Master Teapot Kathryn Lindsey Boston College on preprint My talk - PowerPoint PPT Presentation

Slices of 's Thurston Master Teapot Kathryn Lindsey Boston College

on preprint My talk today based is " Master Teapot " A characterization Thurston 's of Wu Chen xi with joint on a previous preprint , This builds " Master Teapot 's " The shape of Thurston Chen xi Wu Chen Ki Wu Harrison Bray joint with Diana Davis , , a bit Also about a in progress work joint with Giulio Hamson Bray TI 0220 Diana Davis Giulio TI 0220

↳ , unimodal of critically periodic study topological entropy - maps of intervals c. J self . one critical ↳ " point " ± : is :* . Cutting the interval at post critical set ⇒ Markov partition the critical point . • Leading eigenvalue of incidence . quadratic matrix ' is the gt:=e e.g polynomials Perron Frobenius Th m ⇒ growth rate • a wreak ¥3 A positive , algebraic , real is of is z the all integer that norm conjugates its Galois .

Master Teapot the set is Thurston 's ftp.ccx/R:XisthegrowthrateofsomefeF : = { TIP andzisaGaloisconj#tof]}? ) Two plots of finite approximations , critically unimodal of TIP periodic self - map of intervals

Why study the Master Teapot ? Rich + mystery Yasctare . • it's a natural object to study , inherently interesting . condition for • a weak a necessary . unimmodapa ! Teapot shape gives a critically per be the growth rate of • number to Perron - di ni analogues of pseudo - Anosov l Uniform expand ers are • - - - - s¥¥÷ .EE#easoa?::Eohinsex7asnd:rs.inonta'pnuowYsi:e::Yw 'aE ⇒ E. iahatioenisiretiiihifaitorfiiitgowth rate I a weak Perron # - Anosov - what are the growth ' is rates of pseudo i n open question : surface diffeos ? connections to core entropy • analogues of the horizontal slices " iterated functor systems are " • " restricted Mandelbrot set for [ Etc } hey7G { ZEE : Cz , a goal of my talk is to explain this .

Function Systems ( IFS ) Iterated a single function , generated by " iterative dynamics study - actions IN " Normal , fncx ) action of look at the e.g an IFS generated . , fo fi , fz . , you by maps For . . , the fi 's - group generated by the semi the alphabet f. , , fo . words all under in . . . orbits . contractions of a metric space - e i the fi 's to be . . Usually , you want , bounded a unique closed a limit has A contracting IFS , = Ufick ) K . t set K s . length . wider the set of all - n closed , bounded set image of The any in Hausdorff topology . → Imit set words . Cz ? fyofwzo set Glatz ) Equivalently : for a sequence , wzwz . w . . - w - . . . . ) : Gl wit ) doesn't actually depend on z ( Notice a sequence } = { Gcw , z ) limit set : is w .

a critically - conjugate to unimodal , critically periodic semi is map Every F¥÷÷e;÷÷¥÷n f × with the entropy :O " periodic ' ' tent map same fun branch g. wana . ⇒ . . of ons and I 's It , Itineraries : sequence is which sub interval the corresponding to - Xx , > Cx ) under fx fo - orbit of belongs to ' I . - 2- Tx , , x Cx ) if - Ya orbit hits Ambiguity when . = wit for some is periodic : It , For a critically periodic X It , finite word w , choice of is a canonical and there w . has positive cumulative sign ) under ( you choose the word that is minimal land with itinerary from This agrees the twisted lexicographical ordering , BN . . ) . kneading theory so on set of sequences Twisted lexicographic ordering ✓ µ , ⇒ , in . but - Prefix ,dw ) Prefix ( v ) = , if - w , ⇐ , w=w , wzw ] For v = yvzv , then { . . . . . - . E. iri even - if I Thm : 7<72 ⇒ It , Le Itx Kew is odd if WCEV is , .

with positive cumulative sign The Parry polynomial for a word . . we , wz w=w . PwC ⇒ ' = fay , z - I , ,z 4) - Ofw o - - . - for swomordew then It , 4) w = is a critically periodic slope , if X Notice : o fw = O - I Ci ) = f- PwC X ) o . and , , > . . . = O we , × Pw G) - t with integer cuffs . s a polynomial . . is Pw a root of Pw So → growth rate X is . ball Galois conjugates of X ) " ) " kneading polynomial Cand is the product of a cyclo Tomic polynomial and the ( Pw in ZEES not be irreducible may or may PwC # ydotomic factor . of 7 Galois conjugate For , Note a . Ofw : = fu " ' - of w , , y l ' ) , pro f- . o - - we , > - - . - you , . Galois conjugates " reason why " matter can This is a sis fi in fi words , xd 's under orbits , y 's or follow .

