The Equilateral Collision Normal Form of the Jonah Pedal Map

We study the isosceles reduction of the Jonah pedal map, the holomorphic germ G t(s)=Q t(s) s/ the square root of D t(s). For every t, the equilateral point s=1/2 and the Jonah branch s J(t)=(2t minus 1)/(2t) are fixed points, and they collide at t=1. We prove three exact results. The multiplier at the equilateral point is G t prime (1/2)=(3t minus 1)/2 for all t. Writing t=1 minus mu and s= fraction 12+x, the fixed-point function factors as x (x minus x J( mu )) k( mu,x) with x J( mu )= minus mu /(2(1 minus mu )) and k(0,0)= minus 3 is not equal to 0, so near the collision the only fixed points are the two branches x=0 and x=x J( mu ). The normal form ‎G 1 minus mu ( fraction 12+x) minus fraction 12=x minus 3x squared minus fraction 32 mu x+O(3)‎ shows that the multipliers exchange stability through mu =0, a transcritical collision in the holomorphic reduction. The two-dimensional shape map, the boundary endpoint t= fraction 12, and global dynamics are not treated.