Nonrecurrence

Nonrecurrence is a term used in dynamical systems and related fields to describe the property that a state or point does not return, or return arbitrarily close, to its initial position under the evolution of the system.

In formal terms, let f be a function describing the evolution on a space X, and let x be a point in X. A point x is nonrecurrent if there exists a neighborhood U of x and an integer N such that f^n(x) is not in U for all n ≥ N. Equivalently, x is not in the omega-limit set of its forward orbit. Recurrence, by contrast, means that for every neighborhood of x, the orbit returns to that neighborhood infinitely often. Nonrecurrence thus captures transient or escaping behavior.

In many classical settings, the Poincaré recurrence theorem implies that in finite-measure, measure-preserving systems, almost every point is recurrent. Consequently, nonrecurrence is typically associated with exceptional or transient dynamics, such as systems with leakage, dissipative behavior, absorbing states, or infinite invariant measures where orbits can escape to infinity or move away from initial regions.

In stochastic processes, related ideas appear under the notion of transient states, where a process may leave a state and not return with some positive probability, in contrast with recurrent states that are revisited over time.

The concept of nonrecurrence helps describe how long a system may persist in a regime before moving away or settling into an attractor, and it informs the study of transient dynamics, basins of attraction, and long-term behavior before stabilization or exit from a domain. See also recurrence, transient, omega-limit set, attractor, and Poincaré recurrence.