Piecewise-affine permutations of finite fields

Piecewise-affine permutations (p.-a. p.) are defined on any field $\mathrm{GF}(q)$. They are a generalization of piecewise-linear permutations firstly introduced by A. B. Evans. Here some estimates for linear characteristics of p.-a. p. on $\mathrm{GF}(q)$ are given. In some cases, their exact values are pointed. Polynomials representing p.-a. p. are described. Under some conditions on $\sqrt{q-1}$, it is proved that piecewise-affine permutations form the full symmetric group of $\mathrm{GF}(q)$.