Solenoids in Holomorphic Dynamics

Abstract

This Master's thesis is a report on the work of Hubbard and Oberste-Vorth. We study the emergence of solenoidal structures in the dynamics of complex Hénon maps, with particular emphasis on their realization through inductive limit constructions. Starting from a class of solenoidal mappings 𝜏_d : T → T of the solid torus, we construct an associated invariant set Σ_d = ⋂_{m ≥ 0} (𝜏_d^{∘ m}(T)),

and extend τ_d to an orientation-preserving homeomorphism h_d : S³ → S³ admitting Σ_d as an invariant solenoid.

We show that the inductive limit lim→ (T, 𝜏_d) is naturally homeomorphic to the complement S³ \ Σ_d, providing a global topological model for the dynamics. This construction is then related to the dynamics of complex Hénon maps by analyzing the escaping set U₊ and its filtration by preimages of a forward-invariant region V⁺. Using a Böttcher-type coordinate φ⁺, we obtain a conjugacy between the dynamics of the Hénon map in U₊ and the solenoidal model. As a consequence, the level sets of the associated Green’s function G⁺ are shown to be foliated by Riemann surfaces whose topology is governed by the solenoidal structure, and whose leaves are biholomorphic to ℂ and dense in U₊. This establishes a direct connection between inductive limit topology and the global organization of dynamics in ℂ².