Exact packing measure of linear Cantor sets

Let K be the attractor of a linear iterated function system S j x = ρ j x + b j ( j = 1, …, m ) on the real line satisfying the open set condition (where the open set is an interval). It is well known that the packing dimension of K is equal to α , the unique positive solution y of the equation $ \sum _{j = 1} ^m \rho ^y _{j = 1} $ ; and the α –dimensional packing measure α ( K ) is finite and positive. Denote by μ the unique self–similar measure for the IFS $ \{ S _j \} ^m _{j=1} $ with the probability weight $ \{ \rho ^\alpha _j \} ^m _{j=1} $ . In this paper, we prove that α ( K ) is equal to the reciprocal of the so–called “minimal centered density” of μ , and this yields an explicit formula of α ( K ) in terms of the parameters ρ j , b j ( j = 1, …, m ). Our result implies that α ( K ) depends continuously on the parameters whenever $ \sum _j \rho_j < 1 $ .