Metric spaces with unique pretangent spaces. Conditions of the uniqueness

ABDULLAYEV F., Dovgoshey O., Küçükaslan M.

Annales Academiae Scientiarum Fennicae Mathematica, vol.36, no.1, pp.353-392, 2011 (SCI-Expanded, Scopus) identifier

  • Publication Type: Article / Article
  • Volume: 36 Issue: 1
  • Publication Date: 2011
  • Doi Number: 10.5186/aasfm.2011.3623
  • Journal Name: Annales Academiae Scientiarum Fennicae Mathematica
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.353-392
  • Keywords: Cantor set, Logarithmic spiral, Metric spaces, Pretangent spaces, Uniqueness of pretangent metric spaces
  • Uşak University Affiliated: No

Abstract

We find necessary and sufficient conditions for an arbitrary metric space X to have a unique pretangent space at a marked point a ∈ X. Applying this general result we show that each logarithmic spiral has a unique pretangent space at the asymptotic point. Unbounded multiplicative subgroups of C* = C\{0} having unique pretangent spaces at zero are characterized as lying either on the positive real semiaxis or on logarithmic spirals. Our general uniqueness conditions in the case X ⊆ R make it also possible to characterize the points of the ternary Cantor set having unique pretangent spaces.