Reachability problems for PAMs

TL;DR

This paper introduces new techniques using p-adic norms to address reachability problems in one-dimensional piecewise affine maps, establishing decidability for certain classes and exploring their topological and number representation properties.

Contribution

It presents novel methods based on p-adic analysis for solving reachability in PAMs and proves decidability for specific subclasses, linking topological dynamics and number systems.

Findings

Decidability established for two classes of PAMs

Connections made between orbit properties and rational base representations

Example showing non-uniform distribution after shifts and fractional parts

Abstract

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.