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- In mathematics, especially functional analysis, a hypercyclic operator on a Banach space X is a bounded linear operator T: X → X such that there is a vector x ∈ X such that the sequence {Tn x: n = 0, 1, 2, …} is dense in the whole space X. In other words, the smallest closed invariant subset containing x is the whole space. Such an x is then called hypercyclic vector. There is no hypercyclic operator in finite-dimensional spaces, but the property of hypercyclicity in spaces of infinite dimension is not a rare phenomenon: many operators are hypercyclic. The hypercyclicity is a special case of broader notions of topological transitivity (see topological mixing), and universality. Universality in general involves a set of mappings from one topological space to another (instead of a sequence of powers of a single operator mapping from X to X), but has a similar meaning to hypercyclicity. Examples of universal objects were discovered already in 1914 by Julius Pál, in 1935 by Józef Marcinkiewicz, or MacLane in 1952. However, it was not until the 1980s when hypercyclic operators started to be more intensively studied. (en)
- Пусть — топологическое векторное пространство (например, банахово пространство). Линейный непрерывный оператор называется гиперциклическим, если существует элемент , такой что множество плотно в . Этот элемент называется гиперциклическим вектором для оператора . Понятие гиперцикличности является частным случаем более широкого понятия топологической транзитивности. (ru)
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- Пусть — топологическое векторное пространство (например, банахово пространство). Линейный непрерывный оператор называется гиперциклическим, если существует элемент , такой что множество плотно в . Этот элемент называется гиперциклическим вектором для оператора . Понятие гиперцикличности является частным случаем более широкого понятия топологической транзитивности. (ru)
- In mathematics, especially functional analysis, a hypercyclic operator on a Banach space X is a bounded linear operator T: X → X such that there is a vector x ∈ X such that the sequence {Tn x: n = 0, 1, 2, …} is dense in the whole space X. In other words, the smallest closed invariant subset containing x is the whole space. Such an x is then called hypercyclic vector. There is no hypercyclic operator in finite-dimensional spaces, but the property of hypercyclicity in spaces of infinite dimension is not a rare phenomenon: many operators are hypercyclic. (en)
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- Hypercyclic operator (en)
- Гиперциклический оператор (ru)
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