Description:
We consider a system of linear differential equations, x ˙ = A ( ω ⋅ t ) x \dot x = A(\omega \cdot t)x , parametrized by a point ω ∈ T 2 \omega \in {{\mathbf {T}}^2} , the 2 2 -torus, where ( ω , t ) → ω ⋅ t (\omega ,t) \to \omega \cdot t denotes an irrational rotation flow on T 2 {{\mathbf {T}}^2} . We show that if the rotation number of this flow is well approximable by rationals, then residually many equations (with respect to the C k {C^k} -topology on a certain class of matrix valued maps A ( ω ) A(\omega ) on T 2 {{\mathbf {T}}^2} ) exhibit recurrent-proximal behavior. Also the order of differentiability k k of the class in which this generic result holds is related to the "speed" of approximation by rationals.