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

SLIDE 1

Slices

f

Thurston

's

Master Teapot

Kathryn Lindsey

Boston College

SLIDE 2

My talk today

is

based

n preprint

" A characterization

f

Thurston's

Master Teapot

"

joint

with

Chenxi

Wu

This builds

n a previous preprint,

"The shape of Thurston

's

Master Teapot

"

joint with Diana Davis

,

Harrison Bray

,

Chen Ki Wu

Chenxi Wu

Also

a bit

about a

work

in progress

joint with Giulio

TI 0220

Diana Davis

Hamson Bray

Giulio TI 0220

SLIDE 3

study topological entropy

f critically

periodic

, unimodal

self

maps of intervals

.

↳ "

ne critical

point

"

is:*

.

±:

c. J

Cutting the interval at postcritical set

↳

⇒ Markov partition

.

the

critical point

Leading eigenvalue of incidence

matrix e.g

. quadratic '

is the gt:=e

polynomials

Perron Frobenius Th m ⇒ growth

rate

is

a wreak¥3 A positive

, real , algebraic

integer that

is z the norm

f

all

its

Galois

conjugates

.

SLIDE 4

Thurston's

Master Teapot

is

the

set

TIP

: = {

ftp.ccx/R:XisthegrowthrateofsomefeF

andzisaGaloisconj#tof]}?

Two plots of finite approximations

)

unimodal

, critically

f TIP

periodic self

map
f intervals

SLIDE 5

Why study the Master Teapot ?

.

Rich + mysteryYasctare .

it's a natural object to study, inherently interesting
Teapot shape

gives

a necessary

condition for

a weak

Perron

number to

be the growth rate

f

a critically per

. unimmodapa!

Uniform expanders

are

l

di ni analogues of pseudo
Anosov
-
s¥¥÷.EE#easoa?::Eohinsex7asnd:rs.inonta'pnuowYsi:e::Yw'aE⇒

E.iahatioenisiretiiihifaitorfiiitgowth rate is a weak Perron #

I

' i n

pen question

:

what are the growth

rates of pseudo

Anosov -

surface diffeos ?

connections to core entropy
horizontal slices

are

analogues of the

"

Mandelbrot set for

"restricted " iterated functor systems

hey7G { ZEE

: Cz ,

Etc}

[

a goal of my talk is to explain this

.

SLIDE 6

Iterated

Function Systems (IFS)

"Normal " iterative dynamics study

IN

actions

, generated

by

a single function ,

e.g

.

fncx)

For

an IFS generated

by maps

fi , fz

, . . . , fo , you

look at the action of the

semi

group generated

by

the fi 's

. i

e

.

rbits

under

all words

in

the alphabet f. ,

. . . , fo .

Usually , you want

the fi 's to be contractions of a metric space

. A contracting IFS

has

a limit , a unique closed , bounded

set K

s .t .

K

= Ufick) .

The

image of

any

closed , bounded set

wider the set of all

length

n

words

→ Imit set

in Hausdorff topology .

Equivalently: for a sequence

w

w

, wzwz . . .

set Glatz)

fyofwzo

. . . . Cz?

(Notice

: Glwit) doesn't actually depend

n z

.)

limit set

= { Gcw , z) : w is a sequence} .

SLIDE 7

Every

unimodal , critically periodic

map

is

semi

conjugate

to

a critically

f×

periodic

' 'tent map "

with the

same

entropy :O

fun branch

g.wana.⇒

.

F¥÷÷e;÷÷¥÷n

.

Itineraries :

It,

is

sequence

f
ns and

I 's

corresponding to

which sub interval the

rbit of

'I

belongs to

under fx

.

fo

, >Cx)

Xx

Ambiguity

when

rbit hits

Ya

.

if

, , x Cx)

2-Tx

For a critically periodic X

,

It,

is periodic : It, = wit for some

finite word w

and there

is a canonical

choice of

w.

( you choose the word that is minimal land has positive cumulative sign ) under the twisted lexicographical ordering . . This agrees with itinerary from kneading theory .)

Twisted lexicographic ordering

n set of sequences

in

so

, BN . ✓ µ, ⇒ ,

For

v = yvzv ,

. . -

w=w, wzw ]

. . . .

if

Prefix ( v)

Prefix,dw)

but

w ,⇐, = ,

then {

Kew

if

E.iri

is even -

WCEV

if

is

dd

.

I

Thm : 7<72 ⇒ It, Le Itx

,

SLIDE 8

The Parry polynomial for a word

w=w

, wz . . . we

with positive cumulative sign

PwC⇒ ' = fay , z

Ofw

, ,z 4) .

