Invariant subspaces of functions affine equivalent to the finite field inversion

In the paper, we consider affine $\mathbb{F}_{p}$-subspaces of a finite field $\mathbb{F}_{p^n}$, $p$ is prime, such that the function $x^{-1}$ which inverses a field element $x$ (we assume that $0^{-1}$ = 0) maps them to affine subspaces. It is proven that the image of an affine subspace $U$, $|U| > 2$, is an affine subspace as well if and only if $U = q \mathbb{F}_{p^k}$, where $q \in \mathbb{F}^*_{p^n}$ and $k | n$. In other words, these subspaces can be expressed using subfields of $\mathbb{F}_{p^n}$. As a consequence, we propose a sufficent condition providing that a function $A(x^{-1}) + b$ has no invariant affine subspaces $U$ of cardinality $2 < |U| < p^n$, where $A: \mathbb{F}_{p^n} \to \mathbb{F}_{p^n}$ is an invertible $\mathbb{F}_{p}$-linear transformation, $b \in \mathbb{F}^*_{p^n}$. Also, we give examples of functions which have no invariant affine subspaces except for $\mathbb{F}_{p^n}$.