Polynomial Values in Subfields and Affine Subspaces of Finite Fields

For an integer $r$, a prime power $q$, and a polynomial $f$ over a finite field ${\mathbb{F}}_{q^r}$ of $q^r$ elements, we obtain an upper bound on the frequency of elements in an orbit generated by iterations of $f$ which fall in a proper subfield of ${\mathbb{F}}_{q^r}$. We also obtain similar results for elements in affine subspaces of ${\mathbb{F}}_{q^r}$, considered as a linear space over ${\mathbb{F}}_q$.