November  2020, 40(11): 6089-6114. doi: 10.3934/dcds.2020271

Isomorphism and bi-Lipschitz equivalence between the univoque sets

Department of Mathematics, Ningbo University, Ningbo, Zhejiang, China

Received  April 2018 Revised  May 2020 Published  July 2020

In this paper, we consider a class of self-similar sets, denoted by $ \mathcal{A} $, and investigate the set of points in the self-similar sets having unique codings. We call such set the univoque set and denote it by $ U_1 $. We analyze the isomorphism and bi-Lipschitz equivalence between the univoque sets. The main result of this paper, in terms of the dimension of $ U_1 $, is to give several equivalent conditions which describe that the closure of two univoque sets, under the lazy maps, are measure theoretically isomorphic with respect to the unique measure of maximal entropy. Moreover, we prove, under the condition $ U_1 $ is closed, that isomorphism and bi-Lipschitz equivalence between the univoque sets have resonant phenomenon.

Citation: Kan Jiang, Lifeng Xi, Shengnan Xu, Jinjin Yang. Isomorphism and bi-Lipschitz equivalence between the univoque sets. Discrete & Continuous Dynamical Systems - A, 2020, 40 (11) : 6089-6114. doi: 10.3934/dcds.2020271
References:
[1]

Shigeki Akiyama, Cubic Pisot units with finite beta expansions, Algebraic Number Theory and Diophantine Analysis (Graz, 1998), 11–26. de Gruyter, Berlin, 2000.  Google Scholar

[2]

Simon BakerKarma Dajani and Kan Jiang, On univoque points for self-similar sets, Fund. Math., 228 (2015), 265-282.  doi: 10.4064/fm228-3-4.  Google Scholar

[3]

X. ChenK. Jiang and W. Li, Lipschitz equivalence of a class of self-similar sets, Ann. Acad. Sci. Fenn. Math., 42 (2017), 585-591.  doi: 10.5186/aasfm.2017.4238.  Google Scholar

[4]

X. ChenK. Jiang and W. Li, Estimating the Hausdorff dimensions of univoque sets for self-similar sets, Indag. Math. (N.S.), 30 (2019), 862-873.  doi: 10.1016/j.indag.2019.06.001.  Google Scholar

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K. DajaniK. JiangD. Kong and W. Li, Multiple expansions of real numbers with digits set $\{0, 1, q\}$, Math. Z., 291 (2019), 1605-1619.   Google Scholar

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K. Dajani, K. Jiang, D. Kong and W. Li, Multiple codings for self-similar sets with overlaps, arXiv: 1603.09304, 2016. Google Scholar

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M. de Vries and V. Komornik, Unique expansions of real numbers, Adv. Math., 221 (2009), 390-427.  doi: 10.1016/j.aim.2008.12.008.  Google Scholar

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P. ErdősM. Horváth and I. Joó, On the uniqueness of the expansions $1 = \sum q^{-n_i}$, Acta Math. Hungar., 58 (1991), 333-342.  doi: 10.1007/BF01903963.  Google Scholar

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K. J. Falconer and D. T. Marsh, Classification of quasi-circles by Hausdorff dimension, Nonlinearity, 2 (1989), 489-493.  doi: 10.1088/0951-7715/2/3/008.  Google Scholar

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D.-J. Feng and H. Hu, Dimension theory of iterated function systems, Comm. Pure Appl. Math., 62 (2009), 1435-1500.  doi: 10.1002/cpa.20276.  Google Scholar

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P. Glendinning and N. Sidorov, Unique representations of real numbers in non-integer bases, Math. Res. Lett., 8 (2001), 535-543.  doi: 10.4310/MRL.2001.v8.n4.a12.  Google Scholar

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M. Hochman, On self-similar sets with overlaps and inverse theorems for entropy, Ann. of Math. (2), 180 (2014), 773-822.  doi: 10.4007/annals.2014.180.2.7.  Google Scholar

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J. E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J., 30 (1981), 713-747.  doi: 10.1512/iumj.1981.30.30055.  Google Scholar

