{0}^{∞} ( ∫_{0}^{x} \frac{2u^{2}}{(1 + u^{2})^{2}} du)ĝ_{t}(x)G(dx),

γ̃_{t}^{2} := ∫_{−∞}^{0} ( ∫_{0}^{−x} \frac{2u^{2}}{(1 + u^{2})^{2}} du)ĝ_{t}(x)G(dx),

then

1 − λ(t) + itγ_{t} = it(γ̃_{t}^{1} − γ̃_{t}^{2}) + ∫ (1 + \frac{itx}{1 + x^{2}} − exp[itx])g_{t}^{r}(x)G(dx)

+ i ∫ (1 + \frac{itx}{1 + x^{2}} − exp[itx])g_{t}^{+}(x)G(dx) − i ∫ (1 + \frac{itx}{1 + x^{2}} − exp[itx])g_{t}^{−}(x)G(dx)

= c|t|L(1/|t|) + it(H_{1}(1/|t|) − H_{2}(1/|t|)) − it\frac{2βc}{π}CL(1/|t|) + o(|t|L(1/|t|))

where the constants are as in the theorem.

Finally, to complete the proof of theorem 5.1, we note that: γ_{t} ≡ 0 in case 0 < p < 1; in case 1 < p < 2,

γ_{t} := ∫_{ℝ} xĝ_{t}(x)G(dx) = γ + O(|t|) as t → 0;

and in case p = 1,

γ_{t} = ∫_{−∞}^{∞} (\frac{x}{1 + x^{2}} + sgn(x) ∫_{0}^{|x|} \frac{2u^{2}}{(1 + u^{2})^{2}} du)ĝ_{t}(x)G(dx)

= γ + O (\sqrt{|t|L(\frac{1}{|t|}}) as t → 0.

□

§6 Local limit theorems

Again, throughout this section, we let (X, B, m, T, α) be a mixing, probability preserving Gibbs-Markov map. Let

ϕ : X → ℝ

be Lipschitz continuous on each a ∈ α, with D_{α}ϕ := sup_{a∈α} D_{a}ϕ < ∞ and distribution G in the domain of attraction of a stable law with order 0 < p < 2 equivalently ([Fel], [G-K], [I-L], [Le]):

L_{1}(x) := x^{p}(1 − G(x)) = (c_{1} + o(1))L(x), L_{2}(x) := x^{p}G(−x) = (c_{2} + o(1))L(x)

as x → +∞ where L is a slowly varying function on ℝ_{+} and where c_{1}, c_{2} ≥ 0, c_{1} + c_{2} > 0. Throughout this section, we use notations from §4 and write:

N_{j}(t) := N_{j}(P_{t}) (j = 1, 2), λ(t) := λ(P_{t}).

Also, let ϕ_{n} := ∑_{k=0}^{n−1} ϕ ∘ T^{k} (n ≥ 1).

Theorem 6.1 (Distributional limit theorem). Under the above conditions,

\frac{ϕ_{n} − A_{n}}{B_{n}} ⟶^{D} X_{p}

where

nL(B_{n}) = B_{n}^{p}, A_{n} = \{ \begin{matrix}0 0 < p < 1, \\ γn 1 < p < 2, \\ γn + \frac{2n}{π}(H_{1}(B_{n}) − H_{2}(B_{n})) p = 1. \end{matrix}

Proof. We claim first that

n log λ(\frac{t}{B_{n}}) − it\frac{A_{n}}{B_{n}} → log g_{X_{p}}(t) as n → ∞.

This follows from theorem 5.1 as in the independent case.

Using theorem 4.1 (1),

∫_{X} e^{it(\frac{ϕn}{Bn} − \frac{An}{Bn})}dm = e^{−it \frac{An}{Bn}} ∫_{X} P^{n}(e^{it \frac{ϕn}{Bn}} )dm

= e^{−it \frac{An}{Bn}} ∫_{X} P n_{\frac{t}{Bn}} 1dm

= e^{−it \frac{An}{Bn}} λ(\frac{t}{B_{n}})^{n} ∫_{X} g(\frac{t}{B_{n}})dm + O(θ^{n}).

The theorem follows since g(s) →^{L} g(0) ≡ 1 as s → 0. □

Theorem 6.2 (Conditional lattice local limit theorem). Suppose that ϕ : X → ℤ is an aperiodic cocycle whose distribution is in the domain of attraction of X_{p} (p ∈ (0, 2)).

