THE DERIVATIVES OF EQUILIBRIUM STATES

Gonzalo Contreras Matemática. PUC-Rio. Rua Marquês de São Vicente, 225 22453-900 - Rio de Janeiro - Brasil

ABSTRACT: We find some estimates for the derivatives of equilibrium states of subshifts of finite type. We prove the differentiability (with respect to the potential) of integrals of certain discontinuous functions for the equilibrium state of a potential.

To the memory of Ricardo Mañé.

Introduction

In this paper we are interested in certain singular integrals with respect to equilibrium states of a subshift of finite type.

Let σ : Σ ↩ be a topologically transitive subshift of finite type endowed with one usual metric (e.g. diameter(n-cylinder) = 2^{−n}), let ϕ : Σ → ℝ be a Hölder continuous function and let μ_{ϕ} be its equilibrium state.

Let K ⊆ Σ be a compact subset such that μ_{ϕ}(ϵ-neighbourhood of K) ≤ A ϵ^{α}, A, α > 0. Consider a measurable function L : Σ → ℝ having a singular set K of order |L(x)| ≤ B |log d(x, K)|, where B > 0 is a constant and d(·, ·) is the distance in Σ. Then, using the condition on μ_{ϕ} and K, one can prove that L is μ_{ϕ}-integrable. On the other hand, it is known that the map C^{β}(Σ, ℝ) ∋ ϕ ↦ μ_{ϕ} ∈ (C^{β}(Σ, ℝ))^{∗} is real analytic. We will prove (theorem B) that, if moreover, L has local Hölder

Typeset by AMS-TEX

research supported by CNPq, Brasil

e-mail: gonzalo@mat.puc-rio.br; Fax: (55) (21) 274 41 97

constants D(x), |L(x) − L(y)| ≤ D(x) d(x, y)^{γ}, such that D(x) ≤ C d(x, K)^{−δ}, C, γ, δ > 0, then the map C^{β}(Σ, ℝ) ∋ ϕ ↦ ∫ L dμ_{ϕ} ∈ ℝ is C^{∞}.

In order to prove theorem B, one has to show that D_{ϕ}(μ_{ϕ}(L)) exists. For this we need estimates on the derivative D_{ϕ}(μ_{ϕ}(1_{C_{n}})), where 1_{C_{n}} is the characteristic function of an n-cylinder. Observe that one can not expect to have a bound |(D_{ϕ} μ_{ϕ}(C_{n}))(φ)| = |(D_{ϕ} μ_{ϕ}) (φ) · 1_{C_{n}}| ≤ A ∥φ∥ |μ_{ϕ}(C_{n})| (where the first equality is proven in (6)), because for ϕ_{t} = ϕ_{0} + t φ, we would have an estimate |μ_{ϕ_{t}}(C_{n})| ≤ exp(A ∥φ∥ t) μ_{ϕ_{0}}(C_{n}), implying the absolute continuity μ_{ϕ_{t}} ≪ μ_{ϕ_{0}}, which is false. The Theorem A below is the following estimate for n-cylinders C_{n}:

| (D_{ϕ}^{k} μ_{ϕ}(C_{n})) (φ, …^{\underline{k}}, φ) | ≤ n^{k} ∥φ∥^{k} A(ϕ, k) μ_{ϕ}(C_{n}) .

where ∥ ∥ is the α-Hölder norm. We show here two applications of theorem B.

a. One dimensional dynamics.

The first application is in 1-dimensional dynamics. Consider the map f : [−1, 1] ↩, f(x) = 1−2x^{2}. This map is conjugated to the Tent map g : [−1, 1] ↩, g(y) = 1 − 2 |y|. The conjugacy h : [−1, 1] → [−1, 1], f ∘ h = h ∘ g is given by h(y) = sin (\frac{π}{2} y). The full 2-shift is semiconjugated to the Tent map by an α-Hölder semiconjugacy k : Σ_{2} → [−1, 1], for some 0 < α < 1. The function F : [−1, 1] → ℝ, F (p) = log |f′(p)| has at p = 0 a singularity of order |F (x)| ∼ − log |x|, and has local Lipschitz constants of order ∼ F′(x) ∼ 1/|x|. Since h is Lipschitz, if we choose K = k^{−1}{0}, L : Σ_{2} → ℝ, L(x) = F (h ∘ k(x)), then L has a singularity at K of order L(x) ∼ − log d(x, K)^{α} ∼ − log d(x, K) and local α-Hölder constants of order ∼ 1/d(x, K)^{α}. In particular the pair (K, L) satisfies the hypothesis of theorem B. If ϕ : [−1, 1] → ℝ is α-Hölder, then ψ = ϕ ∘ h ∘ k : Σ_{2} → [−1, 1], is β-Hölder for some β = β(α), and the map C^{α}(Σ_{2}, ℝ) → C^{β}(Σ_{2}, ℝ) : ϕ ↦ ψ is real analytic. We get that the Lyapunov exponents of equilibrium states of f depend smoothly with respect to the potential ϕ, i.e.

C^{α}([−1, 1], ℝ) ∋ ϕ ↦ λ_{ϕ} := ∫ log |f′(p)| dμ_{ϕ}(p)

is C^{∞}, where μ_{ϕ} is the equilibrium state for (ϕ, f). b. Average linking numbers.

The second application is to average linking numbers for hyperbolic flows. Let Λ ⊆ S^{3} be a hyperbolic basic set of a flow φ_{t} in S^{3}. Let P O(t) be the set of periodic orbits of φ_{t}|_{Λ} with period ≤ t. Denote by ℓ(γ, η) the linking number of the knots γ and η in S^{3}. Define

L(s, t) := ∑ {ℓ(γ, η) | γ ∈ P O(s) , η ∈ P O(t) , γ ≠ η }

In [3] we proved that identifying S^{3} ≈ ℝ^{3} ∪ {∞}, with ∞ ∉ Λ, we have that

L(Λ) =^{def} lim_{s,t→∞} \frac{1}{st}L(s, t) = ∫_{Λ×Λ} 𝕃(x, y) d(μ × μ)(x, y)

where μ is the measure of maximal entropy of φ_{t}|_{Λ}, and

𝕃(x, y) = −\frac{1}{vol(S^{3})} \frac{F (x) × F (y)}{∥x − y∥} · \frac{(x − y)}{∥x − y∥^{2}}

where × is the vector product in ℝ^{3} ≈ S^{3} − {∞}, · is the inner product in ℝ^{3} and F is the vectorfield of φ_{t}. We show now that there exists a neighbourhood U of F with the C^{3} metric such that the map U ∋ G ↦ L(Λ_{G}) ∈ ℝ is C^{∞}, where Λ_{G} is the continuation of Λ for the vectorfield G.

