2015/12/30 by Roychowdhury, Mrinal Kanti
#28A80 #60Exx #94A34 #Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.1512.09161
Let P be a Borel probability measure on \mathbb R generated by an infinite system of similarity mappings \Sj : j∈ \mathbb N\ such that P=∑j=1^∞ \frac 12j P∘ Sj-1, where for each j∈ \mathbb N and x∈ \mathbb R, Sj(x)=\frac 13jx+1-\frac 1 3j-1. Then, the support of P is the dyadic Cantor set C generated by the similarity mappings f1, f2 : \mathbb R → \mathbb R such that f1(x)=\frac 13 x and f2(x)=\frac 13 x+\frac 23 for all x∈ \mathbb R. In this paper, using the infinite system of similarity mappings \Sj : j∈ \mathbb N\ associated with the probability vector (\frac 12, \frac 122, ⋯), for all n∈ \mathbb N, we determine the optimal sets of n-means and the nth quantization errors for the infinite self-similar measure P. The technique obtained in this paper can be utilized to determine the optimal sets of n-means and the nth quantization errors for more general infinite self-similar measures.