The detailed investigation of the distribution of frequencies of digits of points belonging to attractors K of Infinite iterated functions systems (IIFS’s) is a fundamental and important problem in the study of attractors of IIFS’s. This paper studies the Baire category of different families of sets of points belonging to attractors of IIFS’s characterised by the behaviour of the frequencies of their digits. All our results are of the following form:a typical (in the sense of Baire) point has the following property: the average frequencies of digits of have maximal oscillation.We consider general types of average frequencies, namely, average frequencies associated with general averaging systems. These averages include, for example, all higher order Hölder and Cesaro averages, and Riesz averages. Surprising, for all averaging systems (regardless of how powerful they are) we prove that a typical (in the sense of Baire) point $$x\in K$$ has the following property: the average frequencies of digits of x have maximal oscillation. This substantially extends previous results and provides a powerful topological manifestation of the fact that “points of divergence” are highly visible. Several applications are given, e.g. to continued fraction digits and Lüroth expansion digits.

Given $$\alpha ,\beta ,\gamma \in [0,1]$$ with $$\alpha \le \beta $$, we prove that there exists a subset of $${\mathbb {N}}$$ such that its lower and upper exponential densities and its lower and upper limit ratios are equal to $$\alpha $$, $$\beta $$, $$\gamma $$ and 1, respectively. This result provides an affirmative answer to an open problem posed by Grekos et al. (Unif Distrib Theory 6:117–130, 2011).

Let (X,ℰ,μ ) be a measure space and let f:X→ℝ be a measurable function such that ‖ f‖ _p<∞ for all p≥ 1 and ‖ f‖ _∞>0 . In this paper, we describe the rate of convergence of (‖ f‖ _p/‖ f‖ _∞)^p as p→∞ .

Let X be a compact subset of R-d, and write P(X) for the family of Borel probability measures on X equipped with the weak topology. For a real number q, the lower and upper L-q-dimensions, denoted by (D) under bar (q)(mu) and (D) over bar (q) (mu), of mu is an element of P(X) are defined by (D) under bar (q) (mu) = lim inf (r SE arrow 0)logI(r)(q) (mu)/-logr and (D) over bar (q) (mu) - lim sup(r SE arrow 0) logI(r)(q) (mu) /- logr, where I-r(q)(mu) = 1/r(d) integral mu(B(x,r))(q) dx. Recently, it has been proven that for a typical measure mu is an element of P(X), the moment scaling function appearing in the definition of the L-q-dimensions (D) under bar (q)(mu) and (D) over bar (q) (mu), that is, the function (*) r bar right arrow logI(r)(q)(mu)/-logr for r > 0, diverges in the worst possible way as r SE arrow 0. For example, [Ba1, Ol1] prove that if X is Ahlfors regular and q >= 1, then (D) over bar (q) (mu) = sup(v is an element of P(X))(D) over bar (q) (v) = 0 and (D) under bar (q)(mu) = inf (v is an element of P(X))(D) under bar (q) (v) = dim(H)(X)(1 - q) for a typical measure mu is an element of P(X) where dim(H)(X) denotes the Hausdorlf dimension of X. In this paper we prove that the moment scaling function (*) of a typical measure mu is an element of P(X) is spectacularly more irregular than suggested by the result in [Ba1,Ol1]. In particular, we show the following sur- prising result: not only is the moment scaling function (*) of a typical measure mu is an element of P(X) divergent as r SE arrow 0, but it is so irregular that it re- mains spectacularly divergent as r SE arrow 0 even after being "averaged" or "smoothened out" using arbitrary averaging methods including, for example, all higher-order Holder and Cesaro averages.

Let \((X,{\mathcal {E}},\mu )\) be a measure space and let \(f:X\rightarrow \mathbb R\) be a measurable function such that \(\Vert f\Vert _{p}<\infty \) for all \(p\ge 1\) and \(\Vert f\Vert _{\infty }>0\). In this paper, we describe the rate of convergence of \((\frac{\Vert f\Vert _{p}}{\Vert f\Vert _{\infty }})^{p}\) as \(p\rightarrow \infty \).

