Discrete Holomorphic Local Dynamical Systems

74 Eric Bedford
Fig. 2 Julia set for p(z)=z 2 − 1
in 2D. It is natural to take the corresponding 2-D model to be the complex solenoid.
We set
Σ∗ = {ζ =(ζn)n∈Z : ζn ∈ C∗,ζn+1 = ζ d n }
and we give Σ∗ the infinite product topology induced by Σ∗ ⊂ C ∞ ∗ .Also,Σ∗ is a
group under the operation ζ · η =(ζnηn)n∈Z. Themapσ(ζ) =ζ d is the same as
the (bilateral) shift map, and σ defines a homeomorphism of Σ∗.
Let us write π : Σ∗ → C∗ for the projection π(ζ)=ζ0 and set Σ+ = π −1 (C− ¯D).
Our model dynamical system is now (σ,Σ+); in fact this is the projective limit of
the dynamical system (σ,C− ¯D). The 2-D analogue of C− ¯D is J − + := J − −K.The
following should be clear from the definitions.
Proposition 1.31. If ϕ + extends holomorphically to a neighborhood of J − +,
then it yields a semiconjugacy Φ + : J − + → Σ+ defined at p ∈ J − + by Φ+ (p) =
(ϕ + ( f n p))n∈Z.
Problem: Suppose that f is hyperbolic and that ϕ + extends to J − +.IsthemapΦ +
injective, i.e., does it give a conjugacy between ( f ,J − +) and the model (σ,Σ∗)?
The 2-D version of the circle variable θ is the real solenoid Σ0 := {ζ ∈ Σ∗ :
|ζ0| = 1}. The path components of Σ0 are homeomorphic to R. The “natural” picture
to draw for a complex Hénon map is to start with a saddle point p. The stable and
unstable manifolds W s/u (p) are conformally equivalent to C. Letψ : C → W u (p)
denote a uniformization such that ψ(0) =p. Any other such uniformization ˆψ is
given by ˆψ(ζ) =ψ(αζ) for some α ∈ C∗. A useful picture, then, is to draw the