We give the complete classification of regular projectively Anosov flows on closed three-dimensional manifolds. More precisely, we show that such a flow must be either an Anosov flow or decomposed into a finite union of $T^2\times I$-models. We also apply our method to rigidity problems of some group actions.

This paper concerns projectively Anosov flows ${\phi }^t$ with smooth stable and unstable foliations ${ℱ}^s$ and ${ℱ}^u$ on a Seifert manifold $M$. We show that if the foliation ${ℱ}^s$ or ${ℱ}^u$ contains a compact leaf, then the flow ${\phi }^t$ is decomposed into a finite union of models which are defined on $T^2\times I$ and bounded by compact leaves, and therefore the manifold $M$ is homeomorphic to the 3-torus. In the proof, we also obtain a theorem which classifies codimension one foliations on Seifert manifolds with compact leaves which are incompressible...

A goal of this work is to study the dynamics in the complement of KAM tori with focus on non-local robust transitivity. We introduce $C^r$ open sets ($r=1,2,\cdots ,\infty$) of symplectic diffeomorphisms and Hamiltonian systems, exhibitinglargerobustly transitive sets. We show that the $C^{\infty }$ closure of such open sets contains a variety of systems, including so-calleda priori unstable integrable systems. In addition, the existence of ergodic measures with large support is obtained for all those systems. A main ingredient of...