Complex transfer operators

In this section we extend the definition of the transfer operatorsLf to complex valued functions and also present the complex version of the RPF theorem, which was proved by M. Pollicott [19] (this can also be viewed as an extension of Wielandt’s theorem for matrices). We shall also extend the pressure definition to complex valued functions.

Let f ∈ Cα(X_A+,C), then we can define a complex transfer operator

Lf :Cα(XA+,C) → Cα(X

+

A,C) in similar way to when f ∈Cα(X

+

A,R). Hence we have the following theorem which gives information about the spectrum of the transfer operators whenf is complex valued.

Theorem 2.3.7. (Complex RPF theorem)[17] For f ∈ Cα(X_A+,C), we have

ρ(Lf)≤eP(<(f)). Moreover, we have one of the following cases.

(i) If Lf has an eigenvalueβ such that |β|=eP(<(f)), then it is simple and

unique and Lf = αM L<(f)M−1, where M is a multiplication operator

and α∈_C, |α|= 1 and so ρ(Lf) =eP(<(f)). The rest of the spectrum is

(ii) If Lf has no eigenvalue β with|β|=eP(<(f)), then ρ(Lf)< eP(<(f)).

Remark 2.3.2. If we assume in the above theorem that f is locally constant depending on two coordinates for example, sayf =f(x0, x1), we can deduce the

statement of Wielandt’s theorem for matricesM(i, j) =A(i, j)ef(i,j)_{,1≤i, j≤}

k, stated in the next theorem.

Theorem 2.3.8. (Wielandt’s theorem)([4], p.57) LetN be a positive aperiodic matrix with N(i, j) = |M(i, j)| ≥ 0 and let λ > 0 be the maximal positive eigenvalue for N. Then,

(i) every eigenvalue of M in modulus is less than or equal to the maximal eigenvalue λof N and

(ii) the equality happens if and only if M has the form M = eiaDN D−1, where 0 ≤ a≤ 2π and D is a diagonal matrix with diagonal entries are of unit modulus. In this caseM has an eigenvalue equal toλeia.

Another way to formulate the Complex RPF theorem and so Wielandt’s theorem is by a property related to=(f).

Lemma 2.3.9. [19] For f ∈Cα(X_A+,C)

(i) ρ(Lf) =eP(<(f))iff =(f) is cohomologous to a function of the forma+ψ,

where ψ ∈ C(X_A+,2πZ) and a ∈ R. In fact Lf has a simple eigenvalue β=eia+P(<(f)) _{and ρ(L}

f) =|β|.

(ii) If =(f) is not cohomologous to a function of the form a+ψ, where ψ∈

C(X_A+,2πZ) anda∈R, then ρ(Lf)< eP(<(f)).

Remark 2.3.3. This lemma will be useful in analysing the spectrum of the transfer operator Lf. The condition on =(f) will be connected to a condition

we require on a function, related to =(f), to be not cohomologous to a+φ, where φ∈C(X_A+, dZ), a∈R. As we are going to see in chapter 3, we call this function a non-lattice function.

The pressureP(f) was defined in Lemma 2.3.6 in the case of real valued functionf, where it is characterised by the unique maximal eigenvalue of the

transfer operatorLf. Using perturbation theory this definition can be extended to complex valued functions in a neighbourhood of the real valued functions. To see this, we recall the definition of analytic functions in complex Banach spaces.

Definition 2.3.1. Let B be a complex Banach space and let D ⊂ C be some open domain. A map f : D → B is said to be analytic if, for every bounded linear functional u :B → C, the map u◦f :D → C is analytic in the usual sense. If B1 and B2 are two complex Banach spaces and D0 ⊂ B1 is some

open domain then a map g:D0 → B2 is said to be analytic if the composition

g◦f :D→B2 is analytic for every open domain D⊂Cand any analytic map

f :D→B1 withf(D)⊂D0.

We also have the following perturbation theory lemma from Kato’s book ([8], VII.3).

Lemma 2.3.10. Let B(V) denote the Banach algebra of bounded linear op- erators on a complex Banach space V. If Lg ∈ B(V) has a simple isolated

eigenvalue λg with the corresponding eigenvector ug then for any > 0 there

existsδ >0 such that if Lf ∈B(V) with kLf −Lg|k< δ then

(i) Lf has a simple isolated eigenvalue λ(Lf) and corresponding eigenvector u(Lf) withλ(Lg) =λg, u(Lg) =ug and such that

(ii) Lf 7→λ(Lf), Lf 7→u(Lf) are analytic for kLf −Lgk< δ and

(iii) forkLf−Lgk< δ, we have that|λ(Lf)−λg|< andspec(Lf)\{λ(Lf)} ⊂

{z∈_C:|z−λg|> }.

Moreover, if Σg = spec (Lg)\ {λg} is contained in the interior of a circle C

centred at 0 ∈ _C then provided δ > 0 is sufficiently small, Σf = spec (Lf)\

{λ(Lf)} will also be contained in the interior of C.

The hypothesis of this lemma is satisfied when the transfer operatorLg where g ∈ Cα(X_A+,R), as Lg has a simple isolated eigenvalue λg = eP(g) by the RPF theorem and lemma 2.3.6. Therefore, this lemma implies that forf ∈

Cα(X_A+,C) in a sufficiently small neighbourhood ofg∈Cα(XA+,R) the transfer operatorLf has a simple eigenvalue ateP(f). This extends the definition of the pressure to a neighbourhood off ∈Cα(X_A+,R). So this means that the pressure

can only be defined in the case whenLf has a simple ’maximal’ eigenvalue β and the rest of its spectrum is restricted to a disc of radius smaller then|β|(part (i) of Theorem 2.3.7). For such functionsf, the definition of pressure is given by P(f) = Logβ, where Log is the principal branch of the complex function log. Moreover, by Lemma 2.3.10 the map f 7→ P(f) is analytic. Hence, we have the following lemma.

Lemma 2.3.11. ([17], Proposition 4.7) The domain of the pressure functionP

in Cα_(X+

A,C) is open and the function f 7→P(f) is analytic from this domain

intoC.