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…