## Quantitative recurrence properties and homogeneous self-similar sets

HTML articles powered by AMS MathViewer

- by Yuanyang Chang, Min Wu and Wen Wu
- Proc. Amer. Math. Soc.
**147**(2019), 1453-1465 - DOI: https://doi.org/10.1090/proc/14287
- Published electronically: December 31, 2018
- PDF | Request permission

## Abstract:

Let $K$ be a homogeneous self-similar set satisfying the strong separation condition. This paper is concerned with the quantitative recurrence properties of the natural map $T: K\rightarrow K$ induced by the shift. Let $\mu$ be the natural self-similar measure supported on $K$. For a positive function $\varphi$ defined on $\mathbb {N}$, we show that the $\mu$-measure of the following set: \begin{equation*} R(\varphi ):=\{x\in K: |T^n x-x|<\varphi (n) \text { for infinitely many } n\in \mathbb {N}\} \end{equation*} is null or full according to convergence or divergence of a certain series. Moreover, a similar dichotomy law holds for the general Hausdorff measure, which completes the metric theory of this set.## References

- L. Barreira and B. Saussol,
*Hausdorff dimension of measures via Poincaré recurrence*, Comm. Math. Phys.**219**(2001), no. 2, 443–463. MR**1833809**, DOI 10.1007/s002200100427 - Victor Beresnevich, Detta Dickinson, and Sanju Velani,
*Sets of exact ‘logarithmic’ order in the theory of Diophantine approximation*, Math. Ann.**321**(2001), no. 2, 253–273. MR**1866488**, DOI 10.1007/s002080100225 - Victor Beresnevich, Detta Dickinson, and Sanju Velani,
*Measure theoretic laws for lim sup sets*, Mem. Amer. Math. Soc.**179**(2006), no. 846, x+91. MR**2184760**, DOI 10.1090/memo/0846 - Victor Beresnevich, Felipe Ramírez, and Sanju Velani,
*Metric Diophantine approximation: aspects of recent work*, Dynamics and analytic number theory, London Math. Soc. Lecture Note Ser., vol. 437, Cambridge Univ. Press, Cambridge, 2016, pp. 1–95. MR**3618787** - Victor Beresnevich and Sanju Velani,
*A mass transference principle and the Duffin-Schaeffer conjecture for Hausdorff measures*, Ann. of Math. (2)**164**(2006), no. 3, 971–992. MR**2259250**, DOI 10.4007/annals.2006.164.971 - Michael D. Boshernitzan,
*Quantitative recurrence results*, Invent. Math.**113**(1993), no. 3, 617–631. MR**1231839**, DOI 10.1007/BF01244320 - N. Chernov and D. Kleinbock,
*Dynamical Borel-Cantelli lemmas for Gibbs measures*, Israel J. Math.**122**(2001), 1–27. MR**1826488**, DOI 10.1007/BF02809888 - Kenneth Falconer,
*Fractal geometry*, John Wiley & Sons, Ltd., Chichester, 1990. Mathematical foundations and applications. MR**1102677** - J. L. Fernández, M. V. Melián, and D. Pestana,
*Quantitative mixing results and inner functions*, Math. Ann.**337**(2007), no. 1, 233–251. MR**2262783**, DOI 10.1007/s00208-006-0036-4 - Richard Hill and Sanju L. Velani,
*The ergodic theory of shrinking targets*, Invent. Math.**119**(1995), no. 1, 175–198. MR**1309976**, DOI 10.1007/BF01245179 - Jason Levesley, Cem Salp, and Sanju L. Velani,
*On a problem of K. Mahler: Diophantine approximation and Cantor sets*, Math. Ann.**338**(2007), no. 1, 97–118. MR**2295506**, DOI 10.1007/s00208-006-0069-8 - Bing Li, Bao-Wei Wang, Jun Wu, and Jian Xu,
*The shrinking target problem in the dynamical system of continued fractions*, Proc. Lond. Math. Soc. (3)**108**(2014), no. 1, 159–186. MR**3162824**, DOI 10.1112/plms/pdt017 - Kurt Mahler,
*Some suggestions for further research*, Bull. Austral. Math. Soc.**29**(1984), no. 1, 101–108. MR**732177**, DOI 10.1017/S0004972700021316 - Pertti Mattila,
*Geometry of sets and measures in Euclidean spaces*, Cambridge Studies in Advanced Mathematics, vol. 44, Cambridge University Press, Cambridge, 1995. Fractals and rectifiability. MR**1333890**, DOI 10.1017/CBO9780511623813 - Donald Samuel Ornstein and Benjamin Weiss,
*Entropy and data compression schemes*, IEEE Trans. Inform. Theory**39**(1993), no. 1, 78–83. MR**1211492**, DOI 10.1109/18.179344 - S. Seuret and B.-W. Wang,
*Quantitative recurrence properties in conformal iterated function systems*, Adv. Math.**280**(2015), 472–505. MR**3350227**, DOI 10.1016/j.aim.2015.02.019 - Bo Tan and Bao-Wei Wang,
*Quantitative recurrence properties for beta-dynamical system*, Adv. Math.**228**(2011), no. 4, 2071–2097. MR**2836114**, DOI 10.1016/j.aim.2011.06.034 - B. W. Wang and J. Wu,
*A Survey on the dimension theory in dynamical Diophantine approximation*, in Recent Developments in Fractals and Related Fields Conference on Fractals and Related Fields III, île de Porquerolles, France, 2015, edited by Julien Barral and Stéphane Seuret. - Bao-Wei Wang, Jun Wu, and Jian Xu,
*Dynamical covering problems on the triadic Cantor set*, C. R. Math. Acad. Sci. Paris**355**(2017), no. 7, 738–743 (English, with English and French summaries). MR**3673046**, DOI 10.1016/j.crma.2017.05.014 - V. G. Sprindžuk,
*Metric Theory of Diophantine Approximation*, V.H. Winston & Sons, Washington, DC, 1979, translated by R. A. Silverman. - Jia-An Yan,
*A simple proof of two generalized Borel-Cantelli lemmas*, In memoriam Paul-André Meyer: Séminaire de Probabilités XXXIX, Lecture Notes in Math., vol. 1874, Springer, Berlin, 2006, pp. 77–79. MR**2276890**, DOI 10.1007/978-3-540-35513-7_{7}

## Bibliographic Information

**Yuanyang Chang**- Affiliation: Department of Mathematics, South China University of Technology, Guangzhou, 510640, People’s Republic of China
- MR Author ID: 1223628
- Email: changyy@scut.edu.cn
**Min Wu**- Affiliation: Department of Mathematics, South China University of Technology, Guangzhou, 510640, People’s Republic of China
- MR Author ID: 214816
- Email: wumin@scut.edu.cn
**Wen Wu**- Affiliation: Department of Mathematics, South China University of Technology, Guangzhou, 510640, People’s Republic of China
- Email: wuwen@scut.edu.cn
- Received by editor(s): January 31, 2018
- Received by editor(s) in revised form: April 17, 2018
- Published electronically: December 31, 2018
- Additional Notes: The third author is the corresponding author.

This work was supported by NSFC (Grant No. 11771153), the Fundamental Research Funds for the Central Universities (No. 2017MS110) and the Characteristic innovation project of colleges and universities in Guangdong (No. 2016KTSCX007). - Communicated by: Nimish Shah
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**147**(2019), 1453-1465 - MSC (2010): Primary 28A80, 28D05; Secondary 11K55
- DOI: https://doi.org/10.1090/proc/14287
- MathSciNet review: 3910412