In this paper we prove results about lifting dynamical and ergodic properties of a given smooth dynamical system to its skew-product extensions by smooth cocycles. The classical small divisor argument shows that in general such results are not possible. However, using the notion of the ‘fast periodic approximation’ introduced by A. Katok, we will show that if the dynamical system admits such a ‘fast periodic approximation’ then indeed a certain qualitative behaviour which is prohibited by small divisor type conditions is now in fact generic. The techniques are also applied to show that ‘recurrent-proximal’ behaviour of solutions of linear differential equations with almost periodic coefficients is generic under suitable conditions on the coefficient matrix.
All Science Journal Classification (ASJC) codes
- Applied Mathematics