Geodesic coalescence, or the tendency of geodesics to merge together, is a hallmark phenomenon observed in a variety of planar random geometries involving a random distortion of the Euclidean metric. As a result of this, the union of interiors of all geodesics going to a fixed point tends to form a tree-like structure that is supported on a vanishing fraction of the space. Such geodesic trees exhibit intricate fractal behaviour; for instance, while almost every point in the space has only one geodesic going to the fixed point, there exist atypical points that admit two such geodesics. In this paper, we consider the directed landscape, the recently constructed [ 18] scaling limit of exponential last passage percolation (LPP), with the aim of developing tools to analyse the fractal aspects of the tree of semi-infinite geodesics in a given direction. We use the duality [ 39] between the geodesic tree and the interleaving competition interfaces in exponential LPP to obtain a duality between the geodesic tree and the corresponding dual tree in the landscape. Using this, we show that problems concerning the fractal behaviour of sets of atypical points for the geodesic tree can be transformed into corresponding problems for the dual tree, which might turn out to be easier. In particular, we use this method to show that the set of points admitting two semi-infinite geodesics in a fixed direction a.s. has Hausdorff dimension $4/3$, thereby answering a question posed in [ 12]. We also show that the set of points admitting three semi-infinite geodesics in a fixed direction is a.s. countable.

中文翻译：

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有向景观的对偶性及其在分形几何中的应用

测地线合并，或测地线合并在一起的趋势，是在涉及欧几里得度量随机变形的各种平面随机几何中观察到的标志现象。因此，到达固定点的所有测地线的内部联合往往会形成一个树状结构，该结构由空间的消失部分支撑。这种测地线树表现出复杂的分形行为；例如，虽然空间中几乎每个点只有一条测地线通向固定点，但存在允许两条这样的测地线的非典型点。在本文中，我们考虑有向景观，即最近构建的指数最后通道渗流（LPP）的缩放极限[18]，目的是开发工具来分析给定方向上半无限测地线树的分形方面。我们利用测地线树和指数LPP中的交错竞争界面之间的对偶性[39]来获得测地线树和景观中相应的对偶树之间的对偶性。利用这一点，我们表明，有关测地树非典型点集的分形行为的问题可以转化为对偶树的相应问题，这可能会更容易。特别地，我们使用这种方法来证明在固定方向上接纳两个半无限测地线的点集具有豪斯多夫维数$4/3$，从而回答了[12]中提出的问题。我们还表明，在固定方向上接纳三个半无限测地线的点集是可数的。