Analytic structures and harmonic measure at bifurcation locus

We study conformal quantities at generic parameters with respect to the harmonic measure on the boundary of the connectedness loci M-d for unicritical polynomials f(c)(z) = z(d) + c. It is known that these parameters are structurally unstable and have stochastic dynamics. We prove C1+ alpha/2d+alpha -epsilon-conformality, alpha = 2 - HD (J(c0)), of the parameter-phase space similarity maps Upsilon(c0) (z) : C (sic) C at typical c(0) is an element of partial derivative M-d and establish that globally quasiconformal similarity maps Upsilon(c0) (z), c(0) is an element of partial derivative M-d, are C-1-conformal along external rays landing at c(0) in C \ J(c0) mapping onto the corresponding rays of M-d. This conformal equivalence leads to the proof that the z-derivative of the similarity map Upsilon(c0) (z) at typical c(0) is an element of partial derivative M-d is equal to 1/T(c(0)), where T(c(0)) = Sigma(infinity)(n=0)(D(f(c0)(n))(c(0)))(-1) is the transversality function. The paper builds analytical tools for a further study of the extremal properties of the harmonic measure on partial derivative M-d, [27]. In particular, we will explain how a non-linear dynamics creates abundance of hedgehog neighborhoods in partial derivative M-d effectively blocking a good access of partial derivative M-d from the outside. (c) 2023 Elsevier Inc. All rights reserved.

Automated summary

This paper investigates conformal quantities of unicritical polynomials f(c)(z) = z(d) + c at the harmonic measure on the boundary of the connectedness loci M-d. It proves the C1+ alpha/2d+alpha -epsilon-conformality of the parameter-phase space similarity maps Upsilon(c0) (z), and establishes the C-1-conformality of globally quasiconformal similarity maps Upsilon(c0) (z) along external rays landing at c(0) in C \ J(c0). The conformal equivalence leads to the proof of the z-derivative of the similarity map being equal to 1/T(c(0)).