Geodesic
The limit set of a~Fuchsian group and Dyn'kin's lemma
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 31, Tome 303 (2003), pp. 89-101

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A geometric description (like Dyn'kin's lemma for the porous sets) of the sets $E\subset\mathbb T$ such that $\log\operatorname{dist}^{-1}(x,E)$ has bounded lower oscillation (BLO) is given. As a corollary, a new proof of the Shirokov–Semenova theorem on the limit set of a finitely generated Fuchsian group of the second kind is obtained. 
@article{ZNSL_2003_303_a3,
     author = {A. V. Vasin},
     title = {The limit set of {a~Fuchsian} group and {Dyn'kin's} lemma},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {89--101},
     publisher = {mathdoc},
     volume = {303},
     year = {2003},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2003_303_a3/}
}
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A. V. Vasin. The limit set of a~Fuchsian group and Dyn'kin's lemma. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 31, Tome 303 (2003), pp. 89-101. http://geodesic.mathdoc.fr/item/ZNSL_2003_303_a3/