An introduction to fractal uncertainty principle Semyon Dyatlov (MIT - PowerPoint PPT Presentation

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 action of SL ( 2 , IR ) on Using the z 70 3- = fz E d l Im ' H boundary its and on H = IRV { a } Mobius transformations : by aztbcztd f- ( Ibd ) 8. z = Schottky groups provide interesting nonlinear dynamics on fractal limit sets and appear in many important applications

T o define a Schottky group, we fix: 2. r nonintersecting disks collection of a • with IR centers in ¢ in := Dj AIR . Dzr Ij Q , . . . and • denote A := { 1 . . . ,2r3 . = f Ifa Er if Atr I , if rsaE2r a - r , • fix such that . , Kr 8 , . maps . . ' racial D; )= Da , E- ti • The Schottky Pc SLC 2,112 ) group is the free group generated by ti . . .fr .

⇒ Example of a Schottky group Here is a picture for the case of 4 disks: K ( IC IDE ) = Dz , 8. ( Cil Dj ) = Dz , - Ds , Oy ( IC ID : ) =D , BCCI ID : ) - 83 A

Schottky quotients ) CH ' quotient of Taking the , we get by the action of P a co - compact hyperbolic , surface convex . µ ' - T l H M IIF - plane Three-funneled surface # in G - funnel infinite end

⇒ Words and nested intervals the . . 2r } encodes 91 , Recall that I = . . of the generators P I := at r group • Words of length , n : = far / Vj , ajt , taj } " W . . an ⑨ = A , ' OT . An : = A , . Ah - i • Group . . . . elements : . Jan E 1-7 ta T Ja Ei : = . an a . - . . - . . , ° Intervals ( disks : = Tao . ( Ian ) . ( Dan ) , I @ ta Da • Nesting property : Das C Dois '

Picture of the tree of nested disks and intervals → ← 4 3 2 41¥43 32%534 114274 2%13 11412414 !

The limit set Define the limits of T n ? Ff w IR n Das C Ap = is a compact set with fractal structure It Connection to #sifw on A- THI : M is trapped if A geodesic on lie in Ap both endpoints of its lift to ' H Ei .

T ransfer operator Denote by ACD ) the Hilbert space of L2 functions holomorphic on : = ayy Da D transferoperatorh.si define the For see , - Seco ) → RCD ) z E Db then and If f E fl CD ) ' f Coa Cz ) ) ↳ ffz )=aa¥¥ Ha Cz ) ) '

↳ ZE Dz ① f- Cz ) = Jj Gtf CK GD + K' Eff Ck Cz ) ) + 8 ; Gtf CK , H ) 84 = % Di , Dy , ① Dz D Dy 2 ,

↳ Mapping properties of the transfer operator ↳ f- G) =¥z8a'Hsf CRIED , ZE Db Jacob ) E Da , since is traced ' Il D) → fl #dess?Lg.hffz)=ff ⇒ = { HILL } b : HIDEO , DCO , 7) 0 , hfH=÷if¥¥dt " ' HEEP , fie rank I is

The zeta function selbergzetafuncti.nu Define the - Ls ) s ) : = det CI in terms expressed be can also It " L µ of the lengths of the " set of predge¥ M : on - est He ) e Efm ! ( L - e Scs ) = Re s > 31 when [Borthwick, Spectral theory of infinite area hyperbolic surfaces] length spectrum similarly to how 5 helps count helps count primes the Riemann 5 function

Resonances - is the ) 3G ) = det CI - Ls )=etf µ ( I - e We call the zeroes of 5 Cs ) resonances of M . - - Ls hot invertible resonance ⇐ I Note s a < ⇒ FuEHCD3: ↳ u # f e C- Lm l es -13=0 Cest ) If T sa some 8>0 , then there for are ne M in { Re ( the s > or } yresonances converses ) - is true he Cup to an e ) converse

Resonance free regions that smallest or such is the ⑦ What Res > 8 ? with has zeroes Scs ) no 8 exists , OE 8<1 , ① Such ( i.e 507-0 ) 8 is resonance a . & there other resonances s are no Res on the line = or [Patterson, Sullivan] °

⑦ Is there such that 8 e > 0 is with only the the s > s - e ? resonance 2 disks ( 8=0 ① , if 8>0 → IES elementary - case ) ?/ µ ¥ ⑤ [Naud 2005, using Dolgopyat 1998] Application : remainder exponential in the prime geodesic theorem : - e > O ( not the . . ) F same = life 8T ) to fees . -4T ) # HELM let -13 T sa lick EYE ~ Ex

the smallest a ⇒ ① What is finiklymauyresohsFEEffaiwaiapthaues.IE such that there only are s 72T , ÷¥¥÷÷÷÷¥÷¥:i÷÷÷÷ KNOWN : 8 > o ) ( if - E = E uses spectral theory of Am ° L [Lax–Phillips]

Recent results on spectral gaps in { Re where s 723 < & resonances = Z c ( Ar ) > O - E • a [ = , [Bourgain–D 2018] E (8) > 0 ( whens > o ) = 8 E = • L E , - [Bourgain–D 2017] reduction to • The above use fractal uncertainty principle [D–Zahl 2016] [D–Zworski 2020]

