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.

Let $X=X\left(t\right),t\in {ℝ}^N$ be a Gaussian random field in ${ℝ}^d$ with stationary increments. For any Borel set $E\subset {ℝ}^N$, we provide sufficient conditions for the image X(E) to be a Salem set or to have interior points by studying the asymptotic properties of the Fourier transform of the occupation measure of X and the continuity of the local times of X on E, respectively. Our results extend and improve the previous theorems of Pitt [24] and Kahane [12,13] for fractional Brownian motion.

Let α be an irrational and φ: ℕ → ℝ⁺ be a function decreasing to zero. Let $\omega \left(\alpha \right):=sup\theta \ge 1:limin{f}_{n\to \infty }{n}^{\theta }\left|\right|n\alpha |=0$$.Forany\alpha ...$

We investigate the box dimensions of inhomogeneous self-similar sets. Firstly, we extend some results of Olsen and Snigireva by computing the upper box dimensions assuming some mild separation conditions. Secondly, we investigate the more difficult problem of computing the lower box dimension. We give some non-trivial bounds and provide examples to show that lower box dimension behaves much more strangely than upper box dimension, Hausdorff dimension and packing dimension.

Contractive Markov systems on Polish spaces which arise from graph directed constructions of iterated function systems with place dependent probabilities are considered. It is shown that their stability may be studied using the concentrating methods developed by the second author [Dissert. Math. 415 (2003)]. In this way Werner's results obtained in a locally compact case [J. London Math. Soc. 71 (2005)] are extended to a noncompact setting.

Nonoverlapping contractive self-similar iterated function systems (IFS) have been studied in great detail via the open set condition. On the other hand much less is known about IFS with overlaps. To deal with such systems, a weak separation condition (WSC) has been introduced recently [LN1]; it is weaker than the open set condition and it includes many important overlapping cases. This paper has two purposes. First, we consider the class of self-similar measures generated by such IFS; we give a...

Nous montrons que le tracé d’un kolam indien classique, que l’on retrouve aussi dans la tradition des dessins sur le sable aux îles Vanuatu, peut être engendré par un morphisme de monoïde. La suite infinie morphique ainsi obtenue est reliée à la célèbre suite de Prouhet-Thue-Morse, mais elle n’est $k$-automatique pour aucun entier $k\ge 1$.

We consider the continuous model of log-infinitely divisible multifractal random measures (MRM) introduced in [E. Bacry et al. Comm. Math. Phys. 236 (2003) 449–475]. If M is a non degenerate multifractal measure with associated metric ρ(x,y) = M([x,y]) and structure function ζ, we show that we have the following relation between the (Euclidian) Hausdorff dimension dimH of a measurable set K and the Hausdorff dimension dimHρ with respect to ρ of the same set: ζ(dimHρ(K)) = dimH(K). Our results can...

We consider the continuous model of log-infinitely divisible multifractal random measures (MRM) introduced in [E. Bacry et al. Comm. Math. Phys.236 (2003) 449–475]. If M is a non degenerate multifractal measure with associated metric ρ(x,y) = M([x,y]) and structure function ζ, we show that we have the following relation between the (Euclidian) Hausdorff dimension dimH of a measurable set K and the Hausdorff dimension dimHρ with respect to ρ of the same set: ζ(dimHρ(K)) = dimH(K). Our results can...

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 $L^q$-spectrum, q >0, of the measure. This formula is the first obtained in the case where the open set condition does not hold.