Self-affine Carpets Research Articles

The Assouad dimension of self-affine measures on sponges

AbstractWe derive upper and lower bounds for the Assouad and lower dimensions of self-affine measures in $\mathbb {R}^d$ generated by diagonal matrices and satisfying suitable separation conditions. The upper and lower bounds always coincide for $d=2,3$ , yielding precise explicit formulae for those dimensions. Moreover, there are easy-to-check conditions guaranteeing that the bounds coincide for $d \geqslant 4$ . An interesting consequence of our results is that there can be a ‘dimension gap’ for such self-affine constructions, even in the plane. That is, we show that for some self-affine carpets of ‘Barański type’ the Assouad dimension of all associated self-affine measures strictly exceeds the Assouad dimension of the carpet by some fixed $\delta>0$ depending only on the carpet. We also provide examples of self-affine carpets of ‘Barański type’ where there is no dimension gap and in fact the Assouad dimension of the carpet is equal to the Assouad dimension of a carefully chosen self-affine measure.

New approach to weighted topological entropy and pressure

AbstractMotivated by fractal geometry of self-affine carpets and sponges, Feng and Huang [J. Math. Pures Appl.106(9) (2016), 411–452] introduced weighted topological entropy and pressure for factor maps between dynamical systems, and proved variational principles for them. We introduce a new approach to this theory. Our new definitions of weighted topological entropy and pressure are very different from the original definitions of Feng and Huang. The equivalence of the two definitions seems highly non-trivial. Their equivalence can be seen as a generalization of the dimension formula for the Bedford–McMullen carpet in purely topological terms.

Improved Versions of Some Furstenberg Type Slicing Theorems for Self-Affine Carpets

Abstract 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, $$\begin{align*} & \dim_H (\ell \cap F) \leq \max \left\lbrace 0,\, \frac{\dim_H F}{\dim^* F} \cdot (\dim^* F-1) \right\rbrace\end{align*}$$and $$\begin{align*} & \dim_P (\ell \cap F) \leq \max \left\lbrace 0,\, \frac{\dim_P F}{\dim^* F} \cdot (\dim^* F-1) \right\rbrace,\end{align*}$$where $\dim ^*$ is Furstenberg’s star dimension (maximal dimension of microsets). This improves the state-of-the-art results on Furstenberg type slicing Theorems for affine invariant carpets.

Lipschitz classification of Bedford-McMullen carpets with uniform horizontal fibers

Let Mt,v,r(n,m), 2≤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 Mt,v,r(n,m) are Lipschitz equivalent.

The Assouad spectrum of random self-affine carpets

AbstractWe derive the almost sure Assouad spectrum and quasi-Assouad dimension of one-variable random self-affine Bedford–McMullen carpets. Previous work has revealed that the (related) Assouad dimension is not sufficiently sensitive to distinguish between subtle changes in the random model, since it tends to be almost surely ‘as large as possible’ (a deterministic quantity). This has been verified in conformal and non-conformal settings. In the conformal setting, the Assouad spectrum and quasi-Assouad dimension behave rather differently, tending to almost surely coincide with the upper box dimension. Here we investigate the non-conformal setting and find that the Assouad spectrum and quasi-Assouad dimension generally do not coincide with the box dimension or Assouad dimension. We provide examples highlighting the subtle differences between these notions. Our proofs combine deterministic covering techniques with suitably adapted Chernoff estimates and Borel–Cantelli-type arguments.