We study average Hewitt–Stromberg measures of typical compact metric spaces belonging to the Gromov–Hausdorff space (of all compact metric spaces) equipped with the Gromov–Hausdorff metric.

Let X be a metric space and write K(X) for the family of non-empty compact subsets of X equipped with the Hausdorff metric. The lower and upper box dimensions, denoted by (dim) under bar (B) (E) and (dim) over bar (B) (E), of a subset E of X are defined by (dim) under bar (B) (E) = lim(r SE arrow 0) inf log N-r (E)/-log r, (dim) over bar (B) (E) = lim(r SE arrow 0) sup log N-r (E)/-log r, where N-r (E) is the smallest number of closed balls with centres in E and radii equal to r that are needed to cover E. In the 1980's, Gruber proved that the box counting function (*) log N-r (C)/-log r of a typical compact set C is an element of K(X) diverges in the worst possible way as r SE arrow 0. For example, Gruber proved that (dim) under bar (B) (C) = 0 and (dim) over bar (B) (C) = N for a typical C is an element of K(R-N). In this paper we prove that the box counting function (*) of a typical compact set C is an element of K(X) is spectacularly more irregular than suggested by Gruber's result. In particular, we show the following surprising result: not only is the box counting function (*) of a typical compact set C is an element of K(X) divergent as r SE arrow 0, but it is so irregular that it remains spectacularly divergent as r SE arrow 0 even after being "averaged" or "smoothened out" using powerful averaging methods including, for example, all higher order Holder and Cesaro averages. As an application of our results we obtain strengthened versions of Gruber's result.

We study the average Lq-dimensions of typical Borel probability measures belonging to the Gromov–Hausdorff–Prohoroff space (of all Borel probability measures with compact supports) equipped with the Gromov–Hausdorff–Prohoroff metric.

We show that given any six numbers $r,s,t,u,v,w \in (0,1]$ satisfying $r \leq s \leq \min(t,u) \leq \max(t,u) \leq v \leq w$, it is possible to construct a compact subset of $[0,1]$ with Hausdorff dimension equal to $r$, lower modified box dimension equal to $s$, packing dimension equal to $t$, lower box dimension equal to $u$, upper box dimension equal to $v$ and Assouad dimension equal to $w$. Moreover, the set constructed is an $r$-Hausdorff set and a $t$-packing set.

We study the Hausdorff and packing measures of typical compact metric spaces belonging to the Gromov–Hausdorff space (of all compact metric spaces) equipped with the Gromov–Hausdorff metric.

In this paper we will prove that the box counting function (*) of the graph of a typical function \({f\in C_{\mathsf{u}}(X)}\) is spectacularly more irregular than suggested by the result due to Hyde et al. Namely, we show the following surprising result: not only is the box counting function in (*) divergent as \({\delta\searrow 0}\), but it is so irregular that it remains spectacularly divergent as \({\delta\searrow 0}\) even after being “averaged" or “smoothened out" using exceptionally powerful averaging methods including all higher order Hölder and Cesàro averages and all higher order Riesz–Hardy logarithmic averages. For example, if the box dimension of X exists, then we show that for a typical function \({f\in C_{\mathsf{u}}(X)}\), all the higher order lower Hölder and Cesàro averages of the box counting function (*) are as small as possible, namely, equal to the box dimension of X, and if, in addition, X has only finitely many isolated points, then all the higher order upper Hölder and Cesàro averages of the box counting function (*) are as big as possible, namely, equal to the box dimension of X plus 1.

