Fuchsian groups and ergodic theory

Abstract

Introduction.Let ß be the phase space of a dynamical system.We suppose that every motion can be continued along the entire time-axis.Thus we are concerned with a steady flow in ß.The following concepts are of fundamental significance for the study of dynamical flows.(a) There exists a curve of motion everywhere dense on ß.The existence of such a motion is known under the name of regional transitivity.We now suppose that a measure m in the sense of Lebesgue invariant under the flow exists on ß.Such a measure is usually defined by an invariant phase element dm.The following property is stronger than (a).(b) The curves of motion not everywhere dense on ß form a point set on ß of w-measure zero.Still stronger and more important than (b) is strict ergodicity.We suppose m(ti) to be finite.(c) Let f(P) be an arbitrary w-summable function on ß.The time-average oif(P) along a curve of motion is then,in general, equal to faf(P)dm/m(Q), the exceptional curves forming a point set on ß of m-measure zero.How these concepts are interrelated is seen most clearly if we state them in the following way.(a') Every open point set on ß that is invariant under the flow is everywhere dense on ß. (b') Every open point set on ß that is invariant under the flow has the measure m(ß).(c') Every m-measurable point set on ß that is invariant under the flow has either the m-measure zero orm(û).The latter property of a flow is called metric transitivity.%Its importance rests