FRACTAL UNCERTAINTY FOR TRANSFER OPERATORS

Transcription

1 FRACTAL UNCERTANTY FOR TRANSFER OPERATORS SEMYON DYATLOV AND MACEJ ZWORSK Abstract. We show directly that the fractal uncertainty principle of Bourgain Dyatlov [BD16] implies that there exists σ > 0 for which the Selberg zeta function (1.2) for a convex co-compact hyperbolic surface has only finitely many zeros with Re s 1 2 σ. That eliminates advanced microlocal techniques of Dyatlov Zahl [DZ16] though we stress that these techniques are still needed for resolvent bounds and for possible generalizations to the case of non-constant curvature. The purpose of this note is to give a new explanation of the connection between the fractal uncertainty principle, which is a statement in harmonic analysis, and the existence of zero free strips for Selberg zeta functions, which is a statement in geometric spectral theory/dynamical systems. The connection is proved via the transfer operator which is a well known object in thermodynamical formalism of chaotic dynamics. To explain the fractal uncertainty principle we start with its flat version, given by (1.1) below. Let X [ 1, 1] be a δ-regular set in the following sense: there exists a Borel measure µ supported on X and a constant C R such that for each interval centered on X of size 1, we have C 1 R δ µ() C R δ. Bourgain Dyatlov [BD16, Theorem 4] proved that when δ < 1, there exist β > 0 and C 1 depending only on δ, C R such that for all f L 2 (R) supp F h f X(h) = f L 2 (X(h)) C 1 h β f L 2 (R), (1.1) where F h f := ˆf(ξ/h) is the semiclassical Fourier transform (2.25) and X(h) := X + [ h, h] denotes the h-neighborhood of X. Roughly speaking, (1.1) quantifies the statement that a function and its Fourier transform cannot both concentrate on a fractal set. To explain the spectral geometry side, let M = Γ\H 2 be a convex co-compact hyperbolic surface, that is, a non-compact hyperbolic surface with a finite number of funnel ends and no cusps see [Bo16, 2.4]. The Selberg zeta function is defined by Z M (s) = ( ) 1 e (s+k)l, Re s 1, (1.2) l L M k=0 and it continues as an entire function to C. Here L M denotes the set of lengths of primitive closed geodesics on M with multiplicity see [Bo16, Chapter 10]. The 1

2 2 SEMYON DYATLOV AND MACEJ ZWORSK zeros of Z M enjoy many interpretations, in particular as quantum resonances of the Laplacian on M see [Zw17] for a general introduction and references. n particular, finding resonance free regions has a long tradition and applications in many settings. The limit set, Λ Γ, is defined as the set of accumulation points of orbits of Γ acting on H 2, see also (2.6). t is a subset of the boundary of H 2 at infinity, so in the Poincaré disk model of H 2 we have Λ Γ S 1. The set Λ Γ is δ-regular where δ [0, 1) is the exponent of convergence of Poincaré series, see [Bo16, Lemma 14.13]. The hyperbolic version of fractal uncertainty principle was formulated by Dyatlov Zahl [DZ16, Definition 1.1]. Define the operator B χ = B χ (h) on L 2 (S 1 ) by B χ (h)f(y) = (2πh) 1/2 S 1 y y 2i/h χ(y, y )f(y ) dy, (1.3) where y y is the Euclidean distance between y, y S 1 R 2. Here χ C c (S 1 ), S 1 := {(y, y ) S 1 S 1 y y }. (1.4) We say that Λ Γ satisfies (hyperbolic) fractal uncertainty principle with exponent β 0 if for each ε > 0 there exists ρ < 1 such that for all C 0 > 0, χ Cc ) we have 1l ΛΓ (C 0 h ρ ) B χ (h) 1l ΛΓ (C 0 h ρ ) L 2 (S 1 ) L 2 (S 1 ) = O(h β ε ) as h 0. (1.5) This hyperbolic fractal uncertainty principle with β = β(γ) > 0 was established for arbitrary convex co-compact groups by Bourgain Dyatlov [BD16] by proving the flat version (1.1) and showing that it implies (1.5). t followed earlier partial results of Dyatlov Zahl [DZ16]. With this in place we can now state our main result: Theorem. Assume that M = Γ\H 2 is a convex co-compact hyperbolic surface and the limit set Λ Γ satisfies fractal uncertainty principle with exponent β in the sense of (1.5). Then M has an essential spectral gap of size β, that is for each ε > 0 the Selberg zeta function of M has only finitely many zeroes in {Re s 1 2 β + ε}. Remark. Bourgain Dyatlov [BD17] showed that when δ > 0, the set Λ Γ also satisfies the fractal uncertainty principle with exponent β > 1 δ which only depends on δ. The 2 resulting essential spectral gap is an improvement over the earlier work of Naud [Na05] which gave a gap of size β > 1 δ which also depends on the surface. See also Dyatlov 2 Jin [DJ17]. A stronger theorem was proved in [DZ16] using fine microlocal methods which included second microlocalization and Vasy s method for meromorphic continuation (see [Zw17, 3.1] and references given there). n addition to showing a zero free strip, it provided a bound on the scattering resolvent, see [DZ16, Theorem 3]. Having such bounds is essential to applications see [Zw17, 3.2]. t would be interesting to see if Note the sign change from [DZ16, (1.6)]. t is convenient for us and it does not change the norm. (S 1

3 FRACTAL UNCERTANTY FOR TRANSFER OPERATORS 3 γ D 1 D 2 x 1 x 2 Figure 1. The disks D 1, D 2 and the transformation γ in the trivial case (1.7). one can use the methods of this paper to obtain bounds on the scattering resolvent. However, generalizations to non-constant curvature are likely to be based on the microlocal techniques of [DZ16]. The result of [DZ16] also applies to higher-dimensional hyperbolic quotients. The proof of the present paper can be adapted to higher dimensional cases where the limit set is totally discontinuous (such as Schottky quotients). The present paper instead uses transfer operators see the outline below and 2.3. The identification of M with a quotient by a Schottky group see 2.1 allows to combine simple semiclassical insights with the combinatorial structure of the Schottky data. Outline of the proof. The proof is based on a well known identification of zeros of Z M (s) with the values of s at which the Ruelle transfer operator, L s, has eigenvalue 1 see (2.13). Referring to 2.3 and [Bo16, 15.3] for precise definitions, we have L s u(z) = B (w) s u(w), (1.6) w : Bw=z where B is an expanding Bowen Series map and u is a holomorphic function on a family of disjoint disks symmetric with respect to the real axis. We outline the idea of the proof in the trivial case (the zeros of Z M (s) can be computed explicitly see [Bo16, (10.32)]) when there are only two disks D 1, D 2 and γ SL(2, R), a linear fractional transformation preserving the upper half plane, maps one disk onto the complement of the other disk, see Figure 1: D 2 = Ċ \ γ 1 (D 1), B(z) = { γ 1 (z), z D 1 ; γ(z), z D 2. (1.7) This trivial case can be reduced to γ(z) = kz, k > 1, but we do not want to stress that point.

