Bedford McMullen Carpets Research Articles

Quantitative recurrence problem on some Bedford-McMullen carpets

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.

Lipschitz classification of Bedford-McMullen carpets with locally uniform horizontal fibers

It is well known that the Hausdorff dimension and box dimension of a Bedford-McMullen carpet coincide if and only if the carpet has uniform horizontal fibers. Yang et al. [17] gave a complete Lipschitz classification of such carpets which are totally disconnected and possessing vacant rows. In this paper, we show that if we replace the uniform horizontal fibers condition by a locally uniform horizontal fibers condition, the conclusions of Yang et al. still hold.

THREE PROPERTIES OF FRACTAL NETWORKS BASED ON BEDFORD–MCMULLEN CARPET

In this paper, we consider the networks modeled by several self-affine sets based on the Bedford–Mcmullen carpet. We calculate three properties of the networks, including the cumulative degree distribution, the average clustering coefficient and the average path length. We show that such networks have scale-free and small-world effects.

Constructing bi-Lipschitz maps between Bedford-McMullen carpets via symbolic spaces

Let E=K(n,m,D) be a Bedford-McMullen carpet with expanding factors n,m and digit set D. We call a=(aj)j=0m−1 the distribution sequence of D whereaj=#{i;(i,j)∈D}. Under a certain vertical separation condition, Li et al. (2013) [7] showed that if two totally disconnected Bedford-McMullent carpets share the same distribution sequence, then they are Lipschitz equivalent. In this paper, we define a metric ρ on D∞ and call (D∞,ρ) a half-symbolic space. We show that if E is totally disconnected and satisfies the vertical separation condition, then E is Lipschitz equivalent to (D∞,ρ). Thanks to this result, we extend the result of Li et al. by showing that two Bedford-McMullen carpets K(n,m,D) and K(n,m,D′) are Lipschitz equivalent if a, the distribution sequence of D, is a permutation of a′, the distribution sequence of D′, and that a and a′ satisfy a certain color matchable condition.

An upper bound for the intermediate dimensions of Bedford–McMullen carpets

The intermediate dimensions of a set $\Lambda$, elsewhere denoted by $\dim\_{\theta}\Lambda$, interpolate between its Hausdorff and box dimensions using the parameter $\theta\in\[0,1]$. For a Bedford–McMullen carpet $\Lambda$ with distinct Hausdorff and box dimensions, we show that $\dim\_{\theta}\Lambda$ is strictly less than the box dimension of $\Lambda$ for every $\theta<1$. Moreover, the derivative of the upper bound is strictly positive at $\theta=1$. This answers a question of Fraser; however, determining a precise formula for $\dim\_{\theta}\Lambda$ still remains a challenging problem.

Relations between topological and metrical properties of self-affine Sierpiński sponges

We investigate two Lipschitz invariants of metric spaces defined by δ-connected components, called the maximal power law property and the perfectly disconnectedness. The first property has been studied in literature for some self-similar sets and Bedford-McMullen carpets, while the second property seems to be new. For a self-affine Sierpiński sponge E, we first show that E satisfies the maximal power law if and only if E and all its major projections contain trivial connected components; secondly, we show that E is perfectly disconnected if and only if E and all its major projections are totally disconnected.

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.

On the Convergence Rate of the Chaos Game

Abstract This paper studies how long it takes the orbit of the chaos game to reach a certain density inside the attractor of a strictly contracting IFS of which we only assume that its lower dimension is positive. We show that the rate of growth of this cover time is determined by the Minkowski dimension of the push-forward of the shift invariant measure with exponential decay of correlations driving the chaos game. Moreover, we bound the expected value of the cover time from above and below with multiplicative logarithmic correction terms. As an application, for Bedford–McMullen carpets, we completely characterise the family of probability vectors that minimise the Minkowski dimension of Bernoulli measures. Interestingly, these vectors have not appeared in any other aspect of Bedford–McMullen carpets before.

WHEN DOES A BEDFORD–MCMULLEN CARPET HAVE EQUAL HAUSDORFF AND TOPOLOGICAL HAUSDORFF DIMENSIONS?

