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International Mathematics Research Notices Advance Access originally published online on August 5, 2009
International Mathematics Research Notices (2009) 2010:87-111, doi:10.1093/imrn/rnp115 published on December 22, 2009
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© The Author 2009. Published by Oxford University Press. All rights reserved. For Permissions, please e-mail: journals.permissions@oupjournals.org

Conformal Dimension: Cantor Sets and Fuglede Modulus

Hrant Hakobyan

Department of Mathematics, University of Toronto, Toronto, ON M5S 2E4, Canada

Correspondence: Correspondence to be sent to: hhakob{at}math.toronto.edu

In this paper, we give several conditions for a space to be minimal for conformal dimension. We show that there are sets of zero length and conformal dimension 1, thus answering a question of Bishop and Tyson. Another sufficient condition for minimality is given in terms of a modulus of a system of measures in the sense of Fuglede [7]. It implies in particular that if Formula is minimal for conformal dimension and supports a measure {lambda} such that for every {varepsilon} > 0 there is a constant 0 < C < {infty} such that C–1 r1+{varepsilon} ≤ {lambda} (E{cap} B(x, r)) ≤ C r1–{varepsilon}, then X x Y is minimal for conformal dimension for every compact Y.

Communicated by Prof. Nikolai Makarov


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