Compactness of Transfer Operators and Spectral Representation of Ruelle Zeta Functions for Super-continuous Functions

Transfer operators and Ruelle zeta functions for super-continuous functions on one-sided topological Markov shifts are considered. For every super-continuous function, we construct a Banach space on which the associated transfer operator is compact. Using this Banach space, we establish the trace formula and spectral representation of Ruelle zeta functions for a certain class of super-continuous functions. Our results include, as a special case, the classical trace formula and spectral representation for the class of locally constant functions.


Introduction
Let N ≥ 2 be an integer and A an N × N zero-one matrix. We say that A is aperiodic if there exists a positive integer k such that all entries of A k are positive. In this paper, we always assume that A is aperiodic. We set Σ + A = {ω = (ω m ) m∈N∪{0} ∈ {1, . . . , N } N∪{0} : A(ω m ω m+1 ) = 1, m ∈ N ∪ {0}} and equip Σ + A with the product topology. Then, Σ + A is a compact topological space. We define the shift map σ A : Σ + A → Σ + A by (σ A ω) m = ω m+1 , m ∈ N ∪ {0}.
Then, σ A is a continuous mapping. We call the dynamical system (Σ + A , σ A ) a one-sided topological Markov shift.
We call an element of m≥0 L m a locally constant function on Σ + A . Note that L 0 is the set of constant functions and that L 0 ⊂ L 1 ⊂ · · · . Moreover, for m ∈ N ∪ {0}, L m is a finite-dimensional linear subspace of V and dim L m ≤ N m . There exists a natural topology of V . Indeed, for θ ∈ (0, 1), we define the metric d θ on Σ + A by d θ (ω, ω ′ ) = θ m0 , m 0 = min{m ∈ N ∪ {0} : ω m = ω ′ m } and denote by F θ the set of complex-valued d θ -Lipschitz continuous functions on Σ + A . Then, V = θ∈(0,1) F θ (see Lemma 2.4 below). We denote by · θ the Lipschitz norm with respect to the metric d θ . We equip V with the topology induced by the family of norms { · θ } θ∈(0,1) .
Let f ∈ V . The Ruelle transfer operator L f : V → V of f is defined as follows: We set From [10,Theorem 1], Λ f is a discrete subset of C \ {0} and each eigenvalue has finite multiplicity. Hence, the structure of Λ f is similar to that of the spectrum of a compact operator on a Banach space. In fact, if f is locally constant, then there exists a Banach space B ⊂ V with L f (B) ⊂ B such that L f : B → B is compact and the discrete structure of Λ f comes from the compactness. More precisely, the following assertion holds. Here, for a bounded linear operator T : E → E on a Banach space E, we denote by σ(T : E → E) the spectrum of T .
The first aim of this paper is to extend the above theorem to all super-continuous f . Let {θ m } m∈N satisfy θ 1 ≥ θ 2 ≥ · · · ≥ 0, lim m→∞ θ m = 0. We write θ m (f ) = sup k≥m var k (f ) 1/k for f ∈ V and m ∈ N. It is easy to see that the sequence θ m = θ m (f ), m ∈ N satisfies (2) and (5). Moreover, if m 0 ≥ 2 and f ∈ L m0 \ L m0−1 , then B({θ m (f )}) = L m0−1 . Thus, Theorem 1.1 is an extension of Theorem A to all super-continuous f . We are interested in the connection between the spectrum of the transfer operator L f and the poles of the Ruelle zeta function ζ f (z). Here, the Ruelle zeta function ζ f (z) of f is an exponential of a formal power series defined by It is well known that the radius of convergence of the formal power series is not less than e −P (ℜf ) , where P (ℜf ) denotes the topological pressure of the real part ℜf of f . Let λ 1 (f ), λ 2 (f ), . . . be the sequence of non-zero eigenvalues of L f : V → V , where each eigenvalue is counted according to its multiplicity and |λ n (f )| ≥ |λ n+1 (f )| holds for n ∈ N. (If the number of the eigenvalues is finite, say, M , then we put λ n (f ) = 0 for n > M .) The following theorem is an immediate consequence of [5,Corollary 6].
Theorem B. Let f ∈ V . Then, ζ f (z) −1 admits a holomorphic extension to C and its zeros are exactly {λ n (f ) −1 : n ∈ N, λ n (f ) = 0}. Moreover, the order of each zero coincides with the multiplicity of the corresponding eigenvalue.
