Quasiregular distortion of dimensions

Chrontsios-Garitsis, Efstathios-K.

doi:10.1090/ecgd/398

Quasiregular distortion of dimensions
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by Efstathios-K. Chrontsios-Garitsis;

Conform. Geom. Dyn. 28 (2024), 165-175

DOI: https://doi.org/10.1090/ecgd/398

Published electronically: December 12, 2024

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Abstract:

We investigate the distortion of the Assouad dimension and (regularized) spectrum of sets under planar quasiregular maps. The respective results for the Hausdorff and upper box-counting dimension follow immediately from their quasiconformal counterparts by employing elementary properties of these dimension notions (e.g. countable stability and Lipschitz stability). However, the Assouad dimension and spectrum do not share such properties. We obtain upper bounds on the Assouad dimension and spectrum of images of compact sets under holomorphic and planar quasiregular maps by studying their behavior around their critical points. As an application, the invariance of porosity of compact subsets of the plane under quasiregular maps is established.

References

Patrice Assouad, Plongements lipschitziens dans $\textbf {R}^{n}$, Bull. Soc. Math. France 111 (1983), no. 4, 429–448 (French, with English summary). MR 763553, DOI 10.24033/bsmf.1997
Kari Astala, Area distortion of quasiconformal mappings, Acta Math. 173 (1994), no. 1, 37–60. MR 1294669, DOI 10.1007/BF02392568
Mario Bonk and Juha Heinonen, Smooth quasiregular mappings with branching, Publ. Math. Inst. Hautes Études Sci. 100 (2004), 153–170. MR 2102699, DOI 10.1007/s10240-004-0024-8
Bodil Branner and Núria Fagella, Quasiconformal surgery in holomorphic dynamics, Cambridge Studies in Advanced Mathematics, vol. 141, Cambridge University Press, Cambridge, 2014. With contributions by Xavier Buff, Shaun Bullett, Adam L. Epstein, Peter Haïssinsky, Christian Henriksen, Carsten L. Petersen, Kevin M. Pilgrim, Tan Lei and Michael Yampolsky. MR 3445628, DOI 10.1017/CBO9781107337602
Efstathios K. Chrontsios Garitsis and Jeremy T. Tyson, Quasiconformal distortion of the Assouad spectrum and classification of polynomial spirals, Bull. Lond. Math. Soc. 55 (2023), no. 1, 282–307. MR 4568342, DOI 10.1112/blms.12727
Adrien Douady, Systèmes dynamiques holomorphes, Bourbaki seminar, Vol. 1982/83, Astérisque, vol. 105, Soc. Math. France, Paris, 1983, pp. 39–63 (French). MR 728980
Adrien Douady and John Hamal Hubbard, On the dynamics of polynomial-like mappings, Ann. Sci. École Norm. Sup. (4) 18 (1985), no. 2, 287–343. MR 816367, DOI 10.24033/asens.1491
Alexandre Eremenko, Bloch radius, normal families and quasiregular mappings, Proc. Amer. Math. Soc. 128 (2000), no. 2, 557–560. MR 1641689, DOI 10.1090/S0002-9939-99-05141-2
Kenneth Falconer, Fractal geometry, 3rd ed., John Wiley & Sons, Ltd., Chichester, 2014. Mathematical foundations and applications. MR 3236784
Jonathan M. Fraser, Assouad dimension and fractal geometry, Cambridge Tracts in Mathematics, vol. 222, Cambridge University Press, Cambridge, 2021. MR 4411274, DOI 10.1017/9781108778459
Jonathan M. Fraser and Han Yu, New dimension spectra: finer information on scaling and homogeneity, Adv. Math. 329 (2018), 273–328. MR 3783415, DOI 10.1016/j.aim.2017.12.019
F. W. Gehring and J. Väisälä, Hausdorff dimension and quasiconformal mappings, J. London Math. Soc. (2) 6 (1973), 504–512. MR 324028, DOI 10.1112/jlms/s2-6.3.504
Robert Kaufman, Sobolev spaces, dimension, and random series, Proc. Amer. Math. Soc. 128 (2000), no. 2, 427–431. MR 1670383, DOI 10.1090/S0002-9939-99-05383-6
O. Lehto and K. I. Virtanen, Quasiconformal mappings in the plane, 2nd ed., Die Grundlehren der mathematischen Wissenschaften, Band 126, Springer-Verlag, New York-Heidelberg, 1973. Translated from the German by K. W. Lucas. MR 344463, DOI 10.1007/978-3-642-65513-5
Jouni Luukkainen, Assouad dimension: antifractal metrization, porous sets, and homogeneous measures, J. Korean Math. Soc. 35 (1998), no. 1, 23–76. MR 1608518
Jani Onninen and Kai Rajala, Quasiregular mappings to generalized manifolds, J. Anal. Math. 109 (2009), 33–79. MR 2585391, DOI 10.1007/s11854-009-0028-x
Bruce P. Palka, An introduction to complex function theory, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1991. MR 1078017, DOI 10.1007/978-1-4612-0975-1
Pietro Poggi-Corradini and Kai Rajala, An egg-yolk principle and exponential integrability for quasiregular mappings, J. Lond. Math. Soc. (2) 76 (2007), no. 2, 531–544. MR 2363431, DOI 10.1112/jlms/jdm078
Kai Rajala, A lower bound for the Bloch radius of $K$-quasiregular mappings, Proc. Amer. Math. Soc. 132 (2004), no. 9, 2593–2601. MR 2054784, DOI 10.1090/S0002-9939-04-07405-2
Kai Rajala, Bloch’s theorem for mappings of bounded and finite distortion, Math. Ann. 339 (2007), no. 2, 445–460. MR 2324726, DOI 10.1007/s00208-007-0124-0
Seppo Rickman, Quasiregular mappings, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 26, Springer-Verlag, Berlin, 1993. MR 1238941, DOI 10.1007/978-3-642-78201-5
Jukka Sarvas, The Hausdorff dimension of the branch set of a quasiregular mapping, Ann. Acad. Sci. Fenn. Ser. A I Math. 1 (1975), no. 2, 297–307. MR 396945, DOI 10.5186/aasfm.1975.0121
Mitsuhiro Shishikura, On the quasiconformal surgery of rational functions, Ann. Sci. École Norm. Sup. (4) 20 (1987), no. 1, 1–29. MR 892140, DOI 10.24033/asens.1522
Jussi Väisälä, Porous sets and quasisymmetric maps, Trans. Amer. Math. Soc. 299 (1987), no. 2, 525–533. MR 869219, DOI 10.1090/S0002-9947-1987-0869219-8