Using the Taylor expansion of F one can see that the factor \frac{F (x)×F (y)}{∥x−y∥} is bounded in a neighbourhood of the diagonal x = y, therefore 𝕃(x, y) has order \frac{1}{∥x−y∥} near the diagonal. By dimension arguments, it is seen in [3] that 𝕃(x, y) ∈ L^{1}(μ × μ).

Using a Markov partition, it is shown in [2] that there exists a semiconjugacy π of a suspension S(Σ, τ_{F}) of a subshift of finite type Σ onto Λ, where τ_{F} : Σ →]0, +∞[ is Hölder continuous and S(Σ, τ_{F}) is the quotient space

S(Σ, τ) := {(x, t) | x ∈ Σ , 0 ≤ t ≤ τ(x) } /_{≡}

with the equivalence relation (x, τ(x)) ≡ (σ(x), 0), where σ : Σ ↩ is the shift map. The lift of the measure of maximal entropy μ to the suspension is (π^{∗}μ) = (∫ τ_{F} dν)^{−1} ν × λ where λ is the Lebesgue measure on [0, +∞[ and ν is the equilibrium state of the function ϕ_{F} (x) = h_{top}(F) τ_{F} (x), where h_{top}(F) is the topological entropy of φ|_{Λ}.

Define ℓ : Σ × Σ → ℝ, by

ℓ(x, y) = ∫_{0}^{τ_{F} (x)} ∫_{0}^{τ_{F} (y)} 𝕃 (φ_{s}(π x), φ_{t}(π y)) ds dt

then

L(Λ) = \frac{1}{(∫ τ_{F} dν)^{2}} ∫_{Σ×Σ} ℓ(x, y) d(ν_{F} × ν_{F})

It can be proved (cf. [3b]) that the function ℓ(x, y) has order − log d(x, y) near the diagonal x = y on Σ × Σ and has local Lipschitz constants of order 1/d(x, y). The structural stability for hyperbolic basic sets and the topological invariance of the linking number yields that when we change the vectorfield on a neighbourhood of F, the new average linking can be calculated using the same shift and the same function ℓ. The corresponding orbits under the topological equivalence will have the same linking numbers but their periods will be different. In general the new measures ν_{G} will be singular with respect to ν_{F}.

In [4] it is proven that if U is a small neighbourhood of the vectorfield F_{0} endowed with the C^{2}-metric, then the maps U ∋ F ↦ ϕ_{F} ∈ C^{α}(Σ, ℝ), and U ∋ F ↦ ∫ τ_{F} dν_{F}, are real analytic. So we need to see that U ∋ F ↦ ∫_{Σ×Σ} ℓ d(ν_{F} × ν_{F}) is C^{∞}. Here we apply theorem B to K = {(x, y) ∈ Σ × Σ|x = y} and the function ℓ. For this we observe that the product map σ × σ : Σ × Σ ↩, (σ × σ)(x, y) = (σ(x), σ(y)) is also a subshift of finite type, the measure ν_{F} × ν_{F} is the equilibrium state of ψ_{F} (x, y) := ϕ_{F} (x)+ϕ_{F} (y) for the product shift. Moreover since the measure ν_{F} satisfies a uniform estimate ν_{F} (ϵ-neighbourhood of K) ∼ ϵ^{β} with β > 0, one gets that (ν_{F} × ν_{F} )(ϵ-neighbourhood of K) ∼ ϵ^{β}. Finally, if one constructs the Markov partition using small embedded differentiable transverse sections to the flows, the projection of ℓ, J(p, q) := ℓ(x, y), where πx = p, πy = q, p, q ∈(smooth transversal), is differentiable on the transversal sections and |J| ∼ − log ∥p − q∥, ∥D J∥ ∼ \frac{1}{∥p−q∥}. This gives the required conditions on theorem B.

I want to thank the referee for the detailed suggestions of improvements in the exposition.

Statements of the Results.

Given a matrix A ∈ {0, 1}^{ℓ×ℓ} such that for all 0 ≤ i, j ≤ ℓ there exists M = M(i, j) > 0 such that A_{ij}^{M} > 0, consider the (topologically transitive) subshift of finite type σ: Σ ↩ (resp. σ_{+}: Σ^{+} ↩) where (σ(x))_{i} = x_{i+1},

Σ: = {x = (x_{i})_{i=−∞}^{+∞} ∈ {1, …, ℓ}^{ℤ} | A(x_{i}, x_{i+1}) = 1, ∀i ∈ ℤ}
Σ^{+}: = {x = (x_{i})_{i=0}^{+∞} ∈ {1, …, ℓ}^{ℕ} | A(x_{i}, x_{i+1}) = 1, ∀i ∈ ℕ}

Endow Σ (resp. Σ^{+}) with the metric d(x, y) = b^{⟨x,y⟩} for some fixed 0 < b < 1 and where

⟨x, y⟩: = max({0} ∪ {k | x_{i} = y_{i}, ∀|i| ≤ k}).

For x ∈ Σ (resp. Σ^{+}) let C_{n}(x) be the n−cylinder containing x:

C_{n}(x): = {y ∈ Σ | x_{i} = y_{i}, ∀|i| ≤ n}, (resp. 0 ≤ i ≤ n).

For 0 < α < 1 let C^{α}(Σ, ℝ) (resp. C^{α}(Σ^{+}, ℝ)) be the Banach space of α−Hölder continuous functions on Σ (resp. Σ^{+}).

