A complex Ruelle-Perron-Frobenius theorem for infinite Markov shifts with applications to renewal theory

We prove a complex Ruelle-Perron-Frobenius theorem for Markov shifts over an infinite alphabet, whence extending results by M. Pollicott from the finite to the infinite alphabet setting. As an application we obtain an extension of renewal theory in symbolic dynamics, as developed by S. P. Lalley and in the sequel generalised by the second author, now covering the infinite alphabet case.


Introduction
The core part of the present article is an extension of M. Pollicott's results [Pol84] concerning spectral properties of Perron-Frobenius operators for complex potential functions to the setting of an infinite alphabet (see Thm. 2.14 in Sec. 2.3). In order to obtain this extension we heavily make use of results on the Perron-Frobenius operator for real potential functions and infinite alphabets mainly developed by D. Mauldin and M. Urbański in [MU03] (see Sec. 2.2). Moreover, in Sec. 2.4 we prove analyticity results for complex perturbed resolvents of Perron-Frobenius operators.
Applying M. Pollicott's complex Ruelle-Perron-Frobenius theorem from [Pol84] (see also [PP90]) has lead to various new results, for instance to S. P. Lalley's renewal theorems for counting measures in symbolic dynamics [Lal89, Thms. 1 and 2]. In [Kom15] S. P. Lalley's ideas were generalised to more general measures. By this a setting was found which extends and unifies the setting of several established renewal theorems, namely (i) the above-mentioned theorems by S. P. Lalley [Lal89, Thms. 1 and 2] (ii) the classical key renewal theorem for finitely supported probability measures [Fel71] and (iii) a class of Markov renewal theorems (see e. g. [Als91,Asm03]). By applying our new Thm. 2.14 in Sec. 3 we extend the setting of [Kom15] further by lifting the results from a finite to a countably infinite alphabet leading to a new renewal theorem (see Thm. 3.1). This (i) exhibits new results in the vein of [Lal89] (ii) encompasses the key renewal theorem for arbitrary discrete measures, see Cor. 3.3 and (iii) comprises certain Markov renewal theorems.
Renewal theorems are a useful tool in various areas of mathematics. Of particular interest to us are their applications in geometry (see e.g. [Lal88] or [Fal97,Ch. 7]). Indeed, the new renewal theorem, Thm. 3.1, that is stated and proved in Sec. 3 allows for new results in this area. For instance, it yields statements concerning Minkowski measurability of limit sets of infinitely generated conformal graph directed systems (cGDS). These results will be presented in a forthcoming article [KK16] by the authors. For some previous results on the finite alphabet case we refer to [KK12,KK15]. The class of limit sets of infinitely generated cGDS is very rich and contains the boundary of Apollonian circle packings, limit sets of Fuchsian and Kleinian groups, self-similar and self-conformal sets and restricted continued fraction sets.

Complex Ruelle-Perron-Frobenius theorem
In [Pol84] a Ruelle-Perron-Frobenius theorem for complex potential functions was proven for the case that the underlying alphabet is finite. The aim of this section is to extend these results from [Pol84] to the setting of an infinite alphabet and to obtain analytic properties of resolvents which are associated to Perron-Frobenius operators for a family of complex potential functions. In Sec. 2.1 we introduce the relevant notions and the central object, namely the complex Perron-Frobenius operator. Important results concerning the Perron-Frobenius operator for real potential functions in the setting of an infinite alphabet have been obtained by D. Mauldin and M. Urbański and we collect their relevant results in Sec. 2.2. In Sec. 2.3 we use these statements to extend the findings of [Pol84] to the setting of an infinite alphabet, where we gain information on the spectrum of Perron-Frobenius operators L zξ+η for a family (zξ + η | z ∈ C) of complex potential functions (with real-valued potentials ξ, η). At this point we would like to thank Mariusz Urbański for very valuable discussions on this problem. Finally, in Sec.2.4, we use the statements of Sec. 2.2, 2.3 to obtain analytic properties of the resolvent-valued map z → (Id − L zξ+η ) −1 with Id denoting the identity operator.
