Dimensional Multifractal Analysis

This thesis contains one main topic related to fractal geometry — multifractal analysis. In Chapter One,we develop a theory for the centered Hausdorff measure and the packing measure in metric spaces . Let X be a metric space and μ₁,μ₂ Borel probability measures on X. For q₁,q₂, t ∈ R and E X writeis a centered δ-covering of E},is a centered δ-packing of E},The measures H_μ1,μ2^q₁,q₂,t and P_μ1,μ2^q₁,q₂,t define ,for a fixed q₁ and a fixed q₂ ,in the usual way a generalized Hausdorff dimension dim_μ1,μ2^q₁,q₂(E) and a generalized packing dimension Dim_μ1,μ2^q₁,q₂(E) of subsets E of X. We study the functionsand their relation to the so-called two-dimensional multifractal spectra functions of μ₁,μ₂The part extends the results of L.Olsen (1995),which develop a mathematical rigorous multifractal formalism based on a multifractal generalization of the centered Hausdorff measure and of the packing measure. In Chapter Two,we prove the results stated in Chapter One. In Chapter Three,we give a multifractal analysis of self-similar measures in R^d using our setting. In Chapter Four,we give a multifractal analysis of " cookie-cutter " measures in R using our setting.