Extreme instability of the horocycle flow
Let \(S\) be a compact oriented surface of genus \(\geq 2\) and let \(g\) be a \(C^ 3\)-Riemannian metric on \(S\) of negative curvature. A uniformized version of a result of Feldman and Ornstein for the horocycle flow \(h_ t\) on the unit tangent bundle \(T^ 1S\) of \(S\) with its Margulis- parametrization is presented: Namely there is a number \(\beta>0\) only depending on \(g\) such that for the horocycle flow \(\widehat h_ t\) induced by any metric \(\widehat g\) on \(S\) which is sufficiently close to \(g\) in the \(C^ 2\)-topology and for any two points \(x,y\in T^ 1S\) with \(\text{dist}(x,y)<\beta\) and which are not contained in a single \(\widehat h_ t\)-orbit it holds that \[ \lim_{S\to\infty}{1\over S}\int^ S_ 0\text{dist}(\widehat h_ tx,h_ ty)dt\geq\beta. \] This result is used to show that in contrast to the structural stability of the geodesic flow of \(g\) the horocycle flow \(h_ t\) is instable under small perturbations of \(g\). For \(\widehat g\) close to \(g\) a distance \(\overline d(h_ t,\widehat h_ t)\) between the corresponding horocycle flows \(h_ t\), \(\widehat h_ t\) on \(T^ 1S\) can be defined via stationary coupling which measures the difference in long time behaviour of typical orbits of the flows \(h_ t\), \(\widehat h_ t\) through nearby points. It is shown that \(g(h_ t,\widehat h_ t)\geq A>0\) for a universal constant \(A\) whenever \(\widehat h_ t\) is the horocycle flow of a normalized metric \(\widehat g\) on \(S\) which is sufficiently close to \(g\).
scientific article; zbMATH DE number 221304
scientific article