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 $\{+\mathrm{\infty }\}$ or of the form $[\lambda ,+\mathrm{\infty })$ or $(\lambda ,+\mathrm{\infty })$ for some real number $\lambda \ge 4$. All known examples give either $\{+\mathrm{\infty }\}$ or $[4,+\mathrm{\infty })$. In this paper, we construct infinitely many fields such that the set is an interval, but not equal to $[4,+\mathrm{\infty })$.