On the quantization for self-affine measures on Bedford-McMullen carpets (Q284773)

Let us fix two positive integers \(m\), \(n\) with \(0\leq m\leq n\) and a set \(G\subset [0, 1, \dots, n-1]\times [0, 1, \dots, m-1]\) containing \(N\geq 2\) points. Put \[ f_{ij}: (x,y) \mapsto (n^{-1}x+n^{-1}i, m^{-1}y +m^{-1}j), \quad (i,j)\in G. \] The Bedford-McMullen carpet is a unique non-empty compact set \(E\) satisfying the equality \(E=\bigcup\limits_{(i,j)\in G}f_{ij}(E)\). For a fixed probability vector \((p_{ij})_{(i,j)\in G}\) with positive components, there exists a unique Borel probability measure \(\mu\) on \({\mathbb R}^2\) satisfying \(\mu =\sum\limits_{(i,j)\in G}p_{ij}\mu\circ f^{-1}_{ij}.\) It is a self-affine measure with support \(E\). The authors evaluate the quantization dimension of this measure and obtain certain other results on it.