This is the second paper of the cycle devoted to the new theory of non-conformal mapping. It introduces an important for further investigations concept of dynamical non-conformal mapping and by a simple example shows some new scope available now for the dynamical researchers. This paper was published first in our electronic journal SELF Transactions, 2 (2002), No 1, pp.101-110. We would like to mark for mp_arch readers that here we cannot present the dynamical non-conformal mapping animated, i.e., in its real appearance. All interested are invited to visit our journal and archive at http://angelfire.lycos.com/la3/selftrans We will be pleased if you find there a response to your thoughts and research interests. We will be grateful, receiving your opinion of our results. We are willing to answer your questions and discuss with you any problems having something in common with our developments