Reversors and symmetries for polynomial automorphisms of the complex plane. (English) Zbl 1046.37024

Gómez, A.; Meiss, J. D.

Nonlinearity 17, No. 3, 975-1000 (2004).

Summary: We obtain normal forms for symmetric and reversible polynomial automorphisms (polynomial maps that have polynomial inverses) of the complex and real planes. Our normal forms are based on the Hénon normal form of Friedland and Milnor. We restrict ourselves to the case where the symmetries and reversors are also polynomial automorphisms. We show that each such reversor has finite order and that for nontrivial, real maps, the reversor has order 2 or 4. The normal forms are shown to be unique up to finitely many choices. We investigate some of the dynamical consequences of reversibility, especially, for the case where the reversor is not an involution.

MSC:

37E30	Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces
37C80	Symmetries, equivariant dynamical systems (MSC2010)
37F10	Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
37J15	Symmetries, invariants, invariant manifolds, momentum maps, reduction (MSC2010)
37G05	Normal forms for dynamical systems

Keywords:

Hénon map; normal forms; polynomial automorphisms; polynomial maps; symmetries; reversibility; involution