Charles had a good remark after the lecture: I told that the orbits of the map x(n+1)=2x(n) mod 1has a uniform distribution on the unit circle, but of course it is easy to see that the point x=0 is a fixed point, so it does not hold for all points (in particular it does not hold for any point with rational initial condition. So let me clarify: my comment holds for almost all initial conditions (a set of initial conditions of full arc length or Lebesgue measure). The exceptional points are dense, but have (as a set) zero length. One can interpret this as saying that if one choses an initial condition randomly, there is zero chance that its orbit does not converge to a uniform distribution.
My comments about this example are only tasters for what is to come later and the explanation of all of the above will be central to the course.
My comments about this example are only tasters for what is to come later and the explanation of all of the above will be central to the course.
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