Hostname: page-component-8448b6f56d-gtxcr Total loading time: 0 Render date: 2024-04-18T11:00:12.467Z Has data issue: false hasContentIssue false
  • English
  • Français

Planar self-affine sets with equal Hausdorff, box and affinity dimensions

Published online by Cambridge University Press:  20 October 2016

KENNETH FALCONER  and
TOM KEMPTON
KENNETH FALCONER
Affiliation:
Mathematical Institute, University of St Andrews, North Haugh, St Andrews, KY16 9SS, UK email kjf@st-andrews.ac.uk, tmwk@st-andrews.ac.uk
TOM KEMPTON
Affiliation:
Mathematical Institute, University of St Andrews, North Haugh, St Andrews, KY16 9SS, UK email kjf@st-andrews.ac.uk, tmwk@st-andrews.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

Using methods from ergodic theory along with properties of the Furstenberg measure we obtain conditions under which certain classes of plane self-affine sets have Hausdorff or box-counting dimensions equal to their affinity dimension. We exhibit some new specific classes of self-affine sets for which these dimensions are equal.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 38 , Issue 4 , June 2018 , pp. 1369 - 1388
Copyright
© Cambridge University Press, 2016 

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.)

References

Bárány, B.. On the Ledrappier–Young formula for self-affine measures. Math. Proc. Cambridge Philos. Soc. 159(3) (2015), 405–432.CrossRefGoogle Scholar
Bárány, B., Pollicott, M. and Simon, K.. Stationary measures for projective transformations: the Blackwell and Furstenberg measures. J. Stat. Phys. 148(3) (2012), 393–421.CrossRefGoogle Scholar
Bárány, B. and Rams, M.. Dimension maximizing measures for self-affine systems. Preprint, 2015,arXiv:1507.02829.Google Scholar
Bedford, T.. Crinkly curves, Markov partitions and box dimensions in self-similar sets. PhD Thesis, University of Warwick, 1984.Google Scholar
Birkhoff, G.. Extensions of Jentzsch’s theorem. Trans. Amer. Math. Soc. 85 (1957), 219–227.Google Scholar
Falconer, K. J.. Hausdorff dimension and the exceptional set of projections. Mathematika 29(1) (1982), 109–115.CrossRefGoogle Scholar
Falconer, K. J.. The Hausdorff dimension of self-affine fractals. Math. Proc. Cambridge Philos. Soc. 103(2) (1988), 339–350.CrossRefGoogle Scholar
Falconer, K. J.. The dimension of self-affine fractals. II. Math. Proc. Cambridge Philos. Soc. 111(1) (1992), 169–179.CrossRefGoogle Scholar
Falconer, K.. Dimensions of self-affine sets: a survey. Further Developments in Fractals and Related Fields (Trends in Mathematics) . Birkhäuser/Springer, New York, 2013, pp. 115–134.CrossRefGoogle Scholar
Falconer, K.. Fractal Geometry: Mathematical Foundations and Applications, 3rd edn. John Wiley, Chichester, 2014.Google Scholar
Feng, D.-J. and Shmerkin, P.. Non-conformal repellers and the continuity of pressure for matrix cocycles. Geom. Funct. Anal. 24(4) (2014), 1101–1128.CrossRefGoogle Scholar
Fraser, J. M.. On the packing dimension of box-like self-affine sets in the plane. Nonlinearity 25(7) (2012), 2075–2092.CrossRefGoogle Scholar
Hu, H.. Box dimensions and topological pressure for some expanding maps. Comm. Math. Phys. 191(2) (1998), 397–407.CrossRefGoogle Scholar
Hueter, I. and Lalley, S. P.. Falconer’s formula for the Hausdorff dimension of a self-affine set in R 2 . Ergod. Th. & Dynam. Sys. 15(1) (1995), 77–97.CrossRefGoogle Scholar
Hunt, B. R. and Kaloshin, V. Yu.. How projections affect the dimension spectrum of fractal measures. Nonlinearity 10(5) (1997), 1031–1046.CrossRefGoogle Scholar
Hutchinson, J. E.. Fractals and self-similarity. Indiana Univ. Math. J. 30(5) (1981), 713–747.CrossRefGoogle Scholar
Jordan, T., Pollicott, M. and Simon, K.. Hausdorff dimension for randomly perturbed self affine attractors. Comm. Math. Phys. 270(2) (2007), 519–544.CrossRefGoogle Scholar
Käenmäki, A.. On natural invariant measures on generalised iterated function systems. Ann. Acad. Sci. Fenn. Math. 29(2) (2004), 419–458.Google Scholar
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
Marstrand, J. M.. Some fundamental geometrical properties of plane sets of fractional dimensions. Proc. Lond. Math. Soc. (3) 4 (1954), 257–302.CrossRefGoogle Scholar
Mattila, P.. Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability (Cambridge Studies in Advanced Mathematics, 44) . Cambridge University Press, Cambridge, 1995.CrossRefGoogle Scholar
McMullen, C.. The Hausdorff dimension of general Sierpiński carpets. Nagoya Math. J. 96 (1984), 1–9.CrossRefGoogle Scholar
Pollicott, M. and Vytnova, P.. Estimating singularity dimension. Math. Proc. Cambridge Philos. Soc. 158(2) (2015), 223–238.CrossRefGoogle Scholar
Solomyak, B.. Measure and dimension for some fractal families. Math. Proc. Cambridge Philos. Soc. 124(3) (1998), 531–546.CrossRefGoogle Scholar