De-Jun Feng

• Chinese University of Hong Kong

Let $T_1,\ldots, T_m$ be a family of $d\times d$ invertible real matrices with $\|T_i\|<1/2$ for $1\leq i\leq m$. For ${\bf a}=(a_1,\ldots, a_m)\in {\Bbb R}^{md}$, let $\pi^{\bf a}\colon \Sigma=\{1,\ldots, m\}^{\Bbb N}\to {\Bbb R}^d$ denote the coding map associated with the affine IFS $\{T_ix+a_i\}_{i=1}^m$, and let $K^{\bf a}$ denote the attracto...

The Erdős similarity conjecture asserted that an infinite set of real numbers cannot be affinely embedded into every measurable set of positive Lebesgue measure. The problem is still open, in particular for all fast decaying sequences. In this paper, we relax the problem to the bi-Lipschitz embedding and obtain some sharp criteria about the bi-Lips...

Let $T_1,\ldots, T_m$ be a family of $d\times d$ invertible real matrices with $\|T_i\| <1/2$ for $1\leq i\leq m$. We provide some sufficient conditions on these matrices such that the self-affine set generated by the iterated function system $\{T_ix+a_i\}_{i=1}^m$ on $\Bbb R^d$ has non-empty interior for almost all $(a_1,\ldots, a_m)\in \Bbb R^{md...

Let $T_1,\ldots, T_m$ be a family of $d\times d$ invertible real matrices with $\|T_i\|<1/2$ for $1\leq i\leq m$. For ${\bf a}=(a_1,\ldots, a_m)\in \Bbb R^{md}$, let $\pi^{{\bf a}}:\; \Sigma=\{1,\ldots, m\}^{\Bbb N}\to \Bbb R^d$ denote the coding map associated with the affine IFS $\{T_ix+a_i\}_{i=1}^m$. We show that for every Borel probability mea...

Let $E$ be the attractor of an iterated function system $\{\phi_i(x)=\rho R_ix+a_i\}_{i=1}^N$ on $\Bbb R^d$, where $0<\rho<1$, $a_i\in \Bbb R^d$ and $R_i$ are orthogonal transformations on $\Bbb R^d$. Suppose that $\{\phi_i\}_{i=1}^N$ satisfies the open set condition, but not the strong separation condition. We show that $E$ can not be generated by...

This is the first paper in a two-part series containing some results on dimension estimates for $C^1$ iterated function systems and repellers. In this part, we prove that the upper box-counting dimension of the attractor of any $C^1$ iterated function system (IFS) on ${\Bbb R}^d$ is bounded above by its singularity dimension, and the upper packing...

This is the second part of our study on the dimension theory of $C^1$ iterated function systems (IFSs) and repellers on $\mathbb {R}^d$ . In the first part [D.-J. Feng and K. Simon. Dimension estimates for $C^1$ iterated function systems and repellers. Part I. Preprint , 2020, arXiv:2007.15320], we proved that the upper box-counting dimension of th...

This is the second part of our study of the dimension theory of $C^1$ iterated function systems (IFSs) and repellers on ${\Bbb R}^d$. In the first part we proved that the upper box-counting dimension of the attractor of any $C^1$ IFS on ${\Bbb R}^d$ is bounded above by its singularity dimension, and the upper packing dimension of any ergodic invari...

Let μ be a self-similar measure generated by an IFS Φ={ϕi}i=1ℓ of similarities on Rd (d≥1). When Φ is dimensional regular (see Definition 1.1), we give an explicit formula for the Lq-spectrum τμ(q) of μ over [0, 1], and show that τμ is differentiable over (0, 1] and the multifractal formalism holds for μ at any α∈[τμ′(1),τμ′(0+)]. We also verify th...

