Isometric mappings of the set of all Boolean functions into itself which preserve self-duality and the Rayleigh quotient

In the paper, we study isometric mappings of the set of all Boolean functions in $n$ variables into itself which preserve self-duality and the Rayleigh quotient of Boolean function and generalize known results. It is proved that isometric mapping preserves self-duality if and only if it preserves anti-self-duality. The complete characterization of these mappings is obtained. Based on this result, the set of isometric mappings which preserve the Rayleigh quotient of a Boolean function is described. As a corollary, all isometric mappings which preserve bentness and the Hamming distance between bent function and its dual are given.