On the other hand, if f is locally constant, then we have the next Weierstrass canonical product form of ζ f . Theorem C ([10, Section 3]). Let f ∈ V be locally constant. Then, {n ∈ N : λ n (f ) = 0} is a finite set and (1 − zλ n (f )). (6) Equation (6) means that the entire function ∞ n=1 (1 − zλ n (f )) is a holomorphic extension of ζ f (z) −1 to C. Thus, for a locally constant f , we also obtain an analog of Theorem B by (6). We call (6) the spectral representation of ζ f (z).
The second aim of this paper is to establish the representation (6) for a wider class of f ∈ V . To this end, we consider the following condition for f ∈ V and r ∈ (0, 1): var that is, lim sup m→∞ var m (f ) 1/m /r m < ∞. If f is locally constant, then (7) is valid for any r ∈ (0, 1). (For each r ∈ (0, 1), we give an example of non-locally constant f ∈ V satisfying (7) in Example 5.5 below.) Here is the second maim result of this paper.
Theorem 1.2. Let f ∈ V and let r ∈ (0, 1) satisfy (7). Then, for p > 0 with the following three assertions hold: Then, the infinite product ∞ n=1 E(zλ n (f ), k 0 ) converges uniformly on any compact set of C and we have Note that if p ≤ 1, then k 0 = 0, and hence, (10) yields the spectral representation (6) of ζ f (z).
Equations like (9), which give a connection between the poles of a zeta function and the spectrum of the associated transfer operator, are often called trace formulas. The trace formulas for dynamical zeta functions are widely studied in differentiable dynamical systems; see, e.g., [4,6,12,13].
To prove Theorem 1.2 above, we introduce the operator ideal L (a) p (E). Let E be a Banach space. We denote by L(E) the set of bounded linear operators on E. For p > 0, we set where, for n ∈ N, a n (T ) denotes the n-th approximation number of T defined by a n (T ) = inf{ T − A : A ∈ L(E), rank A < n}.
It is easy to see that any element of L     p (E) to E = B, we prove Theorem 1.2 in Section 5. This paper is organized as follows. In Section 2, we give preliminary definitions and basic facts. In Section 3, we prove some estimates for the proofs of the main results, i.e., Theorems 1.1 and 1.2. In Section 4, we prove Theorem 1.1, and in Section 5, we prove Theorem 1.2. In Appendix A, we study the properties of transfer operators acting on V . It is natural to hope that the transfer operator L f : V → V is a compact operator. However, in Appendix A, we prove that this is not the case for any f ∈ V . Moreover, we give an example of f ∈ V such that ∞ n=1 |λ n (f )| = ∞. In Appendix B, we study the properties of V itself. We will see that V is naturally a nuclear space. Moreover, we prove that V has many nontrivial (i.e., non-locally constant) elements. More precisely, we prove that the set of non-locally constant elements of V is a residual subset of V . In Appendix C, we study the asymptotic behavior of eigenvalues of transfer operators. Using the Weyl inequality in Banach spaces (see, e.g., [7, Theorem 2.a.6]), we obtain an asymptotic behavior of {λ n (f )} n∈N for f ∈ V satisfying (7) for some r ∈ (0, 1). In Appendix C, we give an asymptotic behavior of {λ n (f )} n∈N for arbitrary f ∈ V , using a recent result of Demuth et al. [3].
For m ∈ N ∪ {0} and w ∈ {1, . . . , N } m , we set A is said to be periodic if σ q A ω = ω for some q ∈ N. For a periodic point ω, its period is the smallest q ∈ N such that σ q A ω = ω. We denote by Per q (σ A ) the set of periodic points ω ∈ Σ + A with σ q A ω = ω. We recall that the N × N zero-one matrix A is assumed to be aperiodic, that is, all entries of A k are positive for some positive integer k. The following lemma is needed in Appendices A and B.
Lemma 2.1. At least one row of A has more than two entries which are equal to one, and similarly for columns.
Proof. Assume that every row has just one entry which is equal to 1. Then there exists a permutation τ of the set {1, ..., N } such that A(ij) = 1 (j = τ (i)), = 0 (j = τ (i)). Thus, A k (ij) = 1 (j = τ k (i)), = 0 (j = τ k (i)) for k ≥ 1, and hence, A is not aperiodic. The transpose A T of A is also aperiodic. Hence, the assertion for columns also holds.
We recall from Section 1 that, for θ ∈ (0, 1), F θ denotes the set of complex-valued functions on Σ + A that are Lipschitz continuous with respect to the metric Let us recall the following definition of a super-continuous function.
