5 Local dynamics of holomorphic maps

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Local dynamics of holomorphic maps 5

In this chapter we study the local dynamics of a holomorphic map near a fixed point. The setting will be the following. We have a holomorphic map defined in a neighbourhood of the origin and of the form

f(z) =⁄z+a2z²+a3z³. . . (5.1) We want to find a change of coordinatesÏnear the origin such thatÏ^≠1¶f¶Ï(z) =⁄z (or the first non zero term if⁄= 0). This is called thelinearization problem. Notice that, if we have aperiodicpoint, i.e., a point which is fixed by some iterate of the map, by considering this iterate we can reduce the dynamical study to that of fixed points.

5.1 Attracting and repelling points

In this section, we study the local dynamics at a geometrically attracting or repelling periodic point.

Theorem 5.1.1 (Koenigs). Letf be a holomoprhic map on a neighbourhood of0, fixing the origin. Assume the multiplier⁄at 0 satisfies|⁄|”= 0,1. Then there exists a local holomorphic change of coordinatew=Ï(z), withÏ(0) = 0, so thatÏ¶f¶Ï^≠¹is the linear mapw‘æ⁄w for allwin some neighbourhood of 0. FurthermoreÏis unique up to multiplication by some non-zero constant.

Proof. We will do the proof in the case when|⁄|<1. To prove the theorem when|⁄|>1, apply the same proof to the local inverse off near 0.

We start proving the existence of the mapÏ. Consider the sequence of maps

Ïn(z) = fⁿ(z)

⁄ⁿ .

45

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46 Local dynamics of holomorphic maps Suppose thatÏn converges uniformly to a holomorphic functionÏin a neighbourhood of 0. ThenÏ(0) = 0, and moreover for alln:

Ïn¶f(z)≠⁄Ïn+1(z) = fⁿ⁺¹(z)

⁄ⁿ ≠⁄fⁿ⁺¹(z)

⁄ⁿ⁺¹ = 0.

Hence for the limiting mapÏ(z), we have

Ï¶f(z) =⁄Ï(z),

as required. So we need only show thatÏn converges uniformly in a neighbourhood of 0.

Choose a constant csuch thatc² < ⁄ < c < 1. Choose a neighbourhood D^r of the origin so that|f(z)|< c|z|forzœDr. Thus for any starting pointz0œDr we have

|fⁿ(z0)|< cⁿr.

By Taylor’s Theorem we have that there exists a constantC >0so that

|f(z)≠⁄z|ÆC|z|², zœD^r. Hence

|fⁿ⁺¹(z)≠⁄fⁿ(z)|ÆC|fⁿ(z)|²ÆCr²c²ⁿ. But now

|Ïn+1(z)≠Ïn(z)|=|fⁿ⁺¹(z)

⁄ⁿ⁺¹ ≠fⁿ(z)

⁄ⁿ |Æ 1

⁄ⁿ⁺¹Cr²c²ⁿ= Cr²

⁄ 1c²

⁄ 2n

,

which converges geometrically to 0, sincec²<⁄.Thus theÏnconverge uniformly inDr. Notice that eachÏ^Õ_n(0) = 1, so thatÏis a local conformal isomorphism.

Let us now prove that the mapÏconstructed as above is the only possibility. Suppose thatÂis a second linearization off at 0. Let us show first that

⁄Â¶Ï^≠¹(w) =Â¶Ï^≠¹(⁄w)

Letzbe a point in a neighbourhood of 0 so that

Ï¶f(z) =⁄Ï(z) =⁄w.

ThenÂ¶Ï^≠¹(⁄w) =Â¶Ï^≠¹(Ï¶f(z)) =Â¶f(z) =⁄Â(z) =⁄Â¶Ï^≠¹(w).

Now, Â ¶Ï^≠1 is a holomorphic function defined in a neighbourhood of 0, so we can express it as its power series: Â¶Ï^≠1(w) = b1w+b2w² +b3w³ +. . ., but now,

⁄Â¶Ï^≠1(w) =Â¶Ï^≠1(⁄w)gives up

⁄b₁w+⁄b₂w²+⁄b₃w³+· · ·=⁄b₁w+⁄²b₂w²+⁄³b₃w³+. . . .

