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Register a new userNon-dense orbits of systems with approximate product property
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Authors
Peng Sun
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We show that for any topological dynamical system with approximate product property, the set of points whose forward orbits do not accumulate to any point in a large set carries full topological pressure.
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We give a short discussion about a weaker form of minimality (called quasi-minimality). We call a system quasi-minimal if all dense orbits form an open set. It is hard to find examples which are not already minimal. Since elliptic behaviour makes them minimal, these systems are regarded as parabolic systems. Indeed, we show that a quasi-minimal homeomorphism on a manifold is not expansive (hyperbolic).
It is proved that a certain type of monotone flow has a global period provided periodic points are dense.
Let X be a connected open set in n-dimensional Euclidean space, partially ordered by a closed convex cone K with nonempty interior: y > x if and only if y-x is nonzero and in K. Theorem: If F is a monotone local flow in X whose periodic points are dense in X, then F is globally periodic.
In this paper, we show the existence of non contractible periodic orbits in Hamiltonian systems defined on $T^*T^n$ separating two Lagrangian tori under certain cone assumption. Our result answers a question of Polterovich in cite{P} in a sharp way. As an application, we find periodic orbits of almost all the homotopy types on a dense set of energy level in Lorentzian type mechanical Hamiltonian systems defined on $T^*T^2$. This solves a problem of Arnold in cite{A}.
Katznelsons Question is a long-standing open question concerning recurrence in topological dynamics with strong historical and mathematical ties to open problems in combinatorics and harmonic analysis. In this article, we give a positive answer to Katznelsons Question for certain towers of skew product extensions of equicontinuous systems, including systems of the form $(x,t) mapsto (x + alpha, t + h(x))$. We describe which frequencies must be controlled for in order to ensure recurrence in such systems, and we derive combinatorial corollaries concerning the difference sets of syndetic subsets of the natural numbers.
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