Spectra of expanding maps on Besov spaces

A typical approach to analysing statistical properties of expanding maps is to show spectral gaps of associated transfer operators in adapted function spaces. The classical function spaces for this purpose are H\"older spaces and Sobolev spaces. Natural generalisations of these spaces are Besov spaces, on which we show a spectral gap of transfer operators.


Introduction
Let M be a compact smooth Riemannian manifold endowed with the normalised Lebesgue measure Leb M , and f : M → M a C r expanding map with r > 1. It is well known that individual trajectories of expanding maps tend to have "chaotic behaviour". Therefore, to analyse statistical properties of the expanding map f , it is typical to instead study how densities of points evolve under a so-called transfer operator L f,g induced by f with a given Cr weight function g : M → C with 0 < r ≤ r (the precise definition will be given below). In his celebrated paper [23], Ruelle first showed a spectral gap of the transfer operator of expanding maps on the usual Hölder space C s (M ) with 0 < s ≤r (when g is real-valued and strictly positive), resulting in the demonstration of the existence of a unique equilibrium state µ g of f at g and exponential decay of correlation functions of any C s observables with respect to µ g . (The existence and uniqueness of equilibrium states for C r expanding maps had been proved earlier in his monograph [22] through a thermodynamic approach.) Furthermore, the spectral gap of the transfer operator was used to investigate the dynamical zeta function [24,25], several limit theorems [1,16] and strong stochastic stability [13,14]. As another (deep) development, Gundlach and Latushkin [17] obtained an exact formula of the essential spectral radius of the transfer operator on C s (M ) in thermodynamic expression (see also Remark 6).
Recently, the transfer operator was shown to also have a spectral gap on the Sobolev space W s,p (M ) in Baillif and Baladi [3] (see also Faure [15] for the spectral gap via a semiclassical approach, and Thomine [28] for a spectral gap of the transfer operator of piecewise expanding maps on W s,p (M )) and the "little Hölder space" C s * (M ) in Baladi and Tsujii [12]. Our goal in this paper is to show a spectral gap of the transfer operator on Besov spaces B s pq (M ), which are closely related with the previously-studied function spaces (see Remark 4). We also refer to [5][6][7] and references therein for recent development of Banach spaces adapted to (hyperbolic) dynamical systems.
Our method in the proof is a natural generalisation of the best technology developed in Tsujii and Baladi [11]. Our result gives an answer to Problem 2.40 in the monograph by Baladi [8].
1.1. Definitions and results. Before precisely stating our main result, we introduce some notation. Recall that M is a compact smooth Riemannian manifold and f : M → M is of class C r with r > 1. Let f be an expanding map, i.e., there exist constants C > 0 and λ 0 > 1 such that |Df n (x)v| ≥ Cλ n 0 |v| for each x ∈ M and v ∈ T x M . Let us set the minimal Lyapunov exponent Then χ min > 0 when f is an expanding map. (For the properties of expanding maps, the reader is referred to [19,21,22] e.g.) Let g be a complex-valued Cr function on M with 0 <r ≤ r. We assume a technical conditionr ≥ 1 orr ≤ r − 1. (1.1) Note that (1.1) is satisfied for an important application g = | det Df | −1 , and that if r ≥ 2, then the condition (1.1) always holds (see also Remark 6 for the condition).
For each 0 ≤ s ≤r, the transfer operator L f,g : Standard references for transfer operators are [4,8].
