Consider the family of symmetric towel maps (the towel terminology is due to S. Newhouse):
This looks like a very natural generalization of quadratic interval maps, a step beyond the Viana maps of the form . These maps can also be understood as a coupling of two chaotic interval maps.
One would like to prove things like a two-dimensional version of Jakobson theorem. However little is known about these dynamics, except for their measures of maximal entropy which I was able to study using the entropy-expansion condition (for small enough ).
Now, let , and iterate a random point of forward….
or backward (chosing randomly between the preimages) at each step:
Nice pictures, aren’t they? A towel and its diffraction pattern 😉
More on this later, hopefully…