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Let $F=f^{\left(i\right)}:1\le i\le N$ be a family of Hölder continuous functions and let ${\phi }_i:1\le i\le N$ be a conformal iterated function system. Lindsay and Mauldin’s paper [Nonlinearity 15 (2002)] left an open question whether the lower quantization coefficient for the F-conformal measure on a conformal iterated funcion system satisfying the open set condition is positive. This question was positively answered by Zhu. The goal of this paper is to present a different proof of this result.

The quantization dimension function for the image measure of a shift-invariant ergodic measure with bounded distortion on a self-conformal set is determined, and its relationship to the temperature function of the thermodynamic formalism arising in multifractal analysis is established.

We consider an inhomogeneous measure μ with the inhomogeneous part a self-similar measure ν, and show that for a given r ∈ (0,∞) the lower and the upper quantization dimensions of order r of μ are bounded below by the quantization dimension $D_r\left(\nu \right)$ of ν and bounded above by a unique number ${\kappa }_r\in (0,\infty )$, related to the temperature function of the thermodynamic formalism that arises in the multifractal analysis of μ.

Two invertible dynamical systems (X,,μ,T) and (Y,,ν,S), where X and Y are Polish spaces and Borel probability spaces and T, S are measure preserving homeomorphisms of X and Y, are said to be finitarily orbit equivalent if there exists an invertible measure preserving mapping ϕ from a subset X₀ of X of measure one onto a subset Y₀ of Y of full measure such that (1) ${\varphi |}_{X₀}$ is continuous in the relative topology on X₀ and ${\varphi }^{-1}{|}_{Y₀}$ is continuous in the relative topology on Y₀, (2) $\varphi \left(Or{b}_T\left(x\right)\right)=Or{b}_S\left(\varphi \left(x\right)\right)$ for μ-a.e. x ∈ X. (X,,μ,T) and...