Asymptotics of the Hausdorff dimensions of the Julia sets of McMullen maps with error bounds

For McMullen maps f (lambda) (z) = z (p) + lambda/z (p) , where lambda is an element of C\{0} = 3 and lambda is small enough, then the Julia set J(f (lambda) ) of f (lambda) is a Cantor set of circles. In this paper we show that the Hausdorff dimension of J(f (lambda) ) has the following asymptotic behavior dim(H)J(f(lambda))=1+log2/logp+O(vertical bar lambda vertical bar(2-4/p)),as lambda -> 0. An explicit error estimation of the remainder is also obtained. We also observe a 'dimension paradox' for the Julia set of Cantor set of circles.