THE SHADOW-CURVES OF THE ORBIT DIAGRAM PERMEATE THE BIFURCATION DIAGRAM, TOO

Abstract

The "Q-curves [Formula: see text] 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) = x2 + c. We illustrate that Q-curves also appear implicitly, for a different reason, in a computer-drawn bifurcation diagram of x2 + 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-curve s 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 sin x 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.