Remark 1. Denote by r(A| E ) and r ess (A| E ) the spectral radius and the essential spectral radius of a bounded operator A : E → E on a Banach space E, respectively. Due to Ruelle [23], r(L f,g | C s (M) ) is bounded by exp P top (log |g|), and is equal to exp P top (log g) when g is real-valued and strictly positive (i.e., inf x∈M g > 0), where P top (φ) is the topological pressure of a continuous function φ : M → R (refer to [32] for the definition of topological pressure). It also follows from [8,Lemma 2.16] that R(g) ≥ exp P top (log |g|). Furthermore, in view of [11,Lemma A.1] regarding coincidence of eigenvalues of abstract linear operators in different Banach spaces outside of the essential spectral radii, we get the following: If r ess (L f,g | B s pq (M) ) < exp P top (log g) with real-valued and strictly positive g, then r(L f,g | B s pq (M) ) = exp P top (log g), where B s pq (M ) is the Besov space on M (see below for definition). In particular, the transfer operator on B s pq (M ) is quasi-compact (has a spectral gap), which is our goal in this paper. L f,g can be extended to a bounded operator on L p (M ) with p ∈ [1, ∞]. (Since some Besov spaces do not coincide with the completion of C s (M ) with respect to its Besov norm, this extension would be necessary; see [27, §A.1].) Indeed, a change of variables shows for any u ∈ L ∞ (M ) and ϕ ∈ L 1 (M ). On the other hand, the operator ϕ → Therefore, the transfer operator has a continuous extension to L p (M ) by the duality (1.3). The extension will also be denoted by L f,g . Below we define Besov spaces associated with a partition of unity of M . We first recall the definition of the Besov spaces on R d , where d is the dimension of M . Let ρ : R → R be a C ∞ function such that For each nonnegative integer n, define radial functions ψ n ∈ C ∞ 0 (R d , R) by For a tempered distribution u (that is, u is in the dual space of the set of rapidly decreasing test functions), the operator ∆ n is given by where F is the Fourier transform. Then we have n≥0 ∆ n u = u, called the Littlewood-Paley (dyadic) decomposition. This decomposition was first employed in context of dynamical systems theory to analyse the spectra of transfer operators of Anosov diffeomorphisms by Baladi and Tsujii [10]. They also applied the decomposition to spectral analysis of transfer operators of expanding maps in the little Hölder space C s * (M ) in a survey [12,Subsection 3.2], see Remark 4 for the definition of C s * (M ). For s ∈ R, p, q ∈ [1, ∞], we define the Besov space B s pq (R d ) as a set of tempered distributions u on R d whose norm is finite. We remark that u ∈ B s pq (R d ) if and only if there are a constant C > 0 and a nonnegative sequence {c n } n≥0 ∈ ℓ q with {c n } n≥0 ℓ q ≤ 1 such that for all n ≥ 0. Once we know u ∈ B s pq (R d ), we can take C = u B s pq . Since M is compact, there are a finite open covering {V i } I i=1 and a system of local charts is a partition of unity of M subordinate to the covering In this paper, the support of a continuous function φ : M → R is defined as the closure of {x ∈ M | φ(x) = 0}. Definition 2. The Besov space B s pq (M ) on M is the space of tempered distributions u on M whose norm is finite. This definition does not depend on the choice of charts or the partition of unity, see [30].
Let U ⊂ R d be a nonempty bounded open set. We write C s (U ), L p (U ) and B s pq (U ) for the subspace of C s (R d ), L p (R d ) and B s pq (R d ), respectively, such that the support of each element in these spaces is included in U . Then we have B s pq (U ) ⊂ L p (U ) for s > 0 and p, q ∈ [1, ∞]. 1 Hence, from the argument following (1.3) it holds that L f,g u is in L p (M ) for each u ∈ B s pq (M ) with s > 0 and p, q ∈ [1, ∞]. In particular, L f,g u is a tempered distribution and L f,g u B s pq is well-defined (but possibly takes +∞).
Here we provide our main theorem for an upper bound of the essential spectral radius of the transfer operator on B s pq (M ), which concludes (due to Remark 1) a spectral gap of the transfer operator on B s pq (M ) when g is real-valued and strictly positive and s is sufficiently large: Theorem 3. Let f : M → M be a C r expanding map with r > 1, and g : M → C a Cr function with 0 <r ≤ r satisfying (1.1). Let s ∈ (0,r] and p, q ∈ [1, ∞]. In addition, when q < ∞, assume that s is strictly smaller thanr. Then L f,g can be extended to a bounded operator on B s pq (M ), and it holds r ess (L f,g | B s pq (M) ) ≤ exp(−sχ min ) · R(g), where R(g) is defined in (1.2).