Let A_{n}, B_{n} be as in theorem 6.1, and suppose that k_{n} ∈ ℤ, \frac{k_{n}−A_{n}}{B_{n}} → κ ∈ ℝ as n → ∞, then

∥B_{n}P_{T^{n}}(1_{[ϕ_{n}=k_{n}]}) − f_{X_{p}}(κ)∥_{∞} → 0 as n → ∞,

and, in particular

B_{n}m([ϕ_{n} = k_{n}]) → f_{X_{p}}(κ) as n → ∞.

Proof. By theorem 4.1, ∃ δ > 0, θ ∈ (0, 1) such that ∀ |t| ≤ δ,

∥P_{t}^{n}1 − λ(t)^{n}g(t)∥_{L} = O(θ^{n}) ∀ |t| ≤ δ,

and that

∥P_{y}^{n}1∥_{L} = O(θ^{n}) ∀ δ ≤ |y| ≤ π.

By theorem 5.1 and since L is slowly varying at infinity, by possibly shrinking δ > 0, we can ensure in addition that

−Re log λ(t) ≥ \frac{c}{2}|t|^{p}L(\frac{1}{|t|}) ∀ |t| ≤ δ,

and that ∃ 0 < ϵ = ϵ(δ) such that

\frac{L(\frac{B_{n}}{|t|})}{L(B_{n})} ≥ |t|^{ϵ} ∀ n ≥ 1, |t| ≤ δB_{n}.

It follows that, uniformly on X,

2πB_{n}P_{T^{n}}(1_{[ϕ_{n}=k_{n}]}) = B_{n}P_{T^{n}}( ∫_{−π}^{π} e^{−itk_{n}}e^{itϕ_{n}}dt)

= B_{n} ∫_{−π}^{π} e^{−itk_{n}}P_{T^{n}}(e^{itϕ_{n}})dt

= B_{n} ∫_{−π}^{π} e^{−itk_{n}}P_{t}^{n}1dt

= B_{n} ∫_{|t|≤δ} e^{−itk_{n}}λ(t)^{n}g(t)dt + O(B_{n}θ^{n})

= ∫_{−δB_{n}}^{δB_{n}} e^{−it \frac{An}{Bn}} λ(\frac{t}{B_{n}})^{n}g(\frac{t}{B_{n}})e^{it( \frac{An−kn}{Bn})}dt + o(1)

= ∫_{−δB_{n}}^{δB_{n}} e^{−it \frac{An}{Bn}} λ(\frac{t}{B_{n}})^{n}e^{it( \frac{An−kn}{Bn})}dt + o(1)

→ ∫_{ℝ} g_{X_{p}}(t)e^{−iκt}dt

= 2πf_{X_{p}}(κ)

as n → ∞ by dominated convergence, since for |t| ≤ δB_{n},

|λ(\frac{t}{B_{n}})^{n}| ≤ e^{− \frac{c}{2} |t|^{p} ^{\frac{L( \frac{Bn}{|t|} )}{L(Bn)}}} ≤ e^{− \frac{c}{2} |t|^{p+ϵ}},

which latter function is integrable on ℝ. □

Theorem 6.3 (Conditional non-lattice local limit theorem). Suppose that ϕ : X → ℝ is an aperiodic cocycle whose distribution is in the domain of attraction of X_{p} (p ∈ (0, 2)).

Let A_{n}, B_{n} be as in theorem 6.1, let I ⊂ ℝ be an interval, and suppose that k_{n} ∈ ℤ, \frac{k_{n}−A_{n}}{B_{n}} → κ ∈ ℝ as n → ∞, then

B_{n}P_{T^{n}}(1_{[ϕ_{n}∈k_{n}+I]}) → |I|f_{X_{p}}(κ) as n → ∞

where |I| is the length of I, and in particular

B_{n}m([ϕ_{n} ∈ k_{n} + I]) → |I|f_{X_{p}}(κ) as n → ∞.

Proof. We use the method of Breiman (see [Bre]).

Suppose that h ∈ L^{1}(ℝ), ĥ ∈ L^{1}(ℝ), and that ĥ ≡ 0 off [−M, M].

