Abstract: Understanding Frobenius elements in iterated Galois extensions is a major goal in arithmetic dynamics. In 2012 Boston and Jones conjectured that any quadratic polynomial $f$ over a finite field that is different from $x^2$ is settled, namely the weighted proportion of $f$-stable factors in the factorization of the $n$-th iterate of $f$ tends to $1$ as $n$ tends to infinity. This can be rephrased in terms of Frobenius elements: given a quadratic polynomial $f$ over a number field $K$, an element $\alpha$ in $K$ and the extension $K_\infty$ generated by all the $f^n$-preimages of $\alpha$, the Frobenius elements of unramified primes in $K_\infty$ are settled. In this talk, we will explain how to construct uncountably many non-conjugate settled elements that cannot be the Frobenius of any ramified or unramified prime, for any quadratic polynomial. The key result is a description of the critical orbit modulo squares for quadratic polynomials over local fields. This is joint work with Carlo Pagano.