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I am aware that is it unknown as to whether or not the set $\ \{\ \left(\frac{3}{2}\right)^n \mod 1:n\in\mathbb{N}\ \}\ $ is dense in $\ [0,1].$ It is known that this set has infinitely many limit points in $\ [0,1].$

What about the set of points defined by the recursive sequence:

$a_0=\frac{1}{2}$

$ a_{n+1} = \left( \frac{3}{2}a_n \right) (\mod 1) $

This gives a different set to the first one. It is fundamentally different because we don't sometimes add one half like we do for the first set. Is this set known to be dense or to have infinitely many limit points in $\ [0,1]\ ?$

Adam Rubinson
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  • This is a chaotic map, similar to the doubling map, see https://personalpages.manchester.ac.uk/staff/charles.walkden/magic/lecture01.pdf. – markvs Nov 29 '21 at 02:31
  • @markvs Thank you. – Adam Rubinson Nov 29 '21 at 02:36
  • You should ask ergodic people about your orbit. My understanding is that theoretically you can prove that almost every orbit is dense - with very strong meaning of "almost every" but the question whether a particular orbit is dense is usually too hard. – markvs Nov 29 '21 at 05:33

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