Since⁄is neither 0 nor a root of unity, this implies that b2=b3=b4=· · ·= 0, so thatÂ¶Ï^≠1(w) =b1w, equivalently,Â(z) =b1Ï(z).

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Assume now thatfis not only locally defined near the origin, but is the local expression of a holomorphic mapf :SæSat a fixed pointz0.

Definition 5.1.2. Thetotal basin of attration ofz0A= (z0)is the set of all pointspœS such thatlimnæŒfⁿ(p)exists and is equal toz₀.

Theimmediate basin ofz0 is the connected componentA0ofAcontainingz0. We leave the proof of the following consequence of Theorem 5.1.1 as an exercise.

Corollary 5.1.3 (Global linearization). Let Abe the basin of attraction of an attracting periodic pointz0. There is a holomorphic mapÏ:AæC, such thatÏ¶f(z) =⁄Ï(z), for allzœAandÏmaps a neighbourhood ofz0to a neighbourhood of 0. Furthermore,Ïis unique up to multiplication by a constant.

We now further specialize our study to the case wheref :C^‚ æC^‚ is a rational map. Let

‚

zbe an attracting fixed point.

There exists a holomorphic mapÂ: (A,z)_‚ æ(C,0)that conjugatesfwith multiplication by⁄. SinceÂis a local conformal isomorphism atz‚there existsÁ>0and a mapping ÂÁ:DæA0that is a local inverse toÂ.

Lemma 5.1.4 (Finding a critical point). This local inverseÂÁ:DÁæA0extends to some maximal open diskD^r. Furthermore,Âr extends homeomorphically toˆDr and the image Âr(ˆD^r)contains a critical point off.

Proof. Step 1. There exists some largestr >0, so thatÏÁextends analytically toD^r. If it was possible to extendÂÁinfinitely far in every direction by analytic continuation, we would obtain an entire functionÂwho range omitsJ(f), but this is impossible by Picard’s Little Theorem (which says that the range of such a function can omit at most one point).

Step 2.Ïis well-defined and holomorphicÂ(D^r)µA0andÂextends homeomorphically to D^r. By Step 1, there exists a largestrsuch thatÂÁextends to a holomorphic mapping Â:D^r æA0, so we need only show thatÂextends homeomorphically toD^r. Take any wœD^r,then⁄wœD^⁄r bD^R.Moveoverfis locally invertible, so that we can extendÂ toD^R by the formula,Â(w) =f^≠1(⁄w).

Step 3.ˆU contains a critical point off. First, consider any pointw₀œˆD^r. IfÂ(w0)is not a critical point off, we can extendÂ to a neighbourhood ofw: SinceÂ(w0)is not a critical point off, there exists a local inversegoff nearÂ(w0), now we can extendÂto a neighbourhood ofw0by the formula

Â(w) =g(Â(⁄w)).

If no point inˆD^r is a critical point of offthen sinceˆD^r is compact, we can extendÂ to some larger diskD^r^Õ, but this contradicts Step 1. SoÂ(ˆD^r)contains a critical point of f.

Theorem 5.1.5. Iff is a rational map of degree at least 2, then the immediate basin of attraction of any attracting cycle contains a critical point.

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48 Local dynamics of holomorphic maps Proof. Super attracting cycles contain critical points so their immediate basins of attraction contain critical points to. So suppose thatz0‘æz1‘æ. . .‘æzp≠1 ‘æzp=z0is a geometrically attacting periodic orbit with periodp. Thenf^p(z0) =z0, so thatz0is an attracting fixed point off^p. By Lemma 5.1.4 there exists a critical pointc‚off^pin the immediate basin of attraction of z0, but now Df^p(c) =‚ f^Õ(‚c)f^Õ(f(‚c))f^Õ(f²(c))‚ . . . f^Õ(f^p^≠1(‚c)) = 0.

