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The Directed Landscape, a random directed metric on the plane (where thefirst and the second coordinates are termed spatial and temporal respectively),was constructed in the breakthrough work of Dauvergne, Ortmann, and Vir\'ag,and has since been shown to be the scaling limit of various integrable modelsof Last Passage percolation, a central member of the Kardar-Parisi-Zhanguniversality class. It exhibits several scale invariance properties making it anatural source of rich fractal behavior. Such a study was initiated inBasu-Ganguly-Hammond, where the difference profile i.e., the difference ofpassage times from two fixed points (say $(\pm 1,0)$), was considered. Owing togeodesic geometry, it turns out that this difference process is almost surelylocally constant. The set of non-constancy is connected to disjointness ofgeodesics and inherits remarkable fractal properties. In particular, it hasbeen established that when only the spatial coordinate is varied, the set ofnon-constancy of the difference profile has Hausdorff dimension $1/2$, andbears a rather strong resemblance to the zero set of Brownian motion. Thearguments crucially rely on a monotonicity property, which is absent when thetemporal structure of the process is probed, necessitating the development ofnew methods.  In this paper, we put forth several new ideas, and show that the set ofnon-constancy of the 2D difference profile and the 1D temporal process (whenthe spatial coordinate is fixed and the temporal coordinate is varied) haveHausdorff dimensions $5/3$ and $2/3$ respectively. A particularly crucialingredient in our analysis is the novel construction of a local time processfor the geodesic akin to Brownian local time, supported on the "zero set" ofthe geodesic. Further, we show that the latter has Hausdorff dimension $1/3$ incontrast to the zero set of Brownian motion which has dimension $1/2.$

Fractal geometry of the space-time difference profile in the directed landscape via construction of geodesic local times