Related papers: Hausdorff dimension of particularly non-normal num…
In the present paper we want to focus on this dichotomy of the non-normal numbers -- on the one hand they are a set of measure zero and on the other hand they are residual -- for dynamical system fulfilling the specification property. These…
We show that the set of numbers that are $Q$-distribution normal but not simply $Q$-ratio normal has full Hausdorff dimension. It is further shown under some conditions that countable intersections of sets of this form still have full…
Following in the footsteps of P. Erd\H{o}s and A. R\'enyi we compute the Hausdorff dimension of sets of numbers whose digits with respect to their $Q$-Cantor series expansions satisfy various statistical properties. In particular, we…
We compute the exact Hausdorff and packing measure of sets of real numbers whose digital expansions in a given base are missing the digits beyond a given threshold.
We systematically investigate examples of non-hyperbolic dynamical systems having irregular sets of full topological entropy and full Hausdorff dimension. The examples include some partially hyperbolic systems and geometric Lorenz flows. We…
We establish sharp bounds for the Hausdorff dimension of sets of irrational numbers in $(0,1)$ whose digits in the $N$-expansion are either uniformly bounded or tend to infinity. For sets with digits bounded by an integer $M \ge N$, we…
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 various measure theories using the classical approach and then compute the Hausdorff dimension of some simple objects and self-similar fractals. We then develop a nonstandard approach to these measure theories and examine the…
Non-autonomous iterated function systems are a generalization of iterated function systems. If the contractions in the system are conformal mappings, it is called a non-autonomous conformal iterated function system, and its attractor is…
The classical Hausdorff dimension of finite or countable sets is zero. We define an analog for finite sets, called finite Hausdorff dimension which is non-trivial. It turns out that a finite bound for the finite Hausdorff dimension…
We consider one dimensional maps with several neutral fixed points that do not admit any physical measures. We show that there is simplex of measures so that every measure in this simplex has a basin which has full Hausdorff dimension.
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…
The irrationality exponent of a real number measures how well that number can be approximated by rationals. Real numbers with irrationality exponent strictly greater than $2$ are transcendental numbers, and form a set with rich fractal…
Despite the fact that almost all real numbers are absolutely normal---that is, the digits in their expansions to any base occur in all possible configurations with the expected frequency---not one specific example of an absolutely normal…
In this paper, we study the metrical theory of the growth rate of digits in L\"{u}roth expansions. More precisely, for $ x\in \left( 0,1 \right] $, let $ \left[ d_1\left( x \right) ,d_2\left( x \right) ,\cdots \right] $ denote the…
The digits of the Pierce expansion satisfy the law of large numbers. It is known that the Hausdorff dimension of the set of exceptions to the law of large numbers is 1. We provide an elementary proof of this fact by adapting Jun Wu's…
We introduce {\it (W')-specification} in terms of language decompositions of subshifts, and show that any recurrence set of a subshift with this property has full Hausdorff dimension. Our main result applies to a wide class of subshifts…
The number of distinct symbols appearing in digit expansions generated by full-branch affine countable iterated function systems is studied whose branch weights are regularly varying. The Hausdorff dimensions of the exceptional sets in…
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 paper I explore a nonstandard formulation of Hausdorff dimension. By considering an adapted form of the counting measure formulation of Lebesgue measure, I prove a nonstandard version of Frostman's lemma and show that Hausdorff…