Complex Hénon mappings with (semi-)neutral behaviour

Continued Fractions in Fractals, Ergodic theory and Dynamics
Thematic week Holomorphic Dynamics and related fields, Warsaw, 11 – 15 May 2026

Lecture by: Remus Radu “Complex Hénon mappings with (semi-)neutral behaviour”

In this talk we discuss the dynamics of complex Hénon maps, a class of polynomial automor phisms of C , beyond hyperbolicity, with a focus on (semi-)neutral fixed points, outlining recent progress and future directions in the field. In the dissipative case, we prove the existence of hedgehogs (compact, connected, full, locally invariant sets) relative to the center manifold of a semi-neutral fixed point and study their dynamics via a local transport theorem which states that the Hénon mapping restricted to the center manifold is quasiconformally conjugate to a holomorphic map in the complex plane. This is based on joint work with T. Firsova, M. Lyu bich, and R. Tanase. In the conservative case, we revisit the Hakim-Écalle theory for germs tangent to the identity in the context of the Hénon family. We use currents to study the local structure near a double parabolic fixed point and exhibit a homoclinic parabolic web inside the Julia set. This is based on joint work with J. Raissy, T. Firsova, R. Tanase, and L. Vivas.

This lecture was partially supported by the Simons Foundation grant (award no. SFI-MPS-T-Institutes-00010825) and from State Treasury funds as part of a task commissioned by the Minister of Science and Higher Education under the project “Organization of the Simons Semesters at the Banach Center - New Energies in 2026-2028” (agreement no. MNiSW/2025/DAP/491).

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