I

Notice :

if X

is a critically periodic

slope ,

then It, 4)

=

w

for swomordew

and

PwC X)

= f-

we , ×

.

. .

fw

, , >

Ci)

I

= O .

So

Pw

is

a polynomial

with integer cuffs

. s

t

.

Pw G)

= O .

→ growth

rate X

is

a root of Pw

.

Cand

ball Galois conjugates of X)

( Pw

is the product of a cycloTomic polynomial and the "kneading polynomial ")

PwC# ydotomic factor

may or may

not be irreducible

in ZEES

.

Note

:

For

a

Galois conjugate

f 7

,

f-

we ,>

of w

, , y l ') = fu

,

, pro

. Ofw

. .

"'

.

This

is

a reason why

Galois conjugates

"matter "

you

can

follow

rbits

under

words

in fi

, y 's

r

sis fi , xd 's .

SLIDE 9

I call the

image

f the projection to

Cl of the

Master

Teapot

the

"Thurston

set " and denote

it r?

cp

Ticino proved that RIP n D

is the

R

set of all roots of all power

series

Z

with

coeffs

Equivalently

, it is the set of all

ZED

set

.

the limit set Az

f the IFS

generated by fqz Cx) = Zx

{f ,,zCxI= 2- Zx

contains thepointt

For any word

w -

wi ,
awn

and

Z E G , F- (w , z): = fwm

, z

.

. .

f w

,#-) .

For any

sequence

w = w , wz wz . . . and ZEB ,

Image by

W . Thurston

Gcw ,z) := highs fw

, ,z

ofwniz ")

= dying F( Reverse (Prefix ( w)) , z)

DIP AD

= { ZED :

G (wit)

I for

some sequence

w}

SLIDE 10 I

Some results from

"the Shape of Thurston's

Master Teapot

" (Beag

Dwa
The teapot is

connected

.

The

teapot contains the unit cylinder

S

' x Cl , 23 .

"Persistence Theorem

" :

The part of slices

inside the unit cylinder

are

monotone

increasing with the height a

,

i. e

.

X

< 72

⇒

¥

,

E Hz

.

↳In particular

, ri r ⑤ = Kzn ID .

↳ which

we have seen is

s

' u { ZEID :

Gcw , 2-7=1 for some

seq

w}

SLIDE 11

TheoremILL

Wu

)

:

for any X e Cl

, 27 , the part of the slice Ex inside

the closed disk D-

can be characterized

as

Azn ⑤

= S '

U { ZEB :

GCW

, z

) = I for

somela6k sequence w}

Equivalently ,

Ze tf DD iff the

limit set

f the

"restricted

IFS

" in which you Inly allow

words that are 7-suitable

contains the pt 't .

X

suitability

is

a combinatorial condition

n sequences

.

For each X E Cl , 27

,

let My

= { X

suitable sequences}

.

It

is

a

closed condition

for each 7

. My is closed .

.

For each

7 ,

My

= ?,µ× .

( this implies the Persistence Theorem

)

SLIDE 12

Def

:

A sequence

w is A-suitable if

(1)

Reverse (Prefix CWDEE Prefix (Ita

,)

FX

' > 7

(2) If

equality Uther it has negative cumulate sign

.

then

w

does not contain

KH consecutive Os

.

(3) If

It

= I l

I

y

X

. K O's

4) If AFL

, then a "renormalization of X satisfies these

conditions)

(

i.e .

if K

is the integer so that 52<7%2 ,

then

w = DKCW ') for

some

sequence

w ' ,

where D

is the renormalization

map that

replaces

with

It

and

I

with 01

,

and

for every X

'

> 7,2 "

if It,y

I

I . . .

then

w '

does not

KO

's

contain KH

Os

.

(From

Bray

Davis
L - Wu

:

Renormalization

a point

(Z , X) with

KATZ

is in the teapot ⇐

( zzk

, y

")

is in the

teapot ,

where

k

is

the

integer

s -f .

52 E X'

"

s 2

.)

SLIDE 13

Theorem 2

:

For any XE Cl , 27

, the part of the slice It

,

utside the unit

disk is

74,3 ④

=

{ z e El ID

:

H ( Ita , z)

= 03 where

Hlw

, z) : = Living

1)

⇐

' wi

z

n

. fwmzo . .

fw

, ,z C ')

-

Theorems

I

and 2

give

a way to verify that a point is not in a slice Hy .

This

can be implemented algorithmically .

↳

form

,

For points

z

with

Hal

.

you look at expressions

f this

far bigger

and bigger

n ,

and if the value of the expression

is

ever

"too

big

" ,

then z

can't

be

in X-p

.

