Quantization dimension for condensation systems II: The geometric mean error

Abstract

1 be a family of similitudes on R 1 satisfying the strong separation condition and ν the self-similar measure associated with {f i } N 1 and a probability vector (t 1 , . . . , t N ). Let μ be the attracting measure of a condensation system associated with ν, {f i } N 1 and a probability vector (p 0 , p 1 , . . . , p N ). We establish a relationship between the quantization dimension of μ and its mass distribution on cylinder sets.

Some Recent Developments in Quantization of Fractal Measures

“…After these three steps, for sufficiently "nice" measures, we may additionally assume that ϕ n ≤ ϕ n+1 ≤ Cϕ n for some constant C > 1 (cf. [42,43,44]). To determine the dimension it is then enough to estimate the growth rate of ϕ n .…”

Quantization dimension and stability for infinite self-similar measures with respect to geometric mean error

“…Recently, assuming the same separation property, the quantization dimension D(µ) of a recurrent selfsimilar measure µ with respect to the geometric mean error was determined, and it was proved that D(µ) coincides with the Hausdorff dimension dim * H (µ) of µ (see [RS1]). For other work in this direction one could also see [RS2,Z2,Z3]. To determine the quantization dimension of a fractal probability measure in each of the aforementioned case the number of mappings considered in the iterated function system was finite.…”