Dimension Distortion by Sobolev Mappings in Foliated Metric Spaces

We quantify the extent to which a supercritical Sobolev mapping can increase the dimension of subsets of its domain, in the setting of metric measure spaces supporting a Poincaré inequality. We show that the set of mappings that distort the dimensions of sets by the maximum possible amount is a prevalent subset of the relevant function space. For foliations of a metric space X defined by a David–Semmes regular mapping Π : X → W, we quantitatively estimate, in terms of Hausdorff dimension in W, the size of the set of leaves of the foliation that are mapped onto sets of higher dimension. We discuss key examples of such foliations, including foliations of the Heisenberg group by left and right cosets of horizontal subgroups.

Tectonic evolution of the Proterozoic Itremo Group metasediments in central Madagascar

The major geologic units of the Itremo region in central Madagascar include: (1) upper amphibolite to granulite facies (higher grade) Precambrian rocks, mainly para- and orthogneisses, and migmatites; (2) the newly defined Itremo Nappes, a fold-and-thrust belt containing the Proterozoic Itremo Group sediments, metamorphosed at greenschist to lower amphibolite facies (lower grade) conditions: (3) Middle Neoproterozoic and Late Neoproterozoic-Cambrian intrusives. The stratigraphic succession of the Itremo Group in the eastern part of the Itremo region is, from bottom to top: quartzites, metapelites, metacarbonates and metapelites overlain by metacarbonates. During D1 the Itremo Group sediments were detached from their continental substratum, deformed into a fold-and-thrust nappe (Itremo Nappes), and transported on top of higher grade rocks that are intruded by Middle Neoproterozoic (c. 797–780 Ma) granites and gabbros. A second phase of deformation shortening (D2) affected both the Itremo Sedimentary Nappes and structurally underlying higher-grade rocksunits, and formed large-scale N-S-trending F2 folds. S1 axial plane foliations in Itremo Group sediments are truncated by Late Neoproterozoic-Cambrian granites (c. 570–540 Ma). The age of the formation of the Itremo Nappes is not well constrained: they formed in Neoproterozoic times between 780 and 570 Ma.