Quantization dimension for condensation systems

Abstract

Let µ be the attracting measure of a condensation system associated with a selfsimilar measure ν. We determine the upper and lower quantization dimension of µ under the strong separation condition.

Quantization dimension for condensation systems II: The geometric mean error

“…On the other hand, by (2) and an argument analogous to [17,Lemma 6], one can easily check that the upper and lower quantization dimension are attained at the subsequence (φ(n)). Thus, using the above inequality, we deduce…”

“…[11]) has studied the thin dimension of the attracting measures. The quantization dimension of μ of order r ∈ (0, ∞) has been determined in [17]. As our main result in the present paper, we will establish a relationship between the quantization dimension of μ of order zero and its mass distribution on cylinder sets (cf.…”

The quantization for in-homogeneous self-similar measures with in-homogeneous open set condition

“…We also call the points in such a set α n-optimal points for µ. With the above preparations, we are able to apply the technique in [17] to estimate the number of φ k,r -optimal points lying in the pairwise disjoint neighborhoods of sets in F k,r . Let δ be as defined in (3.8).…”

Quantization dimension of probability measures supported on Cantor-like sets

“…(15) By (14), for σ ∈ Ξ k , h(σ ) = (n 1 · · · n k ) −1 (c 1 · · · c k ) r . Thus by (15), d n,r = s k+1,r for the above k. By Theorem 1, this implies D r (μ) ξ , D r (μ) ξ .…”