Polynomial shift--like maps in ℂ^k

2018/05/08 by Bera, Sayani
#Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1805.03142

Abstract

The purpose of this article is to explore a few properties of polynomial shift-like automorphisms of ℂ^k. We first prove that a ν-shift-like polynomial map (say S_a) degenerates essentially to a polynomial map in ν-dimensions as a → 0. Secondly, we show that a shift-like map obtained by perturbing a hyperbolic polynomial (i.e., S_a, where |a| is sufficiently small) has finitely many Fatou components, consisting of basins of attraction of periodic points and the component at infinity.

Related

• Some aspects of shift-like automorphisms of C^k
• A polynomial automorphism with a wandering Fatou component
• Polynomial Diffeomorphisms of \C². VIII: Quasi-Expansion
• Filtrations, hyperbolicity and dimension for polynomial automorphisms of Cⁿ
• Saddle hyperbolicity implies hyperbolicity for polynomial automorphisms of C²