In this paper, we provide an algorithm to estimate from below the dimension of self-similar measures with overlaps. As an application, we show that for any $ \beta\in(1,2) $, the dimension of the Bernoulli convolution $ \mu_\beta $ satisfies \[ \dim (\mu_\beta) \geq 0.9804085,\] which improves a previous uniform lower bound $0.82$ obtained by Hare...

We compute the Hausdorff dimension of any random statistically self-affine Sierpinski sponge K⊂Rk (k≥2) obtained by using some percolation process in [0,1]k. To do so, we first exhibit a Ledrappier-Young type formula for the Hausdorff dimensions of statistically self-affine measures supported on K. This formula presents a new feature compared to it...

This is the first article in a two-part series containing some results on dimension estimates for $C^1$ iterated function systems and repellers. In this part, we prove that the upper box-counting dimension of the attractor of any $C^1$ iterated function system (IFS) on ${\Bbb R}^d$ is bounded above by its singularity dimension, and the upper packin...

A compact set $E\subset {\Bbb R}^d$ is said to be arithmetically thick if there exists a positive integer $n$ so that the $n$-fold arithmetic sum of $E$ has non-empty interior. We prove the arithmetic thickness of $E$, if $E$ is uniformly non-flat, in the sense that there exists $\epsilon_0>0$ such that for $x\in E$ and $0<r\leq {\rm diam}(E)$, $E\...

Let $\mu$ be a self-similar measure generated by an IFS $\Phi=\{\phi_i\}_{i=1}^\ell$ of similarities on $\mathbb R^d$ ($d\ge 1$). When $\Phi$ is dimensional regular (see Definition~1.1), we give an explicit formula for the $L^q$-spectrum $\tau_\mu(q)$ of $\mu$ over $[0,1]$, and show that $\tau_\mu$ is differentiable over $(0,1]$ and the multifracta...

We compute the Hausdorff dimension of any random statistically self-affine Sierpinski sponge K $\subset$ R k (k $\ge$ 2) obtained by using some percolation process in [0, 1] k. To do so, we first exhibit a Ledrappier-Young type formula for the Hausdorff dimensions of statistically self-similar measures supported on K. This formula presents a new fe...

Let $\mathbf{M}=(M_{1},\ldots ,M_{k})$ be a tuple of real $d\times d$ matrices. Under certain irreducibility assumptions, we give checkable criteria for deciding whether $\mathbf{M}$ possesses the following property: there exist two constants $\unicode[STIX]{x1D706}\in \mathbb{R}$ and $C>0$ such that for any $n\in \mathbb{N}$ and any $i_{1},\ldots...

Let E,F ⊂ ℝ d be two self-similar sets, and suppose that F can be affinely embedded into E. Under the assumption that E is dust-like and has a small Hausdorff dimension, we prove the logarithmic commensurability between the contraction ratios of E and F. This gives a partial affirmative answer to Conjecture 1.2 in [9]. The proof is based on our stu...

We give an expression for the Garsia entropy of Bernoulli convolutions in terms of products of matrices. This gives an explicit rate of convergence of the Garsia entropy and shows that one can calculate the Hausdorff dimension of the Bernoulli convolution $\nu_\beta$ to arbitrary given accuracy whenever $\beta$ is algebraic. In particular, if the G...

We give an expression for the Garsia entropy of Bernoulli convolutions in terms of products of matrices. This gives an explicit rate of convergence of the Garsia entropy and shows that one can calculate the Hausdorff dimension of the Bernoulli convolution $\nu_\beta$ to arbitrary given accuracy whenever $\beta$ is algebraic. In particular, if the G...

Let ${\bf M}=(M_1,\ldots, M_k)$ be a tuple of real $d\times d$ matrices. Under certain irreducibility assumptions, we give checkable criteria for deciding whether ${\bf M}$ possesses the following property: there exist two constants $\lambda\in {\Bbb R}$ and $C>0$ such that for any $n\in {\Bbb N}$ and any $i_1, \ldots, i_n \in \{1,\ldots, k\}$, eit...