Arguing as in the proof of theorem 6.2, we obtain δ > 0, and 0 < θ < 1 such that, uniformly on X:

B_{n}P_{T^{n}}(h(ϕ_{n} − k_{n})) = \frac{B_{n}}{2π} ∫_{−M}^{M} ĥ(x)P_{T^{n}}(e^{ix(ϕ_{n}−k_{n})})dx

= \frac{B_{n}}{2π} ∫_{−M}^{M} ĥ(x)e^{−ik_{n}x}P_{x}^{n}1dx

= \frac{B_{n}}{2π} ∫_{|x|≤δ} ĥ(x)e^{−ik_{n}x}λ(x)^{n}g(x)dx + O(B_{n}θ^{n})

= \frac{1}{2π} ∫_{|x|≤δB_{n}} ĥ(\frac{x}{B_{n}})e^{−i \frac{kn}{Bn} x}λ(\frac{x}{B_{n}}))^{n}g(\frac{x}{B_{n}})dx + o(1)

= \frac{1}{2π} ∫_{|x|≤δB_{n}} ĥ(\frac{x}{B_{n}})e^{−i \frac{kn}{Bn} x}λ(\frac{x}{B_{n}}))^{n}dx + o(1)

→ \frac{1}{2π} ∫_{ℝ} ĥ(0)g_{X_{p}}(x)e^{−iκx}dx

= ∫_{ℝ} h(x)dxf_{X_{p}}(κ)

by dominated convergence as again, for some ϵ > 0, ∀ |t| ≤ δB_{n}, |λ(\frac{t}{B_{n}})^{n}| ≤ e^{− \frac{c}{2} |t|^{p+ϵ}}, which latter function is integrable on ℝ.

Let k(x) = \frac{sin^{2} x}{x^{2}}, then k > 0, k ∈ L^{1}(m_{ℝ}) and k̂ has compact support.

It follows from theorem 10.7 in [Bre] that if U is a vague neighbourhood of m_{ℝ}, then ∃ η > 0 and t_{1}, …, t_{N} ∈ ℝ such that for μ a Radon measure on ℝ:

| ∫_{ℝ} e^{it_{j}x}k(x)dμ − ∫_{ℝ} e^{it_{j}x}k(x)dx| < η (1 ≤ j ≤ N) ⟹ μ ∈ U.

Thus, for h : ℝ → ℝ continuous with compact support,

B_{n}P_{T^{n}}(h(ϕ_{n} − k_{n})) → ∫_{ℝ} h(x)dxf_{X_{p}}(κ)

uniformly on X. The theorem follows from monotone approximation of 1_{I} by non-negative continuous functions with compact support. □

We conclude this section with a local limit theorem for processes with marginals in the domain of attraction of multidimensional symmetric p-stable distributions.

Definition.

A random vector X is called stable if for all a, b > 0 there are c > 0 and v ∈ ℝ such that aX + bX′ =^{d} cX + v where X′ is an independent copy of X.

In this case necessarily a^{p} + b^{p} = c^{p} for some 0 < p ≤ 2, and p is called the order of X.

The stable random vector X is called nondegenerate if its distribution is absolutely continuous on ℝ^{d}.

As is well known (see [S-T]) the random vector X = (X_{1}, …, X_{d}) has symmetric p-stable distribution

if and only if

E(e^{i⟨u,X⟩}) = e^{−c_{p,ν}(u)}.

Here c_{p,ν}(u) := ∫_{S^{d−1}} |⟨u, s⟩|^{p}ν(ds) where ν is a symmetric measure on S^{d−1} (called the spectral measure). It is known that X is nondegenerate iff the support of ν is not contained in any subspace of ℝ^{d} (equivalently c_{p,ν}(u) > 0 ∀ u ∈ ℝ^{d} \ {0}). In this case, we call the spectral measure nondegenerate.

Definition.

The distribution of the symmetric random vector Y = (Y_{1}, …, Y_{d}) is in the strict domain of attraction of a symmetric p-stable distribution (0 < p ≤ 2) if there are constants B_{n} (necessarily regularly varying with index p) such that

dist. \frac{S_{n}}{B_{n}} → dist. Z

where Z has symmetric p-stable distribution, and S_{n} = X^{(1)} + ⋯ + X^{(n)} where {X^{(1)}, …, X^{(n)}} are independent and distributed as X.

As is well known (see [A-D1)] the random vector X = (X_{1}, …, X_{d}) is in the strict domain of attraction of a symmetric p-stable distribution with spectral measure ν

if and only if there is a function L : ℝ_{+} → ℝ_{+}, slowly varying at ∞ such that

E(e^{it⟨u,X⟩}) = e^{−c_{p,ν}(u)|t|^{p}L( \frac{1}{|t|})(1+o(1))} as t → 0 ∀ u ∈ 𝕊^{d−1} \ \{0\}.

