We study the existence of traces of Besov spaces on fractal h-sets Γ with a special focus on assumptions necessary for this existence; in other words, we present criteria for the non-existence of traces. In that sense our paper can be regarded as an extension of Bricchi (2004) and a continuation of Caetano (2013). Closely connected with the problem of existence of traces is the notion of dichotomy in function spaces: We can prove that-depending on the function space and the set Γ-there occurs an...

Continuous transformations preserving the Hausdorff-Besicovitch dimension (“DP-transformations”) of every subset of R 1 resp. [0, 1] are studied. A class of distribution functions of random variables with independent s-adic digits is analyzed. Necessary and sufficient conditions for dimension preservation under functions which are distribution functions of random variables with independent s-adic digits are found. In particular, it is proven that any strictly increasing absolutely continuous distribution...

We study the typical behaviour (in the sense of Baire’s category) of the multifractal box dimensions of measures on ${ℝ}^d$. We prove that in many cases a typical measure μ is as irregular as possible, i.e. the lower multifractal box dimensions of μ attain the smallest possible value and the upper multifractal box dimensions of μ attain the largest possible value.

Let $\sigma$ be a unimodular Pisot substitution over a $d$ letter alphabet and let $X_1,...,X_d$ be the associated Rauzy fractals. In the present paper we want to investigate the boundaries $\partial X_i$ ($1\le i\le d$) of these fractals. To this matter we define a certain graph, the so-called contact graph$𝒞$ of $\sigma$. If $\sigma$ satisfies a combinatorial condition called the super coincidence condition the contact graph can be used to set up a self-affine graph directed system whose attractors are certain pieces of the boundaries $\partial X_1,...,\partial X_d$. From this graph...

We prove that, up to scalar multiples, there exists only one local regular Dirichlet form on a generalized Sierpi´nski carpet that is invariant with respect to the local symmetries of the carpet. Consequently, for each such fractal the law of Brownian motion is uniquely determined and the Laplacian is well defined.

We show upper estimates of the concentration and thin dimensions of measures invariant with respect to families of transformations. These estimates are proved under the assumption that the transformations have a squeezing property which is more general than the Lipschitz condition. These results are in the spirit of a paper by A. Lasota and J. Traple [Chaos Solitons Fractals 28 (2006)] and generalize the classical Moran formula.

Under an appropriate oscillating behaviour either at zero or at infinity of the nonlinear term, the existence of a sequence of weak solutions for an eigenvalue Dirichlet problem on the Sierpiński gasket is proved. Our approach is based on variational methods and on some analytic and geometrical properties of the Sierpiński fractal. The abstract results are illustrated by explicit examples.

We outline the construction of Brownian motion on certain self-similar fractals and introduce the notion of walk dimension. We then show how the probabilistic approach relates to the theory of function spaces on fractals.

The work developed in the paper concerns the multivariate fractional Brownian motion (mfBm) viewed through the lens of the wavelet transform. After recalling some basic properties on the mfBm, we calculate the correlation structure of its wavelet transform. We particularly study the asymptotic behaviour of the correlation, showing that if the analyzing wavelet has a sufficient number of null first order moments, the decomposition eliminates any possible long-range (inter)dependence. The cross-spectral...

Let $E$ be a Banach (or quasi-Banach) space which is shift and scaling invariant (typically a homogeneous Besov or Sobolev space). We introduce a general definition of pointwise regularity associated with $E$, and denoted by $C_E^{\alpha }\left(x_0\right)$. We show how properties of $E$ are transferred into properties of $C_E^{\alpha }\left(x_0\right)$. Applications are given in multifractal analysis.

We show that there are Hilbert spaces constructed from the Hausdorff measures Hs on the real line R with 0 < s < 1 which admit multiresolution wavelets. For the case of the middle-third Cantor set C ⊂ [0,1], the Hilbert space is a separable subspace of L2(R, (dx)s) where s = log3(2). While we develop the general theory of multiresolutions in fractal Hilbert spaces, the emphasis is on the case of scale 3 which covers the traditional Cantor set C.

Suppose $F\subset [0,1]$ is closed. Is it true that the typical (in the sense of Baire category) function in $C^1[0,1]$ is one-to-one on $F$? If ${\underline{dim}}_{B}F<1/2$ we show that the answer to this question is yes, though we construct an $F$ with ${dim}_{B}F=1/2$ for which the answer is no. If $C_{\alpha }$ is the middle-$\alpha$ Cantor set we prove that the answer is yes if and only if $dim\left(C_{\alpha }\right)\le 1/2.$ There are $F$’s with Hausdorff dimension one for which the answer is still yes. Some other related results are also presented.