🤖 AI Summary
This paper addresses the escape problem—determining whether iterates of a continuous map over the reals eventually leave a given closed Euclidean subset—within the bit-model of real computation. We present the first sound, robust, and partially complete decision algorithm: it always terminates on all instances stable under arbitrarily small functional perturbations, and its halting set is dense in the full parameter space. Our approach integrates real computability theory, robustness analysis, dynamical systems theory, and computability-theoretic techniques. The algorithm is applicable to affine linear systems and, conditionally on the Hyperbolicity Density Conjecture, to complex quadratic polynomials. As a consequence, it yields a novel proof strategy for the computability of the Mandelbrot set. Moreover, for both system classes, we achieve conditional decidability—substantially improving upon Hertling’s (2004) resolution of Penrose’s (1989) escape problem, which lacked robustness and density guarantees.
📝 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 $f colon mathbb{R}^d o mathbb{R}^d$ 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.