Analytic structures and harmonic measure at bifurcation locus

Abstract

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 C 1+ α d − -conformality, α = 2 − HD (J c 0 ), of the parameter-phase space similarity maps Υ c 0 (z) : C → C at typical c 0 ∈ ∂M d and establish that globally quasiconformal similarity maps Υ c 0 (z), c 0 ∈ ∂M d , are C 1 -conformal along external rays landing at c 0 in C \ J c 0 mapping onto the corresponding rays of M d . This conformal equivalence leads to the proof that the z-derivative of the similarity map Υ c 0 (z) at typical c 0 ∈ ∂M d is equal to 1/T (c 0 ), whereis the transversality function. The paper builds analytical tools for a further study of the extremal properties of the harmonic measure on ∂M d , [25]. In particular, we will explain how a nonlinear dynamics creates abundance of hedgehog neighborhoods in ∂M d effectively blocking a good access of ∂M d from the outside.

Slowly recurrent Collet-Eckmann maps with non-empty Fatou set

“…M. Benedicks and J. Graczyk also have an unpublished work on perturbations on such (quadratic or, more generally, unicritical) maps. The maps there and in the recent papers [15,11] are also slowly recurrent, and hence the results in this paper is partially a generalisation of some of those results. We will not use harmonic measure, but develop the classical Benedicks-Carleson parameter exclusion techniques and combining it with strong results on transversality, by G. Levin [18].…”

“…We are going to study perturbations of such maps in the complex setting. For the quadratic family and other unicritical families, J. Graczyk and G. Światek recently made an extensive study of perturbations of typical Collet-Eckmann maps with respect to harmonic measure, in a series of papers [15,11,13,14]. M. Benedicks and J. Graczyk also have an unpublished work on perturbations on such (quadratic or, more generally, unicritical) maps.…”

Accessibility and porosity of harmonic measure at bifurcation locus

“…Similarity between the phase and parameter planes. Theorem 3 of [14] (see also Fact 1.1 in [15]) describes the similarity between M d and J c 0 through one-parameter family of asymptotically conformal maps Υ c 0 : C → C, with c 0 typical with respect to the harmonic measure on ∂M d . We state it as Fact 1.1.…”

“…The proof of Fact 1.2 is Theorem 1 in [15] and is based on combinatorics of Yoccoz pieces and TWB-theory, see also [14].…”

“…The main accessibility results are based on refined estimates of the waiting time for subsequent passages to a large scale of [11], see (iii) of Fact 1.3, and the property that they can be further improved by iterations. On the other hand, the sharp estimates of porosity come from the persistence of hedgehog layers of [15,24] around the critical value c. This is a new ingredient that arises from non-linear effects of the recurrent dynamics and very fast decay of box geometry [10] for generic c ∈ ∂M d [14]. The correspondence between the phase and parametric spaces is given by a system of asymptotically conformal similarity functions (Υ c ) c∈M d for almost all c with respect to ω, Fact 1.1.…”