Signature of time-reversal symmetry in polynomial . . .

We investigate the reduction to finite fields of polynomial automorphisms of the plane, which lead to invertible dynamics (permutations) of a finite space. We provide evidence that, if the map of the plane is non-integrable, then the presence or absence of a type of time-reversal symmetry called R-reversibility produces a clear signature in the cycle statistics of the associated permutation. If there is such a time-reversal symmetry, the cycle statistics is conjectured to obey a universal distribution, whereas if no such time-reversal symmetry is present, the cycle statistics is consistent with that of a random permutation. These results furnish necessary conditions for R-reversibility to exist in rational maps of the plane, which can be checked via finite computation in a finite field. This translates into effective tests for the existence of R-reversibility, and a probabilistic algorithm for determining parameter values at which a map has such a property