Fuchsian groups and ergodic theory

Abstract

Introduction.Let 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 .The following concepts are of fundamental significance for the study of dynamical flows.(a) There exists a curve of motion everywhere dense on .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 .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 form a point set on 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 .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 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 that is invariant under the flow is everywhere dense on . (b') Every open point set on that is invariant under the flow has the measure m().(c') Every m-measurable point set on that is invariant under the flow has either the m-measure zero orm().The latter property of a flow is called metric transitivity.%Its importance rests