Full Families on Multimodal Maps on the Circle

E. de Faria, W. de Melo,

P. Salomão, E. Vargas

GOAL

• Find models and parametrize dynamical behaviors.

• Describe each model or behavior.

Multimodal maps on the circle

1mod G

).,...,( typeand degree of :

map modal-2 continuous a oflift a ,

2111 dSSg

mG

)1,2,1(

2,d ,4

m

1

0

1

2

3

-10

G

esential are attractors periodic iv)

points periodic of intervals no iii)

intervals wanderingno ii)

attracting are points turningperiodic i)

Blaschke Products

m

j

k

j

jkiaj

za

azzezb

1

2

1)( 00

values.critical ofposition of control

.1 and ;),,,,,( )1 2222,10 j

ijj

mmm reraRrraa j

. type,and degree of control ),( 2)1

00

m

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. ,,c points critical 2m moreexist may it

but of points critical are ,0,,,,, 3)

121

111

1

Sc

baa

m

amam

open. is R modal -2 is : 4) 2mmb

1a

2a

maj

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j k

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1

1

full is family The b

. attracting are points

turningperiodic and essential are attractors periodic points, periodic

of interval no intervals, wanderingno have which ,, type

of : maps modal - 2 continuous all ofset thebe Let

121

11

m

SSgm

rotations.by

conjugate are maps ingcorrespond theand of essuch valu

most at are there2)(# and finiteially combinator is If

. toconjugatedally topologicis

such that exist there and Given :THEOREM

b

mgPCg

bg

g

.,...,1,2),...,(),...,,( 1212110 mjkkkk jjmm

m

j

k

j

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za

azzezb

1

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1)( 00

1

01222

)).2cos()2sin(()2sin()(m

jjjm

jtjtmtdttT

History

1900). (Poincaré, irrational

isnumber rotation case in the behaviors eparametriz

rotations irrationalrigidy : on hismsDiffeomorp )1 1

S

76). Thurston, andMilnor by y theor

kneading ( behaviors dynamical all esparametriz ]4,0[for

),1()(family quadratic the: ]1,0[on maps Unimodal)2

xxxQ

0 1

4

.boundary togoesboundary and hismdiffeomorp a is 4)()(by given ]1,0[]4,0[: i) cQ

92). Graczyk, -Swiatek (Lyubich, parameter unique a

toscorrespond attractors periodic without mapeach ii)

21c

86). Strien, van - Melo (de behaviors dynamical esparametriz and

modal- is dimension of simplex ain ),,(for

,)(

family polynomial the: ]1,0[on maps Multimodal )3

10

1110

mmaa

xaxaaxP

m

mm

.1,,1,0)()1(:R,, i) 12m

1 mjvvvvV jjj

m

)(1 c

))(( 1 cP

boundary. togoes boundary and hismdiffeomorp

a is ))(,),(()(by given : ii) 1 mcPcPV

03). Strien, van Shen, ,(Kozlovskiparameter unique a

toscorrespond attractors periodic without mapeach iii)

.C of

mautomorphian by n conjugatio toup unique is map rational The

83). (Thurston, cycleLevy no and orbifold hyperbolic with

maps finiteially combinator of case in the behaviors dynamical

eparametriz maps rational : C of coverings Branched 4)

96). Melo, de - (Martens

behaviors dynamical eparametriz schwarzian negative

with maps Lorenz C of families monotone : ]1,0[on maps Lorenz 5) 3

m

j

k

j

jkiaj

za

azzezb

1

2

1)( 00

R modal -2 is : ofTopology 2mmb

).derivative (umbounded

not do valuescritical respectivebut their colapses points critical Two ii)

colapses. valuescritical respecties their and points critical Two i)

ja1ja

ja

i)ii)

spaceparameter The

.1 and ;)(

connected.simply and connected is iii)

j 22 2

2

1

2

1

meraec jj

j

j ijj

im

ja

am

jj

The map of critical values

mimjdvv

vvkvvvvV

m

jjjiiim

m

2,,1 and ,,1 ,

,0 ,0)( )1( :R,, 2)

12

21212

21

m

j

k

j

jkiaj

za

azzezb

1

2

1)( 00

modal -2 is :R),,,,,,(

).1,0( and 1 where ; 2 ,, 2 for

,1 ),1,0[ 1)

2m2210

2

10

mbrraa

reramj

aa

mm

jji

jjj

boundary. togoesboundary and

hismdiffeomorp a is map This . ) )(,),(( )(

by : define and of lift a Take 3)

21 mcBcB

VbB

mammmamm

mama

caccac

caccac

J

122

1

1212

111

1

1111

1

11

1

1 11

1

)( Jac

. , , , , , )(

)( )(

)(det 1212

1211

2,1

21

1 mammma

mjiij

mjiijij

babbabbc

bbcc

J

Realization of finite combinatorics

. and

,...,1 ,2 such that ),,,(any Choose

. and ),,( type, degree of :

map modal -2 finiteially combinator a oflift a be Let 1)

0

1210

12111

j

jjm

m

kdk

mjkkkk

gdSSg

mG

. and )( , typehas : Take 2) | Vb

.,..., if 1)( and ,..., if 1 mod

)(by defined map thebe ,...,1,...,1:Let iii)

.by Zinto mapped points are z ,..., and

of points turning theare ,...,z such that ,..., Choose ii)

Z.)( such that points theand 1 mod iterates

theirall , of points turning thebe 10Let i)

S C I R O TA N I B M O C

21222

)(

t

t21

21

p2m12

21

pmmpmm

jj

t

tpm

ii

s

ttjjttj

zGzss

Gz

Gztt

zGz

Gzzz

m

m

).1 ,2 ,1( typeand

twodegree of map modal-4 a oflift a i)

gG

1z 4z 6z 8z 10z 12z 14z9z

).14,10,8,6,2(),...,( ),12,9,4,1(),...,( ii) 9541 tttt

.10)14( ,6)12( ,10)10( ,14)9(

,2)8( ,6)6( ,2)4( ,2)2( ,8)1( iii)

2z

Thurston map

10:R ),...,( 1s

1 ss xxxxW

).,...,, ,0( ) ,..., ,()(

such that and

of lift a choose ),...,(),...,( define To

211)()()(

11

221 jttt

ss

mxxx

b

ByyxxT

.1 mod )( that so

),(in point unique theis ,For iii)

Z.)( such that points are ,..., ii)

. of points turningare 10 i)

defineThen

)(

1 1

212

21

ii

ttijj

ttt

tt

yBx

yyytit

yByy

Byy

jj

jpmm

m

Uniqueness

rotation. aby n conjugatio toup equal the

thatfinite,ially combinator and that casein that,implies This

sphere.Riemann the

on coverings branched as equivalentThurston are then theycircle on the

conjugateally topologicare and if fixed, For :THEOREM

21

21

bb

bb

trivial

11)( i) m

jm

m xaxaxaxP

11 )1()( ii) m

jm

m xaxaxaxP

11 )1(1)( iii) m

jm

m xaxaxaxP

03). Strien, van Shen, ,(Kozlovskiparameter unique a

toscorrespond attractors periodic without mapeach iii)