REAL ANALYTICITY OF HAUSDORFF DIMENSION OF JULIA SETS OF PARABOLIC POLYNOMIALS fλ(z) = z(1-z-λz~2)

摘要

We prove that D_* , the set of all parameters λ ∈ C {0} for which the cubic polynomial f is parabolic and has no other parabolic or finite attracting periodic cycles, contains a deleted neighborhood Do of the origin 0. Our main result is that if D_0 is sufficiently small then the function D_0 ∈λ→ HD(J(fλ)) ∈ R is real-analytic. This function ascribes to the polynomial f the Hausdorff dimension of its Julia set J(fλ). The theory of parabolic and hyperbolic graph directed Markov systems with infinite number of edges is used in the proofs.