Criterion for \(\text{SL}(2,\mathbb{Z})\)-matrix to be conjugate to its inverse (Q1861020)

Recently, L. Polterovich and Z. Rudnick in a preprint proved that an element \(h\in\text{SL}(2,\mathbb{Z})\) is stably mixing if and only if it is not conjugate to its inverse in \(\text{SL}(2,\mathbb{Z})\). The author of this paper gives a necessary and sufficient condition for a matrix in \(\text{SL}(2,\mathbb{Z})\) to be conjugate to its inverse. This condition reduces the determination of conjugation to solving an indeterminate equation of second degree.