Triple

Given a phrase, classify it is english named entity (e.g., persons, organizations, works of art) in Latin script, or not (e.g., literals, dates, URLs, verbose phrases). For disambiguation, the statement where the phrase occurs as object is also given. Please return a JSON object with `phrase` (string, the phrase being analyzed) and `is_ne` (boolean, indicating whether the phrase is a Named Entity).

Phrase: In a measure-preserving dynamical system with finite total measure, almost every point of a measurable set returns arbitrarily close to its initial position infinitely often. | Statement: [Poincaré recurrence theorem, statement, In a measure-preserving dynamical system with finite total measure, almost every point of a measurable set returns arbitrarily close to its initial position infinitely often.]

Target entity: In a measure-preserving dynamical system with finite total measure, almost every point of a measurable set returns arbitrarily close to its initial position infinitely often.
Context triple: [Poincaré recurrence theorem, statement, In a measure-preserving dynamical system with finite total measure, almost every point of a measurable set returns arbitrarily close to its initial position infinitely often.]

• A. Poincaré recurrence theorem chosen
  The Poincaré recurrence theorem is a fundamental result in dynamical systems and ergodic theory stating that certain systems will, after a sufficiently long but finite time, return arbitrarily close to their initial state.
• B. ergodic theorem
  The ergodic theorem is a fundamental result in dynamical systems and probability theory that links long-term time averages of a system’s evolution to ensemble or space averages, underpinning the statistical behavior of many physical and stochastic processes.
• C. Kakutani equivalence in ergodic theory
  Kakutani equivalence in ergodic theory is a notion of equivalence between measure-preserving dynamical systems based on the isomorphism of their induced transformations on subsets of positive measure.
• D. Kakutani’s random ergodic theorem
  Kakutani’s random ergodic theorem is a fundamental result in ergodic theory that extends classical ergodic theorems to sequences of randomly chosen measure-preserving transformations.
• E. Poincaré map
  The Poincaré map is a mathematical tool in dynamical systems theory that reduces continuous-time dynamics to a discrete map by tracking intersections of trajectories with a lower-dimensional surface.
• F. None of above.
• G. Unsure - the case is ambiguous/there is not enough information to decide.