We define shadowable points for homeomorphism on metric spaces. In the compact case we will prove the following results: The set of shadowable points is invariant, possibly nonempty or noncompact. A homeomorphism has the pseudo-orbit tracing property
if and only if every point is shadowable. The chain recurrent and nonwandering sets coincides when every chain recurrent point is shadowable. Minimal or distal homeomorphisms of compact connected metric spaces have no shadowable points. The space is totally disconnected at every shadowable point for distal homeomorphisms (and conversely for equicontinuous homeomorphisms). A distal homeomorphism has the pseudo-orbit tracing property if and only if the space is totally disconnected (this improves Theorem 4 in cite{mo}).
A {em sectional-Anosov flow} is a vector field on a compact manifold inwardly transverse to the boundary such that the maximal invariant set is sectional-hyperbolic (in the sense of cite{mm}). We prove that any $C^2$ transitive sectional-Anosov flow
has a unique SRB measure which is stochastically stable under small random perturbations.
We analyse the intersection of positively and negatively sectional-hyperbolic sets for flows on compact manifolds. First we prove that such an intersection is hyperbolic if the intersecting sets are both transitive (this is false without such a hypot
hesis). Next we prove that, in general, such an intersection consists of a nonsingular hyperbolic set, finitely many singularities and regular orbits joining them. Afterward we exhibit a three-dimensional star flow with two homoclinic classes, one being positively (but not negatively) sectional-hyperbolic and the other negatively (but not positively) sectional-hyperbolic, whose intersection reduces to a single periodic orbit. This provides a counterexample to a conjecture by Shy, Zhu, Gan and Wen (cite{sgw}, cite{zgw}).
We obtain the curvature form $F^ abla=Pcirc d^ ablacirc abla-d^ ablacirc Pcirc abla+d^ ablacirc ablacirc P$ for a vector bundle pseudoconnection $ abla$, where $d^ abla$ is the exterior derivative associated to $ abla$. We use $F^ abla$ to obtain the
curvature of $ abla$. We also prove that $F^ abla=0$ is a necessary (but not sufficient) condition for $d^ abla$ to be a chain complex. Instead we prove that $F^ abla=0$ and $d^ ablacirc d^ ablacirc abla=0$ are necessary and sufficient conditions for $d^ abla$ to be a {em chain $2$-complex}, i.e., $d^ ablacirc d^ ablacirc d^ abla=0$.
We study homeomorphisms of compact metric spaces whose restriction to the nonwandering set has the pseudo-orbit tracing property. We prove that if there are positively expansive measures, then the topological entropy is positive. Some short applications of this result are included.
We prove that every sectional-hyperbolic Lyapunov stable set contains a nontrivial homoclinic class.
We prove that every sectional-Anosov flow of a compact 3-manifold $M$ exhibits a finite collection of hyperbolic attractors and singularities whose basins form a dense subset of $M$. Applications to the dynamics of sectional-Anosov flows on compact 3
-manifolds include a characterization of essential hyperbolicity, sensitivity to the initial conditions (improving cite{ams}) and a relationship between the topology of the ambient manifold and the denseness of the basin of the singularities.
