DYNAMICS OF TRANSCENDENTAL HENON MAPS

• Lecture 4. (IMUB, November 22, 9h-11h). Escaping and Oscillating Wandering domains. We construct an example of a transcendental Henon map with a wandering domain whose orbits converge to infinity, and of a transcendental Henon map with n oscillating orbit of wandering domains.The first example is inspired by the construction of wandering domains in one variable while the second example is costructed using Runge approximation.

Abstract: I will lecture about ongoing joint work with Arosio, Benini and Peters. This mixes the theories of iteration of entire functions in one complex variable and polynomial Henon maps in two complex variables.

Polynomial dynamics
There exists a radius R > 0 such that D(0, R) is mapped into itself and every orbit starting in D(0, R) goes to infinity. Hence the escaping set I ∞ := {z : f n (z) → ∞} is a Fatou component.

Classification of invariant components [Fatou-Julia]
An invariant Fatou component Ω different from I ∞ is either the basin of attraction of an attracting fixed point |f (p)| < 1 in Ω, the basin of attraction of a parabolic fixed point f (p) = 1 in ∂Ω, a Siegel disk, biholomorphically equivalent to an irrational rotation on the unit disk D.
There is no wandering Fatou component, that is Ω : f n (Ω) = f m (Ω) for all n = m. [Sullivan '85] Transcendental dynamics If f : C → C is transcendental (entire with essential singularity at ∞), there can be escaping wandering domain [Baker '76]: oscillating wandering domain  it is an open question whether there can be orbitally bounded wandering domains.
What about C 2 ?
and δ = 0 is a constant [Hénon '76]. It is an automorphism of C 2 with constant jacobian δ.

Theorem (Han Peters-David Hahn (2018))
There is an invertible polynomial map on C 4 with a wandering domain with bounded orbits.

Definition
We introduce the family of transcendental Hénon maps of the type F (z, w) = (f (z) − δw, z), where f is a transcendental function and δ = 0 is a constant.
Every such F is an automorphism with constant jacobian δ and has nontrivial dynamics: Proof.
If the Julia set is empty, then there is a subsequence F •n k which converges uniformly on compact sets to a holomorphic map G : C 2 → P 2 . Since there is an escaping orbit, G must map at least one point to the line at infinity. The line at infinity is the zero set of a holomorphic function locally. By the Hurwitz theorem it follows that G maps all of C 2 to the line at infinity. However, since F has a periodic point, this is a contradiction.
We explain the main ingredient in the construction of an escaping orbit.
It is similar to the proof in one variable. The key ingredient is Wiman Valiron theory.
Let f (z) = n a n z n be an entire transcendental function. For any radius r , let M(r ) be the maximum value of |f (z)|, |z| = r . Note that a n r n → 0. Hence there is a power n = N(r ) which maximizes |a n |r n . For a given r , pick a point w r , |w r | = r for which |f (w r )| = M(r ). Then in a small disc around w r , f is very close to a monomial, This shows that the image of this disc maps much closer to infinity and the image will cover a very thich annulus. This makes it possible to repeat and thereby construct an escaping orbit. More precisely, the main result in Wiman Valiron Theory is the following, but I wont say anything more about it.

Theorem (Wiman-Valiron estimates)
Let f be entire transcendental, 1 2 < α < 1. Let q be a positive integer. Let r > 0 and let w r be a point of maximum modulus for r , that is, such that |w r | = r and |f (w r )| = M(r ). Let z be such that for all 1 ≤ j ≤ q, where i are functions converging uniformly to 0 in z as r → ∞ provided r stays outside an exceptional set E of finite logarithmic measure. We next discuss the theorem mentioned earlier.
This is a first step towards proving that entire Henon maps have infinite entropy. This is still open.

Example
The map f = e iθ → e 2iθ doubles distance. The iterate f •n (e iθ ) → e 2 n iθ multiply distances by 2 n . The entropy normalizes this to log(2 n ) n = log 2. The map z → z 2 on C has entropy log 2. This comes from the unit circle. The inside of the circle converges to zero and gives no entropy. The same goes for the outside.
The map z → z k has entropy log k .
A polynomial P of degree d has entropy log d. A key property is that if R is large enough, then the image P(∆(0, R)) ⊃ ∆(0, R) and moreover for each w ∈ ∆(0, R), there are d preimages z 1 , . . . , z d ∈ ∆(0, R) (counted with multiplicy)

Example
Let f = k k z n k for a rapidly increasing sequence n k and rapidly decreasing sequence k . Then f has infinite entropy on C. There will be a sequence R k so that f (∆(0, R k )) ⊃ ∆(0, R k ) and moreover for each w ∈ ∆(0, R k ), there are n k preimages z 1 , . . . , z n k ∈ ∆(0, R k ) (counted with multiplicy) Topological Entropy In the case when the space X is not compact, it is not clear how to define entropy. One possibility is to restrict to compact subsets.
Definition (Topological Entropy in the noncompact case) be the maximal cardinality of an (n, δ)-separated set. Then the topological entropy is defined as We show that a similar result as for polynomials (see an above example, point 4) also holds for all entire functions: Theorem Let f be a transcendental entire function, and let n ∈ N. There exists a non-empty bounded open set V ⊂ C so that V ⊂ f (V ) and such that any point in V has at least n preimages for f in V counted with multiplicity.

