The notion of special valuations, i.e. finitely generated valuations inducing klt degenerations, plays an important role in the recent study of K-stability and moduli of Fano varieties. Given an affine log Calabi-Yau variety U, we show that its special skeleton, namely the space of special valuations in the dual complex of U, is independent of the choice of its Fano compactifications. Consequently, the special skeleton of U is invariant under the action of Aut(U). As an example, when U is the complement of a Markov cubic surface in the affine three-space, we show that the action of three Vieta involutions on the special skeleton exhibits a surprising connection with dynamics on the hyperbolic plane.