This subject has several natural points of view, but we shall start with the one that corresponds to the following question: Is there something like Littlewood-Paley theory which is useful for analyzing the geometry of subsets of Rn, in much the same way that traditional Littlewood-Paley theory is good for analyzing functions and operators?

We construct examples of expanding piecewise monotonic maps on the interval which have a closed topologically transitive invariant subset A with Darboux property, Hausdorff dimension d ∈ (0,1) and zero d-dimensional Hausdorff measure. This shows that the results about Hausdorff and conformal measures proved in the first part of this paper are in some sense best possible.

Let A be a topologically transitive invariant subset of an expanding piecewise monotonic map on [0,1] with the Darboux property. We investigate existence and uniqueness of conformal measures on A and relate Hausdorff and conformal measures on A to each other.

We extend the notions of Hausdorff and packing dimension introducing weights in their definition. These dimensions are computed for ergodic invariant probability measures of two-dimensional Lorenz transformations, which are transformations of the type occuring as first return maps to a certain cross section for the Lorenz differential equation. We give a formula of the dimensions of such measures in terms of entropy and Lyapunov exponents. This is done for two choices of the weights using the recurrence...

For a linear solenoid with two different contraction coefficients and box dimension greater than 2, we give precise formulas for the Hausdorff and packing dimensions. We prove that the packing measure is infinite and give a condition necessary and sufficient for the Hausdorff measure to be positive, finite and equivalent to the SBR measure. We also give analogous results, generalizing [P], for affine IFS in ℝ².

We study the Julia sets for some periodic meromorphic maps, namely the maps of the form $f\left(z\right)=h\left(exp\frac{2\pi i}{T}z\right)$ where h is a rational function or, equivalently, the maps $˜f\left(z\right)=exp\left(\frac{2\pi i}{h}\left(z\right)\right)$. When the closure of the forward orbits of all critical and asymptotic values is disjoint from the Julia set, then it is hyperbolic and it is possible to construct the Gibbs states on J(˜f) for -α log |˜˜f|. For ˜α = HD(J(˜f)) this state is equivalent to the ˜α-Hausdorff measure or to the ˜α-packing measure provided ˜α is greater or smaller than 1....

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

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.

There is no non-trivial constraint on the Hausdorff dimension of sums of a set with itself.

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 F : U ⊂ Rn → Rm be a differentiable function and p &lt; m an integer. If k ≥ 1 is an integer, α ∈ [0, 1] and F ∈ Ck+(α), if we set Cp(F) = {x ∈ U | rank(Df(x)) ≤ p} then the Hausdorff measure of dimension (p + (n-p)/(k+α)) of F(Cp(F)) is zero.

We investigate the following question: under which conditions is a σ-compact partial two point set contained in a two point set? We show that no reasonable measure or capacity (when applied to the set itself) can provide a sufficient condition for a compact partial two point set to be extendable to a two point set. On the other hand, we prove that under Martin's Axiom any σ-compact partial two point set such that its square has Hausdorff 1-measure zero is extendable.

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, $2^{-n}\le H^r\left(K\right)\le 2n^{r/2}$.