Reachability in Injective Piecewise Affine Maps
Abstract
One of the most basic, longstanding open problems in the theory of dynamical systems is whether reachability is decidable for onedimensional 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 onedimensional 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.
 Publication:

arXiv eprints
 Pub Date:
 January 2023
 DOI:
 10.48550/arXiv.2301.09752
 arXiv:
 arXiv:2301.09752
 Bibcode:
 2023arXiv230109752G
 Keywords:

 Mathematics  Dynamical Systems;
 Computer Science  Formal Languages and Automata Theory;
 Computer Science  Logic in Computer Science