一句话：证明单射的一维两区间分段仿射映射可达性可判定，并揭示相关问题的丢番图与拓扑性质。
本文研究一维分段仿射映射的可达性问题。该问题在仅有两个区间时长期悬而未决：判定给定轨道是否可达另一状态。作者证明，对于单射分段仿射映射，可达性是可判定的。文章还考察若干相关问题，或给出可判定性结果，或表明它们与某些超越数的丢番图性质密切相关，其形式类似线性递推序列的正性问题。最后，作者研究不限于两个区间的一维分段仿射映射的轨道拓扑性质，并否定回答了 Bournez、Kurganskyy 与 Potapov 关于扩张映射轨道集合的问题。
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.