Related papers: Dimensions of compact invariant sets of some expan…
We show the existence of a bounded Borel measurable saturated compensation function for a factor map between subshifts. As an application, we find the Hausdorff dimension and measures of full Hausdorff dimension for a compact invariant set…
The measure of the full dimension for a general Sierpi\'{n}ski carpet is studied. In the first part of this study, we give a criterion for the measure of the full Hausdorff dimension of a Sierpi\'{n}ski carpet. Meanwhile, it is the…
We determine the Hausdorff, packing and box-counting dimension of a family of self-affine sets generalizing Bara\'nski carpets. More specifically, we fix a Bara\'nski system and allow both vertical and horizontal random translations, while…
Let $m_1 \geq m_2 \geq 2$ be integers. We consider subsets of the product symbolic sequence space $(\{0,\cdots,m_1-1\} \times \{0,\cdots,m_2-1\})^{\mathbb{N}^*}$ that are invariant under the action of the semigroup of multiplicative…
We study the orthogonal projections of a large class of self-affine carpets, which contains the carpets of Bedford and McMullen as special cases. Our main result is that if $\Lambda$ is such a carpet, and certain natural irrationality…
We consider the dimensions of a family of self-affine sets related to the Bedford-McMullen carpets. In particular, we fix a Bedford-McMullen system and then randomise the translation vectors with the stipulation that the column structure is…
For a map $T \colon [0,1] \to [0,1]$ with an invariant measure $\mu$, we study, for a $\mu$-typical $x$, the set of points $y$ such that the inequality $|T^n x - y| < r_n$ is satisfied for infinitely many $n$. We give a formula for the…
The Hausdorff dimension of general Sierpinski carpets, [4] and [20], and the generalization on Lalley-Gatzouras carpets, [10], are today well known results, the formulas being obtain via the variational principle for the dimension. We call…
One of the key challenges in the dimension theory of smooth dynamical systems is in establishing whether or not the Hausdorff, lower and upper box dimensions coincide for invariant sets. For sets invariant under conformal dynamics, these…
In this work we are interested in the self--affine fractals studied by Gatzouras and Lalley and by the author which generalize the famous general Sierpinski carpets studied by Bedford and McMullen. We give a formula for the Hausdorff…
We give a subclass L of Non-linear Lalley-Gatzouras carpets and an open set U in L such that any carpet in U has a unique ergodic measure of full dimension. In particular, any Lalley-Gatzouras carpet which is close to a non-trivial general…
In the present paper we investigate the properties of the Hausdorff mapping $\mathcal{H}$, which takes each compact metric space to the space of its nonempty closed subspaces. It is shown that this mapping is nonexpanding (Lipschitz mapping…
We study the fine scaling properties of planar self-affine carpets. For Gatzouras--Lalley carpets, we give a precise formula for maximal Hausdorff dimension of a tangent in terms of the Hausdorff dimension of the projection and the Assouad…
Mean Hausdorff dimension is a dynamical version of Hausdorff dimension. It provides a way to dynamicalize geometric measure theory. We pick up the following three classical results of fractal geometry. (1) The calculation of Hausdorff…
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 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.
For certain families of complex maps, we give a formula for the Hausdorff dimension of the equilibrium measure. In particular, given an endomorphism $f$ of $\mathbb C\mathbb P^k$ of algebraic degree $d \ge2$, and given the equilibrium…
The notions of shyness and prevalence generalize the property of being zero and full Haar measure to arbitrary (not necessarily locally compact) Polish groups. The main goal of the paper is to answer the following question: What can we say…
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…
Follow-up comment by the author: Theorem 2.2 in this paper is a special case of Theorems 1.1 and 4.1 in the article "Weighted thermodynamic formalism on subshifts and applications", Asian J. Math. 16 (2012), by J. Barral and D. J. Feng. In…