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INVARIANCE OF CAPACITY UNDER QUASISYMMETRIC MAPS OF THE CIRCLE: AN EASY PROOF
 

Summary: INVARIANCE OF CAPACITY UNDER QUASISYMMETRIC MAPS OF
THE CIRCLE: AN EASY PROOF
NICOLA ARCOZZI AND RICHARD ROCHBERG
Abstract. We give a direct, combinatorial proof that the logarithmic capacity is essentially
invariant under quasisymmetric maps of the circle.
It is a known fact that logarithmic capacity of closed sets is essentially invariant under
quasisymmetric maps of the unit circle. An orientation preserving homeomorphism of the
unit disc, identified with an increasing homeomorphism : [0, 1] [0, 1] via the map
e2it
e2i(t)+
, for some real , is quasisymmetric if
1
M

(x + t) - (x)
(x) - (x - t)
M
for some fixed M > 1. The logarithmic capacity of a closed subset of the unit circle, identified
with a subset E of the unit interval, is comparable with its Bessel (2, 1/2)-capacity Cap(E).
Given a positive, Borel measure µ on [0, 1], let

  

Source: Arcozzi, Nicola - Dipartimento di Matematica, Universitą di Bologna

 

Collections: Mathematics