The dynamical system is an area-preserving map of the two-dimensional torus defined by the equations

*x(n+1)=a cos (2 pi x(n))-y(n) mod 1
*

*
y(n+1)=x(n) mod 1*

There is a parameter *a* which can take any real value.
If *a* is taken to be 0, then this is a rotation by a quarter
turn, so the orbits are relatively uninteresting.
As *a* increases, one sees orbits which are closed curves (so called
invariant circles) and then some of these start to break up. This is
familiar in Hamiltonian and area-preserving dynamical systems.
For large *a*, the orbits appear to wander over the whole torus
and this leads one to conjecture that area measure is ergodic for
large values of the parameter. This remains unproven.

Here are some pictures of patterns of orbit for various values of the
parameter.
*a*=0.1

The orbits all seem to be closed curves. Notice that the picture wraps around: the top and bottom are joined, as are the left and right.

*a*=0.15

The solid band in the middle is a single orbit.

*a*=0.2

*a*=0.35

Now, most of the picture seems to be the orbit of a single point.

*a*=0.8

Is all of the picture the orbit of one point? Is Lebesgue measure ergodic? If you know, tell me!

If you want to use the program which generated these images,
or want to discuss these things, mail me.