One of the most basic, longstanding open problems in the theory of dynamical systems is whether reachability is decidable for one-dimensional piecewise affine maps with two intervals. In this paper we prove that for injective maps, it is decidable. We also study various related problems, in each case either establishing decidability, or showing that they are closely connected to Diophantine properties of certain transcendental numbers, analogous to the positivity problem for linear recurrence sequences. Lastly, we consider topological properties of orbits of one-dimensional piecewise affine maps, not necessarily with two intervals, and negatively answer a question of Bournez, Kurganskyy, and Potapov, about the set of orbits in expanding maps.

翻译：动力系统理论中最基本且长期悬而未决的问题之一，是一维两区间分段仿射映射的可达性是否可判定。本文证明，对于单射映射，该问题可判定。我们进一步研究了若干相关命题，在每种情形下或确立了可判定性，或揭示了其与某些超越数丢番图性质之间的密切联系（类似于线性递归序列的正性判定问题）。最后，我们考察了一维分段仿射映射（不限于两区间情形）轨道的拓扑性质，并否证了Bournez、Kurganskyy与Potapov关于扩张映射轨道集的一个猜想。