In this paper we will review several results of rigidity in one-dimensional dynamics and its relation to renormalization. In many one dimensional dynamical systems the existence of a topological conjugacy between two mappings implies that the restriction of the conjugacy to the attractor extends to a smooth mapping. Thus the combinatorics imposes severe restrictions on the geometry of the attractor.
"Rigidity in Dynamics." Bull. Belg. Math. Soc. Simon Stevin 15 (5) 789 - 796, December 2008. https://doi.org/10.36045/bbms/1228486407