In this paper, we provide a complete characterization on when the Hausdorff and topological Hausdorff dimension of a given Bedford–McMullen carpet coincide. These two dimensions have a common value if and only if the carpet is either the Cartesian product of [Formula: see text] and a self-similar set, or it is one of the two diagonals of the unit square.

Hölder Parameterization of Iterated Function Systems and a Self-Affine Phenomenon

Abstract We investigate the Hölder geometry of curves generated by iterated function systems (IFS) in a complete metric space. A theorem of Hata from 1985 asserts that every connected attractor of an IFS is locally connected and path-connected. We give a quantitative strengthening of Hata’s theorem. First we prove that every connected attractor of an IFS is (1/s)-Hölder path-connected, where s is the similarity dimension of the IFS. Then we show that every connected attractor of an IFS is parameterized by a (1/ α)-Hölder curve for all α &gt; s. At the endpoint, α = s, a theorem of Remes from 1998 already established that connected self-similar sets in Euclidean space that satisfy the open set condition are parameterized by (1/s)-Hölder curves. In a secondary result, we show how to promote Remes’ theorem to self-similar sets in complete metric spaces, but in this setting require the attractor to have positive s-dimensional Hausdorff measure in lieu of the open set condition. To close the paper, we determine sharp Hölder exponents of parameterizations in the class of connected self-affine Bedford-McMullen carpets and build parameterizations of self-affine sponges. An interesting phenomenon emerges in the self-affine setting. While the optimal parameter s for a self-similar curve in ℝ n is always at most the ambient dimension n, the optimal parameter s for a self-affine curve in ℝ n may be strictly greater than n.

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.

ASYMPTOTIC ORDER OF THE GEOMETRIC MEAN ERROR FOR SELF-AFFINE MEASURES ON BEDFORD–MCMULLEN CARPETS

Let [Formula: see text] be a Bedford–McMullen carpet associated with a set of affine mappings [Formula: see text] and let [Formula: see text] be the self-affine measure associated with [Formula: see text] and a probability vector [Formula: see text]. We study the asymptotics of the geometric mean error in the quantization for [Formula: see text]. Let [Formula: see text] be the Hausdorff dimension for [Formula: see text]. Assuming a separation condition for [Formula: see text], we prove that the [Formula: see text]th geometric error for [Formula: see text] is of the same order as [Formula: see text].

Slicing theorems and rigidity phenomena for self‐affine carpets

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 are mutually singular, and prove that they do not embed affinely into each other. We obtain these results as an application of a slicing Theorem for products of certain Cantor sets. This Theorem is a generlization of the results of Shmerkin and Wu, that proved Furstenberg's slicing Conjecture.

Shrinking targets on Bedford-McMullen carpets

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

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.

Asymptotic order of the quantization errors for a class of self-affine measures

Let E E be a Bedford-McMullen carpet determined by a set of affine mappings ( f i j ) ( i , j ) ∈ G (f_{ij})_{(i,j)\in G} and μ \mu a self-affine measure on E E associated with a probability vector ( p i j ) ( i , j ) ∈ G (p_{ij})_{(i,j)\in G} . We prove that, for every r ∈ ( 0 , ∞ ) r\in (0,\infty ) , the upper and lower quantization coefficient are always positive and finite in its exact quantization dimension s r s_r . As a consequence, the n n th quantization error for μ \mu of order r r is of the same order as n − 1 s r n^{-\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.

Self-embeddings of Bedford–McMullen 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 zero and one. 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.

Quasi-Assouad dimension of fractals

The Assouad dimension of fractals is not invariant under quasi-Lipschitz mappings, even for Ahlfors–David regular sets. In this manuscript, we shall give a new dimension dim _{qA} of fractals named the quasi-Assouad dimension, which is invariant under any quasi-Lipschitz mapping, satisfying \overline{\mathrm {dim}}_{B}E \leq \mathrm {dim}_{qA}E \leq \mathrm {dim}_{A}E for any compact subset E of a metric space. By virtue of the quasi-Assouad dimension, we show that any Bedford–McMullen carpet F is quasi-Lipschitz Assouad-minimal, i.e., dim _{A}f(F)\geq \mathrm {dim} _{A}F for any quasi-Lipschitz mapping f .

On the quantization for self-affine measures on Bedford–McMullen carpets

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.