In this paper, we study conjugacy invariants for 2-dimensional diffeomorphisms with homoclinic cubic tangencies (two-sided tangencies of the lowest order) under certain open conditions. Ordinary arguments used in past studies of conjugacy invariants associated with one-sided tangencies do not work in the two-sided case. We present a new method which is applicable to the two-sided case.
"Moduli of Surface Diffeomorphisms with Cubic Tangencies." Tokyo J. Math. 42 (2) 371 - 397, December 2019. https://doi.org/10.3836/tjm/1502179278