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We study Bedford--McMullen type carpets whose selected grid rectangles may be reflected in one or both coordinates. The organizing principle is that the Hausdorff dimension is controlled by the entropy of the weak-coordinate projection. When this weak projection is separated, we obtain an explicit McMullen-type formula. This yields stability under arbitrary horizontal reflections and row-compatible weak reflections, and it gives several computable mixed-sign classes, including signed row-branch systems, interval-window systems and finite-block separated systems. We also explain why fully arbitrary weak-coordinate reflection patterns lead instead to a projection-entropy problem.

Tsukamoto (2022) introduced the notion of Bedford-McMullen carpet system, a subsystem of $([0,1]^{\mathbb{N}}\times[0,1]^{\mathbb{N}},shift)$ whose metric mean dimension and mean Hausdorff dimension does not coincide in general. The aim of this paper is to develop the mean dimension theory for Bedford-McMullen sponge system, which is a subsystem of $(([0,1]^r)^{\mathbb{N}},shift)$ with arbitrary $3\leq r\in\mathbb{N}$. In particular, we compute the metric mean dimension and mean Hausdorff dimension of such topological dynamical systems explicitly, extending the results by Tsukamoto. The metric mean dimension is a weighted combination of the standard topological entropy, whereas the mean Hausdorff dimension is expressed in terms of weighted topological entropy. We also exhibit a special situation for which the metric mean dimension and the mean Hausdorff dimension of a sponge system coincide.

We prove that any non-degenerate Bedford-McMullen carpet does not admit oblique self-embedding similitudes; that is, if $f$ is a similitude sending the carpet into itself, then the image of the $x$-axis under $f$ must be parallel to one of the principal axes. This result leads to a logarithmic commensurability result on the contraction ratios of such embeddings, completing a previous study by Algom and Hochman [Ergod. Th. & Dynam. Sys. 39 (2019), 577-603] on Bedford-McMullen carpets generated by multiplicatively independent exponents. Our approach also provides a new proof of their non-obliqueness statement that avoids analyzing the tangent sets. For the self-similar case, however, we construct a generalized Sierpiński carpet that is symmetric with respect to an appropriate oblique line and hence admits a reflectional oblique self-embedding. As a complement, we prove that if a generalized Sierpiński carpet satisfies the strong separation condition and permits an oblique rotational self-embedding similitude, then the tangent of the rotation angle takes values $\pm 1$.

We calculate the Assouad and lower dimensions of graph-directed Bedford-McMullen carpets, which reflect the extreme local scaling laws of the sets, in contrasting with known results on Hausdorff and box dimensions. We also investigate the relationship between distinct dimensions. In particular, we identify an equivalent condition when the box and Assouad dimension coincide, and show that under this condition, the Hausdorff dimension attains the same value.

In this paper, we study the topological properties and the gap sequences of Bedford-McMullen sets. First, we introduce a topological condition, the component separation condition (CSC), and a geometric condition, the exponential rate condition (ERC). Then we prove that the CSC implies the ERC, and that both of them are sufficient conditions for obtaining the asymptotic estimate of gap sequences. We also explore topological properties of Bedford-McMullen sets and prove that all normal Bedford-McMullen sets with infinitely many connected components satisfy the CSC, from which we obtain the asymptotic estimate of the gap sequences of Bedford-McMullen sets without any restrictions. Finally, we apply our result to Lipschitz equivalence.

In this paper, we study the Hausdorff dimension of the quantitative recurrent set of the canonical endomorphism on the Bedford-McMullen carpets whose Hausdorff dimension and box dimension are equal.

Intermediate dimensions were recently introduced to provide a spectrum of dimensions interpolating between Hausdorff and box-counting dimensions for fractals where these differ. In particular, the self-affine Bedford-McMullen carpets are a natural case for investigation, but until now only very rough bounds for their intermediate dimensions have been found. In this paper, we determine a precise formula for the intermediate dimensions $\dim_{\, θ}Λ$ of any Bedford-McMullen carpet $Λ$ for the whole spectrum of $θ\in [0,1]$, in terms of a certain large deviations rate function. The intermediate dimensions exist and are strictly increasing in $θ$, and the function $θ\mapsto \dim_{\, θ}Λ$ exhibits interesting features not witnessed on any previous example, such as having countably many phase transitions, between which it is analytic and strictly concave. We make an unexpected connection to multifractal analysis by showing that two carpets with non-uniform vertical fibres have equal intermediate dimensions if and only if the Hausdorff multifractal spectra of the uniform Bernoulli measures on the two carpets are equal. Since intermediate dimensions are bi-Lipschitz invariant, this shows t

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 $E$ be a Bedford-McMullen carpet determined by a set of affine mappings $(f_{ij})_{(i,j)\in G}$ and $μ$ a self-affine measure on $E$ associated with a probability vector $(p_{ij})_{(i,j)\in G}$. We prove that, for every $r\in(0,\infty)$, the upper and lower quantization coefficient are always positive and finite in its exact quantization dimension $s_r$. As a consequence, the $k$th quantization error for $μ$ of order $r$ is of the same order as $k^{-\frac{1}{s_r}}$. In sharp contrast to the Hausdorff measure for Bedford-McMullen carpets, our result is independent of the horizontal fibres of the carpets.

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 sets in the line.

For a self-affine measure on a Bedford-McMullen carpet we prove that its quantization dimension exists and determine its exact value. Further, we give various sufficient conditions for the corresponding upper and lower quantization coefficient to be both positive and finite. Finally, we compare the quantization dimension with corresponding quantities derived from the multifractal temperature function and show that -- different from conformal systems -- they in general do not coincide.

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.

We describe the shrinking target set for the Bedford-McMullen carpets, with targets being either cylinders or geometric balls.

In this paper we consider the packing spectra for local dimension of Bernoulli measures supported on Bedford-McMullen carpets. We show that typically the packing dimension of the regular set is smaller than the packing dimension of the attractor. We also consider a specific class of measures for which we are able to calculate the packing spectrum exactly and we show that the packing spectrum is discontinuous as a function on the space of Bernoulli measures.

In this paper we compute the multifractal analysis for local dimensions of Bernoulli measures supported on the self-affine carpets introduced by Bedford-McMullen. This extends the work of King where the multifractal analysis is computed with strong additional separation assumptions.