Reachability in Injective Piecewise Affine Maps

TL;DR

This paper proves that reachability is decidable for injective one-dimensional piecewise affine maps with two intervals and explores related problems, connecting some to Diophantine properties and analyzing orbit topologies.

Contribution

It establishes decidability of reachability for injective maps and investigates related problems, linking them to Diophantine properties and orbit topologies.

Findings

Decidability of reachability for injective maps with two intervals.

Connection of certain problems to Diophantine properties of transcendental numbers.

Negative answer to a question about the set of orbits in expanding maps.

Abstract

One of the most basic, longstanding open problems in the theory of dynamical systems is whether reachability is decidable for one-dimensional piecewise affine maps with two intervals. In this paper we prove that for injective maps, it is decidable. We also study various related problems, in each case either establishing decidability, or showing that they are closely connected to Diophantine properties of certain transcendental numbers, analogous to the positivity problem for linear recurrence sequences. Lastly, we consider topological properties of orbits of one-dimensional piecewise affine maps, not necessarily with two intervals, and negatively answer a question of Bournez, Kurganskyy, and Potapov, about the set of orbits in expanding maps.