4 4 SEMYON DYATLOV AND MACEJ ZWORSK Denote j := D j R. n the case (1.7) the limit set Λ Γ consists of the fixed points of γ, Λ Γ = {x 1, x 2 } Ṙ, x j j, γ(x j ) = x j, and the fractal uncertainty principle (1.5) holds with β = 1 2 by a trivial volume bound: 1l ΛΓ (C 0 h ρ ) B χ (h) 1l ΛΓ (C 0 h ρ ) L 2 L 2 1l ΛΓ (C 0 h ρ ) L L 2 B χ(h) L 1 L 1l Λ Γ (C 0 h ρ ) L 2 L 1 Ch ρ 2 h 1 2 h ρ 2 = O(h 1 2 ε ), ε = 1 ρ. (1.8) The operator (1.6) splits as a direct sum over operators on the two disks and hence we only need to consider a simplified transfer operator L s u(z) = γ (z) s u(γ(z)), u H(D 1 ), (1.9) where H(D 1 ) is the Bergman space of holomorphic functions in L 2 (D 1 ). Given (1.8) we want to show, in a complicated way which generalizes, that the equation L s u = u, u H(D 1 ), has no non-trivial solutions for s = 1 2 ν + ih 1 with ν < 1 2 ε. Thus assume that for such an s, L s u = u. The first observation is that u R is semiclassicaly localized to bounded frequencies: that means that for all χ C c ( 1 ), F h (χu)(ξ) = O(h ξ ) for ξ C. (1.10) t is here that holomorphy of u is used: using the maximum principle we show that sup D1 u e C/h sup 1 u and derive the Fourier transform bound from this. See Lemmas 3.2 and 3.3. From now on we work only on the real axis. n the outline we will use concepts from semiclassical analysis but we stress that the actual proofs in the paper are selfcontained. To connect the model used here to B χ (h), which acts on S 1, we identify S 1 with the extended real axis Ṙ see 2.5. Transplanted to Ṙ, B χ(h) is a semiclassical Fourier integral operator with the phase Φ(x, x ) defined in (2.24) and hence associated to the canonical transformation κ : T Ṙ T Ṙ, κ : (x, x Φ) (x, x Φ). We stress that κ is a global diffeomorphism and for (x, ξ) = κ(x, ξ ) we have x x. That means that for the action on compactly microlocalized functions, the singularity removed by the cutoff (1.4) is irrelevant and we can consider a simpler operator B(s) defined by, essentially, removing χ. n the circle model it has the form B(s)f(y) = (2πh) 1/2 S 1 y y 2s f(y )dy.

5 FRACTAL UNCERTANTY FOR TRANSFER OPERATORS 5 See 2.5 for details. This operator has a nice equivariance property which is particularly simple for the operator (1.9): denoting x := 1 + x 2 we have L s B(s) = B(s) L 1 s where Ls := x 2s L s x 2s 2 : L (Ṙ) L2 (Ṙ). (1.11) See Lemma 2.5 for the general version. Here we only say that what lies behind this identity is the following formula valid for linear fractional transformations ϕ: ϕ(x) ϕ(y) 2 ϕ (x) = x y 2 ϕ (y) 1. (1.12) To use (1.11), we put γ N := γ N, the Nth iterate where N log(1/h) is chosen so that γ N ( 1 ) h ρ. Then u(x) = L N s u(x) = γ N(x) s u(γ N (x)), x 1. Choose a cutoff function χ N Cc ((x 1 C 1 h ρ, x 1 + C 1 h ρ )) which is equal to 1 in a neighbourhood of γ N ( 1 ). We then put u N := x 2s χ N u, so that x 2s u = L N s u N on 1. Since ρ < 1, u N remains compactly microlocalized in the sense of (1.10) but in addition it is concentrated at x = x 1 : WF h (u N ) {x = x 1, ξ C}. The operator B(s) is elliptic and we denote its microlocal inverse by B(s) 1 noting that it is a Fourier integral operator associated to κ 1. Hence, u N = B(s)v N + O(h ), v N := B(s) 1 u N, WF h (v N ) κ 1 (WF h (u N )) {x x 1 }. Therefore, changing v N by O(h ), we may assume that supp v N J Ṙ where J is a fixed interval on Ṙ S1 and x 1 / J. Now we use the equivariance property (1.11): modulo an O(h ) error, we have x 2s u = L N s u N = L N s B(s)v N = B(s)w N on 1, w N := L N 1 sv N, supp w N J := γ N (J). Since J lies a fixed distance away from x 1, J = γ N (J) lies in an h ρ sized interval centered at the repelling point, x 2, of the transformation γ. The change of variables in the integrals shows that Now we have w N L 2 h ρν v N L 2 h ρν u N L 2. u N = χ N B(s) 1l J w N + O(h ) so the fractal uncertainty principle (1.8) gives u N L 2 χ N B(s) 1l J L 2 L 2 w N L 2 Ch 1/2 ε ρν u N L 2 + O(h ).

6 6 SEMYON DYATLOV AND MACEJ ZWORSK Since ν < 1 2 ε and we can take ρ = 1 ε, we obtain u N 0 if h is small enough, thus u 0 and the proof is finished. n the case of non-trivial Schottky groups similar ideas work but with combinatorial complications. We only make a general comment that for iterates γ N (or more generally iterates γ a see 2.1), x log γ N (x) for x in a small h-independent neighbourhood of x 1 is essentially equal to 2 x log x γ 1 N (x 0), x 0 x 1. This follows from (1.12) with y := γ 1 N (x 0), ϕ := γ N. Any generalization of our method has to replace this explicit formula by writing x log γ N (x) approximately as xφ(x, y), y = γ 1 N (x 0), x 0 x 1, with Φ generating a canonical tranformation. Notation: We write f = O(g) H for f H g. n particular, f = O(h ) H means that for any N there exists C N such that f H C N h N. The norm refers to the L 2 norm but different L 2 norms are used (L 2 (R), L 2 (S 1 ), L 2 (Ω)) and they are either specified or clear from the context. With some of abuse of notation, C Γ denotes large constants which only depend on a fixed Schottky data of Γ (see 2.1) and whose exact value may vary from place to place. We denote x := 1 + x 2. Acknowledgements. We would like to thank the anonymous referee for suggestions to improve the manuscript. This research was conducted during the period SD served as a Clay Research Fellow. MZ was supported by the National Science Foundation grant DMS and by a Simons Fellowship. 2. ngredients 2.1. Schottky groups. n this section we briefly review properties of Schottky groups, referring the reader to [Bo16, 15.1] for more details. We use notation similar to [BD17, 2.1]. The group SL(2, R) acts on the extended complex plane Ċ := C { } by Möbius transformations: ( ) a b az + b γ = SL(2, R), z Ċ = γ(z) = c d cz + d. Denote by H 2 C the upper half-plane model of the hyperbolic plane. Then SL(2, R) acts on H 2 by isometries. A Schottky group is a convex co-compact subgroup Γ SL(2, R) constructed in the following way (see Figure 2): Fix r N and nonintersecting closed disks D 1,..., D 2r C centered on the real line.

