Deciding Robust Instances of an Escape Problem for Dynamical Systems in Euclidean Space

TL;DR

This paper introduces a decision method for determining whether points escape a set under continuous maps in Euclidean space, emphasizing robustness to perturbations and applicability to specific function families like affine linear systems and quadratic polynomials.

Contribution

It presents a sound partial decision procedure that is complete for robust instances, applicable to general continuous functions and specific families, with implications for complex dynamics and the Mandelbrot set.

Findings

The decision method halts on all robust problem instances.

The halting set of the algorithm is dense among all instances.

Provides an alternative proof related to the computability of the Mandelbrot set.

Abstract

We study the problem of deciding whether a point escapes a closed subset of $\mathbb{R}^d$ under the iteration of a continuous map $f \colon \mathbb{R}^d \to \mathbb{R}^d$ in the bit-model of real computation. We give a sound partial decision method for this problem which is complete in the sense that its halting set contains the halting set of all sound partial decision methods for the problem. Equivalently, our decision method terminates on all problem instances whose answer is robust under all sufficiently small perturbations of the function. We further show that the halting set of our algorithm is dense in the set of all problem instances. While our algorithm applies to general continuous functions, we demonstrate that it also yields complete decision methods for much more rigid function families: affine linear systems and quadratic complex polynomials. In the latter case, completeness is subject to the density of hyperbolicity conjecture in complex dynamics. This in particular yields an alternative proof of Hertling's (2004) conditional answer to a question raised by Penrose (1989) regarding the computability of the Mandelbrot set.