We characterize analytic curves that contain non-trivial self-affine sets. We also prove that compact algebraic surfaces cannot contain non-trivial self-affine sets.

We characterize analytic curves that contain non-trivial self-affine sets. We also prove that compact algebraic surfaces cannot contain non-trivial self-affine sets.

Let $E, F\subset {\Bbb R}^d$ be two self-similar sets, and suppose that $F$ can be affinely embedded into $E$. Under the assumption that $E$ is dust-like and has a small Hausdorff dimension, we prove the logarithmic commensurability between the contraction ratios of $E$ and $F$. This gives a partial affirmative answer to Conjecture 1.2 in \cite{FHR...

Let $\mu$ be a planar Mandelbrot measure and $\pi_*\mu$ its orthogonal projection on one of the main axes. We study the thermodynamic and geometric properties of $\pi_*\mu$. We first show that $\pi_*\mu$ is exactly dimensional, with $\dim(\pi_*\mu)=\min(\dim(\mu),\dim(\nu))$, where~$\nu$ is the Bernoulli product measure obtained as the expectation...

Let $\mu$ be a planar Mandelbrot measure and $\pi_*\mu$ its orthogonal projection on one of the main axes. We study the thermodynamic and geometric properties of $\pi_*\mu$. We first show that $\pi_*\mu$ is exactly dimensional, with $\dim(\pi_*\mu)=\min(\dim(\mu),\dim(\nu))$, where~$\nu$ is the Bernoulli product measure obtained as the expectation...

Let ${\pmb M}$, ${\pmb N}$ and ${\pmb K}$ be $d$-dimensional Riemann manifolds. Assume that ${\bf A}:=(A_n)_{n\in{\Bbb N}}$ is a sequence of Lebesgue measurable subsets of ${\pmb M}$ satisfying a necessary density condition and ${\bf x}:=(x_n)_{n\in {\Bbb N}}$ is a sequence of independent random variables which are distributed on ${\pmb K}$ accordi...

Let ${\pmb M}$, ${\pmb N}$ and ${\pmb K}$ be $d$-dimensional Riemann manifolds. Assume that ${\bf A}:=(A_n)_{n\in{\Bbb N}}$ is a sequence of Lebesgue measurable subsets of ${\pmb M}$ satisfying a necessary density condition and ${\bf x}:=(x_n)_{n\in {\Bbb N}}$ is a sequence of independent random variables which are distributed on ${\pmb K}$ accordi...

A finite subset \(\mathcal{D}\) of \(\mathbb{Z}^{2}\) is called a tile of \(\mathbb{Z}^{2}\), if \(\mathbb{Z}^{2}\) can be tiled by disjoint translates of \(\mathcal{D}\). In this note, we give a simple characterization of tiles of \(\mathbb{Z}^{2}\) with cardinality 4.

Let $\pi:X\to Y$ be a factor map, where $(X,T)$ and $(Y,S)$ are topological dynamical systems. Let ${\bf a}=(a_1,a_2)\in {\Bbb R}^2$ with $a_1>0$ and $a_2\geq 0$, and $f\in C(X)$. The ${\bf a}$-weighted topological pressure of $f$, denoted by $P^{\bf a}(X, f)$, is defined by resembling the Hausdorff dimension of subsets of self-affine carpets. We p...

In this paper, we study the following question raised by Mattila in 1998: what are the self-similar subsets of the middle-third Cantor set $\C$? We give criteria for a complete classification of all such subsets. We show that for any self-similar subset $\F$ of $\C$ containing more than one point every linear generating IFS of $\F$ must consist of...

Soit E et F deux ensembles auto-similaires dans RdRd. Sous des hypothèses raisonnables, on montre qu'il existe un plongement C1C1 de F dans E si et seulement s'il existe un tel plongement affine ; de plus, s'il n'existe pas de plongement affine de F dans E , alors pour tout difféomorphisme C1C1 de RdRd la dimension de Hausdorff de l'intersection E∩...