7 FRACTAL UNCERTANTY FOR TRANSFER OPERATORS 7 γ 4 γ 3 D 2 γ 2 γ D 1 1 D3 D 4 D 12 D 14 D 11 D 33 D 32 D 34 D 21 D 23 D 22 D 44 D 41 D 43 Figure 2. A Schottky structure with r = 2. The real line is dashed. The solid intervals are 1, 3, 2, 4. Define the alphabet A = {1,..., 2r} and for each a A, denote { a + r, 1 a r; a := a r, r + 1 a 2r. Fix group elements γ 1,..., γ 2r SL(2, R) such that for all a A γ a (Ċ \ D a) = D a, γ a = γ 1 a. (2.1) (n the notation of [Bo16, 15.1] we have γ a = S 1 a.) Let Γ SL(2, R) be the group generated by γ 1,..., γ r ; it is a free group on r generators [Bo16, Lemma 15.2]. Every convex co-compact hyperbolic surface M can be viewed as the quotient of H 2 by a Schottky group Γ, see for instance [Bo16, Theorem 15.3]. Note that the complement of a D a in H 2 is a fundamental domain. We fix a Schottky representation for a given hyperbolic surface M and refer to D 1,..., D 2r and γ 1,..., γ 2r as Schottky data. n several places we will use results of [BD17, 2] which can be read independently of the rest of [BD17]. n particular, we use combinatorial notation for indexing the words in the free group Γ and the corresponding disks: For n N 0, define W n, the set of words of length n, as follows: W n := {a 1... a n a 1,..., a n A, a j+1 a j for j = 1,..., n 1}. (2.2) Denote by W := n W n the set of all words. For a W n, put a := n. Denote the empty word by and put W := W \ { }. For a = a 1... a n W, put a := a n... a 1 W. For a set Z W, put Z := {a a Z}. (2.3) For a = a 1... a n W, put a := a 1... a n 1 W. Note that W forms a tree with root and each a W having parent a.

8 8 SEMYON DYATLOV AND MACEJ ZWORSK For a = a 1... a n, b = b 1... b m W, we write a b if either a =, or b =, or a n b 1. Under this condition the concatenation ab is a word. We write a b if a, b W and a n = b 1. n the latter case a b W. For a, b W, we write a b if a is a prefix of b, i.e. b = ac for some c W. Define the following one-to-one correspondence between W and the group Γ: W a 1... a n = a γ a := γ a1 γ an Γ. Note that γ ab = γ a γ b when a b, γ a = γa 1, and γ is the identity. For a = a 1... a n W, define the disk centered on the real line (see Figure 2) D a := γ a (D an ) C. f a b, then D b D a. On the other hand, if a b and b a, then D a D b =. Define the interval a := D a R and denote by a its length (which is equal to the diameter of D a ). For a A, define the interval a a as the convex hull of the union b A, a b ab, see Figure 2. More generally, for a = a 1... a n W define a := γ a ( a n ) a. (2.4) Note that a b for any b W such that a b, a b. Denote D := a A D a C, := a A a = D R, := a A a. (2.5) The limit set is given by Λ Γ := D a R. (2.6) n 1 a W n The fact that Λ Γ R follows from the contraction property [BD17, 2.1] a C Γ (1 C 1 Γ ) a for all a W. (2.7) We finish this section with a few estimates. We start with the following derivative bound which is the complex version of [BD17, Lemma 2.5]: Lemma 2.1. For all a = a 1... a n W and z D an we have C 1 Γ a γ a (z) C Γ a. (2.8) Proof. Define the intervals := an, J := a = γ a (). Let γ, γ J SL(2, R) be the unique affine transformations such that γ ([0, 1]) =, γ J ([0, 1]) = J. Following [BD17, 2.2], we write γ a = γ J γ α γ 1, γ α = ( e α/2 0 e α/2 e α/2 e α/2 ) SL(2, R),

9 FRACTAL UNCERTANTY FOR TRANSFER OPERATORS 9 where α R and α C Γ by [BD17, Lemma 2.4]. For z D an we have γ a (z) = J { γ α(w), w = γ 1 w 1 (z) 1 }. 2 2 We compute γ α(w) = e α (e α 1)w [C 1 Γ, C Γ]. Since C 1 Γ a J / C Γ a, (2.8) follows. The next lemma bounds the number of intervals a of comparable sizes which can contain a given point: Lemma 2.2. For all C 1 2 and τ > 0, we have Proof. Fix x R. We have sup #{a W : τ a C 1 τ, x a } C Γ log C 1. (2.9) x R {a W : τ a C 1 τ, x a } = {a 1,..., a N } for some a 1,..., a N W such that a 1 a 2 a N. We have aj+1 (1 C 1 Γ ) a j, see [BD17, 2.1], and (2.9) follows Functional spaces. For any open Ω C, let H(Ω) be the Bergman space on Ω, consisting of holomorphic functions f : Ω C such that f L 2 (Ω) (with respect to the Lebesgue measure). Endowing H(Ω) with the L 2 norm, we obtain a separable Hilbert space. For general Ω C, we put H(Ω) := H(Ω ). Denote D 2 := a W a =2 D a D (2.10) and let D ± := D {± m z 0}, D ± 2 := D 2 {± m z 0}. See Figure 3. The following basic estimate is needed for the a priori bounds in 3.1 below: Lemma 2.3. There exists c (0, 1] such that for all f H(D ± ) C(D ± ), sup D ± 2 f ( sup f ) c( sup f ) 1 c. (2.11) D ± Proof. The boundary of D ± consists of and a union of half-circles, which we denote S ±. Let F ± : (D ± ) [0, 1] be the harmonic function with boundary values F ± 1, F ± S ± 0.

