Misc

The following animation illustrates equidistribution of preimages of a=2 under the map f(x)=x^2-1. In the nth frame, the roots of f^n (x) - 2 are plotted. This illustrates a result originally due to Brolin in 1965.

The following animation shows roots of the polynomial f(x) = x^(4c) - 4x^(3c) + 3x^(2c) + x^c + 1, where c is an integer that ranges between 1 and 9. This illustrates a more general phenomenon. The roots of a sequence of distinct irreducible integer polynomials with bounded coefficients equidistribute to the unit circle. This is a consequence of Bilu's theorem (for the Mahler measure) and the fact that the Mahler measure of a polynomial is commensurate with the height of the polynomial.

The following animation illustrates equidistribution of n torsion points on an elliptic curve given by a lattice in the plane.

The following picture is a Julia set generated by a fixed sequence of maps, where each term in the sequence is either x^2 or x^2-1. Here is the Sage notebook for this picture. See also here.