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K. Jiang and K. Dajani, Subshifts of finite type and self-similar sets, Nonlinearity, 30 (2017), 659-686.  doi: 10.1088/1361-6544/aa53c7.  Google Scholar

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K. Jiang, S. Wang and L. Xi, On self-similar sets with exact overlaps, Submitted, 2017. Google Scholar

[17]

K. JiangS. Wang and L. Xi, Lipschitz equivalence of self-similar sets with exact overlaps, Ann. Acad. Sci. Fenn. Math., 43 (2018), 905-912.   Google Scholar

[18]

K.-S. Lau and S.-M. Ngai, A generalized finite type condition for iterated function systems, Adv. Math., 208 (2007), 647-671.  doi: 10.1016/j.aim.2006.03.007.  Google Scholar

[19]

B. Li, W. Li and J. Miao, Lipschitz equivalence of McMullen sets, Fractals, 21 (2013), 1350022, 11 pp. doi: 10.1142/S0218348X13500229.  Google Scholar

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P. Mattila and P. Saaranen, Ahlfors-David regular sets and bilipschitz maps, Ann. Acad. Sci. Fenn. Math., 34 (2009), 487-502.   Google Scholar

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R. D. Mauldin and S. C. Williams, Hausdorff dimension in graph directed constructions, Trans. Amer. Math. Soc., 309 (1988), 811-829.  doi: 10.1090/S0002-9947-1988-0961615-4.  Google Scholar

[23]

S.-M. Ngai and Y. Wang, Hausdorff dimension of self-similar sets with overlaps, J. London Math. Soc. (2), 63 (2001), 655-672.  doi: 10.1017/S0024610701001946.  Google Scholar

[24]

William Parry, Intrinsic Markov chains, Trans. Amer. Math. Soc., 112 (1964), 55-66.  doi: 10.1090/S0002-9947-1964-0161372-1.  Google Scholar

[25]

H. RaoH.-J. Ruan and L.-F. Xi, Lipschitz equivalence of self-similar sets, C. R. Math. Acad. Sci. Paris, 342 (2006), 191-196.  doi: 10.1016/j.crma.2005.12.016.  Google Scholar

[26]

H. Rao and Z. Yunjie, Lipschitz equivalence of fractals and finite state automaton, arXiv: 1609.04271, 2016. Google Scholar

[27]

H.-J. RuanY. Wang and L.-F. Xi., Lipschitz equivalence of self-similar sets with touching structures, Nonlinearity, 27 (2014), 1299-1321.  doi: 10.1088/0951-7715/27/6/1299.  Google Scholar

[28]

Peter Walters, An Introduction to Ergodic Theory, volume 79 of Graduate Texts in Mathematics, Springer-Verlag, New York-Berlin, 1982. Google Scholar

[29]

Q. Wang and L. Xi, Quasi-Lipschitz equivalence of Ahlfors-David regular sets, Nonlinearity, 24 (2011), 941-950.  doi: 10.1088/0951-7715/24/3/011.  Google Scholar

[30]

Z. WenZ. Zhu and G. Deng, Lipschitz equivalence of a class of general Sierpinski carpets, J. Math. Anal. Appl., 385 (2012), 16-23.  doi: 10.1016/j.jmaa.2011.06.018.  Google Scholar

[31]

L.-F. Xi and H.-J. Ruan, Lipschitz equivalence of generalized $\{1, 3, 5\}$-$\{1, 4, 5\}$ self-similar sets, Sci. China Ser. A, 50 (2007), 1537-1551.  doi: 10.1007/s11425-007-0113-5.  Google Scholar

[32]

L.-F. Xi and Y. Xiong, Ensembles auto-similaires avec motifs initiaux cubiques, C. R. Math. Acad. Sci. Paris, 348 (2010), 15-20.  doi: 10.1016/j.crma.2009.12.006.  Google Scholar

[33]

