Topological models in holomorphic dynamics  ime.usp.brsylvain/TalkSylvainFloripa2new.pdf ·...
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Topological models in holomorphic dynamics
Sylvain Bonnot (IMEUSP)
Workshop on Dynamics, Numeration and Tilings 2013
1

Summary from last time
Describe the topology of some nice subsets K C:Here nice means:
K is compact,C K is connected,K is connected,K is locally connected.
How to do it?Use Uniformization theorem: C K is a topological disk biholom. : CD C K,Then: locally connected extends continuously (not necess.1 to 1!) to a map : S1 K.
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Summary from last time
Definition
We define on the circle S1 an equivalence relation K by t k t iff(t) = (t).
Some remarks:The equivalence relation has no crossing (unlinked),Such an equivalence relation (i.e graph closed + unlinked) canonly produce loc. connected spaces.
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Summary: application to the case of quadratic dynamicsPc : z 7 z2 + c
Main object to study: Filled Julia set Kc (for a given c)
K+c := {z Corbit of z is bounded}
Strong dichotomy: (recall thatM := {c Corbit of c is bounded})Case c M: then Kc is connected (but not necessarily loc.connected),Case c / M: then Kc is totally disconnected, a Cantor set.
Added bonus: for K connected, the biholomorphism between C Kand CD is given by the Bottcher map (conjugacy with z 7 z2).Question: parameters c for which Kc is locally connected?
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Geometric description of the pinching model forz 7 z2 1
Extension of the conjugacy to the boundary of the disk: here := 1 has a continuous extension CD(r) C int(K),which induces a continuous map : S1 K (a semiconjugacy).Some necessary conditions:
rays cannot cross (if 1 2 and 3 4 then the intervals(1, 2) and (3, 4) are disjoint or nested).Periodic maps to periodic: the map is a semiconjugacy...
Consequences: The unique fixed point of the doubling map hasto go to a fixed point. The unique 2cycle {1/3, 2/3} has to go to a2cycle (impossible, not in K) or a fixed point.Preimages: the other preimage of 1/3 2/3 is 1/6 5/6.Further preimages:f1(1/6 5/6) = 1/12 5/12 7/12 11/12. Thenoncrossing property will force the correct pairings.
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First few pairings for z 7 z2 1
Subshift :(ex : 1/12 = .00(01) = .11(10)) do DE(E)
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More general recipe
Case of periodic critical point: Pick one root angle of the componentof the interior of Kc containing c and split the circle as follows:
Then for any t S1 define the itinerary I(t) under angle doubling.Equivalence relation: t t if and only if t and t have the sameitineraries.
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One more application of the external rays: Yoccozpuzzles
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Douadys Rabbit (pic by A. Cheritat)
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Application to the Mandelbrot set itself
Bottcher coordinate: c(z) for Pc : z 7 z2 + c.
Theorem (DouadyHubbard)
1 The function c(z) is analytic in c and z;2 The function : c 7 c(c) is welldefined in CM;3 : CM CD is a biholomorphism4 M is connected.
Explicit uniformization of CM:
c(c) = c.
n=0
(1 +
cPnc (c)2
) 12n+1
.
Question (MLC conjecture)
The setM is locally connected?10

Pinched disks IV : what is M for Thurston?
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Pinched disks V : description of the lamination associatedto M
Lavaurs Algorithm:1 Angles that are 2periodic under angle doubling map:
( 13 ,
23
)2 Angles that are 3periodic:
( 17 ,
27
),( 3
7 ,47 ,)
,( 5
7 ,67
)3 Angles that are 4periodic:
( 115 ,
215
),( 3
15 ,415 ,)
,( 6
15 ,9
15
), . . .
4 . . .
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Pinched disks VI : Mandelbrot set
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Pinched disks VII : Mandelbrot set (D. Schleicher)
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Moving to higher dimensions
1 Possible dynamical systems: the list is huge. Here are someexamples
Polynomials endomorphisms: (xy) 7 (x3+x.y+4
3x2y )
Analytic selfmaps of P1 P1, of P2(C), . . .Polynomial automorphisms: example (xy) 7 (
P(x)ayx ) (Complex
Henon mappings).etc . . .
2 Polynomial automorphisms of C2:
Theorem (FriedlandMilnor)
Let f Aut(C2). Then either f is conjugated in Aut(C2) to a composition ofHenon maps, or it is conjugated to a product of elementary maps of the typeE(xy) = (
ax+p(y)by+c ) where ab 6= 0 and p is a polynomial.
3 Small perturbations of quadratic polynomials.
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C and C2: they look the same
Same definitions: analytic functions (as power series),holomorphic functions (are holomorphic separately, in eachvariable).Same kind of theorems example: bounded holomorphicfunctions on C2 must be constant.Exercise: Prove that the nonescaping set of F intersects anyhorizontal line {x = constant}.But nothing is straightforward: Find an example of a nonconstant non open analytic map of C2.
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C and C2 are different: FatouBieberbach domains
Given f Aut(C2) such that (00) is an attracting fixed point:
f(
xy
)= L
(xy
)+ h
(xy
)with
L(
xy
)=
(xy
)where 0 < 2 <  <  < 1.
Theorem
Then the basin U of 0 is biholomorphic to C2.
Proof:First: prove f is conjugated to L near the origin,Then extend this conjugacy by the dynamics: given any x U,there exists N such that f N(x) is in the domain of definition of .We can then extend by the formula
(x) = LN(f N)x.17

Some horizontal slices y = constant
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C and C2: they do not look the same...
Attracting fixed points and linearization in dimension 1: Given
f (z) = z( + O(z)) with 0 <  < 1then near 0 we have a conjugacy
f 1(z) = z.This means that f is linearizable: it is conjugate to its linear part.Attracting fixed points and linearization in dimension 2: Assume
f(
xy
)= L
(xy
)+ h
(xy
)with
L(
xy
)=
(xy
)where 0 <   < 1
andh(xy
) C(x2 + y2) for some C. Then f is not alwayslinearizable at the origin: f
(xy
)=
(x/2
y/4 + x2
).
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FatouBieberbach domains (continued)
Case when 0 < 2 <  <  < 1Set n = Ln f n, and then study n+1 n:
n+1 n = L(n+1) h f n.Choose e > 0 so small that (+ e)2 < , and > 0 so small thatthere exists C such that(xy
) < = f (xy) (+ e) (xy
)h(xy) C (xy
)Thenn+1 (xy
) n
(xy
) = L(n+1) h f n (xy)
1n+1 C((+ e)n
)2=
C
((+ e)2
)n.
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Complex Henon mappings in C2
Basic properties: the map (xy) 7 (P(x)ay
x ) has inverse(xy) 7 (
y(1/a)(P(y)x)) and constant jacobian equal to a.
Crude picture of dynamics:
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Basic invariant sets
Filled Julia sets
K+ :={(
xy
) C2 with bounded forward orbit
}Escaping set: U+ := C2 K+. We have also U+ = n0 Hn(V+).Julia sets J+ = K+, also J = K. We can define K = K+ K.Basins of attraction Ws(p) of fixed points.Stable and unstable manifolds of saddle points p: they areisomorphic to C.What is the topology of these sets? Are they connected?(partial) answer: K is always connected.
Theorem (BedfordSmillie)
If p is a sink for f and if B is the basin of attraction of p then B = J+.
If p is a saddle point for f then the stable manifold Ws(p) is dense in J+.
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Examples of horizontal slices of nonescaping sets
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Picture Stable manifold, real slice
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