Attracting fixed points of polynomial dynamical systems in fields of $p$-adic numbers

We consider dynamical systems of the form $h(x)=x+g(x)$, where $g(x)$ is a monic irreducible polynomial with coefficients in the ring of integers of a $\mathfrak p$-adic field $K$. We also study 2-periodic points of some simple polynomials of this form in the case when $K=\mathbb Q_p$.