The purpose of this paper twofold. Firstly, we establish Pi(0)(gamma)-completeness and Sigma(0)(gamma)-completeness of several different classes of multifractal decomposition sets of arbitrary Borel measures (satisfying a mild non-degeneracy condition and two mild "smoothness" conditions). Secondly, we apply these results to study the Pi(0)(gamma)-completeness and Sigma(0)(gamma)-completeness of several multifractal decomposition sets of self-similar measures (satisfying a mild separation condition). For example, a corollary of our results shows if mu is a self-similar measure satisfying the strong separation condition and mu is not equal to the normalized Hausdorff measure on its support, then the classical multifractal decomposition sets of mu defined by {x is an element of R-d vertical bar lim(r SE arrow 0) log mu(B(x, r))/log r = alpha} are Pi(0)(3)-complete provided they are non-empty.

We study several distinct notions of average distances between points belonging to graph-directed self-similar subsets of R. In particular, we compute the average distance with respect to graph-directed self-similar measures, and with respect to the normalised Hausdorff measure. As an application of our main results, we compute the average distance between two points belonging to the Drobot-Turner set T-N(c, m) with respect to the normalised Hausdorff measure, i.e. we compute 1/H-s(T-N(c, m))(2) integral(TN(c, m)2) vertical bar x-y vertical bar d(H-s x H-s)(x, y), where s denotes the Hausdorff dimension of T-N(c, m) and H-s is the s-dimensional Hausdorff measure; here the Drobot-Turner set (introduced by Drobot & Turner in 1989) is defined as follows, namely, for positive integers N and m and a positive real number c, the Drobot-Turner set T-N(c, m) is the set of those real numbers x is an element of[0, 1] for which any m consecutive base N digits in the N-ary expansion of x sum up to at least c. For example, if N = 2, m = 3 and c = 2, then our results show that 1/H-s(T-2(2, 3))(2) integral(T2(2,3)2) vertical bar x-y vertical bar d(H-s x H-s)(x, y) = 4444 lambda(2) + 2071 lambda + 3030/12141 lambda(2) + 5650 lambda + 8281 = 0.36610656..., where lambda = 1.465571232 ... is the unique positive real number such that lambda(3) - lambda(2) - 1 = 0.

The purpose of this paper twofold. Firstly, we establish -completeness and -completeness of several different classes of multifractal decomposition sets of arbitrary Borel measures (satisfying a mild non-degeneracy condition and two mild “smoothness” conditions). Secondly, we apply these results to study the -completeness and -completeness of several multifractal decomposition sets of self-similar measures (satisfying a mild separation condition). For example, a corollary of our results shows if is a self-similar measure satisfying the strong separation condition and is not equal to the normalized Hausdorff measure on its support, then the classical multifractal decomposition sets of defined by are -complete provided they are non-empty.

We introduce multifractal zeta-functions providing precise information of a very general class of multifractal spectra, including, for example, the multifractal spectra of self-conformal measures and the multifractal spectra of ergodic Birkhoff averages of continuous functions. More precisely, we prove that these and more general multifractal spectra equal the abscissae of convergence of the associated zetafunctions.

We analyse the asymptotic behaviour of the mixed moments of Borel probability measures on [0,1]d. In particular, we prove that the asymptotic behaviour of the mixed moments of a measure is intimately related to the local dimensions of the measure.

The purpose of this paper is twofold: (1) we study different notions of the average distance between two points of a self-similar subset of \({\mathbb {R}}\), and (2) we investigate the asymptotic behaviour of higher order average moments of self-similar measures on self-similar subsets of \({\mathbb {R}}\).

We introduce multifractal pressure and dynamical multifractal zeta-functions providing precise information of a very general class of multifractal spectra, including, for example, the fine multifractal spectra of graph-directed self-conformal measures and the fine multifractal spectra of ergodic Birkhoff averages of continuous functions on graph-directed self-conformal sets.

We analyse the asymptotic behaviour of several types of moments of Borel probability measures on R-d. In particular, we prove that the asymptotic behaviour of the moments of a measure is intimately related to the local dimensions of the measure.