On symmetric properties of APN functions

We study symmetric properties of APN functions and the structure of their images. It is proven that there is no permutation of variables which keeps an APN function values. Upper bounds for the number of symmetric coordinate Boolean functions in APN function are obtained. Also, there are proven upper bounds for the number of coordinate Boolean functions of an APN function which are invariant under circular translation of indices. Upper bounds for the maximal number of coincidental values are obtained for $n\le6$. A lower bound for the number of different values of an arbitrary APN function is proven. Bibliogr. 14.