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7) converges (in some domain U) and hence that rh is infinitesimally G-stable. e. z-zkI>Cljil, jEJ, 1 n/2 - 1, E(r) has Lesbegue measure zero and for this reason is called exceptional.

8] D. Sullivan, `Discrete conformal groups and measurable dynamics', Proc. Symp. Pure Math. 39 (1984) 169-185. [9] W. P. Thurston, The geometry and topology of 3-manifolds, (Notes, Princeton, 1983). 4 Symbolic dynamics and Diophantine equations Caroline Series University of Warwick, Coventry, UK §1. The problems Certain classical Diophantine problems have a geometrical interpretation as the height to which geodesics travel up the cusp of the modular surface HI SL(2, Z), where H = {z E C : Im z > 0} is the hyperbolic plane and SL(2, Z) acts by linear fractional transformations.

Let A be a finite set of parabolic cusps of t if there are any, and a finite set of hyperbolic fixed points if there are none. Let E be the set of points x E SN for which there exists a constant c(x)>0sothat forall9E1'andallyEAonehas I Ig(y) - xI I > c(x)/L(0, g(0)). Then E has Hausdorff dimension N. Observe that in these theorems L(x, g(x'))/L(0, g(0)) considered as a function of g on Con(N) is bounded above and below by functions of x, x' alone. Thus the seemingly arbitrary nature of L(0, g(0)) is explained by the fact that in the theorems any of the other L(x, g(x')) could have been used as a measurement of the size of g.

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