This volume collects thirteen expository or survey articles on topics including Fractal Geometry, Analysis of Fractals, Multifractal Analysis, Ergodic Theory and Dynamical Systems, Probability and Stochastic Analysis, written by the leading experts in their respective fields. The articles are based on papers presented at the International Conferenc...

The pressure function $P(A, s)$ plays a fundamental role in the calculation of the dimension of "typical" self-affine sets, where $A=(A_1,\ldots, A_k)$ is the family of linear mappings in the corresponding generating iterated function system. We prove that this function depends continuously on $A$. As a consequence, we show that the dimension of "t...

We examine the interplay between the thermodynamic formalism and the multifractal formalism on the so-called self-affine symbolic spaces, under the specification property assumption. We investigate the properties of a weighted variational principle to derive a new result concerning the approximation of any invariant probability measure $\mu$ by seq...

We study the decay of $\mu(B(x,r)\cap C)/\mu(B(x,r))$ as $r\downarrow 0$ for different kinds of measures $\mu$ on $\R^n$ and various cones $C$ around $x$. As an application, we provide sufficient conditions implying that the local dimensions can be calculated via cones almost everywhere.

We consider the multifractal structure of the Bernoulli convolution $\nu_{\lambda}$, where $\lambda^{-1}$ is a Salem number in $(1,2)$. Let $\tau(q)$ denote the $L^q$ spectrum of $\nu_\lambda$. We show that if $\alpha \in [\tau'(+\infty), \tau'(0+)]$, then the level set $$E(\alpha):={x\in \R:\; \lim_{r\to 0}\frac{\log \nu_\lambda([x-r, x+r])}{\log...

We conduct the multifractal analysis of self-affine measures for "almost all" family of affine maps. Besides partially extending Falconer's formula of $L^q$-spectrum outside the range $1< q\leq 2$, the multifractal formalism is also partially verified.

For a real number $q>1$ and a positive integer $m$, let $Y_m(q):={\sum_{i=0}^n\epsilon_i q^i:\; \epsilon_i\in \{0, \pm 1,..., \pm m\}, n=0, 1,...}.$ In this paper, we show that $Y_m(q)$ is dense in $\R$ if and only if $q<m+1$ and $q$ is not a Pisot number. This completes several previous results and answers an open question raised by Erd\"{o}s, Jo\...

In this note, we show that on certain Gatzouras-Lalley carpet, there exist more than one ergodic measures with full Hausdorff dimension. This gives a negative answer to a conjecture of Gatzouras and Peres.

Let π:X→Y be a factor map, where (X,σX) and (Y,σY) are subshifts over finite alphabets. Assume that X satisfies weak specification. Let a=(a1,a2)∈R2 with a1>0 and a2⩾0. Let f be a continuous function on X with sufficient regularity (Hölder continuity, for instance). We show that there is a unique shift invariant measure μ on X that maximizes . In p...

Let $(X,T)$ be a topological dynamical system. We define the measure-theoretical lower and upper entropies $\underline{h}_\mu(T)$, $\bar{h}_\mu(T)$ for any $\mu\in M(X)$, where $M(X)$ denotes the collection of all Borel probability measures on $X$. For any non-empty compact subset $K$ of $X$, we show that $$\htop^B(T, K)= \sup \{\underline{h}_\mu(T...

Let $\{M_i\}_{i=1}^\ell$ be a non-trivial family of $d\times d$ complex matrices, in the sense that for any $n\in \N$, there exists $i_1... i_n\in \{1,..., \ell\}^n$ such that $M_{i_1}... M_{i_n}\neq {\bf 0}$. Let $P \colon (0,\infty)\to \R$ be the pressure function of $\{M_i\}_{i=1}^\ell$. We show that for each $q>0$, there are at most $d$ ergodic...

