On affine iterated function systems which robustly admit an invariant affine subspace
HTML articles powered by AMS MathViewer
- by Ian D. Morris;
- Proc. Amer. Math. Soc. 151 (2023), 101-112
- DOI: https://doi.org/10.1090/proc/16059
- Published electronically: October 7, 2022
- HTML | PDF | Request permission
Abstract:
In this note we give a simple sufficient condition for an affine iterated function system to admit an invariant affine subspace persistently with respect to changes in the translation parameters. This yields further examples of tuples of contracting linear maps which do not satisfy the conclusions of Falconer’s theorem on the Hausdorff dimension of almost every self-affine set. We also obtain new examples of iterated function systems of similarity transformations which cannot satisfy the open set condition for any choice of translation parameters, and resolve a related question of Peres and Solomyak.References
- Balázs Bárány, Michael Hochman, and Ariel Rapaport, Hausdorff dimension of planar self-affine sets and measures, Invent. Math. 216 (2019), no. 3, 601–659. MR 3955707, DOI 10.1007/s00222-018-00849-y
- Jairo Bochi and Ian D. Morris, Equilibrium states of generalised singular value potentials and applications to affine iterated function systems, Geom. Funct. Anal. 28 (2018), no. 4, 995–1028. MR 3820437, DOI 10.1007/s00039-018-0447-x
- G. A. Edgar, Fractal dimension of self-affine sets: some examples, Rend. Circ. Mat. Palermo (2) Suppl. 28 (1992), 341–358. Measure theory (Oberwolfach, 1990). MR 1183060
- K. J. Falconer, The Hausdorff dimension of self-affine fractals, Math. Proc. Cambridge Philos. Soc. 103 (1988), no. 2, 339–350. MR 923687, DOI 10.1017/S0305004100064926
- D. J. Feng, Dimension of invariant measures for affine iterated function systems, Duke Math. J. (to appear), Preprint, arXiv:1901.0169, 2019.
- De-Jun Feng and Pablo Shmerkin, Non-conformal repellers and the continuity of pressure for matrix cocycles, Geom. Funct. Anal. 24 (2014), no. 4, 1101–1128. MR 3248481, DOI 10.1007/s00039-014-0274-7
- M. Hochman, On self-similar sets with overlaps and inverse theorems for entropy in $\mathbb {R}^d$, Mem. Amer. Math. Soc. 265 (2020), no. 1287.
- Michael Hochman and Ariel Rapaport, Hausdorff dimension of planar self-affine sets and measures with overlaps, J. Eur. Math. Soc. (JEMS) 24 (2022), no. 7, 2361–2441. MR 4413769, DOI 10.4171/jems/1127
- John E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), no. 5, 713–747. MR 625600, DOI 10.1512/iumj.1981.30.30055
- Raphaël Jungers, The joint spectral radius, Lecture Notes in Control and Information Sciences, vol. 385, Springer-Verlag, Berlin, 2009. Theory and applications. MR 2507938, DOI 10.1007/978-3-540-95980-9
- Ian D. Morris, An inequality for the matrix pressure function and applications, Adv. Math. 302 (2016), 280–308. MR 3545931, DOI 10.1016/j.aim.2016.07.025
- Yuval Peres and Boris Solomyak, Problems on self-similar sets and self-affine sets: an update, Fractal geometry and stochastics, II (Greifswald/Koserow, 1998) Progr. Probab., vol. 46, Birkhäuser, Basel, 2000, pp. 95–106. MR 1785622
- F. Przytycki and M. Urbański, On the Hausdorff dimension of some fractal sets, Studia Math. 93 (1989), no. 2, 155–186. MR 1002918, DOI 10.4064/sm-93-2-155-186
- Gian-Carlo Rota, Gian-Carlo Rota on analysis and probability, Contemporary Mathematicians, Birkhäuser Boston, Inc., Boston, MA, 2003. Selected papers and commentaries; Edited by Jean Dhombres, Joseph P. S. Kung and Norton Starr. MR 1944526
- Gian-Carlo Rota and Gilbert Strang, A note on the joint spectral radius, Indag. Math. 22 (1960), 379–381. Nederl. Akad. Wetensch. Proc. Ser. A 63. MR 147922, DOI 10.1016/S1385-7258(60)50046-1
- Károly Simon and Boris Solomyak, On the dimension of self-similar sets, Fractals 10 (2002), no. 1, 59–65. MR 1894903, DOI 10.1142/S0218348X02000963
- Boris Solomyak, Measure and dimension for some fractal families, Math. Proc. Cambridge Philos. Soc. 124 (1998), no. 3, 531–546. MR 1636589, DOI 10.1017/S0305004198002680
Bibliographic Information
- Ian D. Morris
- Affiliation: School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London E1 4NS, United Kingdom
- MR Author ID: 806003
- Email: i.morris@qmul.ac.uk
- Received by editor(s): November 3, 2021
- Received by editor(s) in revised form: November 15, 2021, February 25, 2022, and March 3, 2022
- Published electronically: October 7, 2022
- Additional Notes: This research was partially supported by the Leverhulme Trust (Research Project Grant RPG-2016-194).
- Communicated by: Katrin Gelfert
- © Copyright 2022 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 151 (2023), 101-112
- MSC (2020): Primary 28A80
- DOI: https://doi.org/10.1090/proc/16059
- MathSciNet review: 4504611