Triangular Gatzouras–Lalley-type planar carpets with overlaps

We construct a family of planar self-affine carpets with overlaps using lower triangular matrices in a way that generalizes the original Gatzouras–Lalley carpets (Gatzouras and Lalley 1992 Indiana Univ. Math. J. 41 533–68) defined by diagonal matrices. Assuming the rectangular open set condition, Barański (2008 Discrete Continuous Dyn. Syst. A 21 1015–23) proved for this construction that for typical parameters, which can be explicitly checked, the inequalities between the Hausdorff, box and affinity dimension of the attractor are strict. We generalize this result to overlapping constructions, where we allow complete columns to be shifted along the horizontal axis (as in Fraser and Shmerkin (2016 Ergod. Theor. Dyn. Syst. 36 2463–81) and Pardo-Simón (2019 Ergod. Theor. Dyn. Syst. pp 733–63) or allow parallelograms to overlap within a column in a transversal way. Our main result is to show sufficient conditions under which these overlaps do not cause a drop of the dimension of the attractor. Several examples are provided to illustrate the results, including a self-affine smiley, a family of self-affine continuous curves, examples with overlaps and an application of our results to some three-dimensional systems.

Manhattan property of geodesic paths on self-affine carpets

For any Bedford-McMullen self-affine carpet, the geodesic path on the carpet between points $$(x_{1},y_{1})$$ and $$(x_{2},y_{2})$$ has length greater than or equal to $$|x_{1}-x_{2}|+|y_{1}-y_{2}|.$$ This property fails for self-similar carpets.

Assouad-type spectra for some fractal families

In a previous paper we introduced a new `dimension spectrum', motivated by the Assouad dimension, designed to give precise information about the scaling structure and homogeneity of a metric space. In this paper we compute the spectrum explicitly for a range of well-studied fractal sets, including: the self-affine carpets of Bedford and McMullen, self-similar and self-conformal sets with overlaps, Mandelbrot percolation, and Moran constructions. We find that the spectrum behaves differently for each of these models and can take on a rich variety of forms. We also consider some applications, including the provision of new bi-Lipschitz invariants and bounds on a family of `tail densities' defined for subsets of the integers.

Dimensions of an overlapping generalization of Barański carpets

We determine the Hausdorff, the packing and the box-counting dimensions of a family of self-affine sets generalizing Barański carpets. More specifically, we fix a Barański system and allow both vertical and horizontal random translations, while preserving the structure of the rows and columns. The alignment kept in the construction allows us to give expressions for these fractal dimensions outside of a small set of exceptional translations. Such formulae will coincide with those for the non-overlapping case, and thus provide examples where the box-counting and the Hausdorff dimension do not necessarily agree. These results rely on Hochman’s recent work on the dimensions of self-similar sets and measures, and can be seen as an extension of Fraser and Shmerkin [On the dimensions of a family of overlapping self-affine carpets. Ergod. Th. &amp; Dynam. Sys.doi: 10.1017/etds.2015.21. Published online: 21 July 2015] results for Bedford–McMullen carpets with columns overlapping.

The Assouad dimension of self-affine carpets with no grid structure

Previous study of the Assouad dimension of planar self-affine sets has relied heavily on the underlying IFS having a ‘grid structure’, thus allowing for the use of approximate squares. We study the Assouad dimension of a class of self-affine carpets which do not have an associated grid structure. We find that the Assouad dimension is related to the box and Assouad dimensions of the (self-similar) projection of the self-affine set onto the first coordinate and to the local dimensions of the projection of a natural Bernoulli measure onto the first coordinate. In a special case we relate the Assouad dimension of the Przytycki-Urbański sets to the lower local dimensions of Bernoulli convolutions.

Rigidity of Quasisymmetric Mappings on Self-affine Carpets

We show that the class of quasisymmetric maps between horizontal self-affine carpets is rigid. Such maps can only exist when the dimensions of the carpets coincide, and in this case, the quasisymmetric maps are quasi-Lipschitz. We also show that horizontal self-affine carpets are minimal for the conformal Assouad dimension.

Assouad type dimensions for self-affine sponges

We study the Assouad and lower dimensions of self-affine sponges; the higher dimensional analogue of the planar self-affine carpets of Bedford and McMullen. Our techniques involve the weak tangents of Mackay and Tyson as well as regularity properties of doubling measures in the context studied by Bylund and Gudayol.

