Based on a set of higher order self-similar identities for the Bernoulli convolution measure for (√5-1)/2 given by Strichartz et al., we derive a formula for the -spectrum, q >0, of the measure. This formula is the first obtained in the case where the open set condition does not hold.
We say that a set in a Euclidean space does not contain an angle α if the angle determined by any three points of the set is not equal to α. The goal of this paper is to construct compact sets of large Hausdorff dimension that do not contain a given angle α ∈ (0,π). We will construct such sets in ℝn of Hausdorff dimension c(α)n with a positive c(α) depending only on α provided that α is different from π/3, π/2 and 2π/3. This improves on an earlier construction (due to several authors) that has dimension...
Let be a mapping from a metric space X to a metric space Y, and let α be a positive real number. Write dim (E) and Hs(E) for the Hausdorff dimension and the s-dimensional Hausdorff measure of a set E. We give sufficient conditions that the equality dim (f(E)) = αdim (E) holds for each E ⊆ X. The problem is studied also for the Cantor ternary function G. It is shown that there is a subset M of the Cantor ternary set such that Hs(M) = 1, with s = log2/log3 and dim(G(E)) = (log3/log2) dim (E), for...
This paper studies the geometric structure of graph-directed sets from the point of view of Lipschitz equivalence. It is proved that if and are dust-like graph-directed sets satisfying the transitivity condition, then and are Lipschitz equivalent, and and are quasi-Lipschitz equivalent when they have the same Hausdorff dimension.
Let be a family of Hölder continuous functions and let be a conformal iterated function system. Lindsay and Mauldin’s paper [Nonlinearity 15 (2002)] left an open question whether the lower quantization coefficient for the F-conformal measure on a conformal iterated funcion system satisfying the open set condition is positive. This question was positively answered by Zhu. The goal of this paper is to present a different proof of this result.