Picture Gallery

This is a selection of pictures to illustrate some of my papers. This picture illustrates a map with three thoroughly intermingled attracting basins. The real projective plane is represented as a unit 2-sphere with antipodal points identified. The "S shaped" (or "backward S shaped") white curve represents the Fermat curve with equation x^3   +  y^3   +  z^3  =  0  . This figure shows dynamics on the real projective plane for the elementary Desboves maps with parameters  a= -1,  b=1/3  and  c = 1. In this case every great circle through the north pole [0:1:0] maps to a great circle through the north pole. There are three attractors: the Fermat curve, the equator (y = 0), and the north pole, each marked in white. The corresponding attracting basins are colored red, blue and grey respectively. However, the closely intermingled blue and red yield a purple effect. (Paper: Elliptic Curves as Attractors, Part I)

 Dynamics on the real projective plane for the Desboves map with parameters  a=-1/5  b=7/15  and  c = 17/15.  The same "S shaped" white curve from the picture above, appears in this picture, although it is harder to see it.  In the real case there are two attractors: 1) The basin of the Fermat curve is colored from red to blue according as their orbits converge more rapidly or more slowly toward this Fermat curve. 2) The two small white circles also form an attractor. The corresponding basin is shown in dark grey. In the complex case empirically the only attractor seems to be a cycle of two Herman rings which intersect the real plane in the circles shown in the picture. (Paper: Elliptic Curves as Attractors, Part I)  Intermingled basins for the cylinder map. (Paper: Schwarzian Derivatives and Cylinder Maps)

 Let   F(z) = z^3 - 3a^2z + (2a^3 + v). The curve  S_p  consists of all pairs  (a, v) in the 2-dimensional complex plane such that the marked critical point  +a  has period  p  for this map.  Here  v  denotes the critical value. The curve  S_3  is a torus with 8 punctures. This picture shows the universal covering space of the unpunctured torus. Here  r  stands for the ideal point in the (1/3)-rabbit region (which is given this name since the Julia set for polynomial maps with parameters in this region consists of many copies of the Douady rabbit). The complex conjugate of this region is denoted by  r^* . The airplane region is denoted by  "a" . The 180 degree rotation of this figure corresponds to an involution   I  , of the torus which interchanges the regions  100+  and  100-  and also interchanges the regions  010+  and  010-,  but fixes the remaining four ideal points. (Paper: Cubic Polynomial Maps, Part II)  The curve  S_4  has genus 15 with 20 punctures. One part of  S_4 , with its canonical translation structure is shown here as a plane with 7 slits, each ending at a puncture point. (Paper: Cubic Polynomial Maps, Part II)  This figure shows pictures of a fat Herman ring and a skinny one on the Riemann sphere, projected orthonormally to the plane. Both arise from rational maps which commute with the antipodal map. In the parameter plane for a family of such maps, the Herman ring locus is a very thick set (colored black in the figure at the top of my web page). Each such ring represents one point on a smooth "hair" in this ring locus, consisting of maps with some fixed irrational rotation number. The front (visible) half of the sphere corresponds to the right half-plane, with  z=0  at the bottom and  z  equal to infinity at the top. The boundaries of the rings have been outlined. Otherwise, every point  z   maps to a point of the same color.
(Paper: On Antipode Preserving Cubic Maps)