Let b be a pseudo-Anosov braid whose permutation has a fixed point and let M_b be the mapping torus by the pseudo-Anosov homeomorphism defined on the genus 0 fiber F_b associated with b. We prove that there is a 2-dimensional subcone C₀ contained in the fibered cone C of F_b such that the fiber F_a for each primitive integral class a ∈ C₀ has genus 0. We also give a constructive description of the monodromy φ_a: F_a → F_a of the fibration on M_b over the circle, and consequently provide a construction of many sequences of pseudo-Anosov braids with small normalized entropies. As an application we prove that the smallest entropy among skew-palindromic braids with n strands is comparable to 1/n, and the smallest entropy among elements of the odd/even spin mapping class groups of genus g is comparable to 1/g.

We would like to thank Mitsuhiko Takasawa for helpful conversations and comments. The first author was supported by Grant-in-Aid for Scientific Research (C) (No. 16K05156), Japan Society for the Promotion of Science. The second author was supported by Grant-in-Aid for Scientific Research (C) (No. 18K03299), Japan Society for the Promotion of Science.