10 10 SEMYON DYATLOV AND MACEJ ZWORSK D 1 D3 D 2 D Figure 3. An illustration of Lemma 2.3. The lighter shaded region is D + and the darker region is D + 2. The union of the thick lines is. Since F ± is positive in (D ± ) and D ± 2 lies away from S ±, the infimum of F ± on D ± 2 is positive. Denote this infimum by c := inf D ± F ± (0, 1]. Since log f is subharmonic 2 in D ± and log f ( sup log f ) F ± + ( sup log f ) (1 F ± ) on D ±, D ± the maximum principle implies that for z D ± 2, log f(z) sup log f F ± (z) ( sup log f sup log f ) D ± D ± sup log f c ( sup log f sup log f ). D ± D ± Exponentiating this we obtain (2.11) Transfer operators and resonances. The zeros of the Selberg zeta function, that is the resonances of M = Γ\H 2, are characterized using a dynamical transfer operator, also called the Ruelle operator. Here we follow [Bo16, 15.3],[GLZ04] and consider these operators on Bergman spaces defined in 2.2. We refer to [Ba16] for other approaches to transfer operators and for historical background. Here, for s C we define the transfer operator L s : H(D) H(D) as follows: L s f(z) = a A a b γ a(z) s f(γ a (z)), z D b, b A. (2.12) (We note that in the notation of [Bo16, (15.11)], S i = γ 1 i.) This is the same as (1.6) if we define Bz = γa 1 (z) for z D a. The derivative satisfies γ a(z) > 0 for z b and γ a(z) s is uniquely defined and holomorphic for z D b and s C such that γ a(z) s > 0 when z b, s R. Since γ a takes real values on R, the expression (2.12) additionally gives an operator on L 2 (), also denoted L s. The operators L s are of trace class (see [Bo16, Lemma 15.7]) and depend holomorphically on s C. Hence the determinant det( L s ) defines an entire function of s C. The connection to the Selberg zeta function defined in (1.2) is a special case of Ruelle theory and and can be found in [Bo16, Theorem 15.10]: Z M (s) = det( L s ).

11 FRACTAL UNCERTANTY FOR TRANSFER OPERATORS 11 t also follows that L s is a Fredholm operator of index 0 and its invertibility is equivalent to ker( L s ) = {0}. Hence, Z M (s) = det( L s ) = 0 u H(D), u 0, u = L s u, (2.13) see [Bo16, Theorem A.34] Partitions and refined transfer operators. Our proof uses refined transfer operators which are generalizations of powers of the standard transfer operator L s given by (2.12). To introduce these we use the notion of a partition: A finite set Z W is called a partition if there exists N such that for each a W with a N, there exists unique b Z such that b a. See Figure 4. n terms of the limit set, this means that Λ Γ = b Z( b Λ Γ ). (2.14) The alphabet A is a partition, as is the set W n of words of length n 1. Another important example is the set of words discretizing to some resolution τ > 0: Z(τ) := { a W : a τ < a } (2.15) where we put :=. The set Z(τ) is a partition due to (2.7). Note however that if Z is a partition, this does not imply that the set Z defined in (2.3) is a partition. f Z W is a finite set, we define the refined transfer operator L Z,s : H(D) H(D) as follows: L Z,s f(z) = γ a (z)s f(γ a (z)), a Z z D b, b A. (2.16) a b As in the case of L s, we can also consider L Z,s as an operator on L 2 (). Here are some basic examples of refined transfer operators: f Z = A then L Z,s is the identity operator. f Z = W 2, the set of words of length 2, then L Z,s = L s, the standard transfer operator defined in (2.12). More generally if Z = W N for some N 1, then L Z,s = L N 1 s. Lemma 2.4. Assume that Z is a partition; define Z by (2.3). Then for all u H(D) and s C L s u = u = L Z,s u = u. (2.17) Proof. We argue by induction on b Z b. f Z = A then L Z,s is the identity operator so (2.17) holds. Assume that Z A. Choose a longest word dc Z, where d W and c A. Then Z has the form (see Figure 4)

12 12 SEMYON DYATLOV AND MACEJ ZWORSK Figure 4. An example of a partition Z, with elements of Z shaded grey in the tree of words. The solid black word is one possible choice of d in the proof of Lemma 2.4. Z = ( Z \ {d} ) {da a A, d a} where Z is a partition containing d. By the inductive hypothesis we have L Z,s u = u, thus is remains to prove that We write for each z D b, b A L s u = u = L Z,s u = L Z,su. (2.18) L Z,s u(z) = a Z a b γ a (z)s u(γ a (z)) and similarly for Z. The condition a b simply means b = a 1 where a 1 is the first letter of a. For b d 1 the expressions for L Z,s u(z) and L Z,su(z) are identical. Assume now that b = d 1. Removing identical terms from the conclusion of (2.18) and using that da = ad we reduce (2.18) to γ d (z)s u(γ d (z)) = a A d a γ ad (z)s u(γ ad (z)) for all z D d1. Using the chain rule and dividing by γ d (z)s this reduces to u(γ d (z)) = a A a d γ a(γ d (z)) s u(γ a (γ d (z))) for all z D d1. The latter follows from the equality L s u(w) = u(w) where w := γ d (z) D d An integral operator. We now introduce an integral operator B(s) similar to B χ (h) from (1.3). That operator has a simpler definition than B χ (h) but one pays by introducing singularities. n our approach we use the upper half plane model while B χ (h) acts on functions on a circle rather than a line. Hence, we will use the extended real line Ṙ = R { }

13 FRACTAL UNCERTANTY FOR TRANSFER OPERATORS 13 which is identified with the circle S 1 C by the map x Ṙ y = i x i + x S1. (2.19) The standard volume form on S 1, pulled back by (2.19), is dp (x) = 2 x 2 dx. Denote L 2 (Ṙ) := L2 (Ṙ, dp ) L2 (S 1 ). For x, x Ṙ, let x x S be the Euclidean distance between y(x) and y(x ), namely x x S = 2 x x x x. With this notation in place we define the operator B(s), depending on s C: for Re s < 1, it is a bounded operator on L2 (Ṙ) given by the formula 2 B(s)f(x) = m s 1/2 x x 2s S f(x ) dp (x ). (2.20) 2π Ṙ For general s the integral in (2.20) may diverge however χ 1 B(s)χ 2 : L 2 (Ṙ) L2 (Ṙ), χ 1, χ 2 C (Ṙ), supp χ 1 supp χ 2 =, (2.21) is well defined. n other words, B(s)f can be defined outside of supp f. The following equivariance property of B(s) will be used in the proof of the main theorem in 3.3: Lemma 2.5. Let γ SL(2, R), s C, and consider the operator T γ,s : L 2 (Ṙ) L2 (Ṙ), T γ,sf(x) := γ (x) s Sf(γ(x)). (2.22) Here γ (x) S = x 2 γ(x) 2 γ (x) is the derivative of the action of γ on the circle defined using (2.19). Then 2 Proof. Take f L (Ṙ). We need to show that Ṙ T γ,s B(s) = B(s)T γ,1 s. (2.23) T γ,s B(s)f(x) = B(s)T γ,1 s f(x) for all x when Re s < 1 and for all x / 2 γ 1 (supp f) otherwise. This is equivalent to γ (x) s S γ(x) x 2s S f(x ) dp (x ) = x x 2s S γ (x ) 1 s S f(γ(x )) dp (x ). The latter follows by the change of variables x = γ(x ) using the identity Ṙ γ(x) γ(x ) 2 S = x x 2 S γ (x) S γ (x ) S.