C^{α}(Σ, ℝ): = {ϕ: Σ → ℝ | ∃K > 0, |ϕ(x) − ϕ(y)| < Kd(x, y)^{α}}

with the norm ∥ϕ∥: = |ϕ|_{0} + |ϕ|_{α}, where

|ϕ|_{0}: = sup_{x∈Σ} |ϕ(x)|, |ϕ|_{α}: = sup_{d(x,y)≠0} \frac{|ϕ(x) − ϕ(y)|}{d(x, y)^{α}}

for some fixed a > 0.

It is known (cf. [6]) that the map ϕ ↦ μ_{ϕ} from C^{α}(Σ, ℝ) into the dual space (C^{α}(Σ, ℝ))^{∗} (with the dual norm) is real analytic. For p ∈ Σ, n ≥ 0, let 1_{C_{n}(p)} be the characteristic function of C_{n}(p). Since 1_{C_{n}(p)} is α−Hölder continuous, it follows that the map C^{α}(Σ) → ℝ: ϕ ↦ μ_{ϕ}(C_{n}(p)) is real analytic.

Theorem A.

(a) Let Σ be a topologically transitive two-sided subshift of finite type, and let α > 0, ϕ_{0} ∈ C^{α}(Σ, ℝ). Then there exists a neighbourhood U of ϕ_{0} and D = D(U) > 0 such that

|(D_{ϕ}^{(k)}(μ_{ϕ})(φ_{1}, …, φ_{k})(1_{C_{n}(p)})| ≤ n^{k} k! D(U)^{k} ∥φ_{1}∥ ⋯ ∥φ_{k}∥ μ_{ϕ}(C_{n}(p)) ,

for all n ≥ 0, k ≥ 0, p ∈ Σ, ϕ ∈ U and φ_{1}, …, φ_{k} ∈ C^{α}(Σ, ℝ), where ∥φ∥ is the β−Hölder norm of φ.

(b) The same estimate holds for one-sided topologically transitive subshifts of finite type.

Theorem A will be proven at the end of the paper. In the following theorem we can think on Q_{n} as Q_{n}: = ⋃_{x∈K} C_{n}(x), where K ∈ Σ is a subset of measure 0.

Theorem B. Let Σ be a topologically transitive two sided subshift of finite type. Let ϕ ∈ C^{α}(Σ, ℝ), α > 0, and let μ_{ϕ} be the equilibrium state for ϕ. Let (Q_{n})_{n≥0} be a collection of subsets of Σ such that ∀n ≥ 0, Q_{n+1} ⊂ Q_{n}; Q_{n} is a (disjoint) union of n−cylinders C_{n}(x) and

(a) lim sup_{n→+∞} \frac{1}{n} log μ_{ϕ}(Q_{n}) < 0

Let L: (Σ \ K) → ℝ, K ⊂ ⋂_{n} Q_{n}, be a function such that there exist A, B > 0 and γ > 0 which satisfy

(b) |L(x)| ≤ Bn if x ∉ Q_{n}

(c) |L(x) − L(y)| ≤ A^{n}d(x, y)^{γ} if x, y ∈ Σ \ Q_{n}. Then there exists a neighbourhood U of ϕ in C^{α}(Σ, ℝ) such that for all φ ∈ U the integral μ_{φ}(L): = ∫ L dμ_{φ} exists and the mapping U → ℝ: φ ↦ μ_{φ}(L) is C^{∞}.

Proof of Theorem B

Here we prove theorem B using theorem A. We need the following result.

1. Proposition. Let Σ be a two-sided subshift of finite type. Suppose that we have (a) λ a probability on Σ.

(b) μ_{k} ∈ (C^{α}(Σ, ℝ))^{∗} such that there exists k ≥ 0 and D(k) > 0 such that

∀p ∈ Σ, ∀n ≥ 0: |μ_{k}(1_{C_{n}(p)})| ≤ D(k) n^{k} λ(C_{n}(p)) .

(c) A subset K ⊂ Σ such that if Q_{n}: = ⋃_{x∈K} C_{n}(x) then

lim sup_{n→+∞} \frac{1}{n} log λ(Q_{n}) ≤ −c < 0.

(d) L: (Σ \ K) → [0, +∞[ a non-negative function such that there exist A, B ≥ 1 such that

L(x) ≤ B n if x ∉ Q_{n}
|L(x) − L(y)| ≤ A^{n} d(x, y)^{γ} if x, y ∈ Σ \ Q_{n} .

Then there exists a sequence (g_{n}) of simple non-negative functions which are locally constant on n-cylinders, such that

∑ g_{n} ↑ L outside ⋂_{m} Q_{m} ⊃ K

∑_{n} |μ_{k}(g_{n})| ≤ D(k) G(k) ,

with G(k) > 0, in particular ∑_{n} |μ_{k}(g_{n})| < ∞.

Proof. For simplicity we prove the proposition only for B = 1. Observe that μ_{k} is not necessarily positive and that it can only be applied to Hölder continuous functions. Let D_{N} = C_{N}(p) be an N-cylinder in Q_{N−1}−Q_{N}. Let R: = (\frac{log A}{−γ log b})+1 and choose n = n(N) ∈ ℕ such that

(\frac{logA}{−γ log b}) N < n ≤ R N

so that A^{N}b^{γn} < 1. Let

g_{n}^{D_{N}} (x) : = ∑_{E_{n}} (min_{z∈E_{n}}L(z)) 1_{E_{n}}(x)

where the sum is over all the n-cylinders E_{n} such that E_{n} ⊂ D_{N}. Inductively, for m ≥ 0 let

g_{n+m+1}^{D_{N}}: = ∑_{E_{n+m+1}⊂D_{N}} ( min_{E_{n+m+1}}L) 1_{E_{n+m+1}} − ∑_{r=0}^{m} g_{n+r}^{D_{N}}

where the sum in on all the (n + m − 1)−cylinders E_{n+m+1} ⊂ D_{N}. The g_{m}^{D_{N}} ’s are simple functions of the form

g_{m}^{D_{N}} = ∑_{E_{m}^{r}⊂D_{N}} a_{r} 1_{E_{m}^{r}}.

(where a_{r} ≥ 0 and the E_{m}^{r} are m−cylinders) and hence γ−Hölder continuous. Therefore

μ_{k}(g_{m}^{D_{N}}) = ∑_{E_{m}^{r}⊂D_{N}} a_{r} μ_{k}(1_{E_{m}^{r}}).