Lemma 6.4. Let (X, B, m, T, α) be an probability preserving Gibbs-Markov map, and let

ϕ : X → ℝ^{d}

be Lipschitz continuous on each a ∈ α and aperiodic.

If

E(e^{i \frac{⟨u,ϕn⟩}{B(n)}} ) → e^{−c_{p,ν}(u)}∀ u ∈ ℝ^{d}

where p ∈ (0, 2], c_{p,ν}(u) > 0 ∀ u ∈ ℝ^{d} and B(n) is 1/p-regularly varying at ∞, then

− log λ(u) ∼ \frac{c_{p,ν}(ũ)}{B^{−1}(\frac{1}{|u|})}

as u → 0 where ũ = |u|^{−1}u ∈ S^{d−1}.

Proof.

To see this note first that

E(e^{i⟨u,ϕ_{n}⟩}) = E(P_{u}^{n}1)

and therefore by theorem 4.1,

E(e^{i⟨u,ϕ_{n}⟩}) = λ(u)^{n}E(g(u)) + O(θ^{n})

uniformly on B(0, δ) as n → ∞, whence

E(e^{i \frac{⟨u,ϕn⟩}{B(n)}} ) − λ(u/B_{n})^{n} → 0

as n → ∞ uniformly in u ∈ S^{d−1}.

It follows from our assumption that

E(e^{i \frac{⟨u,ϕn⟩}{B(n)}} ) → e^{−c_{p,ν}(u)}

as n → ∞ uniformly in u ∈ S^{d−1}, whence

λ(u/B_{n})^{n} → e^{−c_{p,ν}(u)}

as n → ∞ uniformly in u ∈ S^{d−1}.

Since ν is nondegenerate, ∃ K ⊂ (0, 1) compact such that e^{−c_{p,ν}(u)} ∈ K ∀ u ∈ S^{d−1}. The function z ↦ log z is uniformly continuous on K, whence

−n log λ(u/B_{n}) → c_{p,ν}(u)

as n → ∞ uniformly in u ∈ S^{d−1}.

It follows that

− log λ(\frac{1}{B(n)}u) ∼ \frac{c_{p,ν}(u)}{n}

as n → ∞, uniformly in u ∈ S^{d−1}.

For s > 0, set N_{s} = [B^{−1}(\frac{1}{|s|})], then

s = \frac{1}{B(B^{−1}(\frac{1}{s}))} = \frac{1}{B(λ_{s}N_{s})} := \frac{μ_{s}}{B(N_{s})}

where λ_{s} → 1 as s → 0+ and hence also μ_{s} → 1 as s → 0+; and we now have that

− log λ(su) ∼ − log λ(\frac{μ_{s}}{B(N_{s})}u) ∼ \frac{c_{p,ν}(u)}{N_{s}} ∼ \frac{c_{p,ν}(u)}{B^{−1}(\frac{1}{s})}

as s → 0+ uniformly in u ∈ S^{d−1}; which proves the lemma. □

Theorem 6.5.

Let (X, B, m, T, α) be a mixing, probability preserving Gibbs-Markov map, and let G be a subgroup of ℝ^{d} of form G = A(ℝ^{k} × ℤ^{ℓ}) where k + ℓ = d and A ∈ GL(d, ℝ).

Suppose that

φ : X → G

is aperiodic, Lipschitz continuous on each a ∈ α, and that dist. φ is in the domain of attraction of a nondegenerate symmetric p-stable distribution with density f_{p} for some 0 < p < 2, then ∀ compact neighbourhood V with m_{G}(∂V) = 0, uniformly on X,

(B_{n})^{d}P_{T^{n}}(1_{[φ_{n}∈V]}) → f_{p}(0)m_{G}(V) uniformly on X as n → ∞.

Proof. It follows from lemma 6.4 that

− log λ(tu) = c_{p,ν}(u)|t|^{p}L(\frac{1}{|t|})(1 + o(1))

as t → 0 uniformly in u ∈ S^{d−1}.

We fix δ, ϵ > 0 and c > 0 such that

− log λ(tu) ≥ c|t|^{p}L(\frac{1}{|t|}) ∀ t < δ, u ∈ S^{d−1}

and

L(λ/r) ≥ r^{ϵ}L(λ) ∀ λ large, and 0 < r ≤ δλ.