Remark 4. The Besov spaces can represent many other function spaces. As an important example, B s ∞∞ (R d ) is known to coincide with the Hölder space C s (R d ) when s is not an integer (whereas B s ∞∞ (R d ) strictly includes C s (R d ) when s is an integer), see [27, §A.1]. Hence Theorem 3 is a generalisation of a well-known result by Ruelle [23,Theorem 3.2]. We note that the little Hölder space C s * (M ) given in [12] is defined as the completion of C ∞ (M ) for the norm · B s ∞∞ (which is smaller than C s (M ), see [27, §A.1]), so that one can see that the proof of Theorem 3 can be translated literally to the case when the functional space where the transfer operator acts is C s * (M ) (this was essentially done in [12, Theorem 3.1] withr = r − 1). Furthermore, for the Sobolev space W s,p (R d ) with s ∈ R and p ∈ (1, 2], it holds [26, p.161] for the cases when s ∈ N). These inclusions imply W s,2 (M ) = B s 22 (M ), thus Theorem 3 recovers a part of the result by Baillif and Baladi [3]. 1 Indeed, by Hölder's inequality, for 1 ≤ q ≤ ∞ it holds where 1/q + 1/q ′ = 1 with a usual modification for q = ∞.
Other famous functional spaces constructed by using the Littlewood-Paley decomposition are Triebel-Lizorkin spaces F s pq (R d ). It seems possible to prove The- in the same manner as in the proof of Theorem 3. We note that W s,p (R d ) = F s p2 (R) for each 1 < p < ∞, see [29].
Remark 5. Theorem 3 is far from optimal: Gundlach and Latushkin [17] showed where Erg(f ) is the set of ergodic f -invariant probability measures, χ − µ is the smallest Lyapunov exponent of µ (for Df over f ), and h µ is the Kolmogorov-Sinai entropy of µ (over f ). We refer to [31,32] for the definitions of Lyapunov exponents and Kolmogorov-Sinai entropy. By the variation principle for topological pressure (see [32, §9] e.g.), we have is bounded by exp(−sχ min + P top (log |g|)). Therefore, when g is real-valued and strictly positive, the transfer operator on C s (M ) is quasi-compact for any 0 < s ≤r, while our s should be large to deduce the quasi-compactness from Theorem 3.
We also give a remark on an exceptional case g = | det Df | −1 : If g = | det Df | −1 , then R(g) = 1 and it follows from the Ruelle inequality and the Pesin identity (see e.g. [20]) that exp P top (log |g|) = 1. Therefore, together with Remark 1, Theorem 3 implies the quasi-compactness of L f,g for any 0 < s <r.
Remark 6. It is a natural question whether one can get a Gundlach-Latushkin type formula of the essential spectral radius on B s pq (M ), but we think that our method does not work for this purpose. Roughly speaking, we avoided the problem of exponential growth of the number of inverse branches of f n by a duality argument (see e.g. (2.4)), and this made our proof a little simpler. Its main shortcoming is that it does not seem that one can use thermodynamic techniques in estimating the essential spectral radius at the final step of the proof, which might be essential to obtain a Gundlach-Latushkin type formula. Baladi [8,Theorem 2.15] recently showed a Gundlach-Latushkin type upper bound of the essential spectral radius of the transfer operator of expanding maps on the Sobolev space W s,p (M ) by using the Littlewood-Paley decomposition, which may be helpful to answering this question.
