Flows with time-reversal symmetric limit sets on surfaces

Abstract

The Long-time behavior of orbits is one of the most fundamental properties in dynamical systems. Poincaré studied the Poisson stability, which satisfies a time-reversal symmetric condition, to capture a property whether points return arbitrarily near the initial positions after a sufficiently long time. Birkhoff introduced and studied the concept of non-wandering points, which is one of time-reversal symmetric conditions. Moreover, minimality and pointwise periodicity satisfy the time-reversal symmetric condition for limit sets. Athanassopoulos characterized a flow that is either irrational or Denjoy by using a time-reversal symmetric condition for limit sets. In this paper, we refine the characterization of irrational or Denjoy flows into one of the flows with the time-reversal symmetric condition for limit sets. Moreover, we demonstrate that limit quasi-circuits can become like the boundary of the lakes of Wada.