The Equilateral Collision Normal Form of the Jonah Pedal Map
Year: 2026
Type: article
Abstract: 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.
Author: Schloss Jonah
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225627302