2.1. The complex Ruelle-Perron-Frobenius operator. In the sequel I ⊂ N shall denote an at most countable alphabet, A : I × I → {0, 1} an incidence matrix and E ∞ := {ω ∈ I N | A ωj ωj+1 = 1 for all j ≥ 1} the space of A-admissible infinite sequences. E n denotes the set of all subwords of E ∞ of length n ≥ 1. The space of A-admissible finite sequences is denoted by where E 0 denotes the set which solely contains the empty word ∅. For ω = ω 1 ω 2 · · · ∈ E ∞ and n ∈ N we write ω| n := ω 1 · · · ω n for the initial subword of ω of length n. For ω, x ∈ E ∞ we write ω ∧ x := max{m ≥ 0 | ω i = x i for i ≤ m} for the length of the longest common initial block of ω and x.
Throughout this paper we assume that the incidence matrix A is finitely irreducible, that is there exists a finite set Λ ⊂ E * such that for all i, j ∈ I there is an ω ∈ Λ with iωj ∈ E * . Note, finitely irreducible is a weaker condition than finitely primitive which is equivalent to the big images and preimages (BIP) property of [Sar03], whenever the shift-dynamical system (E ∞ , σ) (with σ defined next) is topologically mixing. On E ∞ ∪ E * the shift map σ is defined by : ω = ω 1 ω 2 · · · ∈ E ∞ ω 2 ω 3 · · · ω n : ω = ω 1 ω 2 · · · ω n ∈ E n , n ≥ 2 ∅ : For ω ∈ E n we denote the ω-cylinder set by The topological pressure function of u : E ∞ → R with respect to the shift map σ : E ∞ → E ∞ is defined by the well-defined limit where S n f := n−1 j=0 f • σ j for n ≥ 1 and S 0 f := 0 denotes the n-th Birkhoff sum of f : E ∞ → C. Note that since the incidence matrix is finitely irreducible we have that the pressure defined above coincides with the Gurevich pressure (cf. [HU99,JKL14,Sar03]).
We equip I N with the product topology of the discrete topologies on I and equip E ∞ ⊂ I N with the subspace topology. By C(E ∞ ) resp. C(E ∞ , R) we denote the set of continuous compex-resp. real-valued functions on E ∞ . We refer to functions from C(E ∞ ) as potential functions. The set of bounded continuous functions in C(E ∞ ) resp. C(E ∞ , R) with respect to the supremum-norm · ∞ is denoted by Of particular importance to us is the subclass of Hölder continuous functions.
Definition 2.1 (Hölder continuity). For f ∈ C(E ∞ ), θ ∈ (0, 1) and n ∈ N define var n (f ) := sup{|f (x) − f (y)| | x, y ∈ E ∞ and x i = y i for i ≤ n}, var n (f ) θ n and Elements of F θ (E ∞ ) are called θ-Hölder continuous functions on E ∞ . Moreover, by F θ (E ∞ , R) we denote the subclass of real-valued θ-Hölder continuous functions on E ∞ . Note that by our definition a Hölder continuous function is not neccessarily bounded. For the respective spaces of bounded Hölder continuous functions we In order to define the central object of this section, namely the Perron-Frobenius operator of a complex potential function f = u + iv ∈ F θ (E ∞ ), we need to assume that A function u ∈ F θ (E ∞ , R) which satisfies (2.1) is called summable. Notice, when we write f = u + iv for f : E ∞ → C we implicitely assume that u and v are real-valued.
