On the multifractal spectra of mappings of finite distortion
Lauri Hitruhin · Työväentutkimus Vuosikirja · 2018
The thesis” On multifractal spectra of mappings of finite distortion” consist of introductory part and four articles. These articles revolve around stretching and rotational properties of mappings of finite distortion both in pointwise and global situations. In the second and third articles the sharp bounds for pointwise rotation of homeomorphisms with exponentially integrable distortion and homeomorphisms with integrable distortion have been established. The sharpness of these bounds is verified by constructing examples using nested annuli. The proofs for optimal bounds are based on techniques utilizing the modulus of path families. The idea is to find suitable path families and use inequality for moduli to find the correct bounds for rotation. This approach also highlights the correlation between stretching and rotation. Moreover, it can also be used for studying optimal pointwise stretching, where we can sharpen the previously known bound for homeomorphisms with integrable distortion. The method using modulus of path families differs from the methods used for obtaining the optimal bounds for quasiconformal mappings, which was done by Astala, Iwaniec, Prause and Saksman. In the first and fourth articles the maximal size of sets in which mappings of finite distortion can attain some predefined stretching and rotation is studied. Identifying the maximal size for all suitable combinations of stretching and rotation amounts to finding the multifractal spectra. In the first article the previously known bounds for the multifractal spectra of quasiconformal mappings, established by Astala, Iwaniec, Prause and Saksman, were sharpened to the level of the Hausdorff measures. This was done by constructing examples using non self-similar Cantor like sets. In the fourth article the multifractal spectra was studied for homeomorphisms with integrable distortion. Here the optimal bounds for the spectra were given and examples proving sharpness presented. Moreover, as an application a question by Clop and Herron, regarding compression of the Hausdorff measure for homeomorphisms of integrable distortion, was answered. Methods used in the proof for the spectra resemble those used in the pointwise case, but the choice of a suitable path family is more involved in this situation.