For general asymptotically sub-additive potentials (resp. asymptotically additive potentials) on general topological dynamical systems, we establish some variational relations between the topological entropy of the level sets of Lyapunov exponents, measure-theoretic entropies and topological pressures in this general situation. Most of our results...

A generating IFS of a Cantor set F is an IFS whose attractor is F . For a given Cantor set such as the middle-3rd Cantor set we consider the set of its generating IFSs. We examine the existence of a minimal generating IFS, i.e. every other generating IFS of F is an iterating of that IFS. We also study the structures of the semi-group of homogeneous...

Let $\{S_i\}_{i=1}^\ell$ be an iterated function system (IFS) on $\R^d$ with attractor $K$. Let $(\Sigma,\sigma)$ denote the one-sided full shift over the alphabet $\{1,..., \ell\}$. We define the projection entropy function $h_\pi$ on the space of invariant measures on $\Sigma$ associated with the coding map $\pi: \Sigma\to K$, and develop some ba...

For any self-similar measure μ on Rd satisfying the weak separation condition, we show that there exists an open ball U0 with μ(U0)>0 such that the distribution of μ, restricted on U0, is controlled by the products of a family of non-negative matrices, and hence μU0| satisfies a kind of quasi-product property. Furthermore, the multifractal formalis...

Let $\pi:X\to Y$ be a factor map, where $(X,\sigma_X)$ and $(Y,\sigma_Y)$ are subshifts over finite alphabets. Assume that $X$ satisfies weak specification. Let $\ba=(a_1,a_2)\in \R^2$ with $a_1>0$ and $a_2\geq 0$. Let $f$ be a continuous function on $X$ with sufficient regularity (H\"{o}lder continuity, for instance). We show that there is a uniqu...

Let $(X,T)$ and $(Y,S)$ be two subshifts so that $Y$ is a factor of $X$. For any asymptotically sub-additive potential $\Phi$ on $X$ and $\ba=(a,b)\in\R^2$ with $a>0$, $b\geq 0$, we introduce the notions of $\ba$-weighted topological pressure and $\ba$-weighted equilibrium state of $\Phi$. We setup the weighted variational principle. In the case th...

We continue the study in [15, 18] on the upper Lyapunov exponents for products of matrices. Here we consider general matrices. In general, the variational formula about Lyapunov exponents we obtained in part I does not hold in this setting. In any case, we focus our interest on a special case where the matrix function M(x) takes finite values M 1,...

Let $\beta>1$ and let $m>\be$ be an integer. Each $x\in I_\be:=[0,\frac{m-1}{\beta-1}]$ can be represented in the form \[ x=\sum_{k=1}^\infty \epsilon_k\beta^{-k}, \] where $\epsilon_k\in\{0,1,...,m-1\}$ for all $k$ (a $\beta$-expansion of $x$). It is known that a.e. $x\in I_\beta$ has a continuum of distinct $\beta$-expansions. In this paper we pr...

The paper is devoted to the study of the multifractal structure of disintegrations of Gibbs measures and conditional (random) Birkhoff averages. Our approach is based on the relativized thermodynamic formalism, convex analysis and especially, the delicate constructions of Moran-like subsets of level sets.

Let U λ be the union of two unit intervals with gap λ. We show that U λ is a self-similar set satisfying the open set condition if and only if U λ can tile an interval by finitely many of its affine copies (admitting different dilations). Furthermore, each such λ can be characterized as the spectrum of an irreducible double word which represents a...

Refinable functions and distributions with integer dilations have been studied extensively since the pioneer work of Daubechies on wavelets. However, very little is known about refinable functions and distributions with non-integer dilations, particularly concerning its regularity. In this paper we study the decay of the Fourier transform of refina...

We prove that for any self-conformal measures, without any separation conditions, the multifractal formalism partially holds. The result follows by establishing certain Gibbs properties for self-conformal measures.

