Nonmeasurable subgroups of uncountable solvable groups

Abstract

It is well known that basic ideas and concepts of modern theory of dynamical systems come from the classical theory of ordinary differential equations. Indeed, for a given autonomous system of first-order ordinary differential equations in a finite-dimensional phase space (e.g., in Rⁿ), the famous Liouville theorem establishes necessary and sufficient conditions, under which the corresponding phase flow preserves the Lebesgue measure (see, for instance, [196] and Exercise 2 of Chapter 3). Those conditions are trivially fulfilled for a Hamiltonian system in the same phase space. Consequently, we obtain a natural example of a dynamical system which is systematically used in studies of various physical (primarily, evolutional) processes.