About properties Lyapunov exponent on the projector of Baire class

The Lyapunov exponents play a major role in optimal control. The paper is devoted to the following question: If a projection of Baire class one exists, then what are the properties of the Lyapunov exponent? In this research work the analysis of one problem of the existance of the of Baire class one projection, building a uniformly continuous retraction to the convex set, the analysis of the different properties of space ℝ х have been performed.