Logarithm laws for the BCZ map

Logarithm laws for diagonalizable flows were first studied by Sullivan, who showed the logarithm laws for the cusp excursion of geodesic flow. Later, Masur extended the result for geodesic flow in the moduli space. Then, Kleinbock and Margulis studied the actions of one-parameter diagonalizable subgroups on non-compact finite-volume homogeneous spaces in a more general context. Subsequently, Athreya and Margulis extended the previous result to the context of unipotent flows and provided further results on logarithm laws for horocycle flows. We presents the logarithm laws for the BCZ map regarding its itinerary function, which can be considered as an extension of Athreya and Margulis's result mainly due to the relationship between horocycle flow and BCZ map.