Quantization dimension for a generalized inhomogeneous bi-Lipschitz iterated function system

For a given r∈(0,+∞), the quantization dimension of order r, if it exists, denoted by Dr(μ), of a Borel probability measure μ on Rd represents the speed how fast the nth 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. If Dr(μ) does not exists, we call D̲r(μ) and D¯r(μ), the lower and upper quantization dimensions of μ of order r. In this paper, we estimate the quantization dimension of condensation measures associated with condensation systems ({fi}i=1N,(pi)i=0N,ν), where the mappings fi are bi-Lipschitz and the measure ν is an image measure of an ergodic measure with bounded distortion supported on a conformal set. In addition, we determine the optimal quantization for an infinite discrete distribution, and give an example which shows that the quantization dimension of a Borel probability measure can be positive with zero quantization coefficient.