Hence some point in the orbit ‚c ‘æ f(‚c) ‘æ . . . ‘æ f^p≠1(c)‚ must be a critical point of f. Thus there is a critical point in the immediate basin of attraction of the cycle z0‘æz1‘æ. . .‘æzp.

5.2 Superattracting points

Theorem 5.2.1 (Bottcher). Suppose thatf(z) =anzⁿ+an+1zⁿ⁺¹+. . .. There exists a local holomorphic change of coordinatesw=Ï(z), withÏ(0) = 0, which conjugatesf to then-th power mapw‘æwⁿin some neighbourhood of 0. FurthermoreÏis unique up to multiplication by some(n≠1)-st root of unity.

Proof. Sketch of the proof of existence: show that the sequence of functions

Ïk(z) = (f^k(z))^1/n^k =z(1 +n^k≠1b1z+. . .)^1/n^k=z(1 +b1

nz+. . .)

converges uniformly. Notice that the expression on the right gives an explicit choice of then^k-th root.

It is easy to argue as in the proof of Koenigs’ Linearization formula that a uniform limit of theÏksatisfies the conjugacy relation.

First observe that

z^1/n_k+1^k+1

z_k^1/n^k = z_kⁿ(1 +O(zk))^1/n^k+1 z_k^1/n^k

= (1 +O(zk))^1/n^k+1 = 1 +O(zk/n^k+1.) Askæ Œthat converges to 1 superexponentially quickly.

Now,

Ï(z) = lim

kæŒz_k^1/n^k =z ŸŒ k=0

z_k+1^1/n^k+1 z^1/n_k ^k

=z ŸŒ k=0

(1 +O( zk

n^k+1)), which is a convergent infinite product.

The proof of uniqueness is left as an exercise.

Notice that every polynomial map has a superattracting fixed point atŒ. Thus the above theorem applies in this situation. We will make use of this theorem more in detail in the sequel.

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5.3 Parabolic points

In this section we study the dynamics near aparabolicfixed point, i.e., a fixed point where the multiplier is a root of unity. The following elementary results shows that in this case, in general, we have no hope to conjugate the map to its linear part.

Proposition 5.3.1. Letf be a holomorphic map fixing the origin and such that there exists qsuch that(f^Õ(0))^q = 1. Thenf is linearizable at the origin if and only iff^q is equal to the identity.

Proof. Suppose that there exists a mapÏdefined in a neighbourhood of the origin and such thatÏ¶f^q ¶Ï^≠1=z. This immediately implies thatf^q(z) =Ï^≠1¶Ï(z) =z, and one implication follows.

Conversely, assumef^q(z) =z. We need to build a conjugacy forf. Let⁄:=f^Õ(0)as usual. We consider the map

Â(z) := 1 q

q≠1

ÿ

j=0

⁄^≠^jf^j.

It is immediate to check thatÂ(z)is defined and invertible in a neighbourhood of the origin and thatÂ¶f¶Â^≠1(z) =⁄z. This completes the proof.

By considering an iterate of the system, we can assume that the multiplier is equal to 1.

We thus have to study the local dynamics of a map of the form

f(z) =z+a_n+1zⁿ⁺¹+· · ·=z(1 +a_n+1zⁿ+. . .), an+1”= 0.

Notice, in particular, that this time the system is not invertible near the fixed point.

This is source of difficulty with respect to the other cases.

There arencomplex numbersvsuch that na_n+1vⁿ = 1,

andncomplex numbersvsuch that

nan+1vⁿ =≠1.

We label these2nvectors by0Æj <2nso that

vj =e^iﬁⁿ^jv0, and nan+1vj = (≠1)^j.

The even indexed numbers are called therepulsion vectorsforf at 0, and the odd indexed numbers are called theattraction vectorsforfat 0.

