  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 $b$. This paper describes a structure of
the fibered cone $\mathcal{C}$ of $F$ for $M_b$. We prove that there is a
$2$-dimensional subcone $\mathcal{C}_0$ contained in the fibered cone $
\mathcal{C}$ of $F_b$ such that the fiber $F_a$ for each primitive integral
class $a \in \mathcal{C}_0$ has genus $0$. We also give a constructive
description of the monodromy $ \phi_a: F_a \rightarrow 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 $1/g$.