For points

z

with IZKI and Xt

' D :

satisfy

Let UN

, × = { length N words that

conditions

CD - CD of Def . of 7-stability}

Proposition

:

z¢E, ⇐

F N

s .t . for every word w

wi
WN EUN

, > ,

Test bigger and bigger N's

,

fun

,I ' ° . . . Ofw ,,z

" G) > ¥z,

SLIDE 14

The height

I -8 slice

It

,.gr④ plotted 2 ways

constructive plot all roots of Parry

polynomials of critically

pggnwidhirazapsei.us

'ttahd

using

theorem I

algorithm

critical period

E 29 .

white pts

are

shown to

be in

black pts

in teapot

the complement of *

i.so

white pls undetermined

black pts undetermined

.

(Tested all this

, µ

for

N E 18)

SLIDE 15

As

an application

f theorem 1

, we have :

Theorem 3

:

the part of the

Master Teapot

inside the unit cylinder

is not symmetrical

with respect to reflection across the imaginary

axis

.

Galois conjugates

ccur in complex conjugate pacts ,

so it is immediate

that

Cxtiy

, 7) trip ⇐

Cx - iy , x)

c- NiP .

Theorem

3

is surprising because the Thurston set REP

which is

the projection of RCP to e

is symmetrical

under

z to

Z

.

Pf

: RIP ND is the roofs of all power series with

± I

coeffs .

If

Z

is a

root of such a power

series ,

Z

is a root of

the power

series formed

by flipping signs of

dd

degree coefficients

.

Because of renormalization ,

we

know this asymmetry

is confined

to

the part of the teapot

above height

52

.

SLIDE 16

ur first step towards proving th

. I

is to prove that

roots

in

TD of

reducible Parry polynomials don't matter

those

"extra "

roots are

in the teapot

:

Theorem :

Fox K X

c 2 .

For each I > X

,

define

YI

: =

S

'

U { ZED

: Pw Cz) = o for some admissible word

w

Then

Han DT

=

Yj

.

set

.

w- Ee Iti?

This

is useful

because

you don't have to worry about whether roots

are

r

are not

Galois conjugates of the growth rate

.

f Parry polynomials

cronughly speaking) ,

After that

,

a key technical result

is that if

Wo

is

a

"dominant "

word such that

It, E WF

and

v

is a prefix

f

a 7- suitable

sequence ( and

some other conchtons) ,

then

Wo

. Reverse (v)

is

admissible

.

this fact is the motivation for the definition

f

X

suitability

.

The

reason this is useful

is that the beginning of a word determines

the

leading

root

f the Parry polynomial ,

and the end determines roots in D

.

SLIDE 17

Th I says

:

the

slice It,

n D

= { z e ID : l E

limit set of 7- suitable seqs}

analogy

is

It , →

Mandelbrot sets (this

is * the connectedness

locus)

limit set

→

filled Julia set

fo CZ) = Etc

filled Julia set Rlfc)

= Ezek : fo# → a}

f 7-suitable

seq

JCfc)

= 2K#c)

Mandelbrot set M

= Ec :

OE K Cfc)}

.

Theoremasbyymptaoylea.iysas.ysmi.gr?orcaMisiearwiczpt

, M and I

are

at c

.

B

.

Solomyak investigated asymptotic similarity for the connectedness

locus

and

limit sets .

Proved asymptotic self

similarity of connectedness

locus and limit sets for parameters z that are

" like "

Misiearwkz pts.

Verified that

a few " landmark " pts fall in this category .

Conjecture :

For each slice It,

and

each point

z e 2¥ ,

H

at z

is

asymptotically similar to the restricted

IFS

limit set .at I

.

SLIDE 18

slice

limit

set

limit limit set

set

we hope to prove ldlsprore the

slice

conjecture

in future work .

SLIDE 19

A related

, ongoing project

(with

G

. Tiozzo) : "Bottcher

coordinates

" for the Thurston set REPAID

complex dynamics IFS Thurston set

.

For

ZED

,

define

\

/

141mm #it:S :

# Hui're

. . . otwi ,

=

d l l

, Az)

I

Bottcher coordinatesfor

the Mandelbrot set M

"

polar

" coordinates

n Elm

equipotential

measure "rate

at which critical pt → -

"

SLIDE 20

Clear that 114mm G) = O

⇐

ZERIP

.

Working to

show

level sets

f

141mm

are

Jordan curves

,

114mm

is differentable

most places ,

ele

.

SLIDE 21

Thank you !