14 14 SEMYON DYATLOV AND MACEJ ZWORSK We now discuss the properties of B(s) in the semiclassical limit which means that we put s := 1 ν + 2 ih 1, where ν is bounded and 0 < h 1. n the notation of (2.21) we obtain an oscillatory integral representation: χ 1 B(s)χ 2 f(x) = (2πh) 1/2 e i h Φ(x,x ) χ 1 (x)χ 2 (x ) x x 2ν 1 S f(x ) dp (x ) where the phase function Φ is defined by Ṙ Φ(x, x ) := 2 log x x S = 2 log x x + 2 log x + 2 log x log 4. (2.24) Thus χ 1 B(s)χ 2 is a semiclassical Fourier integral operator, see for instance [DZ16, 2.2]. The next two lemmas can be derived from the theory of these operators but we present self-contained proofs in the Appendix, applying the method of stationary phase directly. That first gives Lemma 2.6 (Boundedness of B(s)). Let χ 1, χ 2 C (Ṙ) satisfy supp χ 1 supp χ 2 =. Then there exists C depending only on ν, χ 1, χ 2 such that χ 1 B(s)χ 2 L 2 (Ṙ) L2(Ṙ) C. The next lemma gives partial invertibility of B(s). To state it we recall the definition of semiclassical Fourier transform, F h f(ξ) := f(x)e i h xξ dx, (2.25) R see [Zw12, 3.3] for basic properties. We say that an h-dependent family of functions f = f(h) is semiclassically localized to frequencies ξ M if for every N, F h f(ξ) C N h N ξ N when ξ M. (2.26) We also recall semiclassical quantization a Op h (a) = a(x, hd x ), D x := 1 i x Op h (a)u(x) := 1 e i h (x y)ξ a(x, ξ)u(y) dydξ, (2.27) 2πh stressing that only elementary properties from [Zw12, 4.2,4.3] will be used. R The partial invertibility of B(s) means that for f L 2 (R) which is supported in an interval and semiclassically localized to frequencies ξ (5 ) 1, we have f = B(s)g + O(h ) on for some g which is supported away from : Lemma 2.7 (Partial invertibility of B(s)). Let R be an interval and K 10 satisfy 10K 1. Assume that A = Op h (a), a(x, ξ) C c (R 2 ), supp a {x, ξ < 2K}. Then for all ψ Cc ( ) and ω Cc (Ṙ \ ) satisfying supp a {ψ (x) = 1}, supp(1 ω ) + [ 1 10K, 1 10K ], (2.28)

15 FRACTAL UNCERTANTY FOR TRANSFER OPERATORS 15 there exists an operator Q (s) : L 2 2 (R) L (Ṙ) uniformly bounded in h, such that A = ψ B(s)ω Q (s) + O(h ) L 2 (R) L 2 (R). We recall that the proofs of both lemmas are given in the Appendix. 3. Proof of the main theorem We recall that M = Γ\H 2 is a convex co-compact hyperbolic quotient and that we fix a Schottky representation for Γ as in 2.1. Finally we assume that the limit set Λ Γ satisfies fractal uncertainty principle with exponent β 0 in the sense of (1.5). As seen in (2.13), the main theorem follows from showing that for we have s := 1 2 ν + i, 0 ν β 2ε (3.1) h u H(D), L s u = u = u 0. (3.2) That follows from progressively obtaining more and more bounds on solutions to the equation L s u = u, until we can use the fractal uncertainty principle (1.5) to show that u 0. As shown in [DZ16, 5.1], we always have β 1. This implies the inequality Re s 0 2 which we occasionally use below A priori bounds. We first use the equation in (3.2) to establish some a priori bounds on u. Note that u = L s u implies that u extends holomorphically to a neighbourhood of D and in particular is smooth up to the boundary of D. The prefactors γ a(z) s in (2.12) are exponentially large in h when z is not on the real line. To balance this growth we introduce the weight w K : C (0, ), w K (z) = e K m z /h. The constant K is chosen so that a sufficiently high power of the transfer operator L s is bounded uniformly in h on the weighted space induced by w K : Lemma 3.1. There exist K 10 and n 0 N depending only on the Schottky data so that, with D 2 D defined in (2.10), sup w K L n 0 s f C Γ sup w K f for all f H(D). (3.3) D D 2 Proof. Using Lemma 2.1 and the contraction property (2.7), we choose n 0 such that sup D b γ a 1 2 for all a W n0, a b A.

16 16 SEMYON DYATLOV AND MACEJ ZWORSK (See (2.2) for definitions.) n particular, since γ a maps the real line to itself, we have m γ a (z) 1 2 m z for all a W n 0, a b A, z D b. Take z D b for some b A. Then w K L n 0 s f(z) w K (z) γ a (z) s f(γ a (z)) a Wn 0 a b C Γ a Wn 0 a b a Wn 0 a b e (K m z +2 arg γ a(z))/2h sup D 2 w K f w K (z) γ a(z) s w K (γ a (z)) sup D 2 w K f where in the last inequality we use the bound γ a(z) s C Γ e arg γ a (z)/h. t remains to choose K large enough so that arg γ a(z) 1 2 K m z for all a W n 0, a b A, z D b which is possible since arg γ a(z) = 0 when z R. We next show that for u satisfying (3.2), the norm of u on D is controlled by its norm on = D R: Lemma 3.2. We have sup D w K u C Γ sup u. (3.4) Proof. Let n 0 come from Lemma 3.1. Since u = L n 0 s u, (3.3) shows sup w K u C Γ sup w K u. (3.5) D D 2 Applying Lemma 2.3 to the functions exp(±ikz/h)u(z) in D ±, we get for some c (0, 1] depending only on the Schottky data sup D 2 w K u ( sup Together (3.5) and (3.6) imply (3.4). u ) c( sup w K u ) 1 c. (3.6) D 3.2. Cutoffs and microlocalization. Let Z(h ρ ) W be the partition defined in (2.15), where ρ (0, 1) is fixed as in (1.5). By Lemma 2.4 L s (h ρ )u = u where L s (h ρ ) := L Z(h ρ ),s. (3.7) That is, u(x) = L s (h ρ )u(x) = a b γ a (x)s u(γ a (x)), x b, b A. (3.8)

