The quantization for self-conformal measures with respect to the geometric mean error

Abstract

Let µ be a self-conformal measure on R d associated with a family of contractive conformal mappings {f i } N i=1 and a probability vectori=1 satisfies the strong separation condition, we determine the quantization dimension D(µ) with respect to the geometric mean error and show that D(µ) coincides with the Hausdorff dimension of µ. Various expressions for the Hausdorff dimension dim * H µ of µ are established in terms of cylinder sets for the proof of the main result.

On the quantization for self-affine measures on Bedford–McMullen carpets

“…Let α ∈ C ψj (µ). Using Remark 2.5 and the method in [22,Proposition 3.4], we can find a constant L ∈ N such that l σ (α) ≤ L for all large j and all σ ∈ Λ j . Set L := L + L 1 .…”

Convergence order of the geometric mean errors for Markov-type measures

“…Proof. Since all ν σ , σ ∈ G * , share the properties in (2.6) and (2.4), it suffices to follow the induction in [16,Proposition 3.4] by using (2.2).…”

“…Remark 2.6. For the reader's convenience, let us explain the main idea of the induction in [16] by contradiction: suppose that (2.7) does not hold; we could choose a set β with card(β) ≤ card(α) which is "better" than α.…”

Some Recent Developments in Quantization of Fractal Measures

“…In the following we will focus on the L r -quantization problem with r > 0. For the quantization with respect to the geometric mean error, we refer to [8] for rigorous foundations and [37,33,34,40] for more related results.…”