Topological stability from Gromov-Hausdorff viewpoint
Alexanger Arbieto Carlos Arnoldo Morales Rojas
Discrete & Continuous Dynamical Systems - A 2017, 37(7): 3531-3544

We combine the classical Gromov-Hausdorff metric [5] with the $C^0$ distance to obtain the $C^0$-Gromov-Hausdorff distance between maps of possibly different metric spaces. The latter is then combined with Walters's topological stability [11] to obtain the notion of topologically GH-stable homeomorphism. We prove that there are topologically stable homeomorphism which are not topologically GH-stable. Also that every topological GH-stable circle homeomorphism is topologically stable. Afterwards, we prove that every expansive homeomorphism with the pseudo-orbit tracing property of a compact metric space is topologically GH-stable. This is related to Walters's stability theorem [11]. Finally, we extend the topological GH-stability to continuous maps and prove the constant maps on compact homogeneous manifolds are topologically GH-stable.

keywords: Topological stability topological GH-stability metric space
Topological stability in set-valued dynamics
Roger Metzger Carlos Arnoldo Morales Rojas Phillipe Thieullen
Discrete & Continuous Dynamical Systems - B 2017, 22(5): 1965-1975 doi: 10.3934/dcdsb.2017115

We propose a definition of topological stability for set-valued maps. We prove that a single-valued map which is topologically stable in the set-valued sense is topologically stable in the classical sense [14]. Next, we prove that every upper semicontinuous closed-valued map which is positively expansive [15] and satisfies the positive pseudo-orbit tracing property [9] is topologically stable. Finally, we prove that every topologically stable set-valued map of a compact metric space has the positive pseudo-orbit tracing property and the periodic points are dense in the nonwandering set. These results extend the classical single-valued ones in [1] and [14].

keywords: Topological stability set-valued map metric space
Topological properties of sectional-Anosov flows
Enoch Humberto Apaza Calla Bulmer Mejia Garcia Carlos Arnoldo Morales Rojas
Discrete & Continuous Dynamical Systems - A 2015, 35(10): 4735-4741 doi: 10.3934/dcds.2015.35.4735
We study sectional-Anosov flows on compact $3$-manifolds. First we prove that every periodic orbits represents an infinite order element of the fundamental group outside the strong stable manifolds of the singularities. Next, in the transitive case, we prove that the first Betti number of the manifold is positive, that the number of singularities is given by the Euler characteristic and that every boundary's connected component has nonpositive Euler characteristic. Moreover, there is one component with negative characteristic if and only if the flow has singularities. These results will be used to discuss the existence of transitive sectional-Anosov flows on specific compact 3-manifolds with boundary.
keywords: compact manifold Sectional-Anosov flow transitive flow periodic orbit. handlebody

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