Distortion of Dimension under Quasiconformal Mappings

Quasiconformal mappings are natural generalizations of conformal mappings. They are homeomorphisms with 'bounded distortion' of which there exist several approaches. In this work we study dimension distortion properties of quasiconformal mappings both in the plane and in higher dimensional Euclidean setting. The thesis consists of a summary and three research articles. A basic property of quasiconformal mappings is the local Hölder continuity. It has long been conjectured that this regularity holds at the Sobolev level (Gehring's higher integrabilty conjecture). Optimal regularity would also provide sharp bounds for the distortion of Hausdorff dimension. The higher integrability conjecture was solved in the plane by Astala in 1994 and it is still open in higher dimensions. Thus in the plane we have a precise description how Hausdorff dimension changes under quasiconformal deformations for general sets. The first two articles contribute to two remaining issues in the planar theory. The first one concerns distortion of more special sets, for rectifiable sets we expect improved bounds to hold. The second issue consists of understanding distortion of dimension on a finer level, namely on the level of Hausdorff measures. In the third article we study flatness properties of quasiconformal images of spheres in a quantitative way. These also lead to nontrivial bounds for their Hausdorff dimension even in the n-dimensional case.

Inarinsaamen käsitemetaforat

This study discusses the conceptual metaphors of Inari Saami, an endangered, indigenous, Finno-Ugrian language spoken in northern Finland. The research focuses on systematical mappings between source and target domains in conventional Inari Saami metaphors and metonymies. The research material consists of the Inarinsaamen idiomisanakirja [Inari Saami idiom dictionary] which has been compiled by the author in collaboration with an Inari Saami co-author; the Inarilappisches Wörterbuch; Inarinsaamelaista kansantietoutta [Inari Saami folk knowledge]; and Aanaarkiela čájttuzeh [Inari Saami sample texts]. The metaphors and metonymies found in these literary sources are divided into categories on the basis of the target domains and according to the classic model of Lakoff ja Johnson (1980). This method reveals the systematical recurrence of source domains inside each category and thus discovers the systematical patterns of metaphoric mapping, the conceptual metaphors . As a result 44 conceptual metaphors and 16 conceptual metonymies are presented through approximately 500 glossed examples. These findings are discussed against the background of what is known about the cognitive and neural processing of metaphors on the one hand, and what is known about Inari Saami culture on the other. This theoretical framework highlights culture as the underlying force behind conceptual metaphors. The recurring metonymies seem to follow a culturally salient indexicality. For example, the Inari Saami conceptual metonymy TIME IS NATURE reflects the seasonal changes in the year s cycle, which was the salient index of time in traditional Inari Saami culture. The recurring metaphors, for their part, follow a culturally salient iconicity. The conceptual metaphor PRIDE IS ANTLERS is based on an iconicity which is experienced and interpreted by the Inari Saami. A proud person is associated with a reindeer who shows off his impressive antlers. The conceptual metaphor/metonymy seems to be a reflection of culture rather than a cognitive means of understanding an abstract domain in terms of a concrete domain, as hypothesized by certain theoreticians. Repeating this study with other languages may lead to the possibility of typologizing the metaphorical systems of the world s languages and understanding the diversity of metaphor systems in the endangered languages of the world.