On the Hausdorff dimension of the Gieseking fractal

Warren Dicks, J. Porti

Research output: Contribution to journalArticleResearchpeer-review

1 Citation (Scopus)

Abstract

Let φ:̄ℝ → ̄Cℝ be the Cannon-Thurston map associated to the Gieseking manifold; thus φ is also the Cannon-Thurston map associated to the complement of the figure-eight knot. There are sets ER and Eℂ such that φ decomposes into two maps Eℝ→Eℂ and ℝ-Eℝ→ℂ-Eℂ, where the former is (at-least-two)-to-one, while the latter is a homeomorphism. It is known that Eℝ has Hausdorff dimension zero. Let d denote the Hausdorff dimension of Eℂ. We show that Eℂ is a countable union of Jordan curves of Hausdorff dimension d, and that d=1.35±0.2. In particular, the two-dimensional Lebesgue measure of Eℂ is zero, and the Jordan curves are not rectifiable. We also show that d is the critical exponent of a Poincaré series of a discrete semigroup of isometries of hyperbolic three-space, and describe a computer experiment that suggests that d is near 1.2971. © 2002 Elsevier Science B.V. All rights reserved.
Original languageEnglish
Pages (from-to)169-186
JournalTopology and its Applications
Volume126
DOIs
Publication statusPublished - 30 Nov 2002

Keywords

  • Cannon-Thurston map
  • Fractal curve
  • Hausdorff dimension

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