On Julia Limiting Directions in Higher Dimensions

Symmetries of Julia sets in higher dimensions

Julia directions for holomorphic curves

A theorem of Picard type is proved for entire holomorphic mappings into projective varieties. This theorem has local nature in the sense that the existence of Julia directions can be proved under natural additional assumptions. An example is given which shows that Borel’s theorem on holomorphic curves omitting hyperplanes has no such local counterpart. Let P be complex projective space of dimen...

Dimensions of Julia Sets of Meromorphic Functions

It is shown that for any meromorphic function f the Julia set J(f) has constant local upper and lower box dimensions, d(J(f)) and d(J(f)) respectively, near all points of J(f) with at most two exceptions. Further, the packing dimension of the Julia set is equal to d(J(f)). Using this result it is shown that, for any transcendental entire function f in the class B (that is, the class of function...

Limiting Absorption Principle and Strichartz Estimates for Dirac Operators in Two and Higher Dimensions

In this paper we consider Dirac operators in R, n ≥ 2, with a potential V . Under mild decay and continuity assumptions on V and some spectral assumptions on the operator, we prove a limiting absorption principle for the resolvent, which implies a family of Strichartz estimates for the linear Dirac equation. For large potentials the dynamical estimates are not an immediate corollary of the free...

On Multiwell Liouville Theorems in Higher Dimensions

We consider certain subsets of the space of n × n matrices of the form K = ∪i=1SO(n)Ai, and we prove that for p > 1, q ≥ 1 and for connected Ω ′ ⊂⊂ Ω ⊂ IR, there exists positive constant a < 1 depending on n, p, q,Ω,Ω such that for ε = ‖dist(Du,K)‖ Lp(Ω) we have infR∈K ‖Du−R‖ p Lp(Ω′) ≤ Mε1/p provided u satisfies the inequality ‖D2u‖ Lq(Ω) ≤ aε1−q . Our main result holds whenever m = 2, and als...