  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 which 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 which admit two such geodesics. In this paper, we
consider the directed landscape, the recently constructed 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 (Pimentel '16) 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 Busani-Sepp\"{a}l\"{a}inen-Sorensen '22. We also
show that the set of points admitting three semi-infinite geodesics in a fixed
direction is a.s. countable.
