There has been a lot of interest and activity along the general lines of analysis on metric spaces recently, as in [2], [3], [26], [40], [41], [46], [48], [49], [51], [82], [83], [89], for instance. Of course this is closely related to and involves ideas concerning spaces of homogeneous type, as in [18], [19], [66], [67], [92], as well as sub-Riemannian spaces, e.g., [8], [9], [34], [47], [52], [53], [54], [55], [68], [70], [72], [73], [84], [86], [88]. In the present survey we try to give an introduction...

We obtain a lower bound for the Hausdorff dimension of the graph of a fractal interpolation function with interpolation points $(i/N,y_i):i=0,1,...,N$.

We calculate the almost sure Hausdorff dimension of the random covering set ${lim\; sup}_{n\to \infty }(g_n+{\xi }_n)$ in $d$-dimensional torus ${𝕋}^d$, where the sets $g_n\subset {𝕋}^d$ are parallelepipeds, or more generally, linear images of a set with nonempty interior, and ${\xi }_n\in {𝕋}^d$ are independent and uniformly distributed random points. The dimension formula, derived from the singular values of the linear mappings, holds provided that the sequences of the singular values are decreasing.

The aim of this paper is to calculate (deterministically) the Hausdorff dimension of the scale-sparse Weierstrass-type functions $W_s\left(x\right):={\sum }_{j\ge 1}{\rho }^{-{\gamma }^{j}s}g({\rho }^{{\gamma }^j}x+{\theta }_j)$, where ρ > 1, γ > 1 and 0 < s < 1, and g is a periodic Lipschitz function satisfying some additional appropriate conditions.

For any $x\in [0,1)$, let $x=[{ϵ}_1,{ϵ}_2,\cdots ,]$ be its dyadic expansion. Call $r_n\left(x\right):=max\{j\ge 1:{ϵ}_{i+1}=\cdots ={ϵ}_{i+j}=1$, $0\le i\le n-j\}$ the $n$-th maximal run-length function of $x$. P. Erdös and A. Rényi showed that ${lim}_{n\to \infty }{r}_n\left(x\right)/{log}_{2}n=1$ almost surely. This paper is concentrated on the points violating the above law. The size of sets of points, whose run-length function assumes on other possible asymptotic behaviors than ${log}_{2}n$, is quantified by their Hausdorff dimension.

This is a survey on transformation of fractal type sets and measures under orthogonal projections and more general mappings.

Let X be a complete metric space and write (X) for the family of all Borel probability measures on X. The local dimension $di{m}_{loc}(\mu ;x)$ of a measure μ ∈ (X) at a point x ∈ X is defined by $di{m}_{loc}(\mu ;x)=li{m}_{r\searrow 0}\left(log\mu \left(B(x,r)\right)\right)/\left(logr\right)$ whenever the limit exists, and plays a fundamental role in multifractal analysis. It is known that if a measure μ ∈ (X) satisfies a few general conditions, then the local dimension of μ exists and is equal to a constant for μ-a.a. x ∈ X. In view of this, it is natural to expect that for a fixed x ∈ X, the local dimension...

We classify all homeomorphisms of the double cover of the Sierpiński gasket in n dimensions. We show that there is a unique homeomorphism mapping any cell to any other cell with prescribed mapping of boundary points, and any homeomorphism is either a permutation of a finite number of topological cells or a mapping of infinite order with one or two fixed points. In contrast we show that any compact fractafold based on the level-3 Sierpiński gasket is topologically rigid.

A class of subsets of ℝⁿ is constructed that have certain homogeneity and non-coincidence properties with respect to Hausdorff and box dimensions. For each triple (r,s,t) of numbers in the interval (0,n] with r < s < t, a compact set K is constructed so that for any non-empty subset U relatively open in K, we have $(di{m}_H\left(U\right),{\underline{dim}}_B\left(U\right),{\overline{dim}}_B\left(U\right))=(r,s,t)$. Moreover, $2^{-n}\le H^r\left(K\right)\le 2n^{r/2}$.

We show that almost every function (in the sense of prevalence) in a Sobolev space is multifractal: Its regularity changes from point to point; the sets of points with a given Hölder regularity are fractal sets, and we determine their Hausdorff dimension.