We give a proof in the discrete case only, (the other being analogous and using the method of Breiman as in theorem 6.3).

P_{T_{ϕ}^{n}}(1_{X×{z}})(x, y) = P_{T^{n}}(1_{[ϕ_{n}(·)=z−y]})(x)
= ∫Ĝ \overline{γ}(y − z)P_{γ}^{n}1(x)dγ
= ∫_{B(0,δ)} \overline{γ}(y − z)λ(γ)^{n}g(γ)(x)dγ + O(θ^{n}).

Changing to polar coordinates,

∫_{B(0,δ)} \overline{(y − z)}(γ)λ(γ)^{n}g(γ)(x)dγ

= ∫_{S^{d−1}} ( ∫_{0}^{δ} r^{d−1}\overline{(y − z)}(ru)λ(ru)^{n}g(ru)(x)dr)dμ(u)

= \frac{1}{B_{n}^{d}} ∫_{S^{d−1}} ( ∫_{0}^{δB_{n}} r^{d−1}\overline{(y − z)}(ru/B_{n})λ(ru/B_{n})^{n}g(ru/B_{n})(x)dr)dμ(u)

where μ is Lebesgue measure on S^{d−1}.

For each u ∈ S^{d−1}, n ≥ 1 and r < δB_{n},

|λ(ru/B_{n})^{n}| ≤ e^{−c \frac{nr^{p}}{B^{p}_{n}} L( \frac{Bn}{r})} = e^{−c \frac{r^{p}L( \frac{Bn}{r} )}{L(Bn)}} ≤ e^{−cr^{p+ϵ}} ∀ n large..

The latter function is integrable and so by the dominated convergence theorem,

∫_{S^{d−1}} ( ∫_{0}^{δB_{n}} r^{d−1}\overline{(y − z)}(ru/B_{n})λ(ru/B_{n})^{n}g(ru/B_{n})(x)dr)dμ(u) ⟶
∫_{S^{d−1}} ( ∫_{0}^{∞} r^{d−1}e^{−r^{p}c_{p,ν}(u)}dr)dμ(u)
= ∫_{ℝ^{d}} e^{−c_{p,ν}(u)}dz = f_{p}(0).

□

§7 skew products

Let (X, B, m, T, α) be a mixing, probability preserving Gibbs-Markov map, and let G be a subgroup of ℝ^{d} of form G = A(ℝ^{k} × ℤ^{ℓ}) where k + ℓ = d and A ∈ GL(d, ℝ).

Suppose that

ϕ : X → G

is aperiodic, Lipschitz continuous on each a ∈ α and D_{α}ϕ := sup_{a∈α} D_{a}ϕ < ∞. The skew product (see [Sch]) is T_{ϕ} : X × G → X × G defined by

T_{ϕ}(x, g) = (T x, g + ϕ(x)).

Recall from [A3] that a conservative, ergodic, measure preserving transformation S of a σ-finite measure space (Y, C, ν) is called pointwise dual ergodic if there are constants a_{n} → ∞ such that

\frac{1}{a_{n}} ∑_{k=0}^{n−1} P_{S^{k}}f → ∫_{Y} fdν ∀ f ∈ L^{1}(ν).

The sequence of constants a_{n} is unique up to asymptotic equality and is called the return sequence of S and is denoted a_{n} ∼ a_{n}(S).

Theorem 7.1.

1) Either T_{ϕ} is totally dissipative, or T_{ϕ} is pointwise dual ergodic.

2) If G is discrete and T_{ϕ} is conservative, then T_{ϕ} is exact.

Proof.

1) Fix f ∈ L^{1}(G) such that f, f̂ ≥ 0, and f̂ ∈ C_{C}(Ĝ). Given ϵ > 0, fix δ(ϵ) > 0 such that for γ ∈ B(0, δ):

|g(γ)(x) − 1| < ϵ ∀ x ∈ X, & |γ(y) − 1| < ϵ ∀ y ∈ spt f̂.

