Ergodic Theory and Dynamical Systems

Research Article

Measure of the Julia set of the Feigenbaum map with infinite criticality

GENADI LEVINa1 and GRZEGORZ ŚWIA̧TEKa2

a1 Department of Mathematics, Hebrew University, Jerusalem 91904, Israel (email: levin@math.huji.ac.il)

a2 Wydział MiNI, Politechnika Warszawska, Plac Politechniki 1, 00-661 Warszawa, Poland (email: g.swiatek@mini.pw.edu.pl)

Abstract

We consider fixed points of the Feigenbaum (periodic-doubling) operator whose orders tend to infinity. It is known that the hyperbolic dimension of their Julia sets goes to 2. We prove that the Lebesgue measure of these Julia sets tend to zero. An important part of the proof consists in applying martingale theory to a stochastic process with non-integrable increments.

(Received April 23 2007)

(Revised March 12 2009)