Gaps for finite covers some family of finiteindexsubgroupspg.CI Take , then Mj Tql H2 a finite cover is M= TIKI of a uniform spectral E > otter . F ① Is there : gap in { Re s > 8 - e } only 8 is the resonance yes , sometimes ① Sometimes no . [Bourgain–Gamburd–Sarnak 2011, Oh–Winter 2016, Magee–Oh– Winter 2017, Jakobson–Naud–Soares 2019, Magee–Naud 2019, Magee–Naud–Puder 2020…] a high-frequency 01 ① Always have : Eris :# sinners > ¥7 , Emre , > c , [Magee–Naud 2019] ¢

Patterson–Sullivan measure probability - S The P is a measure measure µ on the limit set Ap F- equivariant - f. flow which is : Dsdulx ) fpfdn - V-8 E T is the transfer operator If Lg Lgfcx ) - ¥ 8 'aCx5fC8akD . × EIb I - LF , kernel of µ spans the : then Safdie f. Cho f) die tf ,

' C- Regularity of the Patterson –Sullivan measure a- C e Here are some basic properties of Schottky groups: 8£ ( x ) It Iast " then • If E- a . . . - an EW . . - ah - a , Ia = FE . ( Ian ) here It • meta ) tucks .CI . . D= faire a , Tofu ex ) . n Italo Therefore µ Cia ) is called 8- regularity of µ This implies that dinge an -_ dimmer ) 'S and € I € Ee Eu - - - - - - - - - ÷÷i¥÷÷÷± .ie tin , . . ± .

Part 2: from fractal uncertainty principle to spectral gap

The standard gap : Lsf G) =¥z Riff fora KD , ZED Recall , then det ( I - Ls ) # 0 Re s > 8 theorem If . Then Fu END ) : Lsu not Assume Proofs - u - his u=u Thus th , Now his Uk ) -¥ .ge?aiHsUHaAD.zEDb so qyksulscsq.PH#wntIals h . But ¥wtIaf~1 and s > 8 , , Re and . So , Ent Iast O FEW Help 0 thus - O . I u -

Improving over the standard gap only finitely many • We want to show there are , forsemeI Res > 2 with resonances • Since resonances form set , discrete a this is equivalent to the highfrequenays.IE#eivso;ess:anEEnssi5 " "i¥E 9 • Assume resonance is a s . u E Llc D ) : Lsu =u F Then for all Liu This implies n - u -

• Take X E Ib C IR Then . ukt-hsuw-E.a.aew.am?faiHsuHaGD Write s=o where of 1 Then uk ) =¥w log Kimura . . . ta Gre ' KTH . If rss then I ran • 8£65 > O oscillates at frequency ~ % • eirbgtal " ( wavelength h ) out smoothens : • Uts ugg reduces frequency by HEH Iet

How fast does u oscillate? u = Lsu = his u u E fl ( D ) egoism , , rains a :# " I e log ki " her . I §¥E¥ . ii. MAN ¥ i.

that expect We CONCLUSION : at frequency ~ % oscillates u h length at i. e wave . . log Kian ① The factors e oscillate at different frequencies for different a → hope . h K 1 can when we So in Iowa to exploit cancellations to get decay of E ( and thus u=o ) This is very roughly how the method even when o = Re s s 8 of Dolgopyat works…

Fractal uncertainty principle ultras ( x ) ) only depends above , • In the sum close to ul Iasb large For n , Iab is ° " . limit set the Ap : - - - . ? ← A , II . - re - , • i a n - n ' - h , h ] n A- Chf A- tf n m h > O , let • For h - fattening of A be the • For XEC.TK ) , Supp X nfx=y7=f define the operator Bx Ch ) : tame Bx ( hlf = anti Xcx , ysftydy Ix - yl

DEFINITION We say ⇒ Ap satisfies the PRINCIPLE UNCERTAINTY FRACTAL , as h → 0 with exponent if FX B , " Inch , Bx = 049 throstle * That is : Supp f C A- Ch ) if FEICK ) and HB , Ch ) ftp.qq.cnfchllflha then . " ? WHY " UNCERTAINTYPRPLE v :=B × ChH is Supp f- CATCH ) in frequency localized HVHecn.cn , localizes position in v

basic form of F UP A more replaces Bx Ch ) by the Fourier transform - Exit fly ) dy thfcx ) = ash ) - e ⇒ F v E EGR ) = Och 1117 × 9112 He , if supp T Ch ' . I Hull patch Hulk then holds with 4=0 • HBxlhllle.EC ⇒ Fop ' - o ⇒ - t ) , If CHI n h = och 419,4114 → no ' ! ENT . Ch ⇒ 11 Daren , Bx Ch ) Barch , Hee STEN ⇒ Fop holds with D= E - 8

Fractal uncertainty principle and spectral gap theorem Assume It satisfies Fop with exponent f M= Titi Then . has only finitely many resonances for any s > I - f ee } { Re E > 0 in -01 ftp.wifptogiaxiauupsaa#4ie ⑤ [D–Zahl 2016, D–Zworski 2020] ' s > I Re - 01 f- up with 9=8 ⇒ PATTERSON - SULLIVAN GAP Re s > or

Proof of Theorem (FUP implies spectral gap) 1. Setup det ( I - Ls ) # O We need the contrary , then Fu END ) : ↳ u=u . Assume s = 5th , where o > I - ft E 04h41 . We have u= Lhs u for z C- Db , i. e . Jai E) Sutra # ) UG ) Ew :* , so that CHOOSE n n h for all I EW " Ital In reality ( Not really possible . is replaced by Lsh " adapted power "of ↳ ) an