Measure-preserving dynamical systems on R3 with all trajectories bounded

Abstract

We present here constructions, both smooth and piecewise-linear, of non-singular, measure-preserving dynamical systems on R3, with each trajectory contained in a bounded set. In the smooth case, we use a sequence of nested subsets of R3, and construct a measure-preserving flow where no trajectory escapes the set in which it originates. In the piecewise-linear case, we again employ a sequence of nested subsets, but rather than defining a flow using vector fields, we construct a measured 1-foliation, which gives rise to a measure-preserving dynamical system.