Since for x ∈ D_{N}, L(x) ≤ N, we have that

(1)  |μ_{k}(g_{n(N)}^{D_{N}})| ≤ D(k) ∑_{E_{n}⊂D_{N}} N n^{k}λ(E_{n}) ≤ D(k) R^{k}N^{k+1}λ(D_{N})

Let E_{n+m−1} ⊂ D_{N} be an (n + m − 1)-cylinder in D_{N}. Since

var_{E_{n+m−1}}L ≤ A^{N} b^{γ(n+m−1)}, we have that

g_{n+m}^{D_{N}}(x) ≤ A^{N} b^{γ(n+m−1)}, ∀x ∈ D_{N}, ∀m ≥ 1.

and then

|μ_{k}(g_{n+m}^{D_{N}})| ≤ D(k) A^{N} b^{γ(n+m−1)} ∑_{E_{n+m}⊂D_{N}}(m + n)^{k} λ(E_{m+n})
≤ D(k) b^{−γ} b^{γm} (m + n)^{k} λ(D_{N})

because A^{N}b^{γn} < 1. Since

∑_{m=1}^{∞} b^{γm} (m + n)^{k} = ∑_{m=1}^{n} b^{γm} (m + n)^{k} + ∑_{m=n}^{∞} b^{γm} (m + n)^{k}

≤ 2^{k} n^{k} (\frac{1}{1 − b^{γ}} ) + ∑_{m=1}^{∞} b^{γm} · 2^{k} m^{k} ,

writing S(k): = ∑_{m=1}^{∞} b^{γm} m^{k}, we have that

∑_{m=1}^{∞} |μ_{k}(g_{n(N)+m}^{D_{N}})| ≤ D(k) (\frac{b^{−γ}}{1 − b^{γ}} ) 2^{k} (R^{k} N^{k} + S(k)) λ(D_{N})

from this and (1) we get

∑_{m=0}^{∞} |μ_{k}(g_{n(N)+m}^{D_{N}})| ≤ D(k) (\frac{b^{−γ}}{1 − b^{γ}} ) (2R)^{k} (S(k) + 2N^{k+1}) λ(D_{N}).

For each N ≥ 1 consider a partition P of Q_{N−1} \ Q_{N} into N-cylinders D_{N} and the corresponding (g_{n(N)+m}^{D_{N}})_{m≥0}, we get

∑_{D_{N} ∈P} ∑_{i≥n(N)} |μ_{k}(g_{i}^{D_{N}})| ≤ D(k) (\frac{b^{−γ}}{1 − b^{γ}} ) (2R)^{k} (S(k) + 2 N^{k+1}) λ(Q_{N−1} \ Q_{N})

Define g_{i} := ∑_{N with n(N)≤i} g_{i}^{D_{N}}. Since ∑_{N} λ(Q_{N−1} \ Q_{N}) ≤ λ(Σ) = 1 and λ(Q_{N−1} \ Q_{N}) ≤ λ(Q_{N−1}), we have that

(2)  ∑_{i} |μ_{k}(g_{i})| ≤ D(k) (\frac{b^{−γ}}{1 − b^{γ}} ) (2R)^{k} (S(k) + 2 ∑_{N} N^{k+1}λ(Q_{N−1}))

where the series F (k) := 2 ∑_{N} N^{k+1}λ(Q_{N−1}) is convergent by hypothesis (c). □

Given ϕ ∈ C^{α}(Σ, ℝ), let P (ϕ) be its topological pressure (cf. [7]).

Proof of theorem B. It is enough to prove theorem B for B = 1 and L nonnegative. There are constants A_{0}, B_{0} > 0, uniform on a neighbourhood of U_{0} of ϕ (cf. [1]) such that for any n-cylinder C_{n}(p) in Σ and all ψ ∈ U_{0}, we have

(3)  μ_{ψ}(C_{n}(p)) ∈ [A_{0}, B_{0}] exp(−2n P (ψ) + S_{n}ψ(p)) ,

where S_{n}ψ(x) := ∑_{k=−n}^{n} ψ(σ^{k}(x)). It is easy to see from the definition of pressure that |P (ϕ + φ) − P (ϕ)| ≤ |φ|_{0}. For ψ = ϕ + φ ∈ U_{0}, we have

0 ≤ μ_{ϕ+φ}(C_{n}(p)) ≤ B_{0} exp(−2n P (ϕ) + S_{n}ϕ(p)) exp(4 |φ|_{0} n)
≤ A_{0}^{−1} B_{0} exp(4 |φ|_{0} n) μ_{ϕ}(C_{n}(p)).

Since Q_{n} is a disjoint union of n-cylinders, we have

lim sup_{n→+∞} \frac{1}{n} log μ_{ϕ+φ}(Q_{n}) ≤ 2 |φ|_{0} + lim sup_{n→+∞} \frac{1}{n} log μ_{ϕ}(Q_{n}) < 0

if |φ|_{0} is sufficiently small. Choose a neighbourhood U_{1} of ϕ and c > 0 such that

(4)  lim sup_{n→+∞} \frac{1}{n} log μ_{ψ}(Q_{n}) < −c < 0

for all ψ ∈ U_{1}.

Using Theorem A(a) and (4), we can apply Proposition 1 to L, λ = μ_{ϕ}, and μ_{k} = (D^{(k)}μ_{ϕ})(φ_{1}, …, φ_{k}) (with D(k) = k! D(U_{2})^{k} ∥φ_{1}∥ ⋯ ∥φ_{k}∥), and a neighbourhood U_{2} ⊂ U_{1}. It follows that there exists a decomposition L = ∑_{n} g_{n} and for any r ≥ 0 a number H(r) > 0, constant on U_{2} and so that

(5)  ∑_{n} |(D(r) ; \overline{ψ}μ_{ψ})(φ_{1}, …, φ_{r}) · g_{n}| ≤ H(r) ∥φ_{1}∥ ⋯ ∥φ_{r}∥

for all φ_{1}, …, φ_{r} in C^{α}(Σ, ℝ) and \overline{ψ} ∈ U_{2}.

