On~the additive cohomological equation and time change for a~linear flow on the torus with a~Diophantine frequency vector

A. V. Rozhdestvenskii

Sbornik. Mathematics, Tome 195 (2004) no. 5, pp. 723-764

Résumé

For a 1-periodic function $f$ of finite smoothness and a Diophantine vector $\alpha$ the solubility problem is studied for the additive cohomological equation on the torus $$ w(T_\alpha x)-w(x)=f(x)-\int_{\mathbb T^d}f(t)\,dt, $$ where $T_\alpha x=x+\alpha\pmod1$ is the shift of the torus $\mathbb T^d$ by the vector $\alpha$ and $w$ is an unknown measurable function. Necessary and sufficient conditions are obtained for the conjugacy of a linear flow on the $(d+1)$-torus to the reparametrized flow $$ \begin{cases} \dot x=\dfrac\alpha{F(x,y)}\,,\\ \dot y=\dfrac1{F(x,y)}\,, \end{cases} $$ where $F(x,y)$ is a positive 1-periodic function of finite smoothness.