We show that an Anosov map has a geodesic axis on the curve graph of the torus. The direct corollary of our result is the stable translation length of an Anosov map on the curve graph is always a positive integer. As the proof is constructive, we also provide an algorithm to calculate the exact translation length for any given Anosov map. The application of our result is threefold: (a) to determine which word realizes the minimal translation length on the curve graph within a specific class of words, (b) to establish the effective bound on the ratio of translation lengths of an Anosov map on the curve graph to that on Teichmüller space, and (c) to estimate the overall growth of the number of Anosov maps which have a sufficient number of Anosov maps with the same translation length .

我们显示Anosov映射在圆环的曲线图上具有测地轴。结果的直接推论是曲线图上Anosov映射的稳定平移长度始终为正整数。由于证明是有建设性的，因此我们还提供了一种算法，可以为任何给定的Anosov映射计算准确的翻译长度。我们的结果的应用是三方面的：（a）确定哪个单词在特定类别的单词内实现曲线图上的最小平移长度，（b）确定有效的Anosov映射平移长度比的界限。将曲线图绘制成Teichmüller空间上的曲线图，并（c）估计具有足够平移长度的足够数量的Anosov映射的Anosov映射的数量的总体增长。