In 1988 Anosov [1] published the construction of an example of a flow (continuous real action) on a cylinder or annulus with a phase portrait strikingly different from our normal experience. It contains orbits whose -limit sets contain a non-periodic orbit along with a simple closed curve of fixed points, but these orbits do not wrap down on this simple closed curve in the usual way. In this paper we modify some of Anosov’s methods to construct a flow on a surface of genus with equally striking behavior that does not occur on a surface of genus or a cylinder. Moreover, our construction is relatively simple and can easily be modified to produce a variety of examples exhibiting similar types of behavior. The key idea that we use from Anosov’s paper can be described in the following way. A flow on a cylinder can always be slowed down near one of the boundary circles so that it becomes fixed. If you slow a flow down very rapidly as you approach a bounding circle, then you can also spin the orbits further and further around the axis of the cylinder as you approach the boundary without destroying the flow. In particular, the boundary circle remains fixed, but orbits that approach even a single point on it now spiral toward it.
@article{bwmeta1.element.bwnjournal-article-cmv84i1p235bwm,
author = {N. Markley and M. Vanderschoot},
title = {An exotic flow on a compact surface},
journal = {Colloquium Mathematicae},
volume = {84/85},
year = {2000},
pages = {235-243},
zbl = {0978.54025},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-cmv84i1p235bwm}
}
Markley, N.; Vanderschoot, M. An exotic flow on a compact surface. Colloquium Mathematicae, Tome 84/85 (2000) pp. 235-243. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-cmv84i1p235bwm/
[000] [1] D. V. Anosov, On the behavior in the Euclidean or Lobachevsky plane of trajectories that cover trajectories of flows on closed surfaces. I, Math. USSR-Izv. 30 (1988), 15-38. | Zbl 0637.58025
[001] [2] S. H. Aranson and V. Z. Grines, On some invariants of dynamical systems on two-dimensional manifolds (necessary and sufficient conditions for the topological equivalence of transitive dynamical systems), Math. USSR-Sb. 19 (1973), 365-393. | Zbl 0281.54023
[002] [3] E. Lima, Common singularities of commuting vector fields on 2-manifolds, Comment. Math. Helv. 39 (1964), 97-110. | Zbl 0124.16101
[003] [4] N. G. Markley, Invariant simple closed curves on the torus, Michigan Math. J. 25 (1978), 45-52. | Zbl 0396.58028
[004] [5] M. H. Vanderschoot, Limit sets for continuous flows on surfaces, Ph.D. thesis, Univ. of Maryland, College Park, 1999.