## Uniformly branching trees

HTML articles powered by AMS MathViewer

- by Mario Bonk and Daniel Meyer PDF
- Trans. Amer. Math. Soc.
**375**(2022), 3841-3897 Request permission

## Abstract:

A quasiconformal tree $T$ is a (compact) metric tree that is doubling and of bounded turning. We call $T$ trivalent if every branch point of $T$ has exactly three branches. If the set of branch points is uniformly relatively separated and uniformly relatively dense, we say that $T$ is uniformly branching. We prove that a metric space $T$ is quasisymmetrically equivalent to the continuum self-similar tree if and only if it is a trivalent quasiconformal tree that is uniformly branching. In particular, any two trees of this type are quasisymmetrically equivalent.## References

- David Aldous,
*The continuum random tree. I*, Ann. Probab.**19**(1991), no. 1, 1–28. MR**1085326** - David Aldous,
*The continuum random tree. III*, Ann. Probab.**21**(1993), no. 1, 248–289. MR**1207226** - Jonas Azzam,
*Hausdorff dimension of wiggly metric spaces*, Ark. Mat.**53**(2015), no. 1, 1–36. MR**3319612**, DOI 10.1007/s11512-014-0197-4 - Mario Bonk,
*Quasiconformal geometry of fractals*, International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 1349–1373. MR**2275649** - Mario Bonk and Daniel Meyer,
*Expanding Thurston maps*, Mathematical Surveys and Monographs, vol. 225, American Mathematical Society, Providence, RI, 2017. MR**3727134**, DOI 10.1090/surv/225 - Mario Bonk and Daniel Meyer,
*Quasiconformal and geodesic trees*, Fund. Math.**250**(2020), no. 3, 253–299. MR**4107537**, DOI 10.4064/fm749-7-2019 - Mario Bonk and Huy Tran,
*The continuum self-similar tree*, in:*Fractal Geometry and Stochastics VI*, Springer, Cham, 2021, 143–189. - Guy David and Stephen Semmes,
*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 measure. MR**1616732** - Kemal Ilgar Eroğlu, Steffen Rohde, and Boris Solomyak,
*Quasisymmetric conjugacy between quadratic dynamics and iterated function systems*, Ergodic Theory Dynam. Systems**30**(2010), no. 6, 1665–1684. MR**2736890**, DOI 10.1017/S0143385709000789 - Kenneth Falconer,
*Fractal geometry*, 2nd ed., John Wiley & Sons, Inc., Hoboken, NJ, 2003. Mathematical foundations and applications. MR**2118797**, DOI 10.1002/0470013850 - Juha Heinonen,
*Lectures on analysis on metric spaces*, Universitext, Springer-Verlag, New York, 2001. MR**1800917**, DOI 10.1007/978-1-4613-0131-8 - David Herron and Daniel Meyer,
*Quasicircles and bounded turning circles modulo bi-Lipschitz maps*, Rev. Mat. Iberoam.**28**(2012), no. 3, 603–630. MR**2949615**, DOI 10.4171/RMI/687 - Jun Kigami,
*Quasisymmetric modification of metrics on self-similar sets*, Geometry and analysis of fractals, Springer Proc. Math. Stat., vol. 88, Springer, Heidelberg, 2014, pp. 253–282. MR**3276005**, DOI 10.1007/978-3-662-43920-3_{9} - Jun Kigami,
*Geometry and analysis of metric spaces via weighted partitions*, Lecture Notes in Mathematics, vol. 2265, Springer, Cham, [2020] ©2020. MR**4175733**, DOI 10.1007/978-3-030-54154-5 - Kyle Kinneberg,
*Conformal dimension and boundaries of planar domains*, Trans. Amer. Math. Soc.**369**(2017), no. 9, 6511–6536. MR**3660231**, DOI 10.1090/tran/6944 - K. Kuratowski,
*Topology. Vol. II*, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe [Polish Scientific Publishers], Warsaw, 1968. New edition, revised and augmented; Translated from the French by A. Kirkor. MR**0259835** - Jean-François Le Gall,
*Random real trees*, Ann. Fac. Sci. Toulouse Math. (6)**15**(2006), no. 1, 35–62 (English, with English and French summaries). MR**2225746**, DOI 10.5802/afst.1112 - Peter Lin and Steffen Rohde,
*Conformal welding of dendrites*, preprint, 2018. - John M. Mackay and Jeremy T. Tyson,
*Conformal dimension*, University Lecture Series, vol. 54, American Mathematical Society, Providence, RI, 2010. Theory and application. MR**2662522**, DOI 10.1090/ulect/054 - Daniel Meyer,
*Quasisymmetric embedding of self similar surfaces and origami with rational maps*, Ann. Acad. Sci. Fenn. Math.**27**(2002), no. 2, 461–484. MR**1922201** - Daniel Meyer,
*Bounded turning circles are weak-quasicircles*, Proc. Amer. Math. Soc.**139**(2011), no. 5, 1751–1761. MR**2763763**, DOI 10.1090/S0002-9939-2010-10634-2 - Sam B. Nadler Jr.,
*Continuum theory*, Monographs and Textbooks in Pure and Applied Mathematics, vol. 158, Marcel Dekker, Inc., New York, 1992. An introduction. MR**1192552** - Steffen Rohde,
*Quasicircles modulo bilipschitz maps*, Rev. Mat. Iberoamericana**17**(2001), no. 3, 643–659. MR**1900898**, DOI 10.4171/RMI/307 - P. Tukia and J. Väisälä,
*Quasisymmetric embeddings of metric spaces*, Ann. Acad. Sci. Fenn. Ser. A I Math.**5**(1980), no. 1, 97–114. MR**595180**, DOI 10.5186/aasfm.1980.0531 - Tamás Vicsek,
*Fractal models for diffusion controlled aggregation*, J. Phys. A: Math. Gen. 16 (1983) L647–L652. - Gordon Thomas Whyburn,
*Analytic topology*, American Mathematical Society Colloquium Publications, Vol. XXVIII, American Mathematical Society, Providence, R.I., 1963. MR**0182943**

## Additional Information

**Mario Bonk**- Affiliation: Department of Mathematics, University of California, Los Angeles, California 90095
- MR Author ID: 39235
- Email: mbonk@math.ucla.edu
**Daniel Meyer**- Affiliation: Department of Mathematical Sciences, University of Liverpool, Mathematical Sciences Building, Liverpool L69 7ZL, United Kingdom
- MR Author ID: 700302
- ORCID: 0000-0003-1881-8137
- Email: dmeyermail@gmail.com
- Received by editor(s): June 2, 2020
- Received by editor(s) in revised form: January 21, 2021
- Published electronically: March 4, 2022
- Additional Notes: The first author was partially supported by NSF grant DMS-1808856
- © Copyright 2022 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**375**(2022), 3841-3897 - MSC (2020): Primary 30L10
- DOI: https://doi.org/10.1090/tran/8404
- MathSciNet review: 4419049