PERIODIC ORBITS OF A DYNAMICAL SYSTEM RELATED TO A KNOT

Following [6] we consider a knot group G, its commutator subgroup K = [G, G], a finite group Σ and the space Hom(K, Σ) of all representations ρ : K → Σ, endowed with the weak topology. We choose a meridian x ∈ G of the knot and consider the homeomorphism σ_x of Hom(K, Σ) onto itself: σ_xρ(a) = ρ(xax^-1) ∀ a ∈ K, ρ ∈ Hom(K, Σ). As proven in [5], the dynamical system (Hom(K, Σ), σ_x) is a shift of finite type. In the case when Σ is abelian, Hom(K, Σ) is finite.

In this paper we calculate the periods of orbits of (Hom(K, ℤ/p), σ_x), where p is prime, in terms of the roots of the Alexander polynomial of the knot. In the case of two-bridge knots we give a complete description of the set of periods.