### Dimensions of the boundaries of self-similar sets.

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Let ${\mathcal{T}}_{X}$ denote the operator-norm closure of the class of convolution operators ${\Phi}_{\mu}:X\to X$ where $X$ is a suitable function space on $R$. Let ${\mathcal{M}}_{r}^{p}$ be the closed subspace of regular functions in the Marinkiewicz space ${\mathcal{M}}^{p}$, $1\le p\<\infty $. We show that the space ${\mathcal{T}}_{{\mathcal{M}}_{r}^{p}}$ is isometrically isomorphic to ${\mathcal{T}}_{{L}^{p}}$ and that strong operator sequential convergence and norm convergence in ${\mathcal{T}}_{{\mathcal{M}}_{r}^{p}}$ coincide. We also obtain some results concerning convolution operators under the Wiener transformation. These are to improve a Tauberian theorem of Wiener on ${\mathcal{M}}^{2}$.

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.

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...

The Ruelle operator and the associated Perron-Frobenius property have been extensively studied in dynamical systems. Recently the theory has been adapted to iterated function systems (IFS) $(X,{{w}_{j}}_{j=1}^{m},{{p}_{j}}_{j=1}^{m})$, where the ${w}_{j}$’s are contractive self-maps on a compact subset $X\subseteq {\mathbb{R}}^{d}$ and the ${p}_{j}$’s are positive Dini functions on X [FL]. In this paper we consider Ruelle operators defined by weakly contractive IFS and nonexpansive IFS. It is known that in such cases, positive bounded eigenfunctions may not exist in general. Our theorems...

The Choquet-Deny theorem and Deny’s theorem are extended to the vector-valued case. They are applied to give a simple nonprobabilistic proof of the vector-valued renewal theorem, which is used to study the ${L}^{p}$-dimension, the ${L}^{p}$-density and the Fourier transformation of vector-valued self-similar measures. The results answer some questions raised by Strichartz.

Let A be a d × d integral expanding matrix and let ${S}_{j}\left(x\right)={A}^{-1}(x+{d}_{j})$ for some ${d}_{j}\in {\mathbb{Z}}^{d}$, j = 1,...,m. The iterated function system (IFS) ${{S}_{j}}_{j=1}^{m}$ generates self-affine measures and scale functions. In general this IFS has overlaps, and it is well known that in many special cases the analysis of such measures or functions is facilitated by expressing them in vector-valued forms with respect to another IFS that satisfies the open set condition. In this paper we prove a general theorem on such representation. The proof is constructive;...

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