Quantization dimensions for inhomogeneous bi-Lipschitz Iterated Function Systems
Amit Priyadarshi, Mrinal K. Roychowdhury, Manuj Verma
TL;DR
This paper addresses the quantization dimension of condensation measures $μ$ generated by an inhomogeneous bi-Lipschitz IFS with a condensation measure $ν$. It develops a framework of lower and upper bounds for the order-$r$ quantization dimension, establishing $\\underline{D}_r(μ) \\\ge \\\max\{k_r, \\\underline{D}_r(ν)\\}$ and $\\overline{D}_r(μ) \\\le \\\max\{l_r, d_r\\}$ under strong separation and open set conditions; in the similarity case, the quantization dimension exists and equals $\\max\{k_r, D_r(ν)\\}$, with $k_r$ and $D_r(ν)$ given by the corresponding pressure-like equations. The results extend prior work by allowing bi-Lipschitz, inhomogeneous systems and general ν, linking the dimension to contraction ratios and measure weights via thermodynamic-like relations. These findings enhance understanding of discretization rates for inhomogeneous fractal measures and contribute to the broader theory of quantization in dynamical systems.
Abstract
Let $ν$ 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 $\{S_j: 1\leq j\leq N\}$ be a set of contractive mappings on $\mathbb{R}^d$. Then, a Borel probability measure $μ$ on $\mathbb R^d$ such that $μ=\sum_{j=1}^N p_jμ\circ S_j^{-1}+p_0ν$ is called an inhomogeneous measure, also known as a condensation measure on $\mathbb{R}^d$. For a given $r\in (0, +\infty)$, the quantization dimension of order $r$, if it exists, denoted by $D_r(μ)$, of a Borel probability measure $μ$ on $\mathbb{R}^d$ represents the speed at which the $n$th quantization error of order $r$ approaches to zero as the number of elements $n$ in an optimal set of $n$-means for $μ$ tends to infinity. In this paper, we investigate the quantization dimension for such a condensation measure.