Core entropy and biaccessibility of quadratic polynomials

Wolf Jung

Published 2014-01-20Version 1

For complex quadratic polynomials, the topology of the Julia set and the dynamics are understood from another perspective by considering the Hausdorff dimension of biaccessing angles and the core entropy: the topological entropy on the Hubbard tree. These quantities are related according to Thurston. Tiozzo [arXiv:1305.3542] has shown continuity on principal veins of the Mandelbrot set M . This result is extended to all veins here, and it is shown that continuity with respect to the external angle theta will imply continuity in the parameter c . Level sets of the biaccessibility dimension are described, which are related to renormalization. H\"older asymptotics at rational angles are found, confirming the H\"older exponent given by Bruin--Schleicher [arXiv:1205.2544]. Partial results towards local maxima at dyadic angles are obtained as well, and a possible self-similarity of the dimension as a function of the external angle is suggested.