DCDS
Entropy-expansiveness for partially hyperbolic diffeomorphisms
Lorenzo J. Díaz Todd Fisher M. J. Pacifico José L. Vieitez
We show that diffeomorphisms with a dominated splitting of the form $E^s\oplus E^c\oplus E^u$, where $E^c$ is a nonhyperbolic central bundle that splits in a dominated way into 1-dimensional subbundles, are entropy-expansive. In particular, they have a principal symbolic extension and equilibrium states.
keywords: symbolic extension. Entropy-expansive dominated splitting equilibrium state partially hyperbolic
DCDS
Lyapunov stability of $\omega$-limit sets
Carlos Arnoldo Morales M. J. Pacifico
We prove that there is a residual subset $C$, in the space of all $\mathcal C^1$ vector fields of a closed $n$-manifold $M$, such that for every $X \in \mathcal R$ the set of points in $M$ with Lyapunov stable $\omega$-limit set is residual in $M$. This improves a result in Arnaud [1] and gives a partial solution to a conjecture in Hurley [8].
keywords: nonwandering set. Lyapunov stable $\omega$-limit set
DCDS
On the intersection of homoclinic classes on singular-hyperbolic sets
S. Bautista C. Morales M. J. Pacifico
We know that two different homoclinic classes contained in the same hyperbolic set are disjoint [12]. Moreover, a connected singular-hyperbolic attracting set with dense periodic orbits and a unique equilibrium is either transitive or the union of two different homoclinic classes [6]. These results motivate the questions of if two different homoclinic classes contained in the same singular-hyperbolic set are disjoint or if the second alternative in [6] cannot occur. Here we give a negative answer for both questions. Indeed we prove that every compact $3$-manifold supports a vector field exhibiting a connected singular-hyperbolic attracting set which has dense periodic orbits, a unique singularity, is the union of two homoclinic classes but is not transitive.
keywords: Homoclinic Class Singular-hyperbolic Set. Attracting Set

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