The Shadow-Curves of the Orbit Diagram Permeate the bifurcation Diagram, Too

The “Q-curves” Q1(c) = c,Q2(c) = c + c, . . . , Qn(c) = (Qn−1(c)) + c = f c (0) have long been observed and studied as the shadowy curves which appear illusively — not explicitly drawn — in the familiar orbit diagram of Myrberg’s map fc(x) = x + c. We illustrate that Q-curves also appear implicitly, for a different reason, in a computer-drawn bifurcation diagram of x + c as well — by “bifurcation diagram” we mean the collection of all periodic points of fc (attracting, indifferent and repelling) — these collections form what we call “P -curves”. We show Q-curves and P -curves intersect in one of two ways: At a superattracting periodic point on a P -curve, the infinite family of Q-curves which intersect there are all tangent to the P -curve. At a Misiurewicz point, no tangencies occur at these intersections; the slope of the P -curve is the fixed point of a linear system whose iterates give the slopes of the Q-curves. We also introduce some new phenomena associated with c sinx illustrating briefly how its two different families of Q-curves interact with P -curves. Our algorithm for finding and plotting all periodic points (up to any reasonable period) in the bifurcation diagram is reviewed in an Appendix.