An introduction to fractal uncertainty principle Semyon Dyatlov (MIT / UC Berkeley) The goal of this minicourse is to give a brief introduction to fractal uncertainty principle and its applications to transfer operators for Schottky groups Part 1: Schottky groups, transfer operators, and resonances Schottky groups of SL ( 2 IR ) on the action , Using ' 70 d z 3- H = fz E l Im its and on boundary H = IRV { a } by Mobius transformations : f- (Ibd ) ⇒ 8. z = aztbcztd Schottky groups provide interesting nonlinear dynamics on fractal limit sets and appear in many important applications To define a Schottky group, we fix: of r disks • 2. a collection nonintersecting IR in ¢ with centers in := . AIR Q . , Dzr Ij Dj := and 1 . • denote A { ,2r3 . Ifa Er Atr if I = , f - r if rsaE2r a , that • 8 . such , . Kr fix maps . , ' )= ti racial D; Da , E- Pc SLC 2,112) • The Schottky group free ti . is the group generated by . .fr Example of a Schottky group Here is a picture for the case of 4 disks: IC = = Cil K ( ) Dz , 8. ( Dj ) Dz , IDE - =D ID ) - Ds ( IC ID:) BCCI : , Oy , 83 ⇒ A Schottky quotients ' of Ta k i n g the quotient CH , ) action of P we a the , get by surface co - . convex compact hyperbolic µ ' - M - T l H IIF plane # Three-funneled surface in G - funnel infinite end Words and nested intervals the = 2r encodes Recall I 91 . } that , . P of the I := at r generators group , n : • Words of length " = W . an far / Vj , ajt, taj } ' = : . ⇒ = . ⑨ . - A , An OT A , Ah i • Group elements : : = E T - . - 1-7 Ei a . an Ja Jan . ta , ° Intervals (disks : = ( . ( I . Ian) Da ta Dan) , @ Ta o • Nesting property : Das C Dois ' Picture of the tree of nested disks and intervals →← 3 4 2 114274 2%13 32%534 41¥43 11412414 ! The limit set of T Define the limits C = IR Ap n Das n? Ff w set with fractal structure It is a compact : on A- THI Connection to #sifw if A on M is geodesic trapped' both endpoints of its lift to H lie in Ap Ei . Transfer operator Denote ) the by ACD Hilbert space of L2 holomorphic functions on D : = ayy Da For define the see , transferoperatorh.si- Seco ) → RCD) then If f E fl CD) and z E Db ' ' ↳ffz)=aa¥¥ Ha Cz)) f Coa Cz )) ZE Dz ① ↳f- Cz ) = Jj GtfCK GD +K' Eff Ck Cz)) + 8 Gtf CK H ; , ) 84 = Di , % Dy , D ① , Dz 2 Dy Mapping properties of the transfer operator ↳f- G) ZE =¥z8a'Hsf CRIED , Db since ) E Da Jacob , ' ↳ Il D) → fl is traced #dess?Lg.hffz)=ff⇒ : 0 DCO = b HIDEO, 7) , , { HILL} hfH=÷if¥¥dt "' HEEP fie , is rank I The zeta function Define the - : = ) selbergzetafuncti.nus) det CI Ls also be in terms It can expressed of the "set " L of the lengths µ M : of predge¥ on - est He = L - e ) Scs) ! ( >31 eEfm when Re s [Borthwick, Spectral theory of infinite area hyperbolic surfaces] to how 5 helps count length spectrum similarly the Riemann 5 function helps count primes Resonances - isthe - 3G) = det CI - Ls)=etfµ ( I e ) We call the zeroes of 5 Cs) M resonances of . - - hot invertible Note s a resonance ⇐ I Ls <⇒FuEHCD3:↳u e C- l es Cest) If #f Lm -13=0Ts a 8>0 then there are ne for some , (the M in { Re s > or } - converses) yresonanceshe is converse true Cup to an e) Resonance free regions the smallest or such that ⑦ What is with Res >8 ? Scs) has no zeroes 8 exists OE 8<1 ① Such , , 8 is a resonance (i.e . 507-0) & there are no other resonances s on the line Res = or ° [Patterson, Sullivan] ⑦ Is there e > 0 such that 8 is with s- the only resonance the s > e? 8=0 → 2 disks if- 8>0 ( ① IES case) , elementary [Naud 2005, using Dolgopyat 1998] Application: remainder exponential theorem: in the -geodesic ?/µ¥⑤ prime ) e > O not the same . F ( -4T 8T ) = ) to # -13 life fees HELMlet Ts a lick EYE ~ Ex ① What is the smallest a such that there are only s 72T finiklymauyresohsFEEffaiwaiapthaues.IE, KNOWN : 8 > o ÷¥¥÷÷÷÷¥÷¥:i÷÷÷÷- E ( if ) ⇒ ° = of L E [Lax–Phillips] uses spectral theory Am Recent results on spectral gaps & where < resonances in { Re s 723 > • = O a = - E [ c (Ar ) Z , [Bourgain–D 2018] whens > o = > 0 ( ) • - E E (8) = 8 E L , [Bourgain–D 2017] to • The above use reduction fractal uncertainty principle [D–Zahl 2016] [D–Zworski 2020] Gaps for finite covers of Ta ke some family then H2 is a finite cover finiteindexsubgroupspg.CI, Mj Tq l of M= TIKI . : F E > ① Is there a uniform spectral gap