A Characterization of BLD-Mappings Between Metric Spaces
- 145 Downloads
We prove a characterization theorem for BLD-mappings between complete locally compact path-metric spaces. As a corollary, we obtain a sharp limit theorem for BLD-mappings.
KeywordsBLD BLD-mappings Metric geometry Path-metric spaces Gromov-Hausdorff convergence Lipschitz quotient mappings Branched covers
Mathematics Subject Classification30L10 30C65 57M12
The author would like to thank, once again, his advisor Pekka Pankka for introducing him to the world of BLD geometry. The ideas presented in this manuscript have been incubating for a long time and have been worked on during several wonderful mathematical events, but the author would like to mention the School in Geometric Analysis, Geometric analysis on Riemannian and singular metric spaces at Lake Como School of Advanced Studies in the fall of 2013 and the Research Term on Analysis and Geometry in Metric Spaces at ICMAT in the spring of 2015 as especially fruitful events concerning the current work. The author is especially grateful for all the interesting discussions with fellow mathematicians at these, and other, events. The contents of this paper were improved further by the author’s discussions with Piotr Hajłasz and his students while visiting University of Pittsburgh in 2016. Finally, the thoroughness and comments of the anonymous referee are gratefully acknowledged and have improved the readability of the manuscript. The author was supported by the Väisälä foundation.
- 4.David, G., Semmes, S.: Fractured Fractals and Broken Dreams. Oxford Lecture Series in Mathematics and its Applications, Vol 7. The Clarendon Press, Oxford University Press, New York (1997). Self-similar geometry through metric and measureGoogle Scholar
- 6.Gromov, M.: Metric Structures for Riemannian and Non-Riemannian Spaces. Progress in Mathematics, Vol 152. Birkhäuser Boston Inc., Boston (1999). Based on the 1981 French original [MR0682063 (85e:53051)], With appendices by M. Katz, P. Pansu and S. Semmes, Translated from the French by Sean Michael BatesGoogle Scholar
- 7.Guo, C.-Y., Williams, M.: Porosity of the Branch set of Discrete Open Mappings with Controlled Linear Dilatation (preprint)Google Scholar
- 9.Hajłasz, P., Malekzadeh, S.: A new characterization of the mappings of bounded length distortion. Int. Math. Res. Not. (2015)Google Scholar
- 12.Kapovich, M.: Hyperbolic Manifolds and Discrete Groups. Modern Birkhäuser Classics. Birkhäuser Boston, Inc., Boston. Reprint of the 2001 edition (2009)Google Scholar
- 13.Kleiner, B., MacKay, J.: Differentiable structures on metric measure spaces: a primer. (preprint)Google Scholar
- 14.Kaczor, W.J., Nowak, M.T.: Problems in Mathematical Analysis. II, Student Mathematical Library, Vol. 12. American Mathematical Society, Providence (2001). Continuity and differentiation, Translated from the 1998 Polish original, revised and augmented by the authorsGoogle Scholar
- 16.Luisto, R.: Note on local-to-global properties of BLD-mappings. Proc. Amer. Math. Soc. 144(2), 599–607 (2016)Google Scholar
- 21.Rickman, S.: Quasiregular Mappings. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], Vol 26. Springer, Berlin, (1993)Google Scholar