Horocycle flows for laminations by hyperbolic Riemann surfaces and Hedlund's theorem (Q316987)

Let \({\mathbb{H}}\) denote the hyperbolic plane with constant curvature \(-1\). Let \(D\),\(U\) and \(B\) denote the subgroups of \({\mathrm{SL}}(2, \mathbb{R})\) given by NEWLINENEWLINE\[NEWLINED = \left\{\begin{pmatrix} e^{t/2} & 0 \\ 0 & e^{- t/2} \end{pmatrix} : t \in \mathbb{R}\right\}, \;NEWLINEU = \left\{\begin{pmatrix} 1 & t \\ 0 & 1 \end{pmatrix} : t \in \mathbb{R}\right\}, \; NEWLINEB = \left\{\begin{pmatrix} a & b \\ 0 & a^{-1} \end{pmatrix} : a > 0, b \in \mathbb{R} \right\}.NEWLINE\]NEWLINENEWLINEThe groups \(D\),\(U\) and \(B\) act on the unit tangent bundle \(T^{1}{\mathbb{H}}\) and the actions of \(D\) and \(U\) are called respectively the geodesic flow \(g^{t}\) and the horocycle flow \(h^{s}\). These flows are related by the equation \(g^{t} \circ h^{s} \circ g^{-t} = h^{s e^{-t}}\) for all \(s,t \in \mathbb{R}\). \newlineNEWLINENEWLINEThe authors study compact metric spaces \(M\) carrying a lamination \(\mathcal{F}\) by hyperbolic surfaces for which all leaves of \(\mathcal{F}\) are dense. Let \(\hat{M}\) denote the union of the unit tangent bundles of the leaves of \(\mathcal{F}\). Note that \(\hat{M}\) is compact since it fibers over \(M\) by circles. The geodesic and horocycle flows extend naturally to \(\hat{M}\) from the unit tangent bundles of the leaves of \(\mathcal{F}\). A central problem is to describe the actions on \(\hat{M}\) of the groups \(D\),\(U\) and \(B\).NEWLINENEWLINEThe authors show that the action of \(B\) on \(\hat{M}\) is topologically transitive, i.e., \(B\) has a dense orbit, and that \(B\) has a unique minimal set \(\mathcal{M}\) in \(\hat{M}\). If \(\mathcal{M} = \hat{M}\) and \(\mathcal{F}\) has a nonsimply connected leaf, then the horocycle flow is topologically transitive on \(\hat{M}\). The authors give examples for which \(\mathcal{M}\) is a proper subset of \(\hat{M}\). Two sufficient conditions for the \(B\)-action on \(\hat{M}\) to be minimal are given. One of these conditions is that \((M,\mathcal{F})\) should admit a transverse holonomy invariant measure.