Conformal Dimension: Theory and Application by John M. Mackay

By John M. Mackay

Conformal measurement measures the level to which the Hausdorff measurement of a metric area may be decreased by way of quasisymmetric deformations. brought via Pansu in 1989, this idea has proved tremendous fruitful in a various diversity of components, together with geometric functionality concept, conformal dynamics, and geometric workforce concept. This survey leads the reader from the definitions and easy thought via to lively examine purposes in geometric functionality thought, Gromov hyperbolic geometry, and the dynamics of rational maps, among different components. It studies the idea of size in metric areas and of deformations of metric areas. It summarizes the fundamental instruments for estimating conformal measurement and illustrates their program to concrete difficulties of self reliant curiosity. various examples and proofs are supplied. operating from simple definitions via to present examine components, this e-book can be utilized as a consultant for graduate scholars drawn to this box, or as a invaluable survey for specialists. heritage wanted for a possible reader of the e-book comprises a operating wisdom of genuine and complicated research at the point of first- and second-year graduate classes

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Example text

A metric space whose Hausdorff and topological dimensions agree) is minimal for conformal dimension, for purely topological reasons. (2) The Cantor set K is not minimal for conformal dimension. Indeed, 2 dim K = log log 3 but Cdim K = 0. To prove the latter result, consider a one-parameter family of Cantor sets Kλ ⊂ [0, 1], 0 < λ < 12 , where Kλ consists of two subsets, one of which contains 0 and the other contains 1, so that each subset is similar to K by scale factor λ. Note that K = K1/3 .

In general, however, it is extremely difficult to write down a simple list of properties characterizing a given conformal gauge. For example, characterizing the conformal gauge of the sphere SN , N ≥ 2, is one of the most difficult and important problems in 21 22 2. CONFORMAL GAUGES AND CONFORMAL DIMENSION the area. 2). 16 states that a conformal gauge G embeds in a Euclidean space if and only if it is doubling. 2. Conformal dimension We now recall one of the most important quasisymmetry invariants, the focus of this survey.

BOUNDARIES OF RANK ONE SYMMETRIC SPACES 39 Proof. 10, d(ξ, η) = exp(−(ξ|η)o ) is a metric in the canonical conformal gauge of ∂∞ M . Thus Cdim ∂∞ M ≤ dim(∂∞ M, d). Let s > h(M, g); we will show that Hds (∂∞ M ) = 0. Choose s > t > h(M, g). By the definition of h(M, g), there exists a constant C so that Volg Bg (o, R) ≤ CetR for sufficiently large R. Cover Bg (o, R) with balls of radius one; the minimal cardinality of such a covering (by volume estimates) is at most C etR for some constant C . Let B(x1 , 1), .

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