We denote by V the set of all super-continuous functions on Σ + A . Remark 2.3. A super-continuous function on a topological Markov shift was first defined by Quas and Siefken in [11] as follows: φ : Σ + A → C is called a supercontinuous function if there exists a positive and non-increasing sequence {A m } m∈N such that var m (φ) ≤ A m for m ∈ N and A m+1 /A m → 0 as m → ∞. Let V ′ be the set of super-continuous functions in the sense of [11].
Recall from Section 1 that V is equipped with the topology induced by the family of norms { · θ } θ∈(0,1) . By the definition of the topology of V , we easily see that L f : V → V is continuous for f ∈ V . Moreover, since · θ ′ ≤ · θ for θ < θ ′ , we see that the topology of V coincides with that induced by the countable subfamily { · 1/(m+1) } m∈N . Hence, we obtain the following proposition: Fix a Borel probability measure µ on Σ + A such that µ(G) > 0 for every nonempty open set G of Σ + A . (The Gibbs measure for a real-valued function in F θ satisfies this condition; see, e.g., [8,Chapter 3].) Let C(Σ + A ) be the set of complexvalued continuous functions on Σ + A . For m ∈ N, we define a finite-rank operator Notice that E m φ = φ for φ ∈ L m . In addition to (1), we write The next lemma will be used in Sections 3-5 and Appendix B.
We note the next easy inequality: The following Lasota-Yorke type inequality is well known and a key tool for the proofs of the main results.

Some estimates for the proofs of the main results
Fix a sequence {θ m } m∈N satisfying (2). Recall from (3) and (4) the definitions of the space B({θ m }) and the norm · B({θm}) , respectively. In this section, we write B = B({θ m }) and · = · B({θm}) for the sake of simplicity. We set We begin with the following easy lemma (we omit the proof): For m ∈ N, the following three assertions hold: Let E m be as in (12).
Proof. First, we consider the case in which θ 1 = 0. Then, B = L 0 from Lemma 3.1 (i), and hence, E m (L 0 ) = L 0 . Next, we consider the case in which θ 1 > 0 and θ m = 0 for some m ≥ 2. Take Thus, (i) follows from the same argument as that in the proof of Lemma 2.6 (iii).
For g ∈ V and m, q ∈ N, we define the two operators K g,m , K (q) g,m by Notice that K g,m = K g,m .
Lemma 3.5. Let g ∈ V . If there exists b > 0 such that var k (g) ≤ bθ k k for k ∈ N, then the following three assertions hold: Proof. (i) follows from Lemma 3.4 immediately.
(ii) Thanks to (i), it is enough to show that K g,m (B) ⊂ B. From Corollary 3.2 and (i), The following lemma plays a key role in the proof of Theorem 1.1.
Lemma 3.6. There exists C 2 > 0, depending only on b 1 and b 2 , such that the following two inequalities hold for m, q ∈ N with θ m+1 ≤ 1 and g ∈ V satisfying (16): Proof. We prove (17). Let C 1 be as in Lemma 3.4. Take φ ∈ B so that φ ≤ 1.
We need the next lemma to prove Theorem 1.2.

Proof of Theorem 1.1
In this section, we prove Theorem 1.1, which is the first main result of this paper.
It is enough to show the following claim: Moreover, for λ ∈ Λ f with |λ| > ρ, the corresponding multiplicities coincide with each other.
For θ ∈ (0, 1), we denote by V θ the completion of V by the norm · θ . Clearly, Therefore, it is enough to show the following claim: Moreover, the corresponding multiplicities coincide with each other.
Here, for a Banach space E and a bounded linear operator T on E, r ess (T : E → E) denotes the essential spectral radius of T , that is, r ess (T : E → E) = inf{r ≥ 0 : any λ ∈ σ(T : E → E) with |λ| > r is an isolated eigenvalue with finite multiplicity}.

Proof of Theorem 1.2
In this section, we prove Theorem 1.2, which is the second main result of this paper.
Let f ∈ V and let r ∈ (0, 1) satisfy (7). Take D > 0 so that In this section, we set When θ m is of the form (26), we can obtain a slightly better estimate than (18).
Lemma 5.1. There exists C 4 > 0, depending only on b 1 and b 2 , such that the following inequality holds for m, q ∈ N with m ≥ 2, Dr m+1 ≤ 1 and g ∈ V satisfying (16): Proof. The outline of the proof is the same as that of the proof of (18). We may prove the inequality only for q = 1. Notice that C in (15) is equal to Dr.
