limposition

Limposition is a theoretical construct used in geometry, computer science, and dynamical systems to denote the limiting position to which a moving point or object converges as time or a parameter tends to infinity. In its simplest form, if a sequence of positions p_n in a metric space (X,d) converges to p*, then p* is the limposition of {p_n}. When the trajectory does not converge, the limposition can be generalized by the omega-limit set, consisting of all limit points of subsequences.

Formally, for a sequence {p_n} in a metric space, the limposition is the limit point p* if d(p_n, p*) → 0 as n → ∞. If this does not occur, one can describe the limiting behavior with the omega-limit set, which includes every point that is the limit of some subsequence {p_{n_k}}. The limposition concept emphasizes the end-state behavior of a process, rather than transient motion alone.

In contractive or convergent systems, a limposition often exists and is unique, coinciding with a fixed point in many models. More generally, limposition may be nonunique or undefined; the limit set may contain multiple accumulation points or be unbounded, depending on the dynamics and time parameterization.

A simple example is the recurrence p_{n+1} = 0.5 p_n with p_0 = 1 in the real line, which yields limposition p* = 0. Applications appear in animation, robotics, and optimization, where predicting the eventual configuration or ensuring stability is valuable. Limitations include dependence on the chosen time scale and the possibility of non-convergent trajectories requiring broader limit-set analysis.

See also: limit, omega-limit set, attractor, fixed point.