17 FRACTAL UNCERTANTY FOR TRANSFER OPERATORS 17 From [BD17, Lemma 2.10] and Lemma 2.1, we have for all a = a 1... a n Z(h ρ ) C 1 Γ hρ a h ρ, (3.9) C 1 Γ hρ a C Γ h ρ, (3.10) C 1 Γ hρ γ a C Γh ρ on a1. (3.11) Let be defined by (2.5); recall also the definition (2.4) of a. Since Re s 0, (3.8) gives sup u C Γ sup u C Γ sup u (3.12) a Z(h ρ ) a where for the first inequality we used u = L s u. That is, the behavior of u on (and thus by Lemma 3.2, on D) is controlled by its behavior on the intervals a for a Z(h ρ ). We now define pieces of u localized to each interval a. a Z(h ρ ) we choose a cutoff function To that aim, for each χ a C c ( a; [0, 1]), supp(1 χ a ) a =, sup j xχ a C Γ,j h ρj. (3.13) This is possible since d R ( a, R \ a ) C 1 Γ hρ by (3.11). We then define u a := x 2s χ a u C c ( a), a Z(h ρ ). (3.14) Equation (3.8) implies that, with T γ,s given by (2.22), x 2s u(x) = T γa,su a (x), x b, b A. (3.15) a b The next lemma shows that u a s are semiclassically localized to a bounded set in ξ. This is where we use that ρ < 1: if we instead put ρ = 1 then multiplication by χ a would inevitably blur the support of the semiclassical Fourier transform. Lemma 3.3. Let K 10 be chosen in Lemma 3.1. Then for each a Z(h ρ ), u a is semiclassically localized to frequencies ξ 3K. More precisely, for all N, 2 F h u a (ξ) C Γ,N h N ξ N sup u when ξ 3K. (3.16) 2 Remark. A finer compact microlocalization statement can be given using the FB transform see [Ji17, Proposition 2.2]. Proof. For each a Z(h ρ ), let χ a C c (D a) be an almost analytic extension of χ a, more precisely for each N χ a R = χ a, h ρ z χ a (z) C Γ,N ( h ρ m z ) N. See for instance [Zw12, Theorem 3.6] for a construction of such extension; here we map D a to the unit disk and use the derivative bounds (3.13). Let z 2s := (1 + z 2 ) s be the holomorphic extension of x 2s to D a. We note that z 2s C Γ exp(2 m z /h), z D a.

18 18 SEMYON DYATLOV AND MACEJ ZWORSK Since u is holomorphic in D a, we have F h u a (ξ) = u(x) x 2s e i h ξx χ a (x) dx R = (sgn ξ) u(z) z 2s e i h ξz z χ a (z) d z dz. D a {(sgn ξ) m z 0} Using Lemma 3.2 and the fact that ξ 3K/2 we estimate for all N, F h u a (ξ) C Γ,N sup u C Γ,N sup u 0 0 (h ρ y) N e (K ξ +2)y/h dy (h ρ y) N e ξ y/5h dy C Γ,N h 1+N(1 ρ) ξ 1 N sup Since ρ < 1 and N can be chosen arbitrary, (3.16) follows. u. Lemma 3.3 implies that the sup-norm of u a can be estimated by its L 2 norm: Lemma 3.4. We have for all N and a Z(h ρ ), sup u a C Γ h 1/2 u a + C Γ,N h N sup u. (3.17) Proof. We use the Fourier inversion formula: u a (x) = (2πh) 1 R eixξ/h F h u a (ξ) dξ. We split this integral into two parts. The first one is the integral over { ξ 3K }, 2 which is bounded by C Γ h 1 F h u a = C Γ h 1/2 u a. The second one, the integral over { ξ 3K }, is bounded by C 2 Γ,Nh N sup u by Lemma 3.3. Since the intervals a, a Z(h ρ ), do not intersect and are contained in, we have #(Z(h ρ )) C Γ h 1. Combining this with (3.12) and (3.17), we get the following bound: ( ) 1/2 sup u C Γ sup u a C Γ h 1 u a 2. (3.18) 3.3. End of the proof. We use (3.15) together with equivariance of the operator B(s) (Lemma 2.5) to obtain a formula for u in terms of B(s), see (3.26) below. Together with the fractal uncertainty bound (1.5) this will give u 0. n order to take advantage of the equivariance of B(s), we approximate u a s appearing in (3.15) by functions in the range of B(s). For that, let K be chosen from Lemma 3.1. Define the partition Z( 1 ) by (2.15). Since h is small and by (3.10), 10K a Z(h ρ )! ã Z ( 1 10K ) : ã a, ã a. (3.19) We stress that 10K ã 1 and that ã lies in Z( 1 ) which is a finite h-independent 10K set. Choose h-independent cutoff functions (see (2.4)) χã C c ( ã), supp(1 χã) ã = ; χ K C c (( 2K, 2K)), supp(1 χ K ) [ 3K 2, 3K 2 ] =

19 FRACTAL UNCERTANTY FOR TRANSFER OPERATORS 19 ã v a u a v a w a γ a = γ 1 a a Figure 5. The supports of u a, v a, w a (shaded) and the intervals ã, a (marked by endpoints). The top picture is mapped to the bottom one by γ a. The supports of u a, w a have size h ρ. and consider semiclassical pseudodifferential operators Aã := χã(x)χ K (hd x ) = Op h ( χã(x)χ K (ξ) ). By Lemma 3.3 and since supp u a a ã, we have for all a Z(hρ ) and all N u a Aãu a C Γ,N h N sup u. (3.20) We now apply Lemma 2.7 with := ã and A := Aã to write Aã = ψãb(s)ωãqã + O(h ) L 2 (R) L 2 (R) (3.21) where Qã : L 2 2 (R) L (Ṙ) is bounded uniformly in h and Define (see Figure 5) ψã C c ( ã), supp χã {ψã = 1}, ωã C (Ṙ \ ã). (3.22) v a := ωãqãu a L 2 (Ṙ), v a L 2 (Ṙ) C Γ u a L 2 (R). (3.23) Then (3.20) and (3.21) imply that for all N, u a ψãb(s)v a C Γ,N h N sup u. (3.24) To approximate u by an element of the image of B(s) we use the equation (3.15) and the equivariance property (2.23). Let a Z(h ρ ), b A, a b. By (3.22), we have ψã = 1 on γ a ( b ) = a ã. Thus by (2.23) T γa,sψãb(s)v a = B(s)T γa,1 sv a on b. (3.25) Substituting (3.24) into (3.15) and using (3.25), we obtain the following approximate formula for u, true for each N and b A: x 2s u B(s) a b L w a C Γ,N h N sup u (3.26) 2 ( b )

