Related papers: The quantization for in-homogeneous self-similar m…

We study the quantization for a class of in-homogeneous self-similar measures $\mu$ supported on self-similar sets. Assuming the open set condition for the corresponding iterated function system, we prove the existence of the quantization…

Let $(f_i)_{i=1}^N$ be a family of contractive similitudes on $\mathbb{R}^q$ satisfying the open set condition. Let $(p_i)_{i=0}^N$ be a probability vector with $p_i>0$ for all $i=0,1,\ldots,N$. We study the asymptotic geometric mean errors…

We further study the asymptotics of quantization errors for two classes of in-homogeneous self-similar measures $\mu$. We give a new sufficient condition for the upper quantization coefficient for $\mu$ to be finite. This, together with our…

Let $P:=\frac 1 3 P\circ S_1^{-1}+\frac 13 P\circ S_2^{-1}+\frac 13\nu$, where $S_1(x)=\frac 15 x$, $S_2(x)=\frac 1 5 x+\frac 45$ for all $x\in \mathbb R$, and $\nu$ be a Borel probability measure on $\mathbb R$ with compact support. Such a…

Let $\{f_i\}_{i=1}^N$ be a set of equi-contractive similitudes on $\mathbb{R}^1$ satisfying the finite-type condition. We study the asymptotic quantization error for self-similar measures $\mu$ associated with $\{f_i\}_{i=1}^N$ and a…

For a given $r \in (0, +\infty)$, the quantization dimension of order $r$, if it exists, denoted by $D_r(\mu)$, represents the rate at which the $n$th quantization error of order $r$ approaches to zero as the number of elements $n$ in an…

Let $\nu$ be a Borel probability measure on a $d$-dimensional Euclidean space $\mathbb{R}^d$, $d\geq 1$, with a compact support, and let $(p_0, p_1, p_2, \ldots, p_N)$ be a probability vector with $p_j>0$ for $0\leq j\leq N$. Let $\{S_j:…

For a given $r\in (0, +\infty)$, the quantization dimension of order $r$, if it exists, denoted by $D_r(\mu)$, of a Borel probability measure $\mu$ on ${\mathbb R}^d$ represents the speed how fast the $n$th quantization error of order $r$…

We investigate quantization coefficients for self-similar probability measures \mu on limit sets which are generated by systems S of infinitely many contractive similarities and by probabilistic vectors. The theory of quantization…

In this paper, we investigate the quantization dimension of self-similar measures, particularly when the IFS does not satisfy the separation condition, but the sub-IFS at some level satisfies the separation condition. Further, we study the…

We show that if $X$ is a uniformly perfect complete metric space satisfying the finite doubling property, then there exists a fully supported measure with lower regularity dimension as close to the lower dimension of $X$ as we wish.…

We consider condensation measures of the form $P:=\frac 13 P\circ S_1^{-1}+ \frac 13 P\circ S_2^{-1}+ \frac 13 \nu $ associated with the system $(\mathcal{S}, (\frac 13, \frac 13, \frac 13), \nu) , $ where $\mathcal{S}=\{S_i\}_{i=1}^2 $ are…

Let $\mu$ be the self-similar measure supported on the self-similar set $K$ with open set condition. In this article, we discuss the packing dimension of the set $\{x\in K: A(\frac{\log\mu(B(x,r))}{\log r})=I\}$ for $I\subseteq\mathbb{R}$,…

Let $\mu$ be a Borel probability measure associated with an iterated function system consisting of a countably infinite number of contracting similarities and an infinite probability vector. In this paper, we study the quantization…

We consider extensions of quasiconformal maps and the uniformization theorem to the setting of metric spaces $X$ homeomorphic to $\mathbb R^2$. Given a measure $\mu$ on such a space, we introduce $\mu$-quasiconformal maps $f:X \to \mathbb…

Let K be a self-similar or self-affine set in R^d, let \mu be a self-similar or self-affine measure on it, and let G be the group of affine maps, similitudes, isometries or translations of R^d. Under various assumptions (such as separation…

We study the dimension of self-similar measures associated to a homogeneous iterated function system of three contracting similarities on $\bf R$ and other more general IFS's. We extend some of the theory recently developed for Bernoulli…

The primary objective of the present paper is to develop the theory of quantization dimension of an invariant measure associated with an iterated function system consisting of finite number of contractive infinitesimal similitudes in a…

Quantization dimension has been computed for many invariant measures of dynamically defined fractals having well separated cylinders, that is, in the cases when the so-called Open Set Condition (OSC) holds. To attack the same problem in…

We provide a complete picture of the upper quantization dimension in terms of the R\'enyi dimension by proving that the upper quantization dimension $\bar{D}_{r}(\nu)$ of order $r>0$ for an arbitrary compactly supported Borel probability…