Grenzmenge

Grenzmenge, literally “limit set” in German mathematical terminology, denotes the collection of limit points that a sequence, a trajectory, or more generally a net can approach in a given topological space. In standard analysis, the Grenzmenge of a sequence (x_n) in a metric space X is the set of all points x ∈ X for which some subsequence (x_{n_k}) converges to x. Equivalently, it is the set of accumulation points of {x_n}. The Grenzmenge is always a closed subset of X. If X is complete and the sequence is bounded, the Grenzmenge is nonempty and compact in metric spaces. If the sequence converges to a point x, the Grenzmenge is the singleton {x}.

In dynamical systems and continuous-time flows, the concept generalizes to limit sets describing asymptotic behavior. For a discrete-time orbit {f^n(x)} of a continuous map f, the omega-limit set ω(x) consists of all y that are limits of some subsequence f^{n_k}(x). The alpha-limit set α(x) describes limits as n → −∞. These limit sets are closed, nonempty in compact spaces, and invariant under the dynamics.

Examples: The sequence x_n = 1/n in R has Grenzmenge {0}. The sequence x_n = (-1)^n has Grenzmenge {−1, 1}. For sin(n), the limit set is the interval [−1, 1] because the sequence of values is dense in that interval.

Grenzmenge is closely related to accumulation (Häufung) points and to the closure of the orbit. In many texts, terms like Häufungspunktemenge or Akkumulationspunktemenge are used synonymously.