The conjugate operator L * f acting on C * b (E ∞ ) can be restricted to the subset of finite Borel measures. In fact, for any finite Borel measure µ the functional L * f (µ) given by for all g ∈ C b (E ∞ ), is tight in the following sense. For every ε > 0 there exists a compact set K ⊂ E ∞ such that for all Here, 1 B denotes the indicator function on a set B, that is 1 B (x) = 1 if x ∈ B and 0 otherwise. To verify this condition we exclude the trivial measure and first choose an integer M ∈ N such that ε M := e∈I,e>M exp(sup(u| [e] )) ≤ ε/(2µ(E ∞ )). Since E i,ℓ := {ω ∈ E ∞ : ω i ≥ ℓ} ↓ ∅, for ℓ → ∞ we find an increasing sequence (ℓ k ) of integers with ℓ 1 ≥ M and µ(E k,ℓ k ) < ε2 −k−1 /C u . Then for the compact set K := E ∞ \ k∈N E k,ℓ k we have µ(E ∞ \ K) < ε/(2C u ) and eω ∈ K for all e < M and ω ∈ K with A eω1 = 1. For later use let us set K M := K M (ε, µ) := K. Hence we have which proves our claim of L * f (µ) being tight. Applying an analogue of Riesz representation theorem for non-compact spaces stated in [Bog07, Thm. 7.10.6] yields that the functional L * f (µ) can be represented uniquely by a finite Radon measure.

2.2.
A real Ruelle-Perron-Frobenius theorem and Gibbs measures. A Borel probability measure µ on E ∞ is said to be a Gibbs state for u ∈ C(E ∞ , R) if there exists a constant c > 0 such that for every ω ∈ E ∞ and n ∈ N. of finite irreducibility that any convergent subsequence of (n −1 n−1 k=0 L k u−P (u) (1)) n in the proof of [MU03, Thm. 2.4.3] is uniformly bounded away from zero. Here, Theorem 2.4 (Real Ruelle-Perron-Frobenius theorem for infinite alphabets, . Moreover, the following hold. (i) There is a unique Borel probability eigenmeasure ν u of the conjugate Perron-Frobenius operator L * u and the corresponding eigenvalue is equal to e P (u) . Moreover, ν u is a Gibbs state for u.
has an eigenfunction h u which is bounded from above and which satisfies´h u dν u = 1. Further, there exists an The function u has a unique ergodic σ-invariant Gibbs state µ u . (iv) There exist constants M > 0 and γ ∈ (0, 1) such that for every Note that for our purposes it is important in Thm. 2.4(ii) that h u is uniformly bounded from below by R rather than just positive, see e. g. proof of Prop. 2.6. Directly from (2.3) we infer the following: Corollary 2.5. In the setting of Thm. 2.4(iv), e P (u) is a simple isolated eigenvalue is contained in a disc centred at zero of radius at most γe P (u) < e P (u) .

Spectral Properties of the complex Ruelle-Perron-Frobenius operator and Complex
Ruelle-Perron-Frobenius theorems. Important spectral properties of the Perron-Frobenius operator in the case of a finite alphabet have been obtained by W. Parry and M. Pollicott in [PP90,Pol84]. In this section we are extending some of their results to the setting of an infinite alphabet.
Suppose that u is summable. For 0 ≤ a < 2π the following are equivalent: takes values only in 2πZ.
Proof. For the implication "(ii) ⇒ (i)" one readily sees that e ia+P (u) is an eigenvalue corresponding to the eigenfunction e −iζ h u with h u as in Thm. 2.4.
where h is continuous (note, h is unique only up to mod(2π)) and thus we have for all From (2.4) and Thm. 2.4 we infer Thus, |h| is a version of the unique strictly positive eigenfunction h u of L u to the eigenvalue e P (u) . We deduce from Thm. 2.4 that ν u -almost surely, |h| ≥ R.
(Here, it is important that h u ≥ R > 0 holds true (see Thm. 2.4(ii)) rather than h u > 0.) Thus, by Rem. 2.2 L u |h| is ν u -almost surely bounded away from zero. The equations (2.5) and (2.6) together imply for ν u -almost every The above equation represents a (countable) convex combination of points on the unit circle which lies on the unit circle. Thus, all the points on the unit circle need to coincide. As moreover the left hand side is equal to 1 it follows that for all y with σy = x and ν u -almost all x ∈ E ∞ . Now, this set is dense in E ∞ , since being a Gibbs measure, ν u assigns positive mass to every cylinder set. Since v and h are both continuous functions we obtain Definition 2.7. If f = u + iv ∈ F θ (E ∞ ) satisfies one (and hence both) of the conditions of Prop. 2.6 then f is called an a-function. If f is not an a-function (for any a) then f is called regular.
