Mera Rivas, María EugeniaLlorente Comi, MartaMorán Cabré, Manuel2024-01-242024-01-242023Morán M., Llorente M., Mera M.E. Local geometry of self-similar sets: typical balls, tangent measures and asympotic spectra. Fractals Vol. 31, No. 05, 2350059 (2023)0218-348X10.1142/s0218348x23500597https://hdl.handle.net/20.500.14352/94994We analyze the local geometric structure of self-similar sets with open set condition through the study of the properties of a distinguished family of spherical neighborhoods, the typical balls. We quantify the complexity of the local geometry of self-similar sets, showing that there are uncountably many classes of spherical neighborhoods that are not equivalent under similitudes. We show that at a tangent level, the uniformity of the Euclidean space is recuperated in the sense that any typical ball is a tangent measure of the measure at mu-a.e. point, where mu is any self-similar measure. We characterize the spectrum of asymptotic densities of metric measures in terms of the packing and centered Hausdorff measures. As an example, we compute the spectrum of asymptotic densities of the Sierpiński gasket.engjournal article1793-6543https://www.worldscientific.com/worldscinet/fractalsopen access5Self-similar SetsHausdorff MeasuresTangent MeasuresDensity of MeasuresComputability of Fractal MeasuresComplexity of Topological SpacesSierpiński GasketMatemáticas (Matemáticas)GeometríaAnálisis matemático12 Matemáticas1204 Geometría1202 Análisis y Análisis Funcional