On the classification of groups of orientation-preserving homeomorphisms of $\mathbb R$. I. Invariant measures

Groups of orientation-preserving homeomorphisms of $\mathbb R$ are studied. Such metric invariants as invariant and projectively-invariant measures are investigated. The approach taken results in the classification of groups of homeomorphisms by the complexity of the set of all fixed points of the group elements. In each of the classes of groups thus distinguished a finer classification is carried out in terms of the complexity of the topological structure of orbits and the combinatorial properties of the group.