A refinable spline is a compactly supported refinable function that is piece-wise polynomial. Refinable splines, such as the well known B-splines, play a key role in computer aided geometric designs. Refinable splines have been studied in several papers, most noticably in [7] for integer dilations and [3] for real dilations. There are general chara...

The topological pressure is defined for sub-additive potentials via separated sets and open covers in general compact dynamical systems. A vari-ational principle for the topological pressure is set up without any additional assumptions. The relations between different approaches in defining the topo-logical pressure are discussed. The result will h...

A refinable spline is a compactly supported refinable function that is piecewise polynomial. Refinable splines, such as the well-known B-splines, play a key role in computer aided geometric design. So far all studies on refinable splines have focused on positive integer dilations and integer translations, and under this setting a rather complete cl...

Finite tight frames are used widely for many applications. An important problem is to construct finite frames with prescribed norm for each vector in the tight frame. In this paper we provide a fast and simple algorithm for such purpose. Our algorithm employs the Householder transformations. For a finite tight frame consisting m vectors in Rn or Cn...

In Part I we showed that the $L^q$-spectrum of the 3-fold convolution of the Cantor measure has a non-differentiable point at a $q_0 < 0$ [LW], therefore the standard multifractal formalism does not hold. In this Part II, we prove a modified multifractal formalism for the measure.

In this paper, we give a systematical study of the local structures and fractal indices of the limited Rademacher functions and Bernoulli convolutions associated with Pisot numbers. For a given Pisot number in the interval (1,2), we construct a finite family of non-negative matrices (maybe non-square), such that the corresponding fractal indices ca...

The Bernoulli convolution νλ measure is shown to be absolutely continuous with L2 density for almost all 1/2 <λ<1, and singular if λ-1 is a Pisot number. It is an open question whether the Pisot type Bernoulli convolutions are the only singular ones. In this paper, we construct a family of non-Pisot type Bernoulli convolutions νλ such that their de...

Let ((A , σ) be a subshift of finite type and let M(x) be a continuous function on A taking values in the set of non-negative matrices. We set up the variational principle between the pressure function, entropy and Lyapunov exponent for M on A . We also present some properties of equilibrium states.

Using the theory of random closed sets, we extend the statistical framework introduced by Schreiber(11) for inference based on set-valued observations from the case of finite sample spaces to compact metric spaces with continuous distributions.

For a given expanding d-fold covering transformation of the one-dimensional torus, the notion of weak Gibbs measure is defined by a natural generalization of the classical Gibbs property. For these measures, we prove that the singularity spectrum and the $L^q$-spectrum form a Legendre transform pair. The main difficulty comes from the possible exis...

Let $\mu$ be the self-similar measure for a linear function system $S_jx=\rho x+b_j$ ($j=1,2,\ldots,m$) on the real line with the probability weight $\{p_j\}_{j=1}^m$. Under the condition that $\{S_j\}_{j=1}^m$ satisfies the finite type condition, the $L^q$-spectrum $\tau(q)$ of $\mu$ is shown to be differentiable on $(0,\infty)$; as an application...

Let (Σ,σ) be a full shift space on an alphabet consisting ofm symbols and letM: Σ→L +(ℝ d , ℝ d ) be a continuous function taking values in the set ofd×d positive matrices. Denote by λ M (x) the upper Lyapunov exponent ofM atx. The set of possible Lyapunov exponents is just an interval. For any possible Lyapunov exponentα, we prove the following va...

Let K be the attractor of a linear iterated function system Sjx = ρjx +bj (j = 1, …, m) on the real line satisfying the open set condition (where the open set is an interval). It is well known that the packing dimension of K is equal to , the unique positive solution y of the equation ; and the –dimensional packing measure (K) is finite and positiv...

Let $\mu$ be a self-similar measure on $\mathbb{R}$ generated by an equicontractive iterated function system. We prove that the Hausdorff dimension of $\mu^{*n}$ tends to $1$ as $n$ tends to infinity, where $\mu^{*n}$ denotes the $n$-fold convolution of $\mu$. Similar results hold for the $L^q$ dimension and the entropy dimension of $\mu^{*n}$.

