Self-affine sets are attractors of IFSs consisting of affine contractions. They are a well-studied and important class of fractal and are notoriously difficult to study since
affine maps can distort by different amounts in different directions. This is a particular case of the more general class of non-conformal attractors. I have worked a lot with self-affine sets
and associated self-affine measures. I am especially interested in planar carpet type sets, such as those introduced by Bedford and McMullen. I wrote a survey article on Bedford-McMullen carpets, which contains some open problems:
► Fractal geometry of Bedford-McMullen carpets, arXiv
Proceedings of the Fall 2019 Jean-Morlet Chair programme, Springer Lecture Notes Series, 2021, (Eds. M. Pollicott & S. Vaienti).
Some work on box and Hausdorff dimensions of self-affine sets includes:
► The box dimensions of exceptional self-affine sets in ℝ^{3} (with N. Jurga), arXiv
Advances in Mathematics, 385, (2021), 107734.
► On the dimensions of a family of overlapping self-affine carpets (with P. Shmerkin), arXiv
Ergodic Theory and Dynamical Systems, 36, (2016), 2463-2481.
► On the packing dimension of box-like self-affine sets in the plane, arXiv
Nonlinearity, 25, (2012), 2075-2092.
Some work on the Assouad type dimensions of (random and deterministic) self-affine sets includes:
► The Assouad spectrum of random self-affine carpets (with S. Troscheit), arXiv
Ergodic Theory and Dynamical Systems, 41, (2021), 2927-2945.
► Assouad type spectra for some fractal families (with H. Yu), arXiv
Indiana University Mathematics Journal, 67, (2018), 2005-2043.
► The Assouad dimension of self-affine carpets with no grid structure (with T. Jordan), arXiv
Proceedings of the American Mathematical Society, 145, (2017), 4905-4918.
► Assouad type dimensions for self-affine sponges (with D. Howroyd), arXiv
Annales Academiæ Scientiarum Fennicæ Mathematica, 42, (2017), 149-174.
► Assouad type dimensions and homogeneity of fractals, arXiv
Transactions of the American Mathematical Society, 366, (2014), 6687-6733.
Some work on self-affine measures, especially the Lq spectra and Hausdorff dimension:
► L^{q}-spectra of self-affine measures: closed forms, counterexamples, and split binomial sums (with L. D. Lee, I. D. Morris & H. Yu), arXiv
Nonlinearity, 34, (2021), 6331-6357.
► On the upper regularity dimensions of measures (with D. C. Howroyd), arXiv
Indiana University Mathematics Journal, 69, (2020), 685-712.
► Dimensions of equilibrium measures on a class of planar self-affine sets (with T. Jordan & N. Jurga), arXiv
Journal of Fractal Geometry, 7, (2020), 87-111.
► On the L^{q} Dimensions of Measures on Hueter-Lalley Type Self-Affine Sets (with T. Kempton), arXiv
Proceedings of the American Mathematical Society, 146, (2018), 161-173.
► On the L^{q}-spectrum of planar self-affine measures, arXiv
Transactions of the American Mathematical Society, 368, (2016), 5579-5620.
Here are some slides from talks I've given on self-affine sets and measures:
On the left is a self-affine carpet of the type introduced by Bedford and McMullen and on the right is another planar self-affine set.