3.3 Holomorphic dynamics
3.3.1 The dynamical partition
A special property of holomorphic dynamical systems is the splitting of the dynamical space induced by the concept of normal families.
Definition 3.3.1 (Normal family of holomorphic maps (compact case)). Let U ⊂ bC be a domain and F a family of holomorphic maps from U to bC. We say that F is a normal family in U if any infinite sequence of elements of F contains a subsequence which converges uniformly on compact sets of U to some continuous limit map.
In order to be able to consider the case of a holomorphic mapping f : U → T , where T may not be compact, we need the following definitions. A sequence of points (tn) in a non-compact surface T diverges from T if for any compact subset K ⊂ T , we have tn ∈ K, for n suffficiently large. In a similar way, a/ family of maps fn: U → T diverges locally uniformly from T if for any compact subset K ⊂ S i K0 ⊂ T , we have fn(K) ∩ K0 = ∅, for n sufficiently large (of course, this can never happen if T itself is compact).
Example 3.3.1. The qualification “from T ” in the above definition is essential.
For example, the sequence of points i, i/2, i/3, . . . diverges from the upper half plane H, but converges to 0 within its closure H ⊂ C.
Definition 3.3.2 (Normal family of holomorphic maps (general case)). Let U ⊂ bC be a domain and F a family of holomorphic maps from U to a (possibly non-compact) Riemann surface T . We say that F is a normal family in U if any
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infinite sequence of elements of F contains either a subsequence which converges locally uniformly or a subsequence which diverges locally uniformly from T .
The condition of normality can be phrased in terms of equicontinuity, asking the family to be locally equicontinuous in U with respect to the spherical metric in bC (see the Arzelà-Ascoli Theorem). An easy way to check normality is to apply Montel’s Theorem.
Theorem 3.3.1 (Montel’s Theorem). Let U ⊂ bC be a domain and F a family of holomorphic maps from U to bC. If there exist three distinct points a, b, c ∈ bC such that f (U ) ⊂ bC \ {a, b, c} for all f ∈ F , then F is a normal family in U . In particular, if there exists R > 0 such that |f (U )| ≤ R for all f ∈ F , then F is a normal family in U .
In holomorphic dynamics the concept of normality is applied to the family of iterates of the given map f . If {fn}n≥0is normal in a domain U , then orbits of points in U behave in a similar manner. The concept of normality defines the dynamical partition explained in the following definition.
Definition 3.3.3 (Fatou set and Julia set). Given a holomorphic map f : S → S as above, we define the Fatou set of f as
Ff..= {z ∈ S : {fn(z)}n forms a normal family of maps in a neighbourhood of z}, and the Julia set of f as its complement
Jf = S \ Ff.
For f ∈ Ent ∪ Ent∗ (see the List of Symbols), the essential singularities are sometimes considered part of the Julia set, which is then defined in bC. By definition, the Fatou set is open and the Julia set is closed in S, and both sets are totally invariant (i.e. f (Jf) ⊂ Jf and f (Ff) ⊂ Ff). Intuitively, a point z0
belongs to the Fatou set if the dynamics in a neighbourhood of z0 is in some sense tame or controlled, while it belongs to the Julia set if the dynamics in any neighbourhood of z0 shows sensitive dependence on initial conditions1.
Figures 3.2 and 3.3 show typical examples of rational functions. In each case, the Julia set is painted in white and the Fatou set is painted in black. Figure 3.2 shows five examples of polynomial maps of degree two. In Figure 3.2a we can see a Jordan curve. In Figure 3.2b there is a dendrite, that is a compact connected set with void interior and which does not separate the plane. In Figure 3.2c we observe a totally disconnected set, and two last examples are Julia sets whose complementary have infinitely many connected components.
The Fatou and the Julia sets are preserved under topological conjugacy in the sense that if ϕ is a homeomorphism such that ϕ ◦ f = g ◦ ϕ, for f, g holomorphic, then J (g) = ϕ(J (f )) and F (g) = ϕ(F (f )).
Example 3.3.2 (Fatou and Julia sets of f (z) = zd for d ≥ 2). Since any orbit inside D converges to 0 while every orbit in bC \ D converges to infinity, the Fatou
1A mapping f : S → S has sensitive dependence on initial conditions if there exists δ > 0 such that, for any x ∈ X and any neighbourhood U containing x, there exists y ∈ U and n ≥ 0 such that |fn(x) − fn(y)| > δ
(a) A simple closed curve, z 7→
z2+ (.99 + .14i)z.
(b) A “dendrite”, z 7→ z2+ i.