The Assouad dimension of randomly generated fractals

We consider several different models for generating random fractals including random self-similar sets, random self-affine carpets, and Mandelbrot percolation. In each setting we compute either thealmost sureor theBaire typicalAssouad dimension and consider some illustrative examples. Our results reveal a phenomenon common to each of our models: the Assouad dimension of a randomly generated fractal is generically as big as possible and does not depend on the measure-theoretic or topological structure of the sample space. This is in stark contrast to the other commonly studied notions of dimension like the Hausdorff or packing dimension.

Inhomogeneous self-affine carpets

We investigate the dimension theory of inhomogeneous self-affine carpets. Through the work of Olsen, Snigireva and Fraser, the dimension theory of inhomogeneous self-similar sets is now relatively well-understood, however, almost no progress has been made concerning more general non-conformal inhomogeneous attractors. If a dimension is countably stable, then the results are immediate and so we focus on the upper and lower box dimensions and compute these explicitly for large classes of inhomogeneous self-affine carpets. Interestingly, we find that the `expected formula' for the upper box dimension can fail in the self-affine setting and we thus reveal new phenomena, not occurring in the simpler self-similar case.

On the dimensions of a family of overlapping self-affine carpets

We consider the dimensions of a family of self-affine sets related to the Bedford–McMullen carpets. In particular, we fix a Bedford–McMullen system and then randomize the translation vectors with the stipulation that the column structure is preserved. As such, we maintain one of the key features in the Bedford–McMullen set-up in that alignment causes the dimensions to drop from the affinity dimension. We compute the Hausdorff, packing and box dimensions outside of a small set of exceptional translations, and also for some explicit translations even in the presence of overlapping. Our results rely on, and can be seen as a partial extension of, Hochman’s recent work on the dimensions of self-similar sets and measures.

Scaling scenery of [formula omitted] invariant measures

We study the scaling scenery and limit geometry of invariant measures for the non-conformal toral endomorphism (x,y)↦(mxmod1,nymod1) that are Bernoulli measures for the natural Markov partition. We show that the statistics of the scaling can be described by an ergodic CP-chain in the sense of Furstenberg. Invoking the machinery of CP-chains yields a projection theorem for Bernoulli measures, which generalises in part earlier results by Hochman–Shmerkin and Ferguson–Jordan–Shmerkin. We also give an ergodic theoretic criterion for the dimension part of Falconer's distance set conjecture for general sets with positive length using CP-chains and hence verify it for various classes of fractals such as self-affine carpets of Bedford–McMullen, Lalley–Gatzouras and Barański class and all planar self-similar sets.

Variational formula related to the self-affine Sierpinski carpets

We consider those subsets of the self-affine Sierpinski carpets that are the union of an uncountable number of sets each of which consists of the points with their location codes having prescribed group frequencies. It is proved that their Hausdorff dimensions equal to the supremum of the Hausdorff dimensions of the sets in the union. The main advantage is that we treat these subsets in a unified manner and the value of the Hausdorff dimensions do not need to be guessed a priori.

Assouad dimension of self-affine carpets

We calculate the Assouad dimension of the self-affine carpets of Bedford and McMullen, and of Lalley and Gatzouras. We also calculate the conformal Assouad dimension of those carpets that are not self-similar.

Multifractal analysis for Bedford–McMullen carpets

AbstractIn 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.

The Hausdorff dimension of the projections of self-affine carpets

We study the orthogonal projections of a large class of self-affine carpets, which contains the carpets of Bedford and McMullen as special cases. Our main result is that if Lambda is such a carpet, and certain natural irrationality conditions hold, then every orthogonal projection of Lambda in a non-principal direction has Hausdorff dimension min(gamma, 1), where gamma is the Hausdorff dimension of Lambda. This generalizes a recent result of Peres and Shmerkin on sums of Cantor sets.