Using theorem 4.1,

P_{T_{ϕ}^{n}}(1 ⊗ f)(x, y) = P_{T^{n}}(f(y − ϕ_{n}(·))(x)
= ∫_{spt f̂} \overline{γ}(y)f̂(γ)P_{γ}^{n}1(x)dγ
= ∫_{B(0,δ)} \overline{γ}(y)f̂(γ)P_{γ}^{n}1(x)dγ + O(θ^{n})
= ∫_{B(0,δ)} \overline{γ}(y)f̂(γ)λ(γ)^{n}g(γ)(x)dγ + O(θ^{n})
= (1 ± ϵ)^{2} ∫_{B(0,δ)} f̂(γ)λ(γ)^{n}dγ + O(θ^{n})
= (1 ± ϵ)^{2} ∫_{X} f(−ϕ_{n})dm + O(θ^{n}).

It follows that T_{ϕ} is totally dissipative or conservative according to whether the series

∑_{n=1}^{∞} ∫_{X} f(−ϕ_{n})dm

converges or diverges (respectively), and in case of divergence that

\frac{∑_{n=1}^{N} P_{T_{ϕ}^{n}}(1 ⊗ f)}{∑_{n=1}^{N} ∫_{X} f(−ϕ_{n})dm} → 1 a.e. as N → ∞.

Pointwise dual ergodicity follows from the Hurewicz ergodic theorem.

2) In case G is discrete, we have that T_{ϕ} is an aperiodic Markov fibred system with the Renyi property, and hence exact if conservative (by theorem 3.2 in [A-D-U]). □

Now suppose in addition that dist. ϕ is in the strict domain of attraction of a nondegenerate p-stable distribution for some 0 < p ≤ 2, specifically suppose that

\frac{ϕ_{n}}{B_{n}} →^{d.} X_{p}

where B_{n} > 0 and X_{p} is nondegenerate p-stable.

Theorem 7.2.

T_{ϕ} is conservative iff

∑_{n=1}^{∞} \frac{1}{B_{n}^{d}} = ∞.

In this case, T_{ϕ} is pointwise dual ergodic with return sequence

a_{n}(T_{ϕ}) ∼ f_{X_{p}}(0) ∑_{k=0}^{n} \frac{1}{B_{k}^{d}} .

Proof. Let V ⊂ G be a compact neighbourhood with m_{G}(∂V) = 0, then

P_{T_{ϕ}^{n}}(1 ⊗ 1_{V})(x, y) = P_{T^{n}}(1_{V} (y − ϕ_{n}(·)))(x) ∼ \frac{f_{X_{p}}(0)m_{G}(V)}{(B_{n})^{d}}

uniformly in x ∈ X and y ∈ any compact subset of G by theorem 6.5. The theorem evidently follows from this. □

We finish this section with some

Remarks about the periodic case.

Let (X, B, m, T, α) be a mixing Gibbs-Markov map with respect to the partition α and invariant probability m, and let ϕ : X → ℤ^{d} be Lipschitz continuous on each a ∈ α with Dϕ := sup_{a∈α} D_{a}ϕ < ∞.

By theorem 4.1, there are constants ϵ > 0, K > 0 and θ ∈ (0, 1); and functions λ : B(0, ϵ) → B_{ℂ}(0, 1), g : B(0, ϵ) → L such that

∥P_{t}^{n}h − λ(t)^{n}g(t) ∫_{X} hdm∥_{L} ≤ Kθ^{n}∥h∥_{L} ∀ |t| < ϵ, n ≥ 1, h ∈ L

and

∥g(t) − 1∥_{L} ≤ K(|t| + E(|e^{itϕ} − 1|)).

We consider T_{ϕ} for ϕ periodic (i.e. not aperiodic).

It follows from proposition 3.7 that ∥P_{t}∥_{Hom(L,L)} = |λ(t)| = 1 ∀ |t| small iff ϕ is cohomologous to a constant.

Now suppose that ϕ : X → ℤ^{d}, t_{0} ∈ 𝕋^{d}, t_{0} ≠ 0 and |λ(t_{0})| = 1. Let γ : X → 𝕋 be Lipschitz continuous and z ∈ 𝕋 such that χ_{t_{0}}(ϕ) = z\overline{γ}γ ∘ T. We have for small t ∈ 𝕋 that

P_{t_{0}+t}(f) = P (χ_{t}(ϕ)χ_{t_{0}}(ϕ)f) = P (χ_{t}(ϕ)z\overline{γ}γ ∘ T f) = zγP_{t}(\overline{γ}f)

whence λ(t_{0} + t) = zλ(t) and

∥P_{t_{0}+t}^{n}h − γz^{n}λ(t)^{n}g(t) ∫_{X} \overline{γ}hdm∥_{L} ≤ Kθ^{n}∥h∥_{L} ∀ |t| < ϵ, n ≥ 1. h ∈ L

Assume now (as in §5 and §6) that |λ(t)| < 1 ∀ 0 < |t| < ϵ.

