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-dimensional 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.
This research is supported by EPSRC grant “Reachability problems for words, matrices and maps” (EP/M00077X/1).
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Notes
- 1.
It will be clear from the context if brackets are used in other conventional ways, for example, to indicate a set of numbers.
- 2.
Also in a similar way it is possible to define set-to-point and set-to-set reachability problems.
- 3.
In particularly the continuous piecewise affine mapping of degree two.
- 4.
I.e. with linear coefficients that are greater than 1.
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Kurganskyy, O., Potapov, I. (2016). Reachability Problems for PAMs. In: Freivalds, R., Engels, G., Catania, B. (eds) SOFSEM 2016: Theory and Practice of Computer Science. SOFSEM 2016. Lecture Notes in Computer Science(), vol 9587. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-49192-8_29
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DOI: https://doi.org/10.1007/978-3-662-49192-8_29
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