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Homework Help: Irrational Flow yields dense orbits.

  1. Mar 19, 2012 #1
    I have the folloring problem:

    Given the following flow on the torus (θ_1)' = ω_1 and (θ_2)' = ω_2, where ω_1 /ω_2 is irrational then I am asked to show that each trajectory is DENSE. So I need to prove that Given any point p on the torus, any initial condition q, and any ε > 0, then there exists t finite such that the trajectory starting at q passes within a distance epsilon of p, that is to say find a t such that |q - p| < ε.

    My problem is how can I find such a t? Can I prove this by contradiction?

    Thank you very much for your help,

  2. jcsd
  3. Mar 28, 2012 #2


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