Reachability Problems for One-Dimensional Piecewise Affine Maps

Potapov, I, Bournez, Olivier and Kurganskyy, Oleksiy
(2018) Reachability Problems for One-Dimensional Piecewise Affine Maps. International Journal of Foundations of Computer Science, 29 (4).

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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 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 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 3/2 problem.

Item Type: Article
Uncontrolled Keywords: Reachability problems; piecewise affine maps (PAMs); β-expansion; p-adic analysis
Depositing User: Symplectic Admin
Date Deposited: 02 Jul 2018 06:31
Last Modified: 10 Dec 2021 06:11
DOI: 10.1142/S0129054118410046