(c) A Cantor set, z 7→ z2+ (−.765 + .12i). (d) The Douady rabbit z 7→ z2+ (−.122 + .745i).
(e) The plane, z 7→ z2− 1.75488 . . . Figure 3.2: Julia sets for quadratic maps.
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(a) z 7→ 1 − 1/z2. (b) z 7→ (c + z2)/(1 − z2).
(c) A Sierpinsky carpet, z 7→ −.138(z + 1/z) − .303.
(d) z 7→ (z5− .00001)/z3.
Figure 3.3: Julia sets for rational maps.
set is equal to bC \ S1 and Jf = S1. Moreover, orbits of points in S1 stay in S1 and follow the dynamics of the map Md: T → T defined by t 7→ d · t (mod 1).
However, the asymptotic behaviour of points arbitrarily close to the unit circle is radically different from the behaviour of those on the circle. Although Julia sets are in general fractals, the dynamics on a Julia set share most of the properties of the map Md on the unit circle.
In the example above, 0 and ∞ are attracting fixed points. In general, if O(z0) = {z0, z1, . . . , zp−1} form a p−cycle, we define the multiplier of the cycle as
λ..= (fp)0(zi) = f0(z0) · f0(z1) · · · f0(zp−1),
for any i ∈ {0, 1 . . . , p − 1}, if the orbit lies in C. If the orbit includes the point at infinity, the multiplier is defined as above after a change of variables that moves the orbit into C. The periodic orbits are classified as
attracting if |λ| < 1. In case λ = 0, we say that it is superattracting.
repelling if |λ| > 1.
neutral (also indifferent ) if |λ| = 1. In case λn = 1 for some integer n,
we call it a rationally neutral or parabolic. Otherwise, it is irrationally neutral.
In Section 3.3.2 we describe how these definitions relate to the local dynamics around the periodic orbits. Observe that the multiplier is zero if and only if the derivative vanishes at one of the points in the cycle. Let us recall that the points z for which f0(z) = 0 are called critical points, and their dynamical behaviour plays an important role in the study of the dynamics of f .
One particular case appears when the fixed point belongs to a set of orbits whose iterates converge towards the point. Such sets are in general called basins of attraction.
Definition 3.3.4 (Basin of attraction of an attracting cycle). Given an at-tracting p−cycle O(z0) of f , we define its basin of attraction A = Af as the set of points z ∈ S such that fnp(z) converges to some zi ∈ O(z0) as n tends to infinity.
The following theorem summarises some of the basic properties of Julia sets and Fatou sets. All proofs can be found in [28, §4, §12, §14] or [8]. In the statement we use the concept of grand orbit.
Definition 3.3.5 (Grand orbit and exceptional set). Given f : S → S and a point z ∈ S, its grand orbit consists of all points in S which are related forwards or backwards with z under iteration of f . More precisely,
GO(z)..= {w ∈ bC : fp(z) = fq(w) for some p, q ∈ N}.
Grand orbits are totally invariant. We define the exceptional set E (f ) as the set of points with a finite grand orbit. Since there is no possible confusion with this notation in this context, we shall as well write E when the reference to the map f can be omitted.
Theorem 3.3.2 (Properties of Julia and Fatou sets). Consider the map f : S → S as above. Then
(i) For any k > 0, the Julia set Jfk coincides with Jf.
(ii) Every attracting cycle and its basin of attraction belong to Ff. (iii) If A is the basin of attraction of an attracting cycle, then Jf= ∂A.
(iv) Every repelling and parabolic cycle belongs to Jf.
(v) The Julia set is non-empty. If f is a rational function, then Jf contains a repelling fixed point or a parabolic fixed point with multiplier 1.
(vi) The set E (f ) has at most two, one or no points if f is in Rat, Ent or Ent ∗, respectively.
(vii) If z0 ∈ Jf and U is a neighbourhood of z0 disjoint from E (f ), then S \ E(f ) ⊂ ∪n(fn(U )). Consequently, if K is a compact set disjoint from E(f ), there exists N > 0 such that K ⊂ fn(U ) for all n ≥ N . If f is in Rat, then fn(Jf∩ U ) = Jf for all n ≥ N .
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(viii) Either Jf has no interior point, or Jf= S.
(ix) If z0∈ S \ E(f ), then Jf ⊂ ∪n>0f−n(z0).
(x) Jf has no isolated points.
(xi) Jf is either connected or has uncountably many components.
(xii) For z in a generic set of points in Jf, the forward orbit of z is everywhere dense in Jf.
(xiii) Repelling periodic points are dense in Jf.