Let $ T:[0,1]\to [0,1] $ be an expanding Markov map with a finite partition. Let $ \mu _\phi $ be the invariant Gibbs measure associated with a Hölder continuous potential $ \phi $. For $ x\in [0,1] $ and $ \kappa>0 $, we investigate the size of the uniform approximation set

$$ \begin{align*}\mathcal U^\kappa(x):=\{y\in[0,1]:\text{ for all } N\gg1, \text{ there exists } n\le N, \text{ such that }|T^nx-y|<N^{-\kappa}\}.\end{align*} $$

The critical value of $ \kappa $ such that $ \operatorname {\mathrm {\dim _H}}\mathcal U^\kappa (x)=1 $ for $ \mu _\phi $-almost every (a.e.) $ x $ is proven to be $ 1/\alpha _{\max } $, where $ \alpha _{\max }=-\int \phi \,d\mu _{\max }/\int \log |T'|\,d\mu _{\max } $ and $ \mu _{\max } $ is the Gibbs measure associated with the potential $ -\log |T'| $. Moreover, when $ \kappa>1/\alpha _{\max } $, we show that for $ \mu _\phi $-a.e. $ x $, the Hausdorff dimension of $ \mathcal U^\kappa (x) $ agrees with the multifractal spectrum of $ \mu _\phi $.

Start with a compact set K ⊂ Rd. This has a random number of daughter sets, each of which is a (rotated and scaled) copy of K and all of which are inside K. The random mechanism for producing daughter sets is used independently on each of the daughter sets to produce the second generation of sets, and so on, repeatedly. The random fractal set F is the limit, as n goes to ∞, of the union of the nth generation sets. In addition, K has a (suitable, random) mass which is divided randomly between the daughter sets, and this random division of mass is also repeated independently, indefinitely. This division of mass will correspond to a random self-similar measure on F. The multifractal spectrum of this measure is studied here. Our main contributions are dealing with the geometry of realisations in Rd and drawing systematically on known results for general branching processes. In this way we generalise considerably the results of Arbeiter and Patzschke (1996) and Patzschke (1997).

On the boundary of a Galton-Watson tree we can define the visibility measure by splitting mass equally between the children of each vertex, and the branching measure by splitting unit mass equally between all vertices in the nth generation and then letting n go to infinity. The multifractal structure of each of these measures is well studied. In this paper we address the question of a joint multifractal spectrum, i.e. we ask for the Hausdorff dimension of the boundary points which simultaneously have an unusual local dimension for both these measures. The resulting two-parameter spectrum exhibits a number of surprising new features, among them the emergence of a swallowtail-shaped spectrum for the visibility measure in the presence of a nontrivial condition on the branching measure.

We obtained the Hα images of some solar active regions and prominences. Our astronomical observatory has the telescope-coronagraph which was equipped with the birefringent Halle Hα filter and CCD camera Apogee U4 (2048x2048 pixels). This paper presents multifractal spectra of images of some solar active regions in Hα line obtained with the coronagraph. The Pointwise Hoelder exponents (α) and Hausdorff spectrum fn(α) for a part of chromosphere with active region and without it, have been obtained. It is visible, that curves fn(α) for quiet and active regions of chromosphere differ very strongly. In particular, fn(α) for region with a sunspot and flare has very complicated form, and for region with filament the curves do not considerably differ from quiet chromosphere. The multifractal spectrum of quiet chromosphere shows that the quiet chromosphere is very well described by fractals with different dimensions. We analyze this result.

Suppose that μ is the branching measure on the boundary of a supercritical Galton–Watson tree with offspring distribution N such that E[N log N] < ∞ and P{N = 1} > 0. We determine the multifractal spectrum of μ using a method different from that proposed by Shieh and Taylor, which is flawed.

If Z is the branching mechanism for a supercritical Galton-Watson tree with a single progenitor and E[ZlogZ] < ∞, then there is a branching measure μ defined on ∂Γ, the set of all paths ξ which have a unique node ξ|n at each generation n. We use the natural metric ρ(ξ,η) = e−n, where n = max{k : ξ|k = η|k}, and observe that the local dimension index is d(μ,ξ) = limn→∞ log(μB(ξ|n))/(-n) = α = logm, for μ-almost every ξ. Our objective is to consider the exceptional points where the above display may fail. There is a nontrivial ‘thin’ spectrum for ̄d(μ,ξ) when p1 = P{Z = 1} > 0 and Z has finite moments of all positive orders. Because ̱d(μ,ξ) = a for all ξ, we obtain a ‘thick’ spectrum by introducing the ‘right’ power of a logarithm. In both cases, we find the Hausdorff dimension of the exceptional sets.