Abstract In this article we study dynamical behaviour of generic Lebesgue measure-preserving interval maps. We show that for each k ⩾ 1 the set periodic points period at least is a Cantor Hausdorff dimension zero and upper box one. Moreover, obtain analogous results also in context circle Furthermore, building on former results, there dense collection transitive maps whose have full measure pos...

A method is proposed for detecting whether two CCITT group 4 images were scanned from the same document. Features are extracted from rectangular patches of text and compared with a modified Hausdorff distance measure. Two images are said to be ‘‘equivalent’’ (i.e., they were scanned from the same document) if the Hausdorff measure finds that a specified number of features are located within a g...

For a closed subset K of a compact metric space A possessing an α-regular measure μ with μ(K) > 0, we prove that whenever s > α, any sequence of weighted minimal Riesz s-energy configurations ωN = {x i,N} N i=1 on K (for ‘nice’ weights) is quasi-uniform in the sense that the ratios of its mesh norm to separation distance remain bounded as N grows large. Furthermore, if K is an α-rectifiable com...

We show that if compact set $$E\subset \mathbb {R}^d$$ has Hausdorff dimension larger than $$\frac{d}{2}+\frac{1}{4}$$ , where $$d\ge 4$$ is an even integer, then the distance of E positive Lebesgue measure. This improves previously best known result towards Falconer’s conjecture in dimensions.

How many fractals exist in nature or the virtual world In this work, we partially answer second question using Mandelbrots fundamental definition of and their quantities Hausdorff dimension Lebesgue measure. We prove existence beth-two with a bivariate function them given The remains unanswered for other fractal dimensions.

We prove that the 1-dimensional Hausdorff measure restricted to a simple real analytic curve γ : R → R , N ≥ 2, is locally 1-monotone.

Let \(c \geq 2\) be any fixed real number. Matomäki [4] inverstigated the set of \(A > 1\) such that integer part \( A^{c^k}\) is a prime number for every \(k \in \mathbb{N}\). She proved uncountable, nowhere dense, and has Lebesgue measure 0. In this article, we show Hausdorff dimension 1.