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 and Continuous Dynamical Systems, 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.

[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.

[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.

[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.

[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. 

[6]

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

[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.

[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.

[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.

[10]

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

[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.

[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.

[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.

[14]

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

[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.

[16]

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

[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. 

[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.

[19]

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

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

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

[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.

[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.

[24]

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

[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.

[26]

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

[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.

[28]

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

[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.

[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.

[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.

[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.

[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.

[34]

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

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.

[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.

[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.

[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.

[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. 

[6]

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

[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.

[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.

[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.

[10]

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

[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.

[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.

[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.

[14]

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

[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.

[16]

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

[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. 

[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.

[19]

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

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

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

[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.

[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.

[24]

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

[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.

[26]

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

[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.

[28]

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

[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.

[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.

[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.

[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.

[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.

[34]

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

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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