With the above proposition we conclude with the same argument as in [Pol84,p.139] that the spectrum of L f is precisely the spectrum of L u rotated through the angle a, when f is an a-function. Together with Cor. 2.5 this yields the following analogue of [Pol84, Prop. 3].
Proposition 2.8 (Complex Ruelle-Perron-Frobenius theorem for a-functions). If f = u + iv ∈ F θ (E ∞ ) with summable u is an a-function then exp(ia + P (u)) is a simple eigenvalue for L f and the rest of the spectrum is contained in a disc of radius strictly smaller than | exp(ia + P (u))| = exp(P (u)). Now, we study the spectrum of L f when f is regular and show that it is disjoint from the circle with centre at the origin and radius exp(P (u)). For this, we adapt the arguments in [Pol84,p. 139]. For simplicity, we will often assume that u is normalised so that P (u) = 0 and L u 1 = 1. This is possible, since for any summable when u is summable by the bounded distortion property stated next.
Here, ( * ) follows from the mean value theorem with some z ∈ E ∞ and in ( * * ) we used the Gibbs property of ν u with constant c, see (2.2) and Thm. 2.4. Setting C := f θ cθ 1−θ , we obtain var m L n f g ≤ θ m C g L 1 νu + (C + 1)θ n g θ which shows the assertion.
Choose a point e it on the unit circle. For h ∈ F θ (E ∞ ) which satisfies and for each N ∈ N we write Lemma 2.11. Let f = u + iv ∈ F θ (E ∞ , C) be regular and u summable (with P (u) = 0 and L u 1 = 1). Then h N L 1 νu tends to zero when N → ∞.
Proof. Suppose for a contradiction that h N L 1 νu does not tend to zero when N → ∞. Under this assumption lim sup N →∞ h N L 1 νu =: s ∈ (0, 1]. Thus, there is a sequence (N k ) k∈N such that lim k→∞ h N k L 1 νu = s. We now show that there exists a subsequence of (N k ) k∈N along which (h N ) N converges ν u -almost surely to a function h * ∈ C(E ∞ ). For this, we assume without loss of generality that I = N and need the following two ingredients.
(i) Recall that on p. 4 we constructed for every Borel measure µ and every ε > 0, M ∈ N a compact set K = K M = K M (ε, µ) for which µ(E ∞ \ K) < ε/(2C u ) and eω ∈ K for every e < M and ω ∈ K with A eω1 = 1. Moreover, by construction As ν u is a Gibbs state for u, it assigns positive mass to every cylinder set. Thus, (i) implies that Y M is dense in E ∞ and we can uniquely extend h * continuously to E ∞ . We denote the extension of h * to E ∞ by h * as well and show in the following that h * is a non-zero eigenfunction of L f to the eigenvalue e it , a contradiction to the regularity of f . Then, for all x ∈ M∈N Y M . Since L f h * and e it h * are both uniformly continuous and as they coincide on a dense subset of E ∞ , they need to coincide on E ∞ , that is (i) and (ii) together with (iii) show that e it is an eigenvalue of L f with non-zero eigenfunction h * ∈ C b (E ∞ ). This together with Prop. 2.6 is a contradiction to the assumption that f is regular, whence lim n→∞ h N L 1 νu = 0.
Proof. For any n ≤ N we deduce the following inequalities from Lem. 2.10.
The above inequalities are in particular valid for n = √ N . Additionally using that |||h||| θ ≤ 1 we obtain Applying Lem. 2.11 now finishes the proof.
