In this paper we analyze the dynamics of one-dimensional piecewise maps. We show that one-dimensional piecewise affine maps are equivalent to pseudo-billiard or so called “strange billiard” systems. We also show that use of more general classes of functions lead to undecidability of reachability problem for one-dimensional piecewise maps.

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Piecewise affine maps (PAMs) are frequently used as a reference model to discuss the frontier between known and open questions about the decidability for reachability questions. In particular, the reachability problem for one-dimensional PAM is still an open problem, even if restricted to only two intervals. As the main contribution of this paper we introduce new techniques for solving reachability problems based on [Formula: see text]-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 construct an example where the distribution properties of well studied sequences can be significantly disrupted by taking fractional parts after regular shifts. The study of such sequences could help with understanding similar sequences generated in PAMs or in well known Mahler’s [Formula: see text] problem.

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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.