Hausdorff dimension of invariant sets and positive linear operators

In this thesis we obtain theorems which give the Hausdorff dimension of the invariant set for a family of contraction mappings on a complete, perfect metric space. We obtain results for the graph-directed systems similar to that of Mauldin and Williams [17] but for more general contraction mappings which are "infinitesimal similitudes" rather than just "similitudes". Also the underlying spaces are not assumed to be finite dimensional. For finite graph-directed systems our results are valid on any bounded, complete, perfect metric space. We use the theory of positive linear operators and generalizations of the Krein-Rutman theorem to characterize the Hausdorff dimension as the unique value of δ > 0 for which r(L δ) = 1, where Lδ , δ > 0, is a naturally associated family of positive linear operators and r(Lδ ) denotes the spectral radius of Lδ . We also obtain theorems for infinite graph-directed systems (with finitely many vertices and countably many edges) on compact, perfect metric spaces. In this case too we have a family of positive linear operators Lδ for δ > δ₀ > 0, and the Hausdorff dimension of the invariant set is given by the infimum of δ > δ₀ for which r(L δ) < 1. We discuss an infinte iterated function system given by complex continued fractions and obtain lower bound for the Hausdorff dimension of the invariant set. Our estimate improves the lower bound to 1.787 from 1.2484 obtained in [16]. Finally we also give a theorem proving the continuity of the Hausdorff dimension with respect to the contractions in the graph-directed systems.