Topological invariants in period-doubling cascades

Topological characterization is a useful description of dynamical behaviour s as exemplified by templates which synthesize the topological properties o f very dissipative chaotic attractors embedded in tri-dimensional phase spa ces. Such a description relies on topological invariants such as linking nu mbers between two periodic orbits which may be viewed as knots. These invar iants may, therefore, be used to understand the structure of dynamical beha viours. Nevertheless, as an example, the celebrated period-doubling cascade is usually investigated by using total twists which are not topological in variants. Instead, we introduce linking numbers between an orbit, viewed as the core of a small ribbon, and the edges of the ribbon. Such a linking nu mber (which is in fact the Calugareanu invariant) is related to the total t wist number and the number of writhes of the ribbon. A second topological i nvariant, called the effective twist number, is also introduced and is usef ul for investigating period-doubling cascades. In the case of a trivial sus pension of a horseshoe map, this topological invariant may be predicted fro m a symbolic dynamics with the aid of framed braid representations.