Now we prove that for all k ≥ 0, one has

(6)  (D_{ϕ}^{(k)}μ_{φ})(φ_{1}, …, φ_{k}) · g_{n} = (D_{ϕ}^{(k)}μ_{ϕ}(g_{n})) (φ_{1}, …, φ_{k}).

Observe that if (μ_{t})_{t≥0} ⊂ (C^{α}(Σ, ℝ))^{∗} and ω = lim_{t→0} μ_{t} in the dual α−Hölder norm on (C^{α}(Σ, ℝ))^{∗}, then for all g ∈ C^{α}(Σ, ℝ) we have that ω(g) = lim_{t→0} μ_{t}(g). For k = 0, (6) is trivial. Suppose by induction that (6) is true for k ≥ 0, then

[(D_{ϕ}^{(k+1)}μ_{ϕ})(φ_{1}, …, φ_{k+1})](g_{n}) =

= lim_{t→0} \frac{1}{t} [(D_{ϕ+tφ_{k+1}}^{(k)}μ_{ϕ+tφ_{k+1}})(φ_{1}, …, φ_{k}) − (D_{ϕ}^{(k)}μ_{ϕ})(φ_{1}, …, φ_{k})] (g_{n})

but since g_{n} is α−Hölder, we have

= lim_{t→0} \frac{1}{t}[(D_{ϕ+tφ}^{(k)}μ_{ϕ+tφ_{k+1}})(φ_{1}, …, φ_{k})(g_{n}) − (D_{ϕ}^{(k)}μ_{ϕ})(φ_{1}, …, φ_{k})(g_{n})]

and by the induction hypothesis

= lim_{t→0} \frac{1}{t}[ (D_{ϕ+tφ}^{(k)}μ_{ϕ+tφ}(g_{n})) (φ_{1}, …, φ_{k}) − (D_{ϕ}^{(k)}μ_{ϕ}(g_{n})) (φ_{1}, …, φ_{k})]

= (D_{ϕ}^{(k+1)}μ_{ϕ}(g_{n})) (φ_{1}, …, φ_{k+1}).

Moreover, for k ≥ 0,we can write

(D_{ψ+f}^{(k)}μ_{ψ}(g_{n}))(φ_{1}, …, φ_{k}) = (D_{ψ}^{(k)}μ_{ψ}(g_{n}))(φ_{1}, …, φ_{k})
+ (D_{ψ}^{(k+1)}μ_{ψ}(g_{n}))(φ_{1}, …, φ_{k}, f)

+ ∫_{0}^{1} s (D_{ψ+sf}^{(k+2)}μ_{ψ}(g_{n})) (φ_{1}, …, φ_{k}, f, f)ds (7)

From proposition 1 we get that μ_{ψ}(L) = ∑_{n} μ_{ψ}(g_{n}) is finite. Equation (7) for k = 0 and equations (5) and (6) for r = 0, 1, 2, prove the continuity and differentiability of ψ ↦ μ_{ψ}(L) in U_{2}. By induction we assume that ψ ↦ μ_{ψ}(L) is k-times differentiable, k ≥ 1, and that

(D_{ψ}^{(k)}(μ_{ψ}(L)))(φ_{1}, …, φ_{k}) = ∑_{n}(D_{ψ}^{(k)}μ_{ψ}(g_{n}))(φ_{1}, …, φ_{k}).

Then (7), (5) and (6) for r = k, k + 1, k + 2, prove that ψ ↦ μ_{ψ}(L) is (k + 1)-times differentiable in ψ ∈ U_{2}. □

Observe that since S(k) and F (k) in proposition 1 have order k!, the estimate (2) in proposition 1 is not enough to prove the analyticity of ϕ ↦ ∫ L dμ_{ϕ}.

Proof of Theorem A.

In order to prove Theorem A we need some estimates. Let Σ^{+} be a topologically transitive one-sided subshift of finite type. From now on fix β = α/2 > 0, ϕ_{0} ∈ C^{β}(Σ^{+}, ℝ), p ∈ Σ^{+} and n ≥ 0. Consider the β−Hölder norm, ∥ ∥, in C^{β}(Σ^{+}, ℝ), we have that

|f · g|_{β} ≤ |f|_{0} |g|_{β} + |f|_{β} |g|_{0}
∥f · g∥ ≤ ∥f∥ ∥g∥

Given ϕ ∈ C^{β}(Σ^{+}, ℝ) let P (ϕ) be its topological pressure (cf. [7]).

2. Lemma. For all ϕ ∈ C^{β}(Σ^{+}, ℝ) there exists A(ϕ) > 0, constant on a neighbourhood of ϕ_{0}, such that for all φ ∈ C^{β}(Σ^{+}, ℝ) and all n ≥ 0, k ≥ 0 we have

|D_{ϕ}^{(k)} exp(−P (ϕ)n)(φ_{1}, …, φ_{k})| ≤ n^{k} k! A(ϕ)^{k} exp(−P (ϕ)n) ∥φ_{1}∥ ⋯ ∥φ_{k}∥ .

Proof. We know that (cf. [6] 5.27) the map C^{β}(Σ^{+}, ℝ) → ℝ; ϕ ↦ P (ϕ) is analytic. Hence the coefficients of its Taylor series can grow at most exponentially. Therefore there exists B(ϕ) > 0 such that, for all k ≥ 0,

\frac{1}{k!} ∥D_{ϕ}^{(k)}P (ϕ)∥ ≤ B(ϕ)^{k} .

Moreover, B(ϕ) can be chosen constant on a neighbourhood of ϕ. Since the map ϕ ↦ exp(−P (ϕ) n) is also analytic, we have that D_{ϕ}^{(k)}(exp(−P (ϕ)n))(φ_{1}, …, φ_{k}) =

= ∑_{s=1}^{k} ∑_{A_{1}+···+A_{s}=ℕ_{k}_{#A_{i}≥1}} exp(−P (ϕ)n)(−1)^{s} n^{s} (D_{ϕ}^{(#A_{1})}P ) (φ_{i})_{i∈A_{1}} ⋯ (D_{ϕ}^{(#A_{s})}P ) (φ_{j})_{j∈A_{s}} ,

where ℕ_{k} := {1, 2, …, k} and + denotes the disjoint union of sets. Therefore

|D_{ϕ}^{(k)} (exp(−P (ϕ)n)) (φ_{1}, …, φ_{k})| ≤

≤ ∑_{s=1}^{k} ∑_{A_{1}+···+A_{s}=ℕ_{k}} exp(−P (ϕ)n) n^{k} (#A_{1})! ⋯ (#A_{s})! B(ϕ)^{k} ∥φ_{1}∥ ⋯ ∥φ_{k}∥

≤ ∑_{s=1}^{k} exp(−P (ϕ)n) n^{k} k! B(ϕ)^{k} ∥φ_{1}∥ ⋯ ∥φ_{k}∥

≤ n^{k} k k! B(ϕ)^{k} exp(−P (ϕ)n) ∥φ_{1}∥ ⋯ ∥φ_{k}∥ .

