Semidynamical Systems and Hilbert’s Fifth Problem

g : [0,∞)×X → X and g(t, x) = T (t)x, t ≥ 0, x ∈ X, then g is continuous. For many semigroups strong continuity implies joint continuity. A function gx as in (1) is called a trajectory of T . If T has domain all of R instead of just [0,∞) we refer to T as a ‘group’. In his Fifth Problem, Hilbert asked whether assumptions of differentiability that Sophus Lie made were actually a consequence of Lie’s algebraic and continuity assumptions. It was shown (see for example[10]) that this was the case. It is less widely recognized that Hilbert raised some additional, rather open ended, questions of a similar nature (see translation in [7], for example). Some of these questions relate to ‘functional equations’ of which our present semigroups are examples. It seems reasonable to think that Hilbert was interested in algebraic and topological assumptions on a system that imply enough differentiability for an analysis of the system. The open ended parts of Hilbert’s problems will continue to give inspiration for a long time after specific questions are settled.