of the projection to Cl of the Master I call the image it r ? set " and denote " Thurston Teapot the Ticino proved that RIP n D is the cp set of all roots of all power series R Z co effs with , it is the set of all Equivalently of the IFS the limit set Az set ZED . generated by fqz Cx ) = Zx { f , ,zCxI= 2- Zx contains thepointt Z E G , and - awn For any word - wi , w - - , #-) F- ( w , z ) : = fwm o f w o . . . , z . . and ZEB w , wz wz For any w = sequence . , . Image by . Thurston - ofwniz " ) Gcw , z ) : = highs fw W , ,z - , z ) = dying F ( Reverse ( Prefix ( w ) ) = { ZED DIP AD w } G ( wit ) I for some sequence : - -

" ( Beag I ⇒ Master Teapot - Dwa " the Shape of Thurston 's - Some results from connected The teapot is ' teapot contains the unit cylinder . x Cl , 23 • S . The • " " Persistence Theorem : inside the unit cylinder • monotone are The part of slices increasing with the height a , E Hz < 72 ¥ X i. e . . , ↳ In particular = Kzn ID , ri r ⑤ . is ↳ which we have seen Gcw , 2-7=1 for some w } ' seq u { ZEID : s

, the part of the slice Ex inside for any X e Cl , 27 ) Theorem ILL - Wu : be characterized as disk D- the closed can some la6k sequence w} ) = I for ' U { ZEB : GCW Azn ⑤ = S , z " " restricted IFS of the limit set Ze tf DD iff the contains the pt 't . Equivalently , words that are 7- suitable in which you Inly allow on sequences a combinatorial condition - suitability . - suitable sequences } is X = { X let My For each X E Cl , 27 . is closed . My , for each 7 condition . closed ) - = ? , µ× It is a ( this implies the Persistence Theorem • My For each 7 , . .

is A- suitable if A sequence Def w ' : , ) > 7 FX Reverse ( Prefix CWDEE Prefix ( Ita (1) equality Uther it has negative cumulate sign (2) If . KH consecutive Os does not contain then (3) If w It I . l = - I - - X y conditions ) K O 's . " renormalization of X satisfies these ( 4) If AFL , then a so that 52<7%2 then is the integer if K , i.e . is the renormalization where D w ' w = DKCW ' ) for sequence some , with 01 and It I with map that replaces o , - does not > 7,2 w ' " ' then if It ,y I I for every X - and - . . . contain KH Os 's KO . ( Z , X ) with a point KATZ Renormalization - L - Wu ( From - Davis Bray - : in the teapot ⇐ " ) ( zzk where teapot , in the , y is is " . ) 52 E X' s - f s 2 integer the k is .

↳ ⇐ , the part of the slice It outside the unit For any XE Cl , 27 Theorem 2 , : = 03 H ( Ita , z ) { z e El ID disk 74,3 ④ is : = o fw . fwmzo C ' ) , z ) : = Living - 1) ' wi - n Hlw where z , ,z . - . - - a slice Hy a point a way to verify that is not in give . and 2 I Theorems can be implemented algorithmically . This far bigger form of this you look at expressions , For points Hal with " . big then the value of the expression z " too • ever is , and if and bigger n , in X-p can't be z . ' D : and Xt For points with IZKI z satisfy . of 7- stability } = { length N words that Let UN CD - CD of Def , × conditions - WN EUN . for every word - wi - w Proposition z¢E , ⇐ s .t - - , > , F N " G ) > ¥z , : . Ofw ' fun , I ° Test bigger and bigger N 's . , ,z . ,

, .gr ④ plotted 2 ways The height It I -8 slice constructive plot all roots of Parry polynomials of critically ' ttahd pggnwidhirazapsei.us algorithm theorem I using critical period be in shown to E 29 white pts are . in teapot black pts the complement of * white pls undetermined i. so black pts undetermined . ( Tested all this N E 18 ) for , µ

an application we have of theorem 1 : As , inside the unit cylinder Master Teapot the part of the Theorem 3 with respect to reflection across the imaginary : is not symmetrical axis . is immediate occur in complex conjugate pacts , so it Galois conjugates c- NiP Cx - iy , x ) , 7) trip ⇐ that Cxtiy . is surprising because the Thurston set REP - which is is symmetrical 3 Theorem under - Z z to the projection of RCP to . e co effs - series with ± I is the roofs of all power . : RIP ND a root of Pf - Z root of such a power is series , degree coefficients If odd by flipping signs of Z is a series formed . the power know this asymmetry is confined Because of renormalization , we above height 52 the part of the teapot . to