The Kobayashi metric
A key property of the Kobayashi metric is that it is distance decreasing under holomorphic maps.
This implies that if f : ∆(0, 1) → C \ {0, 1}, then if |f (0)| is very large, then |f (z)| is very large for all |z| < 1/2. The reason is that the Kobayashi metric is distance decreasing. More generally, if C ⊂⊂ D ⊂ C and f : D → C \ {0, 1} and |f (p)| is very large for some p ∈ C, then |f (p)| is very large for all p ∈ C.
Note on entire transcendental functions f : The max value M(R) for f on the circle of radius R goes to infinity faster then any power R j of R. Another important fact: The Picard theorem says that all values in C except at most 1 are taken infinitely many times. This has an important consequence:

Lemma
There exist for any j arbitrarily large R so that M(R) > R j and the minimum m(R) on the circle is less than 1.
In fact, we can prove a stronger result: The point 1 can be replaced by any value α ∈ A R .

Corollary
Let f be entire, transcendental. Then there exist arbitrarily large R so This suffices to prove that nonvanishing entire transcendental functions have infinite entropy.
Note that if we replace A R by two halves, D R , midpoints θ = θ R , then f will have roots because D R is simply connected.

Corollary
Let f be entire, transcendental. Let n be an integer. Then there exist arbitrarily large R so that if f = 0 on A R , then f (A R ) ⊃ A R and covers A R at least n times.
We can finally do the same argument, replacing 0 by any point in A R .

Theorem
Let f be a transcendental function. Let n ∈ N. Then there exist arbitrarily large R and j large and θ ∈ [0, 2π] so that either . In the latter case, each β ∈ A R \ ∆(α, 1 R j/2 ) has at least n distinct and uniformly separated preimages in D R . The oscillating wandering domain Let 0 < a < 1. We construct a sequence of maps F k (z, w) = (f k (z) + aw, az) → F with oscillating orbit (P n ) and diam F n (B(P 0 , 1)) → 0. (4) We ensure that every F k has a saddle fixed point at the origin. Assume that we defined F k with an orbit P 1 , . . . , P n k . First step: use the Lambda Lemma to construct a new oscillation Q 0 , . . . , Q N coming in along the stable manifold of F k and going out along the unstable manifold of F k .
Second step: use Runge approximation to obtain F k +1 connecting the old orbit P 0 , . . . , P n k with the new oscillation (Q j ) via a contracting detour T 0 , . . . , T M , long enough to neutralize (possible) expansion on (Q j ). We modify only the 1-dimensional function f k . Finally we send Q N far away and obtain the point P n k +1 .

Example
Let F (z, w) = 2(z, w). (1) Choose two real numbers 0 < δ << < 1. Let A = {δ < z < }. Let U be the connected component of the Fatou set which is punctured at the origin. If is small enough, A will divide U into three connected components, A, B, C where B = {0 < z ≤ δ} and C = U \ (A ∪ B). If there exists R so that F n (A) ⊂ B(0, R) for all n, then by the maximum principle F n (B) ⊂ B(0, R) for all n and then 0 is in the Fatou set, a contradiction. Hence there must exist a sequence n k so that F n k converges uniformly on A to the line at infinity. In particular there is an n so that f n (A) ∩ { z < } = ∅. We also have that U = F n (A) ∪ F n (B) ∪ F n (C) which again divides U into three disjoint connected sets. Clearly F n (B) contains a punctured neighborhood of the origin. It follows that {0 < z < } ⊂ F n (B). This implies that F −n ({ z < }) ⊂ { z < δ}. Hence both eigenvalues of (F −n ) (0) are strictly less than one. Hence the same is true for (F −1 ) (0) so indeed 0 is a repelling fixed point for F .
(2) Suppose that 0 is an isolated repelling fixed point in the Julia set and let U be the Fatou component with a puncture at 0. Since the Jacobian is larger than one, all limits of F n must be in the line at infinity. Let V be the subset of C 2 consisting of those points for which F −n (z) → 0. This is a Fatou Bieberbach domain. Since F −1 has an escaping point, V is not the whole space. So V has a boundary point p. Let A = {δ < z < } for 0 < δ << << 1. Then the sequence F n (A) converges uniformly to infinity, and hence cannot cluster at p. But there are points q arbitrarily close to p so that F −n (q) → 0. Hence for some n, F −n (q) ∈ A. Contradiction.