20 20 SEMYON DYATLOV AND MACEJ ZWORSK where 2 w a := T γa,1 sv a L (Ṙ). (3.27) We now establish a few properties of w a, starting with its support: Lemma 3.5. Let a = a 1... a n Z(h ρ ) and put a := a 2... a n W. Then where (τ) = + [ τ, τ] denotes the τ-neighbourhood of. supp w a a (C Γ h ρ ) (3.28) Proof. From the definition of w a and (3.22) we have (see Figure 5) supp w a γ 1 (supp v a a ) γ a (Ṙ \ ã). Recall that ã a, thus ã = a n a n 1... a n m for some m C Γ, in particular m n. Then by (2.1) we have γ an m (Ṙ \ a n m ) = a n m and thus γ a (Ṙ \ ã) = γ a2 γ an m (Ṙ \ a n m ) = a 2...a n m. We have a2...a n m a and by [BD17, Lemma 2.7] we have a2...a n m C Γ a C Γ h ρ. Therefore a2...a n m a (C Γ h ρ ), finishing the proof. We next estimate the norm of w a. from (3.1) enters the argument. This is where the value of the parameter ν Lemma 3.6. We have w a C Γ h ρν u a. (3.29) Proof. The definition (3.27) of w a gives (recalling (2.22)) w a (x) = γ a (x) 1 s S v a (γ a (x)) for x Ṙ. Using the notation of the proof of Lemma 3.5, we can restrict to x a2...a n m. Since m C Γ we have We have for x a2...a n m C 1 Γ γ a n m 1...a 2 (x) S γ a (x) S C Γ γ a n m 1...a 2 (x) S. γ a n m 1...a 2 (x) S = γ a 2...a n m 1 (x ) 1 S, x := γ an m 1...a 2 (x) an m. By Lemma 2.1 and [BD17, Lemma 2.7] we have C 1 Γ hρ γ a 2...a n m 1 (x ) S C Γ h ρ, therefore C 1 Γ h ρ γ a (x) S C Γ h ρ, x a2...a n m, which using the change of variables formula and (3.1) implies w a L 2 (R) C Γ h ρν v a L 2 (Ṙ). Together with (3.23) this gives (3.29).

21 FRACTAL UNCERTANTY FOR TRANSFER OPERATORS 21 We next rewrite the formula (3.26) in terms of the operator B χ (h) defined in (1.3). Let χ 0 C (Ṙ Ṙ) be such that and Put supp χ 0 {(x, x) x Ṙ} = a, b A, a b = supp(1 χ 0 ) ( a b ) =. χ(x, x ) := x x 2ν 1 S χ 0 (x, x ). (3.30) For each b A define the following compactly supported function w (b) := w a L 2 (R). a b Then by (3.28) we have (identifying Ṙ with S1 using (2.19)) B χ (h)w (b) = B(s) w a on b. Thus (3.26) implies a b x 2s u B χ (h)w (b) C L 2 ( Γ,Nh N sup u. (3.31) b ) The supports of w a, as well as the supports of u a, lie in a C Γ h ρ neighbourhood of the limit set and have bounded overlaps. Analysing this closely will give us Lemma 3.7. Denote by Λ Γ (τ) = Λ Γ + [ τ, τ] the τ-neighbourhood of the limit set Λ Γ R. Then supp w (b) Λ Γ (C Γ h ρ ), (3.32) b A max b A w(b) 2 C Γ w a 2, (3.33) u a 2 C Γ 1l ΛΓ (C Γ h ρ ) x 2s u 2. (3.34) Proof. By Lemma 3.5, for each a Z(h ρ ) we have supp w a a (C Γ h ρ ). By [BD17, Lemma 2.7] the interval a has length C Γ h ρ and it intersects the limit set, therefore supp w a Λ Γ (C Γ h ρ ). This proves (3.32). Next, we have the following multiplicity estimates: sup #{a Z(h ρ ) x supp w a } C Γ, (3.35) x R #{a Z(h ρ ) x supp χ a } C Γ. (3.36) sup x R

22 22 SEMYON DYATLOV AND MACEJ ZWORSK Both of these follow from Lemma 2.2. ndeed, from the proof of Lemma 3.5 we see that for a = a 1... a n Z(h ρ ), we have supp w a a2...a n m where m is fixed large enough depending only on the Schottky data. By [BD17, Lemma 2.7] we have C 1 Γ hρ a2...a n m C Γ h ρ. Therefore, the number of intervals a2...a n m containing a given point x is bounded, which gives (3.35). To prove (3.36), we use that supp χ a a by (3.13) and C 1 Γ hρ a C Γ h ρ by (3.10). Now, (3.35) immediately gives (3.33): max b A w(b) 2 L 2 (R) R ( ) 2 w a (x) dx CΓ R w a (x) 2 dx. To show (3.34), we first note that for all a Z(h ρ ) we have supp χ a a Λ Γ (C Γ h ρ ), since a is an interval of size C Γ h ρ intersecting the limit set. We now recall (3.14): u a 2 L 2 (R) = χ a (x) 2 x 2s u(x) 2 dx Λ Γ (C Γ h ρ ) C Γ Λ Γ (C Γ h ρ ) where in the last inequality we used (3.36). x 2s u(x) 2 dx We are now ready to prove (3.2) and hence finish the proof of the main theorem. From (3.31) and the fact that Λ Γ (C Γ h ρ ) b A b we obtain 1l ΛΓ (C Γ h ρ ) x 2s u 2 C Γ max 1l Λ Γ (C Γ h b A ρ ) B χ (h)w (b) 2 + O(h ) sup u 2. (3.37) The fractal uncertainty bound (1.5) and the support property (3.32) show that max b A 1l Λ Γ (C Γ h ρ ) B χ (h)w (b) 2 C Γ h 2(β ε) max b A w(b) 2. (3.38) The estimate (3.33) and Lemma 3.6 give h 2(β ε) max b A w(b) 2 C Γ h 2(β ε ρν) u a 2. (3.39) Due to (3.34), the left hand side of (3.37) bounds the sum on the right hand side of (3.39). Putting (3.37), (3.38), and (3.39) together and using (3.18) to remove O(h ) sup u 2, we obtain for h small enough u a 2 C Γ h 2(β ε ρν) u a 2. (3.40) From (3.1) we have 2(β ε ρν) > 0, thus (3.40) implies that u a 2 = 0 if h is small enough. Recalling (3.14), analyticity of u shows that u 0.