L.-F. Xi and Y. Xiong, Lipschitz equivalence of fractals generated by nested cubes, Math. Z., 271 (2012), 1287-1308.  doi: 10.1007/s00209-011-0916-5.  Google Scholar

[34]

L.-F. Xi and Y. Xiong, Lipschitz equivalence class, ideal class and the gauss class number problem, arXiv: 1304.0103, 2013. Google Scholar

show all references

References:
[1]

Shigeki Akiyama, Cubic Pisot units with finite beta expansions, Algebraic Number Theory and Diophantine Analysis (Graz, 1998), 11–26. de Gruyter, Berlin, 2000.  Google Scholar

[2]

Simon BakerKarma Dajani and Kan Jiang, On univoque points for self-similar sets, Fund. Math., 228 (2015), 265-282.  doi: 10.4064/fm228-3-4.  Google Scholar

[3]

X. ChenK. Jiang and W. Li, Lipschitz equivalence of a class of self-similar sets, Ann. Acad. Sci. Fenn. Math., 42 (2017), 585-591.  doi: 10.5186/aasfm.2017.4238.  Google Scholar

[4]

X. ChenK. Jiang and W. Li, Estimating the Hausdorff dimensions of univoque sets for self-similar sets, Indag. Math. (N.S.), 30 (2019), 862-873.  doi: 10.1016/j.indag.2019.06.001.  Google Scholar

[5]

K. DajaniK. JiangD. Kong and W. Li, Multiple expansions of real numbers with digits set $\{0, 1, q\}$, Math. Z., 291 (2019), 1605-1619.   Google Scholar

[6]

K. Dajani, K. Jiang, D. Kong and W. Li, Multiple codings for self-similar sets with overlaps, arXiv: 1603.09304, 2016. Google Scholar

[7]

M. de Vries and V. Komornik, Unique expansions of real numbers, Adv. Math., 221 (2009), 390-427.  doi: 10.1016/j.aim.2008.12.008.  Google Scholar

[8]

P. ErdősM. Horváth and I. Joó, On the uniqueness of the expansions $1 = \sum q^{-n_i}$, Acta Math. Hungar., 58 (1991), 333-342.  doi: 10.1007/BF01903963.  Google Scholar

[9]

K. J. Falconer and D. T. Marsh, Classification of quasi-circles by Hausdorff dimension, Nonlinearity, 2 (1989), 489-493.  doi: 10.1088/0951-7715/2/3/008.  Google Scholar

[10]

K. J. Falconer and D. T. Marsh, On the Lipschitz equivalence of Cantor sets, Mathematika, 39 (1992), 223-233.  doi: 10.1112/S0025579300014959.  Google Scholar

[11]

D.-J. Feng and H. Hu, Dimension theory of iterated function systems, Comm. Pure Appl. Math., 62 (2009), 1435-1500.  doi: 10.1002/cpa.20276.  Google Scholar

[12]

P. Glendinning and N. Sidorov, Unique representations of real numbers in non-integer bases, Math. Res. Lett., 8 (2001), 535-543.  doi: 10.4310/MRL.2001.v8.n4.a12.  Google Scholar

[13]

M. Hochman, On self-similar sets with overlaps and inverse theorems for entropy, Ann. of Math. (2), 180 (2014), 773-822.  doi: 10.4007/annals.2014.180.2.7.  Google Scholar

[14]

J. E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J., 30 (1981), 713-747.  doi: 10.1512/iumj.1981.30.30055.  Google Scholar

[15]

K. Jiang and K. Dajani, Subshifts of finite type and self-similar sets, Nonlinearity, 30 (2017), 659-686.  doi: 10.1088/1361-6544/aa53c7.  Google Scholar

[16]

K. Jiang, S. Wang and L. Xi, On self-similar sets with exact overlaps, Submitted, 2017. Google Scholar

[17]

K. JiangS. Wang and L. Xi, Lipschitz equivalence of self-similar sets with exact overlaps, Ann. Acad. Sci. Fenn. Math., 43 (2018), 905-912.   Google Scholar

[18]

