Dynamic properties of an exact algorithm for square root calculation

Abstract

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 H defined in the semi-closed interval [0, 100). We prove that H is chaotic and topologically conjugated to the shift map in the Bernoulli space on 10 symbols. We also exhibit a natural measure for H 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 H, advantageously due to its dynamic properties.