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 Incoming links: Alpha-limit set

Limit set 2026-10-13

A trajectory's limit set consists of its accumulation points along sequences of times tending to positive or negative infinity. The forward-time set is its omega-limit set and the backward-time set its alpha-limit set.

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Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 1 / 31E / Solution 2026-09-29

Let

ϕ_{t} (x)

denote the flow through

x

. Its alpha-limit set and omega-limit set are

α (x) = ⋂_{T < 0} \overline{{ϕ_{t} (x) : t < T}}, ω (x) = ⋂_{T > 0} \overline{{ϕ_{t} (x) : t > T}} .

Equivalently, they contain the limits of

ϕ_{t_{j}} (x)

along sequences

t_{j} \to - \infty

and

t_{j} \to + \infty

, respectively.

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