Lemma 2.13. If u is summable then the closed unit ball Proof. It suffices to show that H is a sequentially compact subset of the Banach space of L 1 νu (E ∞ )-functions. Since H is equicontinuous, we can use the arguments of the proof of Lem. 2.11 to conclude that any sequence (f n ) n∈N in H posesses a subsequence (f nm ) m∈N which converges pointwise to a uniformly continuous limiting function f * . We use the notation from Lem. 2.11. Write δ m (x) := |f * (x) − f nm (x)| and ε M m := sup x∈YM |f * (x) − f nm (x)|. Uniform convergence of (f nm ) m on Y M implies lim m→∞ ε M m = 0. Further let ε M := e∈I,e>M exp(sup(u| [e] )) be as above. We again assume without loss of generality that I = N. Because of and that f nm ∞ implies f * ∞ ≤ 1, which gives Combining Lem. 2.11, 2.13 and (2.9) yields that we can choose N ∈ N such that |||h N ||| θ < 1 for all h ∈ H with H defined in Lem. 2.13. The arguments of [Pol84,p. 140] imply that e it is not in the spectrum of L f and finally the following theorem.
Theorem 2.14 (Complex Ruelle-Perron-Frobenius theorem for regular functions). Let f = u + iv ∈ F θ (E ∞ ). Suppose that u is summable. If f is regular then the spectrum of L f is contained in a disc of radius strictly smaller than exp(P (u)).

Analyticity of
We suppose that there exists a unique δ ∈ R for which P (η−δξ) = 0. This condition in particular implies that ξ cannot be identically zero. Moreover, by Rem. 2.2 In order to study the analytic properties of the operator-valued function z → (Id − L η+zξ ) −1 , we let B(F b θ (E ∞ )) denote the set of all bounded linear opera- , · op ) a Banach space. Here, · op denotes the operator norm.
for z in some punctured neighbourhood of z = −δ. Here, The factor e P (η+zξ) of the first summand of (2.10) is missing in [Lal89]. However, this does not make a difference, since the z-value of interest is z = −δ, where P (η + zξ) = 0.
Corollary 2.17. If −δ < t * and´−(η + tξ)dµ η−δξ < ∞ for all t in an open neighbourhood of −δ then, for each χ ∈ F θ (E ∞ , R) and x ∈ E ∞ , we have that the residue of z → (Id − L η+zξ ) −1 χ(x) at the simple pole z = −δ is equal to By using the same arguments as in [Lal89]  (i) If ξ is non-lattice, then z → (Id − L η+zξ ) −1 is holomorphic in a neighbourhood of every z on the line Re(z) = −δ except for z = −δ. (ii) If ξ is integer-valued but there does not exist any ψ ∈ C(E ∞ ) such that the range of ξ − ψ + ψ • σ is contained in a proper subgroup of Z, then z → (Id − L η+zξ ) −1 is 2πi-periodic, and holomorphic at every z on the line Re(z) = −δ such that Im(z)/(2π) is not an integer.

Renewal theory
In [Lal89] renewal theorems for counting measures in symbolic dynamics were established, where the underlying symbolic space is based on a finite alphabet. (Given a measurable space (Ω, A), a counting measure µ A on A ∈ A is defined through µ A (B) = #A ∩ B for B ∈ A.) These renewal theorems were extended to more general measures in [Kom11,Kom15]. The renewal theorems of [Kom15] generalise and unify (i) [Lal89, Thms. 1 and 2] (ii) the classical key renewal theorem for finitely supported probability measures [Fel71] and (iii) a class of Markov renewal theorems (see e. g. [Als91,Asm03]). In the present section we show how the results of Sec. 2.2 and 2.3 lead to an extension of the generalised versions from [Kom15] to the setting of an underlying countable alphabet. Moreover, we explain that the classical key renewal theorem for arbitrary discrete measures [Fel71] is a simple special case of the new renewal theorem. Having the results of Sec. 2, the proof of the newly extended renewal theorem follows along the lines of proof in [Kom15,Lal89]. Therefore, we only present an outline of proof focusing on the necessary modifications, in Sec. 3.1.
For x ∈ E ∞ we study the asymptotic behaviour as t → ∞ of the renewal function x ∈ E ∞ and t ∈ R, where N abs (t, x) := ∞ n=0 y:σ n y=x χ(y)|f y (t − S n ξ(y))|e Snη(y) .
