Folding Transformations and HKY Mappings

We present a new method for the derivation of mappings HKY type. These are second-order which do not have biquadratic invariant like QRT mappings, but rather an degree higher than two in at least one variables. Our is based on folding transformations exist some discrete Painleve equations. They relate variable equation to square other one. By considering autonomous limit these relations we derive folding-like ones. construct invariants latter and show how they can be extended beyond ones given by strict application transformation.