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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 linearity linear growth of $T$ and an irrational coefficient are both necessary. Are there functions other than irrational rotation that satisfies question 1 or 2?)

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# ModularModulo dynamics on [0,1)

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 linearity and an irrational coefficient are both necessary. Are there functions other than irrational rotation that satisfies question 1 or 2?)

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# Modular dynamics on [0,1)

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 $\theta \in [0,1)$?