23 FRACTAL UNCERTANTY FOR TRANSFER OPERATORS 23 Appendix We give proofs of the two lemmas about the operator B(s) introduced in 2.5. Proof of Lemma 2.6. For simplicity we assume that supp χ 1, supp χ 2 R; same proof applies to the general case, identifying Ṙ with the circle by (2.19). The operator (χ 1 B(s)χ 2 ) χ 1 B(s)χ 2 has integral kernel K(x, x ) = (2πh) 1 e i h (Φ(x,x ) Φ(x,x)) x x 2ν 1 S x x 2ν 1 S Ṙ (A.1) χ 1 (x ) 2 χ 2 (x)χ 2 (x ) dp (x ). We have x Φ(x, x) = 2(x x ) 1 + 2x x 2, thus x ( Φ(x, x ) Φ(x, x) ) x x /C for x, x supp χ 2, x supp χ 1. ntegrating by parts N times in (A.1) we see that K(x, x ) C N h N 1 x x N. Therefore sup x Ṙ K(x, x ) dp (x ) C. (Here we use the case N = 0 for x x h and the case N = 2 for x x h.) By Schur s Lemma (see for instance [Zw12, Proof of Theorem 4.21]) we see that the operator (χ 1 B(s)χ 2 ) χ 1 B(s)χ 2, and thus χ 1 B(s)χ 2, 2 is bounded on L (Ṙ) uniformly in h. The following technical lemma is useful to establish invertibility of B(s) in Lemma 2.7. To state it we use the standard (left) quantization procedure (2.27). Lemma A.1. Assume that χ 1, χ 3 C c (R) and χ 2 C (Ṙ) satisfy supp χ 1 supp χ 2 = supp χ 2 supp χ 3 =. Let q(x, ξ) C c (R 2 ). Define the operator B = χ 1 B(s)χ 2 B(1 s)χ 3 Op h (q) : L 2 (R) L 2 (R). Then B = Op h (b) where b(x, ξ; h) S (R 2 ) (uniformly in h) admits an asymptotic expansion: ( b(x, ξ; h) χ 1 (x) h j L j χ2 (x )χ 3 (x)q(x, ξ) ), x =x (x,ξ) j=0 x (x, ξ) := x + 2 x 2 x 2 ξ 2x Ṙ, where L j is an order 2j differential operator on R x Ṙx on x, ξ, ν, and L 0 = 1/2. (A.2) with coefficients depending Proof. For simplicity we assume that supp χ 2 R. The general case is handled similarly, identifying Ṙ with the circle.

24 24 SEMYON DYATLOV AND MACEJ ZWORSK For ξ R, define the function e ξ (x) = exp(ixξ/h). By oscillatory testing [Zw12, Theorem 4.19], we have B = Op h (b) where b is defined by the formula (Be ξ )(x) = b(x, ξ; h)e ξ (x). t remains to show that b S and had the expansion (A.2). We compute b(x, ξ; h) = (2πh) 1 e i h Ψ(x,x ;x,ξ) p(x, x ; x, ξ) dx dx R 2 where Ψ = Φ(x, x ) Φ(x, x ) + (x x)ξ, p = 4χ 1 (x)χ 2 (x )χ 3 (x )q(x, ξ) x x 2ν 1 S x x 2ν 1 S x 2 x 2. We have 2(x x Ψ x) = (x x )(x x ), 2x x Ψ = ξ x x x. t follows that Ψ is a Morse function on {x x, x x } with the unique critical point given by x = x, x = x (x, ξ) where x (x, ξ) is defined in (A.2). We have supp b {x supp χ 1 }. Next, for (x, x ) supp p and large ξ, we have x Ψ 1 ξ. Therefore, repeated integration by parts in 2 x shows that b = O(h ) S (R 2 ) when x + ξ is large. The expansion (A.2) follows from the method of stationary phase. To compute L 0 we use that the Hessian of Ψ at the critical point has signature 0 and determinant 4(x x (x, ξ)) 4. Proof of Lemma 2.7. Choose ψ, ω satisfying the conditions in the statement of the lemma. t follows from the definition of x (x, ξ) in (A.2) that x x (x, ξ) 2/(1 + ξ ). Since 10K 1 we have We now put where q(x, ξ; h) C c and has an asymptotic expansion x (x, ξ) / supp(1 ω ) for all x, ξ 2K. (A.3) Q (s) := ω B(1 s)ψ Op h (q) : L 2 (R) L 2 (Ṙ) (R 2 ), to be chosen later, satisfies the support condition supp q supp a {ψ (x) = 1, ξ 2K} q(x, ξ; h) h j q j (x, ξ) as h 0. j=0 The L 2 boundedness of Q (s) follows from Lemma 2.6, whose proof applies to B(1 s). By Lemma A.1, we have ψ B(s)ω Q (s) = Op h (b) where the symbol b satisfies b(x, ξ; h) h j b j (x, ξ), b j (x, ξ) = 1 2 q j(x, ξ) +... j=0

25 FRACTAL UNCERTANTY FOR TRANSFER OPERATORS 25 and... denotes terms depending on q 0,..., q j 1 according to (A.2), and in particular supported in supp a. Here we use that on supp q j supp a we have ψ (x) = 1 and ω (x (x, ξ)) = 1 by (A.3). We can now iteratively construct q 0, q 1,... such that b = a + O(h ) S (R 2 ), finishing the proof. References [Ba16] Viviane Baladi, Dynamical determinants and dynamical zeta functions for hyperbolic maps, [Bo16] David Borthwick, Spectral theory of infinite-area hyperbolic surfaces, 2nd edition, Birkhäuser, [BD16] Jean Bourgain and Semyon Dyatlov, Spectral gaps without the pressure condition, to appear in Ann. Math. (2), arxiv: [BD17] Jean Bourgain and Semyon Dyatlov, Fourier dimension and spectral gaps for hyperbolic surfaces, Geom. Funct. Anal. 27(2017), [Do98] Dmitry Dolgopyat, On decay of correlations in Anosov flows, Ann. Math. (2) 147(1998), [DJ17] Semyon Dyatlov and Long Jin, Dolgopyat s method and the fractal uncertainty principle, to appear in Analysis&PDE, arxiv: [DZ16] Semyon Dyatlov and Joshua Zahl, Spectral gaps, additive energy, and a fractal uncertainty principle, Geom. Funct. Anal. 26(2016), [GLZ04] Laurent Guillopé, Kevin K. Lin, and Maciej Zworski, The Selberg zeta function for convex co-compact Schottky groups, Comm. Math. Phys. 245(2004), [Ji17] Long Jin, Semiclassical Cauchy estimates and applications, Trans. AMS 369(2017), [Na05] Frédéric Naud, Expanding maps on Cantor sets and analytic continuation of zeta functions, Ann. de l ENS (4) 38(2005), [Zw12] Maciej Zworski, Semiclassical analysis, Graduate Studies in Mathematics 138, AMS, [Zw17] Maciej Zworski, Mathematical study of scattering resonances, Bull. Math. Sci. 7(2017), address: dyatlov@math.mit.edu Department of Mathematics, Massachusetts nstitute of Technology, Cambridge, MA Department of Mathematics, University of California, Berkeley, CA address: zworski@math.berkeley.edu Department of Mathematics, University of California, Berkeley, CA 94720