Iterated corner associate dynamics and a quadratic pedal map attracted to the incenter

For a triangle A1B1C1 inscribed in ABC, the linear part of the affine map ABC → A1B1C1 has characteristic polynomial x^2 + x + δ, and its finite fixed point (the ATFF point) has barycentrics (1 − αA : 1 − αB : 1 − αC), where the α are the corner area ratios. The corner-centroid map equals Id/3 + 2T/3; its square is a homothety of ratio (1 − 4δ)/9. The map M → ATFF(ABC, pedal triangle of M) is a quadratic map of the plane whose fixed points are the incenter and excenters; the incenter is an attracting node for every triangle, with multipliers (2R − r ± 2·OI)/(4R + r), and numerical evidence supports global attraction (conjecture). The map sends the Euler line onto the Thomson–Gibert–Moses hyperbola.