Dynamic properties of an exact algorithm for square root calculation

The digits of the square root of any real number can be consecutively calculated by hand with the use of a very popular exact algorithm. We show that the application of that algorithm defines a dynamic system in the sense that it can be reduced to the consecutive iteration of a map HH defined in the semi-closed interval [0,100). We prove that HH is chaotic and topologically conjugated to the shift map in the Bernoulli space on 10 symbols. We also exhibit a natural measure for HH which is mixing and of maximum entropy. Finally, we adapt the cryptography method proposed by Baptista [M.S. Baptista, Cryptography with chaos, Phys. Lett. A 240 (1998) 50–54] to the dynamics associated with HH, advantageously due to its dynamic properties.