Related papers: No-shadowing for singular hyperbolic sets with a s…
We prove new lower bounds on the crossing number of a complete graphs assuming that it is drawn in such a way that it contains a Hamiltonian cycle with no crossings.
This Part establishes the geometric theory of uniformly hyperbolic sets with explicit quantitative bounds throughout, and contains five main theorems. The Stable Manifold Theorem is proved via the backward graph transform, with a complete…
Addressing a question of Zaremsky, we give conditions on a finite simplicial graph which guarantee that the associated matching arc complex is connected and hyperbolic.
We study strong hyperbolicity of first order partial differential equations for systems with differential constraints. In these cases, the number of equations is larger than the unknown fields, therefore, the standard Kreiss necessary and…
We prove that Lyapunov exponents of typical H\"older continuous fiber bunched linear cocycles over singular hyperbolic flows have multiplicity 1: the subspace of exceptional cocycles has infinite codimension.
We study a partially hyperbolic and topologically transitive local diffeomorphism $F$ that is a skew-product over a horseshoe map. This system is derived from a homoclinic class and contains infinitely many hyperbolic periodic points of…
We prove that for generic three-dimensional vector fields, domination implies singular hyperbolicity.
The dynamics of many important high-dimensional dynamical systems are both chaotic and complex, meaning that strong reducing hypotheses are required to understand the dynamics. The highly influential chaotic hypothesis of Gallavotti and…
We show there is a residual set of non-Anosov $C^{\infty}$ Axiom A diffeomorphisms with the no cycles property whose elements have trivial centralizer. If $M$ is a surface and $2\leq r\leq \infty$, then we will show there exists an open and…
We define the concept of $(\mathscr{F},\mathscr{G})-$shadowing property on uniform space and say it as a topological $(\mathscr{F},\mathscr{G})-$shadowing property. We show that topological shadowing, topological…
For linear control systems in discrete time controllability properties are characterized. In particular, a unique control set with nonvoid interior exists and it is bounded in the hyperbolic case. Then a formula for the invariance pressure…
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…
If we perturb a completely integrable Hamiltonian system with two degrees of freedom, the perturbed flow might display, on every energy level, invariant sets that are laminations over Aubry-Mather sets of a Poincar\'e section of the flow.…
In this paper we prove the infinitesimal uniqueness theorem for the Newton potential of non simply connected bodies using the singularity theory approach. We consider the Newtonian potentials of the domains in ${\bf R}^n$ boundaries of…
This paper proves that any monoid presented by a confluent context-free monadic rewriting system is word-hyperbolic. This result then applied to answer a question asked by Duncan & Gilman by exhibiting an example of a word-hyperbolic monoid…
We study $C^1$-generic vector fields on closed manifolds without points accumulated by periodic orbits of different indices and prove that they exhibit finitely many sinks and sectional-hyperbolic transitive Lyapunov stable sets with…
We describe unicorn paths in the arc graph and show that they form 1-slim triangles and are invariant under taking subpaths. We deduce that all arc graphs are 7-hyperbolic. Considering the same paths in the arc and curve graph, this also…
We prove that for any $C^1$-stably weakly shadowing transitive set $\Lambda$, either $\Lambda$ is a sink or a source, or $\Lambda$ admits a dominated splitting.
We obtain sufficient conditions for an invariant splitting over a compact invariant subset of a $C^1$ flow $X_t$ to be dominated. In particular, we reduce the requirements to obtain sectional hyperbolicity and hyperbolicity.
We consider a dynamical system which has the hyperbolic structure along an attracting invariant manifold $M$. The problem is whether every motion starting in a neighborhood of $M$ possesses an asymptotic phase, i.e. eventually approaches a…