This paper examines self-similar sets and some of their properties, including the natural equivalence relation found in bilipschitz equivalence. Both dimension and preservation of paths are determined to be invariant under this equivalence. Also, sophisticated techniques, one involving the use of directed graphs, show the equivalence of two spaces. / Master of Science

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This paper will examine analogues of Cantor sets, called fractal squares, and some of the geometric ways in which fractal squares raise issues not raised by Cantor sets. Also discussed will be a technique using directed graphs to prove bilipschitz equivalence of two fractal squares. / Ph. D.

Primary objective of the thesis is proof of the statement that if for ∈ ℕ a ≥ 1 a bilipschitz mapping belongs to +1, loc ∩ ,∞ loc then also its inverse −1 belongs to +1, loc . We prove a similar statement also for spaces loc . For this purpose we construct a new ordering of -th partial derivatives to generalized Jacobian matrix. Thanks to this matrix we are able to differentiate matrices in an applicable way. Generalized Jacobian matrix is projected so that there still holds the Chain rule and, in some way, also rules for matrices product differentiation. 1

The aim of this thesis is to expand on parts of the work of Björn–Björn–Shanmugalingam [2] and in particular on the construction and properties of hyperbolic fillings of nonempty bounded metric spaces. In light of [2], we introduce two new parameters λ and ξ to the construction while relaxing a specific maximal-condition. With these modifications we obtain a slightly more flexible model that generates a larger family of hyperbolic fillings. We then show that every hyperbolic filling in this family possess the desired property of being Gromov hyperbolic. Next, we uniformize an arbitrary hyperbolic filling of this type and show that, under fairly weak conditions, the boundary of the uniformization is snowflake-equivalent to the completion of the metric space it corresponds to. Finally, we show that this unifomized hyperbolic filling is a uniform space. In summary, our construction generates hyperbolic fillings which satisfy the necessary conditions for it to serve its intended purpose of an analytical tool for further studies in [2, Chapters 9-13 ] or similar. As such, it can be regarded as an improvement to the reference model.

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In this thesis we consider an open question of Feige that asks whether there always exists a constantly Lipschitz bijection of an n2 -point subset of Z2 onto a regular grid [n] × [n] for every n ∈ N. We relate this question to an already resolved problem of the existence of a bounded positive measurable density in R2 that is not the Jacobian of any bilipschitz map. This problem was resolved by Burago and Kleiner [1], and independently, by McMullen [12]. We present the work of Burago and Kleiner, analyze its relation to Feige's problem and sug- gest a continuous formulation of Feige's question in a special case. Then we present the Burago-Kleiner density, make several observation about the properties of this density, and after that we construct a density that is everywhere nonrealizable as the Jacobian of a bilipschitz map. Subsequently, we discuss our continuous variant of Feige's question, provide several observation concerning it, and finally, we try to use the everywhere nonrealizable density constructed before to answer our continuous variant of Feige's question. However, this last task still remains incomplete. 1

The thesis deals with two separate problems. In the first part we show that the regular n×n grid of points in Z2 cannot be recovered from an arbitrary n2 -element subset of Z2 using only mappings with prescribed maximum stretch independent of n. This provides a negative answer to a question of Uriel Feige from 2002. The present approach builds on the work of Burago and Kleiner and McMullen from 1998 on bilipschitz non-realisable densities and bilipschitz non-equivalence of separated nets in the plane. We describe a procedure that takes a positive, measurable function and encodes it into a sequence of discrete sets. Then we show that applying this procedure to a typical positive, continuous function on the unit square yields a counter-example to Feige's question. Along the way we provide a new proof of a result on bilipschitz decomposition for Lipschitz regular mappings, which was originally proved by Bonk and Kleiner in 2002. In the second part we provide a constructive proof for the strong Hanani- Tutte theorem on the projective plane. In contrast to the previous proof by Pelsmajer, Schaefer and Stasi from 2009, the presented approach does not rely on characterisation of embeddability into the projective plane via forbidden minors. 1