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For a hyperbolic polynomial automorphism of C^2 with a disconnected Juliaset, and under a mild dissipativity condition, we give a topologicaldescription of the components of the Julia set. Namely, there are finitely many"quasi-solenoids" that govern the asymptotic behavior of the orbits of allnon-trivial components. This can be viewed as a refined Spectral Decompositionfor a hyperbolic map, as well as a two-dimensional version of the (generalized)Branner-Hubbard theory in one-dimensional polynomial dynamics. An importantgeometric ingredient of the theory is a John-like property of the Julia set inthe unstable leaves.

Structure of hyperbolic polynomial automorphisms of C^2 with disconnected Julia sets