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A measure of maximal entropy of the rational fraction 1/(z^2+d) on the Riemann sphere according to Arnaud Chéritat. It describes the distribution of preimages of almost all points.

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Consider the family of symmetric towel maps (the towel terminology is due to S. Newhouse):

f(x,y)=(1-ax^2-cy^2,1-ay^2-cx^2).

This looks like a very natural generalization of quadratic interval maps, a step beyond the Viana maps of the form f(x,y)=(dx \mod 1,1-ay^2+c\sin(2\pi x). 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 |c|).

Now, let a=1.8, c=0.2 and iterate a random point of [0,1]^2 forward….

Forward iteration of a towel map

or backward (chosing randomly between the preimages) at each step:

Backward iteration of a towel map

Nice pictures, aren’t they? A towel and its diffraction pattern 😉

More on this later, hopefully…

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Fall in Orsay

The Orsay campus has beautiful and diverse trees which take on gorgeous colors as can be guessed from the header picture (no post-processing, mind you!) taken in the fall of 2007 on the path betweeen the train station and the math dept.

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