This figure is a rather powerful blow-up. The discontinuty at x = -1 is shown as the diagonal line at the left.
Window: -1.127 < xx < -1.107, .492 < y < .508, where xx = x - a/2 y. Color i.c. millions of iterations Comments green (-1, .5105) 210 Orbit ranges over whole rhombus. brown (-1, .509) 13 Eventually escapes to join green. magenta (-.984149,.0742484) 1 Pete's orbit (b) in the paper. red (-.99991472, .5061287) 1 Close to an invariant curve. cyan (-1, .5058) 51 dark blue (-1, .505) 10 blue (-.99860194,.50405996) 2 note the three-fold pattern salmon (-1, .503) 20 purple (-1, .4995) 5Plus, there are many circular orbits that are not listed here.