Hausdorffen dimentsioa talde profinituetan

Abstract

The notions of Hausdorff dimension and box dimension are basic concepts in the field of fractal geometry. These concepts generalise the idea of the traditional topological dimension since, while the fractal dimension of a common geometric object coincides with the value of its typical dimension, certain pathological sets which, intuitively, do not have such a clear dimension, may have non-integer fractal dimension. In the last decades, the concept of Hausdorff dimension has provided fruitful and interesting applications in the context of countably based profinite groups, as these groups can always be seen as metric spaces. In this paper, on the one hand, we will give a general introduction to the theory of Hausdorff and box dimensions. On the other hand, we will see different significant results concerning Hausdorff dimension in profinite groups, focusing on two specific areas: the Hausdorff spectrum and R-analytic groups.; Hausdorffen dimentsioa eta kutxa-dimentsioa geometria fraktalaren esparruko oinarrizko kontzeptuak dira. Horiek ohiko dimentsio topologikoaren ideia orokortzen dute; izan ere, objektu geometriko arrunten dimentsioari ohiko balio bera ematen dieten bitartean, intuitiboki hain dimentsio argia ez duten zenbait multzo patologikori dimentsio ez oso bat esleitzen diote. Bestalde, azken hamarkadetan, Hausdorffen dimentsioaren kontzeptuak aplikazio emankor eta interesgarriak eman ditu oinarri zenbakigarriko talde profinituen testuinguruan, talde horiei espazio metriko egitura eman baitakieke. Artikulu honetan, alde batetik, Hausdorffen dimentsioaren eta kutxa-dimentsioaren teoriaren sarrera orokor bat egingo dugu. Beste alde batetik, Hausdorffen dimentsioak talde profinituetan eman dituen zenbait emaitza esanguratsu azalduko dira, arreta bi alor konkretutara bideratuz: Hausdorffen espektroa eta talde R-analitikoak.