Now let A(ϕ) = 2 B(ϕ). □

Let f_{n} be the branch of the inverse of σ_{+}^{n} which sends f_{n}: C_{0}(σ_{+}^{n}(p)) → C_{n}(p). For φ ∈ C^{β}(Σ^{+}, ℝ), n ≥ 0, p ∈ Σ^{+}, write S̃_{n}φ: = S_{n}φ ∘ f_{n}, S̃_{n}φ: C_{0}(σ_{+}^{n}p) → ℝ, where S_{n}φ(z): = ∑_{k=0}^{n−1} φ(σ_{+}^{k}(z)). Let ∥| |∥ be the β-Hölder norm in C^{β}(C_{0}(σ_{+}^{n}(p))), i.e.

∥|ψ|∥ := [ψ]_{0} + [ψ]_{β} , [ψ]_{0} := sup_{x∈C_{0}(σ_{+}^{n}(p))} |ψ(x)|

[ψ]_{β} := sup \{\frac{|ψ(x) − ψ(y)|}{d(x, y)^{β}} | x, y ∈ C_{0}(σ_{+}^{n}(p)) , d(x, y) ≠ 0\}

Since d(f_{n}(x), f_{n}(y)) ≤ b^{n}d(x, y), we have that

(8)  |∥φ ∘ f_{n}∥| ≤ ∥φ∥  for all φ ∈ C^{β}(Σ^{+}, ℝ)

3. Lemma. The map C^{β}(Σ^{+}, ℝ) → C^{β}(C_{0}(σ_{+}^{n}(p)), ℝ): ϕ ↦ exp(S̃_{n}ϕ) is analytic. Moreover, there exist B(ϕ) > 0 and E(ϕ) > 0, constant on a neighbourhood of

ϕ_{0}, such that for all ϕ ∈ C^{β}(Σ^{+}, ℝ) and k ≥ 0 we have

(a)  |∥(D_{ϕ}^{(k)}h̃_{ϕ})(φ_{1}, …, φ_{k})∥| ≤ k! E(ϕ)^{k} ∥φ_{1}∥ ⋯ ∥φ_{k}∥ .
(b) ∥|(D_{ϕ}^{(k)} exp(S̃_{n}ϕ))(φ_{1}, …, φ_{k})|∥ ≤ B(ϕ) n^{k} ∥φ_{1}∥ ⋯ ∥φ_{k}∥ exp(S_{n}ϕ(p)) .

Proof.

The estimate (8) shows that the map C^{β}(Σ^{+}, ℝ) → C^{β}(C_{0}(σ_{+}^{n}(p), ℝ), ψ ↦ ψ∘f_{n} is a bounded linear map. Since the map C^{β}(Σ^{+}, ℝ) ↩: ϕ ↦ h_{ϕ} is analytic, we get that ϕ ↦ h̃_{ϕ} = h_{ϕ} ∘ f_{n} is analytic. This implies the existence of E(ϕ) > 0.

We have that

∥|S̃_{n}ϕ|∥ ≤ ∑_{k=0}^{n−1} ∥|(ϕ ∘ σ_{+}^{k}) ∘ f_{n}|∥ = ∑_{k=0}^{n−1} ∥|ϕ ∘ f_{n−k}|∥ ≤ n ∥φ∥ .

Therefore S̃_{n} is a bounded linear operator S̃_{n} : C^{β}(Σ^{+}, ℝ) → C^{β}(C_{0}(σ_{+}^{n}(p)), ℝ) and hence analytic. We have that

(D_{ϕ}^{(k)} exp(S̃_{n}ϕ))(φ_{1}, …, φ_{k}) = (exp S̃_{n}ϕ)(S_{n}φ_{1}) ⋯ (S_{n}φ_{k}) ,
∥|(D_{ϕ}^{(k)} exp(S̃_{n}ϕ))(φ_{1}, …, φ_{k})|∥ ≤ ∥| exp(S̃_{n}ϕ)∥| n^{k} ∥φ_{1}∥ ⋯ ∥φ_{k}∥

If z, w ∈ C_{0}(σ_{+}^{n}(p)) and x: = f_{n}(z), y: = f_{n}(w), we have that

|S̃_{n}ϕ(z) − S̃_{n}ϕ(w)| ≤ ∑_{k=0}^{n−1} |ϕ(σ_{+}^{k}x) − ϕ(σ_{+}^{k}y)|

≤ |ϕ|_{β} ∑_{k=0}^{n−1} d(σ_{+}^{k}x, σ_{+}^{k}y)^{β} ≤ |ϕ|_{β} ∑_{k=0}^{n−1} b^{−kβ}d(x, y)^{β}

≤ |ϕ|_{β} B_{0} d(z, w)^{β} ≤ B_{0} |ϕ|_{β} , where B_{0}: = (b^{−β} − 1)^{−1}. Thus, taking z = σ_{+}^{n}(p) we obtain

[exp S̃_{n}ϕ]_{0} ≤ (exp S_{n}ϕ(p)) exp(B_{0} |ϕ|_{β}) ,
|exp S̃_{n}ϕ(z) − exp S̃_{n}ϕ(w)| = |exp S̃_{n}ϕ(z)| |1 − exp(S̃_{n}ϕ(w) − S̃_{n}ϕ(z))|

