Hostname: page-component-76fb5796d-vfjqv Total loading time: 0 Render date: 2024-04-25T14:45:58.054Z Has data issue: false hasContentIssue false
  • English
  • Français

Slices of the Takagi function

Part of: Real functions Classical measure theory

Published online by Cambridge University Press:  20 December 2023

ROOPE ANTTILA ,
BALÁZS BÁRÁNY  and
ANTTI KÄENMÄKI Open the ORCID record for ANTTI KÄENMÄKI [Opens in a new window]
ROOPE ANTTILA
Affiliation:
Research Unit of Mathematical Sciences, University of Oulu, P.O. Box 8000, FI-90014 Oulu, Finland (e-mail: roope.anttila@oulu.fi)
BALÁZS BÁRÁNY
Affiliation:
Department of Stochastics, Institute of Mathematics, Budapest University of Technology and Economics, Műegyetem rkp. 3, H-1111 Budapest, Hungary (e-mail: balubsheep@gmail.com)
ANTTI KÄENMÄKI*
Affiliation:
Research Unit of Mathematical Sciences, University of Oulu, P.O. Box 8000, FI-90014 Oulu, Finland
*
e-mail: antti.kaenmaki@oulu.fi
Get access Rights & Permissions [Opens in a new window]

Abstract

We show that the Hausdorff dimension of any slice of the graph of the Takagi function is bounded above by the Assouad dimension of the graph minus one, and that the bound is sharp. The result is deduced from a statement on more general self-affine sets, which is of independent interest. We also prove that Marstrand’s slicing theorem on the graph of the Takagi function extends to all slices if and only if the upper pointwise dimension of every projection of the length measure on the x-axis lifted to the graph is at least one.

Keywords

Takagi functionself-affine setMarstrand’s slicing theoremAssouad dimension

MSC classification

Primary: 28A80: Fractals
Secondary: 28A75: Length, area, volume, other geometric measure theory 26A27: Nondifferentiability (nondifferentiable functions, points of nondifferentiability), discontinuous derivatives
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , First View , pp. 1 - 38
Check for updates
Copyright
© The Author(s), 2023. Published by Cambridge University Press