Let q be a Pisot number and m a positive integer. Consider the increasing sequence of those real numbers y which have at least one representation of the form with some integer n ≥ 0 and coefficients e i εo {0, 1, ..., m}. When m ≥ q - 1, we will determine the structure of the difference sequence {y k+1 - y k) k ≥ 0, that is, it is the image of a se...

For each 0 < s < 1, define c(s) = inf(E) P-s(E)/H-s(E), where P-s, H-s denote respectively the s-dimensional packing measure and Hausdorff measure, and the infimum is taken over all the sets E subset of R with 0 < H-s(E) < infinity. In this paper we give a nontrivial estimation of c(s), namely, 2(s) (1 + v(s))(s) less than or equal to c(s) less tha...

Let J be the repeller of an expanding, C1+δ-conformal topological mixing map g. Let Φ:J→Rd be a continuous function and let α(x)=limn→∞1n∑n−1j=0Φ(gjx) (when the limit exists) be the ergodic limit. It is known that the possible α(x) are just the values ∫Φ dμ for all g-invariant measures μ. For any α in the range of the ergodic limits, we prove the f...

Let $(\Sigma_A, \sigma)$ be a subshift of finite type and let $M(x)$ be a continuous function on $\Sigma_A$ taking values in the set of non-negative matrices. We extend the classical scalar pressure function to this new setting and prove the existence of the Gibbs measure and the differentiability of the pressure function. We are especially interes...

Let (Σ,σ) be the one-sided shift space on m symbols. For any x = (xi)i≥1Σ and positive integer n, define We prove that for each pair of numbers α,β[0,∞] with α≤β, the following recurrent set has Hausdorff dimension one.

Let be the classical middle-third Cantor set and let μ be the Cantor measure. Set s = log 2/log 3. We will determine by an explicit formula for every point x ∈ the upper and lower s-densities Θ*s(μ, x), Θ∗s(μ, x) of the Cantor measure at the point x, in terms of the 3-adic expansion of x. We show that there exists a countable set F ⊂ such that 9(Θ*...

The pressure was studied in a rather abstract theory as an important notion of the thermodynamic formalism. The present paper gives a more concrete account in the case of symbolic spaces, including subshifts of finite type. We relate the pressure of an interaction function to its long-term time averages through the Hausdorff and packing dimensions...

Let K be a compact subset of ℝ n, 0 ≤ s ≤ n. Let P s0, ℘ s denote s-dimensional packing premeasure and measure, respectively. We discuss in this paper the relation between P s0 and ℘ s. We prove: if P s0(K) < ∞, then ℘ s(K) = P s0(K.); and if P s0(K) = ∞, then for any ε > 0, there exists a compact subset F of K such that ℘ s(F) = P s0(F) and ℘ s(F)...

Some properties of the upper and lower box-counting dimensions and their geometrical explanations are presented. There exists an essential difference between the upper and the lower box-counting dimension. In particular, the upper box-counting dimension plays an important role in the dimension theory.

Let (X, T) be a dynamical system. The recurrence of a point x related to a function Φ : X → Rd is described by the limit of ergodic means n−1∑j=0n−1Φ(Tjx. We are interested in estimating the size of the set of points with a prescribed recurrence. The dimensions of such sets are computed in the case of symbolic spaces.

Let (X,T) be a dynamical system. The recurrence of a point x related to a function Φ: X → Rd is described by the limit of ergodic means n-1 ∑n-1j=0 Φ(Tjx). We are interested in estimating the size of the set of points with a prescribed recurrence. The dimensions of such sets are computed in the case of symbolic spaces.

The equivalence between the Hausdorff measure induced by a natural covering net and the Hausdorff measure in usual meaning has been obtained for one-dimensional symmetric Cantor sets. As an application, the Hausdorff dimensions of such sets are determined.