It follows that

𝔔 := {t ∈ 𝕋^{d} : χ_{t}(ϕ) is cohomologous to a constant}

is a finite subgroup of 𝕋^{d}. Write 𝔔 = {0, a, 2a, … (q − 1)a}, then ∃ p ∈ ℕ such that z(ja) = e^{2πi \frac{jp}{q}}.

Fix γ : X → 𝕋 such that χ_{a}(ϕ) = z(a)\overline{γ}γ ∘ T. It follows that χ_{ja}(ϕ) = z(ja)\overline{γ}^{j}γ^{j} ∘ T, whence γ^{q} ∘ T = γ^{q} and γ^{q} is constant. We may assume without loss of generality that γ^{q} ≡ 1. Thus

P^{k}(1_{[ϕ_{k}=0]}) = ∫_{𝕋^{d}} P_{t}^{k}1dt = ∑_{j=0}^{q−1} γ^{j}z(a)^{jk} ∫_{X} \overline{γ}^{j}dm ∫_{B(0,ϵ)} λ(t)^{k}g(t)dt + o(ρ^{k}).

Writing k = qκ + ℓ (κ ≥ 0, 0 ≤ ℓ < q),

P^{qκ+ℓ}(1_{[ϕ_{qκ+ℓ}=0]}) =^{±o(ρ^{κ})} ∑_{j=0}^{q−1} γ^{j}z(a)^{j(qκ+ℓ)} ∫_{X} \overline{γ}^{j}dm ∫_{B(0,ϵ)} λ(t)^{qκ+ℓ}g(t)dt

= ∑_{j=0}^{q−1} γ^{j}z(a)^{jℓ} ∫_{X} \overline{γ}^{j}dm ∫_{B(0,ϵ)} λ(t)^{qκ+ℓ}g(t)dt

= ∫_{X} ( ∑_{j=0}^{q−1}(γz(a)^{ℓ}\overline{γ}(y))^{j}dm(y)) ∫_{B(0,ϵ)} λ(t)^{qκ+ℓ}g(t)dt

= qm({y ∈ X : γ(y) = γz(a)^{ℓ}}) ∫_{B(0,ϵ)} λ(t)^{qκ+ℓ}g(t)dt

Suppose that the distribution of ϕ is in the strict domain of attraction of a p-stable law with normalising constants B_{n}, then by theorem 5.1 and the proof of theorem 6.2, B_{n} ∫_{B(0,ϵ)} P_{t}^{n}1dt ≈ B_{n}u_{n} → f_{X_{p}}(0) where u_{n} := ∫_{B(0,ϵ)} λ(t)^{n}dt.

Noting that ∑_{ℓ=0}^{q−1} qm({y ∈ X : γ(y) = γz(a)^{ℓ}}) > 0, ∑_{k=0}^{∞} P^{k}1_{[ϕ_{k}=0]} converges, or diverges a.e. on X together with ∑_{κ=0}^{∞} u_{qκ} and in case of divergence,

∑_{k=0}^{qn−1} P^{k}1_{[ϕ_{k}=0]} = ∑_{κ=0}^{n−1} ∑_{ℓ=0}^{q−1} P^{qκ+ℓ}(1_{[ϕ_{qκ+ℓ}=0]})

∼ ∑_{κ=0}^{n−1} u_{qκ} ∑_{ℓ=0}^{q−1} qm({y ∈ X : γ(y) = γz(a)^{ℓ}}).

We have established

Theorem 7.3. Suppose that ϕ : X → ℤ^{d} is Lipschitz continuous on each a ∈ α and D_{α}ϕ < ∞. Suppose that u_{n} := ∫_{B(0,ϵ)} λ(t)^{n}dt is regularly varying as n → ∞, then T_{ϕ} is either totally dissipative or conservative according to whether ∑_{n=0}^{∞} u_{n} converges or diverges (respectively).

If T_{ϕ} is conservative then each of its ergodic components is pointwise dual ergodic with asymptotic type a_{n}(T_{ϕ}) ∝ ∑_{k=0}^{n} u_{k}.