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

Algom, A.. Slicing theorems and rigidity phenomena for self affine carpets. Proc. Lond. Math. Soc. (3) 121 (2020), 312–353.CrossRefGoogle Scholar
Allaart, P. C.. The finite cardinalities of level sets of the Takagi function. J. Math. Anal. Appl. 388(2) (2012), 1117–1129.Google Scholar
Allaart, P. C.. How large are the level sets of the Takagi function? Monatsh. Math. 167(3–4) (2012), 311–331.Google Scholar
Allaart, P. C.. Level sets of signed Takagi functions. Acta Math. Hungar. 141(4) (2013), 339–352.CrossRefGoogle Scholar
Allaart, P. C.. Hausdorff dimension of level sets of generalized Takagi functions. Math. Proc. Cambridge Philos. Soc. 157(2) (2014), 253–278.CrossRefGoogle Scholar
Allaart, P. C. and Kawamura, K.. The Takagi function: a survey. Real Anal. Exchange 37(1) (2011/12), 1–54.CrossRefGoogle Scholar
Bárány, B., Hochman, M. and Rapaport, A.. Hausdorff dimension of planar self-affine sets and measures. Invent. Math. 216(3) (2019), 601–659.Google Scholar
Bárány, B., Käenmäki, A. and Morris, I. D.. Domination, almost additivity, and thermodynamic formalism for planar matrix cocycles. Israel J. Math. 239(1) (2020), 173–214.Google Scholar
Bárány, B., Käenmäki, A. and Rossi, E.. Assouad dimension of planar self-affine sets. Trans. Amer. Math. Soc. 374(2) (2021), 1297–1326.Google Scholar
Bárány, B., Käenmäki, A. and Yu, H.. Finer geometry of planar self-affine sets. Preprint, 2021, arXiv:2107.00983.Google Scholar
Bishop, C. J. and Peres, Y.. Fractals in Probability and Analysis (Cambridge Studies in Advanced Mathematics, 162). Cambridge University Press, Cambridge, 2017.Google Scholar
Bochi, J. and Gourmelon, N.. Some characterizations of domination. Math. Z. 263(1) (2009), 221–231.Google Scholar
Bochi, J. and Morris, I. D.. Continuity properties of the lower spectral radius. Proc. Lond. Math. Soc. (3) 110(2) (2015), 477–509.Google Scholar
Buczolich, Z.. Irregular 1-sets on the graphs of continuous functions. Acta Math. Hungar. 121(4) (2008), 371–393.Google Scholar
de Amo, E., Bhouri, I., Díaz Carrillo, M. and Fernández-Sánchez, J.. The Hausdorff dimension of the level sets of Takagi’s function. Nonlinear Anal. 74(15) (2011), 5081–5087.Google Scholar
de Amo, E., Díaz Carrillo, M. and Fernández Sánchez, J.. The Hausdorff dimension of the generalized level sets of Takagi’s function. Real Anal. Exchange 38(2) (2012/13), 421–423.Google Scholar
Falconer, K. J.. Techniques in Fractal Geometry. John Wiley & Sons, Chichester, 1997.Google Scholar
Fraser, J. M.. Assouad type dimensions and homogeneity of fractals. Trans. Amer. Math. Soc. 366(12) (2014), 6687–6733.Google Scholar
Fraser, J. M.. Assouad Dimension and Fractal Geometry (Cambridge Tracts in Mathematics, 222). Cambridge University Press, Cambridge, 2020.Google Scholar
Fraser, J. M., Howroyd, D. C., Käenmäki, A. and Yu, H.. On the Hausdorff dimension of microsets. Proc. Amer. Math. Soc. 147(11) (2019), 4921–4936.CrossRefGoogle Scholar
Furstenberg, H.. Intersections of Cantor sets and transversality of semigroups. Problems in Analysis (Symposium in Honor of Salomon Bochner (PMS-31)). Ed. Gunning, R. C.. Princeton University Press, Princeton, NJ, 1970, pp. 41–59.Google Scholar
Hutchinson, J. E.. Fractals and self-similarity. Indiana Univ. Math. J. 30(5) (1981), 713–747.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., Koivusalo, H. and Rossi, E.. Self-affine sets with fibred tangents. Ergod. Th. & Dynam. Sys. 37(6) (2017), 1915–1934.Google Scholar
Käenmäki, A. and Nissinen, P.. Non-invertible planar self-affine sets. Preprint, 2022, arXiv:2205.07351.Google Scholar
Käenmäki, A., Ojala, T. and Rossi, E.. Rigidity of quasisymmetric mappings on self-affine carpets. Int. Math. Res. Not. IMRN 2018(12) (2018), 3769–3799.CrossRefGoogle Scholar
Lagarias, J. C.. The Takagi function and its properties. Functions in Number Theory and their Probabilistic Aspects (RIMS Kôkyûroku Bessatsu, B34). Eds. K. Matsumoto, S. Akiyama, K. Fukuyama, H. Nakada, H. Sugita and A. Tamagawa. Research Institute for Mathematical Sciences (RIMS), Kyoto, 2012, pp. 153–189.Google Scholar
Lagarias, J. C. and Maddock, Z.. Level sets of the Takagi function: generic level sets. Indiana Univ. Math. J. 60(6) (2011), 1857–1884.Google Scholar
Lagarias, J. C. and Maddock, Z.. Level sets of the Takagi function: local level sets. Monatsh. Math. 166(2) (2012), 201–238.CrossRefGoogle Scholar
Ledrappier, F.. On the dimension of some graphs. Symbolic Dynamics and Its Applications (New Haven, CT, 1991) (Contemporary Mathematics, 135). Ed. Williams, S. G.. American Mathematical Society, Providence, RI, 1992, pp. 285–293.Google Scholar
Liu, C. and Li, H.. Hausdorff dimension of local level sets of Takagi’s function. Monatsh. Math. 177(1) (2015), 101–117.Google Scholar
Mackay, J. M.. Assouad dimension of self-affine carpets. Conform. Geom. Dyn. 15 (2011), 177–187.Google Scholar
Maddock, Z.. Level sets of the Takagi function: Hausdorff dimension. Monatsh. Math. 160(2) (2010), 167–186.Google Scholar
Manning, A. and Simon, K.. Dimension of slices through the Sierpinski carpet. Trans. Amer. Math. Soc. 365(1) (2013), 213–250.Google 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
Mishura, Y. and Schied, A.. On (signed) Takagi–Landsberg functions: $p$ th variation, maximum, and modulus of continuity. J. Math. Anal. Appl. 473(1) (2019), 258–272.Google Scholar
Rossi, E.. Visible part of dominated self-affine sets in the plane. Ann. Fenn. Math. 46(2) (2021), 1089–1103.Google Scholar
Shmerkin, P.. On Furstenberg’s intersection conjecture, self-similar measures, and the ${L}^q$ norms of convolutions. Ann. of Math. (2) 189(2) (2019), 319–391.Google Scholar
Shmerkin, P.. ${L}^q$ dimensions of self-similar measures and applications: a survey. New Trends in Applied Harmonic Analysis. Volume 2: Harmonic Analysis, Geometric Measure Theory, and Applications (Applied and Numerical Harmonic Analysis). Eds. Aldroubi, A., Cabrelli, C., Jaffard, S. and Molter, U.. Birkhäuser/Springer, Cham, 2019, pp. 257–292.Google Scholar
Wu, M.. A proof of Furstenberg’s conjecture on the intersections of $\times p$ - and $\times q$ -invariant sets. Ann. of Math. (2) 189(3) (2019), 707–751.Google Scholar
Yu, H.. Weak tangent and level sets of Takagi functions. Monatsh. Math. 192(1) (2020), 249–264.Google Scholar