For x ∈ R, ⌊x⌋ denotes the largest integer less than or equal to x. Recall, from (11) in Section 1, the definition of approximation numbers of a bounded linear operator acting on a Banach space. We estimate the approximation numbers of transfer operators acting on the Banach space B.
Recall from Section 1 the definition of the operator ideal L (a) p (E). The following corollary plays a key role in the proof of Theorem 1.2.
Corollary 5.4. Let g ∈ V satisfy (16). For p > 0 with (8) and m, q ∈ N with p ≥ q, the following two assertions hold (we write g m = E m g): 1 (B) and ∞ n=1 a n (L q g − L q gm ) → 0 as m → ∞. Proof. We prove (i) and the former part of (ii). From (i) and (ii) in Lemma 2.6, g m also satisfies (16). Thus, it is enough to prove the assertions only for g. Take R > e htop(σA) with r 2p R < 1. Then, Hence, L g ∈ L (a) p (B) and L q g ∈ L (a) 1 (B) follow from Corollary 5.3. We prove the latter part of (ii). Let N 1 ∈ N be as in Corollary 5.3. Let ǫ > 0. By (29), we can take n 0 ≥ N 1 so that n≥n0 (C 4 /r 2 ) q (n − 1) 2q/ log r R < ǫ. Moreover, by Lemma 3.7, we can take m 0 ∈ N so that θ m+1 ≤ 1 and n 0 L q g − L q gm B→B < ǫ for m ≥ m 0 .
Let m ≥ m 0 . We show ∞ n=1 a n (L q g − L q gm ) < 6ǫ. We easily have 1≤n<2n0 a n (L q g − L q gm ) ≤ 2n 0 a 1 (L q g − L q gm ) = 2n 0 L q g − L q gm B→B < 2ǫ.
On the other hand, we have n≥2n0 a n (L q g − L q gm ) ≤ l≥n0 n:2l≤n<2(l+1) a n (L q g − L q gm ) ≤ 2 l≥n0 a 2l (L q g − L q gm ).
By [9,Theorem 2.3.3], a 2l (L q g − L q gm ) ≤ a 2l−1 (L q g − L q gm ) ≤ a l (L q g ) + a l (L q gm ). Thus, by Corollary 5.3, we have n≥2n0 a n (L q g − L q gm ) ≤ 2 as desired.
We are ready to prove Theorem 1.2.
Proof of Theorem 1.2. (i) follows from Corollary 5.4 (i) and (I) in Section 1.
Appendix A. Transfer operators on V A metrizable topological vector space is said to be complete if every Cauchy sequence converges. Note that our space V is metrizable and complete since V is a Fréchet space (see Proposition 2.5). We recall the following definition of a compact operator on a metrizable and complete topological vector space.
Definition A.1. Let X be a metrizable and complete topological vector space and T a continuous linear operator on X. We say that T is a compact operator if the closure T (N ) of the image T (N ) is compact for some neighborhood N of zero.
In this appendix, we prove the following two theorems:

Appendix B. Some properties of V
We recall the following definitions of a nuclear operator and a nuclear space.
Definition B.1. Let E, F be Banach spaces and T : E → F a bounded linear operator. We say that T is a nuclear operator if T can be written in the form T x = ∞ n=1 λ n x, x ′ n y n , where the sequence {λ n } n∈N ⊂ C is summable and both of the two sequences {x n } n∈N ⊂ E ′ and {y n } n∈N ⊂ F are bounded. (E ′ denotes the dual Banach space of E.) Definition B.2. Let X be a locally convex Hausdorff topological vector space. We say that X is a nuclear space if, for every continuous seminorm p on X, there exists a continuous seminorm q on X such that p ≤ q and the natural embedding X q → X p is a nuclear operator. Here, X p denotes the completion of X/ ker p by p.
Let E, F be Banach spaces and T : E → F a bounded linear operator. We extend the definition (11) of the approximation numbers of T to the case E = F . For n ∈ N, the n-th approximation number a n (T : E → F ) of T is defined by a n (T : In this appendix, we prove the following two theorems: Theorem B.4. Let R be the set of φ ∈ V such that φ is not cohomologous with any locally constant function, that is, To prove Theorem B.3, we need the following lemma: Lemma B.5. Let θ, θ ′ ∈ (0, 1). If θ < θ ′ /N , then the natural embedding ι : F θ → F θ ′ is nuclear.
Recall from Section 1 that, for q ∈ N, φ : Σ + A → C and ω ∈ Σ + A , we write A ω). To prove Theorem B.4, we need some lemmas.
Here is a key lemma.