Some Smooth Ergodic Systems

Exponential Decay of Correlations in Multi-Dimensional Dispersing Billiards

“…To avoid confusion we mention that Condition A.6 is not explicitly stated in this form in [5]. However, the relevant statement, Theorem 5.9 of [5] is known to imply the absolute continuity property of Condition A.6, based on a classical argument by Anosov and Sinai from [1]. Furthermore, we may allow for a little more flexibility: the manifolds W 1 and W 2 that appear in Condition A.6 may be arbitrary u-manifolds (manifolds with tangent planes in the unstable cone).…”

On contact Anosov flows

“…Remark 4.2. In the following we will never need to investigate the duals B * , B * w ; it will suffice to consider elements of D β and D 1 . Accordingly we will always use (4.3).…”

“…The proof there is group theoretical in nature and therefore ill suited to generalizations of the nonconstant curvature case. 1 The conjecture that all Axiom A mixing flows exhibit exponential decay of correlations had already been proven false by Ruelle [34], [27] who produced piecewise constant ceiling suspensions with arbitrarily slow rates of decay.…”

“…While many results were obtained through the years for maps, some positive results have been established for Anosov flows only recently. Notwithstanding the proof of ergodicity, and mixing, for geodesic flows on manifolds of negative curvature [15], [1], [35], the first quantitative results consisted in the proof of exponential decay of correlations for geodesic flows on manifolds of constant negative curvature in two [4], [23], [30] and three [26] dimensions. The proof there is group theoretical in nature and therefore ill suited to generalizations of the nonconstant curvature case.…”

Rigidity properties of Anosov optical hypersurfaces

“…Since the flows of two such vector fields are reparametrizations of one another, the property of being Anosov is independent of the chosen vector field (cf. [1]) and is a property of Σ.…”