Uniformly fat sets

On Uniformly Approximable Sidon Sets

Let G be a compact abelian group and let T be the character group of G. Suppose £ is a subset of T. A trigonometric polynomial f on G is said to be an ^-polynomial if its Fourier transform / vanishes off E. The set E is said to be a Sidon set if there is a positive number B such that 2^xeb |/(X)| á-B||/||u for all E-polynomials /; here, ||/||„ = sup{ |/(x)| : xEG}. In this note we shall discuss...

Julia Sets Are Uniformly Perfect

We prove that Julia sets are uniformly perfect in the sense of Pommerenke (Arch. Math. 32 (1979), 192-199). This implies that their linear density of loganthmic capacity is strictly positive, thus implying that Julia sets are regular in the sense of Dinchlet. Using this we obtain a formula for the entropy of invanant harmonic measures on Julia sets. As a corollary we give a very short proof of ...

Uniformly Quasiregular Maps with Toroidal Julia Sets

The iterates of a uniformly quasiregular map acting on a Riemannian manifold are quasiregular with a uniform bound on the dilatation. There is a Fatou-Julia type theory associated with the dynamical system obtained by iterating these mappings. We construct the first examples of uniformly quasiregular mappings that have a 2-torus as the Julia set. The spaces supporting this type of mappings incl...

Big Piece approximations of Uniformly rectifiable sets

This talk will give a short introduction of how one may put a “dyadic cube” structure on spaces of homogeneous type (see Coifman-Weiss ’77) and how when this dyadic cube structure is coupled with the Whitney decomposition it allows one to construct approximating NTA domains for uniformly rectifiable (UR) sets. (The term uniformly rectifiable is quantitative, scale invariant version of rectifiab...

Diets for Fat Sets