On the torus automorphisms: analytic solution, computability and quantization

The exact solution of the classical torus automorphism, which partial case is Arnold Cat map is obtained and compared with the numerical solution. The torus, considered as the classical phase space admits the quantization in terms of the Weyl pair. The remarkable fact is that quantum map, as the evolution with respect to the discrete time, preserves the Weyl commutation relation. We have obtained also the operator solution of this quantum torus automorphism.