An introduction to 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 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, 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 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 . So snap lwk.ul.CC Sgp implies 3. Cutting into pieces :

ul . Recall From now on we only study µ H) UHH ← :b uh Ew...taNsuC0a tdhepisends ,

. Define ue = Xan u ECT (Zz) hEI, is still up localized Not " not this) at frequencies Sh (XE does spoil Fop the featured in Recall operator - -

I - - fly Xcx, - g) B×Ch) f Cx) Chih) fplx yl )dy related We use a operator

closely' - " Bf Cx) - Guhl :{ Ix -yl fly> dy (recall s = Ot %) Claim we can write

u = to Chd for some , Xp Bras )

' ' Vei . , Ven Ia Huastec, Cllueilheqpg Supp , =¢ ee me ##i-> bit micro . . of . The proof uses a local analysis " " similar Fourier transform • is to B unitary , - I -In 't Fhflxt Chih) f e fly) dy " '' ' so put Vas : = B- uan is far from • Ias The fact that Supp Vas follows from Uas being localized to frequencies Ef : is close to u Bre if Supp Fa ta then E- oscillates too fast 4. Manipulating the sum htt uh )) By , oailxlsuacralx =¥w. . . Bre ) ( ta Cx)) = Z Xa Cra Wraiths ( "

- " Fromtithe definition Bfcxj-czu-ht.IS/x-yl-2sfIy3dyW-

: we get an aquiproperty 's = . ) 8£ HP CB Fa)Cta Cx)) Blais Ciao 8£)) Cx the relation To show this we use property ' '

. - - . x) 8 ) 18k) Nyse Ix yr 8 ( Cy - in B which is where the choice of Ix yl becomes important ' -s Denote wa Cxk 8£63 Vas Has KD Then at) gives Cup to Ocho )) Cx) Uk Cta . ? a- Bwa )=¥wn. of was : -Properties ' c = ) • ( Supp we ta Supp Van IE sifpvanza.at

- where . . - . . . E- Here . I a an E a- n , .

=- OE - ta - p tf - E - Supp Vas Op

• x E so ja (x) n ht for Supp was ,

Hwalle~hkes-tzllvallu~ho-tzllua.tl e

- (recall that s - Ot ith ) 5. Using FUP to finish the proof

- Denote = W . V Iwata , wwe Then 1*7 gives Cup to Ocho )) ftp.XBXW where - X ~ B Bxlh ) ZwXa Ataru, . - - "well H n Kaali 11Wh : And VIII. E , ; E

. Thus 11 with what Huett .

) # . - K¥1101 HEIKE Atta, a,B×Ch mallee ' to Ei he . . I -4Wh sinkage't . Part 3: Fourier decay and fractal uncertainty principle TODO