§8 An illustration

Consider T : ℝ → ℝ defined by T x = x + v(x) where v : ℝ → ℝ is: odd (v(−x) = −v(x)), 1-periodic (v(x+1) = v(x)), and piecewise C^{2}, increasing and onto.

Noting that T (x + 1) = T (x) + 1 we can write I = [−\frac{1}{2}, \frac{1}{2}] and

ℝ ≅ I × ℤ by (x, n) ⟶ n + x.

T is conjugate to

τ_{ϕ} : I × ℤ → I × ℤ defined by τ_{ϕ}(x, n) = (τx, n + ϕ(x))

where τ : I → I is defined by

τx = \{(x + v(x)) + \frac{1}{2}\} − \frac{1}{2},

and ϕ : I → ℤ is defined by

ϕ(x) = [(x + v(x)) + \frac{1}{2}].

Here { · }, and [ · ] denote fractional and integral parts respectively.

Evidently the map τ : I → I is piecewise onto.

Theorem 8.1.

Suppose that v′ ≥ c > 0 and that sup \frac{|v′′|}{(1+v′)^{2}} < ∞, then there is a probability p ∼ Lebesgue measure such that τ is a piecewise onto Gibbs-Markov map, and ϕ : (−\frac{1}{2}, \frac{1}{2}) → ℤ is aperiodic.

Proof. We have that τ′ = 1 + v′ ≥ 1 + c > 1, and τ′′ = v′′, whence τ is an expanding, piecewise onto interval map satisfying Adler’s condition (see example 2 in §1). As in example 2, τ is a (piecewise onto) Gibbs-Markov map.

There is an invariant probability p ∼ Lebesgue measure with Lipschitz continuous logdensity, whence by proposition 1.1, τ is a (piecewise onto) Gibbs-Markov map with respect to p (see e.g. lemma 2.1 in [A-D-U]).

Since ϕ : I → ℤ is onto, its aperiodicity follows from corollary 3.2. □

Corollary 8.2.

1) ∃ μ ∼ Lebesgue measure with periodic, Lipschitz continuous log-density.

2) Either T is totally dissipative, or T is pointwise dual ergodic and exact.

Proof. By theorems 7.1 and 8.1. Note that μ(A) = ∑_{n∈ℤ} p(A+n) where p is the invariant measure for τ. □

44 JON. AARONSON AND MANFRED DENKER

Theorem 8.3. Suppose that

m([v ≥ t]) ∼ \frac{L(t)}{t^{p}} as t → ∞

where L is slowly varying at ∞ and p ∈ (0, 2) and set

B_{n}^{p} = nL(B_{n}),

then T is conservative iff ∑_{n=1}^{∞} \frac{1}{B_{n}} = ∞

and in this case

a_{n}(T) ∝ ∑_{k=1}^{n} \frac{1}{B_{k}} .

Proof. By theorem 7.2. □ Example. Let a > 0 and

v(x) = a tan πx,

then

tan′ x = \frac{1}{cos^{2} x} = 1 + tan^{2} x, tan′′ x = 2 tan x(1 + tan^{2} x),

whence

tan′ x ≥ 1, \frac{| tan′′ x|}{(1 + a tan′ x)^{2}} ≤ \frac{2| tan x|(1 + tan^{2} x)}{(1 + a(1 + tan^{2} x))^{2}} ≤ M < ∞

and v satisfies the conditions of theorem 8.1.

Consider the transformation T with its invariant measure μ. The p-distribution of ϕ is symmetric round 0, and

|a tan πx − ϕ(x)| ≤ 1 ∀ x ∈ I

so

p([ϕ > y]) ∼ p(\{x : tan πx > y/a\}) ∼ \frac{p′(1/2)}{π} ∫_{y/a}^{∞} \frac{dx}{1 + x^{2}} ∝ \frac{1}{y} .

It follows that T is exact and pointwise dual ergodic with return sequence a_{n}(T) ∝ log n.

It is known that μ= Lebesgue measure, and it can be seen from theorem 7.2 that a_{n}(T) ∼ \frac{log n}{aπ}.

These examples were first treated in [A1], where this was shown using complex methods.

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Aaronson: School of Mathematical Sciences, Tel Aviv University, 69978 Tel Aviv, Israel. E-mail: aaro@math.tau.ac.il

Denker: Institut für Mathematische Stochastik, Universität Göttingen, Lotzestr. 13, 37083 Göttingen, Germany E-mail: denker@namu01.gwdg.de