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LIPSCHITZ EQUIVALENCE OF CANTOR SETS AND IRREDUCIBILITY OF POLYNOMIALS

Part of: Classical measure theory Algebraic number theory: global fields

Published online by Cambridge University Press:  26 June 2018

Jun Jason Luo ,
Huo-Jun Ruan  and
Yi-Lin Wang
Jun Jason Luo
Affiliation:
College of Mathematics and Statistics, Chongqing University, 401331 Chongqing, China Institut für Mathematik, Friedrich-Schiller-Universität Jena, 07743 Jena, Germany email [email protected]
Huo-Jun Ruan
Affiliation:
School of Mathematical Science, Zhejiang University, Hangzhou 310027, China email [email protected]
Yi-Lin Wang
Affiliation:
College of Mathematics and Statistics, Chongqing Uinversity, Chongqing 401331, China email [email protected]
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Abstract

In the paper, we provide an effective criterion for the Lipschitz equivalence of two-branch Cantor sets and three-branch Cantor sets by studying the irreducibility of polynomials. We also find that any two Cantor sets are Lipschitz equivalent if and only if their contraction vectors are equivalent provided one of the contraction vectors is homogeneous.

MSC classification

Primary: 28A80: Fractals
Secondary: 11R09: Polynomials (irreducibility, etc.)
Type
Research Article
Information
Mathematika , Volume 64 , Issue 3 , 2018 , pp. 730 - 741
Copyright
Copyright © University College London 2018 

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Footnotes

The research of Luo and Wang is supported in part by NSFC (no. 11301322), the Fundamental and Frontier Research Project of Chongqing (no. cstc2015jcyjA00035), the Fundamental Research Funds for the Central Universities (no. 106112017CDJXY100005). The research of Ruan is supported in part by NSFC (nos. 11271327, 11771391) and ZJNSFC (no. LR14A010001).

References

Cooper, D. and Pignataro, T., On the shape of Cantor sets. J. Differential Geom. 28(2) 1988, 203–221.Google Scholar
Dubickas, A., Nonreciprocal algebraic numbers of small measure. Comment. Math. Univ. Carolin. 45(4) 2004, 693–697.Google Scholar
Falconer, K. J., Fractal Geometry, Mathematical Foundation and Applications, Wiley (New York, 2003).Google Scholar
Falconer, K. J. and Marsh, D. T., Classification of quasi-circles by Hausdorff dimension. Nonlinearity 2 1989, 489–493.Google Scholar
Falconer, K. J. and Marsh, D. T., On the Lipschitz equivalence of Cantor sets. Mathematika 39 1992, 223–233.Google Scholar
Finch, C. and Jones, L., On the irreducibility of {-1, 0, 1}-quadrinomials. Integers 6 2006, #A16, 4 pp.Google Scholar
Lau, K. S. and Luo, J. J., Lipschitz equivalence of self-similar sets and hyperbolic boundaries. Adv. Math. 235 2013, 555–579.Google Scholar
Ljunggren, W., On the irreducibility of certain trinomials and quadrinomials. Math. Scand. 8 1960, 65–70.Google Scholar
Llorente, M. and Mattila, P., Lipschitz equivalence of subsets of self-conformal sets. Nonlinearity 23 2010, 875–882.Google Scholar
Mills, W. H., The factorization of certain quadrinomials. Math. Scand. 57 1985, 44–50.Google Scholar
Rao, H., Ruan, H. J. and Wang, Y., Lipschitz equivalence of Cantor sets and algebraic properties of contraction ratios. Trans. Amer. Math. Soc. 364 2012, 1109–1126.Google Scholar
Rao, H., Ruan, H. J. and Wang, Y., Lipschitz equivalence of self-similar sets: algebraic and geometric properties. Contemp. Math. 600 2013, 349–364.Google Scholar
Rao, H., Ruan, H. J. and Xi, L.-F., Lipschitz equivalence of self-similar sets. C. R. Acad. Sci. Paris, Ser. I 342 2006, 191–196.Google Scholar
Rao, H. and Zhang, Y., Higher dimensional Frobenius problem and Lipschitz equivalence of Cantor sets. J. Math. Pures Appl. (9) 104 2015, 868–881.Google Scholar
Xi, L.-F., Lipschitz equivalence of self-conformal sets. J. Lond. Math. Soc. (2) 70 2004, 369–382.Google Scholar
Xi, L.-F. and Ruan, H. J., Lipschitz equivalence of generalized {1, 3, 5}-{1, 4, 5} self-similar sets. Sci. China Ser. A 50 2007, 1537–1551.Google Scholar
Xi, L.-F. and Xiong, Y., Self-similar sets with initial cubic patterns. C. R. Acad. Sci. Paris, Ser. I 348 2010, 15–20.Google Scholar
Xi, L.-F. and Xiong, Y., Lipschitz equivalence of fractals generated by nested cubes. Math. Z. 271 2012, 1287–1308.Google Scholar