Smoothness of holonomy maps derived from unstable foliation (Q2778796)

It is proved that the unstable foliation holonomy on an area-preserving Anosov diffeomorphism on a two-dimensional torus has the derivative with bounded variation only if the diffeomorphism is smoothly conjugate to a linear map. Let \(f\) be an Anosov, area-preserving diffeomorphism of a two-dimensional torus \(T^2\) onto itself. For every point \(p\in T^2\) we can define new local coordinates \((\xi_p(x,y)\), \(\psi_p(x,y))\) such that: (1) They are \(C^\infty\) in \(x,y\) and functions \(\xi_p(x,y)\), \(\psi_p(x,y)\) as well as their all derivatives are \(C^1\) with respect to \(p\). (2) The change of coordinates is area-preserving for any \(p\). (3) After the change of coordinates at \(p\) and \(f(p)\) the map \(f\) has in some \(r\)-neighborhood of \(p\) \((r\) does not depend on \(p)\) the following form \(f(x',y')= (\lambda_p x'\Phi_p(x',y')\), \((\lambda_p)^{-1}y'(\Phi_p)^{-1} (x',y'))\), where \(\lambda_p\) is the contraction rate for a stable direction for \(f\) at the point \(p\), \(\Phi_p(0)=1\), \(\Phi_p(z)\) is \(C^\infty\) in \(z\) and \(\Phi_p(z)\) as well as its all derivatives are \(C^1\) with respect to \(p\). (4) In new coordinates the curves \(L_p^s\) and \(L^u_p\) being the stable and -- respectively -- unstable manifolds for \(f\) at \(p\) have the form \((x',0)\) and \((0,y')\) in the \(r\)-neighborhood of \(p\).NEWLINENEWLINENEWLINEPut \(\varphi(p)= (\Phi_p)'(0)\) and observe that \(\varphi(p)\) is a cocycle, called Anosov cocycle. Having two \(C^2\) curves \(l(t)\) and \(k(t)\) defined for \(0\leq t\leq 1\) transversal to the unstable foliation we define a holonomy map \(H(t)\). The author proves the following:NEWLINENEWLINENEWLINETheorem. Let \(p=l(t_0)\) and \(q=H(p)\) where \(t_0\) is in the interval \((0,1)\). If \(\sum^n_{i=1} \varphi(f^{-1} (q))\) is not bounded over \(n\), then \(H''(t_0)\) does not exist.NEWLINENEWLINENEWLINECorollary. If \(\varphi(p)\) is not a coboundary, then \(H''(t)\) does not exist a.e. Thus \(H'(t)\) is not absolutely continuous and the variation of \(H'(t)\) is unbounded for any segment.NEWLINENEWLINEFor the entire collection see [Zbl 0973.00044].