For $T: \mathbb{R} \mapsto \mathbb{{R}_{+}}$, we have $\{ {T}^{n}(\theta)\ mod \ 1\} \subset [0,1)$. (where ${T}^{n}(\theta)$ means applying $T$ $n$ times on $\theta$, not the $n$th power of $T(\theta)$)

Question 1: Does there exist (preferably elementary) $T$ such that $\{{T}^{n}(\theta)\ mod \ 1\}$ is dense in $[0,1)$ for all irrational $\theta \in [0,1)$?

Question 2: Does there exist $T$ such that $\{{T}^{n}(\theta)\ mod \ 1\}$ is non-periodic (i.e., contains infinite elements) for all $\theta \in [0,1)$?

Edit: @Nikita's comment made me realize that I just asked about a well-known result: See Equidistribution theorem. But this was NOT my intention. Weirdly, I somehow just left the irrational rotation behind my mind when considering $T$. So I would still like to know whether the irrational rotation $T(\theta)=\theta+\alpha$ is the only class of functions that satisfies question 1 or 2?

(I considered $T(\theta)=2\theta$, for example, and they fail both questions. It seems linear growth of $T$ and an irrational coefficient are both necessary. Are there functions other than irrational rotation that satisfies question 1 or 2?)

  • $\begingroup$ what about taking $T^n(θ)$= n number of digits in the decimal expansion of an irrational number say $\alpha \epsilon [0,1)$, starting from $(\frac{(n(n−1)}{2}+1)^th$ digit? $\endgroup$
    – nb1
    Jan 4, 2012 at 6:07
  • $\begingroup$ @Nikhil: Well, $T$ is only a function of $\theta$ by definition. $\endgroup$
    – Eric
    Jan 4, 2012 at 8:12
  • 1
    $\begingroup$ If it is not critical that $T(x)\geq0$, then any irrational rotation will do, i.e., $T(\theta)=\theta+\alpha$, where $\alpha$ is irrational. Every orbit of $T$ is known to be dense modulo 1 and aperiodic. $\endgroup$ Jan 4, 2012 at 9:37

1 Answer 1

  1. Interval Exchange Transformations provide a vast class of further examples.

  2. Let $f:[0,1] \to [0,1]$ be increasing, then the conjugation $f^{-1} \circ T \circ f$ is another example for $T$ being an example.

  3. Let $\mathbb{T} = \mathbb{R}/\mathbb{Z}$. Then for $f:\mathbb{T} \to \mathbb{T}$ twice continuously differentiable with irrational rotation number and transitive, then it is of the previous form. This is known as Denjoy Theorem, Theorem 12.1.1. in Katok, Hasselblatt, Introduction to the modern theory of dynamical systems.


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