This is the first of two papers in which we investigate the properties of the displacement functions of automorphisms of free groups (more generally, free products) on Culler-Vogtmann Outer space and its simplicial bordification - the free splitting complex - with respect to the Lipschitz metric. The theory for irreducible automorphisms being well-developed, we concentrate on the reducible case. Since we deal with the bordification, we develop all the needed tools in the more general setting of deformation spaces, and their associated free splitting complexes. In the present paper we study the local properties of the displacement function. In particular, we study its convexity properties and the behaviour at bordification points, by geometrically characterising its continuity-points. We prove that the global-simplex-displacement spectrum of $Aut(F_n)$ is a well-ordered subset of $\mathbb R$, this being helpful for algorithmic purposes. We introduce a weaker notion of train tracks, which we call {\em partial train tracks} (which coincides with the usual one for irreducible automorphisms) and we prove that, for any automorphism, points of minimal displacement - minpoints - coincide with the marked metric graphs that support partial train tracks. We show that any automorphism, reducible or not, has a partial train track (hence a minpoint) either in the outer space or its bordification. We show that, given an automorphism, any of its invariant free factors is seen in a partial train track map. In a subsequent paper we will prove that level sets of the displacement functions are connected, and we will apply that result to solve certain decision problems.

中文翻译：

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自由群自同构的位移 I：位移函数、最小点和火车轨道

这是两篇论文中的第一篇，我们研究了 Culler-Vogtmann 外层空间上自由群（更一般地说，自由积）的自同构的位移函数的性质及其单纯边界化——自由分裂复形——关于 Lipschitz公制。不可约自同构的理论得到了很好的发展，我们专注于可约的情况。由于我们处理边界化，因此我们在更一般的变形空间及其相关的自由分裂复合体中开发了所有需要的工具。在本文中，我们研究了位移函数的局部特性。特别是，我们通过几何表征其连续点来研究其凸性特性和边界点处的行为。我们证明了 $Aut(F_n)$ 的全局单纯形位移谱是 $\mathbb R$ 的有序子集，这有助于算法目的。我们引入了一个较弱的火车轨道概念，我们称之为 {\em 部分火车轨道}（这与通常的不可约自同构重合）并且我们证明，对于任何自同构，最小位移点 - minpoints - 与标记的支持部分火车轨道的度量图。我们证明了任何自同构，无论可约与否，在外层空间或其边界中都有部分火车轨道（因此是最小点）。我们表明，给定自同构，其任何不变的自由因子都可以在部分火车轨道图中看到。在随后的论文中，我们将证明位移函数的水平集是连通的，