Shift-like Automorphisms of Ck

• Main Author: Bera, Sayani
• Other Authors: Verma, Kaushal
• Language: en_US
• Published: 2018
• Subjects: Automorphisms; Endomorphism; Polynomial Shift-Like Maps; Unstable Manifolds; Yoccoz Inequality; Manifolds (Mathematics); Mappings (Mathematics); Polynomial Automorphisms; Polynomial Shifts; Mathematics
• Online Access: http://etd.iisc.ernet.in/handle/2005/2959; http://etd.ncsi.iisc.ernet.in/abstracts/3821/G26702-Abs.pdf

We use transcendental shift-like automorphisms of Ck, k > 2 to construct two examples of non-degenerate entire mappings with prescribed ranges. The first example exhibits an entire mapping of Ck, k>2 whose range avoids a given polydisc but contains the complement of a slightly larger concentric polydisc. This generalizes a result of Dixon-Esterle in C2. The second example shows the existence of a Fatou-Bieberbach domain in Ck,k > 2 that is constrained to lie in a prescribed region. This is motivated by similar results of Buzzard and Rosay-Rudin. In the second part we compute the order and type of entire mappings that parametrize one dimensional unstable manifolds for shift-like polynomial automorphisms and show how they can be used to prove a Yoccoz type inequality for this class of automorphisms.