The flow of the vector field (x,y) |-> (cos(x),sin(x)sin(y))

This is an example of a strictly discontinuous action of a group G on a space X and a compact representing subset A&subeteq;X
for which the natural continuous bijection A/~→ X/G is not a homeomorphism,
since there exists a compact saturated subset K⊆A whose saturated hull in X is not closed.

Radius of neighborhood

Time

Infos

The space X is the gray square [-π/2,π/2]×[0,π] without its corners.
The action of the group of integers is given by the flow of the vector field.
The dots form the orbit of a given initial dark red point, which can be choosen by sliders near the boundary.
A neighborhood showing the strict discontinuity of the action can be specified by the slider for its radius. The strict discontinuity can be verified by applying the flow using the time slider.
A compact representing subspace A is formed by the closed rectangle together with the two closed vertical intervals on the boundary. That it is indeed representative for the action can be seen by applying the flow to it (the light green and light blue parts).
A compact and in A saturated subset K is formed by the rectangle. Its orbit in X is not closed since it has the vertical sides of the square in its closure.

This java script numerically solves the flow equation using Runge Kutta 4 (RK4). It uses a fixed number of iterations per point, no matter what t is.
It is based on Orbifold.html written by Tobias Beran and licensed under MIT license.

Config

interpolation points for neighborhood

iterations