Another shortcoming of our approach is the technical condition (1.1) for the regularity of weight functions. This problem is also solved in the Sobolev case in the approach of [8, Theorem 2.15] (see Remark 8 also), and it is highly likely that one can remove the condition (1.1) by a similar argument.
is larger than 1. Let G be a Cr function whose support is included in U ′ ∩ F −1 (U ). Furthermore, we assume that there are finitely many compact subsets Then, in a manner similar to one below (1.3), for each p ∈ [1, ∞] we can extend L F,G to a bounded operator from L p (R d ) to L p (U ). The only essential difference from the argument below (1.3) is that in the case 1 < p ≤ ∞, we need to notice Furthermore, it follows from this argument that the operator norm of L F,G : We define C 1 as the operator norm of ∆ n on L p (R d ), Theorem 3 will follow from the next Lasota-Yorke type inequality.
Theorem 7. Assume that (s, p, q) satisfies the condition in Theorem 3, and let F and G be as above. Then, L F,G can be extended to a bounded operator from B s pq (U ′ ) to B s pq (U ). Furthermore, there are a constantĈ F,G > 0 and σ ∈ (0, s) such that where the constants are defined in (2.1), (2.5), and (2.7).
Remark 8. The proof of Theorem 7 works even when λ ≤ 1, although it may give no spectral information for the original dynamical system f (in our setting (1.1) about r): the role of F will played by f n in local chart and λ ≤ 1 corresponds to χ min ≤ 0, so that Theorem 3 only means r ess (L f,g | B s pq (M) ) is bounded by exp(−sχ min ) · R(g) with exp(−sχ min ) ≥ 1 and R(g) ≥ exp P top (log |g|) (see Remark 1). However yet, Theorem 7 with F = id (λ = 1) seems to be actually helpful to demonstrate a spectral gap withr in full generality (that is, withr being not in the range of (1.1)), see [8,Theorem 2.15] for the Sobolev case.
Proof. We follow [12] and [8]. Let u ∈ B s pq (U ′ ). It follows from (1.6) that both ℓ:ℓ֒→n ∆ ℓ u and ℓ:ℓ ֒→n ∆ ℓ u are in L p (R d ) for any n ≥ 0. Thus, if we define L 0,n u and L 1,n u with n ≥ 0 by then both L 0,n u and L 1,n u are in L p (U ) (due to (2.5) and (2.6)), where the summations are taken over nonnegative integers ℓ and we write ℓ ֒→ n if 2 n ≤ λ −1 2 ℓ+4 and ℓ ֒→ n if 2 n > λ −1 2 ℓ+4 . It is obvious from the fact ℓ≥0 ∆ ℓ u = u that ∆ n L F,G u = L 0,n u + L 1,n u, so we have by Minkowski's inequality that in the case q < ∞ (a corresponding inequality also holds for q = ∞). Therefore, it suffices to show the following two inequalities for each q < ∞ (and corresponding two inequalities for q = ∞): and with some constantC F,G > 0 and σ ∈ (0, s). First we show (2.8). We focus on the case when q is finite, but the other case is analogous. Using (1.6), (2.5) and (2.6), we can find a nonnegative ℓ q sequence To estimate it, we recall Young's inequality for the locally compact group Z. We define ℓ q (Z) as the set of (two-sided) nonnegative sequences a (·) ≡ {a ℓ } ℓ∈Z such that a (·) ℓ q (Z) := ℓ∈Z a q ℓ 1/q is finite. Then, it follows from [2, Lemma 1.4] that is the convolution of a (·) and b (·) given by a (·) * b (·) (n) = ℓ∈Z a n−ℓ b ℓ for n ∈ Z. Thus, if we let {c ℓ } ℓ∈Z be asc ℓ = c ℓ for ℓ ≥ 0 and = 0 for ℓ < 0, then we have n≥0 ℓ≥0 On the other hand, since we have that (2.13) By (2.10), (2.11) and (2.13), we obtain (2.8).
