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 ,
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mG
)1,2,1(
2,d ,4
m
1
0
1
2
3
-10
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esential are attractors periodic iv)
points periodic of intervals no iii)
intervals wanderingno ii)
attracting are points turningperiodic i)
Blaschke Products
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values.critical ofposition of control
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of interval no intervals, wanderingno have which ,, type
of : maps modal - 2 continuous all ofset thebe Let
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rotations.by
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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
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not do valuescritical respectivebut their colapses points critical Two ii)
colapses. valuescritical respecties their and points critical Two i)
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i)ii)
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connected.simply and connected is iii)
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modal -2 is :R),,,,,,(
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boundary. togoesboundary and
hismdiffeomorp a is map This . ) )(,),(( )(
by : define and of lift a Take 3)
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Realization of finite combinatorics
. and
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. and ),,( type, degree of :
map modal -2 finiteially combinator a oflift a be Let 1)
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. 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
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).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
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11
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.1 mod )( that so
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Z.)( such that points are ,..., ii)
. of points turningare 10 i)
defineThen
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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
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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)
