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Bilipschitz homogeneity and Jordan curves
| Content Provider | Semantic Scholar |
|---|---|
| Author | Freeman, David |
| Copyright Year | 2009 |
| Abstract | We analyze Jordan curves in the plane that are bilipschitz homogeneous with respect to Euclidean distance and/or inner diameter distance. We begin our analysis from the Euclidean vantage point. In this setting, we produce a quantitative bound on the bounded turning constant for unbounded curves. We then construct a catalogue of curves that accounts for all unbounded bilipschitz homogeneous Jordan curves in the plane, up to bilipschitz equivalence. Some techniques utilized in this construction are implemented to characterize doubling conformal densities on the upper half plane. Finally, the interaction between bilipschitz homogeneity and dimension is examined, and fractal chordarc curves are characterized in terms of their invariance under Möbius maps. Our analysis proceeds to the inner diameter distance setting, where we again demonstrate that bilipschitz homogeneity implies a bounded turning condition, quantitatively. In this setting we obtain a very explicit bound on the bounded turning constant that is essentially best possible. Moreover, this bound holds for both bounded and unbounded curves. We then provide a quantitative link between the above catalogue for Euclidean bilipschitz homogeneous curves and inner distance bilipschitz homogeneous curves. We conclude with a characterization of Riemann maps onto domains whose boundaries are bilipschitz homogeneous in the inner distance. |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | https://etd.ohiolink.edu/!etd.send_file?accession=ucin1251229498&disposition=inline |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
