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The equivalence of two conditions on the primitive elements in an $SL(2,\mathbb C)$ representation of the free group $F_2 = $, namely Minsky's condition of primitive stability and the $BQ$-conditions introduced by Bowditch and generalised by Tan, Wong and Zhang, has been proved by Lee and Xu and independently by the author in arXiv:1901.01396. This note is a revised version of our original proof, which is greatly simplified by incorporating some of the ideas introduced by Lee and Xu, combined with the language of the Bowditch tree., Comment: 20 pages , 3 figures. This paper is a substantial revision and simplification of arXiv:1901.01396

We prove the equivalence of two conditions on the primitive elements in an $SL(2,\mathbb C)$ representation of the free group $F_2$ on two generators, which may hold even when the image of $F_2$ is not discrete. One is Minsky's condition of primitive stability and the other is the $BQ$-condition introduced by Bowditch and generalised by Tan, Wong and Zhang., Comment: 20 pages, 2 figures

Pointwise convergence of spherical averages is proved for a measure-preserving action of a Fuchsian group. The proof is based on a new variant of the Bowen-Series symbolic coding for Fuchsian groups that, developing a method introduced by Wroten, simultaneously encodes all possible shortest paths representing a given group element. The resulting coding is self-inverse, giving a reversible Markov chain to which methods previously introduced by the first author for the case of free groups may be applied., Comment: 71 pages, 14 figures; a new proof for the result on the structure of thickened paths (see Sections 3 and 4), many minor improvements and clarifications