Reachability problems for PAMs
For researchers in dynamical systems and verification, this work advances the understanding of reachability for a fundamental class of systems, though the results are for specific subclasses.
The paper introduces new techniques based on p-adic norms and weights to solve reachability problems for piecewise affine maps, proving decidability for two classes of maps and establishing connections between topological properties, reachability, and rational base representations.
Piecewise affine maps (PAMs) are frequently used as a reference model to show the openness of the reachability questions in other systems. The reachability problem for one-dimentional PAM is still open even if we define it with only two intervals. As the main contribution of this paper we introduce new techniques for solving reachability problems based on p-adic norms and weights as well as showing decidability for two classes of maps. Then we show the connections between topological properties for PAM's orbits, reachability problems and representation of numbers in a rational base system. Finally we show a particular instance where the uniform distribution of the original orbit may not remain uniform or even dense after making regular shifts and taking a fractional part in that sequence.