Contributions to the ergodic theory of hyperbolic flows: unique ergodicity for quasi-invariant measures and equilibrium states for the time-one map (Q6594739)

The authors consider a codimension-one hyperbolic flow \(\Phi = \{\Phi_t : M \to M\}\), where \(M\) is a closed Riemannian manifold with splitting \(TM = E^s \oplus E^c \oplus E^u\). By the Hadamard-Perron stable manifold theorem, it follows that \(E^u\) integrates to a flow \(\Phi^u = \{\Phi_t^u : M \to M\}_{t \in \mathbb{R}}\), called the horocyclic flow.\N\NUnder the former assumptions, the authors are able to prove the following result:\N\NTheorem. If \(\Phi\) is a (transitive) codimension-one Anosov flow that is not a suspension, then \(\Phi^u\) is uniquely ergodic. That is, there exists only one (probability) measure invariant under \(\Phi^u\).\N\NConsider the set of Hölder multiplicative cocycles \N\[\N\mathrm{Coc}(\Phi) = \left\{h : M \to \mathbb{R}_+ : h(x) = \mathrm{exp}\left({\int_0^1 \varphi(\Phi_t x) dt}\right) \text{ for some Hölder function } \varphi\right\}.\N\]\NThe authors prove the following:\N\NTheorem A. Let \(\Phi\) be a codimension-one Anosov flow of class \(\mathcal{C}^2\) that is not a suspension, and consider \(h \in \mathrm{Coc}(\Phi)\). Assume that the unstable bundle of \(\Phi\) is orientable and denote by \(\Phi^u\) the induced horocyclic flow. Then there exists \(\mu^{cs} = \{\mu^{cs}_x\}_x\), a transverse measure for \(\Phi^u\) such that \(\mu^{cs}\) is the unique quasi-invariant measure with Jacobian given by the family \({h}\) determined by \(h\).\N\NAs a consequence, the authors obtain:\N\NCorollary. Under the same hypotheses of the above theorem, the horocyclic flow \(\Phi^u\) has a unique \({h}\)-conformal measure.\N\NThe authors also prove the following result:\N\NTheorem B. Assume the same hypotheses of Theorem A. Then there exists a family of measures \(\{\nu_x^u\}_{x \in M}\), where each \(\nu_x^u\) is a Radon measure on \(W^u(x)\), such that for any probability \(m\) on \(M\) (not necessarily invariant under any \(\Phi_t\)) whose conditionals in some \(m\) SLY (after Strelcyn, Ledrappier and Young) partition \(\xi\) are of the form\N\[\Nm_x^\xi = \frac{\nu_x^u}{\nu_x^u(\xi(x))} \;,\N\]\Nnecessarily satisfies \(m = m_{\Phi, \varphi}\).