≤ (exp S_{n}ϕ(p)) exp(B_{0} |ϕ|_{β}) A_{0} B_{0} |ϕ|_{β} d(z, w)^{β} ,

where A_{0}: = exp(B_{0} |ϕ|_{β}) ≥ sup\{|\frac{d}{dx}e^{x}|_{x_{0}} | : |x_{0}| ≤ B_{0} |ϕ|_{β}\}. And then ∥| exp S̃_{n}ϕ|∥ ≤ (A_{0} + A_{0}^{2} B_{0} |ϕ|_{β}) exp(S_{n}ϕ(p)). Therefore

|∥(D_{ϕ}^{(k)} exp(S̃_{n}ϕ))(φ_{1}, …, φ_{k})∥| ≤ B̃(ϕ) n^{k} ∥φ_{1}∥ ⋯ ∥φ_{k}∥ exp(S_{n}ϕ(p))

where B̃(ϕ): = A_{0} + A_{0}^{2}B_{0} |ϕ|_{β}. We can choose B(ϕ) = B̃(ϕ_{0}) + 1 for all ϕ on a small neighbourhood of ϕ_{0}. □

Let L_{ϕ}: C^{β}(Σ^{+}, ℝ) ↩ be the Perron-Frobenius operator:

(L_{ϕ}φ)(x): = ∑_{y∈σ_{+}^{−1}(x)} φ(y) exp(ϕ(y)).

We use the following notation from Ruelle’s theorem ([1] (1.7)):

L_{ϕ}(h_{ϕ}) = λ_{ϕ} h_{ϕ} , ∫ h_{ϕ} dν_{ϕ} = 1 , λ_{ϕ} = e^{P (ϕ)}.

L_{ϕ}^{∗}(ν_{ϕ}) = λ_{ϕ} ν_{ϕ} , μ_{ϕ} = h_{ϕ} ν_{ϕ}.

A proof that the maps C^{β}(Σ^{+}, ℝ) → ℝ : ϕ ↦ P (ϕ), C^{β}(Σ^{+}, ℝ) → (C^{β}(Σ^{+}, ℝ))^{∗} : ϕ ↦ μ_{ϕ}, ϕ ↦ ν_{ϕ} and C^{β}(Σ^{+}, ℝ) → C^{β}(Σ^{+}, ℝ) : ϕ ↦ h_{ϕ} are real analytic can be found in [5].

Define  J_{ϕ}^{n}(z): = λ_{ϕ}^{−n} L_{ϕ}^{n}(h_{ϕ} · 1_{C_{n}(p)})(z)

= \{ \begin{matrix}exp(−P (ϕ)n + S_{n}ϕ(x)) h_{ϕ}(x) & if σ_{+}^{n}x = z, x ∈ C_{n}(p) \\ 0 & if z ∉ C_{0}(σ_{+}^{n}p) \end{matrix}

Thus the measure of a one-sided cylinder is given by

μ_{ϕ}(C_{n}(p)) = ν_{ϕ}(h_{ϕ} · 1_{C_{n}(p)}) = λ_{ϕ}^{−n} ν_{ϕ}(L_{ϕ}^{n}(h_{ϕ} · 1_{C_{n}(p)})) = ∫_{C_{0}(σ_{+}^{n}p)} J_{ϕ}^{n} dν_{ϕ} .

Given p ∈ Σ^{+}, n ≥ 0, denote by f_{n} the branch of the inverse of σ_{+}^{n} sending f_{n} : C_{0}(σ_{+}^{n}(p)) → C_{n}(p). Given ϕ ∈ C^{β}(Σ^{+}, ℝ), let

h̃_{ϕ} := h_{ϕ} ∘ f_{n} ∈ C^{β}(C_{0}(σ_{+}^{n}(p)), ℝ) ,
J̃_{ϕ}^{n} := exp(−P (ϕ)n + S̃_{n}ϕ) h̃_{ϕ} .

4. Lemma. The map C^{β}(Σ^{+}, ℝ) ⟶ C^{β}(C_{0}(σ_{+}^{n}(p)), ℝ) : ϕ ↦ J̃_{ϕ}^{n} is analytic and there exists D_{0}(ϕ) > 0, constant on a neighbourhood of ϕ_{0}, such that for all φ ∈ C^{β}(Σ^{+}, ℝ) we have

|∥(D_{ϕ}^{(k)}J̃_{ϕ}^{n})(φ_{1}, …, φ_{k})∥| ≤ k! D_{0}(ϕ)^{k} n^{k} exp(−P (ϕ)n + S_{n}ϕ(p)) ∥φ_{1}∥ ⋯ ∥φ_{k}∥ .

Proof. We have that

(D_{ϕ}^{(k)}J̃_{ϕ}^{n}) (φ_{1} … φ_{k}) = ∑_{A+B+C=ℕ_{k}} ((D_{ϕ}^{(#A)} exp(−P (ϕ)n))(φ_{a})_{a∈A}) ·
· ((D_{ϕ}^{(#B)} exp(S̃_{n}ϕ))(φ_{b})_{b∈B}) ((D_{ϕ}^{(#C)}h̃_{ϕ})(φ_{c})_{c∈C})

By lemmas 2 and 3 we have that

|∥(D_{ϕ}^{k}J̃_{ϕ}^{n})(φ_{1}, … φ_{k})∥| ≤ ∑_{A+B+C=ℕ_{k}}(#A)! n^{#A} A(ϕ)^{#A} exp(−P (ϕ)n) ∏_{a∈A} ∥φ_{a}∥
· B(ϕ) n^{#B} exp(S_{n}ϕ(p)) ∏_{b∈B} ∥φ_{b}∥
· (#C)! E(ϕ)^{#C} ∏_{c∈C} ∥φ_{c}∥ .

|∥(D_{ϕ}^{(k)}J̃_{ϕ}^{n})(φ_{1}, …, φ_{k})∥| ≤ k! n^{k} (D_{0}(ϕ))^{k} exp(−P (ϕ)n + S_{n}ϕ(p)) ∥φ_{1}∥ ⋯ ∥φ_{k}∥ ,

where D_{0}(ϕ): = max{A(ϕ), B(ϕ), E(ϕ), 2}. □

Proof of Theorem A(b). Let E_{0} be the 0-cylinder E_{0}: = C_{0}(σ_{+}^{n}(p)). Consider the map R: C^{β}(E_{0}, ℝ) → C^{β}(Σ^{+}, ℝ) given by R(φ)(z): = φ(z) if z ∈ E_{0}, R(φ)(z): = 0 if z ∉ E_{0}. We claim that for φ ∈ C^{β}(E_{0}, ℝ), we have

∥R(φ)∥ ≤ (1 + b^{−β}) |∥φ|∥ .

