Invariant set theory is an important tool in the control theory. It has rich history that goes back over a century, yet it is still an active research topic both in pure mathematics and theoretical engineering. It is easy to reduce a traditional discrete-time control dynamical system to an iterative dynamics of one endomorphism in the phase space. It is not easy to do the same in the presence o...

In the context of Mather’s theory of Lagrangian systems, we study the decomposition in chain-transitive classes of the Mather invariant sets. As an application, we prove, under appropriate hypotheses, the semi-continuity of the so-called Aubry set as a function of the Lagrangian. In the study of Lagrangian systems, John Mather introduced several invariant sets composed of globally minimizing ex...

/ d e i \ (is) y — r*?W' = -/"Y'. \dx* c h / The factor — r on the right-hand side of this equation shows that the Dirac equation for an electron is not invariant under inversions. However, if we set 0? = O and JU = 0, then equation (15) is numerically invariant under inversions. This is done in the neutrino theory of light. Hence that theory has the same invariance properties as the Maxwell th...