Proposition 5.3.2. Letfbe a holomorphic map defined on a neighbourhood of0with f(z) =z+an+1zⁿ⁺¹+. . . , an+1”= 0,

with attraction and repulsion vectors vj defined as above. Then if an orbit zn = fⁿ(z0) converges to 0, then either

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50 Local dynamics of holomorphic maps

• the sequence is eventually identically equal to 0 or

• limkæŒk^1/nzk exists and is equal to one of the attracting vectorsvj, j odd.

Similarly, if an inverse orbit

z0 Ω[z≠1 Ω[z≠2Ω[. . . off^≠1tends to 0, then either

• the sequence is eventually identically 0 or

• lim_kæŒk^1/nzkexists and is equal to one of the repulsion vectorsvj,j even.

Finally, each of the attraction and repulsion vectors occurs as a limit.

Associated to each directionvj, there is a “petal”Pj, which is an open neighbourhood of points which are attracted to the origin tangentially to the attraction vector (or repelled tangentially to the repulsion vector).

Theorem 5.3.3 (Parabolic Flower Theorem). Ifz_‚is a fixed point of multiplicityn+ 1Ø2, there exist simply connected petalsPj,j = 1, . . . ,2n, attractign or repelling depends onjodd or even. The petals ca be chosen so that their union with{z‚}is an open neighbourhood ofz._‚ Whenn >1, eachPj intersects each of its immediate neighboroods in a simply connected region but is disjoint from all remaining petals.

When⁄ = 1, each (attracting) petal is invariant for the dynamicsF : P æ P (the repelling petals are invariant for the local inverse). Inside of each petal, it is possible to linearize the dynamics.

Proposition 5.3.4 (Parabolic Linearization). Letf:C^‚ æC^‚ be a rational map with degree at least 2. Assume thatf(p) =‚ p‚andf^Õ(p) = 1. Suppose that‚ P is a petal forp. Then there_‚ is a mapping–:P æCsuch that

–(f(z)) = 1 +–(z), zœPﬂf^≠1(P).

This proposition tells us that the dynamics inside a petal is conjugate with dynamics of translation by 1.

Sketch of proof. Assume thatn= 1andp_‚= 0, so that f(z) =z+az²+. . . , a”= 0.

First conjugatef byz ‘æaz, so that we can assume that f(z) =z+z²+. . . ,

now move 0 to infinity by conjugatingf withz‘æ ≠1/z, to obtain g(z) =z+ 1 +b/z+. . . .

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Whenzis very big, this is evidently like translation by 1. To get the precise conjugation, one sets

Ïn(z) =gⁿ(z)≠n≠blogn,

and proves thatÏnconverges uniformly to a functionÏin a neighbourhood ofŒ, and that for the limiting mapÏ,Ï¶f(z) =Ï(z) + 1.

• Whenf(z) =z+an+1zⁿ⁺¹+. . . , an+1”= 0,f is conjugate to

z ‘æz+ 1 + b1

z^1/p + b2

z^2/p +. . . .

• Iff(z) =⁄z+an+1zⁿ⁺¹+. . ., where⁄is a root of unity, then for somep,f^p(z) = z+. . . near zero, and the petals (fixed byf^p) are permuted in disjoint cycles of orderpbyf.

• Just as for attracting periodic points, the conjugacy inside the petal can be extended to the entire basin of attraction of a parabolic point.

• The basin of attraction is contained in the Fatou set off.

• The boundary of the basin of attraction, which contains the parabolic point, is contained inJ(f).

Arguing as in the case of immediate basins of attraction for attracting cycles one can show:

Theorem 5.3.5. Every parabolic basin of attraction contains a critical point off.

This implies the following bound on the number of attracting and parabolic cycles.

Theorem 5.3.6. For a rational map of degree at least 2, the number of attracting cycles plus the number of parabolic cycles is at most2d≠2(the number of critical points of the rational map).

Corollary 5.3.7. For each attracting and repelling basin, the quotient manifoldP/f is conformally isomorphic to the cylinderC/Z.

5.4 Indifferent points

Only sketched, not in the exam.