Julia Robinson numbers and arithmetical dynamic of quadratic polynomials

For rings $\mathcal{O}_K$ of totally real algebraic integers, J. Robinson defined a set which is always $\{+\infty\}$ or of the form $[\lambda,+\infty)$ or $(\lambda,+\infty)$ for some real number $\lambda\ge4$. All known examples give either $\{+\infty\}$ or $[4,+\infty)$. In this paper, we construct infinitely many fields such that the set is an interval, but not equal to $[4,+\infty)$.