  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.

Dynamical Systems