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Quasisymmetries of finitely ramified Julia sets (2310.17442v2)
Published 26 Oct 2023 in math.DS and math.GR
Abstract: We develop a theory of quasisymmetries for finitely ramified fractals, with applications to finitely ramified Julia sets. We prove that certain finitely ramified fractals admit a naturally defined class of "undistorted metrics" that are all quasi-equivalent. As a result, piecewise-defined homeomorphisms of such a fractal that locally preserve the cell structure are quasisymmetries. This immediately gives a solution to the quasisymmetric uniformization problem for topologically rigid fractals such as the Sierpi\'nski triangle. We show that our theory applies to many finitely ramified Julia sets, and we prove that any connected Julia set for a hyperbolic unicritical polynomial has infinitely many quasisymmetries, generalizing a result of Lyubich and Merenkov. We also prove that the quasisymmetry group of the Julia set for the rational function $1-z{-2}$ is infinite, and we show that the quasisymmetry groups for the Julia sets of a broad class of polynomials contain Thompson's group $F$.
- C. Bandt and T. Retta, Topological spaces admitting a unique fractal structure. Fundamenta Mathematicae 141 (1992): 257–268. Crossref.
- J. Belk and B. Forrest, Rearrangement groups of fractals. Transactions of the American Mathematical Society 372 (2019): 4509–4552. arXiv. Crossref.
- A. Beurling and L. Ahlfors, The boundary correspondence under quasiconformal mappings. Acta Mathematica 96 (1956): 125–142. Crossref.
- M. Bonk, Uniformization of Sierpiński carpets in the plane. Inventiones Mathematicae 186.3 (2011): 559–665. arXiv. Crossref.
- M. Bonk and B. Kleiner, Quasisymmetric parametrizations of two-dimensional metric spheres. Inventiones Mathematicae 150 (2002): 127–183. arXiv. Crossref.
- M. Bonk and B. Kleiner, Conformal dimension and Gromov hyperbolic groups with 2-sphere boundary. Geometry and Topology 9 (2005): 219–246. arXiv. Crossref.
- M. Bonk and S. Merenkov, Quasisymmetric rigidity of square Sierpiński carpets. Annals of Mathematics 177 (2013): 591–643. arXiv. Crossref.
- M. Bonk and D. Meyer, Quasiconformal and geodesic trees. Fundamenta Mathematicae 250 (2020): 253–299. arXiv. Crossref
- M. Bonk and D. Meyer, Uniformly branching trees. Transactions of the American Mathematical Society 375.6 (2022): 3841–3897. arXiv Crossref.
- M. Bonk and O. Schramm, Embeddings of Gromov hyperbolic spaces. Geometric and Functional Analysis 10 (2000): 266–306. Crossref.
- A. Douady and J. H. Hubbard, Exploring the Mandelbrot set: the Orsay notes. Publications Mathématiques d’Orsay (1984). Cornell.
- L. Geyer and K. Wildrick, Quantitative quasisymmetric uniformization of compact surfaces. Proceedings of the American Mathematical Society 146 (2018): 281–293. arXiv. Crossref
- A. Kameyama, Julia sets of postcritically finite rational maps and topological self-similar sets. Nonlinearity 13.1 (2000): 165–188. Crossref.
- J. Kelingos, Boundary correspondence under quasiconformal mappings. Michigan Mathematical Journal 13.2 (1966): 235–249. Crossref.
- J. Kigami, Harmonic calculus on P.C.F. self-similar sets. Transactions of the American Mathematical Society 335.2 (1993): 721–755. Crossref.
- J. Louwsma, Homeomorphism groups of the Sierpinski carpet and Sierpinski gasket. Preprint (2004).
- M. Lyubich and S. Merenkov, Quasisymmetries of the basilica and the Thompson group. Geometric and Functional Analysis 28.3 (2018): 727–754. arXiv. Crossref.
- C. T. McMullen, Frontiers in complex dynamics. Bulletin of the American Mathematical Society 31.2 (1994): 155–172. arXiv. Crossref.
- S. Merenkov and K. Wildrick, Quasisymmetric Koebe uniformization. Revista Matemática Iberoamericana 29.3 (2013): 859–910. arXiv. Crossref.
- J. Milnor, Remarks on iterated cubic maps. Experimental Mathematics 1.1 (1992): 5–24. arXiv. Project Euclid.
- Y. Neretin, On the group of homeomorphisms of the Basilica. Preprint (2023). arXiv.
- C. L. Petersen, On the Pommerenke–Levin–Yoccoz inequality. Ergodic Theory and Dynamical Systems 13.4 (1993): 785–806. Roskilde University.
- K. M. Pilgrim, Rational maps whose Fatou components are Jordan domains. Ergodic Theory and Dynamical Systems 16.6 (1996): 1323–1343. Crossref.
- C. Pommerenke, On conformal mapping and iteration of rational functions. Complex Variables and Elliptic Equations 5.2–4 (1986): 117–126. Crossref.
- W. Qiu and F. Yang, Quasisymmetric uniformization and Hausdorff dimensions of Cantor circle Julia sets. Transactions of the American Mathematical Society 374.7 (2021): 5191–5223. arXiv. Crossref.
- S. Semmes, Good metric spaces without good parameterizations. Revista Matemática Iberoamericana 12.1 (1996): 187–275. Crossref.
- S. Semmes, On the nonexistence of bilipschitz parameterizations and geometric problems about A∞subscript𝐴A_{\infty}italic_A start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT-weights. Revista Matemática Iberoamericana 12.2 (1996): 337–410. Crossref.
- R. S. Strichartz, Analysis on fractals. Notices of the American Mathematical Society 46.10 (1999): 1199–1208. AMS.
- R. S. Strichartz, Fractafolds based on the Sierpinski gasket and their spectra. Transactions of the American Mathematical Society 355.10 (2003): 4019–4043. Crossref.
- D. Sullivan, Quasiconformal homeomorphisms and dynamics I. Solution of the Fatou–Julia problem on wandering domains. Annals of Mathematics 122.2 (1985): 401–418. Crossref.
- M. Tarocchi, On Thompson groups for Ważewski dendrites. Preprint (2023). arXiv.
- A. Teplyaev, Harmonic coordinates on fractals with finitely ramified cell structure. Canadian Journal of Mathematics 60.2 (2008): 457–480. arXiv. Crossref.
- P. Tukia and J. Väisälä, Quasisymmetric embeddings of metric spaces. Ann. Acad. Sci. Fenn. Ser. A I Math. 5 (1980): 97–114. Crossref.
- J. Väisälä, Quasi-symmetric embeddings in Euclidean spaces. Transactions of the American Mathematical Society 264.1 (1981): 191–204. Crossref.
- K. Wildrick, Quasisymmetric parametrizations of two-dimensional metric planes. Proceedings of the London Mathematical Society 97.3 (2008): 783–812. arXiv. Crossref