Hostname: page-component-6bb9c88b65-6scc2 Total loading time: 0 Render date: 2025-07-17T15:01:44.194Z Has data issue: false hasContentIssue false
  • English
  • Français

Recurrence to Shrinking Targets on Typical Self-Affine Fractals

Part of: Smooth dynamical systems: general theory Classical measure theory

Published online by Cambridge University Press:  15 February 2018

Henna Koivusalo  and
Felipe A. Ramírez
Henna Koivusalo*
Affiliation:
Faculty of Mathematics, University of Vienna, Oskar Morgensternplatz 1, 1090 Vienna, Austria (henna.koivusalo@univie.ac.at)
Felipe A. Ramírez
Affiliation:
Wesleyan University, Middletown, CT, USA (framirez@wesleyan.edu)
*
*Corresponding author.
Get access Rights & Permissions [Opens in a new window]

Abstract

We explore the problem of finding the Hausdorff dimension of the set of points that recur to shrinking targets on a self-affine fractal. To be exact, we study the dimension of a certain related symbolic recurrence set. In many cases, this set is equivalent to the recurring set on the fractal.

Keywords

self-affine setsshrinking targetHausdorff dimension

MSC classification

Primary: 37C45: Dimension theory of dynamical systems
Secondary: 28A80: Fractals

Information

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 61 , Issue 2 , May 2018 , pp. 387 - 400
Copyright
Copyright © Edinburgh Mathematical Society 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

1.Bedford, T., Crinkly curves, Markov partitions and box dimensions in self-similar sets. PhD Thesis, The University of Warwick, 1984.Google Scholar
2.Beresnevich, V. and Velani, S., A mass transference principle and the Duffin-Schaeffer conjecture for Hausdorff measures, Ann. of Math. (2) 164(3) (2006), 971–992.CrossRefGoogle Scholar
3.Bowen, R., Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Mathematics, Volume 470 (Springer, Berlin, New York, 1975).CrossRefGoogle Scholar
4.Bugeaud, Y., Harrap, S., Kristensen, S. and Velani, S., On shrinking targets for ℤm actions on tori, Mathematika 56(2) (2010), 193–202.CrossRefGoogle Scholar
5.Chernov, N. and Kleinbock, D., Dynamical Borel-Cantelli lemmas for Gibbs measures, Israel J. Math. 122 (2001), 1–27.CrossRefGoogle Scholar
6.Falconer, K., The Hausdorff dimension of self-affine fractals, Math. Proc. Cambridge Philos. Soc. 103(2) (1988), 339–350.CrossRefGoogle Scholar
7.Falconer, K. J., The dimension of self-affine fractals. II, Math. Proc. Cambridge Philos. Soc. 111(1) (1992), 169–179.CrossRefGoogle Scholar
8.Fayad, B., Mixing in the absence of the shrinking target property, Bull. Lond. Math. Soc. 38(5) (2006), 829–838.CrossRefGoogle Scholar
9.Ferguson, A., Jordan, T. and Rams, M., Dimension of self-affine sets with holes, Ann. Acad. Sci. Fenn. Math. 40(1) (2015), 63–88.CrossRefGoogle Scholar
10.Hill, R. and Velani, S. L., The ergodic theory of shrinking targets, Invent. Math. 119(1) (1995), 175–198.CrossRefGoogle Scholar
11.Hill, R. and Velani, S. L., The shrinking target problem for matrix transformations of tori, J. Lond. Math. Soc. (2) 60(2) (1999), 381–398.CrossRefGoogle Scholar
12.Hill, R. and Velani, S. L., A zero-infinity law for well-approximable points in Julia sets, Ergodic Theory Dynam. Systems 22(6) (2002), 1773–1782.CrossRefGoogle Scholar
13.Hutchinson, J., Fractals and self-similarity, Indiana Univ. Math. J. 30(5) (1981), 713–747.CrossRefGoogle Scholar
14.Käenmäki, A., On natural invariant measures on generalised iterated function systems, Ann. Acad. Sci. Fenn. Math. 29(2) (2004), 419–458.Google Scholar
15.Käenmäki, A. and Reeve, H. W. J., Multifractal analysis of Birkhoff averages for typical infinitely generated self-affine sets, J. Fractal Geom. 1(1) (2014), 83–152.CrossRefGoogle Scholar
16.Käenmäki, A. and Shmerkin, P., Overlapping self-affine sets of Kakeya type, Ergodic Theory Dynam. Systems 29(3) (2009), 941–965.CrossRefGoogle Scholar
17.Käenmäki, A. and Vilppolainen, M., Separation conditions on controlled Moran constructions, Fund. Math. 200(1) (2008), 69–100.CrossRefGoogle Scholar
18.Käenmäki, A. and Vilppolainen, M., Dimension and measures on sub-self-affine sets, Monatsh. Math. 161(3) (2010), 271–293.CrossRefGoogle Scholar
19.Kurzweil, J., On the metric theory of inhomogeneous diophantine approximations, Studia Math. 15 (1955), 84–112.CrossRefGoogle Scholar
20.Li, B., Wang, B.-W., Wu, J. and Xu, J., The shrinking target problem in the dynamical system of continued fractions, Proc. Lond. Math. Soc. (3) 108(1) (2014), 159–186.CrossRefGoogle Scholar
21.McMullen, C., The Hausdorff dimension of general Sierpiński carpets, Nagoya Math. J. 96 (1984), 1–9.CrossRefGoogle Scholar
22.Persson, T. and Rams, M., On shrinking targets for piecewise expanding rational maps, available at http://arxiv.org/abs/1406.6785.Google Scholar
23.Rajala, T. and Vilppolainen, M., Weakly controlled moran constructions and iterated function systems in metric spaces, Illinois J. Math. 55(3) (2011), 1015–1051.CrossRefGoogle Scholar
24.Reeve, H., Shrinking targets for countable Markov maps, available at http://arxiv.org/abs/1107.4736.Google Scholar
25.Solomyak, B., Measure and dimension for some fractal families, Math. Proc. Cambridge Philos. Soc. 124(3) (1998), 531–546.CrossRefGoogle Scholar