Hostname: page-component-7479d7b7d-8zxtt Total loading time: 0 Render date: 2024-07-13T09:22:35.427Z Has data issue: false hasContentIssue false
  • English
  • Français

On the dimension of triangular self-affine sets

Published online by Cambridge University Press:  28 November 2017

BALÁZS BÁRÁNY ,
MICHAŁ RAMS  and
KÁROLY SIMON
BALÁZS BÁRÁNY
Affiliation:
Budapest University of Technology and Economics, MTA-BME Stochastics Research Group, P.O. Box 91, 1521 Budapest, Hungary email balubsheep@gmail.com
MICHAŁ RAMS
Affiliation:
Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00-656 Warszawa, Poland email rams@impan.pl
KÁROLY SIMON
Affiliation:
Budapest University of Technology and Economics, Department of Stochastics, Institute of Mathematics, 1521 Budapest, P.O. Box 91, Hungary email simonk@math.bme.hu
Get access Rights & Permissions [Opens in a new window]

Abstract

As a continuation of a recent work [Bárány et al, On the dimension of self-affine sets and measures with overlaps. Proc. Amer. Math. Soc.144 (2016) 4427–4440] of the same authors, in this note we study the dimension theory of diagonally homogeneous triangular planar self-affine iterated function systems.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 39 , Issue 7 , July 2019 , pp. 1751 - 1783
Copyright
© Cambridge University Press, 2017 

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.. Dimension of the generalized 4-corner set and its projections. Ergod. Th. & Dynam. Sys. 32(04) (2012), 1190–1215.Google Scholar
Bárány, B.. On the Ledrappier–Young formula for self-affine measures. Math. Proc. Cambridge Philos. Soc. 159(3) (2015), 405–432.Google Scholar
Bárány, B. and Käenmäki, A.. Ledrappier-Young formula and exact dimensionality of self-affine measures. Adv. in Math. 318 (2017), 88–129.Google 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.Google Scholar
Bárány, B. and Rams, M.. Dimension maximizing measures and local dimension spectrum for self-affine systems. Trans. Amer. Math. Soc. (2017), to appear. https://doi.org/10.1090/tran/7103.Google Scholar
Bárány, B., Rams, M. and Simon, K.. On the dimension of self-affine sets and measures with overlaps. Proc. Amer. Math. Soc. 144 (2016), 4427–4440.Google Scholar
Falconer, K. and Kempton, T.. The dimension of projections of self-affine sets and measures. Ann. Acad. Sci. Fenn. Math. 42(1) (2017), 473–486.Google Scholar
Falconer, K. and Miao, J.. Dimensions of self-affine fractals and multifractals generated by upper-triangular matrices. Fractals 15(03) (2007), 289–299.Google Scholar
Falconer, K. J.. The Geometry of Fractal Sets. Cambridge University Press, Cambridge, 1986.Google Scholar
Falconer, K. J.. The Hausdorff dimension of self-affine fractals. Math. Proc. Cambridge Philos. Soc. 103(3) (1988), 339–350.Google Scholar
Feng, D.-J. and Hu, H.. Dimension theory of iterated function systems. Comm. Pure Appl. Math. 62(11) (2009), 1435–1500.Google Scholar
Furstenberg, H. and Kifer, Y.. Random matrix products and measures on projective spaces. Israel J. Math. 46(1–2) (1983), 12–32.Google Scholar
Hochman, M.. On self-similar sets with overlaps and inverse theorems for entropy. Ann. of Math. (2) 180(2) (2014), 773–822.Google Scholar
Hochman, M. and Solomyak, B.. On the dimension of Furstenberg measure for SL2(ℝ) random matrix products. Invent. Math. (2017), to appear. https://doi.org/10.1007/s00222-017-0740-6.Google Scholar
Hunt, B. R. and Kaloshin, V. Yu.. How projections affect the dimension spectrum of fractal measures. Nonlinearity 10(5) (1997), 1031–1046.Google Scholar
Jordan, T., Pollicott, M. and Simon, K.. Hausdorff dimension for randomly perturbed self affine attractors. Comm. Math. Phys. 270(2) (2007), 519–544.Google Scholar
Lau, K.-S.. Self-similarity, L p -spectrum and multifractal formalism. Fractal Geometry and Stochastics (Finsterbergen, 1994) (Progress in Probability, 37) . Birkhäuser, Basel, 1995, pp. 55–90.Google Scholar
Mattila, P.. Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability. Cambridge University Press, Cambridge, 1999.Google Scholar
Peres, Y. and Schlag, W.. Smoothness of projections, Bernoulli convolutions, and the dimension of exceptions. Duke Math. J. 102(2) (2000), 193–251.Google Scholar
Rapaport, A.. On self-affine measures with equal Hausdorff and Lyapunov dimension. Trans. Amer. Math. Soc. (2017), to appear. https://doi.org/10.1090/tran/7099.Google Scholar
Shmerkin, P.. On Furstenberg’s intersection conjecture, self-similar measures, and the $L^{q}$ norms of convolutions. Preprint, 2016, arXiv:1609.07802.Google Scholar
Shmerkin, P. and Solomyak, B.. Absolute continuity of self-similar measures, their projections and convolutions. Trans. Amer. Math. Soc. 368(7) (2016), 5125–5151.Google Scholar
Simon, K.. The dimension theory of almost self-affine sets and measures. Fractals, Wavelets, and their Applications (Springer Proceedings in Mathematics & Statistics, 92) . Springer, Cham, 2014, pp. 103–127.Google Scholar
Simon, K. and Tóth, H. R.. The absolute continuity of the distribution of random sums with digits {0, 1, …, m - 1}. Real Anal. Exchange 30(1) (2004/05), 397–409.Google Scholar
Solomyak, B.. On the random series ∑ ±𝜆 n (an Erdös problem). Ann. of Math. (2) 142(3) (1995), 611–625.Google Scholar
Solomyak, B.. Measure and dimension for some fractal families. Math. Proc. Cambridge Philos. Soc. 124 (1998), 531–546.Google Scholar
Walters, P.. An Introduction to Ergodic Theory (Graduate Texts in Mathematics, 79) . Springer, New York, 1982.Google Scholar