The examples in this website are of invariant currents for smooth self maps of a Torus. The Theorem being illustrated holds in much greater generality, but it is easiest to make illustrations for a Torus. Each example comes from a specific self map of the Torus. In order to depict the specific self map used we always use a pair of images under the heading "The Map". The right hand image in every case is the same, it is a grey grid insided a colored square. In every case the left hand image is the preimage of the grey grid and the colored square. One can then get a quick sense of the map since each of the deformed squares in the left hand images maps to the square shown in the right hand image. Also, the colored boundary lines are actually made of two colors each, one boundary line is green and blue, the other is yellow and red. The preimage lines are drawn so that the red side of the red-yellow curves in the left image are mapped to the red side of the straight red-yellow line in the right image. Similarly the blue side of the blue-green curves in the left image maps to the blue side of the straight line in the right image. Since we are on a torus, the blue-green line on the top of the square and on the bottom of the square in the right hand image are really the same line. Also the red-yellow line on the left and right hand sides of the square in the right hand image are really the same line. Clicking on either image in the examples pages will give a higher resolution picture in which the grey grid is 100 by 100 instead of 10 by 10. (By default most browsers seem to resize these high resolution images to fit in their viewing window, presenting instead extremely low quality low resolution images. Clicking on the low quality version shows each image at proper size and quality.)