Topological obstructions to smoothness for infinitely renormalizable maps of the disc

We analyse the signature type of a cascade of periodic orbits associated with period-doubling renormalizable maps of the two-dimensional disc. The signature is a sequence of rational numbers invariant with respect to orientation-preserving topological conjugacies, which describes how periodic orbits are linked around each other. We prove that in the class of area-contracting maps the signature cannot be a monotone sequence. This explains why classical examples of infinitely renormalizable maps due to Bowen, Franks and Young cannot be achieved by smooth dissipative maps, which shows that there are topological obstructions to realizing infinitely renormalizable maps in the area-contracting case.