Next, we shall show (2.9). Let {ψ ℓ } ℓ≥1 be a family of smooth functions on R d given byψ for ξ ∈ R d , where ρ is defined in (1.4). Notice thatψ n ≡ 1 on the support of ψ n for each n ≥ 0. For a tempered distribution u, we define∆ n u by∆ n u = F −1 [ψ n F u] for each n ≥ 0. Then it is straightforward to see that We borrow the following crucial lemma by Baladi [8] (from the proof of Lemma 2.21 of [8]), which has been proven in the special caser = r − 1 and p = ∞ in the survey [12] by Baladi and Tsujii (inequality (15) and a comment following inequality (19) of [12]). Since it is a key estimate in the proof of Theorem 7, we give a full proof in Appendix A.

2.2.
Completion of the proof. Now we can complete the proof of Theorem 3 by reducing the transfer operator L f n ,g (n) = L n f,g to a family of transfer operators on the local charts and applying Theorem 7 for each n ≥ 1.
Proof of Theorem 3. For the time being, we shall fix n ≥ 1 and suppress it from the notation. Let ι be an isometric embedding from B s pq (M ) to Furthermore, let G ij ≡ G n,ij be a compactly supported Cr function on R d such that (2. 19) In particular, L n f,g u B s pq = L • ι(u) I i=1 B s pq (Ui) for each u ∈ B s pq (M ). It follows from Theorem 7 that one can find a constantĈ n > 0 and σ ∈ (0, s) such that for each 1 ≤ i, j ≤ I satisfying U ij = ∅ and each u ∈ B s pq (U j ), we get the bound of L ij u B s pq by where the second factor of the first term is replaced with 1 when p = 1. On the other hand, it is straightforward to see that there is a constant C 2 > 0 independently of n, i, j such that the first term of (2.20) is bounded by Moreover, it follows from the estimate (4.5) in [9] that L n f,| det Df | −1 1 M L ∞ is bounded by a constant C 3 > 1 which is independent of n.
Therefore, by (2.19), we obtain the bound of L n f,g u B s pq by This completes the proof of Theorem 3.
, and that |T j (x) − T j (y)| > c j |x − y| for any x, y ∈ R d with some constant c j > 0. Then we have .
On the other hand, for j such that Hence, we get (A.1), and it follows from Minkowski's inequality that • T i and a C r mapping T i . Therefore, it suffices to show the following local version of Lemma 9: Let T : R d → R d be a C r mapping such that |T (x) − T (y)| >c|x − y| for any x, y ∈ R d with some constantc > 0, and that T : V → T (V ) is a C r diffeomorphism on a bounded open set V . LetG be a Cr function whose support is included in V . Define (with slight abuse of notation) L T,G : Also, we use the notation ℓ ֒→ n when it holds 2 n > Λ2 ℓ+4 , with where DT tr (w) is the transpose matrix of DT (w). Lemma 11. There exists a constant C T,G > 0 such that for any p ∈ [1, ∞], u ∈ L p (R d ), n ≥ 0 and ℓ ֒→ n, it holds whenr ≤ r − 1 and ∆ n L T,G∆ ℓ u L p ≤ C T,G 2 min{n,ℓ}−r max{n,ℓ} u L p (A.3) whenr ≥ 1.
Proof. We start the proof by noticing that ∆ n L T,G∆ ℓ : , whose operator norm is bounded by a constant C T > 0 (see (2.4)), and an integral operator We will give a bound of V ℓ n by a convolution kernel together with the factor 2 −r max{n,ℓ} : Let b : R d → (0, 1] be the integrable function given by Then, we will show that there is a constantC T,G > 0 such that for all ℓ ֒→ n, and |V ℓ n (x, y)| ≤C T,G 2 min{n,ℓ}−r max{n,ℓ} b min{n,ℓ} (x − y) ifr > 1. Step 1: We first prove (A.7). We further divide the proof of (A.7) according to whetherr is an integer.
Therefore, in a manner similar to one in the case (1-a) (replacing V ℓ n with V (0,L) n,ℓ and V (1,L) n,ℓ ), one can get (A.7).
Then in view of the cases (1-b) and (2-a), we obtain (A.8).