Topological invariants in period-doubling cascades

Topological characterization is a useful description of dynamical behaviours as exemplified by templates which synthesize the topological properties of very dissipative chaotic attractors embedded in tri-dimensional phase spaces. Such a description relies on topological invariants such as linking numbers between two periodic orbits which may be viewed as knots. These invariants may, therefore, be used to understand the structure of dynamical behaviours. Nevertheless, as an example, the celebrated period-doubling cascade is usually investigated by using total twists which are not topological invariants. 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 number (which is in fact the Cãlugãreanu invariant) is related to the total twist number and the number of writhes of the ribbon. A second topological invariant, called the effective twist number, is also introduced and is useful for investigating period-doubling cascades. In the case of a trivial suspension of a horseshoe map, this topological invariant may be predicted from a symbolic dynamics with the aid of framed braid representations.