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Bilipschitz Homogeneity and Jordan Curves

Freeman, David M.
http://rave.ohiolink.edu/etdc/view?acc_num=ucin1251229498

Abstract Details

2009, PhD, University of Cincinnati, Arts and Sciences : Mathematical Sciences.
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.
David Herron, PhD (Committee Chair)
Carl Minda, PhD (Committee Member)
Nageswari Shanmugalingam, PhD (Committee Member)
171 p.
Mathematics
bilipschitz homogeneity; jordan curves; bounded turning; fractals; inner distance

Recommended Citations

Citations

  • Freeman, D. M. (2009). Bilipschitz Homogeneity and Jordan Curves [Doctoral dissertation, University of Cincinnati]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1251229498

    APA Style (7th edition)

  • Freeman, David. Bilipschitz Homogeneity and Jordan Curves. 2009. University of Cincinnati, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=ucin1251229498.

    MLA Style (8th edition)

  • Freeman, David. "Bilipschitz Homogeneity and Jordan Curves." Doctoral dissertation, University of Cincinnati, 2009. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1251229498

    Chicago Manual of Style (17th edition)

ucin1251229498
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© 2009, all rights reserved.
This open access ETD is published by University of Cincinnati and OhioLINK.