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Conformal trace theorem for Julia sets of quadratic polynomials

Published online by Cambridge University Press:  04 December 2017

A. CONNES
Affiliation:
Collège de France, IHES, 3, rue d’Ulm, 75231 Paris cedex 05, France email alain@connes.org
E. MCDONALD
Affiliation:
School of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052, Australia email edward.mcdonald@unsw.edu.au, f.sukochev@unsw.edu.au, d.zanin@unsw.edu.au
F. SUKOCHEV
Affiliation:
School of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052, Australia email edward.mcdonald@unsw.edu.au, f.sukochev@unsw.edu.au, d.zanin@unsw.edu.au
D. ZANIN
Affiliation:
School of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052, Australia email edward.mcdonald@unsw.edu.au, f.sukochev@unsw.edu.au, d.zanin@unsw.edu.au

Abstract

If $c$ is in the main cardioid of the Mandelbrot set, then the Julia set $J$ of the map $\unicode[STIX]{x1D719}_{c}:z\mapsto z^{2}+c$ is a Jordan curve of Hausdorff dimension $p\in [1,2)$. We provide a full proof of a formula for the Hausdorff measure on $J$ in terms of singular traces announced by the first named author in 1996.

Type
Original Article
Copyright
© Cambridge University Press, 2017 

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