Computability for Axiom A Polynomial Skew Products of $\mathbb{C}^2$

TL;DR

This paper investigates the computability of dynamical sets for a class of polynomial maps in two complex variables, establishing that Axiom A skew products have computable chain recurrent sets and semi-decidable hyperbolicity properties.

Contribution

It introduces the first computability results for multi-dimensional complex maps, specifically polynomial skew products, and provides algorithms for identifying hyperbolic and chain recurrent sets.

Findings

Chain recurrent sets of Axiom A skew products are computable.

Hyperbolic sets of various types can be identified algorithmically.

The hyperbolicity locus of polynomial skew products is semi-decidable.

Abstract

The computability of Julia sets of rational maps on the Riemann sphere has been intensively studied in recent years (see, e.g. https://doi.org/10.17323/1609-4514-2008-8-2-185-231, https://doi.org/10.1090/conm/797/15936) for an overview. For example, by Braverman's results (https://doi.org/10.1016/j.entcs.2004.06.031, https://doi.org/10.1088/0951-7715/19/6/009), hyperbolic and parabolic Julia sets are computable in polynomial time. In this paper, we present the first work on computability related to maps of more than one complex dimension. We examine a family of polynomial endomorphisms of $\mathbb{C}^2$, the polynomial skew products; i.e., maps of the form $f(z,w) = (p(z), q(z,w)),$ where $p$ and $q$ are complex polynomials of the same degree $d\geq 2$. We show that if a polynomial skew product is Axiom A, then its chain recurrent set, which is equal to its non-wandering set and also equal to the closure of the periodic orbits, is computable. Our algorithm also identifies the various hyperbolic sets of different types, i.e., expanding, attracting, and hyperbolic sets of saddle-type. One consequence of our results is that Axiom A is a semi-decidable property on the closure of the Axiom A polynomial skew product locus. Finally, we introduce an algorithm that establishes the lower semi-computability of the hyperbolicity locus of polynomial skew products of a fixed degree.