Cantat and Dujardin prove that loxodromic automorphisms of C^2 have no smooth or rectifiable Julia slices, preserve no invariant real-analytic foliation, and are rigid under real-analytic conjugacy when a saddle point has non-real eigenvalues.
Topological and geometric hyperbolicity criteria for polynomial automor- phisms of C2
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Some rigidity results for polynomial automorphisms of C^2
Cantat and Dujardin prove that loxodromic automorphisms of C^2 have no smooth or rectifiable Julia slices, preserve no invariant real-analytic foliation, and are rigid under real-analytic conjugacy when a saddle point has non-real eigenvalues.