Self-affine Sponges Research Articles

Weighted topological pressure revisited

Abstract Feng and Huang [Variational principle for weighted topological pressure. J. Math. Pures Appl. (9)106 (2016), 411–452] introduced weighted topological entropy and pressure for factor maps between dynamical systems and established its variational principle. Tsukamoto [New approach to weighted topological entropy and pressure. Ergod. Th. & Dynam. Sys.43 (2023), 1004–1034] redefined those invariants quite differently for the simplest case and showed via the variational principle that the two definitions coincide. We generalize Tsukamoto’s approach, redefine the weighted topological entropy and pressure for higher dimensions, and prove the variational principle. Our result allows for an elementary calculation of the Hausdorff dimension of affine-invariant sets such as self-affine sponges and certain sofic sets that reside in Euclidean space of arbitrary dimension.

Distribution of δ-connected components of self-affine sponges of Lalley-Gatzouras type

Let (E,ρ) be a metric space and let hE(δ) be the cardinality of the set of δ-connected components of E. In literature, in case of that E is a self-conformal set satisfying the open set condition or E is a self-affine Sierpiński sponge, necessary and sufficient condition is given for the validity of the relationhE(δ)≍δ−dimB⁡E, when δ→0. In this paper, we generalize the above result to self-affine sponges of Lalley-Gatzouras type; actually in this case, we show that there exists a Bernoulli measure μ such that for any cylinder W, it holds thathW(δ)≍μ(W)δ−dimB⁡E, when δ→0.

Dimensions of ‘self-affine sponges’ invariant under the action of multiplicative integers

AbstractLet $m_1 \geq m_2 \geq 2$ be integers. We consider subsets of the product symbolic sequence space $(\{0,\ldots ,m_1-1\} \times \{0,\ldots ,m_2-1\})^{\mathbb {N}^*}$ that are invariant under the action of the semigroup of multiplicative integers. These sets are defined following Kenyon, Peres, and Solomyak and using a fixed integer $q \geq 2$ . We compute the Hausdorff and Minkowski dimensions of the projection of these sets onto an affine grid of the unit square. The proof of our Hausdorff dimension formula proceeds via a variational principle over some class of Borel probability measures on the studied sets. This extends well-known results on self-affine Sierpiński carpets. However, the combinatoric arguments we use in our proofs are more elaborate than in the self-similar case and involve a new parameter, namely $j = \lfloor \log _q ( {\log (m_1)}/{\log (m_2)} ) \rfloor $ . We then generalize our results to the same subsets defined in dimension $d \geq 2$ . There, the situation is even more delicate and our formulas involve a collection of $2d-3$ parameters.

Dimension drop of connected part of slicing self-affine sponges

The connected part of a metric space E is defined to be the union of non-trivial connected components of E. We proved that for a class of self-affine sets called slicing self-affine sponges, the connected part of E either coincides with E, or is essentially contained in the attractor of a proper sub-IFS of an iteration of the original IFS. This generalizes an early result of Huang and Rao (2021) [9] on a class of self-similar sets called fractal cubes. Moreover, we show that the result is no longer valid if the slicing property is removed. Consequently, for a Barański carpet E possessing trivial points, the Hausdorff dimension and the box dimension of the connected part of E are strictly less than that of E, respectively. For slicing self-affine sponges in Rd with d≥3, whether the attractor of a sub-IFS has strictly smaller dimensions is an open problem.

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 > 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.

Assouad type dimensions for self-affine sponges with a weak coordinate ordering condition

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 their more subtle details along the way.

Badly approximable points on self-affine sponges and the lower Assouad dimension

We highlight a connection between Diophantine approximation and the lower Assouad dimension by using information about the latter to show that the Hausdorff dimension of the set of badly approximable points that lie in certain non-conformal fractals, known as self-affine sponges, is bounded below by the dynamical dimension of these fractals. For self-affine sponges with equal Hausdorff and dynamical dimensions, the set of badly approximable points has full Hausdorff dimension in the sponge. Our results, which are the first to advance beyond the conformal setting, encompass both the case of Sierpiński sponges/carpets (also known as Bedford–McMullen sponges/carpets) and the case of Barański carpets. We use the fact that the lower Assouad dimension of a hyperplane diffuse set constitutes a lower bound for the Hausdorff dimension of the set of badly approximable points in that set.

The Hausdorff and dynamical dimensions of self-affine sponges: a dimension gap result

We construct a self-affine sponge in mathbb {R}^3 whose dynamical dimension, i.e. the supremum of the Hausdorff dimensions of its invariant measures, is strictly less than its Hausdorff dimension. This resolves a long-standing open problem in the dimension theory of dynamical systems, namely whether every expanding repeller has an ergodic invariant measure of full Hausdorff dimension. More generally we compute the Hausdorff and dynamical dimensions of a large class of self-affine sponges, a problem that previous techniques could only solve in two dimensions. The Hausdorff and dynamical dimensions depend continuously on the iterated function system defining the sponge, implying that sponges with a dimension gap represent a nonempty open subset of the parameter space.

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.