Dynamics of semigroups of Hénon maps

2022/02/14 by Bera, Sayani
#32H50 #37F80 #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.2202.06522

Abstract

The goal of this article is two fold. Firstly, we explore the dynamics of a semigroup of polynomial automorphisms of ℂ², generated by a finite collection of Hénon maps. In particular, we construct the positive and negative dynamical Green's functions G_\mathscrG^± and the corresponding dynamical Green's currents μ_\mathscrG^± for a semigroup S, generated by a collection \mathscrG. Using them, we show that the positive (or negative) Julia set of the semigroup S, i.e., J_S⁺ (or J_S^-) is equal to the closure of the union of individual positive (or negative) Julia sets of the maps, in the semigroup S. Furthermore, we prove that μ_\mathscrG⁺ is supported on the whole of J_S⁺ and is also the unique positive closed (1,1)-current supported on J_S⁺, satisfying a semi-invariance relation that depends on the generating set \mathscrG. Secondly, we study the dynamics of a non-autonomous sequence of Hénon maps, say \h_k\, contained in the semigroup S. Similarly, as above, here too, we construct the non-autonomous dynamical positive and negative Green's function and the corresponding dynamical Green's currents. Further, we use the properties of Green's function to conclude that the non-autonomous attracting basin of any such sequence \h_k\, sharing a common attracting fixed point, is biholomorphic to ℂ².

Related

• Dynamics of semigroups of entire maps of ℂ^k
• Dynamics of transcendental Hénon maps-II
• Dynamics of postcritically bounded polynomial semigroups III: classification of semi-hyperbolic semigroups and random Julia sets which are Jordan curves but not quasicircles
• Dynamical properties and structure of Julia sets of postcritically bounded polynomial semigroups
• Dynamics of postcritically bounded polynomial semigroups I: connected components of the Julia sets