Related papers: Robust transitivity of singular hyperbolic attract…
In this article, we generalize Eberlein's Rigidity Theorem to the singular case, namely, one of the spaces is only assumed to be a CAT(0) topological manifold. As a corollary, we get that any compact irreducible but locally reducible…
There are few explicit examples in the literature of vector fields exhibiting complex dynamics that may be proved analytically. We construct explicitly a {two parameter family of vector fields} on the three-dimensional sphere $\EU^3$, whose…
We prove that any diffeomorphism of a compact manifold can be approximated in topology C1 by another diffeomorphism exhibiting a homoclinic bifurcation (a homoclinic tangency or a heterodimensional cycle) or by one which is essentially…
The aim of this paper is to provide a discussion on current directions of research involving typical singularities of 3D nonsmooth vector fields. A brief survey of known results is presented. The main purpose of this work is to describe the…
This paper is concerned with the analysis of a typical singularity of piecewise smooth vector fields on $R^3$ composed by two zones. In our object of study, the cusp-fold singularity, we consider the simultaneous occurrence of a cusp…
An important, if relatively less well known aspect of the singularity theorems in Lorentzian Geometry is to understand how their conclusions fare upon weakening or suppression of one or more of their hypotheses. Then, theorems with modified…
We present several results suggesting that the concept of $C^1$-inverse limit stability is free of singularity theory. We describe an example of a $C^1$-inverse stable endomorphism which is robustly transitive with persistent critical set.…
In this work, we study ergodic properties of certain partially hyperbolic attractors whose central direction has a neutral behavior, the main feature is a condition of transversality between unstable leaves when projected by the stable…
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…
We prove that any $C^{1+\alpha}$ transitive conservative partially hyperbolic diffeomorphism of a closed 3-manifold with virtually solvable fundamental group is ergodic. Consequently, in light of \cite{FP-classify}, this establishes the…
We prove that a C2 Hamiltonian system H in M is globally hyperbolic if any of the following statements holds: H is robustly topologically stable; H is stably shadowable; H is stably expansive; and H has the stable weak specification…
We obtain sufficient conditions for the existence of physical/SRB measures for asymptotically sectionally hyperbolic attracting sets with any finite co-dimension, extending the co-dimension two case. We provide examples of such attractors,…
In this work a homoclinic-like loop of a piecewise smooth vector field passing through a typical singularity is analyzed. We have shown that such a loop is robust in one-parameter families of Filippov systems. The basin of attraction of…
The {\it two-fold singularity} has played a significant role in our understanding of uniqueness and stability in piecewise smooth dynamical systems. When a vector field is discontinuous at some hypersurface, it can become tangent to that…
We prove that any C1-stably weakly shadowable volume-preserving diffeomorphism defined on a compact manifold displays a dominated splitting E + F. Moreover, both E and F are volume-hyperbolic. Finally, we prove the version of this result…
We consider a discrete dynamical system on a pseudo-Riemannian manifold and we determine the concept of a hyperbolic set for it. We insert a condition in the definition of a hyperbolic set which implies to the unique decomposition of a part…
The aim of this paper is twofold. First, we introduce standard blenders (special hyperbolic sets) and their variations, and prove their fundamental properties on the generation of $C^1$-robust tangencies. In particular, these blenders…
We provide conditions on the coupling function such that a system of 4 globally coupled identical oscillators has chaotic attractors, a pair of Lorenz attractors or a 4-winged analogue of the Lorenz attractor. The attractors emerge near the…
It is known that a generic star vector field $X$ on a $3$ or $4$-dimensional manifold is such that its chain recurrence classes are either hyperbolic, or singular hyperbolic ([MPP] and [GSW]). Palis conjectured that every vector field must…
We find the explicit local models of isolated singularities of conformal hyperbolic metrics by Complex Analysis, which is interesting in its own and could potentially be extended to high-dimensional case.