otter resonance in Re s > 8 - e} 8 is the only { Sometimes sometimes no . ① yes , [Bourgain–Gamburd–Sarnak 2011, Oh–Winter 2016, Magee–Oh– Winter 2017, Jakobson–Naud–Soares 2019, Magee–Naud 2019, Magee–Naud–Puder 2020…] : 01 ① Always have a high-frequency Eris:#sinners> ¥7,Emre, > c , [Magee–Naud 2019] ¢ Patterson–Sullivan measure The P- S measure is a probability limit set measure µ on the Ap which is F- equivariant : - ) - flow Dsdulx fpfdn f. V-8 E T If Lg is the transfer operator ×EIb ) - 8'aCx5fC8akD . Lgfcx ¥, : the kernel of I - LF then µ spans tf Chof) die , Safdie f. ' Regularity of the Patterson –Sullivan measure C- e a- C Here are some basic properties of Schottky groups: " • It then x E- a .. - an EW ( ) Iast If . 8£ = . ( a - here It - ) , ah a Ia FE Ian , • ex) . .CI . D= To f u meta) tucks . faire Therefore µ Cia) n Italo This is called 8- of regularity µ and implies that dinge an -_ dimmer)'S € I - - - - - - - - € Ee Eu - tin , .. ± .ie . ÷÷i¥÷÷÷± Part 2: from fractal uncertainty principle to spectral gap The standard gap KD ZED fora , Recall : LsfG) =¥z Riff , then - theorem If Re s > 8 det (I Ls) #0 - Then Fu ) : - u Proofs Assume not . END Lsu Thus th his u=u , Now hisUk) -¥.ge?aiHsUHaAD.zEDb h so qyksulscsq.PH#wntIals . But 8 and Re s > , ¥wtIaf~1 , So O and 0 . Iast Help ,Ent FEW - thus u - I O. Improving over the standard gap are • there We want to show only finitely many resonances with Res > 2 , forsemeI • Since resonances form a discrete set , this is equivalent to the highfrequenays.IE#eivso;ess:anEEnssi5""i¥E 9 resonance . • Assume s is a D) : =u Then F u E Llc Lsu - This implies Liu - u for all n X •Ta ke C E Ib IR . Then ukt-hsuw-E.a.aew.am?faiHsuHaGD Write where s=o of 1 log Then uk) ta Gre Kimura =¥w . ' . then KTH • > O ran 8£65 . If rss I " at ~ % • eirbgtal oscillates frequency (wavelength h) smoothens out : • Uts ugg reduces frequency by HEHIet How fast does u oscillate? u E fl (D) u = = his u , Lsu " rains egoism I , a:# ki " e log her. §¥E¥ ¥ii. MAN I i. CONCLUSION: We expect that ~ % u oscillates at frequencyh i. at wave . e . length log Kian ① The factors e oscillate at different frequencies for different a→ . K 1 we can So when h hope to exploit cancellations in Iowa to of E (and thus u=o) get decay This is very roughly how the method even when 8 o = Re s s of Dolgopyat works… Fractal uncertainty principle • sum above (x)) In the , ultras only depends " ° ul n is close to . For large Iasb , Iab the limit set Ap : - - - re - -, • . ← II .? A i n a n - , ' n n m - h h ] • h > O let Chf A- tf , For , A- be the h- fattening of A • For ) X nfx=y7=f XEC.TK , Supp define the operator Bx Ch ) : tame (hlf = anti Ix- Xcx Bx yl ,ysftydy We satisfies the DEFINITION say Ap FRACTAL UNCERTAINTY PRINCIPLE with FX as h→0 exponent B , if , " = 049 Inch, Bx throstle * That is : f C Ch ) if FEICK) and Supp A- then HB Ch) . , ftp.qq.cnfchllflha " " WHY UNCERTAINTYPRPLE ? is f- CATCH) ⇒ v :=B×ChH Supp localized in frequency localizes v in HVHecn.cn, position A more basic form of F UP replaces BxCh ) the Fourier transform - by - Exit thfcx) = ash) e fly) dy F v E EGR) ⇒ = Och ' if T Ch . I 1117×9112 He , supp then Hull patch Hulk ⇒ holds with • Fop 4=0 HBxlhllle.EC ' - o - n ⇒ = och t ) If CHI h , 419,4114 → no '! . Ch ENT ⇒ Ch) STEN 11 Daren,Bx Barch, Hee ⇒ Fop holds with D= E - 8 Fractal uncertainty principle and spectral gap satisfies Fop theorem Assume It M= with exponent f . Then Titi has only finitely many resonances E >0 - for in { Re s > I f ee } any ⑤ -01 [D–Zahl 2016, D–Zworski 2020] ' Re s > ftp.wifptogiaxiauupsaa#4ieI - SULLIVAN ⇒ - f-up with 9=8 PATTERSON 01 GAP Re s > or Proof of Theorem (FUP implies spectral gap) 1. Setup det I- # O We need ( Ls ) . Assume the then Fu END):↳u=u contrary, = where s o > - E 04h41 , ft 5th I . We have z C- u= u i. e . for Db Lhs , UG) Jai E)Sutra #) Ew:*, n so that CHOOSE " all I EW Ital n h for In ( Not really possible . reality " Lsh is replaced by an adapted power"of↳) 2. Rough localization in frequency E i - : lives at e . ul % , Claim µ frequencies - o for 1313C tu ( %) I = Ocho 131 ) , D Prot : put D= aehh.BA Dz=¥wzDa¥¥¥¥¥a" Since ulzt-a.EE?aEPul8ak 771 we ) E for K get and Tracz Da , suplwk.ulscs.me/wk.ulsCCsup1uDd.Csup1wk.uDiTfazz lul the which claim .