The intermediate dimensions of a set $Λ$, elsewhere denoted by $\dim_θΛ$, interpolates between its Hausdorff and box dimensions using the parameter $θ\in[0,1]$. Determining a precise formula for $\dim_θΛ$ is particularly challenging when $Λ$ is a Bedford-McMullen carpet with distinct Hausdorff and box dimension. In this direction, answering a question of Fraser, we show that $\dim_θΛ$ is strictly less than the box dimension of $Λ$ for every $θ<1$, moreover, the derivative of the upper bound is strictly positive at $θ=1$. We also improve on the lower bound obtained by Falconer, Fraser and Kempton.
In 1984 Bedford and McMullen independently introduced a family of self-affine sets now known as \emph{Bedford-McMullen carpets}. Their work stimulated a lot of research in the areas of fractal geometry and non-conformal dynamics. In this survey article we discuss some aspects of Bedford-McMullen carpets, focusing mostly on dimension theory.
Let $F \subseteq \mathbb{R}^2$ be a Bedford-McMullen carpet defined by multiplicatively independent exponents, and suppose that either $F$ is not a product set, or it is a product set with marginals of dimension strictly between $0$ and $1$. We prove that any similarity $g$ such that $g(F) \subseteq F$ is an isometry composed of reflections about lines parallel to the axes. Our approach utilizes the structure of tangent sets of $F$, obtained by "zooming in" on points of $F$, projection theorems for products of self-similar sets, and logarithmic commensurability type results for self similar se…
Let $E$ be a Bedford-McMullen carpet associated with a set of affine mappings $\{f_{ij}\}_{(i,j)\in G}$ and let $μ$ be the self-affine measure associated with $\{f_{ij}\}_{(i,j)\in G}$ and a probability vector $(p_{ij})_{(i,j)\in G}$. We study the asymptotics of the geometric mean error in the quantization for $μ$. Let $s_0$ be the Hausdorff dimension for $μ$. Assuming a separation condition for $\{f_{ij}\}_{(i,j)\in G}$, we prove that the $n$th geometric error for $μ$ is of the same order as $n^{-1/s_0}$.
Let ${\cal M}_{t,v,r}(n,m)$, $2\leq m<n$, be the collection of self-affine carpets with expanding matrix $\diag(n,m)$ which are totally disconnected, possessing vacant rows and with uniform horizontal fibers. In this paper, we introduce a notion of structure tree of a metric space, and thanks to this new notion, we completely characterize when two carpets in ${\cal M}_{t,v,r}(n,m)$ are Lipschitz equivalent.
Let $F$ be a Bedford-McMullen carpet defined by independent exponents. We prove that $\overline{\dim}_B (\ell \cap F) \leq \max \lbrace \dim^* F -1,0 \rbrace$ for all lines $\ell$ not parallel to the principal axes, where $\dim^*$ is Furstenberg's star dimension (maximal dimension of a microset). We also prove several rigidity results for incommensurable Bedford-McMullen carpets, that is, carpets $F$ and $E$ such that all defining exponents are independent: Assuming various conditions, we find bounds on the dimension of the intersection of such carpets, show that self affine measures on them a…
Let $F$ be a Bedford-McMullen carpet defined by independent integer exponents. We prove that for every line $\ell \subseteq \mathbb{R}^2$ not parallel to the major axes, $$ \dim_H (\ell \cap F) \leq \max \left\lbrace 0,\, \frac{\dim_H F}{\dim^* F} \cdot (\dim^* F-1) \right\rbrace$$ and $$ \dim_P (\ell \cap F) \leq \max \left\lbrace 0,\, \frac{\dim_P F}{\dim^* F} \cdot (\dim^* F-1) \right\rbrace$$ where $\dim^*$ is Furstenberg's star dimension (maximal dimension of microsets). This improves the state of art results on Furstenberg type slicing Theorems for affine invariant carpets.
Recently self-affine sponges have been shown to be interesting counter-examples to several previously open problems. One class of recently studied sponges are Bedford-McMullen sponges with a weak coordinate ordering condition, that is, sponges with several coordinates having the same contraction ratio. The Assouad type dimensions of such sets cannot be calculated using the same formula as the regular Bedford-McMullen sponges. We calculate the Assouad type dimensions for such sponges and the more general Lalley-Gatzouras sponges with a weak coordinate ordering condition, discussing some of thei…
Mean Hausdorff dimension is a dynamical version of Hausdorff dimension. It provides a way to dynamicalize geometric measure theory. We pick up the following three classical results of fractal geometry. (1) The calculation of Hausdorff dimension of homogeneous sets in the circle. (2) The coincidence of Hausdorff and Minkowski dimensions for self-similar sets. (3) The calculation of Hausdorff dimension of Bedford--McMullen carpets. We develop their analogues for mean Hausdorff dimension: (1) The calculation of mean Hausdorff dimension of homogeneous systems in the infinite dimensional torus. (2)…
We introduce a continuum of dimensions which are `intermediate' between the familiar Hausdorff and box dimensions. This is done by restricting the families of allowable covers in the definition of Hausdorff dimension by insisting that $|U| \leq |V|^θ$ for all sets $U, V$ used in a particular cover, where $θ\in [0,1]$ is a parameter. Thus, when $θ=1$ only covers using sets of the same size are allowable, and we recover the box dimensions, and when $θ=0$ there are no restrictions, and we recover Hausdorff dimension. We investigate many properties of the intermediate dimension (as a function of $…
