Measure-preserving diffeomorphisms of the torus (Q2773057)

Let \((T^2,\lambda)\) be the probability space formed by the two-dimensional torus and the Lebesgue measure. Let \(f:(T^2,\lambda)\rightarrow (T^2,\lambda)\) be a measure preserving diffeomorphism. The paper deals with bounded properties of the sequence \(\{ Df^n\} _n\). This sequence has linear growth if there are positive constants \(c\) and \(C\) such that \(0\leq c\leq \|Df^n(x)\|/n\leq C\) for any \(x\in T^2\) and any positive integer \(n\). This definition is preserved under algebraic conjugation.NEWLINENEWLINENEWLINEIn a direct way, the author shows that if a measure preserving \(C^3\)-diffeomorphism \(f:(T^2,\lambda)\rightarrow (T^2,\lambda)\) is conjugate to a skew product of an irrational rotation on the circle and a circle \(C^3\)-cocycle with nonzero degree, then \(f\) is ergodic, \(f\) has linear growth of the derivative and the sequence \(\{ n^{-1}Df^n\} _n\) is bounded in the Banach space \(C^2(T^2,M_2(R))\) of \(C^2\) maps from \(T^2\) into the set of real matrices \(2\times 2\). The paper is mainly devoted to prove the converse result.