(D) Exponential decay of N on the negative half-axis. There exist c > 0, s > 0 and t 0 ∈ R such that e −tδ N abs (t, x) ≤ ce st for all t ≤ t 0 .
The asymptotic behaviour of N as t → ∞ depends on whether the potential ξ is lattice or non-lattice. Two functions f, g : R → R are called asymptotic as t → ∞, written f (t) ∼ g(t) as t → ∞, if for all ε > 0 there exists t ∈ R such that for all t ≥ t the value f (t) lies between (1 − ε)g(t) and (1 + ε)g(t). For t ∈ R we define ⌊t⌋ := max{k ∈ Z | k ≤ t} and {t} := t − ⌊t⌋ ∈ [0, 1). Note that ⌊t⌋ and {t} respectively are the integer and the fractional part of t if t ≥ 0.
Assume that x → f x (t) is θ-Hölder continuous for every t ∈ R and that Conditions (A) to (D) hold.
(i) If ξ is non-lattice and f x is monotonic for every x ∈ E ∞ , then as t → ∞, uniformly for x ∈ E ∞ . (ii) Assume that ξ is lattice and let ζ, ψ ∈ C(E ∞ , R) satisfy the relation where ζ is a function whose range is contained in a discrete subgroup of R. Let a > 0 be maximal such that ζ(E ∞ ) ⊆ aZ. Then as t → ∞, uniformly for x ∈ E ∞ , where G x is periodic with period a and (iii) We always have Remark 3.2. The monotonicity condition in Thm. 3.1 (i) can be substituted by other conditions (see [Kom15]). One such condition is that there exists n ∈ N for which S n ξ is bounded away from zero and that the family (t → e −tδ |f x (t)| | x ∈ E ∞ ) is equi directly Riemann integrable; a condition which is motivated by the key renewal theorem, see [Fel71].
Our renewal theorem deals with renewal functions which act on the product space R × E ∞ . When making the restrictions (i) to (iv) below the renewal function is independent of the second component and we obtain the classical key renewal theorem for discrete measures (see [Kom15] for details).
(ii) f x = f is independent of x ∈ I N (iii) χ = 1 (iv) ξ and η are constant on cylinder sets of length one.
Notice, any probability vector (p 1 , p 2 , . . .) with p i ∈ (0, 1) and every (s 1 , s 2 , . . .) with s i ≥ 0 determine η, ξ ∈ F θ (E ∞ , R) via η(ω 1 ω 2 · · · ) := log(p ω1 e δsω 1 ) and ξ(ω 1 ω 2 · · · ) := s ω1 . Setting Z(t) := e −δt N (t, x), which now is independent of x, and z(t) := e −δt f (t) (which is directly Riemann integrable by Rem. 3.2) we deduce from (3.2) that Z solves the classical renewal equation: for t ∈ R or equivalently, Z = Z ⋆ F + z, where F is the distribution which assigns mass p i to s i and where ⋆ denotes the convolution operator. Thm. 3.1 implies (ii) If {s 1 , s 2 , . . .} ⊂ a · Z and a > 0 is maximal, then as t → ∞ (iii) We always have 3.1. The ideas of proof. In proving Thm. 3.1 we use the methods of proof which were developed in [Lal89] and extended in [Kom15]. Besides using our new results of Sec. 2 only small modifications are necessary. Thus, below, we only provide an outline of the main steps of the proof and focus on the necessary modifications. For more details we refer the reader to [Kom15], where similar notation is used.
The spectral radius formula now implies, for z ∈ Z, that Note that χ ∞ ℓ=−∞ e ℓz f · (aℓ + β − ψ) ∞ is finite because of Conditions (C), (D). Since a −1 ζ is integer-valued but not co-homologous to any function valued in a proper subgroup of the integers, we can apply Prop. 2.18. Therefore, z → (Id − L η+a −1 zζ ) −1 is holomorphic at each z = −aδ + iθ, for 0 < |θ| ≤ π. Moreover, it has a simple pole at z = −aδ with residue as t → ∞. Since in all instances where t occurs only the fractional part is involved, it is clear that G x is periodic with period a, which finishes the proof.