Indeed, clearly |R(φ)|_{0} = [φ]_{0}. Now, if x, y ∈ E_{0}, then |R(φ)(x) − R(φ)(y)| = |φ(x) − φ(y)| ≤ [φ]_{β} d(x, y)^{β}. If x, y ∉ E_{0}, then |R(φ)(x) − R(φ)(y)| = 0. If x ∈ E_{0}, y ∉ E_{0}, then d(x, y) = b and |R(φ)(x) − R(φ)(y)| = |R(φ)(x)| ≤ [φ]_{0} = [φ]_{0} b^{−β} d(x, y)^{β}. Therefore |R(φ)|_{β} ≤ b^{−β} [φ]_{0}.

We have that μ_{ϕ}(1_{C_{n}(p)}) = ν_{ϕ}(J_{ϕ}^{n}) = ν_{ϕ}(R(J̃_{ϕ}^{n})). Since the map (C^{β}(Σ^{+}, ℝ))^{∗}× C^{β}(E_{0}, ℝ) → ℝ: (ν, J) ↦ ν(R(J)) is bilinear, we have that

(D_{ϕ}^{(k)}μ_{ϕ}(1_{C_{n}(p)}))(φ_{1}, …, φ_{k}) =

= ∑_{A+B=ℕ_{k}} ((D_{ϕ}^{(#A)}ν_{ϕ})(φ_{a})_{a∈A}) R ((D_{ϕ}^{(#B)}J̃_{ϕ}^{n})(φ_{b})_{b∈B})

Since ϕ ↦ ν_{ϕ} ∈ (C^{β}(Σ^{+}, ℝ))^{∗} is analytic, there exists C(ϕ) > 0, constant on a neighbourhood of ϕ_{0}, such that ∥(D_{ϕ}^{(r)}ν_{ϕ})(φ_{1}, …, φ_{r})∥_{∗} ≤ r! C(ϕ)^{r} ∥φ_{1}∥ ⋯ ∥φ_{r}∥, where ∥ ∥_{∗} is the dual norm in (C^{β}(Σ^{+}, ℝ))^{∗}. Therefore, by lemma 4, we have that

|D_{ϕ}^{(k)}μ_{ϕ}(1_{C_{n}(p)})(φ_{1}, …, φ_{k})| ≤

≤ ∑_{A+B=ℕ_{k}}(#A)! C(ϕ)^{#A} · (1 + b^{−β})(#B)! ∥|\frac{1}{(#B)!} (D_{ϕ}^{(s)}J̃_{ϕ}^{n}) (φ_{b})_{b∈B}|∥

(9) ≤ k! n^{k} D_{1}(ϕ)^{k}(1 + b^{−β}) exp(−P (ϕ)n + S_{n}ϕ(p)) ∥φ_{1}∥ ⋯ ∥φ_{k}∥ ,

where D_{1}(ϕ) = (1 + b^{β}) max{ C(ϕ) , D_{0}(ϕ) }. Part (b) of Theorem A follows from (9) and recalling (cf. [1]) that there are constants A, B > 0, uniform on a neighbourhood of ϕ_{0}, such that

μ_{ϕ}(C_{n}(p)) ∈ [A, B] exp(−P (ϕ)n + S_{n}ϕ(p)) . □

In order to prove part (a) of Theorem A we need some definitions. We say that two functions ϕ, φ ∈ C^{β}(Σ, ℝ) are homologous if there exists h ∈ C^{β}(Σ, ℝ) such that ϕ = φ + h ∘ σ − h. Homologous functions have the same pressure and the same equilibrium state. There is a natural embedding C^{β}(Σ^{+}, ℝ) ↪ C^{β}(Σ, ℝ) : ϕ ↦ \overline{ϕ} by considering \overline{ϕ}(x) := ϕ(x_{0}, x_{1}, …). The proof of the following lemma appears in pg. 11 of [1].

5. Lemma. There exists a continuous linear map A: C^{α}(Σ, ℝ) → C^{β}(Σ^{+}, ℝ),

β = α/2, such that A(ψ) is homologous to ψ. In particular A^{∗}: C^{β}(Σ^{+}, ℝ)^{∗} →

C^{α}(Σ, ℝ)^{∗}, (A^{∗}μ)(φ) = μ(Aφ) is a continuous linear map.

Proof of Therorem A(a). We first claim that there exists E(ϕ) > 0, constant

on a neighbourhood of ϕ_{1} such that for all n ≥ 0, k ≥ 0, p ∈ Σ, we have

|(D_{ϕ}^{(k)}μ_{ϕ}(1_{C_{n}(p)}))(φ_{1}, …, φ_{k})| ≤

≤ k! (2n)^{k} E(ϕ) exp(−2n P (ϕ) + S_{n}φ(p)) ∥φ_{1}∥ ⋯ ∥φ_{k}∥ , (10)

where S_{n}φ(p) = ∑_{k=−n}^{n} φ(σ^{k}p).

The proof of this claim consists on using (9) (for 1-sided subshifts) and observing

that μ_{ϕ}(C_{n}(p)) = μ+ ; \overline{ϕ}(σ^{−n}C_{n}(p)), where σ^{−n}(C_{n}(p)) is considered as a cylinder in

Σ^{+} and μ+ ; \overline{ϕ} is the equilibrium state for \overline{ϕ}: = A(ϕ) ∈ C^{α/2}(Σ^{+}, ℝ) which is homol-

ogous to ϕ on Σ. In particular P_{Σ^{+}}(\overline{ϕ}) = P_{Σ}(ϕ) and |S_{n}ϕ(p) − S_{n}\overline{ϕ}(p)| ≤ 2 |u|,

where \overline{ϕ} = ϕ + u ∘ σ − σ. This introduces a factor exp(2 |u|) in the constant E(ϕ).

The corollary follows from (10) and recalling the Gibbs Property, stated in

formula (3). □

References

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