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We estimate the error in the semiclassical trace formula for the Sinai billiard under the assumption that the largest source of error is due to Penumbra diffraction, that is diffraction effects for trajectories passing within a distance R O((kR)^(-2/3) to the disk and trajectories being scattered in very forward directions. Here k is the momentum and R the radius of the scatterer. The semiclassical error is estimated by perturbing the Berry-Keating formula. The analysis necessitates an asymptotic analysis of very long periodic orbits. This is obtained within an approximation originally due to Baladi Eckmann and Ruelle. We find that the average error, for sufficiently large value of kR, will exceed the mean level spacing., 30 pages, 12 postscript figures

The topological pressure is obtained as the leading zero of a dynamical zeta function. We consider the problem of computing this zero when it is close to a singularity. In particular we study a family of intermittent maps, which we argue exhibit a branch point singularity in its zeta functions. The convergence of the cycle expansion close to this point is extremely slow. To deal with this problem we consider a resummation of the cycle expansion. The idea is quite similar to that of Pade approximants, but the ansatz is a generalized series expansion around the branch point rather than a rational function. The improvement on convergence of the leading zero is considerable. We also briefly discuss the relation between correlation decay and the nature of the branch point.

We show that a regular curve in Minkowski space with a light-like tangent will have Hausdorff dimension D = 1/2. We present a light-like curve with D = 1/2 exhibiting complete self-similarity. We also present a Lorentz invariant formulation of the neighbourhood of a curve with "diameter" l with a volume scaling as l 4-D, where D is the fractal dimension of the curve. The connection to the theory of a massless relativistic string is briefly discussed.