| In this thesis, we study the fractal dimensions of the space of geodesic laminations on hyperbolic surfaces. For a compact hyperbolic surface S, the space GL (S) of geodesic laminations on S is naturally endowed with the Hausdorff metric dH. We prove that the fractal dimension of the space ( GL (S), dH) is equal to 0. We then introduce a new metric dlog on GL (S), which induces the same topology as d H. The primary motivation to study dlog is that the metric dlog depends only on the topology of S up to Lipschitz equivalence, so that its fractal dimension is well-defined. We prove that the fractal dimension of the space of the chain recurrent geodesic laminations on the once-punctured torus with metric dlog is equal to 2. |
