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 [60] 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 θ will imply continuity in the parameter c. Level sets of the biaccessibility dimension are described, which are related to renormalization. Hölder asymptotics at rational angles are found, confirming the Hölder exponent given by Bruin-Schleicher [11]. 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.
Proposition 3.8 (β -type Misiurewicz points)Suppose c is a β -type Misiurewicz point of preperiod k ≥ 1. 1. The Markov matrix A is irreducible and primitive. 2. The largest eigenvalue satisfies λ k ≥ 2, so h(c) ≥ log 2 k , with strict inequality for k ≥ 2. 3. For the Thurston matrix F according to Proposition 3.5, the arc [c, β c ] is generating an irreducible primitive block B of F, which corresponds to the largest eigenvalue λ .Proof: Consider the claim that the union), 1 ≤ j ≤ k, provides arcs from α c to all endpoints except β c . When an arc from α c is crossing z = 0, then either its endpoint is behind −α c and the part before −α c is chopped away before the next iteration. Or this arc is branching off between 0 and −α c , and further iterates will be branching from arcs at α c constructed earlier. This proves the claim. 2. Now the arc [0, β c ] is containing preimages of itself, so f k c ([0, β c ]) is the Hubbard tree T c . Since f k c (z) is even, every arc of the Hubbard tree has at least two preimages underEvery edge e of T c covers z = 0 in finitely many iterations. Then it is iterated to an arc at β c , so the edge e contains an arc iterated to [0, β c ]. Now f n c (e) = T c for n ≥ n e , and A n is strictly positive for n ≥ max{n e | e ⊂ T c }, thus A is irreducible and primitive. (Alternatively, this follows from Lemma 3.9.4, since f c is not renormalizable.) Finally, if k ≥ 2 and we had λ k = 2, the characteristic polynomial of A would contain the irreducible factor x k − 2 and A would be imprimitive. 3. Denote the postcritical points by c j = f j−1 c (c) again. The iteration of any arc is giving two image arcs frequently, but at least one of these has an endpoint with index j growing steadily, so reaching β c = c k+1 . Further iterations produce the arc [c, β c ] after the other endpoint was in part A of K c \ {0}, which happens when it becomes −β c or earlier. So the forward-invariant subspace generated by [c, β c ] corresponds to an irreducible block B of F. It is primitive by the same arguments as above, since [c, β c ] → [c, β c ] + [c, c 2 ].Consider the lift F c : X c → X c from the proof o...
