\(p\)-adic dynamics and angle doubling (Q358952)

The authors consider the squaring map over the \(p\)-adic numbers for an odd prime \(p\) and study its symbolic dynamics on the unit circle \(\mathbb{Z}_p\). When restricted to the set of squares, an equivalence is presented with angle double (mod 1) for rational angles. For primes \(p \equiv 3 \pmod 4\), this map may be represented as a unitary permutation matrix of the type used in quantum phase estimation. Explicit computations are included for Mersenne primes, Fermat primes as well the unitary matrices associated to primes of the form \(2^t+3\). Some ideas of symbolic dynamics are covered in Section 2 of the paper.