Related papers: Hausdorff Dimension of Sets of Generic Points for …
In this paper, we study the Hausdorff dimension of self-similar measures and sets on the real line, where the generating iterated function system consists of some maps that share the same fixed point. In particular, we will show that out of…
In this short note, we show that, in any given metric space, every Lipschitz open-map image of every subset of a given metric space whose boundary is Hausdorff-null is Hausdorff-measurable with respect to the same dimension. The main…
We review the motivation and fundamental properties of the Hausdorff dimension of metric spaces and illustrate this with a number of examples, some of which are expected and well-known. We also give examples where the Hausdorff dimension…
We study the multifractal analysis for smooth dynamical systems in dimension one. It is characterized the Hausdorff dimension of the level set obtained from the Birkhoff averages of a continuous function by the local dimensions of…
In this paper, we consider non-normal numbers occurring in dynamical systems fulfilling the specification property. It has been shown that in this case the set of non-normal numbers has measure zero. In the present papers we show that a…
In this paper, we consider spectral properties of Riesz product measures supported on homogeneous Cantor sets and we show the existence of spectral measures with arbitrary Hausdorff dimensions, including non-atomic zero-dimensional spectral…
We study the set of irregular points for topologically mixing subshifts of finite type. It is well known that despite the irregular set having zero measure for every invariant measure, it has full topological entropy and full Hausdorff…
We are going to introduce a new algebraic, analytic structure that is a kind of generalization of the Hausdorff dimension and measure. We give many examples and study the basic properties and relations of such systems.
For all $n \geq 2$, we construct a metric space $(X,d)$ and a quasisymmetric mapping $f\colon [0,1]^n \rightarrow X$ with the property that $f^{-1}$ is not absolutely continuous with respect to the Hausdorff $n$-measure on $X$. That is,…
We give a complete characterization of the size of Borel sets that are mid-point convex but not (essentially) convex, in terms of their Hausdorff dimensions and Hausdorff measures.
In this paper, we introduce the notion of asymptotic self-similar sets on general doubling metric spaces by extending the notion of self-similar sets, and determine their Hausdorff dimensions, which gives an extension of Balogh and Rohner…
We give explicit bounds for the Hausdorff dimension of the unique invariant measure of $C^3$ multicritical circle maps without periodic points. These bounds depend only on the arithmetic properties of the rotation number.
If $(X,d)$ is a metric space then the map $f\colon X\to X$ is defined to be a weak contraction if $d(f(x),f(y))<d(x,y)$ for all $x,y\in X$, $x\neq y$. We determine the simplest non-closed sets $X\subseteq \mathbb{R}^n$ in the sense of…
We prove bounds for the almost sure value of the Hausdorff dimension of the limsup set of a sequence of balls in $\mathbf{R}^d$ whose centres are independent, identically distributed random variables. The formulas obtained involve the rate…
In this paper we study the dimension of a family of sets arising in open dynamics. We use exponential mixing results for diagonalizable flows in compact homogeneous spaces $X$ to show that the Hausdorff dimension of set of points that lie…
We study the Hausdorff dimension of Gibbs measures with infinite entropy with respect to maps of the interval with countably many branches. We show that under simple conditions, such measures are symbolic-exact dimensional, and provide an…
The set of badly approximable $m \times n $ matrices is known to have Hausdorff dimension $mn $. Each such matrix comes with its own approximation constant $c$, and one can ask for the dimension of the set of badly approximable matrices…
Basic properties of Hausdorff content, dimension, and measure of subsets of metric spaces are discussed, especially in connection with Lipschitz mappings and topological dimension.
The Hausdorff dimension of the set of points that are covered infinitely many times by a sequence of randomly distributed balls in the unit cube can be expressed in terms of the sizes of the balls. This note presents a new proof of the…
A fundamental problem in the dimension theory of self-affine sets is the construction of high-dimensional measures which yield sharp lower bounds for the Hausdorff dimension of the set. A natural strategy for the construction of such…