Narrow Results

Measures were constructed on symbolic spaces that satisfy an extended multifractal formalism, where Olsen’s functions $b$ and $B$ differ, and their Legendre transforms have the expected interpretation in terms of dimensions. These measures were composed with a Gray code and projected onto the unit interval to obtain doubling measures. It was demonstrated that the projected measure retains the same Olsen’s functions as the original and also satisfies the extended multifractal formalism. In this paper, we show that the use of a Gray code is not essential to achieve these results, even when dealing with non-doubling measures. Moreover, general results on multifractal analysis of inhomogeneous multinomial measures with their non-doubling projections are obtained. The key points of the proof include two main components: the study of weak doubling properties and the method of constructing auxiliary measure to get sharp bound for the dimension under consideration.

Let the self-similar set C in R be defined by with a disjoint union, where the h_j's are similitude mappings with ratios 0 < a_j < 1. Let μ on C be the self-similar probability measure corresponding to the probability vector , where ξ = dim_HC is the Hausdorff dimension of C. Let S be the set of points at which the probability distribution function F of μ has no derivative, finite or infinite. We prove that dim_H S = (dim_HC)² and dim_P S = dim_B S = dim_HC.

By prescribing their code run behavior, we consider some subsets of Moran fractals. Fractal dimensions of these subsets are exactly obtained. Meanwhile, an interesting decomposition of Moran fractals is given.

Given metric spaces $E$ and $F$, it is well known that

Given two metric spaces $E,F$, it is well known that,

In this paper, we prove that a large class of homogeneous perfect sets of Hausdorff dimension 1 is quasisymmetrically Hausdorff minimal. We also obtain the similar result for quasisymmetrically packing minimality.

For an infinite sequence $t={({𝜀}_k)}_k$ of $1$ and $-1$ with probability $p$ and $1-p$, we mainly study the multifractal analysis of one-dimensional biased walks. Let $H(t)=-t{log}_{2}t-(1-t){log}_2(1-t)$ and $S_n(t)={\sum }_{k=1}^n{𝜀}_k$. The Hausdorff and packing dimensions of the sets $E_b=\{t\in {\{\pm 1\}}^{\mathbb{N}}:{lim}_{n\to \infty }\frac{1}{n}[S_n(t)-𝔼(S_n(t))]=b\}$ are ${dim}_H{E}_b={dim}_P{E}_b=H(\frac{2p+b}{2})$, which is the development of the theorem of Besicovitch [On the sum of digits of real numbers represented in the dyadic system, Math. Ann.110 (1934) 321–330] on random walk, saying that: For any $0<b<1$, the set $\{t\in {\{\pm 1\}}^{\mathbb{N}}:\underset{n\to \infty }{lim}\frac{S_n(t)}{n}=b\}$ has Hausdorff dimension $H(\frac{1+b}{2})$.

The $p$-adic number field ${\mathbb{Q}}_p$ and the $p$-adic analogue of the complex number field ${ℂ}_p$ have a rich algebraic and geometric structure that in some ways rivals that of the corresponding objects for the real or complex fields. In this paper, we attempt to find and understand geometric structures of general sets in a $p$-adic setting. Several kinds of fractal measures and dimensions of sets in ${ℂ}_p$ are studied. Some typical fractal sets are constructed. It is worthwhile to note that there exist some essential differences between $p$-adic case and classical case.

In this paper, we prove that a large class of Moran sets on the line with packing dimension 1 is quasisymmetrically packing-minimal.

In this paper, two large classes of Moran sets with packing dimension 1 are shown to be quasisymmetrically minimal for packing dimension.

For regular continued fraction, if a real number $x$ and its rational approximation $p/q$ satisfying $|x-p/q|<1/q^2$, then, after deleting the last integer of the partial quotients of $p/q$, the sequence of the remaining partial quotients is a prefix of that of $x$. In this paper, we show that the situation is completely different if we consider the Hurwitz continued fraction expansions of a complex number and its rational approximations. More specifically, we consider the set $E(\psi )$ of complex numbers which are well approximated with the given bound $\psi$ and have quite different Hurwitz continued fraction expansions from that of their rational approximations. The Hausdorff and packing dimensions of such set are determined. It turns out that its packing dimension is always full for any given approximation bound $\psi$ and its Hausdorff dimension is equal to that of the $\psi$-approximable set $W(\psi )$ of complex numbers. As a consequence, we also obtain an analogue of the classical Jarník theorem in real case.

There are only two kinds of measures in which the mixed multifractal formalism applies, which are self-similar and self-conformal measures. This paper studies the validity and non-validity of the mixed multifractal formalism of other kinds of measures, called irregular/homogeneous Moran measures.

In this work, we are interested in characterizing typical (generic) dimensional properties of invariant measures associated with the full-shift system, $T$, in a product space whose alphabet is a perfect Polish metric space (thus, uncountable). More specifically, we show that the set of invariant measures with upper Hausdorff dimension equal to zero and lower packing dimension equal to infinity is a dense $G_{\delta }$ subset of ${ℳ}(T)$, the space of $T$-invariant measures endowed with the weak topology. We also show that the set of invariant measures with upper rate of recurrence equal to infinity and lower rate of recurrence equal to zero is a $G_{\delta }$ subset of ${ℳ}(T)$. Furthermore, we show that the set of invariant measures with upper quantitative waiting time indicator equal to infinity and lower quantitative waiting time indicator equal to zero is residual in ${ℳ}(T)$.