We study countable partitions for measurable maps on measure spaces such that, for every point $x$, the set of points with the same itinerary as that of $x$ is negligible. We prove in nonatomic probability spaces that every strong generator (Parry, W., Aperiodic transformations and generators, J. London Math. Soc. 43 (1968), 191–194) satisfies this property (but not conversely). In addition, measurable maps carrying partitions with this property are aperiodic and their corresponding spaces are nonatomic....

We consider projectively Anosov flows with differentiable stable and unstable foliations. We characterize the flows on $T^2$ which can be extended on a neighbourhood of $T^2$ into a projectively Anosov flow so that $T^2$ is a compact leaf of the stable foliation. Furthermore, to realize this extension on an arbitrary closed 3-manifold, the topology of this manifold plays an essential role. Thus, we give the classification of projectively Anosov flows on $T^3$. In this case, the only flows on $T^2$ which extend to $T^3$...

Two dynamical deformation theories are presented – one for surface homeomorphisms, called pruning, and another for graph endomorphisms, called kneading – both giving conditions under which all of the dynamics in an open set can be destroyed, while leaving the dynamics unchanged elsewhere. The theories are related to each other and to Thurston’s classification of surface homeomorphisms up to isotopy.