K.-S. Lau and S.-M. Ngai, A generalized finite type condition for iterated function systems, Adv. Math., 208 (2007), 647-671.  doi: 10.1016/j.aim.2006.03.007.  Google Scholar

[19]

B. Li, W. Li and J. Miao, Lipschitz equivalence of McMullen sets, Fractals, 21 (2013), 1350022, 11 pp. doi: 10.1142/S0218348X13500229.  Google Scholar

[20] D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, Cambridge, 1995.  doi: 10.1017/CBO9780511626302.  Google Scholar
[21]

P. Mattila and P. Saaranen, Ahlfors-David regular sets and bilipschitz maps, Ann. Acad. Sci. Fenn. Math., 34 (2009), 487-502.   Google Scholar

[22]

R. D. Mauldin and S. C. Williams, Hausdorff dimension in graph directed constructions, Trans. Amer. Math. Soc., 309 (1988), 811-829.  doi: 10.1090/S0002-9947-1988-0961615-4.  Google Scholar

[23]

S.-M. Ngai and Y. Wang, Hausdorff dimension of self-similar sets with overlaps, J. London Math. Soc. (2), 63 (2001), 655-672.  doi: 10.1017/S0024610701001946.  Google Scholar

[24]

William Parry, Intrinsic Markov chains, Trans. Amer. Math. Soc., 112 (1964), 55-66.  doi: 10.1090/S0002-9947-1964-0161372-1.  Google Scholar

[25]

H. RaoH.-J. Ruan and L.-F. Xi, Lipschitz equivalence of self-similar sets, C. R. Math. Acad. Sci. Paris, 342 (2006), 191-196.  doi: 10.1016/j.crma.2005.12.016.  Google Scholar

[26]

H. Rao and Z. Yunjie, Lipschitz equivalence of fractals and finite state automaton, arXiv: 1609.04271, 2016. Google Scholar

[27]

H.-J. RuanY. Wang and L.-F. Xi., Lipschitz equivalence of self-similar sets with touching structures, Nonlinearity, 27 (2014), 1299-1321.  doi: 10.1088/0951-7715/27/6/1299.  Google Scholar

[28]

Peter Walters, An Introduction to Ergodic Theory, volume 79 of Graduate Texts in Mathematics, Springer-Verlag, New York-Berlin, 1982. Google Scholar

[29]

Q. Wang and L. Xi, Quasi-Lipschitz equivalence of Ahlfors-David regular sets, Nonlinearity, 24 (2011), 941-950.  doi: 10.1088/0951-7715/24/3/011.  Google Scholar

[30]

Z. WenZ. Zhu and G. Deng, Lipschitz equivalence of a class of general Sierpinski carpets, J. Math. Anal. Appl., 385 (2012), 16-23.  doi: 10.1016/j.jmaa.2011.06.018.  Google Scholar

[31]

L.-F. Xi and H.-J. Ruan, Lipschitz equivalence of generalized $\{1, 3, 5\}$-$\{1, 4, 5\}$ self-similar sets, Sci. China Ser. A, 50 (2007), 1537-1551.  doi: 10.1007/s11425-007-0113-5.  Google Scholar

[32]

L.-F. Xi and Y. Xiong, Ensembles auto-similaires avec motifs initiaux cubiques, C. R. Math. Acad. Sci. Paris, 348 (2010), 15-20.  doi: 10.1016/j.crma.2009.12.006.  Google Scholar

[33]

L.-F. Xi and Y. Xiong, Lipschitz equivalence of fractals generated by nested cubes, Math. Z., 271 (2012), 1287-1308.  doi: 10.1007/s00209-011-0916-5.  Google Scholar

[34]

L.-F. Xi and Y. Xiong, Lipschitz equivalence class, ideal class and the gauss class number problem, arXiv: 1304.0103, 2013. Google Scholar

Figure 1.  First iteration of $ K $
Figure 3.  First iteration of $ K_1 $
Figure 4.  First iteration of $ K_2 $
Figure 2.  First iteration of $ K $
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