The Transverse Multiplier of the Jonah Saddle

We study the transverse multiplier ‎lambda s(t)=(3t minus 1)(6t squared minus 6t+1)/(2t cubed )‎ of the Jonah saddle, a formula imported from the companion paper. We check it numerically against the Jacobian of the two-dimensional pedal shape map at eight values of t, agreeing to 10 to the minus 8. Conditional on this formula, we compute exact values at its zero t 0=(3+ the square root of 3)/6: the derivative lambda s prime (t 0)=18(2 minus the square root of 3), the longitudinal multiplier lambda u(t 0)=3 minus the square root of 3, and the shape parameter s *(t 0)=( the square root of 3 minus 1)/2. We prove | lambda s(t)|<1 on (1/2,1), so the transverse direction is contracting throughout. The sign of lambda s changes at t 0; this is a sign change of a multiplier, not a change of topological type, since the linear maps with plus or minus lambda s are conjugate. The linearization is degenerate at t 0 and the nonlinear terms are not computed. No global dynamics, Julia set or arithmetic claim is made.