Asymptotically Brownian skew products give non-loosely Bernoulli K- automorphisms
The cocycle \(f(n,\omega)\) generated by the real-valued function f on the Lebesgue probability space \((\Omega,{\mathfrak F},\mu)\) and a measure preserving ergodic map T is said to be asymptotically Brownian if \(\Omega\) can be joined to a space \({\tilde \Omega}\) on which is defined a sequence of random variables \(X(t,{\tilde \omega}\)) so that both \(X^+(t)=X(t)\), \(t\geq 0\), \(X^-(t)=X(t)\), \(t\leq 0\) are Brownian motions and \[ \lim_{| t| \to \infty}| f([t],.)-X(t,.)| /| t|^{1/2-\delta_ 0}=0\quad a.s. \] A quite natural exponential mixing condition implies that the cocycle is asymptotically Brownian motion to a degree better than \(t^{1/2}\). One of the main results states that for the measure preserving flow \(T_ t\) on \((\Omega_ 1,{\mathfrak F}_ 1,\mu_ 1)\) having positive entropy, the \(f,T_ t\)- extension \(\hat T\) is not loosely Bernoulli, where \(\hat T\) is the standard skew product \(\tilde T(\omega,\omega_ 1)= (T(\omega),T_{f(\omega)}(\omega_ 1))\). This is the complete analog to \textit{S. Kalikov}'s T, \(T^{-1}\) argument [Ann. Math., II. Ser. 115, 393- 409 (1982; Zbl 0523.28018)]. Some open problems are formulated, too.
scientific article; zbMATH DE number 4069892